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✓ 18 results · all verified · 7 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 11 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Evolutionary incompressible flow discretization

1 · Prerequisites

2 · Summary

Evolutionary incompressible flow discretization

This draft treats Newtonian incompressible Navier–Stokes on bounded connected Lipschitz polygonal or polyhedral domains in dimensions two and three, with viscosity $ν > 0$ , homogeneous Dirichlet velocity data, and pressure in the zero-mean gauge $L_{0}^{2} (D)$ . The bilinear viscosity convention is $a_{ν} (u, v) = 2 ν (ε (u), ε (v))$ and $b (v, p) = - (p, div v)$ . The nonlinear term is the real skew form in The skew convective form for incompressible flow; its cancellation is independent of pointwise discrete incompressibility.

Evolution problem and discrete step

The regular-pressure energy weak formulation and its time-space integrability are stated in Energy weak solutions of incompressible Navier–Stokes and Integrability of the energy weak Navier–Stokes formulation. That definition specifies a regular-pressure solution class; it does not assert existence in that class for all energy data. The weak MINI kernel $Z_{h}$ is distinguished from the exact divergence-free kernel; the scheme uses the stable continuous P1-bubble/P1 pair supplied by mixed-finite-elements-and-stokes-stability.

The mixed mini-element backward Euler Navier–Stokes scheme defines the mixed backward Euler equations with cell-average forcing. Backward Euler MINI step existence and discrete energy bound proves finite-step existence by an outward-pointing Brouwer argument and records both the work-form energy inequality and its mesh-independent consequence. The coercivity, pressure reconstruction at each finite step, skew cancellation, and forcing estimate are given in the local proof.

Velocity compactness, limits, and uniqueness

The Fortin map estimates in The mini-element Fortin map approximates in L2, the second-order $L^{2}$ estimate on $H^{2}$ data in Second-order L2 approximation by the MINI Fortin map, and the finite-probe argument in Strong L2 compactness for weakly solenoidal MINI trajectories give strong $L_{t}^{2} L_{x}^{2}$ compactness without treating weakly divergence-free MINI velocities as pointwise divergence-free. The local Scalar weak equations give a weakly continuous Hilbert-space representative turns the scalar test identities into a weakly continuous $H_{σ}$ representative. MINI backward Euler converges subsequentially to a solenoidal energy weak velocity proves the subsequential solenoidal velocity equation, weak initial trace, and work-form energy inequality for arbitrary vanishing mesh and time-step sizes. The additive Integrated pressure recovery and MINI pressure-primitive convergence reconstructs a unique mean-zero pressure distribution with a bounded weakly continuous primitive for that limit, and proves bounded discrete pressure primitives and convergence of the discrete pressures as time distributions. The original design additionally required every such limit to have pressure in $L_{t}^{4 / 3} L_{0}^{2}$ . That universal conclusion is false under its original hypotheses: the exact nonlinear square counterexample on the companion page has $L_{t}^{2} L_{x}^{2}$ forcing and an $H_{σ}$ initial datum, but no pressure in that class. The primitive result retains the exact criterion: regular pressure exists if and only if the actual full distributional derivative $u_{t}$ belongs to $L^{4 / 3} (0, T; V^{*})$ .

The pressure-free Uniqueness and strong trace for two-dimensional solenoidal energy velocities proves strong initial trace and uniqueness for two-dimensional solenoidal energy velocities. Two-dimensional uniqueness and conditional three-dimensional weak–strong uniqueness proves unconditional uniqueness in two dimensions and weak–strong uniqueness in three dimensions only under its explicit strong-solution regularity and equation hypotheses. In each case velocity equality also forces equality of the zero-mean pressures.

Smooth branch and instantaneous pressure refutations

Approximation by the MINI solenoidal energy projection supplies the finite-dimensional energy projection and its MINI approximation rate. Smooth-branch velocity error for backward Euler MINI proves a fully discrete velocity estimate on a smooth no-slip branch, including the required time-step restriction and explicit viscosity dependence. Its Gronwall factor contains $ν^{- 3}$ ; it makes no Reynolds-uniform claim. The additive First-order smooth pressure-primitive error for backward Euler MINI proves the corresponding first-order estimate for the pressure primitive and a precise negative time norm under the same initialization.

The PE06 design originally asked for both universal instantaneous pressure regularity and an instantaneous smooth pressure-error rate. The first clause is refuted by the square example; the second is refuted by the independent smooth graded-mesh construction on the companion page, under its original data, arbitrary shape-regular grading, mass initialization and time-partition hypotheses. PE06 instantaneous pressure refutations and retained primitive estimates retains the original formulation, the exact generic derivative criterion, and the distinction between full-test derivative estimates and restricted solenoidal bounds. The proposed uniform $H^{1}$ stability of the $L^{2}$ -orthogonal projector onto $Z_{h}$ is false on the square MINI family and is not used as a supplier. The square construction supplies only the generic refutation. The smooth construction supplies its own actual pressure observation and nonlinear backward Euler realization; it preserves the positive velocity and primitive-pressure theorems.

Worked cases

The square energy-data pressure counterexample, the smooth graded-mesh pressure-rate counterexample, the periodic Taylor–Green solution, the no-forcing discrete energy budget, the zero-mean pressure gauge, a matching unit-square mesh family, and a conditional three-dimensional computation are on energy-stable-incompressible-navier-stokes-discretization-examples. The periodic vortex uses a torus and is explicitly separate from the no-slip theorem branch.

3 · Logical flowchart

4 · Definitions, theorems and proofs

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Logarithmic halfline heat mass loss at the pressure endpoint

Statement

Assume the Axiom of Choice. Fix $ν > 0$ , $1 / 2 < α \leq 3 / 4$ , and nonnegative $η \in C^{\infty} ([0, \infty))$ supported in $[0, 1 / 16)$ and equal to $1$ on $[0, 1 / 32]$ . Define $w_{0} (y) = η (y) y^{- 1 / 2} (\log (1 / y))^{- α}$ for $0 < y < 1 / 16$ , and $w_{0} (y) = 0$ for $y \geq 1 / 16$ . Put $G_{t} (s) = (4 π ν t)^{- 1 / 2} e^{- s^{2} / (4 ν t)},$ $w (t, y) = \int_{0}^{\infty} [G_{t} (y - z) - G_{t} (y + z)] w_{0} (z) d z .$ Then $w_{0} \in L^{1} \cap L^{2} (0, \infty)$ , $w \geq 0$ , $w (t, 0) = 0$ , $w_{t} = ν w_{y y}$ , and $w$ is smooth for positive time, including at $y = 0$ . Moreover $\sup_{t > 0} ∥ w (t) ∥_{j} \leq ∥ w_{0} ∥_{j} (j = 1, 2), w (t) \to w_{0} in L^{2} as t ↓ 0,$ $\int_{0}^{T} ∥ w_{y} (t) ∥_{2}^{2} d t \leq (2 ν)^{- 1} ∥ w_{0} ∥_{2}^{2}, \int_{0}^{T} ∥ w (t) ∥_{4}^{4} d t < \infty .$ Write $M (t) = \int_{0}^{\infty} w (t, y) d y$ , $M_{0} = \int_{0}^{\infty} w_{0}$ , $m (t) = M (t) - M_{0}$ , and $h (t, y) = - \int_{y}^{\infty} w (t, z) d z$ . Then $h_{y} = w$ , $h_{t} = ν h_{y y}$ , $h (t, 0) = - M (t)$ , and $m (0) = 0$ . For some $t_{*} > 0$ and $c, C > 0$ , writing $L (t) = \log (1 / t)$ , $c t^{- 3 / 4} L (t)^{- α} \leq - m^{'} (t) \leq C t^{- 3 / 4} L (t)^{- α}, ∣ m (t) ∣ \leq C t^{1 / 4} L (t)^{- α} (0 < t < t_{*}) .$ Consequently $m^{'} \in L^{1} (0, T) ∖ L^{4 / 3} (0, T)$ and $\int_{0}^{T} (∣ m^{'} (t) ∣ t^{1 / 4} + ∣ m (t) ∣ t^{- 3 / 4})^{2} d t < \infty$ for every finite $T > 0$ . On each bounded $y$ interval, $h (t, \cdot) \to h_{0} (\cdot) = - \int_{\cdot}^{\infty} w_{0}$ uniformly as $t ↓ 0$ . On each closed interval separated from $supp w_{0}$ , all spatial derivatives of $w$ are bounded for small positive time by $C_{N} t^{- N} e^{- c / t}$ , with constants allowed to depend on the derivative.

Facts & Assumptions

Given: The Axiom of Choice and the data in the Statement.

[A1]

The Axiom of Choice supplies the Countable and Dependent Choice hypotheses of the integration and approximation suppliers. (The Axiom of Choice, AC supplies the countable and dependent choices used in Banach integration)

[F1]

Gaussian kernels have unit mass, solve the heat equation, and have Gaussian spatial derivative bounds. They form an approximate identity, which converges in $L^{2}$ ; convolution obeys Young's inequality. (Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel, Gaussian kernels form an approximate identity, Every $L^{1}$ approximate identity converges to the identity in $L^{p}$ for $1 \leq p < \infty$ , Young's convolution inequality under Countable Choice)

[F2]

Dominated convergence and Holder's inequality apply to the displayed Lebesgue integrals. (Dominated convergence, Holder's inequality for integrals, including the endpoint cases)

Proof

1.1A1F1F2givenalgebra

The square of $w_{0}$ near zero is $y^{- 1} (\log (1 / y))^{- 2 α}$ , integrable because $2 α > 1$ ; its first power is also integrable. For $y, z > 0$ , $G_{t} (y - z) \geq G_{t} (y + z)$ , proving positivity. Its integral in either positive variable is at most $1$ . Thus the positive kernel is an $L^{1}$ contraction; Jensen's inequality in its subprobability measure, followed by integration in $y$ , proves the $L^{2}$ contraction. Equivalently the latter follows by Young applied to the odd extension ${\tilde{w}}_{0}$ on the line, whose heat evolution remains odd; restricting the equal two halfline norms gives the same bound. The Gaussian derivative bounds justify differentiating the convolution for every $t > 0$ , and give $w_{t} = ν w_{y y}$ and the zero boundary value. Approximate-identity convergence applied to ${\tilde{w}}_{0}$ gives strong $L^{2}$ initial convergence.

2.1F1F2step 1.1construct

Fix a smooth $β : [0, \infty) \to [0, 1]$ equal to $0$ on $[0, 1]$ and $1$ on $[2, \infty)$ . The explicit $w_{0, j} (y) = β (j y) w_{0} (y)$ are smooth and compactly supported away from zero and converge to $w_{0}$ in $L^{2}$ by dominated convergence. Evolve their odd extensions. Integration by parts on the line, with Gaussian tails, gives $∥ w (t) ∥_{2}^{2} + 2 ν \int_{0}^{t} ∥ w_{y} ∥_{2}^{2} = ∥ w_{0} ∥_{2}^{2}$ for each approximant. Applying this identity to differences makes the evolutions Cauchy in $L_{t}^{\infty} L_{y}^{2}$ and their derivatives Cauchy in $L_{t, y}^{2}$ . Their limit is the kernel evolution by its contraction, and its weak derivative is the derivative limit by integration against compactly supported tests. This gives the stated energy inequality. For a smooth halfline function vanishing at zero and decaying at infinity, the fundamental theorem gives $∥ w ∥_{\infty}^{2} \leq 2 ∥ w ∥_{2} ∥ w_{y} ∥_{2}$ , hence $∥ w ∥_{4}^{4} \leq 2 ∥ w ∥_{2}^{3} ∥ w_{y} ∥_{2}$ . Time integration, the energy bound and Holder prove the $L^{4}$ assertion; approximation passes the inequality to this $w$ .

2.2F1F2step 1.1algebra

Integrating the Gaussian kernel first in $y$ gives $M (t) = \int_{0}^{\infty} \erf (z / (2 \sqrt{ν t})) w_{0} (z) d z$ , where $\erf (r) = 2 π^{- 1 / 2} \int_{0}^{r} e^{- s^{2}} d s$ . Dominated convergence gives $M (t) \to M_{0}$ , and differentiation for positive time gives $m^{'} (t) = - \frac{t^{- 3 / 2}}{2 \sqrt{π ν}} \int_{0}^{\infty} z e^{- z^{2} / (4 ν t)} w_{0} (z) d z .$ After $z = \sqrt{t} r$ , the prefactor is $t^{- 3 / 4}$ . The integral on $1 \leq r \leq 2$ is bounded below by $c L (t)^{- α}$ for small $t$ . For $r \leq t^{- 1 / 4}$ , $\log (1 / (\sqrt{t} r)) \geq L (t) / 4$ , so the integral is bounded above by $C L (t)^{- α} \int_{0}^{\infty} r^{1 / 2} e^{- r^{2} / (4 ν)} d r$ . For $r > t^{- 1 / 4}$ the Gaussian tail is exponentially small; on the support of $η (\sqrt{t} r)$ the logarithmic factor is uniformly bounded above, which gives the same upper bound. This proves both derivative bounds. Integrating the upper bound from zero gives $∣ m (t) ∣ \leq C t^{1 / 4} L (t)^{- α}$ .

3.1step 2.2algebra

The upper derivative bound is integrable. The lower bound gives $\int_{0}^{t_{*}} ∣ m^{'} ∣^{4 / 3} \geq c \int_{0}^{t_{*}} t^{- 1} L (t)^{- 4 α / 3} d t = \infty$ , since $4 α / 3 \leq 1$ . In contrast the squared critical-corrector expression is at most $C t^{- 1} L (t)^{- 2 α}$ , which is integrable since $2 α > 1$ . Away from time zero every quantity is smooth, proving all stated time-integral conclusions.

4.1F1F2step 1.1step 2.2algebra∎

Differentiating the tail integral gives $h_{y} = w$ and $h_{t} = ν w_{y} = ν h_{y y}$ ; the boundary term at infinity vanishes by Gaussian decay. Also $h (t, y) = \int_{0}^{y} w (t, z) d z - M (t)$ . Thus on $[0, B]$ its difference from $h_{0}$ is at most $\sqrt{B} ∥ w (t) - w_{0} ∥_{2} + ∣ M (t) - M_{0} ∣$ , which tends to zero. Finally a positive separation between $y$ and $supp w_{0}$ puts every differentiated Gaussian convolution under the bound $C_{N} t^{- N} e^{- c / t} ∥ w_{0} ∥_{1}$ . These arguments prove the remaining assertions.

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A shrinking clamped-stream correction on the square

Statement

Assume the Axiom of Choice. Let $D = (0, 1)^{2}$ , $b \in C^{\infty} (\overline{D})$ vanish on $\partial D$ , and choose $ρ \in C^{\infty} ([0, \infty))$ equal to $1$ on $[0, 1 / 3]$ and $0$ on $[1, \infty)$ . Set $d_{1} = x$ , $d_{2} = 1 - x$ , $d_{3} = y$ , $d_{4} = 1 - y$ , and $R_{ℓ} (x, y) = 1 - \prod_{i = 1}^{4} (1 - ρ (d_{i} / ℓ)), 0 < ℓ < 1 / 16.$ This function is $1$ near the boundary and supported in strips of total area at most $4 ℓ$ . There is $C$ , independent of $ℓ$ , such that $∥ \nabla (b R_{ℓ}) ∥_{\infty} \leq C, ∥ D^{k} (b R_{ℓ}) ∥_{2} \leq C ℓ^{3 / 2 - k} (k = 1, 2, 3) .$ For $ℓ (t) = \sqrt{t}$ , $0 < t < T < 1 / 256$ , $∥ \partial_{t} \nabla (b R_{\sqrt{t}}) ∥_{2} \leq C t^{- 1} t^{1 / 4} .$ If $m \in C^{1} ((0, T))$ , $rot ψ = (ψ_{y}, - ψ_{x})$ , and $r = - m rot (b R_{\sqrt{t}})$ , then $∥ r ∥_{\infty} \leq C ∣ m ∣, ∥ r ∥_{2} \leq C ∣ m ∣ t^{1 / 4}, ∥ \nabla r ∥_{2} \leq C ∣ m ∣ t^{- 1 / 4},$ $∥ r_{t} ∥_{2} + ∥ Δ r ∥_{2} \leq C (∣ m^{'} ∣ t^{1 / 4} + ∣ m ∣ t^{- 3 / 4}) .$ Moreover $b R_{ℓ} = 0$ and $\partial_{n} (b R_{ℓ}) = \partial_{n} b$ on each side. Thus if a smooth stream $Ψ$ has zero boundary value and $\partial_{n} Ψ = m \partial_{n} b$ , then $Ψ - m b R_{ℓ}$ has both zero value and zero normal derivative there, and its rotation has zero velocity trace. On any fixed $J ⋐ (0, 1)$ and sufficiently small $t$ , the additional estimate $∥ \partial_{x} r ∥_{L^{\infty} (J \times (0, 1))} \leq C_{J} ∣ m ∣$ holds.

Facts & Assumptions

Given: The Axiom of Choice, the square, smooth $b, ρ$ , and the parameters of the Statement.

[F1]

The product and chain rules apply to these finite products of smooth scalar functions. (Chain sum product and composition rules for Banach derivatives)

[F2]

A function supported in a set of area at most $A$ and bounded by $B$ has $L^{2}$ norm at most $A^{1 / 2} B$ . (Holder's inequality for integrals, including the endpoint cases)

Proof

1.1F1givenalgebra

The product formula shows that $R_{ℓ} = 1$ whenever some $d_{i} < ℓ / 3$ , and $R_{ℓ} = 0$ whenever every $d_{i} \geq ℓ$ . The supporting union has area at most $4 ℓ$ . Smoothness follows from the finite product. Because $b = 0$ on each side, the fundamental theorem in its normal coordinate gives $∣ b (x, y) ∣ \leq C d_{i} (x, y)$ for every side $i$ , and $\nabla b$ and all fixed higher derivatives are bounded on the closed square.

2.1F1F2step 1.1algebra

Expand $D^{k} (b R_{ℓ})$ by the product rule. Every spatial derivative on a cutoff costs at most $C ℓ^{- 1}$ . If at least one cutoff is differentiated, choose that side $i$ ; on that term's support $d_{i} \leq ℓ$ , so an undifferentiated $b$ supplies a factor $C ℓ$ . Thus the terms with no derivative on $b$ have magnitude at most $C ℓ^{1 - k}$ . If $j \geq 1$ derivatives hit $b$ , the other factors cost at most $C ℓ^{- (k - j)} \leq C ℓ^{1 - k}$ . Terms with no differentiated cutoff have the same bound for $k \geq 1$ . All terms are supported in the union from step 1.1. Therefore $∥ D^{k} (b R_{ℓ}) ∥_{2} \leq C ℓ^{3 / 2 - k}$ for $k = 1, 2, 3$ ; for $k = 1$ the pointwise bound also gives $∥ \nabla (b R_{ℓ}) ∥_{\infty} \leq C$ . Intersections of strips require no disjointness: the bound uses any differentiated side and the total union area.

3.1F1F2step 2.1algebra

With $ℓ = \sqrt{t}$ , a time derivative of $ρ (d_{i} / ℓ)$ is bounded by $C t^{- 1}$ and is supported where $d_{i} \leq ℓ$ . One extra spatial derivative costs at most $C ℓ^{- 1}$ , offset by $b = O (ℓ)$ if it does not hit $b$ . If it hits $b$ , its derivative is bounded. Hence $∣ \partial_{t} \nabla (b R_{\sqrt{t}}) ∣ \leq C t^{- 1}$ on the same union of strips, and its $L^{2}$ norm is at most $C t^{- 1} ℓ^{1 / 2}$ . Differentiating $r$ and applying step 2.1 with $k = 1, 2, 3$ gives every asserted bound for $r$ , $r_{t}$ and $Δ r$ .

4.1F1step 1.1step 2.1givenalgebra∎

Near each boundary point $R_{ℓ} = 1$ , so the boundary value of $b R_{ℓ}$ is zero and its normal derivative equals that of $b$ . Subtraction therefore cancels the stated stream trace and normal derivative. The tangential derivative of a zero boundary stream is zero, so its rotation has both zero normal and zero tangential velocity on each side. For $x \in J$ and sufficiently small $t$ , the cutoffs of the vertical sides vanish identically and $R$ depends only on $y$ . Since $b (x, 0) = b (x, 1) = 0$ , also $b_{x}, b_{x x}$ vanish on those edges and satisfy $∣ b_{x} ∣ + ∣ b_{x x} ∣ \leq C \min (y, 1 - y)$ . Consequently $\partial_{x} r = - m rot (b_{x} R)$ is bounded by $C_{J} ∣ m ∣$ : the only factor $R_{y} = O (ℓ^{- 1})$ is multiplied by $b_{x} = O (ℓ)$ in its supporting strip. This proves the last estimate and completes the proof.

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Scalar weak equations give a weakly continuous Hilbert-space representative

Statement

Assume the Axiom of Choice. Let $0 < T < \infty$ , let $H$ be a separable real Hilbert space, let $X \subseteq H$ be a dense real linear subspace, and let $u_{0} \in H$ . Suppose $u \in L^{2} (0, T; H)$ and $∥ u (t) ∥_{H} \leq M$ for almost every $t$ , for some finite $M$ . Suppose also that for every $v \in X$ there is $h_{v} \in L^{1} (0, T)$ such that, for every $η \in C^{1} ([0, T])$ vanishing near $T$ , $- \int_{0}^{T} (u (t), v)_{H} η^{'} (t) d t = \int_{0}^{T} h_{v} (t) η (t) d t + (u_{0}, v)_{H} η (0) .$ Then there is a unique weakly continuous map $\tilde{u} : [0, T] \to H$ equal to $u$ almost everywhere and satisfying $\tilde{u} (0) = u_{0}$ . Here weak continuity means that $t \mapsto (\tilde{u} (t), w)_{H}$ is continuous for every $w \in H$ .

Facts & Assumptions

Given: The Axiom of Choice, $0 < T < \infty$ , a separable real Hilbert space $H$ , a dense real linear subspace $X \subseteq H$ , $u_{0} \in H$ , and the scalar weak equations in the Statement.

[F1]

If a scalar $g \in L^{1} (0, T)$ has distributional derivative $h \in L^{1} (0, T)$ , then it has a unique absolutely continuous representative on $[0, T]$ with derivative $h$ almost everywhere. (One-dimensional $W^{1, p}$ functions have unique absolutely continuous representatives)

[F2]

A locally integrable function that induces the zero distribution is zero almost everywhere; thus an $L^{1}$ distributional derivative is unique almost everywhere. (Locally integrable functions embed in distributions)

[F3]

Every bounded real linear functional on a real Hilbert space is inner product with a unique vector. (Riesz representation for Hilbert spaces)

[F4]

A separable space has a countable dense subset. (Separability: the existence of an at most countable dense subset)

[F5]

A countable union of null sets is null. (A countable union of measure-zero sets has measure zero, by countable choice)

[A1]

The Axiom of Choice supplies Countable Choice, which selects a point from each nonempty member of the countable family of balls used to construct a dense subset of $X$ . (The Axiom of Choice, AC supplies the countable and dependent choices used in Banach integration, The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

1.1A1F4givenchoose

Choose a countable dense set $D \subseteq H$ by [F4]. For every $d \in D$ and positive rational $r$ for which $B_{H} (d, r) \cap X \neq \emptyset$ , choose one point of that intersection using [A1]; the resulting countable set $W \subseteq X$ is dense in $H$ . Indeed, given $y \in H$ and $ϵ > 0$ , choose $d \in D$ with $∥ y - d ∥_{H} < ϵ / 4$ , then a rational $r$ with $∥ y - d ∥_{H} < r < ϵ - ∥ y - d ∥_{H}$ ; density of $X$ makes the selected point available and it lies within $ϵ$ of $y$ . The rational linear span $W_{0}$ of $W$ is still countable and dense in $H$ .

1.2A1F1given

For $w \in W_{0}$ , set $g_{w} (t) = (u (t), w)_{H}$ almost everywhere. Since $T < \infty$ and $u \in L^{2} (0, T; H)$ , $g_{w} \in L^{1} (0, T)$ ; the scalar weak equation implies its distributional derivative is $h_{w}$ . By [F1] it has a unique absolutely continuous representative $G_{w}$ on $[0, T]$ . Integration by parts for $G_{w}$ and comparison with the given identity give $G_{w} (0) = (u_{0}, w)_{H}$ . Moreover $∣ g_{w} (t) ∣ \leq M ∥ w ∥_{H}$ almost everywhere, so continuity gives $∣ G_{w} (t) ∣ \leq M ∥ w ∥_{H}$ for every $t \in [0, T]$ .

2.1F2step 1.1step 1.2algebra

For $w, z \in W_{0}$ and rational $q$ , the identities $g_{w + z} = g_{w} + g_{z}$ and $g_{q w} = q g_{w}$ hold almost everywhere. The corresponding identities for the continuous representatives hold everywhere, since two continuous functions equal almost everywhere on $[0, T]$ are equal everywhere. Thus for each fixed $t$ , $w \mapsto G_{w} (t)$ is rational-linear on $W_{0}$ and bounded by $M ∥ w ∥_{H}$ . It extends uniquely by continuity from the dense set $W_{0}$ to a bounded real-linear functional $L_{t}$ on $H$ ; real linearity follows by approximating any real scalar by rationals.

3.1F3F5step 1.1step 1.2step 2.1algebra

By [F3], there is a unique $\tilde{u} (t) \in H$ with $(\tilde{u} (t), w)_{H} = L_{t} (w)$ for every $w \in H$ . For $w \in W_{0}$ , $t \mapsto (\tilde{u} (t), w)_{H} = G_{w} (t)$ is continuous, and $∥ \tilde{u} (t) ∥_{H} \leq M$ . For any $z \in H$ , approximate it in norm by $w_{k} \in W_{0}$ ; then $\sup_{t} ∣ (\tilde{u} (t), z - w_{k})_{H} ∣ \leq M ∥ z - w_{k} ∥_{H}$ , so $t \mapsto (\tilde{u} (t), z)_{H}$ is a uniform limit of continuous functions and is continuous. Since $G_{w} (0) = (u_{0}, w)_{H}$ on the dense set $W_{0}$ , Riesz uniqueness gives $\tilde{u} (0) = u_{0}$ . Choose a strongly measurable representative of $u$ . For each $w \in W_{0}$ , $(\tilde{u} (t), w)_{H} = (u (t), w)_{H}$ outside a null set; [F5] makes their countable union null. Outside that union the two vectors have the same inner products with the dense set $W_{0}$ , hence are equal. Thus $\tilde{u} = u$ almost everywhere as $H$ -valued functions.

4.1step 3.1algebra∎

If ${\tilde{u}}_{1}$ and ${\tilde{u}}_{2}$ are two weakly continuous representatives of $u$ , then for every $z \in H$ their scalar pairings are continuous and agree almost everywhere, hence agree for every $t$ . Since this holds for every $z$ , the Hilbert inner product separates vectors and ${\tilde{u}}_{1} (t) = {\tilde{u}}_{2} (t)$ for every $t$ .

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The skew convective form for incompressible flow

Definition

Let $D \subset R^{d}$ be a bounded connected Lipschitz domain with $d \in {2, 3}$ , and let $V = H_{0}^{1} (D; R^{d})$ as in Stokes velocity and pressure spaces and weak forms. For $a, b, c \in V$ , define the real skew convective form $c (a; b, c) : = \frac{1}{2} \int_{D} ((a \cdot \nabla) b \cdot c - (a \cdot \nabla) c \cdot b) d x .$ The middle and last slots are displayed separately to emphasize $c (a; b, c) = - c (a; c, b)$ . In particular $c (a; b, b) = 0$ . The integrals are well-defined on $V$ by Continuity, energy cancellation, and Sobolev bounds for skew convection ↗. If additionally $div a = 0$ , then $c (a; b, c) = \int_{D} (a \cdot \nabla) b \cdot c$ .

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Continuity, energy cancellation, and Sobolev bounds for skew convection

Statement

Assume Countable Choice. For $d \in {2, 3}$ and $V = H_{0}^{1} (D; R^{d})$ on a bounded Lipschitz domain, the form in The skew convective form for incompressible flow is a continuous real trilinear map $V^{3} \to R$ , is skew in its last two slots, and satisfies $∣ c (a; b, w) ∣ \leq C_{D} ∥ a ∥_{L^{2}}^{1 / 2} ∥ \nabla a ∥_{L^{2}}^{1 / 2} ∥ \nabla b ∥_{L^{2}} ∥ \nabla w ∥_{L^{2}} .$ Consequently, if $u \in L^{\infty} (0, T; L^{2}) \cap L^{2} (0, T; V)$ , then $c (u; u, \cdot) \in L^{4 / 3} (0, T; V^{*})$ . If $div u = 0$ and $d = 2$ , the sharper estimate $∣ c (u; u, w) ∣ \leq C_{D} ∥ u ∥_{L^{2}} ∥ \nabla u ∥_{L^{2}} ∥ \nabla w ∥_{L^{2}}$ holds. If $div a = 0$ , the skew form agrees with the advective form $\int_{D} (a \cdot \nabla) b \cdot w$ .

Facts & Assumptions

Given: Countable Choice, a bounded connected Lipschitz domain $D \subset R^{d}$ , $d = 2$ or $3$ , the zero-boundary space $V$ , and $a, b, w \in V$ .

[F1]

$V = H_{0}^{1} (D; R^{d})$ is the Sobolev closure of compactly supported smooth fields; its gradient seminorm is a norm equivalent to the full $H^{1}$ norm on bounded $D$ . (Stokes velocity and pressure spaces and weak forms, The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction)

[F2]

Under Countable Choice, the endpoint Sobolev inequality is $∥ z ∥_{d / (d - 1)} \leq C_{d} ∥ \nabla z ∥_{1}$ for compactly supported smooth scalar functions on $R^{d}$ . Holder controls products and finite-measure inclusions. (The p=1 Gagliardo-Nirenberg-Sobolev inequality, Holder's inequality for integrals, including the endpoint cases)

[F3]

The form is defined by the skew difference in its definition. (The skew convective form for incompressible flow)

[F4]

For $z \in H_{0}^{1} (D)$ in dimension two, linewise fundamental-theorem estimates and the $H_{0}^{1}$ closure give $∥ z ∥_{4}^{2} \leq C_{D} ∥ z ∥_{2} ∥ \nabla z ∥_{2}$ . (Zero-boundary Sobolev space as a norm closure, Stokes velocity and pressure spaces and weak forms)

[F5]

The one-dimensional vector-valued fundamental theorem applies on each horizontal and vertical line for the smooth compactly supported field. (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz)

[A1]

Countable Choice selects the zero-boundary smooth approximation sequence used to pass inequalities to $H_{0}^{1}$ . (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Zero-boundary Sobolev space as a norm closure)

Proof

1.1A1F1F2algebra

For real $ψ \in C_{c}^{\infty} (D)$ extended by zero, in $d = 3$ apply [F2] to $ψ^{4}$ : $∥ ψ ∥_{6}^{4} \leq C \int ∣ ψ ∣^{3} ∣ \nabla ψ ∣ \leq C ∥ ψ ∥_{6}^{3} ∥ \nabla ψ ∥_{2}$ . In $d = 2$ apply it to $ψ^{3}$ : $∥ ψ ∥_{6}^{3} \leq C \int ∣ ψ ∣^{2} ∣ \nabla ψ ∣ \leq C ∥ ψ ∥_{4}^{2} ∥ \nabla ψ ∥_{2} \leq C ∣ D ∣^{1 / 6} ∥ ψ ∥_{6}^{2} ∥ \nabla ψ ∥_{2}$ . Divide by the indicated power of $∥ ψ ∥_{6}$ when nonzero; the zero case is immediate. Thus $∥ ψ ∥_{6} \leq C_{D} ∥ \nabla ψ ∥_{2}$ in both dimensions. Choose smooth approximants by [A1]; applying the bound to their differences and identifying their $L^{6}$ and $L^{2}$ limits gives the same bound on $H_{0}^{1} (D)$ , componentwise also for vector fields. Holder interpolation then gives $∥ a ∥_{3} \leq ∥ a ∥_{2}^{1 / 2} ∥ a ∥_{6}^{1 / 2} \leq C_{D} ∥ a ∥_{2}^{1 / 2} ∥ \nabla a ∥_{2}^{1 / 2}$ , with $∥ b ∥_{6} + ∥ w ∥_{6} \leq C_{D} (∥ \nabla b ∥_{2} + ∥ \nabla w ∥_{2})$ .

1.2F3algebra

Setting $w = b$ in [F3] makes its two integrands identical, hence $c (a; b, b) = 0$ for all $a, b \in V$ . Trilinearity follows from linearity of the weak gradient and integral.

1.3A1F1F2F4F5

For a scalar $ψ \in C_{c}^{\infty} (D)$ , extend by zero to $R^{2}$ . The one-dimensional fundamental theorem [F5] on horizontal and vertical lines gives $∣ ψ (x, y) ∣^{2} \leq 2 \int_{R} ∣ ψ (s, y) \partial_{x} ψ (s, y) ∣ d s$ and $∣ ψ (x, y) ∣^{2} \leq 2 \int_{R} ∣ ψ (x, t) \partial_{y} ψ (x, t) ∣ d t$ . Multiplying, integrating, and applying Holder [F2] yields $∥ ψ ∥_{4}^{4} \leq 4 ∥ ψ ∥_{2}^{2} ∥ \partial_{x} ψ ∥_{2} ∥ \partial_{y} ψ ∥_{2} \leq 2 ∥ ψ ∥_{2}^{2} ∥ \nabla ψ ∥_{2}^{2}$ . Countable Choice [A1] selects a smooth sequence converging in $H^{1}$ to any $H_{0}^{1}$ input, so the estimate passes by norm convergence; componentwise Holder gives [F4] for vector fields.

2.1F2F3step 1.1

Apply Holder with exponents $3, 2, 6$ from [F2] to each integral in [F3]. This gives $∣ c (a; b, w) ∣ \leq \frac{1}{2} ∥ a ∥_{3} (∥ \nabla b ∥_{2} ∥ w ∥_{6} + ∥ \nabla w ∥_{2} ∥ b ∥_{6})$ , so step 1.1 proves the stated bound and continuity.

3.1A1F1F3step 2.1algebra

If $div a = 0$ , [A1] selects smooth compactly supported sequences $b_{k} \to b$ and $w_{k} \to w$ in $H_{0}^{1}$ . For every $k$ , the distributional definition of $div a = 0$ gives $\int_{D} a \cdot \nabla (b_{k} \cdot w_{k}) = 0$ . Expanding by the product rule gives $\int_{D} (a \cdot \nabla w_{k}) \cdot b_{k} = - \int_{D} (a \cdot \nabla b_{k}) \cdot w_{k}$ . The trilinear bound in step 2.1 passes both sides to the limit, and substitution into [F3] yields $c (a; b, w) = \int_{D} (a \cdot \nabla b) \cdot w$ .

4.1F1F2F4step 2.1step 3.1algebra∎

The bound in the Statement with $a = b = u$ gives $∥ c (u; u, \cdot) ∥_{V^{*}} \leq C_{D} ∥ u ∥_{2}^{1 / 2} ∥ \nabla u ∥_{2}^{3 / 2}$ . The continuous quadratic map $v \mapsto c (v; v, \cdot)$ sends simple-function approximants of the Bochner $V$ representative of $u$ to simple $V^{*}$ -valued approximants, proving strong measurability. Raising to $4 / 3$ and integrating in time is finite because $∥ u ∥_{2}$ is essentially bounded and $\nabla u \in L^{2} (0, T; L^{2})$ . If additionally $div u = 0$ in $d = 2$ , step 3.1 and [F4] yield $∣ c (u; u, w) ∣ = ∣ \int_{D} (u \cdot \nabla w) \cdot u ∣ \leq C_{D} ∥ u ∥_{2} ∥ \nabla u ∥_{2} ∥ \nabla w ∥_{2}$ , which is an $L^{2} (0, T; V^{*})$ bound.

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Energy weak solutions of incompressible Navier–Stokes

Definition

Assume Countable Choice, as in the Stokes-space conventions of Stokes velocity and pressure spaces and weak forms. Let $D \subset R^{d}$ be a nonempty bounded connected Lipschitz domain, $d \in {2, 3}$ , and fix $0 < T < \infty$ and $ν > 0$ . Use $V = H_{0}^{1} (D; R^{d})$ and $Q = L_{0}^{2} (D)$ from Stokes velocity and pressure spaces and weak forms, the viscosity form $a_{ν} (v, w) = 2 ν \int_{D} ε (v) : ε (w)$ , and the form $c$ from The skew convective form for incompressible flow. Set $V_{σ} = {v \in V : div v = 0 in L^{2} (D)}, H_{σ} = {\overline{V_{σ}}}^{L^{2} (D)} .$ For $f \in L^{2} (0, T; L^{2} (D; R^{d}))$ and $u_{0} \in H_{σ}$ , an energy weak solution is a velocity $u \in L^{\infty} (0, T; H_{σ}) \cap L^{2} (0, T; V_{σ})$ , weakly continuous into $H_{σ}$ , and a pressure $p \in L^{4 / 3} (0, T; Q)$ with zero spatial mean a.e., such that $div u = 0$ a.e. and, for every $φ \in C^{1} ([0, T]; V)$ vanishing near $T$ , $- \int_{0}^{T} (u, \partial_{t} φ)_{L^{2}} d t + \int_{0}^{T} (a_{ν} (u, φ) + c (u; u, φ) + b (φ, p)) d t = \int_{0}^{T} (f, φ)_{L^{2}} d t + (u_{0}, φ (0))_{L^{2}} .$ It also satisfies, for almost every $t \in (0, T)$ , $\frac{1}{2} ∥ u (t) ∥_{L^{2}}^{2} + ν \int_{0}^{t} ∥ \nabla u (s) ∥_{L^{2}}^{2} d s \leq \frac{1}{2} ∥ u_{0} ∥_{L^{2}}^{2} + \int_{0}^{t} (f (s), u (s))_{L^{2}} d s .$ No uniqueness is included in this definition. The formulation uses the real duality convention of Banach evolution triples and weak time derivatives.

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Integrability of the energy weak Navier–Stokes formulation

Statement

Assume Countable Choice. For the energy class and data in Energy weak solutions of incompressible Navier–Stokes, every term in its time-tested momentum identity is integrable, and the nonlinear map $t \mapsto c (u (t); u (t), \cdot)$ belongs to $L^{4 / 3} (0, T; V^{*})$ .

Facts & Assumptions

Given: Countable Choice, $u \in L^{\infty} (0, T; H_{σ}) \cap L^{2} (0, T; V_{σ})$ , $p \in L^{4 / 3} (0, T; Q)$ , $f \in L^{2} (0, T; L^{2})$ , $ν > 0$ , and a test $φ \in C^{1} ([0, T]; V)$ .

[F1]

The energy class and weak-form conventions are those in the definition, including the Hilbert evolution triple used for the time-dependent pairings. (Energy weak solutions of incompressible Navier–Stokes, Banach evolution triples and weak time derivatives)

[F2]

The convective form obeys $∥ c (u; u, \cdot) ∥_{V^{*}} \leq C ∥ u ∥_{2}^{1 / 2} ∥ \nabla u ∥_{2}^{3 / 2}$ in both dimensions, and the viscosity form is bounded on $V \times V$ . (Continuity, energy cancellation, and Sobolev bounds for skew convection, Stokes velocity and pressure spaces and weak forms)

[A1]

Countable Choice is inherited from the weak-solution and Stokes-space conventions; the Bochner integrability criterion itself applies to the strongly measurable functions stated here. (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Energy weak solutions of incompressible Navier–Stokes, Bochner-integrable function, Bochner integrability criterion)

[F3]

Holder's inequality controls time and space pairings at conjugate exponents. (Holder's inequality for integrals, including the endpoint cases)

Proof

1.1A1F1F2algebra

The Bochner $L^{2} (0, T; V)$ hypothesis makes $u$ strongly measurable in $V$ . The continuous trilinear bound [F2] makes $v \mapsto c (v; v, \cdot)$ a continuous map $V \to V^{*}$ , so applying it to simple-function approximants of $u$ shows that $t \mapsto c (u (t); u (t), \cdot)$ is strongly measurable. Since $∥ u (t) ∥_{2}$ is essentially bounded and $\nabla u \in L^{2} (0, T; L^{2})$ , [F2] gives $\int_{0}^{T} ∥ c (u; u, \cdot) ∥_{V^{*}}^{4 / 3} d t \leq C ∥ u ∥_{L^{\infty} L^{2}}^{2 / 3} \int_{0}^{T} ∥ \nabla u ∥_{2}^{2} d t < \infty$ .

1.2A1F1F3algebra

The time derivative term pairs $u \in L^{\infty} (0, T; H_{σ})$ with $\partial_{t} φ \in L^{\infty} (0, T; L^{2})$ ; the viscosity term is bounded by $2 ν ∥ \nabla u ∥_{L^{2} L^{2}} ∥ \nabla φ ∥_{L^{2} L^{2}}$ . Since $∣ div φ ∣ \leq \sqrt{d} ∣ \nabla φ ∣$ , the pressure term is finite by Holder from $p \in L^{4 / 3} L^{2}$ and $φ \in L^{\infty} V$ . The forcing term is finite because $f \in L^{2} L^{2}$ and $φ$ is bounded in $L^{\infty} L^{2}$ .

2.1A1step 1.1step 1.2∎

Each spatial pairing is therefore finite, and step 1.1 gives the stated Bochner integrability of the nonlinear functional; the time-tested weak formulation is well-defined.

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Solenoidal energy weak velocity for incompressible Navier–Stokes

Definition

Assume Countable Choice as inherited from Energy weak solutions of incompressible Navier–Stokes. Let $D$ , $V$ , $Q$ , $V_{σ}$ , $H_{σ}$ , $a_{ν}$ , and $c$ be as in Energy weak solutions of incompressible Navier–Stokes; the Stokes velocity and pressure spaces are $V = H_{0}^{1} (D)^{d}$ and $Q = L_{0}^{2} (D)$ as specified in Stokes velocity and pressure spaces and weak forms. For $f \in L^{2} (0, T; L^{2} (D)^{d})$ and $u_{0} \in H_{σ}$ , a solenoidal energy weak velocity is a velocity $u \in L^{\infty} (0, T; H_{σ}) \cap L^{2} (0, T; V_{σ})$ that is weakly continuous into $H_{σ}$ , satisfies $div u = 0$ almost everywhere, and obeys the scalar-test weak equation $- \int_{0}^{T} (u (t), v)_{L^{2}} η^{'} (t) d t + \int_{0}^{T} (a_{ν} (u (t), v) + c (u (t); u (t), v)) η (t) d t = \int_{0}^{T} (f (t), v)_{L^{2}} η (t) d t + (u_{0}, v)_{L^{2}} η (0)$ for every fixed $v \in V_{σ}$ and $η \in C^{1} ([0, T])$ vanishing near $T$ . It also satisfies, for almost every $t \in (0, T)$ , $\frac{1}{2} ∥ u (t) ∥_{L^{2}}^{2} + ν \int_{0}^{t} ∥ \nabla u (s) ∥_{L^{2}}^{2} d s \leq \frac{1}{2} ∥ u_{0} ∥_{L^{2}}^{2} + \int_{0}^{t} (f (s), u (s))_{L^{2}} d s .$ This solenoidal definition does not itself assert a pressure exists in $L^{4 / 3} (0, T; L_{0}^{2} (D))$ ; that is a separate regularity conclusion.

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The mixed mini-element backward Euler Navier–Stokes scheme

Definition

Assume Countable Choice as inherited from the energy-space conventions. Interpret each interval forcing average as a Bochner integral; its existence follows from the stated L2 forcing and the Bochner integrability criterion (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Bochner integrability criterion). Let $D$ be as in Energy weak solutions of incompressible Navier–Stokes, let $T_{h}$ be a conforming shape-regular affine simplicial mesh of $D$ , and let $V_{h}, Q_{h}, Z_{h}$ be the MINI spaces and weak divergence kernel of The conforming P1-bubble/P1 mini element and Weak and pointwise divergence-free discrete velocities. For a partition $0 = t_{0} < \dots < t_{N} = T$ , write $τ_{n} = t_{n} - t_{n - 1}$ and $f_{h}^{n} = τ_{n}^{- 1} \int_{t_{n - 1}}^{t_{n}} f (t) d t$ . Given $u_{h}^{0} \in Z_{h}$ , a backward Euler mini-element solution is a sequence $(u_{h}^{n}, p_{h}^{n}) \in V_{h} \times Q_{h}$ satisfying, for $n = 1, \dots, N$ , ${(\frac{u_{h}^{n} - u_{h}^{n - 1}}{τ_{n}}, v_{h})}_{L^{2}} + a_{ν} (u_{h}^{n}, v_{h}) + c (u_{h}^{n}; u_{h}^{n}, v_{h}) + b (v_{h}, p_{h}^{n}) = (f_{h}^{n}, v_{h})_{L^{2}} (v_{h} \in V_{h}),$ $b (u_{h}^{n}, q_{h}) = 0 (q_{h} \in Q_{h}) .$ All pressure variables use the zero-mean gauge $Q_{h} \subset L_{0}^{2} (D)$ . Restriction to $v_{h} \in Z_{h}$ eliminates pressure. The equations are well-defined by Well-definedness of the finite-dimensional mini backward Euler equations ↗; this definition claims no solver termination or error rate.

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Uniqueness and strong trace for two-dimensional solenoidal energy velocities

Statement

Assume the Axiom of Choice. Let $D \subset R^{2}$ be bounded, connected and Lipschitz, $T > 0$ , $ν > 0$ , $f \in L^{2} (0, T; L^{2} (D)^{2})$ and $u_{0} \in H_{σ}$ . There is at most one solenoidal energy weak velocity in the sense of Solenoidal energy weak velocity for incompressible Navier–Stokes for these data. Every such velocity belongs to $L^{4} (0, T; L^{4} (D)^{2}) \cap C ([0, T]; H_{σ})$ and has its actual restricted derivative in $L^{2} (0, T; V_{σ}^{*})$ . The continuity is strong, its initial value is $u_{0}$ , and its solenoidal evolution equation satisfies the Hilbert-triple energy chain rule. No full nonsolenoidal $V^{*}$ derivative or regular instantaneous pressure is asserted.

Facts & Assumptions

Given: The Axiom of Choice and the data and spaces in the Statement.

[A1]

Full Choice supplies Countable and Dependent Choice, including the approximation and countable selections below. (The Axiom of Choice, AC supplies the countable and dependent choices used in Banach integration)

[F1]

The velocity is in $L_{t}^{\infty} H_{σ} \cap L_{t}^{2} V_{σ}$ , is weakly continuous in $H_{σ}$ with initial value $u_{0}$ , and satisfies the fixed-solenoidal-test momentum equation. The skew convection agrees with the divergence-free convective form, and its energy pairing cancels by the proved skew-convection supplier. The strain identity polarizes to $2 (ε (u), ε (v)) = (\nabla u, \nabla v) + (div u, div v)$ ; for solenoidal inputs the Newtonian viscosity is therefore $ν (\nabla u, \nabla v)$ . (Solenoidal energy weak velocity for incompressible Navier–Stokes, Stokes velocity and pressure spaces and weak forms, The skew convective form for incompressible flow, Continuity, energy cancellation, and Sobolev bounds for skew convection, The homogeneous Dirichlet strain identity)

[F2]

$L^{2}$ is Hilbert, $H^{1}$ is complete, the zero-boundary gradient norm is equivalent by Poincare, and Hilbert spaces are reflexive. Rational-box step functions give separability of scalar $L^{2}$ . ( $L^{2}$ with the integral pairing is a Hilbert space, Integer-order Sobolev spaces are Banach, The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction, Hilbert spaces are reflexive, Rational box-step functions form a countable dense subset of $L^{p} (R^{n})$ for $1 \leq p < \infty$ )

[F3]

A separable reflexive energy space continuously and densely embedded in its Hilbert pivot forms the canonical evolution triple, and its $L_{t}^{2} V$ functions with $L_{t}^{2} V^{*}$ derivative are strongly continuous in the pivot and satisfy the energy chain rule. (Banach evolution triples and weak time derivatives, The Banach evolution-space energy identity and H trace)

[F4]

Holder and dominated convergence apply. In particular, for a nonnegative $B \in L^{1} (0, T)$ the integrable majorant $B$ gives $\int_{{B > K}} B \to 0$ as $K \to \infty$ . (Holder's inequality for integrals, including the endpoint cases, Dominated convergence)

Proof

1.1F1F4givenalgebra

For scalar $z \in C_{c}^{\infty} (R^{2})$ the coordinate fundamental theorem gives $∣ z (x, y) ∣^{2} \leq 2 \int_{R} ∣ z z_{x} ∣ (s, y) d s = : A (y)$ and $∣ z (x, y) ∣^{2} \leq 2 \int_{R} ∣ z z_{y} ∣ (x, s) d s = : B (x)$ . Thus $∣ z ∣^{4} \leq A (y) B (x)$ , and integration and Holder give $∥ z ∥_{4}^{4} \leq 4 ∥ z ∥_{2}^{2} ∥ z_{x} ∥_{2} ∥ z_{y} ∥_{2} \leq 2 ∥ z ∥_{2}^{2} ∥ \nabla z ∥_{2}^{2}$ . Apply this to components of a vector and to differences of compact smooth approximants defining $H_{0}^{1} (D)^{2}$ , extended by zero. The inequality makes the approximants Cauchy in $L^{4}$ , so their limit is the same $L^{2}$ function; consequently $∥ z ∥_{4}^{2} \leq C ∥ z ∥_{2} ∥ \nabla z ∥_{2}$ on $H_{0}^{1} (D)^{2}$ . Each energy velocity therefore belongs to $L_{t}^{4} L_{x}^{4}$ .

1.2A1F1F2F3constructalgebra

With its $H^{1}$ inner product the complete space $H^{1} (D)^{2}$ is Hilbert. Its closed subspace $H_{0}^{1} (D)^{2}$ is Hilbert, as is $V_{σ}$ , the kernel of bounded divergence therein. Poincare gives its equivalent gradient norm and [F2] gives reflexivity. Separability follows by mapping $H^{1} (D)^{2}$ isometrically, in its full norm, into the finite product of scalar $L^{2}$ spaces listing components and first derivatives. This product is separable by [F2]; every metric subspace of a separable metric space is separable: restrict a countable ambient base and, using Countable Choice, select a point from each nonempty restricted basic set. This proves separability of $V_{σ}$ . The closure $H_{σ}$ is Hilbert as a closed $L^{2}$ subspace. Its inclusion of $V_{σ}$ is continuous by Poincare, injective and dense by definition. Thus the canonical pivot map $J h (v) = (h, v)_{2}$ is bounded and injective and the triple has all hypotheses of [F3].

2.1F1F3F4step 1.1step 1.2algebra

For a solenoidal energy velocity $u$ , its convection functional on $V_{σ}$ is $N (u) (v) = - \int_{D} (u \otimes u) : \nabla v$ , since the divergence and boundary terms vanish by approximation. Holder bounds its norm by $∥ u ∥_{4}^{2}$ , which belongs to $L^{2} (0, T)$ by step 1.1. This is a strongly measurable dual-valued function: the map $L^{4} (D)^{2} \to V_{σ}^{*}$ is continuous and quadratic, and $u$ is strongly measurable in $L^{4}$ as a continuous image of its Bochner $V_{σ}$ representative. The viscous functional is a bounded linear image of $u \in L_{t}^{2} V_{σ}$ , and the force functional is a bounded linear image of $f \in L_{t}^{2} L^{2}$ . The fixed-test equation [F1] therefore says that the actual restricted derivative is the $L_{t}^{2} V_{σ}^{*}$ function $f - ν (\nabla u, \nabla \cdot) - N (u)$ . Apply [F3] to obtain a unique strong $C ([0, T]; H_{σ})$ representative and its energy chain rule. This representative agrees with the given weakly continuous one (fixed dense tests identify them), and so its initial value is $u_{0}$ .

3.1F1F3F4step 1.1step 2.1algebra∎

For two such velocities $u, \tilde{u}$ with the same data let $z = u - \tilde{u}$ . Their difference has restricted derivative in $L_{t}^{2} V_{σ}^{*}$ by step 2.1, so [F3] permits pairing its equation with $z$ and differentiating its squared pivot norm. The trilinear cancellation leaves, with either velocity named $u$ , $\frac{1}{2} \frac{d}{d t} ∥ z ∥_{2}^{2} + ν ∥ \nabla z ∥_{2}^{2} \leq ∣ \int_{D} (z \cdot \nabla) u \cdot z ∣ \leq C ∥ \nabla u ∥_{2} ∥ z ∥_{2} ∥ \nabla z ∥_{2} \leq \frac{ν}{2} ∥ \nabla z ∥_{2}^{2} + C ν^{- 1} ∥ \nabla u ∥_{2}^{2} ∥ z ∥_{2}^{2} .$ The first estimate uses the planar bound in step 1.1; the final one is $a b \leq ν a^{2} / 2 + C ν^{- 1} b^{2}$ . The initial difference is zero by step 2.1. Put $Y (t) = ∥ z (t) ∥_{2}^{2}$ and $B (t) = C ν^{- 1} ∥ \nabla u (t) ∥_{2}^{2}$ , increasing the constant to absorb the factor $2$ . The chain rule makes $Y$ absolutely continuous and gives $Y (t) \leq Y (s) + \int_{s}^{t} B (r) Y (r) d r$ , with $B \geq 0$ in $L^{1}$ . Choose $K > 0$ with $\int_{{B > K}} B < 1 / 4$ by [F4], and choose $δ > 0$ with $K δ < 1 / 4$ . Partition $[0, T]$ into finitely many intervals of length at most $δ$ ; each has $\int_{I} B < 1 / 2$ . On the first interval $Y (0) = 0$ and continuity gives $\sup_{I} Y \leq (\sup_{I} Y) \int_{I} B \leq \frac{1}{2} \sup_{I} Y$ , hence $Y = 0$ . Inductively the same argument applies on every next interval since its initial value is zero. Thus $z = 0$ throughout $[0, T]$ . This proves uniqueness and, together with the earlier steps, all stated regularity and trace assertions.

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Well-definedness of the finite-dimensional mini backward Euler equations

Statement

Assume Countable Choice. For each fixed finite mesh and time partition in The mixed mini-element backward Euler Navier–Stokes scheme, every term of its velocity equation is a continuous finite-dimensional functional of the trial and test vectors, the pressure constraints are linear, and the forcing averages exist.

Facts & Assumptions

Given: Countable Choice, a bounded Lipschitz domain, a finite matching mesh, its mini spaces, a finite partition, $f \in L^{2} (0, T; L^{2})$ , and $u_{h}^{n - 1} \in Z_{h}$ .

[F1]

The MINI spaces are finite-dimensional subspaces of $H_{0}^{1} (D)^{d}$ and the weak kernel $Z_{h}$ is a finite-dimensional linear subspace. (The conforming P1-bubble/P1 mini element, Weak and pointwise divergence-free discrete velocities)

[F2]

The skew convective form is continuous on $V^{3}$ . (The skew convective form for incompressible flow, Continuity, energy cancellation, and Sobolev bounds for skew convection)

[A1]

Countable Choice is inherited from the MINI/Stokes space conventions; the Bochner criterion itself only uses the asserted strong measurability and integrable norm. (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The conforming P1-bubble/P1 mini element, Bochner integrability criterion)

[F3]

A strongly measurable $L^{2} (0, T; L^{2})$ function is Bochner integrable on the finite interval, since Holder makes its norm integrable. (Bochner-integrable function, Bochner integrability criterion, Holder's inequality for integrals, including the endpoint cases)

Proof

1.1F1givenalgebra

Every finite-dimensional subspace inherits the continuous inclusions and bilinear forms from $V$ and $H$ ; thus $(u_{h}^{n} - u_{h}^{n - 1}, v_{h})_{H}$ , $a_{ν} (u_{h}^{n}, v_{h})$ , and $b (v_{h}, p_{h}^{n})$ are continuous in their coordinates.

2.1A1F1F2F3step 1.1∎

The nonlinear term is continuous by [F2], and the force average is a finite vector in $L^{2} (D)^{d}$ by [F3] because each interval has $τ_{n} > 0$ . The divergence constraint is a finite set of linear equations in $V_{h}$ and $Q_{h}$ . These facts make the scheme a well-defined finite-dimensional system.

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Two-dimensional uniqueness and conditional three-dimensional weak–strong uniqueness

Statement

Assume the Axiom of Choice. Let $D \subset R^{d}$ be bounded, connected and Lipschitz, with $d \in {2, 3}$ , let $ν > 0$ , and use the energy weak-solution definition in Energy weak solutions of incompressible Navier–Stokes. For fixed data $(u_{0}, f)$ , if $d = 2$ , energy weak solutions are unique. If $d = 3$ and one velocity $v$ additionally belongs to $C^{1} ([0, T]; V_{σ})$ , has $\nabla v \in L^{1} (0, T; L^{\infty} (D))$ , and its solenoidal equation holds as an identity in $V_{σ}^{*}$ almost everywhere in time, then every other energy weak-solution velocity with the same data equals $v$ . This three-dimensional conclusion is conditional and asserts no uniqueness between arbitrary energy weak-solution velocities. In both cases the associated zero-mean pressures also agree almost everywhere.

Facts & Assumptions

Given: The Axiom of Choice, a common bounded connected Lipschitz domain, $ν > 0$ , forcing $f \in L^{2} (0, T; L^{2})$ , initial datum $u_{0} \in H_{σ}$ , and one or two energy weak solutions with these data.

[F1]

The velocity component of every pressure-bearing energy weak solution is a solenoidal energy weak velocity: testing with $η (t) v$ for fixed $v \in V_{σ}$ removes the pressure term because $div v = 0$ . (Energy weak solutions of incompressible Navier–Stokes, Solenoidal energy weak velocity for incompressible Navier–Stokes, Stokes velocity and pressure spaces and weak forms)

[F2]

On a bounded connected Lipschitz domain in two dimensions, solenoidal energy weak velocities with the same data are unique under the Axiom of Choice. (Uniqueness and strong trace for two-dimensional solenoidal energy velocities)

[F3]

The skew form satisfies $c (a; b, b) = 0$ and, for exactly divergence-free $a$ , $c (a; b, w) = \int_{D} (a \cdot \nabla) b \cdot w$ . Its continuous trilinear bound makes the time pairings integrable for the energy-class velocity and the stated strong branch. (Continuity, energy cancellation, and Sobolev bounds for skew convection)

[F4]

For $w \in V_{σ}$ , the homogeneous Korn identity gives $a_{ν} (w, w) \geq ν ∥ \nabla w ∥_{2}^{2}$ . (Stokes velocity and pressure spaces and weak forms, The homogeneous Dirichlet strain identity)

[F5]

Absolutely continuous scalar functions satisfy the fundamental theorem of calculus and are closed under products. If $B$ is absolutely continuous, then $e^{- B}$ is also absolutely continuous because the exponential is Lipschitz on the compact range of $- B$ ; its almost-everywhere derivative follows from the scalar chain rule. (Fundamental theorem of calculus for absolutely continuous functions, The product of two absolutely continuous functions is absolutely continuous, The exponential function is smooth and $(\exp)^{'} = \exp$ , The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ )

[F6]

The real exponential is strictly positive. (The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ )

[F7]

A scalar $W^{1, 1} (0, T)$ class has an absolutely continuous representative; continuous representatives equal almost everywhere agree everywhere. (One-dimensional $W^{1, p}$ functions have unique absolutely continuous representatives)

[A1]

The Axiom of Choice is assumed by the two-dimensional solenoidal uniqueness input and the scalar integration facts. (The Axiom of Choice)

[F8]

There is a bounded divergence right inverse $B_{D} : Q \to V$ . Scalar locally integrable functions are determined by their distributions, and a countable union of null sets is null. $L^{2} (D)$ is separable, by restriction of the dense rational-box step functions on $R^{d}$ . (Divergence right inverse on bounded connected Lipschitz domains, Locally integrable functions embed in distributions, A countable union of measure-zero sets has measure zero, by countable choice, Rational box-step functions form a countable dense subset of $L^{p} (R^{n})$ for $1 \leq p < \infty$ )

Proof

1.1A1F1F2given

For $d = 2$ , [F1] identifies each velocity component with a solenoidal energy weak velocity, and [F2] gives equality of those velocities for the common data.

1.2F1F3F4F7givenalgebra

Now let $d = 3$ , let $u$ be any energy weak-solution velocity, and assume $v$ satisfies the additional hypotheses in the Statement. Testing the weak equation for $u$ by time cutoffs of $v$ is allowed because $v \in C^{1} ([0, T]; V_{σ})$ ; the pressure term vanishes. The scalar pairing $(u, v)_{2}$ has an integrable distributional derivative by [F3], so [F7] makes it absolutely continuous; its derivative is $(u, v_{t})_{2} + (f, v)_{2} - a_{ν} (u, v) - c (u; u, v)$ . Both weak solutions have initial trace $u_{0}$ . Testing the strong equation for $v$ by $v$ gives its energy equality since $c (v; v, v) = 0$ . Combining that equality, the cross-pairing identity, and the weak energy inequality for $u$ yields, for almost every $t$ , $\frac{1}{2} ∥ u (t) - v (t) ∥_{2}^{2} + ν \int_{0}^{t} ∥ \nabla (u - v) ∥_{2}^{2} d s \leq \int_{0}^{t} ∣ c (u - v; v, u - v) ∣ d s$ . Here $a_{ν} (u, u) + a_{ν} (v, v) - 2 a_{ν} (u, v) = a_{ν} (u - v, u - v)$ , which is bounded below by the viscous term using [F4]. The convection terms reduce as $c (v; v, u) + c (u; u, v) = c (w; u, v) = c (w; v + w, v) = - c (w; v, w)$ for $w = u - v$ , by skewness and [F3]; the strong equation for $v$ may be paired with $w \in L^{2} (0, T; V_{σ})$ and integrated.

2.1F3step 1.2algebra

By [F3], $∣ c (w; v, w) ∣ = ∣ \int_{D} (w \cdot \nabla) v \cdot w ∣ \leq ∥ \nabla v ∥_{L^{\infty}} ∥ w ∥_{2}^{2}$ . Dropping the nonnegative viscous term in step 1.2 gives $y (t) \leq 2 \int_{0}^{t} b (s) y (s) d s$ for almost every $t$ , where $y (t) = ∥ w (t) ∥_{2}^{2}$ and $b (t) = ∥ \nabla v (t) ∥_{L^{\infty}} \in L^{1} (0, T)$ .

3.1F5F6step 2.1givenalgebra

Since $u, v \in L^{\infty} (0, T; H_{σ})$ , $y \in L^{\infty} (0, T)$ ; thus $Y (t) = \int_{0}^{t} b (s) y (s) d s$ and $B (t) = 2 \int_{0}^{t} b (s) d s$ are absolutely continuous, with $Y (0) = 0$ . Step 2.1 gives $Y^{'} (t) = b (t) y (t) \leq 2 b (t) Y (t) = B^{'} (t) Y (t)$ almost everywhere. By [F5], $e^{- B} Y$ is absolutely continuous and has derivative $e^{- B} (Y^{'} - B^{'} Y) \leq 0$ almost everywhere. It starts at zero and is nonnegative by [F6], so it is identically zero; hence $Y = 0$ and step 2.1 implies $y = 0$ almost everywhere. The difference $w = u - v$ is weakly continuous into $H_{σ}$ ; for any $t$ , choose $t_{n} \to t$ from the full-measure set where $w (t_{n}) = 0$ . Weak continuity gives $(w (t), h)_{2} = 0$ for every $h \in H_{σ}$ , so $w (t) = 0$ .

4.1A1F8step 1.1step 3.1algebra∎

In either case let $p_{1}, p_{2}$ be the associated pressures. Subtracting the full weak equations after velocity equality gives $\int_{0}^{T} b (v, p_{1} - p_{2}) η d t = 0$ for every $v \in V$ and $η \in C_{c}^{\infty} (0, T)$ . For any $q \in Q$ , choose $v = B_{D} q$ using [F8]. Then $(p_{1} - p_{2}, q)_{2}$ is zero as a time distribution and hence almost everywhere. The space $Q$ is separable: take a countable dense family in $L^{2} (D)$ from [F8] and subtract each mean. Outside the union of the null sets for that family all these pairings vanish; density implies $p_{1} (t) = p_{2} (t)$ in $Q$ . Thus the zero-mean pressures agree almost everywhere, completing both uniqueness claims.

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Mean-zero pressure recovery from a solenoidal residual

Statement

Assume Countable Choice and use the Stokes spaces $V = H_{0}^{1} (D; R^{d})$ and $Q = L_{0}^{2} (D)$ of Stokes velocity and pressure spaces and weak forms: $D \subset R^{d}$ is nonempty, bounded, connected and Lipschitz, with $d \in {2, 3}$ . Fix $0 < T < \infty$ . If $1 < s < \infty$ and $R \in L^{s} (0, T; V^{*})$ satisfies $R (t) (v) = 0$ for every $v \in V_{σ} = \ker (div : V \to L_{0}^{2} (D))$ outside one common null set, then there is a unique $p \in L^{s} (0, T; L_{0}^{2} (D))$ such that $R (t) (v) = b (v, p (t))$ for every $v \in V$ almost everywhere. Moreover, $∥ p ∥_{L^{s} L^{2}} \leq C_{d i v} (D) ∥ R ∥_{L^{s} V^{*}}$ .

Facts & Assumptions

Given: Countable Choice, $V = H_{0}^{1} (D)^{d}$ , $Q = L_{0}^{2} (D)$ , a bounded divergence right inverse $B_{D} : Q \to V$ , and a Bochner functional $R \in L^{s} (0, T; V^{*})$ annihilating $V_{σ}$ .

[F1]

Every $q \in Q$ has $B_{D} q \in V$ with $div B_{D} q = q$ and $∥ B_{D} q ∥_{V} \leq C_{d i v} (D) ∥ q ∥_{2}$ . (Divergence right inverse on bounded connected Lipschitz domains)

[F2]

Every $v \in V$ has $div v \in Q$ . (Zero mean of the divergence of a zero-boundary field)

[F3]

The pressure gauge is $Q = L_{0}^{2} (D)$ and $b (v, p) = - \int_{D} p div v$ . (Stokes velocity and pressure spaces and weak forms)

[F4]

Every bounded linear functional on the real Hilbert space $Q$ is represented by a unique $L^{2}$ inner-product vector. (Riesz representation for Hilbert spaces)

[A1]

Countable Choice is assumed by the divergence-lift and Hilbert Riesz suppliers, including the Bochner measurability of the represented pressure. (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Divergence right inverse on bounded connected Lipschitz domains, Riesz representation for Hilbert spaces)

Proof

1.1A1F1F4givenconstruct

For almost every $t$ , define a bounded linear functional $ℓ_{t}$ on $Q$ by $ℓ_{t} (q) = R (t) (B_{D} q)$ . From [F1], $∥ ℓ_{t} ∥_{Q^{*}} \leq C_{d i v} (D) ∥ R (t) ∥_{V^{*}}$ . By [F4], there is a unique $\hat{p} (t) \in Q$ such that $(\hat{p} (t), q)_{L^{2}} = ℓ_{t} (q)$ for all $q \in Q$ . The map $R (t) \mapsto \hat{p} (t)$ is bounded linear, so $\hat{p} \in L^{s} (0, T; Q)$ with the corresponding norm bound.

2.1F1F2F3step 1.1algebra

For $v \in V$ , put $q = div v \in Q$ by [F2]. Then $v - B_{D} q \in V_{σ}$ , so the annihilator assumption gives $R (t) (v) = R (t) (B_{D} q) = (\hat{p} (t), q)_{L^{2}} = - b (v, \hat{p} (t)) .$ Set $p = - \hat{p}$ to obtain $R (t) (v) = b (v, p (t))$ .

3.1F1F3step 2.1algebra∎

If $p_{1}, p_{2} \in Q$ both represent the same residual, then $b (B_{D} q, p_{1} - p_{2}) = 0$ for every $q \in Q$ . By [F1], $div B_{D} q = q$ , so $(p_{1} - p_{2}, q)_{L^{2}} = 0$ for all $q \in Q$ ; choosing $q = p_{1} - p_{2}$ proves equality. This also confirms uniqueness in the zero-mean gauge.

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The mini-element Fortin map approximates in L2

Statement

Let $Π_{h} : H_{0}^{1} (D; R^{d}) \to V_{h}$ be the MINI Fortin map constructed in Uniform Fortin stability of the mini element. On every shape-regular matching mesh, there is a constant $C$ independent of $h$ such that $∥ v - Π_{h} v ∥_{L^{2} (D)} \leq C h ∥ \nabla v ∥_{L^{2} (D)} (v \in H_{0}^{1} (D; R^{d})) .$ For every fixed $v \in H_{0}^{1} (D)^{d}$ , $Π_{h} v \to v$ in $H_{0}^{1}$ as $h \to 0$ . If $v \in V_{σ}$ , exact Fortin preservation also gives $Π_{h} v \in Z_{h}$ .

Facts & Assumptions

Given: Countable Choice, a bounded connected polyhedral domain, a shape-regular matching mesh, $v \in H_{0}^{1} (D)^{d}$ , its face-average interpolant $I_{h}$ , and the MINI bubbles.

[F1]

The Fortin construction is $Π_{h} v = I_{h} v + \sum_{K} b_{K} c_{K}$ , with $c_{K} = (\int_{K} b_{K})^{- 1} \int_{K} (v - I_{h} v)$ , and it maps $H_{0}^{1}$ into $V_{h}$ . (Uniform Fortin stability of the mini element)

[F2]

The interpolant satisfies $∥ v - I_{h} v ∥_{L^{2} (K)} \leq C h_{K} ∥ \nabla v ∥_{L^{2} (ω_{K})}$ , with uniformly bounded patch overlap. (Boundary-preserving face-average P1 interpolation)

[F3]

Reference scaling gives $\int_{K} b_{K} ≍ h_{K}^{d}$ and $∥ b_{K} ∥_{L^{2} (K)} \leq C h_{K}^{d / 2}$ . (The conforming P1-bubble/P1 mini element, Shape regularity controls affine simplex geometry)

[F4]

The Fortin map preserves the MINI pressure pairing: $b (Π_{h} v, q_{h}) = b (v, q_{h})$ for every $q_{h} \in Q_{h}$ . (Uniform Fortin stability of the mini element)

[F5]

The face-average interpolant reproduces affine fields, has uniformly bounded facet dual moments under shape scaling, and satisfies the global $H^{1}$ approximation estimate for $H^{2}$ fields. (Boundary-preserving face-average P1 interpolation, Shape regularity controls affine simplex geometry)

[F6]

The multivariable Taylor formula on a convex open set gives a second-order pointwise remainder along each segment from the Taylor base point. (Multivariable Taylor formula with a Lagrange remainder along a line segment)

[F7]

$H_{0}^{1} (D)$ is the closure of $C_{c}^{\infty} (D)$ , and the Fortin map is uniformly bounded in $H_{0}^{1}$ . (Zero-boundary Sobolev space as a norm closure, Uniform Fortin stability of the mini element)

[F8]

Holder's inequality bounds scalar-product integrals by the product of the L2 norms. (Holder's inequality for integrals, including the endpoint cases)

[A1]

Countable Choice is assumed through the Fortin construction and the stated well-definedness inputs. (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Uniform Fortin stability of the mini element)

Proof

1.1F1F3F8

Put $e_{K} = v - I_{h} v$ . By Holder and [F3], $∣ c_{K} ∣ \leq \frac{∣ K ∣^{1 / 2}}{\int_{K} b_{K}} ∥ e_{K} ∥_{L^{2} (K)} \leq C h_{K}^{- d / 2} ∥ e_{K} ∥_{L^{2} (K)} .$ Therefore $∥ b_{K} c_{K} ∥_{L^{2} (K)} \leq C ∥ e_{K} ∥_{L^{2} (K)}$ .

1.2F1F4given

If $v \in V_{σ}$ , then $b (v, q_{h}) = 0$ for all $q_{h} \in Q_{h}$ by definition of $V_{σ}$ and $b$ . Preservation [F4] gives $b (Π_{h} v, q_{h}) = 0$ for every $q_{h}$ , so $Π_{h} v \in Z_{h}$ .

1.3F3F5F6

Let $v \in C_{c}^{\infty} (D)^{d}$ and extend it by zero to $R^{d}$ . On each cell $K$ , choose a vertex $x_{K}$ and let $p_{K}$ be the affine Taylor polynomial of $v$ there. Every selected facet used in $I_{h}$ for a vertex of $K$ lies within $C h_{K}$ of $x_{K}$ by the patch geometry in the face-average interpolation supplier. Apply [F6] on the convex open set $R^{d}$ ; the pointwise Taylor error on those facets is at most $C h_{K}^{2} ∥ D^{2} v ∥_{\infty}$ . The scaled dual moments in [F5] then bound the coefficients of $I_{h} (v - p_{K})$ on $K$ by the same amount. Since $I_{h}$ reproduces $p_{K}$ , this yields $∥ v - I_{h} v ∥_{L^{\infty} (K)} \leq C h_{K}^{2} ∥ D^{2} v ∥_{\infty}$ and $∣ c_{K} ∣ \leq C h_{K}^{2} ∥ D^{2} v ∥_{\infty}$ . Bubble scaling [F3] gives $∥ \nabla (b_{K} c_{K}) ∥_{L^{2} (K)} \leq C h_{K}^{d / 2 + 1} ∥ D^{2} v ∥_{\infty}$ . Also $h_{K}^{d} \leq C ∣ K ∣$ by [F3], so summing squares gives $∥ \nabla (Π_{h} v - I_{h} v) ∥_{2} \leq C h ∥ D^{2} v ∥_{\infty}$ . The H2-to-H1 estimate in [F5] gives $∥ I_{h} v - v ∥_{V} \to 0$ , hence $Π_{h} v \to v$ in $V$ .

2.1F2step 1.1algebra

On each cell, $v - Π_{h} v = e_{K} - b_{K} c_{K}$ , so step 1.1 and the triangle inequality give $∥ v - Π_{h} v ∥_{L^{2} (K)} \leq C ∥ v - I_{h} v ∥_{L^{2} (K)}$ . Square, sum over cells, apply [F2], and use bounded patch overlap to obtain $∥ v - Π_{h} v ∥_{L^{2} (D)} \leq C h ∥ \nabla v ∥_{L^{2} (D)}$ .

3.1A1F7step 1.3choose∎

For general $v \in V$ , choose $ϕ \in C_{c}^{\infty} (D)^{d}$ with $∥ v - ϕ ∥_{V} < ϵ$ , using [F7]. Uniform Fortin stability [F7] gives $∥ Π_{h} v - v ∥_{V} \leq (C_{F} + 1) ϵ + ∥ Π_{h} ϕ - ϕ ∥_{V}$ . Let $h \to 0$ by step 1.3 and then $ϵ ↓ 0$ . This proves strong $H_{0}^{1}$ convergence for every fixed $v$ .

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Backward Euler MINI step existence and discrete energy bound

Statement

Assume Countable Choice, the mini-element hypotheses of The mixed mini-element backward Euler Navier–Stokes scheme, $f \in L^{2} (0, T; L^{2} (D)^{d})$ , and an initial value $u_{h}^{0} \in Z_{h}$ . On every finite positive time partition there is at least one mixed solution $(u_{h}^{n}, p_{h}^{n})_{n = 1}^{N}$ of the backward Euler scheme. Every such solution satisfies, for $0 \leq m \leq N$ , the work-form inequality $\frac{1}{2} ∥ u_{h}^{m} ∥_{2}^{2} + \frac{1}{2} \sum_{n = 1}^{m} ∥ u_{h}^{n} - u_{h}^{n - 1} ∥_{2}^{2} + ν \sum_{n = 1}^{m} τ_{n} ∥ \nabla u_{h}^{n} ∥_{2}^{2} \leq \frac{1}{2} ∥ u_{h}^{0} ∥_{2}^{2} + \sum_{n = 1}^{m} τ_{n} (f_{h}^{n}, u_{h}^{n})_{L^{2}} .$ Consequently it satisfies the mesh-independent estimate $\frac{1}{2} ∥ u_{h}^{m} ∥_{2}^{2} + \frac{1}{2} \sum_{n = 1}^{m} ∥ u_{h}^{n} - u_{h}^{n - 1} ∥_{2}^{2} + \frac{ν}{2} \sum_{n = 1}^{m} τ_{n} ∥ \nabla u_{h}^{n} ∥_{2}^{2} \leq \frac{1}{2} ∥ u_{h}^{0} ∥_{2}^{2} + \frac{C_{P} (D)^{2}}{2 ν} ∥ f ∥_{L^{2} (0, T; L^{2})}^{2} .$ The constant is independent of mesh size and time-step sizes.

Facts & Assumptions

Given: Countable Choice, the MINI spaces, $ν > 0$ , a finite partition, $f \in L^{2} (0, T; L^{2})$ , and $u_{h}^{0} \in Z_{h}$ .

[A1]

Countable Choice is inherited from the MINI/Stokes stability supplier; no arbitrary-index choice is used inside a single finite partition. (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Uniform Fortin stability of the mini element)

[F1]

The velocity mass form is real Hilbert; $a_{ν} (z, z) = 2 ν ∥ ε (z) ∥_{2}^{2} \geq ν ∥ \nabla z ∥_{2}^{2}$ on $H_{0}^{1}$ . (Stokes velocity and pressure spaces and weak forms, The homogeneous Dirichlet strain identity)

[F2]

$c (z; z, z) = 0$ for every $z \in V$ , and $c$ is continuous on the finite-dimensional MINI space. (The skew convective form for incompressible flow, Continuity, energy cancellation, and Sobolev bounds for skew convection)

[F3]

A continuous finite-dimensional map with positive outward radial pairing on a sphere has a zero in the ball. (An outward-pointing finite-dimensional map has a zero)

[F4]

The stable MINI pair has a positive discrete inf-sup constant, so the pressure map is injective and the velocity constraint map is onto. (Uniform Fortin stability of the mini element, Weak and pointwise divergence-free discrete velocities)

[F5]

The identity $2 (x - y, x) = ∥ x ∥^{2} - ∥ y ∥^{2} + ∥ x - y ∥^{2}$ holds in $L^{2}$ , and Poincare gives $∥ z ∥_{2} \leq C_{P} (D) ∥ \nabla z ∥_{2}$ . (The Hilbert-space discrete energy identity, The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction)

[F6]

Holder and Young absorb $∥ f_{h}^{n} ∥_{2} ∥ u_{h}^{n} ∥_{2}$ into $ν ∥ \nabla u_{h}^{n} ∥_{2}^{2} / 2$ with remainder $C_{P} (D)^{2} ∥ f_{h}^{n} ∥_{2}^{2} / (2 ν)$ . (Holder's inequality for integrals, including the endpoint cases, Young's inequality for conjugate real exponents)

[F7]

The norm of a Bochner integral is at most the integral of the norm. (Bochner integral norm inequality)

Proof

1.1A1F2givenconstruct

Fix a time cell and the previous value $u_{h}^{n - 1} \in Z_{h}$ . On $Z_{h}$ , define $F_{n} (z) \in Z_{h}^{*}$ by $⟨ F_{n} (z), w_{h} ⟩ = τ_{n}^{- 1} (z - u_{h}^{n - 1}, w_{h})_{L^{2}} + a_{ν} (z, w_{h}) + c (z; z, w_{h}) - (f_{h}^{n}, w_{h})_{L^{2}} .$ The map is continuous by [F2] and finite dimensionality. If $Z_{h} = {0}$ its only velocity is zero; otherwise represent $F_{n}$ in coordinates of a gradient-orthonormal basis of $Z_{h}$ .

2.1F1F2F3givenstep 1.1algebra

By [F1]–[F2], $⟨ F_{n} (z), z ⟩ \geq ν ∥ \nabla z ∥_{2}^{2} - C_{P} (D) (τ_{n}^{- 1} ∥ u_{h}^{n - 1} ∥_{2} + ∥ f_{h}^{n} ∥_{2}) ∥ \nabla z ∥_{2} .$ Choose $R$ so the right side is strictly positive whenever $∥ \nabla z ∥_{2} = R$ . The outward-zero result [F3] supplies $u_{h}^{n} \in Z_{h}$ with $F_{n} (u_{h}^{n}) = 0$ .

3.1F4step 2.1algebra

Let $G_{n} (v_{h})$ be the velocity residual (time term plus viscosity and convection minus forcing). Step 2.1 gives $G_{n} = 0$ on $Z_{h} = \ker B$ , where $B v_{h} (q_{h}) = b (v_{h}, q_{h})$ . Since [F4] makes $B$ onto $Q_{h}^{*}$ , $B^{*}$ is injective and its range is contained in $(\ker B)^{\circ}$ . Rank-nullity gives both spaces dimension $\dim Q_{h}$ , so they are equal. Thus $- G_{n} = B^{*} p_{h}^{n}$ for a unique $p_{h}^{n} \in Q_{h}$ , which gives the stated mixed equation with the correct residual sign. Repeating the finite step for $n = 1, \dots, N$ gives a mixed solution.

4.1F1F2F5givenstep 3.1

Test the mixed equation with $u_{h}^{n}$ . Its pressure term is zero since $u_{h}^{n} \in Z_{h}$ , and its convective term is zero by [F2]. Apply [F5] to the time term and [F1] to the viscosity term, obtaining $\frac{1}{2} (∥ u_{h}^{n} ∥_{2}^{2} - ∥ u_{h}^{n - 1} ∥_{2}^{2} + ∥ u_{h}^{n} - u_{h}^{n - 1} ∥_{2}^{2}) + ν τ_{n} ∥ \nabla u_{h}^{n} ∥_{2}^{2} \leq τ_{n} (f_{h}^{n}, u_{h}^{n})_{L^{2}} .$ The right side is retained with its sign; it follows directly from the tested equation, skew cancellation, and the lower Korn bound.

5.1F5F6F7step 4.1algebra∎

Before applying Young, retain the actual work term in step 4.1 and sum over $n = 1, \dots, m$ ; telescoping gives the first displayed inequality. For the coarser estimate, Cauchy–Schwarz, Poincare and Young give $τ_{n} (f_{h}^{n}, u_{h}^{n}) \leq \frac{ν}{2} τ_{n} ∥ \nabla u_{h}^{n} ∥_{2}^{2} + \frac{C_{P} (D)^{2}}{2 ν} τ_{n} ∥ f_{h}^{n} ∥_{2}^{2} .$ For each interval, [F7] and Holder [F6] give $τ_{n} ∥ f_{h}^{n} ∥_{2}^{2} = τ_{n}^{- 1} {∥ \int_{t_{n - 1}}^{t_{n}} f (t) d t ∥}_{2}^{2} \leq τ_{n}^{- 1} {(\int_{t_{n - 1}}^{t_{n}} ∥ f (t) ∥_{2} d t)}^{2} \leq \int_{t_{n - 1}}^{t_{n}} ∥ f (t) ∥_{2}^{2} d t .$ Summing proves $\sum_{n} τ_{n} ∥ f_{h}^{n} ∥_{2}^{2} \leq ∥ f ∥_{L^{2} L^{2}}^{2}$ . Substitution into the work-form inequality proves the second displayed estimate.

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Second-order L2 approximation by the MINI Fortin map

Statement

Assume Countable Choice. Let $D$ be a bounded connected Lipschitz polygonal or polyhedral domain in dimension $d \in {2, 3}$ , with a matching affine shape-regular simplicial mesh and $h = \max_{K} h_{K}$ . Let $I_{h}$ be the face-average interpolant of Boundary-preserving face-average P1 interpolation, and let $Π_{h}$ be exactly the MINI Fortin map constructed in Uniform Fortin stability of the mini element: $Π_{h} v = I_{h} v + \sum_{K} b_{K} c_{K}, c_{K} = \frac{\int_{K} (v - I_{h} v)}{\int_{K} b_{K}} .$ For $ω_{K}$ the vertex patch of $K$ and every $v \in H^{2} (D)^{d} \cap H_{0}^{1} (D)^{d}$ , $∥ v - Π_{h} v ∥_{L^{2} (K)} \leq C h_{K}^{2} ∥ D^{2} v ∥_{L^{2} (ω_{K})}, ∥ \nabla (v - Π_{h} v) ∥_{L^{2} (K)} \leq C h_{K} ∥ D^{2} v ∥_{L^{2} (ω_{K})} .$ Consequently, $∥ v - Π_{h} v ∥_{2} \leq C h^{2} ∥ D^{2} v ∥_{2}, ∥ v - Π_{h} v ∥_{V} \leq C h ∥ D^{2} v ∥_{2}, ∥ w ∥_{V} : = ∥ \nabla w ∥_{2} .$ The constants depend only on dimension, the fixed domain and the mesh shape-regularity bound. They are independent of the smallest element diameter and require no quasiuniformity.

Let $V_{σ}$ and $Z_{h}$ be the exact solenoidal space and weak MINI kernel, and let $P_{Z_{h}} : L^{2} (D)^{d} \to Z_{h}$ be the $L^{2}$ -orthogonal projector. For every $v \in H^{2} (D)^{d} \cap V_{σ}$ , $Π_{h} v \in Z_{h}, ∥ (I - P_{Z_{h}}) v ∥_{2} \leq C h^{2} ∥ D^{2} v ∥_{2} .$ In particular, if $u \in W^{1, \infty} (0, T; H^{2} (D)^{d} \cap V_{σ})$ , then $∥ (I - P_{Z_{h}}) u ∥_{L^{\infty} (0, T; L^{2})} \leq C h^{2} ∥ u ∥_{L^{\infty} (0, T; H^{2})}, ∥ (I - P_{Z_{h}}) u_{t} ∥_{L^{\infty} (0, T; L^{2})} \leq C h^{2} ∥ u_{t} ∥_{L^{\infty} (0, T; H^{2})} .$

Facts & Assumptions

Given: Countable Choice, the domain, mesh, spaces, face-average interpolant and existing Fortin map in the Statement.

[A1]

Countable Choice is inherited from the face-average interpolant and Fortin construction. No point evaluations or additional choice principle are used. (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Boundary-preserving face-average P1 interpolation, Uniform Fortin stability of the mini element)

[F1]

The face-average interpolant is defined on $H^{1} (D)$ , reproduces global affine polynomials, and satisfies $∥ w - I_{h} w ∥_{L^{2} (K)} \leq C h_{K} ∥ \nabla w ∥_{L^{2} (ω_{K})}$ . Its proof step 2.2 establishes that for $v \in H^{2} (D)$ there is an affine polynomial $p$ on each patch with $∥ v - p ∥_{L^{2} (ω_{K})} \leq C h_{K}^{2} ∥ D^{2} v ∥_{L^{2} (ω_{K})}$ and $∥ \nabla (v - p) ∥_{L^{2} (ω_{K})} \leq C h_{K} ∥ D^{2} v ∥_{L^{2} (ω_{K})}$ . It also gives $∥ \nabla (v - I_{h} v) ∥_{L^{2} (K)} \leq C h_{K} ∥ D^{2} v ∥_{L^{2} (ω_{K})}$ , uniformly bounded patch overlap and comparable cell diameters within each patch. (Boundary-preserving face-average P1 interpolation)

[F2]

The existing Fortin construction has the formula in the Statement, maps $H_{0}^{1} (D)^{d}$ into $V_{h}$ , and preserves $b (Π_{h} v, q_{h}) = b (v, q_{h})$ for every $q_{h} \in Q_{h}$ . (Uniform Fortin stability of the mini element)

[F3]

MINI bubble scaling gives $\int_{K} b_{K} \geq c ∣ K ∣$ , $∥ b_{K} ∥_{L^{2} (K)} \leq C ∣ K ∣^{1 / 2}$ , and $∥ \nabla b_{K} ∥_{L^{2} (K)} \leq C h_{K}^{- 1} ∣ K ∣^{1 / 2}$ . (The conforming P1-bubble/P1 mini element, Shape regularity controls affine simplex geometry)

[F4]

The spaces satisfy $V_{σ} = {v \in H_{0}^{1} (D)^{d} : b (v, q) = 0 for all q \in L_{0}^{2} (D)}$ and $Z_{h} = {v_{h} \in V_{h} : b (v_{h}, q_{h}) = 0 for all q_{h} \in Q_{h}}$ . The finite-dimensional kernel $Z_{h}$ is closed in Hilbert $L^{2} (D)^{d}$ ; its orthogonal projection is the unique best $L^{2}$ approximation and agrees on $V_{h}$ with the mass projector of the discrete Helmholtz theorem. (Stokes velocity and pressure spaces and weak forms, Weak and pointwise divergence-free discrete velocities, $L^{2}$ with the integral pairing is a Hilbert space, Orthogonal decomposition by a closed subspace, Discrete Helmholtz projection and pressure Schur identities)

[F5]

Holder's inequality gives $∣ \int_{K} e ∣ \leq ∣ K ∣^{1 / 2} ∥ e ∥_{L^{2} (K)}$ for vector-valued $e$ , by Euclidean duality and the scalar inequality. (Holder's inequality for integrals, including the endpoint cases)

Proof

Given: The inputs and the same operators $I_{h}, Π_{h}$ specified in the Statement.

1.1A1F1givenchoose

Fix a cell $K$ and first take one scalar component of $v$ . Let $p$ be the affine patch approximant from [F1], regarded as a global affine polynomial on $D$ . This causes no boundary restriction, because $I_{h}$ is defined on all of $H^{1} (D)$ . Set $w = v - p$ . Affine reproduction gives $v - I_{h} v = w - I_{h} w$ on $K$ . Apply the first-order local $L^{2}$ estimate in [F1] to $w$ and its patch gradient estimate to obtain $∥ v - I_{h} v ∥_{L^{2} (K)} \leq C h_{K} ∥ \nabla (v - p) ∥_{L^{2} (ω_{K})} \leq C h_{K}^{2} ∥ D^{2} v ∥_{L^{2} (ω_{K})}$ . Apply this componentwise and sum squares for vector $v$ . The local gradient bound $∥ \nabla (v - I_{h} v) ∥_{L^{2} (K)} \leq C h_{K} ∥ D^{2} v ∥_{L^{2} (ω_{K})}$ is also [F1]. All these estimates use facet averages and require no representative with point values.

1.2F2F3F5algebra

Write $e_{K} = v - I_{h} v$ on $K$ . The actual numerator of the Fortin coefficient is the unweighted integral of $e_{K}$ , so [F3] and [F5] give $∣ c_{K} ∣ \leq ∣ K ∣^{1 / 2} (\int_{K} b_{K})^{- 1} ∥ e_{K} ∥_{L^{2} (K)} \leq C ∣ K ∣^{- 1 / 2} ∥ e_{K} ∥_{L^{2} (K)}$ . Hence $∥ b_{K} c_{K} ∥_{L^{2} (K)} \leq C ∥ e_{K} ∥_{L^{2} (K)}$ and $∥ \nabla (b_{K} c_{K}) ∥_{L^{2} (K)} \leq C h_{K}^{- 1} ∥ e_{K} ∥_{L^{2} (K)}$ . No bound involving a global inverse mesh scale is used.

2.1F2step 1.1step 1.2algebra

Bubble interiors are disjoint, so on $K$ the existing map satisfies $v - Π_{h} v = e_{K} - b_{K} c_{K}$ . The triangle inequality, step 1.2 and step 1.1 imply $∥ v - Π_{h} v ∥_{L^{2} (K)} \leq C h_{K}^{2} ∥ D^{2} v ∥_{L^{2} (ω_{K})}$ and $∥ \nabla (v - Π_{h} v) ∥_{L^{2} (K)} \leq C h_{K} ∥ D^{2} v ∥_{L^{2} (ω_{K})}$ . The zero-boundary membership of $Π_{h} v$ is already ensured by [F2].

3.1F1step 2.1algebra

Square and sum the two estimates in step 2.1. Since $h_{K} \leq h$ and the patch overlap is uniformly bounded by [F1], $\sum_{K} h_{K}^{4} ∥ D^{2} v ∥_{L^{2} (ω_{K})}^{2} \leq C h^{4} ∥ D^{2} v ∥_{2}^{2}$ and $\sum_{K} h_{K}^{2} ∥ D^{2} v ∥_{L^{2} (ω_{K})}^{2} \leq C h^{2} ∥ D^{2} v ∥_{2}^{2}$ . Taking square roots proves the two global approximation estimates, with the asserted dependence of constants.

4.1F2F4step 3.1given

Suppose $v \in H^{2} (D)^{d} \cap V_{σ}$ . Because $Q_{h} \subset L_{0}^{2} (D)$ , [F4] gives $b (v, q_{h}) = 0$ for every $q_{h} \in Q_{h}$ . Fortin preservation [F2] therefore puts $Π_{h} v \in Z_{h}$ . Best approximation [F4] and step 3.1 give $∥ (I - P_{Z_{h}}) v ∥_{2} \leq ∥ v - Π_{h} v ∥_{2} \leq C h^{2} ∥ D^{2} v ∥_{2}$ . This uses only $L^{2}$ orthogonality, with no $H^{1}$ estimate for $P_{Z_{h}}$ .

5.1step 2.1step 3.1step 4.1given∎

Finally let $X = H^{2} (D)^{d} \cap V_{σ}$ with its $H^{2}$ norm and suppose $u \in W^{1, \infty} (0, T; X)$ . By this Bochner Sobolev hypothesis, $u (t)$ and $u_{t} (t)$ belong to $X$ for almost every $t$ , and their $H^{2}$ norms have the stated essential bounds. Apply step 4.1 separately to these two fields at such times and take essential suprema. The two claimed time-dependent estimates follow. Together with steps 2.1 and 3.1 this proves every assertion.

Remarks

This result concerns the existing face-average Fortin map and the mass-orthogonal projector onto the weak MINI kernel. It supplies neither uniform $H^{1}$ stability of that mass projector nor an instantaneous evolutionary pressure estimate. Its second-order $L^{2}$ conclusion is obtained directly from local interpolation and cell moments, with no elliptic duality or domain Stokes regularity assumption.

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Strong L2 compactness for weakly solenoidal MINI trajectories

Statement

Assume the Axiom of Choice. Let $D \subset R^{d}$ be bounded, connected and polyhedral, $d = 2$ or $3$ . Let $T_{j}$ be matching shape-regular meshes with $h_{j} \to 0$ , let $Z_{j}$ be the weak MINI divergence kernels, and let $0 = t_{0}^{j} < \dots < t_{N_{j}}^{j} = T$ have maximum step tending to zero. Suppose $U_{j}^{n} \in Z_{j}$ , their piecewise-constant reconstructions ${\overline{U}}_{j}$ satisfy $\sup_{j} (\max_{n} ∥ U_{j}^{n} ∥_{L^{2}} + ∥ {\overline{U}}_{j} ∥_{L^{2} (0, T; H_{0}^{1})}) < \infty,$ and the following coefficient condition holds: for every finite $L^{2}$ -orthonormal family $e_{1}, \dots, e_{m} \in V_{σ}$ , the $E$ -valued piecewise-constant functions $Y_{j} (t) = \sum_{i = 1}^{m} (U_{j}^{n}, Π_{j} e_{i})_{L^{2}} e_{i} (t \in (t_{n - 1}^{j}, t_{n}^{j}])$ have uniformly bounded $L^{2} (0, T; E)$ norms and their backward difference quotients are bounded in $L^{4 / 3} (0, T; E)$ ; here $Π_{j}$ is the MINI Fortin map and $E = span {e_{i}}$ . Then $({\overline{U}}_{j})$ is relatively compact in $L^{2} (0, T; L^{2} (D)^{d})$ .

Facts & Assumptions

Given: The shape-regular matching mesh family, $Z_{j}$ , the energy bounds, the coefficient condition, and the Axiom of Choice.

[F1]

Under the Axiom of Choice, $H_{0}^{1} (D)$ embeds compactly into $L^{2} (D)$ on bounded open $D$ ; applying the scalar result successively to the finitely many components gives the same compact embedding for $H_{0}^{1} (D)^{d}$ . (Compactness of $W_{0}^{1, p} (Ω) ↪ L^{p} (Ω)$ on bounded open sets)

[F2]

$\int_{D} div z = 0$ for $z \in H_{0}^{1} (D)^{d}$ , and $b (z, q_{h}) = 0$ for $z \in Z_{j}$ , $q_{h} \in Q_{j}$ . (Zero mean of the divergence of a zero-boundary field, Weak and pointwise divergence-free discrete velocities)

[F3]

For $q \in H^{1} (D)$ , the face-average interpolant approximates it in $L^{2}$ ; subtracting the mean gives a continuous piecewise-affine zero-mean pressure approximation. (Boundary-preserving face-average P1 interpolation, Stokes velocity and pressure spaces and weak forms)

[F4]

The MINI Fortin map satisfies $b (Π_{j} v, q_{h}) = b (v, q_{h})$ and $∥ v - Π_{j} v ∥_{2} \leq C h_{j} ∥ v ∥_{V}$ . (Uniform Fortin stability of the mini element, The mini-element Fortin map approximates in L2)

[F5]

The piecewise-constant discrete Aubin–Lions theorem gives compactness in $L^{2} (0, T; E)$ from a uniform $L^{2} (E)$ bound and an $L^{4 / 3} (E)$ difference-quotient bound when the time steps vanish. (Piecewise-constant discrete Aubin–Lions compactness)

[F6]

The scalar space $H_{0}^{1} (D)$ with its Sobolev norm is a separable real Hilbert space; its finite vector product is therefore also separable and Hilbert. (The zero-boundary Sobolev energy space is a separable Hilbert space)

[F7]

Poincare makes the gradient norm on $H_{0}^{1} (D)^{d}$ equivalent to its Sobolev norm, so a sequence bounded in $V$ is bounded in the separable Hilbert norm and weak convergence is the same for both norms. Every bounded sequence in a separable real Hilbert space has a weakly convergent subsequence. (The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction, A bounded sequence in a separable Hilbert space has a weakly convergent subsequence)

[F8]

The Sobolev norm is weakly lower semicontinuous; the equivalent gradient norm on $V$ is also weakly lower semicontinuous. (Weak lower semicontinuity of the Sobolev norm, The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction)

[F9]

The Bochner space $L^{2} (0, T; L^{2} (D)^{d})$ is complete. (Bochner Lp is complete for a Banach value space)

[A1]

The compactness and subsequence extractions use the Axiom of Choice. (The Axiom of Choice)

Proof

1.1A1F1F2F3givenalgebra

Consider any sequence $z_{j} \in Z_{j}$ with $∥ z_{j} ∥_{V} \leq R$ and $h_{j} \to 0$ . By [F1], a subsequence converges strongly in $L^{2}$ and weakly in $H_{0}^{1}$ to some $z \in H_{0}^{1}$ . For any $q \in C_{c}^{\infty} (D)$ , put $q^{0} = q - ∣ D ∣^{- 1} \int_{D} q$ , interpolate $q^{0}$ by [F3], and subtract the interpolant's mean to obtain $q_{j} \in Q_{j}$ with $q_{j} \to q^{0}$ in $L^{2}$ . Since $b (z_{j}, q_{j}) = 0$ and $\int_{D} div z_{j} = 0$ by [F2], $\int_{D} q div z_{j} = \int_{D} (q^{0} - q_{j}) div z_{j} \to 0$ . Weak convergence of the gradients gives $div z = 0$ , hence $z \in V_{σ}$ .

1.2A1F4F5givenalgebra

For an $L^{2}$ -orthonormal basis $e_{1}, \dots, e_{m}$ of $E$ , [F4] and the uniform $L^{\infty} L^{2}$ bound imply $∥ P_{E} {\overline{U}}_{j} - Y_{j} ∥_{L^{\infty} (0, T; E)}^{2} = \sum_{i} ∣ (U_{j}^{n}, e_{i} - Π_{j} e_{i})_{L^{2}} ∣^{2} \leq C_{E} h_{j}^{2} .$ The coefficient condition and [F5], applied to the finite-dimensional evolution triple $E ↪ E ↪ E$ , make $(Y_{j})$ relatively compact in $L^{2} (0, T; E)$ .

2.1A1F1F6F7F8step 1.1choose

For each $R > 0$ , $K_{R} = {z \in V_{σ} : ∥ z ∥_{V} \leq R}$ is totally bounded in $L^{2}$ by [F1], so choose a finite $ϵ / 2$ -net with centers $e_{i} \in K_{R}$ and let $E$ be their span. If arbitrarily fine meshes had $z_{j} \in Z_{j}$ , $∥ z_{j} ∥_{V} \leq R$ , and distance at least $ϵ$ from $E$ , [A1, F1, F6, F7] would give a subsequence strongly convergent in $L^{2}$ and weakly in $H_{0}^{1}$ . Step 1.1 puts its limit in $K_{R}$ , and [F8] retains the norm bound; this contradicts the $ϵ / 2$ -net. Thus every such $z_{j}$ is within $ϵ$ of $E$ for all sufficiently large $j$ .

3.1givenstep 2.1algebra

Let $P_{E}$ be the $L^{2}$ -orthogonal projection. On times when $∥ {\overline{U}}_{j} (t) ∥_{V} \leq R$ , step 2.1 gives $∥ {\overline{U}}_{j} - P_{E} {\overline{U}}_{j} ∥_{2} \leq ϵ$ . The measure of the remaining times is at most $R^{- 2} ∥ {\overline{U}}_{j} ∥_{L^{2} V}^{2}$ , and there $∥ {\overline{U}}_{j} - P_{E} {\overline{U}}_{j} ∥_{2} \leq 2 \sup_{j, n} ∥ U_{j}^{n} ∥_{2}$ . First choose $R$ large, then $ϵ$ small; this makes the $L^{2} (0, T; L^{2})$ projection remainder uniformly small for sufficiently large $j$ .

4.1F5F9step 3.1step 1.2algebra∎

Given any tolerance, steps 3.1 and 1.2 approximate the tail of $({\overline{U}}_{j})$ within that tolerance by a relatively compact family $Y_{j}$ in $L^{2} (0, T; E)$ . Hence the tail is totally bounded in $L^{2} (0, T; L^{2})$ ; adjoining finitely many initial terms preserves total boundedness. Completeness [F9] gives relative compactness.

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Approximation by the MINI solenoidal energy projection

Statement

Assume Countable Choice and the MINI mesh hypotheses. For $z \in V_{σ}$ , define $S_{h} z \in Z_{h}$ by $a_{ν} (S_{h} z, w_{h}) = a_{ν} (z, w_{h}) (w_{h} \in Z_{h}) .$ This defines a unique linear map, the restriction to $V_{σ}$ of the $a_{ν}$ -orthogonal projection $V \to Z_{h}$ , and $∥ z - S_{h} z ∥_{V} \leq \sqrt{2} \inf_{w_{h} \in Z_{h}} ∥ z - w_{h} ∥_{V} .$ If additionally $z \in H^{2} (D)^{d}$ and $q \in H^{1} (D) \cap L_{0}^{2} (D)$ , then $∥ z - S_{h} z ∥_{V} \leq C_{0} (1 + ν^{- 1}) h (∥ z ∥_{H^{2} (D)} + ∥ q ∥_{H^{1} (D)}),$ where $C_{0}$ is independent of $h$ and $ν$ and depends on $D$ , shape regularity and the uniform MINI inf-sup constant.

Facts & Assumptions

Given: A bounded connected Lipschitz polyhedral domain, its shape-regular MINI mesh, $z \in V_{σ}$ , and viscosity $ν > 0$ .

[F1]

$Z_{h}$ is a finite-dimensional subspace of $V$ and $a_{ν}$ is symmetric; Korn and $∣ ε (w) ∣ \leq ∣ \nabla w ∣$ give $ν ∥ w ∥_{V}^{2} \leq a_{ν} (w, w) \leq 2 ν ∥ w ∥_{V}^{2}$ . (Weak and pointwise divergence-free discrete velocities, Stokes velocity and pressure spaces and weak forms, The homogeneous Dirichlet strain identity)

[F2]

Every bounded linear functional on a real Hilbert space has a unique Riesz representative. (Riesz representation for Hilbert spaces)

[F3]

For a continuous Stokes pair in $H^{2} (D)^{d} \times (H^{1} (D) \cap L_{0}^{2} (D))$ , the MINI inf-sup and mixed Galerkin results apply, with discrete velocity in $Z_{h}$ ; the Scott--Zhang interpolant has the stated $H^{1}$ and pressure approximation bounds. (Uniform Fortin stability of the mini element, Mixed Galerkin error bound retaining beta_h, Mini-element H1 velocity and L2 pressure error estimate, Boundary-preserving face-average P1 interpolation, Weak and pointwise divergence-free discrete velocities)

[A1]

Countable Choice is assumed as in the Stokes approximation and Hilbert representation inputs. (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Mini-element H1 velocity and L2 pressure error estimate)

Proof

1.1A1F1F2givenalgebra

For any $z \in V$ , equip $Z_{h}$ with the inner product $a_{ν}$ . It is positive definite by [F1]. The functional $w_{h} \mapsto a_{ν} (z, w_{h})$ is bounded on this finite-dimensional Hilbert space, so [F2] gives a unique $S_{h} z \in Z_{h}$ satisfying the displayed equation. Linearity follows from uniqueness and linearity of the right-hand side. For $z \in Z_{h}$ the equation gives $S_{h} z = z$ , so this extension to $V$ is a projection. For any $w_{h} \in Z_{h}$ , orthogonality gives $a_{ν} (z - w_{h}, z - w_{h}) = a_{ν} (z - S_{h} z, z - S_{h} z) + a_{ν} (S_{h} z - w_{h}, S_{h} z - w_{h}) .$ Using $ν ∥ r ∥_{V}^{2} \leq a_{ν} (r, r) \leq 2 ν ∥ r ∥_{V}^{2}$ yields the stated best-approximation bound.

2.1F3step 1.1algebra∎

Let $F (v) = a_{ν} (z, v) + b (v, q)$ for $v \in V$ . It is bounded on $V$ , and $b (z, r) = 0$ for every $r \in Q$ because $z \in V_{σ}$ ; hence $(z, q)$ solves the continuous Stokes problem with data $(F, 0)$ . Let $w_{h} = I_{h} z$ and $r_{h} = I_{h} q - ∣ D ∣^{- 1} \int_{D} I_{h} q$ . By [F3], $w_{h} \in V_{h}$ , $r_{h} \in Q_{h}$ , and $δ_{u} : = ∥ z - w_{h} ∥_{V} \leq C h ∥ z ∥_{H^{2}}$ , $δ_{q} : = ∥ q - r_{h} ∥_{Q} \leq C h ∥ q ∥_{H^{1}}$ . The MINI mixed problem has a solution $(z_{h}, q_{h})$ with $z_{h} \in Z_{h}$ . Its inf-sup constant satisfies $β_{h}^{- 1} \leq C$ independently of $h$ and $ν$ , while $M_{a} = 2 ν$ , $M_{b} = \sqrt{d}$ , and $α_{*} = ν$ . The mixed error formula therefore gives $R_{h} \leq C δ_{u}$ and $U_{h} = δ_{u} + ν^{- 1} {2 ν (δ_{u} + R_{h}) + \sqrt{d} δ_{q}} + R_{h} \leq C (δ_{u} + ν^{- 1} δ_{q})$ . Step 1.1 with $w_{h} = z_{h}$ now yields $∥ z - S_{h} z ∥_{V} \leq \sqrt{2} U_{h} \leq C_{0} (1 + ν^{- 1}) h (∥ z ∥_{H^{2}} + ∥ q ∥_{H^{1}})$ , with $C_{0}$ independent of $h$ and $ν$ .

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MINI backward Euler converges subsequentially to a solenoidal energy weak velocity

Statement

Assume the Axiom of Choice. Let $D \subset R^{d}$ , $d \in {2, 3}$ , be nonempty, bounded, connected, Lipschitz and polyhedral, and let $0 < T < \infty$ . Let ${T_{j}}$ be matching shape-regular affine simplicial meshes with $h_{j} \to 0$ , and let the time partitions of $[0, T]$ have maximum step tending to zero, with no relation required between the two limits. Let $f \in L^{2} (0, T; L^{2} (D)^{d})$ and $u_{0} \in H_{σ}$ . On each mesh take $u_{j}^{0} = P_{Z_{j}} u_{0}$ , the $L^{2}$ -orthogonal projection onto the weak MINI kernel, and choose any backward Euler MINI solution from Backward Euler MINI step existence and discrete energy bound. Then a subsequence of piecewise-constant velocities converges strongly in $L^{2} (0, T; L^{2} (D)^{d})$ and weakly in $L^{2} (0, T; H_{0}^{1} (D)^{d})$ to a velocity $u$ satisfying the solenoidal weak momentum equation, initial trace and energy inequality in Solenoidal energy weak velocity for incompressible Navier–Stokes. This theorem supplies the velocity limit and its energy inequality. It does not assert the pressure regularity and pressure-bearing formulation required by the PE06 plan; the remaining pressure obligation is explicitly escalated on the A page.

Facts & Assumptions

Given: The Axiom of Choice, the domain and viscosity hypotheses, the shape-regular mesh/time families, $f \in L^{2} (0, T; L^{2})$ , and $u_{0} \in H_{σ}$ .

[A1]

The Axiom of Choice is used for subsequence extraction in the PE12 discrete Aubin–Lions input. (The Axiom of Choice, Piecewise-constant discrete Aubin–Lions compactness)

[F1]

Each finite mixed backward Euler problem has a solution and the uniform energy bound in Backward Euler MINI step existence and discrete energy bound.

[F2]

The local coefficient-probe compactness lemma applies to weak MINI kernels, shape-regular meshes, and vanishing maximum time steps. (Strong L2 compactness for weakly solenoidal MINI trajectories)

[F3]

The MINI Fortin map for the P1-bubble/P1 pair in The conforming P1-bubble/P1 mini element and the weak kernel in Weak and pointwise divergence-free discrete velocities preserves the divergence pairing, maps $V_{σ}$ into $Z_{j}$ , is uniformly $H_{0}^{1}$ -stable, and converges to the identity in $H_{0}^{1}$ for each fixed input. (Uniform Fortin stability of the mini element, The mini-element Fortin map approximates in L2)

[F4]

For the weak MINI kernel $Z_{j}$ defined from the P1-bubble/P1 pair and weak divergence constraint in The conforming P1-bubble/P1 mini element and Weak and pointwise divergence-free discrete velocities, the discrete L2-orthogonal projector exists; its best-approximation property is used only to initialize the scheme. (Discrete Helmholtz projection and pressure Schur identities)

[F5]

$H_{0}^{1} (D) ↪ L^{6} (D)$ in $d = 3$ ; the same $L^{6}$ bound in $d = 2$ follows from the $p = 3 / 2$ zero-boundary Sobolev inequality and Holder. (The Sobolev inequality for zero-boundary Sobolev closures on open sets, Holder's inequality for integrals, including the endpoint cases)

[F6]

Every zero-boundary velocity has zero integral divergence. (Zero mean of the divergence of a zero-boundary field)

[F7]

Cell averages of a Bochner $L^{2}$ function converge strongly in $L^{2}$ as the maximum cell length tends to zero. (Time translations and mollification in Bochner Lp)

[F9]

For a sequence bounded in $L^{2} (0, T; H_{0}^{1} (D))$ and strongly convergent in $L^{2} (0, T; L^{2} (D))$ , the PE12 weak-compactness lemma gives a subsequence weakly convergent in the Bochner Sobolev space and weak convergence of its gradients; apply it successively to the finitely many vector components. The Sobolev norm is weakly lower semicontinuous. (Weak compactness of bounded space-time Sobolev gradients, Weak lower semicontinuity of the Sobolev norm)

[F10]

Strong $L^{2} (0, T; L^{2})$ convergence yields a subsequence converging in $L^{2} (D)^{d}$ for almost every time. (Assuming Countable Choice, $L^{p}$ -convergent sequences have almost-everywhere convergent subsequences, The Axiom of Choice)

[F11]

The skew trilinear form is continuous on $V^{3}$ and has the energy-class $L^{4 / 3} (0, T; V^{*})$ bound used for coefficient differences. (Continuity, energy cancellation, and Sobolev bounds for skew convection)

[F12]

$H_{σ}$ is defined as the $L^{2}$ -closure of $V_{σ}$ . (Energy weak solutions of incompressible Navier–Stokes)

[F13]

The discrete mixed equation on the MINI spaces has the mass, viscosity, skew-convection, pressure and interval-average forcing terms in The mixed mini-element backward Euler Navier–Stokes scheme, with its spaces supplied by The conforming P1-bubble/P1 mini element and Weak and pointwise divergence-free discrete velocities.

[F14]

$L^{2} (D)^{d}$ is a separable real Hilbert space: extend scalar classes by zero to $R^{d}$ , restrict the countable dense rational-box step family there, and take finite products; the integral pairing gives its Hilbert structure. A closed subspace of a Hilbert space is Hilbert. ( $L^{2}$ with the integral pairing is a Hilbert space, Rational box-step functions form a countable dense subset of $L^{p} (R^{n})$ for $1 \leq p < \infty$ , Separability: the existence of an at most countable dense subset, Orthogonal decomposition by a closed subspace)

[F15]

For a separable real Hilbert space $H$ , dense subspace $X$ , and scalar weak equations with integrable derivatives and a common essential bound, the corresponding $L^{2} (0, T; H)$ function has a unique weakly continuous representative with the prescribed initial value. (Scalar weak equations give a weakly continuous Hilbert-space representative)

Proof

1.1A1F3F4givenchoose

Since $u_{0} \in H_{σ} = {\overline{V_{σ}}}^{L^{2}}$ , for any $ϵ > 0$ choose $v \in V_{σ}$ with $∥ u_{0} - v ∥_{2} < ϵ$ . By [F3], $Π_{j} v \in Z_{j}$ and $∥ Π_{j} v - v ∥_{2} \to 0$ . The best-approximation property [F4] gives $∥ P_{Z_{j}} u_{0} - u_{0} ∥_{2} \leq ∥ Π_{j} v - u_{0} ∥_{2} \leq ∥ Π_{j} v - v ∥_{2} + ϵ$ . Letting $j \to \infty$ and then $ϵ ↓ 0$ proves strong convergence of the discrete initial data.

1.2F1given

Apply [F1] to every mesh and partition. The energy inequality gives uniform bounds for $\max_{n} ∥ u_{j}^{n} ∥_{2}$ and $∥ {\overline{u}}_{j} ∥_{L^{2} V}$ ; it also gives $∥ {\overline{u}}_{j} ∥_{L^{\infty} H} \leq M$ .

1.3F1F2F3F5F11F13givenalgebra

Fix a finite orthonormal family $e_{1}, \dots, e_{m} \in V_{σ}$ . For each mesh, $Π_{j} e_{i} \in Z_{j}$ by [F3], so testing the discrete equation [F13] with $Π_{j} e_{i}$ removes pressure. The backward difference of $a_{j, i}^{n} = (u_{j}^{n}, Π_{j} e_{i})_{2}$ is therefore bounded by $∣ (D_{τ} a_{j, i}^{n}) ∣ \leq C_{e_{i}} (∥ f_{j}^{n} ∥_{2} + ν ∥ u_{j}^{n} ∥_{V} + ∥ u_{j}^{n} ∥_{2}^{1 / 2} ∥ u_{j}^{n} ∥_{V}^{3 / 2}) .$ The first two terms are bounded in $L^{2} (0, T)$ by [F1], and the last in $L^{4 / 3} (0, T)$ by [F11], because its $4 / 3$ power is at most $M^{2 / 3} C ∥ u_{j}^{n} ∥_{V}^{2}$ . Thus the coefficient condition of [F2] holds for every finite family.

1.4A1F1F2F9F10

By [F2], after a subsequence ${\overline{u}}_{j} \to u$ strongly in $L^{2} (0, T; L^{2})$ . The energy bounds [F1] and [F9] give, after a further subsequence, ${\overline{u}}_{j} ⇀ u$ in $L^{2} (0, T; V)$ . Apply [F10] to the scalar functions $t \mapsto ∥ {\overline{u}}_{j} (t) - u (t) ∥_{2}$ to obtain almost-everywhere-in-time strong convergence along a further subsequence. The uniform $L^{\infty} H$ bound then implies $u \in L^{\infty} (0, T; H)$ .

2.1F1F6step 1.4F5

For any $q \in C_{c}^{\infty} (D)$ , apply the face-average interpolant to $q - ∣ D ∣^{- 1} \int_{D} q$ and subtract the mean of its interpolant to obtain $q_{j} \in Q_{j}$ converging in $L^{2}$ , using the approximation estimate in Boundary-preserving face-average P1 interpolation. Since $b (u_{j}^{n}, q_{j}) = 0$ , [F6] gives $\int_{D} q div u_{j}^{n} = 0$ for the $q_{j}$ part and $\int_{D} div u_{j}^{n} = 0$ for the mean part. The $L^{2}$ approximation error times the uniformly bounded $L^{2} (0, T; V)$ divergence norm tends to zero by Holder. Passing to the weak gradient limit proves $div u = 0$ a.e.; hence $u \in L^{2} (0, T; V_{σ})$ .

2.2F1F3F7F13step 1.1step 1.4algebra

Let $v \in V_{σ}$ and $η \in C^{1} ([0, T])$ vanish near $T$ . Test each time step [F13] by $Π_{j} v \in Z_{j}$ , multiply by the left-endpoint value $η (t_{n - 1}^{j})$ , and sum. Discrete summation by parts gives $- (u_{j}^{0}, Π_{j} v)_{2} η (0) - \sum_{n = 1}^{N_{j} - 1} (u_{j}^{n}, Π_{j} v)_{2} (η (t_{n}^{j}) - η (t_{n - 1}^{j}))$ ; the terminal term is zero for all fine partitions because $η$ vanishes near $T$ . The sum is exactly the integral of $- {\overline{u}}_{j} \cdot Π_{j} v η^{'}$ on the nonterminal cells, so the time contribution converges to $- (u_{0}, v)_{2} η (0) - \int_{0}^{T} (u, v)_{2} η^{'}$ . Moving its initial term to the right side gives the required positive initial trace. Uniform convergence of the left-endpoint step approximation to $η$ , [F3], and step 1.1 justify these limits. The viscosity term converges by weak $L^{2} V$ convergence and $Π_{j} v \to v$ in $V$ ; the force averages converge by [F7] and the same uniform time-test approximation.

2.3A1F1F7F9step 1.1step 1.2step 1.4algebra

For almost every $t$ , choose the first time node $t_{j} (t) \geq t$ and sum the work-form energy inequality of [F1] through that node. Then ${\overline{u}}_{j} (t) = u_{j}^{n_{j} (t)}$ , so step 1.4 gives the required strong convergence of the kinetic-energy term for almost every $t$ . By [F7], $f_{j} \to f$ in $L^{2} H$ and ${\overline{u}}_{j} \to u$ in $L^{2} H$ , so their work integrals converge on $[0, t]$ . The overshoot interval $[t, t_{j} (t)]$ contributes at most $∥ f_{j} ∥_{L^{2} H} M \sqrt{τ_{\max}^{}}$ by Cauchy--Schwarz and the uniform $L^{\infty} H$ bound, hence tends to zero. The gradient sum through $t_{j} (t)$ is bounded below by its integral through $t$ ; weak $L^{2} V$ convergence and [F9] give lower semicontinuity on $[0, t]$ . The initial projections converge by step 1.1. This proves the energy inequality for almost every $t$ .

3.1F2F3F5F11step 1.4step 2.2

The test interpolation error in convection vanishes: continuity from [F11] bounds its integral by $C ∥ Π_{j} v - v ∥_{V} ∥ {\overline{u}}_{j} ∥_{L^{2} V}^{2} ∥ η ∥_{\infty}$ . The uniform left-endpoint test approximation from step 2.2 permits replacing the discrete weights by $η$ . For fixed $v \in V_{σ}$ , strong $L^{2} L^{2}$ convergence and the uniform $L^{2} L^{6}$ bound imply strong $L^{2} L^{3}$ and strong $L^{2} L^{4}$ convergence of velocities by Holder interpolation. In the skew formula, $u_{j} \otimes v$ then converges strongly in $L^{2} L^{2}$ while $\nabla u_{j}$ converges weakly in $L^{2} L^{2}$ ; $u_{j} \otimes u_{j}$ converges strongly in $L^{1} L^{2}$ . Thus the convective term passes to $c (u; u, v)$ . This proves the limiting momentum equation for all solenoidal tests.

4.1A1F1F11F12F14F15step 1.4step 2.2step 2.3step 3.1

By [F14], $L^{2} (D)^{d}$ is separable, so balls with centers in a countable dense set and positive rational radii form a countable base. The space $H_{σ} = {\overline{V_{σ}}}^{L^{2}}$ from [F12] is a closed subspace and hence Hilbert by [F14]; intersecting that countable base with $H_{σ}$ gives a countable base there, and [A1] selects one point from each nonempty member to make a dense set. For every $v \in V_{σ}$ , steps 2.2 and 3.1 give the scalar weak equation with $h_{v} (t) = (f (t), v)_{L^{2}} - a_{ν} (u (t), v) - c (u (t); u (t), v)$ . This function is integrable: its three terms lie in $L^{2} (0, T)$ , $L^{2} (0, T)$ , and $L^{4 / 3} (0, T)$ respectively by the data, energy bound [F1], and convection bound [F11]; finite $T$ embeds these spaces into $L^{1}$ . The uniform $L^{\infty} H$ bound follows from step 1.4. Apply [F15] with $H = H_{σ}$ and $X = V_{σ}$ to obtain a unique weakly continuous representative with initial value $u_{0}$ . Its energy inequality is already proved in step 2.3.

5.1step 1.4step 2.1step 2.2step 2.3step 3.1step 4.1∎

Steps 1.4, 2.1–2.3, 3.1, and 4.1 give the strong and weak convergence, divergence constraint, solenoidal momentum equation, weakly continuous initial trace, and energy inequality required by Solenoidal energy weak velocity for incompressible Navier–Stokes. The stronger pressure-bearing PE06 claim is not inferred here and remains escalated as stated.

Remarks

The pressure obligation mentioned in the Statement and final proof step records the original PE06 design. Its universal $L_{t}^{4 / 3} L_{0}^{2}$ pressure conclusion is refuted under those exact energy-data hypotheses by the contextual An energy Navier–Stokes velocity on the square without endpoint-integrable pressure ↗. This does not change the solenoidal velocity conclusion proved here. Distributional pressure and pressure-primitive recovery remain valid, and the independent smooth instantaneous pressure-error obligation remains unresolved. The counterexample is contextual evidence and is not a logical dependency of this theorem.

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Smooth-branch velocity error for backward Euler MINI

Statement

Assume Countable Choice. Assume the MINI mesh and time-partition hypotheses of The mixed mini-element backward Euler Navier–Stokes scheme. Suppose $(u, p)$ is a strong solution of the no-slip Navier–Stokes equations with forcing $f \in L^{2} (0, T; L^{2} (D)^{d})$ and $u \in W^{1, \infty} (0, T; H^{2} (D)^{d} \cap V_{σ})$ and $p \in W^{1, \infty} (0, T; H^{1} (D) \cap L_{0}^{2} (D))$ . Set $M : = 1 + ∥ u ∥_{W^{1, \infty} H^{2}} + ∥ p ∥_{W^{1, \infty} H^{1}}, M_{u} : = ∥ u ∥_{L^{\infty} V},$ $M_{\infty} : = (∥ u_{0} ∥_{2}^{2} + C_{P} (D)^{2} ν^{- 1} ∥ f ∥_{L^{2} L^{2}}^{2})^{1 / 2}, M_{E} : = ν^{- 1 / 2} ∥ u_{0} ∥_{2} + C_{P} (D) ν^{- 1} ∥ f ∥_{L^{2} L^{2}},$ and $Λ : = 1 + C_{D} ν^{- 3} M_{u}^{4}$ , where $C_{D}$ depends only on the domain, dimension and the trilinear-form constant. For sufficiently fine time partitions satisfying $τ_{\max} Λ \leq \frac{1}{2}$ , initialize by $u_{h}^{0} = P_{Z_{h}} u_{0}$ and choose any backward Euler MINI solution. Then $\max_{0 \leq n \leq N} ∥ u (t_{n}) - u_{h}^{n} ∥_{2} + ν^{1 / 2} {(\sum_{n = 1}^{N} τ_{n} ∥ u (t_{n}) - u_{h}^{n} ∥_{V}^{2})}^{1 / 2} \leq C_{*} (h M + τ_{\max} M^{2}),$ where one may take $C_{*} : = C_{D, s h a p e} (1 + ν)^{2} (1 + ν^{- 4}) (1 + T)^{2} (1 + M_{\infty} + M_{E} + M)^{4} \exp (2 T Λ) .$ Here $f$ is the forcing of the strong solution and $C_{D, s h a p e}$ is independent of $h$ , all time steps, and $ν$ . The polynomial factor displays the positive and inverse viscosity powers produced by Young's inequalities and static Stokes projection. The explicit factor $ν^{- 3}$ in $Λ$ and exponential dependence on $T Λ$ are retained; no Reynolds-uniform estimate is asserted. The separate instantaneous pressure assertion is addressed in Remarks.

Facts & Assumptions

Given: Countable Choice, a bounded connected Lipschitz polyhedral domain, a shape-regular MINI mesh family, a partition of $[0, T]$ , viscosity $ν > 0$ , and the strong solution and initialization in the Statement.

[F1]

The exact solution satisfies the time-integrated mixed weak equation; for tests $z_{h} \in Z_{h}$ its pressure contribution is $b (z_{h}, {\overset{ˉ}{p}}_{n})$ . (The mixed mini-element backward Euler Navier–Stokes scheme, Energy weak solutions of incompressible Navier–Stokes)

[F2]

The linear solenoidal energy projection $S_{h}$ satisfies $∥ u - S_{h} u ∥_{V} \leq C_{0} (1 + ν^{- 1}) h (∥ u ∥_{H^{2}} + ∥ p ∥_{H^{1}})$ and the same estimate applies to $(u_{t}, p_{t})$ . (Approximation by the MINI solenoidal energy projection)

[F3]

Mean-adjusted face-average interpolation maps $H^{1} (D)$ to $Q_{h}$ with $∥ p - r_{h} ∥_{2} \leq C h ∥ p ∥_{H^{1}}$ . (Boundary-preserving face-average P1 interpolation, The conforming P1-bubble/P1 mini element)

[F4]

The skew form is trilinear and continuous, obeys $∣ c (a; b, w) ∣ \leq C_{D} ∥ a ∥_{2}^{1 / 2} ∥ a ∥_{V}^{1 / 2} ∥ b ∥_{V} ∥ w ∥_{V}$ , and $c (a; w, w) = 0$ . (Continuity, energy cancellation, and Sobolev bounds for skew convection, The skew convective form for incompressible flow)

[F5]

The discrete solution obeys $\max_{n} ∥ u_{h}^{n} ∥_{2} \leq M_{\infty}$ and $\sum_{n} τ_{n} ∥ u_{h}^{n} ∥_{V}^{2} \leq M_{E}^{2}$ . (Backward Euler MINI step existence and discrete energy bound)

[F6]

The L2 projector $P_{Z_{h}}$ onto the weak MINI kernel defined in Weak and pointwise divergence-free discrete velocities exists and is a best approximation in $L^{2}$ . (Discrete Helmholtz projection and pressure Schur identities)

[F7]

On the Stokes velocity space $V = H_{0}^{1} (D)^{d}$ , Poincare, Korn, Holder and Young give $∥ z ∥_{2} \leq C_{P} ∥ z ∥_{V}$ and $a_{ν} (z, z) \geq ν ∥ z ∥_{V}^{2}$ . (Stokes velocity and pressure spaces and weak forms, The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction, The homogeneous Dirichlet strain identity, Holder's inequality for integrals, including the endpoint cases, Young's inequality for conjugate real exponents)

[A1]

Countable Choice is assumed through the finite-dimensional projection and scheme-existence inputs. (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Approximation by the MINI solenoidal energy projection, Backward Euler MINI step existence and discrete energy bound)

[F8]

The static mixed projection contributes at most $1 + ν^{- 1}$ , pressure and trilinear Young estimates contribute inverse powers no worse than $ν^{- 3}$ , and the viscous consistency term contributes one factor $ν$ before it is balanced by coercivity. Thus all residual constants are bounded by $C_{D, s h a p e} (1 + ν + ν^{- 3})$ ; all such constants depend otherwise only on the domain, mesh-shape bound and dimension. (Mixed Galerkin error bound retaining beta_h, Approximation by the MINI solenoidal energy projection, Young's inequality for conjugate real exponents)

Proof

1.1A1F2F6F7given

Let $S_{h}$ be the projection of [F2] and set $η (t) = u (t) - S_{h} u (t)$ and $M_{ν} = (1 + ν^{- 1}) M$ . Linearity and time independence of $S_{h}$ give $η_{t} = u_{t} - S_{h} u_{t}$ . Applying [F2] to $(u (t), p (t))$ and $(u_{t} (t), p_{t} (t))$ yields $∥ η ∥_{L^{\infty} V} + ∥ η_{t} ∥_{L^{\infty} V} + ∥ η ∥_{L^{\infty} L^{2}} \leq C h M_{ν},$ where the last term follows from Poincare [F7]. Define $θ^{n} = S_{h} u (t_{n}) - u_{h}^{n} \in Z_{h}$ . Since $P_{Z_{h}}$ is a best $L^{2}$ approximation and $S_{h} u_{0} \in Z_{h}$ , [F2], [F6] and Poincare give $∥ θ^{0} ∥_{2} \leq 2 ∥ u_{0} - S_{h} u_{0} ∥_{2} \leq C h M_{ν}$ .

2.1F1F2step 1.1algebra

Write $u_{n} = u (t_{n})$ and ${\overset{ˉ}{u}}_{n} = τ_{n}^{- 1} \int_{t_{n - 1}}^{t_{n}} u (t) d t$ , with analogous notation ${\overset{ˉ}{p}}_{n}$ and ${\overset{ˉ}{c}}_{n} = τ_{n}^{- 1} \int c (u (t); u (t), \cdot) d t$ . Average the exact equation in time and subtract the discrete equation. For every $z_{h} \in Z_{h}$ this gives $(\frac{θ^{n} - θ^{n - 1}}{τ_{n}}, z_{h}) + a_{ν} (θ^{n}, z_{h}) = - (\frac{η (t_{n}) - η (t_{n - 1})}{τ_{n}}, z_{h}) - a_{ν} (η (t_{n}) + {\overset{ˉ}{u}}_{n} - u_{n}, z_{h}) - b (z_{h}, {\overset{ˉ}{p}}_{n}) + c (u_{h}^{n}; u_{h}^{n}, z_{h}) - {\overset{ˉ}{c}}_{n} (z_{h}) .$ The identity uses $a_{ν} (S_{h} u_{n} - u_{n}, z_{h}) = 0$ from the defining projection; all forcing averages cancel exactly.

3.1F2F3F7step 2.1

Test step 2.1 with $z_{h} = θ^{n}$ and multiply by $τ_{n}$ . For the pressure choose the mean-adjusted interpolant $r_{h} (t)$ from [F3] and set ${\overset{ˉ}{r}}_{h}^{n} = τ_{n}^{- 1} \int_{t_{n - 1}}^{t_{n}} r_{h} (t) d t \in Q_{h}$ ; then $b (θ^{n}, {\overset{ˉ}{r}}_{h}^{n}) = 0$ . Linearity and the triangle inequality for the Bochner average give $∥ {\overset{ˉ}{p}}_{n} - {\overset{ˉ}{r}}_{h}^{n} ∥_{2} \leq C h M$ , and $∣ div θ^{n} ∣ \leq \sqrt{d} ∥ θ^{n} ∥_{V}$ , so $∣ b (θ^{n}, {\overset{ˉ}{p}}_{n}) ∣ \leq C_{D} h M ∥ θ^{n} ∥_{V}$ . Also $∥ {\overset{ˉ}{u}}_{n} - u_{n} ∥_{V} \leq τ_{n} M$ . The time derivative estimate [F2] gives $∥ η (t_{n}) - η (t_{n - 1}) ∥_{2} \leq C τ_{n} h M_{ν}$ . Cauchy–Schwarz and Young, with viscous fractions $ν τ_{n} / 16$ for each of the viscous and pressure terms, yield $∣ (η (t_{n}) - η (t_{n - 1}), θ^{n}) ∣ + τ_{n} ∣ a_{ν} (η (t_{n}) + {\overset{ˉ}{u}}_{n} - u_{n}, θ^{n}) ∣ + τ_{n} ∣ b (θ^{n}, {\overset{ˉ}{p}}_{n} - {\overset{ˉ}{r}}_{h}^{n}) ∣ \leq \frac{τ_{n}}{8} ∥ θ^{n} ∥_{2}^{2} + \frac{ν τ_{n}}{8} ∥ θ^{n} ∥_{V}^{2} + C_{D, ν} τ_{n} (h^{2} M_{ν}^{2} + τ_{n}^{2} M^{2}) .$ Here $C_{D, ν}$ is bounded by $C_{D, s h a p e} (1 + ν + ν^{- 1})$ for these linear residuals.

3.2F4F5F7step 1.1step 2.1

The identity $c (u_{h}^{n}; u_{h}^{n}, θ^{n}) - c (u_{n}; u_{n}, θ^{n}) = - c (u_{h}^{n}; η (t_{n}), θ^{n}) - c (η (t_{n}); u_{n}, θ^{n}) - c (θ^{n}; u_{n}, θ^{n})$ follows by bilinearity in the first two slots and skew cancellation $c (u_{h}^{n}; θ^{n}, θ^{n}) = 0$ from [F4]. Apply [F4], Poincare and Young after multiplication by $τ_{n}$ . Each of these three terms is bounded by $ν τ_{n} ∥ θ^{n} ∥_{V}^{2} / 64$ plus respectively $C_{D, ν} τ_{n} M_{\infty} ∥ u_{h}^{n} ∥_{V} h^{2} M_{ν}^{2}, C_{D, ν} τ_{n} ∥ u_{n} ∥_{V}^{2} h^{2} M_{ν}^{2}, C_{D} ν^{- 3} τ_{n} M_{u}^{4} ∥ θ^{n} ∥_{2}^{2} .$ For the time-averaging remainder, [F4] and $∥ u (t) - u_{n} ∥_{V} \leq τ_{n} M$ give $τ_{n} ∣ c (u_{n}; u_{n}, θ^{n}) - {\overset{ˉ}{c}}_{n} (θ^{n}) ∣ \leq C_{D} τ_{n}^{2} M^{2} ∥ θ^{n} ∥_{V} \leq ν τ_{n} ∥ θ^{n} ∥_{V}^{2} / 64 + C_{D, ν} τ_{n}^{3} M^{4}$ .

4.1F4F5F8step 2.1step 3.1step 3.2algebra

The Hilbert identity $2 (x - y, x) = ∥ x ∥^{2} - ∥ y ∥^{2} + ∥ x - y ∥^{2}$ applied to the time term in step 2.1 converts the tested equation to an energy recurrence. The viscous Young fractions in steps 3.1–3.2 sum to $3 / 16$ of $ν τ_{n} ∥ θ^{n} ∥_{V}^{2}$ before doubling, so at least $5 / 8$ remains and in particular the displayed half remains. Doubling also changes the time residual in step 3.1 into a coefficient at most $τ_{n} / 4$ of $E_{n}$ ; the convection coefficient is at most $C_{D} ν^{- 3} τ_{n} M_{u}^{4}$ . Thus, enlarging $C_{D}$ if necessary, $E_{n} - E_{n - 1} + \frac{ν}{2} τ_{n} ∥ θ^{n} ∥_{V}^{2} \leq τ_{n} Λ E_{n} + C_{D, ν} τ_{n} {h^{2} M_{ν}^{2} (1 + M_{\infty} ∥ u_{h}^{n} ∥_{V} + M_{u}^{2}) + τ_{n}^{2} M^{4}}, E_{n} : = ∥ θ^{n} ∥_{2}^{2},$ with $Λ = 1 + C_{D} ν^{- 3} M_{u}^{4}$ . The residual coefficient satisfies $C_{D, ν} \leq C_{D, s h a p e} (1 + ν + ν^{- 3})$ by [F8].

5.1F5F7F8step 1.1step 4.1algebra

Since $τ_{n} Λ \leq 1 / 2$ , rearrange step 4.1 and use $(1 - x)^{- 1} \leq e^{2 x}$ for $0 \leq x \leq 1 / 2$ . Iterating gives $\max_{n} E_{n} \leq e^{2 T Λ} (E_{0} + C_{D, ν} \sum_{n} τ_{n} {h^{2} M_{ν}^{2} (1 + M_{\infty} ∥ u_{h}^{n} ∥_{V} + M_{u}^{2}) + τ_{n}^{2} M^{4}}) .$ By Cauchy–Schwarz and [F5], $\sum_{n} τ_{n} ∥ u_{h}^{n} ∥_{V} \leq \sqrt{T} M_{E}$ and $\sum_{n} τ_{n}^{3} \leq T τ_{\max}^{}$ . Summing step 4.1 and dropping the nonnegative final $E_{N}$ also gives $\frac{ν}{2} \sum_{n} τ_{n} ∥ θ^{n} ∥_{V}^{2} \leq E_{0} + T Λ \max_{n} E_{n} + C_{D, ν} \sum_{n} τ_{n} {h^{2} M_{ν}^{2} (1 + M_{\infty} ∥ u_{h}^{n} ∥_{V} + M_{u}^{2}) + τ_{n}^{2} M^{4}} .$ Since $M_{u} \leq M$ , these bounds and $E_{0} \leq C h^{2} M_{ν}^{2}$ imply $\max_{n} ∥ θ^{n} ∥_{2} + ν^{1 / 2} (\sum_{n} τ_{n} ∥ θ^{n} ∥_{V}^{2})^{1 / 2} \leq C_{D, s h a p e} (1 + ν)^{2} (1 + ν^{- 4}) (1 + T)^{2} (1 + M_{\infty} + M_{E} + M)^{4} e^{2 T Λ} (h M + τ_{\max} M^{2})$ . Here $(1 + ν)^{2} (1 + ν^{- 4})$ dominates $1 + ν^{- 1}$ from $M_{ν}$ and the residual inverse powers through $ν^{- 3}$ from [F8]; the extra $\sqrt{Λ}$ in the energy norm combines with those factors to give no power worse than $ν^{- 4}$ .

6.1F2step 5.1algebra∎

Finally $u_{n} - u_{h}^{n} = η (t_{n}) + θ^{n}$ . The projection estimate [F2] bounds $\max_{n} ∥ η (t_{n}) ∥_{2}$ by $C h M_{ν}$ and $\sum_{n} τ_{n} ∥ η (t_{n}) ∥_{V}^{2}$ by $C T h^{2} M_{ν}^{2}$ . Combine with step 5.1; the explicit viscosity factor in $C_{*}$ absorbs $M_{ν} / M = 1 + ν^{- 1}$ . This proves the velocity estimate. Its $ν$ dependence is explicit and contains an exponential in $ν^{- 3} M_{u}^{4} T$ ; it makes no Reynolds-uniform claim.

Remarks

The positive velocity estimate above retains the original mesh, data, mass initialization, arbitrary-step and any-root scope. The originally paired unrestricted instantaneous $L_{t}^{4 / 3} L_{x}^{2}$ pressure-error rate is false: the contextual Graded MINI mass initialization can destroy the smooth instantaneous pressure rate ↗ constructs an exact-scope graded-mesh counterexample with smooth data and genuine backward Euler roots. That counterexample is not a logical premise of the velocity proof. Pressure-primitive estimates remain compatible with it.

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Integrated pressure recovery and MINI pressure-primitive convergence

Statement

Assume the Axiom of Choice. Let $D \subset R^{d}$ be nonempty, bounded, connected and Lipschitz, with $d \in {2, 3}$ , let $0 < T < \infty$ and $ν > 0$ . Use $V = H_{0}^{1} (D; R^{d})$ , $Q = L_{0}^{2} (D)$ , $V_{σ} = \ker (div : V \to Q)$ , $H_{σ} = {\overline{V_{σ}}}^{L^{2} (D)}$ , the bilinear form $a_{ν} (v, w) = 2 ν \int_{D} ε (v) : ε (w)$ , the pressure form $b (v, q) = - (q, div v)_{L^{2}}$ , and the real skew convective form $c$ from Stokes velocity and pressure spaces and weak forms, The skew convective form for incompressible flow and Solenoidal energy weak velocity for incompressible Navier–Stokes. Let $C_{P}$ be a Poincare constant, $C_{d i v}$ a bounded divergence-right-inverse constant, and choose $C_{c}$ large enough that $∣ c (a; b, v) ∣ \leq C_{c} ∥ a ∥_{V} ∥ b ∥_{V} ∥ v ∥_{V}, ∥ c (a; a, \cdot) ∥_{V^{*}} \leq C_{c} ∥ a ∥_{L^{2}}^{1 / 2} ∥ a ∥_{V}^{3 / 2} .$ For $f \in L^{2} (0, T; L^{2} (D)^{d})$ and $u_{0} \in H_{σ}$ , let $u$ be a solenoidal energy weak velocity. Put $U = ∥ u ∥_{L^{\infty} L^{2}}$ , $E = ∥ u ∥_{L^{2} V}$ and $F = ∥ f ∥_{L^{2} L^{2}}$ . There is a unique $P \in L^{\infty} (0, T; Q) \cap C_{w} ([0, T]; Q)$ with $P (0) = 0$ such that, for every $t \in [0, T]$ and $v \in V$ , $(u (t) - u_{0}, v)_{L^{2}} + \int_{0}^{t} (a_{ν} (u (s), v) + c (u (s); u (s), v)) d s + b (v, P (t)) = \int_{0}^{t} (f (s), v)_{L^{2}} d s .$ Here $C_{w}$ means continuity when tested against every fixed element of the Hilbert space $Q$ . Moreover, $∥ P ∥_{L^{\infty} Q} \leq C_{d i v} (C_{P} (U + ∥ u_{0} ∥_{L^{2}}) + C_{P} \sqrt{T} F + 2 ν \sqrt{T} E + C_{c} T^{1 / 4} U^{1 / 2} E^{3 / 2}) .$ Define the $Q$ -valued time distribution $p = \partial_{t} P$ by $⟨ p, η ⟩ = - \int_{0}^{T} P (t) η^{'} (t) d t (η \in C_{c}^{\infty} (0, T)) .$ The bound $∥ ⟨ p, η ⟩ ∥_{Q} \leq ∥ P ∥_{L^{\infty} Q} ∥ η^{'} ∥_{L^{1}}$ makes this a continuous linear map from scalar test functions to $Q$ . Then $p$ is the unique mean-zero distributional pressure for $u$ , and for every $φ \in C^{1} ([0, T]; V)$ vanishing near $T$ , $- \int_{0}^{T} (u, φ^{'})_{L^{2}} d t + \int_{0}^{T} (a_{ν} (u, φ) + c (u; u, φ)) d t - \int_{0}^{T} b (φ^{'}, P) d t = \int_{0}^{T} (f, φ)_{L^{2}} d t + (u_{0}, φ (0))_{L^{2}} .$ This pressure is a distribution; the assertion does not place $p$ in $L^{4 / 3} (0, T; Q)$ .

For the discrete assertions, further assume that $D$ is polyhedral. Let $T_{j}$ be matching shape-regular MINI meshes with $h_{j} \to 0$ , and let $0 = t_{0}^{j} < \dots < t_{N_{j}}^{j} = T$ have maximum step $τ_{\max, j} \to 0$ . On each mesh take $u_{h, j}^{0} = P_{Z_{j}} u_{0}$ and any actual scheme solution $(u_{h, j}^{n}, p_{h, j}^{n})$ . Write ${\overline{u}}_{h, j} (t) = u_{h, j}^{n}$ and ${\overline{p}}_{h, j} (t) = p_{h, j}^{n}$ on $(t_{n - 1}^{j}, t_{n}^{j}]$ . For any subsequence for which ${\overline{u}}_{h, j}$ converges to $u$ as in MINI backward Euler converges subsequentially to a solenoidal energy weak velocity, define $P_{h, j}^{0} = 0$ , $P_{h, j}^{m} = \sum_{n = 1}^{m} τ_{n}^{j} p_{h, j}^{n}$ , and let $P_{h, j}$ be the continuous piecewise-affine interpolant of these nodal values. If $\max_{n} ∥ u_{h, j}^{n} ∥_{L^{2}} \leq U_{j}$ and $∥ {\overline{u}}_{h, j} ∥_{L^{2} V} \leq E_{j}$ , then the uniform MINI inf-sup lower bound $β > 0$ gives $\sup_{t \in [0, T]} ∥ P_{h, j} (t) ∥_{L^{2}} \leq β^{- 1} (2 C_{P} U_{j} + C_{P} \sqrt{T} F + 2 ν \sqrt{T} E_{j} + C_{c} T^{1 / 4} U_{j}^{1 / 2} E_{j}^{3 / 2}) .$ Along a further subsequence, $P_{h, j} ⇀ P$ in $L^{2} (0, T; Q)$ . The piecewise-constant pressures converge to $p$ as $Q$ -valued time distributions: for every $q \in Q$ and $η \in C_{c}^{\infty} (0, T)$ , $\int_{0}^{T} ({\overline{p}}_{h, j} (t), q)_{L^{2}} η (t) d t ⟶ (⟨ p, η ⟩, q)_{L^{2}} .$ No relation between mesh size and maximum time step is imposed. These conclusions give pressure primitives and time-distribution convergence, not an instantaneous pressure regularity estimate.

Finally, let $u_{t}^{f u l l}$ denote the distributional derivative of $u$ as a $V^{*}$ -valued distribution, using the embedding $L^{2} (D)^{d} ↪ V^{*}$ . For this $u$ , the pressure-bearing $L^{4 / 3} (0, T; Q)$ clause holds if and only if $u_{t}^{f u l l}$ is represented by an $L^{4 / 3} (0, T; V^{*})$ Bochner function. This criterion concerns the actual full derivative, not an extension of its restriction to $V_{σ}$ .

Facts & Assumptions

Given: The Axiom of Choice, the domain, viscosity and time interval in the Statement, a solenoidal energy weak velocity $u$ , data $f, u_{0}$ , and, for the discrete clauses, the stated mesh/time families and mixed MINI solutions.

[A1]

The Axiom of Choice supplies Countable Choice and the dependent-choice interfaces used by the scalar absolutely-continuous, Bochner and Hilbert subsequence results below. (The Axiom of Choice, AC supplies the countable and dependent choices used in Banach integration, The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The solenoidal energy velocity has a representative weakly continuous in $H_{σ}$ , with $u (0) = u_{0}$ , $u \in L^{\infty} (0, T; H_{σ}) \cap L^{2} (0, T; V_{σ})$ , and it satisfies the scalar time-tested equation on every fixed $v \in V_{σ}$ . (Solenoidal energy weak velocity for incompressible Navier–Stokes)

[F2]

$V = H_{0}^{1} (D; R^{d})$ has norm $∥ v ∥_{V} = ∥ \nabla v ∥_{2}$ , $Q = L_{0}^{2} (D)$ , $b (v, q) = - (q, div v)$ and $a_{ν} (v, w) = 2 ν (ε (v), ε (w))$ . Every $v \in V$ has $div v \in Q$ and $∥ v ∥_{2} \leq C_{P} ∥ v ∥_{V}$ . (Stokes velocity and pressure spaces and weak forms, Zero mean of the divergence of a zero-boundary field, The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction)

[F3]

The real skew form is trilinear and its two bounds in the Statement hold. In $d = 3$ the $L^{6}$ estimate uses the zero-boundary Sobolev inequality at exponent $p = 2$ . In $d = 2$ it uses that inequality at exponent $p = 3 / 2$ , whose Sobolev conjugate is $6$ , followed by finite-measure Holder to bound the $L^{3 / 2}$ gradient norm by the $L^{2}$ gradient norm. Thus the same $V ↪ L^{6}$ bound is available in both dimensions; no $d = 2, p = 2$ critical-exponent formula is used. (The skew convective form for incompressible flow, Continuity, energy cancellation, and Sobolev bounds for skew convection, The Sobolev inequality for zero-boundary Sobolev closures on open sets, Holder's inequality for integrals, including the endpoint cases)

[F4]

There is a bounded linear $B : Q \to V$ with $div B q = q$ and $∥ B q ∥_{V} \leq C_{d i v} ∥ q ∥_{2}$ ; this applies to the stated bounded connected Lipschitz domain in dimensions two and three. (Divergence right inverse on bounded connected Lipschitz domains)

[F5]

Strongly measurable functions with integrable norm have Bochner integrals; the norm of an integral is at most the integral of the norm, bounded linear maps commute with integration, and scalar Fubini applies to the integrable functions used below. (Bochner-integrable function, Strongly measurable Banach-valued function, Bochner integrability criterion, Bochner integral norm inequality, Bounded linear maps commute with Bochner integration, Fubini's theorem for L^1 functions on a sigma-finite product)

[F6]

Scalar weak equations with an $L^{1}$ derivative have absolutely-continuous representatives and obey the fundamental theorem; locally integrable functions that define the zero distribution vanish almost everywhere. Distributional derivatives use the negative test-derivative convention. (One-dimensional $W^{1, p}$ functions have unique absolutely continuous representatives, Fundamental theorem of calculus for absolutely continuous functions, Locally integrable functions embed in distributions, Distributional derivative, A countable union of measure-zero sets has measure zero, by countable choice)

[F7]

$L^{2} (D)$ is a separable real Hilbert space. Its closed mean-zero subspace $Q$ and the closed subspace $H_{σ}$ are Hilbert; a closed-subspace orthogonal projection exists, and Riesz represents bounded real-linear functionals on $Q$ . The Hilbert space $Q$ and $L^{2} (0, T; Q)$ are separable, and bounded sequences in the latter have weakly convergent subsequences under Countable Choice. ( $L^{2}$ with the integral pairing is a Hilbert space, Rational box-step functions form a countable dense subset of $L^{p} (R^{n})$ for $1 \leq p < \infty$ , Separability: the existence of an at most countable dense subset, Orthogonal decomposition by a closed subspace, Riesz representation for Hilbert spaces, Bounded sequences in a separable Hilbert space have weakly convergent subsequences)

[F8]

The mixed MINI scheme holds on all $V_{h}$ tests; its energy theorem gives uniform velocity and gradient bounds and a jump-square bound. The MINI Fortin map preserves the pressure pairing, is uniformly $V$ -stable, approximates each fixed $V$ vector in $V$ , and the discrete $L^{2}$ projector onto $Z_{h}$ is best-approximating. (The mixed mini-element backward Euler Navier–Stokes scheme, Backward Euler MINI step existence and discrete energy bound, Uniform Fortin stability of the mini element, The mini-element Fortin map approximates in L2, Discrete Helmholtz projection and pressure Schur identities)

[F9]

On the subsequence in the Statement, the piecewise-constant velocities converge strongly in $L^{2} (0, T; L^{2})$ and weakly in $L^{2} (0, T; V)$ to $u$ . The cell averages of $f$ converge strongly in $L^{2} (0, T; L^{2})$ . (MINI backward Euler converges subsequentially to a solenoidal energy weak velocity, Time translations and mollification in Bochner Lp)

[F10]

An $L^{4 / 3} (0, T; V^{*})$ residual that annihilates $V_{σ}$ outside one common null set has a unique representing pressure in $L^{4 / 3} (0, T; Q)$ . (Mean-zero pressure recovery from a solenoidal residual)

Proof

technique · direct

1.1A1F1givenchoose

The weakly continuous representative of $u$ obeys $∥ u (t) ∥_{2} \leq U$ for every $t \in [0, T]$ . Indeed, if $∥ u (t_{0}) ∥_{2} > U$ , take $w = u (t_{0}) / ∥ u (t_{0}) ∥_{2} \in H_{σ}$ . Weak continuity makes $(u (t), w)$ continuous and greater than $U$ on a relative neighborhood of $t_{0}$ with positive time measure, contradicting the essential bound $U$ .

1.2A1F1F2F3F5givenconstruct

Define $g (t) \in V^{*}$ by $g (t) (v) = (f (t), v)_{2} - a_{ν} (u (t), v) - c (u (t); u (t), v)$ . The first two terms are strongly measurable bounded linear images of Bochner functions; the last is a continuous quadratic map $V \to V^{*}$ by the trilinear bound in [F3], hence is strongly measurable. Poincare and [F3] give $∥ g (t) ∥_{V^{*}} \leq C_{P} ∥ f (t) ∥_{2} + 2 ν ∥ u (t) ∥_{V} + C_{c} ∥ u (t) ∥_{2}^{1 / 2} ∥ u (t) ∥_{V}^{3 / 2} .$ Holder makes $g \in L^{4 / 3} (0, T; V^{*}) \subset L^{1} (0, T; V^{*})$ and yields $∥ g ∥_{L^{1} V^{*}} \leq C_{P} \sqrt{T} F + 2 ν \sqrt{T} E + C_{c} T^{1 / 4} U^{1 / 2} E^{3 / 2}$ .

1.3F2F3F5F8givenalgebra

For a mixed MINI solution define ${\hat{u}}_{h}$ by affine interpolation of the velocity nodes and let bars denote right-constant reconstructions. On each time cell, $\partial_{t} P_{h} = {\overline{p}}_{h}$ and $\partial_{t} {\hat{u}}_{h} = D_{τ} u_{h}^{n}$ . Integrating the discrete equation over complete or partial cells gives, for every $t$ and $v_{h} \in V_{h}$ , $({\hat{u}}_{h} (t) - u_{h}^{0}, v_{h})_{2} + \int_{0}^{t} (a_{ν} ({\overline{u}}_{h}, v_{h}) + c ({\overline{u}}_{h}; {\overline{u}}_{h}, v_{h})) d s + b (v_{h}, P_{h} (t)) = \int_{0}^{t} (f_{h}, v_{h})_{2} d s .$ Because $P_{h} (t) \in Q_{h}$ , the uniform inf-sup estimate applies. Convexity gives $∥ {\hat{u}}_{h} (t) ∥_{2} \leq U_{h}$ ; [F2], Holder, and [F3] bound the mass, force, viscosity and convection terms by $2 C_{P} U_{h}$ , $C_{P} \sqrt{T} F$ , $2 ν \sqrt{T} E_{h}$ , and $C_{c} T^{1 / 4} U_{h}^{1 / 2} E_{h}^{3 / 2}$ times $∥ v_{h} ∥_{V}$ . This proves the displayed uniform primitive bound.

1.4A1F8givenchoose

Let $u_{h, j}^{0} = P_{Z_{j}} u_{0}$ . For any $ϵ > 0$ , choose $v \in V_{σ}$ with $∥ u_{0} - v ∥_{2} < ϵ$ ; the Fortin map satisfies $Π_{j} v \in Z_{j}$ and $∥ Π_{j} v - v ∥_{2} \to 0$ . Best approximation of the orthogonal projector gives $∥ u_{h, j}^{0} - u_{0} ∥_{2} \leq ∥ Π_{j} v - u_{0} ∥_{2} \leq ∥ Π_{j} v - v ∥_{2} + ϵ$ . Letting $j \to \infty$ and then $ϵ ↓ 0$ proves $u_{h, j}^{0} \to u_{0}$ strongly.

2.1F5step 1.2construct

Set $G (t) = \int_{0}^{t} g (s) d s$ . By [F5], $G$ is norm-continuous in $V^{*}$ , $G (0) = 0$ , and $\sup_{t} ∥ G (t) ∥_{V^{*}} \leq ∥ g ∥_{L^{1} V^{*}}$ .

2.2A1F7F8step 1.3construct

The energy theorem [F8] gives $\sum_{n} ∥ u_{h, j}^{n} - u_{h, j}^{n - 1} ∥_{2}^{2} \leq C$ uniformly. Therefore $∥ {\hat{u}}_{h, j} - {\overline{u}}_{h, j} ∥_{L^{2} L^{2}}^{2} = \frac{1}{3} \sum_{n} τ_{n}^{j} ∥ u_{h, j}^{n} - u_{h, j}^{n - 1} ∥_{2}^{2} \leq C τ_{\max, j} ⟶ 0.$ The pressure bound in step 1.3 is uniform in $j$ , so $(P_{h, j})$ is bounded in the separable Hilbert space $L^{2} (0, T; Q)$ . By [A1] and [F7], a further subsequence converges weakly to some $P_{*} \in L^{2} (0, T; Q)$ . If $K$ is the uniform $L^{\infty} Q$ bound and $A_{ϵ} = {t : ∥ P_{*} (t) ∥_{2} > K + ϵ}$ had positive measure, testing weak convergence against $1_{A_{ϵ}} P_{*}$ gives $\int_{A_{ϵ}} ∥ P_{*} (t) ∥_{2}^{2} d t = \lim_{j} \int_{A_{ϵ}} (P_{h, j} (t), P_{*} (t))_{2} d t \leq K \int_{A_{ϵ}} ∥ P_{*} (t) ∥_{2} d t .$ The left side is at least $(K + ϵ) \int_{A_{ϵ}} ∥ P_{*} (t) ∥_{2} d t$ , a contradiction. Hence every $A_{ϵ}$ is null and $P_{*} \in L^{\infty} Q$ with bound $K$ .

2.3A1F1F5F6F7step 1.2construct

Conversely suppose the actual $u_{t}^{f u l l}$ is represented in $L^{4 / 3} (0, T; V^{*})$ and set $r = g - u_{t}^{f u l l}$ . For each $z \in V_{σ}$ , the scalar weak equation [F1] gives $r (t) (z) = 0$ as a time distribution. The space $V$ is separable: with its equivalent full $H^{1}$ norm it embeds isometrically into a finite product of scalar $L^{2} (D)$ spaces by listing each velocity component and its first derivatives; restricting rational-box step functions makes that product separable. The subspace $V_{σ} = \ker div$ is closed, hence separable (a countable base for the ambient metric space restricts to a countable base on $V_{σ}$ , and Countable Choice selects one point from each nonempty base member). For a countable dense family $(z_{k})$ in $V_{σ}$ , each scalar $r (\cdot) (z_{k}) \in L^{1} (0, T)$ is zero almost everywhere by [F6]; outside the countable union of these null sets, continuity of $r (t) \in V^{*}$ makes it vanish on all of $V_{σ}$ .

3.1F1F2F7step 2.1construct

For every $t$ , define the full-test residual $R (t) \in V^{*}$ by $R (t) (v) = G (t) (v) - (u (t) - u_{0}, v)_{2}$ . This functional is bounded with $∥ R (t) ∥_{V^{*}} \leq C_{P} (U + ∥ u_{0} ∥_{2}) + ∥ g ∥_{L^{1} V^{*}}$ . For each fixed $v \in V$ , the scalar map $t \mapsto R (t) (v)$ is continuous: orthogonally project $v$ onto the closed subspace $H_{σ}$ ; since $u (t) \in H_{σ}$ , its pairing with $v$ equals its pairing with that projection and is continuous by [F1].

3.2F3F8F9step 2.2algebra

For the convergence of convection, use the separate Sobolev routes in [F3]: $H_{0}^{1} ↪ L^{6}$ by the $p = 2$ Sobolev exponent in dimension three, and by the $p = 3 / 2$ inequality plus finite-measure Holder in dimension two. Interpolation gives $∥ w ∥_{L^{2} L^{3}} \leq ∥ w ∥_{L^{2} L^{2}}^{1 / 2} ∥ w ∥_{L^{2} L^{6}}^{1 / 2}$ and $∥ w ∥_{L^{2} L^{4}} \leq ∥ w ∥_{L^{2} L^{2}}^{1 / 4} ∥ w ∥_{L^{2} L^{6}}^{3 / 4}$ . The energy bounds give uniform $L^{2} L^{6}$ norms, while strong $L^{2} L^{2}$ convergence in [F9] therefore implies strong $L^{2} L^{3}$ and $L^{2} L^{4}$ convergence. For fixed $v \in V$ , the gradient term in the skew form pairs weakly convergent gradients with ${\overline{u}}_{h, j} \otimes v$ converging strongly in $L^{2} L^{2}$ ; its other term pairs $\nabla v$ with ${\overline{u}}_{h, j} \otimes {\overline{u}}_{h, j}$ converging strongly in $L^{1} L^{2}$ , since $∥ {\overline{u}}_{h, j} \otimes {\overline{u}}_{h, j} - u \otimes u ∥_{L^{1} L^{2}} \leq (∥ {\overline{u}}_{h, j} ∥_{L^{2} L^{4}} + ∥ u ∥_{L^{2} L^{4}}) ∥ {\overline{u}}_{h, j} - u ∥_{L^{2} L^{4}} \to 0.$ Replacing $v$ by $Π_{j} v$ changes the integral by at most $C ∥ Π_{j} v - v ∥_{V} ∥ {\overline{u}}_{h, j} ∥_{L^{2} V}^{2}$ , which tends to zero. No divergence-free property of the test $v$ is used.

4.1F1F5F6step 1.2step 3.1

Fix $z \in V_{σ}$ . The scalar weak equation [F1] says that $(u (t), z)_{2}$ has distributional derivative $g (t) (z)$ and initial trace $(u_{0}, z)_{2}$ . Since $g (z) \in L^{1}$ , [F6] gives an absolutely-continuous representative equal to $(u_{0}, z)_{2} + G (t) (z)$ . Weak continuity in [F1] identifies that representative with $(u (t), z)_{2}$ for every $t$ . Hence $R (t) (z) = 0$ for every $z \in V_{σ}$ and every $t$ , and $R (0) = 0$ .

5.1F4F5F7step 3.1step 4.1construct

For each $t$ , the map $q \mapsto - R (t) (B q)$ is a bounded real-linear functional on $Q$ of norm at most $C_{d i v} ∥ R (t) ∥_{V^{*}}$ . Riesz representation [F7] therefore defines a unique $P (t) \in Q$ by $(P (t), q)_{2} = - R (t) (B q)$ for all $q \in Q$ . The uniform bound from step 3.1 gives the stated $L^{\infty} Q$ estimate. For fixed $q$ , $t \mapsto (P (t), q)_{2} = - R (t) (B q)$ is continuous, so $P \in C_{w} ([0, T]; Q)$ and $P (0) = 0$ . Choose a countable orthonormal basis of the separable Hilbert space $Q$ . Each finite-coordinate projection of $P$ is strongly measurable because its coordinates are continuous; the projections converge in norm to $P (t)$ at every $t$ . Thus $P$ is strongly measurable and belongs to $L^{\infty} (0, T; Q)$ .

6.1F2F4step 4.1step 5.1algebra

For $v \in V$ , let $q = div v \in Q$ by [F2]. Then $v - B q \in V_{σ}$ , so step 4.1 gives $R (t) (v) = R (t) (B q) = - (P (t), q)_{2} = b (v, P (t))$ . Substitution of the definition of $R$ proves the integrated identity for every $t$ and $v$ . If $P_{1}, P_{2}$ both satisfy it, set $v = B q$ for arbitrary $q \in Q$ to obtain $(P_{1} (t) - P_{2} (t), q)_{2} = 0$ for every $q$ , hence $P_{1} (t) = P_{2} (t)$ .

7.1F2F5step 6.1construct

The formula $⟨ p, η ⟩ = - \int_{0}^{T} P (t) η^{'} (t) d t$ defines a continuous $Q$ -valued distribution because its norm is bounded by $∥ P ∥_{L^{\infty} Q} ∥ η^{'} ∥_{L^{1}}$ . Differentiating step 6.1 against scalar tests gives $u_{t} + a_{ν} (u, \cdot) + c (u; u, \cdot) + b (\cdot, p) = f$ in $V^{*}$ -valued distributions. If another $Q$ -valued distribution has this property, test the difference against $B q$ for each $q \in Q$ ; since $div B q = q$ , the $Q$ inner product of the difference with every $q$ vanishes, so the difference is zero.

7.2F5F7F8F9step 6.1step 1.3step 1.4step 2.2step 3.2

Fix $v \in V$ and $η \in C_{c}^{\infty} (0, T)$ . Test the integrated identity of step 1.3 with $Π_{j} v$ and integrate in $t$ against $η$ . Scalar Fubini rewrites each primitive integral using the bounded weight $s \mapsto \int_{s}^{T} η (t) d t$ . Strong convergence of ${\hat{u}}_{h, j}$ from [F9] and step 2.2, strong convergence of initial projections from step 1.4, weak $L^{2} V$ convergence from [F9], convergence of force averages from [F9], Fortin convergence $Π_{j} v \to v$ in $V$ , the convection convergence in step 3.2, and weak convergence $P_{h, j} ⇀ P_{*}$ pass every term to the limit. In the pressure term, weak convergence of $P_{h, j}$ is paired with the strongly convergent tests $η div Π_{j} v \to η div v$ in $L^{2} (0, T; Q)$ . The result is the continuous integrated identity of step 6.1 with $P_{*}$ in place of $P$ , as a scalar time distribution.

8.1F1F2F5F6step 6.1step 7.1algebra

For fixed $v \in V$ , step 6.1 says $(u (t), v)_{2} + b (v, P (t)) = (u_{0}, v)_{2} + \int_{0}^{t} g (s) (v) d s$ . The right side is absolutely continuous. Multiply this scalar identity by a $C^{1}$ time function vanishing near $T$ and integrate by parts; for a finite sum of tests $φ (t) = \sum_{k = 1}^{m} η_{k} (t) v_{k}$ this gives the stated initial-trace formula. Such finite sums are dense in $C^{1} ([0, T]; V)$ for tests vanishing near $T$ : since $φ$ vanishes on a terminal interval, so does $φ'$ . On successively finer partitions, piecewise-linear interpolation of this uniformly continuous derivative has finite-dimensional range, converges uniformly to $φ'$ , and remains zero on a terminal interval. Write an interpolant as $\sum_{k = 1}^{m} χ_{k} (t) v_{k}$ with fixed $v_{k} \in V$ and continuous piecewise-linear scalar functions $χ_{k}$ . Then $φ_{m} (t) = - \sum_{k = 1}^{m} (\int_{t}^{T} χ_{k} (s) d s) v_{k}$ is a finite sum with $C^{1}$ scalar coefficients, vanishes near $T$ , and converges to $φ$ in $C^{1}$ . Every term in the weak-form identity is continuous in that norm by [F1]–[F5], so the identity extends to every stated $φ$ .

8.2F4F5F6F7step 6.1step 7.1step 7.2algebra

Subtract step 6.1 from the limit identity in step 7.2 and set $v = B q$ for each $q \in Q$ . Then $(P_{*} - P, q)_{2} = 0$ as a scalar time distribution. Since $P_{*} - P \in L^{2} (0, T; Q)$ and $Q$ has a countable dense subset, [F6] and the countable-union result make $P_{*} = P$ almost everywhere. For $η \in C_{c}^{\infty} (0, T)$ , the identity $\partial_{t} P_{h, j} = {\overline{p}}_{h, j}$ gives $\int_{0}^{T} ({\overline{p}}_{h, j}, q)_{2} η d t = - \int_{0}^{T} (P_{h, j}, q)_{2} η^{'} d t ⟶ - \int_{0}^{T} (P, q)_{2} η^{'} d t = (⟨ p, η ⟩, q)_{2} .$ This is the claimed weak convergence as $Q$ -valued time distributions.

8.3F2F5step 1.2step 7.1

To prove the forward direction of the instantaneous-pressure criterion, suppose $p \in L^{4 / 3} (0, T; Q)$ . The full equation in step 7.1 gives $u_{t}^{f u l l} = g - b (\cdot, p)$ as a $V^{*}$ -valued distribution. Since $∣ div v ∣ \leq \sqrt{d} ∣ \nabla v ∣$ , the map $q \mapsto b (\cdot, q)$ is bounded from $Q$ to $V^{*}$ ; [F5] therefore makes $b (\cdot, p)$ Bochner $L^{4 / 3} V^{*}$ . As $g \in L^{4 / 3} V^{*}$ by step 1.2, the actual full derivative has that same regularity.

9.1F2F4F10step 7.1step 2.3∎

The continuous residual-recovery lemma [F10] applies to $r \in L^{4 / 3} (0, T; V^{*})$ and gives $p_{s} \in L^{4 / 3} (0, T; Q)$ with $r (t) (v) = b (v, p_{s} (t))$ for all $v \in V$ almost everywhere. Thus $p_{s}$ is a full pressure distribution for $u$ . Uniqueness in step 7.1 identifies $p_{s}$ with $p$ , proving the reverse direction. Therefore the exact necessary-and-sufficient condition is the stated regularity of the actual full derivative.

Remarks

The source's pressure-free weak formulation and the primitive conclusions above do not establish the stronger PE06 instantaneous pressure regularity or instantaneous smooth pressure-error rate. The derivative criterion isolates the actual missing continuous input; an extension of $u_{t}$ known only on $V_{σ}$ is not that input.

TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

First-order smooth pressure-primitive error for backward Euler MINI

Statement

Assume Countable Choice and every domain, mesh, time-partition, smooth-solution, initialization and time-step hypothesis of Smooth-branch velocity error for backward Euler MINI. Thus $D$ is a bounded connected Lipschitz polyhedral domain in dimension $d \in {2, 3}$ , the matching affine simplicial meshes are shape regular, $ν > 0$ , $f \in L^{2} (0, T; L^{2} (D)^{d})$ , and $(u, p)$ is a strong no-slip solution with $u \in W^{1, \infty} (0, T; H^{2} (D)^{d} \cap V_{σ}), p \in W^{1, \infty} (0, T; H^{1} (D) \cap Q), Q = L_{0}^{2} (D) .$ The point values of these classes mean their unique continuous representatives, constructed in the proof using Countable Choice. Set $M = 1 + ∥ u ∥_{W^{1, \infty} H^{2}} + ∥ p ∥_{W^{1, \infty} H^{1}}, M_{u} = ∥ u ∥_{L^{\infty} V}, M_{E} = ν^{- 1 / 2} ∥ u (0) ∥_{2} + C_{P} ν^{- 1} ∥ f ∥_{L^{2} L^{2}} .$ Let $C_{*}$ and $Λ$ be exactly the explicit viscosity-dependent constants in Smooth-branch velocity error for backward Euler MINI and impose its restriction $τ_{\max} Λ \leq 1 / 2$ . Use the same initialization $u_{h}^{0} = P_{Z_{h}} u (0)$ and any mixed backward Euler MINI solution. Define $e_{n} = u (t_{n}) - u_{h}^{n}, E_{0} = \max_{0 \leq n \leq N} ∥ e_{n} ∥_{2}, E_{1} = {(\sum_{n = 1}^{N} τ_{n} ∥ e_{n} ∥_{V}^{2})}^{1 / 2}, H = h M + τ_{\max} M^{2} .$ Let $β > 0$ be the uniform MINI inf-sup lower bound, $M_{b} = \sqrt{d}$ , and let $C_{I}$ be a mean-adjusted continuous P1 pressure interpolation constant satisfying $\inf_{q_{h} \in Q_{h}} ∥ q - q_{h} ∥_{2} \leq C_{I} h ∥ q ∥_{H^{1} (D)} (q \in H^{1} (D) \cap Q) .$ Choose $C_{c}$ with $∣ c (a; b, v) ∣ \leq C_{c} ∥ a ∥_{V} ∥ b ∥_{V} ∥ v ∥_{V}$ , put $P (t) = \int_{0}^{t} p (s) d s$ , $P_{h}^{0} = 0$ , $P_{h}^{m} = \sum_{n = 1}^{m} τ_{n} p_{h}^{n}$ , and let $P_{h}$ be the continuous piecewise-affine interpolant through $P_{h}^{m}$ . Then $\max_{0 \leq m \leq N} ∥ P (t_{m}) - P_{h}^{m} ∥_{2} \leq (1 + M_{b} / β) C_{I} T h M + β^{- 1} (2 C_{P} E_{0} + [2 ν \sqrt{T} + C_{c} (M_{E} + \sqrt{T} M_{u})] E_{1} + 2 ν T τ_{\max} M + 2 C_{c} T τ_{\max} M^{2}) .$ In particular, with $K = (1 + M_{b} / β) C_{I} T + β^{- 1} (2 C_{P} + [2 ν \sqrt{T} + C_{c} (M_{E} + \sqrt{T} M_{u})] ν^{- 1 / 2}) C_{*} + β^{- 1} (2 ν T + 2 C_{c} T),$ one has $\max_{m} ∥ P (t_{m}) - P_{h}^{m} ∥_{2} \leq K H, ∥ P - P_{h} ∥_{L^{\infty} (0, T; Q)} \leq (K + 1) H .$ For the right-constant pressure reconstruction ${\overline{p}}_{h} (t) = p_{h}^{n}$ on $(t_{n - 1}, t_{n}]$ , every $η \in C_{c}^{\infty} (0, T)$ also satisfies ${∥ \int_{0}^{T} (p (t) - {\overline{p}}_{h} (t)) η (t) d t ∥}_{2} \leq (K + 1) H ∥ η^{'} ∥_{L^{1} (0, T)} .$ The constants are independent of $h$ and the individual time steps; their viscosity dependence includes the displayed powers and the exponential contained in $C_{*}$ . No additional mesh/time relation is imposed. This estimate controls accumulated pressure and the stated negative time norm; it does not claim an instantaneous $L_{t}^{4 / 3} L_{x}^{2}$ pressure-error rate.

Facts & Assumptions

Given: Countable Choice, the smooth solution, MINI scheme solution, constants, initialization and partition in the Statement.

[A1]

Countable Choice is the only choice principle needed for the local representative argument below; it is also inherited by the smooth velocity, MINI stability and Bochner suppliers. No use of the Axiom of Choice or Dependent Choice is added here. (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Smooth-branch velocity error for backward Euler MINI)

[F1]

The strong solution satisfies the full time-tested mixed equation, with $b (v, q) = - (q, div v)$ and $∣ a_{ν} (v, w) ∣ \leq 2 ν ∥ v ∥_{V} ∥ w ∥_{V}$ ; every term is integrable. (Energy weak solutions of incompressible Navier–Stokes, Integrability of the energy weak Navier–Stokes formulation, Stokes velocity and pressure spaces and weak forms)

[F2]

The current smooth velocity theorem gives $E_{0} + \sqrt{ν} E_{1} \leq C_{*} H$ . The discrete energy theorem gives $\sum_{n} τ_{n} ∥ u_{h}^{n} ∥_{V}^{2} \leq M_{E}^{2}$ . (Smooth-branch velocity error for backward Euler MINI, Backward Euler MINI step existence and discrete energy bound)

[F3]

The MINI pressure space $Q_{h}$ from The conforming P1-bubble/P1 mini element is finite-dimensional and has inf-sup constant $β > 0$ . The face-average P1 interpolant in Boundary-preserving face-average P1 interpolation maps $H^{1} (D)$ to the continuous P1 space and has $L^{2}$ error at most $C h ∥ q ∥_{H^{1}}$ on the stated shape-regular meshes. Also $∣ b (v, q) ∣ \leq \sqrt{d} ∥ v ∥_{V} ∥ q ∥_{2}$ . (Uniform Fortin stability of the mini element, Boundary-preserving face-average P1 interpolation, The conforming P1-bubble/P1 mini element, Stokes velocity and pressure spaces and weak forms)

[F4]

The skew form is trilinear and satisfies the $V^{3}$ bound in the Statement under Countable Choice, as proved directly in the skew-form supplier. Poincare gives $∥ v ∥_{2} \leq C_{P} ∥ v ∥_{V}$ , and Holder gives the weighted time Cauchy–Schwarz estimates used below. (The skew convective form for incompressible flow, Continuity, energy cancellation, and Sobolev bounds for skew convection, The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction, Holder's inequality for integrals, including the endpoint cases)

[F5]

The Bochner norm inequality and bounded-linear-map interchange apply to interval averages and primitives. Scalar Fubini applies to the time integrals. (Bochner-integrable function, Strongly measurable Banach-valued function, Bochner integrability criterion, Bochner integral norm inequality, Bounded linear maps commute with Bochner integration, Fubini's theorem for L^1 functions on a sigma-finite product)

[F6]

$L^{2} (D)$ is a separable real Hilbert space, and every closed subspace has an orthogonal decomposition by Orthogonal decomposition by a closed subspace. $H^{1} (D)$ and $H^{2} (D)^{d}$ embed into finite products of $L^{2} (D)$ by listing their weak derivatives and are separable as subspaces under Countable Choice. A locally integrable scalar function defining the zero distribution vanishes almost everywhere, and countable unions of null sets are null. ( $L^{2}$ with the integral pairing is a Hilbert space, Rational box-step functions form a countable dense subset of $L^{p} (R^{n})$ for $1 \leq p < \infty$ , Separability: the existence of an at most countable dense subset, Locally integrable functions embed in distributions, A countable union of measure-zero sets has measure zero, by countable choice)

Proof

technique · direct

Given: The strong solution and all MINI scheme inputs in the Statement.

1.1A1F5F6givenconstruct

To justify pointwise time estimates without strengthening Countable Choice, let $X$ be either $H^{2} (D)^{d}$ or $H^{1} (D) \cap Q$ , let $w \in W^{1, \infty} (0, T; X)$ , and let $g = w^{'} \in L^{\infty} (0, T; X)$ be its weak Bochner derivative. The separability of $X$ is [F6]. Define $G (t) = \int_{0}^{t} g (s) d s$ . By [F5], $∥ G (t) - G (s) ∥_{X} \leq ∥ g ∥_{L^{\infty} X} ∣ t - s ∣$ . For $ξ \in X$ and $ϕ \in C_{c}^{\infty} (0, T)$ , scalar Fubini gives $- \int_{0}^{T} (G (t), ξ)_{X} ϕ^{'} (t) d t = - \int_{0}^{T} \int_{0}^{t} (g (s), ξ)_{X} d s ϕ^{'} (t) d t = \int_{0}^{T} (g (s), ξ)_{X} ϕ (s) d s .$ Thus $G^{'} = g$ in scalar distributions and $(w - G, ξ)$ has zero distributional derivative.

1.2F1givenalgebra

For $v \in V$ , Cauchy–Schwarz for the matrix trace gives $∣ div v ∣ = ∣ tr \nabla v ∣ \leq \sqrt{d} ∣ \nabla v ∣$ almost everywhere. Integrating and applying Cauchy–Schwarz yields $∥ b (v, q) ∥ \leq \sqrt{d} ∥ v ∥_{V} ∥ q ∥_{2} = M_{b} ∥ v ∥_{V} ∥ q ∥_{2}$ , as used in the Statement.

1.3F3F6givenconstructalgebra

For $q \in H^{1} (D) \cap Q$ , let $I_{h} q$ be the face-average P1 interpolant from [F3] and set $r_{h}^{0} q = I_{h} q - ∣ D ∣^{- 1} \int_{D} I_{h} q$ . This belongs to $Q_{h}$ . Since $q$ has mean zero, Cauchy–Schwarz gives $∣ D ∣^{1 / 2} ∣ ∣ D ∣^{- 1} \int_{D} I_{h} q ∣ \leq ∥ I_{h} q - q ∥_{2}$ , hence $∥ q - r_{h}^{0} q ∥_{2} \leq 2 ∥ q - I_{h} q ∥_{2} \leq 2 C h ∥ q ∥_{H^{1} (D)}$ . Thus some finite uniform best-approximation constant exists, as required by the Statement. Now let $r_{h}$ be the $L^{2}$ -orthogonal projection onto $Q_{h}$ . The finite-dimensional space $Q_{h}$ is closed in Hilbert $Q$ ; [F6] gives its unique orthogonal decomposition. Uniqueness makes $r_{h}$ linear, and Pythagoras gives $∥ r_{h} q ∥_{2} \leq ∥ q ∥_{2}$ , so it is bounded. The Statement's selected best-approximation constant therefore yields $∥ q - r_{h} q ∥_{2} = \inf_{q_{h} \in Q_{h}} ∥ q - q_{h} ∥_{2} \leq C_{I} h ∥ q ∥_{H^{1} (D)}$ .

2.1F6step 1.1algebra

A locally integrable scalar function with zero distributional derivative is constant almost everywhere. Indeed, fix $ρ \in C_{c}^{\infty} (0, T)$ with $\int ρ = 1$ . For any $ϕ \in C_{c}^{\infty} (0, T)$ , $ψ = ϕ - (\int ϕ) ρ$ has integral zero, so $θ (t) = \int_{0}^{t} ψ (s) d s$ belongs to $C_{c}^{\infty} (0, T)$ and $θ^{'} = ψ$ . The zero-derivative identity gives $\int (w - G, ξ) ψ = 0$ , hence $\int (w - G, ξ) ϕ = c_{ξ} \int ϕ$ with $c_{ξ} = \int (w - G, ξ) ρ$ . Injectivity of the locally integrable embedding into distributions [F6] yields $(w - G, ξ) = c_{ξ}$ almost everywhere.

3.1A1F6step 1.1step 2.1construct

Choose a countable dense set $(ξ_{k})$ in $X$ . By [F6], the scalar identities in step 2.1 hold outside one common null set. Choose $t_{0}$ outside that set where the Bochner representative of $w$ is defined, and put $c = w (t_{0}) - G (t_{0})$ . Density gives $w (t) - G (t) = c$ for almost every $t$ . Thus $w^{*} (t) = G (t) + c$ is the unique continuous representative and is Lipschitz with constant $∥ w^{'} ∥_{L^{\infty} X}$ . Apply this to $u$ and $p$ : for all $s, t$ , $∥ u (t) - u (s) ∥_{V} \leq M ∣ t - s ∣$ and $∥ p (t) - p (s) ∥_{H^{1}} \leq M ∣ t - s ∣$ . Continuity and their essential bounds then give $∥ u (t) ∥_{V} \leq M_{u}$ and $∥ p (t) ∥_{H^{1}} \leq M$ for every $t$ . Also $M \geq M_{u}$ , since the $H^{2}$ norm controls the $V$ norm. These representatives are the ones used for the nodal errors in the existing smooth velocity theorem.

4.1F1F5F6step 3.1givenalgebra

Write $u_{n} = u (t_{n})$ , ${\overline{u}}_{n} = τ_{n}^{- 1} \int_{t_{n - 1}}^{t_{n}} u (t) d t$ , and use bars similarly for exact pressure, forcing and convection. For each vector in a basis of the finite-dimensional $V_{h}$ , [F1] and step 3.1 imply that the full scalar momentum residual is the zero distribution. All residual terms are integrable, so [F6] identifies that residual with zero almost everywhere; integrating over each time cell and using $u (t_{n}) - u (t_{n - 1}) = \int_{t_{n - 1}}^{t_{n}} u_{t}$ from step 3.1 gives the exact interval equation. Subtract the mixed backward Euler equations and sum through $m$ . The identical forcing averages cancel and the time derivatives telescope, giving for every $v_{h} \in V_{h}$ $b (v_{h}, P (t_{m}) - P_{h}^{m}) = - (e_{m} - e_{0}, v_{h})_{2} - \sum_{n \leq m} τ_{n} a_{ν} ({\overline{u}}_{n} - u_{h}^{n}, v_{h}) - \sum_{n \leq m} τ_{n} ({\overline{c (u; u, v_{h})}}_{n} - c (u_{h}^{n}; u_{h}^{n}, v_{h})) .$ No projection onto the discrete kernel is used in this identity.

5.1F3F5step 1.2step 1.3step 4.1constructalgebra

Set $R_{h}^{m} = \int_{0}^{t_{m}} r_{h} p (t) d t \in Q_{h}$ , where $r_{h}$ is the $L^{2}$ -orthogonal pressure-space projection chosen in step 1.3. Its bounded linearity and [F5] give $∥ P (t_{m}) - R_{h}^{m} ∥_{2} \leq C_{I} \int_{0}^{t_{m}} h ∥ p (t) ∥_{H^{1}} d t \leq C_{I} T h M .$ Apply inf-sup to $R_{h}^{m} - P_{h}^{m}$ , and use $∣ b (v_{h}, R_{h}^{m} - P (t_{m})) ∣ \leq M_{b} ∥ v_{h} ∥_{V} ∥ R_{h}^{m} - P (t_{m}) ∥_{2}$ . The triangle inequality yields $∥ P (t_{m}) - P_{h}^{m} ∥_{2} \leq (1 + M_{b} / β) C_{I} T h M + β^{- 1} \sup_{v_{h} \neq 0} \frac{∣ b (v_{h}, P (t_{m}) - P_{h}^{m}) ∣}{∥ v_{h} ∥_{V}} .$

5.2F1F2F4step 3.1step 4.1algebra

Poincare bounds $∣ (e_{m} - e_{0}, v_{h})_{2} ∣ \leq 2 C_{P} E_{0} ∥ v_{h} ∥_{V}$ . Step 3.1 gives $∥ {\overline{u}}_{n} - u_{n} ∥_{V} \leq τ_{n} M$ , hence $\sum_{n \leq m} τ_{n} ∥ {\overline{u}}_{n} - u_{h}^{n} ∥_{V} \leq \sqrt{T} E_{1} + T τ_{\max} M .$ The viscosity contribution in step 4.1 is therefore at most $2 ν (\sqrt{T} E_{1} + T τ_{\max} M) ∥ v_{h} ∥_{V}$ .

5.3F2F4step 4.1algebra

Trilinearity gives $c (u_{n}; u_{n}, v_{h}) - c (u_{h}^{n}; u_{h}^{n}, v_{h}) = c (e_{n}; u_{n}, v_{h}) + c (u_{h}^{n}; e_{n}, v_{h}) .$ Its absolute value is at most $C_{c} (∥ u_{n} ∥_{V} + ∥ u_{h}^{n} ∥_{V}) ∥ e_{n} ∥_{V} ∥ v_{h} ∥_{V}$ . Weighted Cauchy–Schwarz, $∥ u_{n} ∥_{V} \leq M_{u}$ , and [F2] show that its summed contribution is at most $C_{c} (\sqrt{T} M_{u} + M_{E}) E_{1} ∥ v_{h} ∥_{V}$ .

6.1F4step 3.1step 4.1step 5.3algebra

For $t \in (t_{n - 1}, t_{n})$ , split $c (u (t); u (t), v_{h}) - c (u_{n}; u_{n}, v_{h}) = c (u (t) - u_{n}; u (t), v_{h}) + c (u_{n}; u (t) - u_{n}, v_{h}) .$ By step 3.1, $∥ u (t) - u_{n} ∥_{V} \leq τ_{n} M$ , and $M \geq M_{u}$ . The $V^{3}$ estimate bounds the integrand by $2 C_{c} τ_{n} M^{2} ∥ v_{h} ∥_{V}$ . Integrating over the cells through $m$ gives at most $2 C_{c} T τ_{\max} M^{2} ∥ v_{h} ∥_{V}$ , completing the convection bound in step 4.1.

7.1F2step 5.1step 5.2step 5.3step 6.1algebra

Insert the estimates in steps 5.2, 5.3, and 6.1 into step 5.1. This gives the first displayed pressure-primitive inequality, uniformly in $m$ . Substitute $E_{0} \leq C_{*} H$ and $E_{1} \leq ν^{- 1 / 2} C_{*} H$ from [F2]. Since $M \geq 1$ , both $h M$ and $τ_{\max} M$ are at most $H$ , and $τ_{\max} M^{2} \leq H$ . The resulting coefficient is exactly $K$ , so $\max_{m} ∥ P (t_{m}) - P_{h}^{m} ∥_{2} \leq K H$ .

8.1F5step 3.1step 7.1algebra

The exact primitive $P$ is Lipschitz in $Q$ with constant at most $∥ p ∥_{L^{\infty} Q} \leq M$ , by step 3.1. Let $\tilde{P}$ interpolate its exact endpoint values affinely on the partition. For $t \in [t_{n - 1}, t_{n}]$ and $α = (t - t_{n - 1}) / τ_{n}$ , $P (t) - \tilde{P} (t) = (1 - α) (P (t) - P (t_{n - 1})) + α (P (t) - P (t_{n})),$ so $∥ P (t) - \tilde{P} (t) ∥_{2} \leq 2 α (1 - α) τ_{n} M \leq τ_{\max} M$ . Convexity gives $∥ \tilde{P} - P_{h} ∥_{L^{\infty} Q} \leq \max_{m} ∥ P (t_{m}) - P_{h}^{m} ∥_{2}$ . Together with step 7.1 this proves $∥ P - P_{h} ∥_{L^{\infty} Q} \leq (K + 1) H$ , because $τ_{\max} M \leq H$ .

9.1F5step 8.1algebra∎

On each time cell, $P_{h}^{'} = {\overline{p}}_{h}$ . Scalar integration by parts against $η \in C_{c}^{\infty} (0, T)$ and cancellation of the node terms give the $Q$ -valued identity $\int_{0}^{T} (p - {\overline{p}}_{h}) η d t = - \int_{0}^{T} (P - P_{h}) η^{'} d t .$ Scalarization and [F5] justify the vector identity; its norm is at most $∥ P - P_{h} ∥_{L^{\infty} Q} ∥ η^{'} ∥_{L^{1}}$ . Step 8.1 proves the stated negative-time-norm estimate.

Remarks

This is an additive pressure estimate under the existing initialization and smooth-solution hypotheses. Differentiating a small primitive does not give a small instantaneous $L_{t}^{4 / 3} Q$ pressure error. The original unrestricted instantaneous pressure-rate clause is separately refuted by the smooth graded-mesh counterexample on the companion page; this does not change any hypothesis or conclusion here. The proof uses uniform inf-sup after time integration and neither assumes nor derives uniform $H^{1}$ stability of the mass-orthogonal projector onto $Z_{h}$ .

RemarkRemark: AI-adaptedProof: Not applicableOpen item page →

PE06 instantaneous pressure refutations and retained primitive estimates

Escalation

The original PE06 plan required two instantaneous pressure conclusions: a pressure-bearing subsequential weak solution with $p \in L^{4 / 3} (0, T; L_{0}^{2} (D))$ , and the pressure part of the smooth-branch velocity/pressure error estimate. The unchanged MINI backward Euler converges subsequentially to a solenoidal energy weak velocity proves the solenoidal velocity equation and its energy inequality; it does not state a pressure-bearing solution in the regular-pressure class Energy weak solutions of incompressible Navier–Stokes. The unchanged Smooth-branch velocity error for backward Euler MINI proves the velocity estimate only. The generic regular-pressure conclusion is now refuted under its exact original hypotheses by the square construction described in Remarks below; the independent smooth instantaneous rate is also refuted under its unchanged hypotheses by the graded mass-initialization construction in Remarks. Both original formulations are retained here as the claims refuted, rather than silently replaced by primitive estimates.

The additive Integrated pressure recovery and MINI pressure-primitive convergence now gives a unique mean-zero time-distribution pressure and bounded weakly continuous pressure primitive for the limiting velocity, plus uniformly bounded discrete primitives and weak convergence of reconstructed pressures as time distributions. It also proves the exact criterion: the regular $L_{t}^{4 / 3} L_{0}^{2}$ pressure exists precisely when the actual full distributional derivative $u_{t} \in L^{4 / 3} (0, T; V^{*})$ . The current limit argument establishes the derivative only after restriction to $V_{σ}$ . Extending that restricted functional by a divergence lift does not prove that the extension is the actual full derivative. The sufficient condition $u \in W^{1, 4 / 3} (0, T; L^{2} (D)^{d})$ would imply the criterion, but this regularity is not supplied by the stated energy hypotheses. The primitive result does not establish it. The square counterexample proves that this extra regularity can actually fail for the original energy data, so the generic expectation cannot be supplied without a genuine additional hypothesis. The regular-pressure Definition remains a valid conditional solution class.

The additive First-order smooth pressure-primitive error for backward Euler MINI gives a first-order estimate for the accumulated pressure $P (t) = \int_{0}^{t} p (s) d s$ , its piecewise-affine discrete primitive, and the specified negative time norm. It does not bound the instantaneous $L_{t}^{4 / 3} L_{x}^{2}$ pressure error. Applying the discrete inf-sup inequality to the unintegrated pressure equation leaves the full-test derivative residual $d_{n} = \sup_{v_{h} \neq 0} \frac{∣ (D_{τ} e_{n}, v_{h})_{L^{2}} ∣}{∥ v_{h} ∥_{V}}, Δ = {(\sum_{n} τ_{n} d_{n}^{4 / 3})}^{3 / 4} .$ The nodal velocity estimate does not itself control this derivative norm uniformly as the smallest time step tends to zero, and the primitive estimate cannot be differentiated to supply it. The exact-scope smooth counterexample now shows that the original unrestricted instantaneous rate cannot be recovered by another argument under the unchanged hypotheses. This is a refutation of that universal rate, not a claim that every stronger initialization or more restricted mesh class fails.

The formerly proposed uniform $H^{1}$ bound for the $L^{2}$ -orthogonal projector $P_{Z_{h}} : V_{h} \to Z_{h}$ is false. On the square, the continuous orthogonal projection of an explicit smooth no-slip velocity onto the Leray space has a nonzero tangential boundary trace, so it is not in $H_{0}^{1}$ . If the proposed discrete bound held on a refining MINI family, composition with the actual uniformly $H^{1}$ -stable Fortin maps and weak compactness would force that false continuous mapping property. The full-test and restricted- $V_{h}$ versions are therefore unavailable. The projector obstruction is a refutation of that proposed supplier only; it proves neither that the original pressure claims are impossible nor that another route cannot prove them.

Affected claims: the PE06 pressure-bearing regularity clause following MINI backward Euler converges subsequentially to a solenoidal energy weak velocity, including $p \in L^{4 / 3} (0, T; L_{0}^{2} (D))$ , and the instantaneous pressure-rate clause paired with Smooth-branch velocity error for backward Euler MINI.

Source limits: Süli–Tscherpel explicitly use a pressure-free weak formulation and discuss a distributional-in-time pressure expectation for their Lipschitz no-slip setting. Their problem has a more general stress law and a different approximation/limit route, so that statement is not a proof of the exact Newtonian, $L^{2}$ -forcing PE06 claim or of its impossibility. Guermond and Boffi–Brezzi–Fortin supply static divergence lifting and mixed inf-sup context, not either missing evolutionary estimate.

Current decision: retain the original generic formulation and record its exact-scope refutation. Preserve the complete velocity and distributional/primitive pressure results and their exact full-derivative criterion. Retain the original smooth instantaneous formulation and its exact-scope counterexample; the primitive estimate is an independent positive result and does not substitute for that formulation. Do not classify either authoring obligation as an open problem or use the false projector route.

Remarks

The contextual An energy Navier–Stokes velocity on the square without endpoint-integrable pressure ↗ gives an actual two-dimensional square no-slip trajectory with $u_{0} \in H_{σ}$ and $f \in L_{t}^{2} L_{x}^{2}$ , including the nonlinear force, whose unique mean-zero pressure lies in $L_{t}^{1} L_{x}^{2}$ but not $L_{t}^{4 / 3} L_{x}^{2}$ . Its energy equality and pressure-free two-dimensional uniqueness identify every MINI velocity limit for those data. Thus it refutes the original generic pressure-bearing conclusion rather than merely a proposed proof route. It is contextual counterexample evidence, not a logical premise of this remark or of any pressure-recovery theorem. Its singular initial profile does not meet the original smooth-branch assumptions, so the smooth clause needs the independent construction below.

The contextual Graded MINI mass initialization can destroy the smooth instantaneous pressure rate ↗ refutes the separate instantaneous smooth pressure-error assertion on one fixed convex polygon and one fixed-shape family of arbitrarily graded actual MINI meshes. Its smooth exact data have uniformly bounded original norms, its initialization is the actual weak-kernel orthogonal mass projection, and its pressures arise from genuine backward Euler roots with arbitrarily small steps satisfying the original upper-step condition. The proof retains the actual symmetric-gradient operator, full nonlinear force, mean-zero gauge and arbitrary-root quantifier. It does not infer pressure growth from arbitrary fractional velocity roughness: a positive pressure observation for four specifically realized coarse sources and a finite real-amplitude nonlinear argument supply the actual pressure channel. This contextual generated counterexample is not a logical dependency of a positive theorem.

5 · Examples, counterexamples and false statements

None yet.