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The higher-order energy inequality and a common Galerkin lifespan with $∥ u_{N} (t) ∥_{H^{s}} \leq 2 Y_{0}$

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), used through the Fourier-coefficient suppliers. Fix $N \geq 1$ , $ν \geq 0$ and a real $s > 5 / 2$ , and let $u_{N} : I \to X_{N}$ be the maximal solution of the Galerkin system $u_{N}^{'} = F_{N} (u_{N})$ of The Galerkin coefficient system: finite-dimensional invariance, local Lipschitz property, and the unique maximal approximation on the torus with initial datum $u_{N} (0) = P_{N} u_{0}$ , where $u_{0} \in H_{σ}^{s}$ and $Y_{0} : = ∥ u_{0} ∥_{H^{s}}$ .

Energy inequality. For every real $m \geq s$ and every $t \in I$ , $\frac{1}{2} \frac{d}{d t} ∥ u_{N} (t) ∥_{H^{m}}^{2} + ν ∥ \nabla u_{N} (t) ∥_{H^{m}}^{2} \leq C_{m} ∥ u_{N} (t) ∥_{H^{s}} ∥ u_{N} (t) ∥_{H^{m}}^{2},$ where $C_{m}$ is the commutator constant of The transport term is orthogonal to itself and the tame commutator estimate $∥ [Λ^{m}, u \cdot \nabla] u ∥_{L^{2}} \leq C_{m} ∥ u ∥_{H^{m}} ∥ u ∥_{H^{s}}$ (claim 2), and $∥ \nabla u_{N} ∥_{H^{m}}^{2} : = \sum_{j = 1}^{3} ∥ \partial_{j} u_{N} ∥_{H^{m}}^{2}$ .
Common lifespan. With $T : = {\begin{cases} (2 C_{s} Y_{0})^{- 1}, & Y_{0} > 0, \\ 1, & Y_{0} = 0, \end{cases}$ one has $[0, T] \subseteq I$ and $∥ u_{N} (t) ∥_{H^{s}} \leq 2 Y_{0} (0 \leq t \leq T);$ if $Y_{0} = 0$ then $u_{N} \equiv 0$ on $I$ . In particular the same interval length $T$ , depending only on $Y_{0}$ and $s$ (and not on $N$ or on the finer profile of $u_{0}$ ), is a common interval of existence for all Galerkin approximations.

Facts & Assumptions

Given: The Galerkin solution $u_{N}$ of The Galerkin coefficient system: finite-dimensional invariance, local Lipschitz property, and the unique maximal approximation on the torus (whose values are trigonometric polynomials, hence belong to every $H^{m}$ ), the operators $Λ^{m}$ , $\partial_{j}$ , $P$ , $P_{N}$ and the weights of Periodic vector Sobolev spaces on the three-torus, the weight family, and the Leray projection.

[F1]

$H^{m}$ is a real Hilbert space with weighted inner product $⟨ f, g ⟩_{H^{m}} = \sum_{k} w_{k}^{2 m} \hat{f} (k) \cdot \overline{\hat{g} (k)}$ and $∥ f ∥_{H^{m}}^{2} = ⟨ f, f ⟩_{H^{m}}$ . Bare $⟨ f, g ⟩$ denotes the unweighted $L^{2}$ pairing, so for polynomials $⟨ f, g ⟩_{H^{m}} = ⟨ f, Λ^{2 m} g ⟩$ ; $Λ^{2 m}$ is self-adjoint and positive and $Λ^{2 m} = Λ^{m} Λ^{m}$ ; $∥ \partial_{j} v ∥_{H^{m}} \leq ∥ v ∥_{H^{m + 1}}$ (Periodic vector Sobolev spaces on the three-torus, the weight family, and the Leray projection).

[F2]

$P$ and $P_{N}$ are self-adjoint orthogonal projections on $H^{m}$ , norm nonincreasing, commuting with $Δ$ ; $P_{N}$ is the identity on $X_{N}$ and $u_{N}^{'} \in X_{N}$ (Periodic vector Sobolev spaces on the three-torus, the weight family, and the Leray projection, The Galerkin coefficient system: finite-dimensional invariance, local Lipschitz property, and the unique maximal approximation on the torus).

[F3]

Claims 1-3 of The transport term is orthogonal to itself and the tame commutator estimate $∥ [Λ^{m}, u \cdot \nabla] u ∥_{L^{2}} \leq C_{m} ∥ u ∥_{H^{m}} ∥ u ∥_{H^{s}}$ : $\int ((u \cdot \nabla) w) \cdot w d x = 0$ for $u \in H^{m}$ divergence-free and $w \in H^{m}$ ; $∥ [Λ^{m}, u \cdot \nabla] u ∥_{L^{2}} \leq C_{m} ∥ u ∥_{H^{m}} ∥ u ∥_{H^{s}}$ ; and $⟨ (u \cdot \nabla) u, Λ^{2 m} u ⟩ = ⟨ [Λ^{m}, u \cdot \nabla] u, Λ^{m} u ⟩$ for $u \in H^{2 m}$ divergence-free.

[F5]

If $G : [0, t] \to R$ is differentiable with integrable derivative $G^{'}$ then $\int_{0}^{t} G^{'} = G (t) - G (0)$ (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ), and if $f \leq g$ pointwise then $\int f \leq \int g$ (If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ ).

Proof

technique · direct; differentiate the squared $H^m$-norm along the Galerkin flow, bound the transport term by the commutator estimate, and integrate the resulting scalar differential inequality by a regularization of the reciprocal

Given: The Galerkin solution $u_{N}$ and facts [F1]-[F5].

1.1F1F2

Since $t \mapsto u_{N} (t)$ is differentiable in $H^{m}$ with derivative $u_{N}^{'} (t)$ and the squared norm is the quadratic form $∥ u_{N} ∥_{H^{m}}^{2} = ⟨ u_{N}, u_{N} ⟩_{H^{m}}$ , the derivative of $t \mapsto ∥ u_{N} (t) ∥_{H^{m}}^{2}$ is $2 ⟨ u_{N}^{'}, u_{N} ⟩_{H^{m}}$ , and $⟨ u_{N}^{'}, u_{N} ⟩_{H^{m}} = ⟨ - P_{N} P ((u_{N} \cdot \nabla) u_{N}) + ν Δ u_{N}, u_{N} ⟩_{H^{m}}$ . By [F2], $P_{N}$ is self-adjoint and the identity on $X_{N} ∋ u_{N}$ , and $P$ is self-adjoint with $P Λ^{2 m} u_{N} = Λ^{2 m} u_{N}$ (as $u_{N}$ is divergence-free); by [F1], $⟨ Δ u_{N}, u_{N} ⟩_{H^{m}} = - ∥ \nabla u_{N} ∥_{H^{m}}^{2}$ . Hence with $w : = Λ^{m} u_{N}$ one has the identity $\frac{1}{2} \frac{d}{d t} ∥ u_{N} ∥_{H^{m}}^{2} + ν ∥ \nabla u_{N} ∥_{H^{m}}^{2} = - ⟨ (u_{N} \cdot \nabla) u_{N}, Λ^{2 m} u_{N} ⟩$ .

2.1step 1.1F3

By claim 3 and claim 2 of [F3], $∣ ⟨ (u_{N} \cdot \nabla) u_{N}, Λ^{2 m} u_{N} ⟩ ∣ = ∣ ⟨ [Λ^{m}, u_{N} \cdot \nabla] u_{N}, Λ^{m} u_{N} ⟩ ∣ \leq C_{m} ∥ u_{N} ∥_{H^{m}} ∥ u_{N} ∥_{H^{s}} ∥ Λ^{m} u_{N} ∥_{H^{0}} = C_{m} ∥ u_{N} ∥_{H^{s}} ∥ u_{N} ∥_{H^{m}}^{2}$ . Inserting this into step 1.1 gives the energy inequality of claim 1 for every $m \geq s$ , the case $m > 5 / 2$ being the hypothesis of [F3] and $m \geq s > 5 / 2$ .

3.1step 2.1algebra

Put $Z (t) : = ∥ u_{N} (t) ∥_{H^{s}}^{2}$ and $Y (t) : = ∥ u_{N} (t) ∥_{H^{s}} = Z (t)^{1 / 2}$ . Step 2.1 with $m = s$ gives $\frac{1}{2} Z^{'} \leq C_{s} Z^{3 / 2}$ , that is $Z^{'} \leq 2 C_{s} Z^{3 / 2}$ . For $ϵ > 0$ define $Y_{ϵ} : = (Z + ϵ^{2})^{1 / 2} > 0$ ; then $Y_{ϵ}^{'} = Z^{'} / (2 Y_{ϵ}) \leq C_{s} Z^{3 / 2} / Y_{ϵ} \leq C_{s} Y_{ϵ}^{2}$ because $Z \leq Y_{ϵ}^{2}$ and $Z^{3 / 2} \leq Y_{ϵ}^{3}$ .

4.1step 3.1F5

Since $(1 / Y_{ϵ})^{'} = - Y_{ϵ}^{'} / Y_{ϵ}^{2} \geq - C_{s}$ on any interval where $Y_{ϵ}$ is defined, the fundamental theorem [F5] and monotonicity of the integral give $\frac{1}{Y_{ϵ} (t)} \geq \frac{1}{Y_{ϵ} (0)} - C_{s} t$ for $t \in I$ , $t \geq 0$ , as long as the right-hand side is positive; hence $Y_{ϵ} (t) \leq Y_{ϵ} (0) (1 - C_{s} Y_{ϵ} (0) t)^{- 1}$ whenever the denominator is positive.

5.1step 4.1given

Letting $ϵ ↓ 0$ in step 4.1 (with $Y_{ϵ} (0) \to ∥ P_{N} u_{0} ∥_{H^{s}} \leq Y_{0}$ and $Y_{ϵ} (t) \to Y (t)$ ; the function $y \mapsto y / (1 - C_{s} y t)$ is increasing while its denominator is positive) gives, for $t \in I$ with $t \geq 0$ and $C_{s} Y_{0} t < 1$ , the bound $Y (t) \leq Y_{0} (1 - C_{s} Y_{0} t)^{- 1}$ . In particular, for the time $T$ of the statement with $Y_{0} > 0$ , the denominator at $t = T$ is $1 / 2$ , so $Y (t) \leq 2 Y_{0}$ for $t \in I \cap [0, T]$ ; if $Y_{0} = 0$ , the zero field solves the IVP with the same initial datum, so IVP uniqueness gives $u_{N} \equiv 0$ on the whole maximal interval.

6.1step 5.1given∎

The bound $∥ u_{N} (t) ∥_{H^{s}} \leq 2 Y_{0}$ on $[0, T]$ precludes a finite maximal right endpoint $T_{\max} \leq T$ : by the blow-up alternative of The Galerkin coefficient system: finite-dimensional invariance, local Lipschitz property, and the unique maximal approximation on the torus (claim 3), $∥ u_{N} (t) ∥_{H^{s}}$ would have to tend to $+ \infty$ as $t ↑ T_{\max}$ , contradicting the bound. Hence $T_{\max} > T$ , that is, $[0, T] \subseteq I$ ; this proves claim 2.

Remarks

The constant $C_{s}$ is the commutator constant of order $s$ ; the lifespan bound is therefore explicit in terms of the commutator estimate, with no hidden compactness.
If $Y_{0} = 0$ then $P_{N} u_{0} = 0$ in $H^{s}$ , so the Galerkin solution is identically zero; the stated choice $T = 1$ is a harmless convention to make the common-lifespan statement unconditional.

The higher-order energy inequality and a common Galerkin lifespan with $∥ u_{N} (t) ∥_{H^{s}} \leq 2 Y_{0}$

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

The higher-order energy inequality and a common Galerkin lifespan with ∥uN(t)∥Hs≤2Y0∥uN​(t)∥Hs​≤2Y0​

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

The higher-order energy inequality and a common Galerkin lifespan with $∥ u_{N} (t) ∥_{H^{s}} \leq 2 Y_{0}$