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Bounded-domain Stokes residual estimate — conditional physical prediction

Physical scope

Conditional estimate for already-existing smooth constant-density incompressible Newtonian and restricted steady Stokes fields with the same prescribed wall velocity, zero body acceleration, and an explicit zero-trace Poincare bound; includes the periodic channel constant h. It asserts no solution existence, exterior-domain estimate, approximation-error estimate against the full momentum equation retaining material acceleration beyond these conditional residual bounds, or empirical applicability.

Statement

Assume ${A C}_{ω}$ , meaning countable choice for a sequence of nonempty sets. Work in the nonrelativistic SI fluid model: position is measured in metres, time in seconds, velocity in metres per second, pressure in pascals, density in kilograms per cubic metre, dynamic viscosity $μ > 0$ in pascal-seconds, and kinematic viscosity $ν = μ / ρ_{0}$ in square metres per second for constant density $ρ_{0} > 0$ . Use the tension-positive Cauchy stress convention. Adopt the smooth single-component continuum balances, the selected no-slip wall data, and the constant-coefficient incompressible Newtonian branch. The pointwise strong equations below are hypotheses on already-existing fields; no existence claim or inference from the integral balances alone is made.

Let $Ω \subset R^{3}$ be a bounded connected open $C^{1}$ domain, meaning that each boundary point has a neighborhood in which the boundary is a $C^{1}$ graph and $Ω$ lies on one side of that graph after a rigid coordinate change. Let $C_{Ω} > 0$ have units of metres and satisfy $∥ v ∥_{L^{2} (Ω)} \leq C_{Ω} ∥ \nabla v ∥_{L^{2} (Ω)}$ for every $v \in C^{1} (\overline{Ω}; R^{3})$ with zero boundary trace. Suppose $u_{N S}, u_{S} \in C^{2} (\overline{Ω}; R^{3})$ and $p_{N S}, p_{S} \in C^{1} (\overline{Ω})$ are divergence-free velocity fields with the same prescribed no-slip boundary velocity, zero body acceleration, and pointwise equations

$ρ_{0} (u_{N S} \cdot \nabla) u_{N S} + \nabla p_{N S} - μ Δ u_{N S} = 0, \nabla p_{S} - μ Δ u_{S} = 0.$

The first equation is the stationary strong incompressible Navier–Stokes system. The second is the separately adopted restricted steady Stokes comparison system, with its convective term omitted only for $u_{S}$ . Define $w = u_{N S} - u_{S}$ , $π = p_{N S} - p_{S}$ , and $f = - ρ_{0} (u_{N S} \cdot \nabla) u_{N S}$ . Then

$∥ \nabla w ∥_{L^{2} (Ω)} \leq \frac{C_{Ω}}{μ} ∥ f ∥_{L^{2} (Ω)}, ∥ w ∥_{L^{2} (Ω)} \leq \frac{C_{Ω}^{2}}{μ} ∥ f ∥_{L^{2} (Ω)} .$

For a periodic channel $Q = T_{L_{x}} \times (0, h) \times T_{L_{z}}$ , with $L_{x}, L_{z}, h > 0$ measured in metres and $T_{L} = R / (L Z)$ , take periodic fields in the two torus coordinates and integrate over one fundamental cell using Lebesgue measure. Assume the same stationary systems and boundary data on the periodic lift $R \times (0, h) \times R$ , with $u_{N S}, u_{S} \in C^{2} (\overline{Q}; R^{3})$ and $p_{N S}, p_{S} \in C^{1} (\overline{Q})$ . Define $w, π, f$ by the same formulas. The direct channel Poincare bound has constant $h$ , so

$∥ \nabla w ∥_{L^{2} (Q)} \leq \frac{h}{μ} ∥ f ∥_{L^{2} (Q)}, ∥ w ∥_{L^{2} (Q)} \leq \frac{h^{2}}{μ} ∥ f ∥_{L^{2} (Q)} .$

The periodic cell is not treated as a bounded $C^{1}$ Euclidean domain. Both estimates are conditional a-posteriori bounds for fields that already exist; neither asserts existence, an exterior-domain estimate, an approximation-error estimate against the full momentum equation retaining material acceleration beyond the stated conditional residual bounds, nor empirical validity of either model.

Statement references

Guermond, Part I, Chapter 3, §§3.3–3.4.4, energy and Poincare-Steklov context

Tong, Chapter 3, §3.4, Stokes-flow dissipation context

Stewart, MIT 8.09, §6.8, restricted low-Reynolds-number Stokes formulation and sphere calculation

Facts & Assumptions

Given: Assume ${A C}_{ω}$ , the physical model and SI conventions in the Statement, the stated domains and regularity, the no-slip boundary data, the common boundary velocity, zero body acceleration, the strong-form equations, and the explicit Poincare bound on $Ω$ .

[A1]

Single-component nonpolar continuum and material balances: the smooth single-component continuum model adopts mass and linear/angular momentum balances on material subregions; these are physical assumptions and do not by themselves supply a solution or its existence.

[A2]

Specified wall, thermal, periodic and initial data: no-slip is the separately selected equality of fluid and wall velocities on the wall; periodic faces are identified and are not physical walls.

[A3]

Newtonian stress and Fourier conductive closure: the constitutive branch adopts $σ = - p I_{3} + 2 μ D_{0} + ζ θ I_{3}$ with stated state laws. Here the theorem selects a constant $μ > 0$ and incompressibility $θ = 0$ ; no coefficient value is inferred from measurement.

[F1]

Classical frames, primitive quantities and SI units: time is absolute in Newtonian spacetime; the fields use a Cartesian inertial chart; velocity is an Eulerian vector field, stress is a tensor with traction $σ n$ , and $μ, ρ_{0}, p, u$ have the SI types and units stated in the Definition.

[F2]

Formal parameterized fluid systems and observables: for constants $ρ_{0}, μ > 0$ , the formal pointwise incompressible Navier–Stokes equation is $\partial_{t} u + (u \cdot \nabla) u + ρ_{0}^{- 1} \nabla p - (μ / ρ_{0}) Δ u = 0$ , $div u = 0$ ; its steady Stokes system omits the time derivative and convection. This is mathematical notation and asserts no physical solution.

[F3]

Bounded-domain Stokes residual estimate is a draft mathematical theorem with a proof present; no independent proof acceptance is recorded. Under ${A C}_{ω}$ , on bounded $C^{1}$ Euclidean domains, a smooth divergence-free zero-trace field satisfying $- μ Δ w + \nabla q = f$ and the stated zero-trace Poincare inequality obeys both residual bounds in the Statement. The same result separately proves $∥ v ∥_{L^{2} (Q)} \leq h ∥ \nabla v ∥_{L^{2} (Q)}$ for smooth periodic channel fields vanishing at $y = 0, h$ ; its proof does not regard $Q$ as a bounded $C^{1}$ Euclidean domain. Its current Step-1 record is owner-resolved ready for authoring only, not proof acceptance.

[F4]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ means every sequence of nonempty sets has a choice function; no stronger choice principle is asserted.

[F5]

Restricted incompressible creeping-flow Stokes model: the current Definition selects $div u_{S} = 0$ , $σ_{S} = - p_{S} I_{3} + 2 μ D (u_{S})$ , and $div σ_{S} + ρ_{0} b = 0$ , equivalently $- \nabla p_{S} + μ Δ u_{S} + ρ_{0} b = 0$ . For this consumer $b = 0$ , so it gives the stated restricted Stokes equation. Its current carrier was inspected and the interface matches this zero-body-acceleration specialization.

[A5]

Alternative restricted creeping-flow force balance: the current Postulate adopts the alternative steady incompressible balance $div σ + ρ_{0} b = 0$ with constant $ρ_{0} > 0$ and Newtonian stress; for this comparison set $b = 0$ . This replaces, and is not simultaneous with, the full material-acceleration momentum balance for $u_{S}$ . The adopted interface matches this specialization; its formulation notes explain the algebraically equivalent equation without presenting a deductive proof of the physical balance.

[A4]

The compared fields are explicitly assumed to exist with the stated regularity and to satisfy the two pointwise systems in the Statement. This is a strong-solution hypothesis, not a deduction from [A1] and not a claim that either adopted model describes nature.

Proof

1.1givenA1A2A3A4A5F1F2F3F5algebra

Within the adopted continuum, wall, and Newtonian assumptions [A1]–[A3] and Cartesian SI conventions [F1], take the full Navier–Stokes equation as the explicit strong-solution hypothesis [A4], not as a consequence of the integral balances alone. For $u_{S}$ , the current model Definition [F5] and zero-body-force Postulate specialization [A5] give $div u_{S} = 0$ and $div σ_{S} = 0$ . With constant $μ$ , componentwise differentiation gives $(div σ_{S})_{i} = - \partial_{i} p_{S} + μ \sum_{j} (\partial_{j j} u_{S, i} + \partial_{j} \partial_{i} u_{S, j}) = - \partial_{i} p_{S} + μ Δ u_{S, i} + μ \partial_{i} (div u_{S}) = - \partial_{i} p_{S} + μ Δ u_{S, i}$ , so $- μ Δ u_{S} + \nabla p_{S} = 0$ . Set $w = u_{N S} - u_{S}$ and $π = p_{N S} - p_{S}$ . The shared boundary data give $w ∣_{\partial Ω} = 0$ , and the divergence-free assumptions give $\nabla \cdot w = 0$ . Rearranging the Navier–Stokes equation gives $- μ Δ u_{N S} + \nabla p_{N S} = f$ . Subtracting yields $- μ Δ w + \nabla π = f$ , with the exact smoothness and Poincare hypotheses of [F3].

2.1givenA4step 1.1algebra

On one periodic channel cell, integrate $(- μ Δ w + \nabla π) \cdot w = f \cdot w$ . Integration by parts in $y$ has zero wall terms because $w = 0$ there; the paired periodic-face terms in $x, z$ cancel. Also $\int_{Q} \nabla π \cdot w = - \int_{Q} π \nabla \cdot w = 0$ by the same boundary conditions and incompressibility. Thus $μ ∥ \nabla w ∥_{L^{2} (Q)}^{2} = \int_{Q} f \cdot w$ . This uses the fundamental theorem of calculus on a fundamental cell, not the bounded-domain divergence theorem.

2.2givenF3F4step 1.1algebra

Apply [F3] under the exact countable-choice hypothesis [F4] to the residual system in step 1.1 and the stated $C_{Ω}$ inequality. It gives $∥ \nabla w ∥_{L^{2} (Ω)} \leq C_{Ω} ∥ f ∥_{L^{2} (Ω)} / μ$ and then $∥ w ∥_{L^{2} (Ω)} \leq C_{Ω} ∥ \nabla w ∥_{L^{2} (Ω)} \leq C_{Ω}^{2} ∥ f ∥_{L^{2} (Ω)} / μ$ .

3.1givenstep 2.1algebra

Write $a = ∥ f ∥_{L^{2} (Q)}$ , $b = ∥ w ∥_{L^{2} (Q)}$ , and $c = \int_{Q} ∣ f ∣ ∣ w ∣$ . Pointwise, if $w = 0$ then $f \cdot w = 0$ ; if $w \neq 0$ , $0 \leq ∣ f - ((f \cdot w) / ∣ w ∣^{2}) w ∣^{2}$ expands to $∣ f \cdot w ∣ \leq ∣ f ∣ ∣ w ∣$ . Therefore $∣ \int_{Q} f \cdot w ∣ \leq c$ . If $b = 0$ , continuity makes $w = 0$ and $c = 0$ . If $b > 0$ , then for every real $s$ , $0 \leq \int_{Q} (∣ f ∣ - s ∣ w ∣)^{2} = a^{2} - 2 s c + s^{2} b^{2}$ ; choosing $s = c / b^{2}$ gives $c^{2} \leq a^{2} b^{2}$ . Hence $∣ \int_{Q} f \cdot w ∣ \leq a b$ .

4.1givenF3F4step 2.1step 3.1algebra∎

Let $r = ∥ \nabla w ∥_{L^{2} (Q)}$ . By [F3], $b \leq h r$ . Steps 2.1 and 3.1 give $μ r^{2} \leq a b \leq h a r$ . If $r = 0$ the gradient estimate is immediate; if $r > 0$ , divide by $μ r$ to get $r \leq h a / μ$ . Applying $b \leq h r$ again gives $b \leq h^{2} a / μ$ . These are the two periodic-channel estimates.

The deduction is conditional on the strong-solution hypotheses and the adopted restricted Stokes definition recorded in Facts & Assumptions. The zero-inertia force balance is a physical assumption; its algebraically equivalent form does not prove that assumption. No pointwise equations are inferred from the integral balance postulate alone, and no empirical claim is made.

Bounded-domain Stokes residual estimate — conditional physical prediction

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources