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Conditional nonlinear Couette energy stability — conditional physical prediction

Physical scope

Conditional energy estimate for an already-existing smooth incompressible constant-density Newtonian solution in a horizontally periodic channel with Couette wall velocities and zero perturbation trace. The sufficient decay threshold applies only on the stated solution interval; no global existence, optimal transition Reynolds number, or empirical applicability is asserted.

Statement

Assume ${A C}_{ω}$ , meaning countable choice for a sequence of nonempty sets. Work in the nonrelativistic SI fluid model with absolute time in seconds and Euclidean spatial coordinates in metres. Let $L_{x}, L_{z}, h > 0$ be lengths in metres, let $T > 0$ be a time in seconds, let $ρ_{0} > 0$ be a constant density in kilograms per cubic metre, and let $μ > 0$ be a constant dynamic viscosity in pascal-seconds on the selected state branch. Set $ν = μ / ρ_{0}$ in square metres per second and take a constant shear rate $S \in R$ in inverse seconds. Define $Q = T_{L_{x}} \times (0, h) \times T_{L_{z}}$ , where $T_{L} = R / (L Z)$ ; cell integrals use Lebesgue measure on one horizontal fundamental cell.

Let $u \in C^{1} ([0, T]; C^{2} (\overline{Q}; R^{3}))$ and $p \in C ([0, T]; C^{1} (\overline{Q}))$ be periodic in the two torus coordinates. Suppose $u$ is an already-existing strong incompressible solution with zero body acceleration of

$\partial_{t} u + (u \cdot \nabla) u + ρ_{0}^{- 1} \nabla p - ν Δ u = 0, \nabla \cdot u = 0,$

with no-slip wall values $u ∣_{y = 0} = 0$ and $u ∣_{y = h} = S h (1, 0, 0)$ . The wall data and strong solution are hypotheses, not existence conclusions. Set $U_{b} (x, y, z) = (S y, 0, 0)$ , $w = u - U_{b}$ , and $E (t) = \int_{Q} ∣ w (t, x, y, z) ∣^{2} d x d y d z$ . If $∣ S ∣ h^{2} / ν < 2$ , then for every $t \in [0, T]$ ,

$E (t) \leq \exp ⁣ [- (\frac{2 ν}{h^{2}} - ∣ S ∣) t] E (0) .$

The condition is sufficient and is not asserted to be sharp. This model deduction holds only on the interval where the assumed smooth solution exists; it proves neither existence on all time nor global three-dimensional regularity nor a physical transition threshold.

Statement references

Tong, Chapter 3, §3.2.2, Couette flow and its prescribed wall velocities

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

Facts & Assumptions

Given: Assume ${A C}_{ω}$ , the positive parameters and SI units in the Statement, the periodic channel, the stated regularity and periodicity, the existing strong solution of the displayed equations, and the prescribed no-slip wall values.

[A1]

Single-component nonpolar continuum and material balances: the smooth single-component continuum model adopts mass and momentum balances on material subregions; these physical premises do not assert that a solution exists.

[A2]

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

[A3]

Newtonian stress and Fourier conductive closure: the adopted Newtonian/Fourier closure specifies the state-dependent shear and bulk coefficients and the tension-positive stress; this item selects a constant positive shear coefficient $μ$ and incompressibility, without treating the coefficient as measured or universal.

[F1]

Classical frames, primitive quantities and SI units: the model uses Newtonian spacetime and a Cartesian inertial chart; velocity is an Eulerian vector field, $D_{t} = \partial_{t} + u \cdot \nabla$ , and the listed physical quantities have their stated SI types and units.

[F2]

Formal parameterized fluid systems and observables: the formal constant-density incompressible Navier–Stokes system is $\partial_{t} u + (u \cdot \nabla) u + ρ_{0}^{- 1} \nabla p - (μ / ρ_{0}) Δ u = 0$ , $div u = 0$ , with componentwise vector Laplacian. This formal definition asserts no physical solution.

[F3]

Conditional nonlinear Couette energy stability is a draft mathematical theorem with a proof present; no independent proof acceptance is recorded. Under ${A C}_{ω}$ , for an existing smooth solution on the periodic channel with Couette wall values, the sufficient condition $∣ S ∣ h^{2} / ν < 2$ implies $E (t) \leq e^{- (2 ν / h^{2} - ∣ S ∣) t} E (0)$ on its assumed interval. Its current Step-1 record is owner-resolved ready; this is authoring readiness only and does not accept the mathematical proof.

[F4]

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

[A4]

The displayed pointwise equations and existence of $u, p$ on $[0, T]$ are explicit strong-solution hypotheses. They are not derived here from the integral balance postulate and do not imply global existence.

Proof

1.1givenA1A2A3A4F1F2F3F4

The field in the Statement has the exact parameters $L_{x}, L_{z}, h, ν, ρ_{0}, S, T$ , periodicity, regularity, zero-body-acceleration strong equations, divergence constraint, Couette wall values, and countable-choice hypothesis [F4] required by [F3]. The perturbation in [F3] is $w = u - U_{b}$ , and its energy is exactly the stated integral $E (t)$ . The adopted postulates [A1]–[A3] give the physical model context; the displayed pointwise strong solution remains an explicit hypothesis [A4].

2.1givenF3F4step 1.1algebra∎

Apply [F3] with the exact inherited countable-choice assumption [F4] to these matched hypotheses. It gives the claimed estimate for all $t \in [0, T]$ when $∣ S ∣ h^{2} / ν < 2$ . Since this is a deduction from the strong model equations and assumed smooth existence, it gives no existence or empirical conclusion.

The mathematical supplier remains draft despite having a proof present. Its current Step-1 record is owner-resolved ready for authoring only; this is not independent proof acceptance. The conditional argument does not treat readiness as proof approval.

Conditional nonlinear Couette energy stability — conditional physical prediction

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources