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Conditional nonlinear Couette energy stability

Statement

Assume ${A C}_{ω}$ . Let $L_{x}, L_{z}, h, ν, ρ_{0} > 0$ , $S \in R$ , and $T > 0$ . For $L > 0$ , define $T_{L} = R / (L Z)$ and set $Q = T_{L_{x}} \times (0, h) \times T_{L_{z}}$ , with Lebesgue integration over one fundamental cell. Define $μ : = ρ_{0} ν > 0$ , so $ν = μ / ρ_{0}$ as in the formal convention of definition def-fd-x00. Let $U_{b} (x, y, z) = (S y, 0, 0)$ . Let $u \in C^{1} ([0, T]; C^{2} (\overline{Q}; R^{3}))$ and $p \in C ([0, T]; C^{1} (\overline{Q}))$ , periodic in $x, z$ . Regard these fields as periodic lifts $\tilde{u} : [0, T] \times R \times [0, h] \times R \to R^{3}$ and $\tilde{p} : [0, T] \times R \times [0, h] \times R \to R$ , with periods $L_{x}, L_{z}$ in $x, z$ ; all spatial derivatives and equations are interpreted on these lifts. For $a = (a_{1}, a_{2}, a_{3}) \in R^{3}$ , use the Euclidean norm $∣ a ∣ = (a_{1}^{2} + a_{2}^{2} + a_{3}^{2})^{1 / 2}$ as in the norm convention in theorem thm-fd-x07. With ordinary Cartesian derivatives, $\nabla p = (\partial_{x} p, \partial_{y} p, \partial_{z} p)$ , $\nabla \cdot u = \partial_{x} u_{1} + \partial_{y} u_{2} + \partial_{z} u_{3}$ , $(u \cdot \nabla) u = \sum_{j = 1}^{3} u_{j} \partial_{x_{j}} u$ , and $(Δ u)_{i} = \partial_{x x} u_{i} + \partial_{y y} u_{i} + \partial_{z z} u_{i}$ for $i = 1, 2, 3$ . Suppose $u$ is an already-existing solution of the constant-density incompressible Navier-Stokes equation $u_{t} + (u \cdot \nabla) u + ρ_{0}^{- 1} \nabla p - ν Δ u = 0$ , $\nabla \cdot u = 0$ , with wall values $u ∣_{y = 0} = 0$ and $u ∣_{y = h} = S h (1, 0, 0)$ . Define $w = u - U_{b}$ 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. The estimate holds only on the interval where the assumed smooth solution exists; it proves neither existence on all time nor global three-dimensional regularity or a physical transition threshold.

Statement references

Tong, Chapter 2, §3.2.2 Couette flow

Guermond, Part I, Chapter 3, energy method and Poincare-Steklov context

Falsaperla, Mulone, and Perrone, §§2.1 and 3.1; nonlinear energy treatment for streamwise perturbations

Facts & Assumptions

Given: The parameters, periodic channel, existing smooth solution, equations and wall data in [A1]. The exact formal model convention is [F1]; the channel Poincare estimate is [F2].

[A1]

Assume ${A C}_{ω}$ , $L_{x}, L_{z}, h, ν, ρ_{0} > 0$ , $S \in R$ , $T > 0$ , $u$ periodic in $x, z$ , $u_{t}$ and its spatial derivatives through order two continuous up to $y = 0, h$ , $u_{t} + (u \cdot \nabla) u + ρ_{0}^{- 1} \nabla p - ν Δ u = 0$ , $\nabla \cdot u = 0$ , $u ∣_{y = 0} = 0$ , and $u ∣_{y = h} = S h (1, 0, 0)$ .

[F1]

Formal parameterized fluid systems and observables on $f d - x 01$ defines, for constants $ρ_{0} > 0$ and $μ > 0$ with $ν = μ / ρ_{0}$ , the pointwise constant-density incompressible equation $u_{t} + (u \cdot \nabla) u + ρ_{0}^{- 1} \nabla p - ν Δ u = 0$ , $\nabla \cdot u = 0$ . Its vector Laplacian is componentwise; the fields here have the stated stronger channel regularity.

[F2]

Bounded-domain Stokes residual estimate states that for the periodic channel and any $C^{1}$ field $v$ periodic in $x, z$ and zero at $y = 0, h$ , $∥ v ∥_{L^{2} (Q)}^{2} \leq h^{2} ∥ \nabla v ∥_{L^{2} (Q)}^{2}$ . Its full conditional proof interface includes ${A C}_{ω}$ .

[F3]

Divergence on a bounded C1 Euclidean domain assumes ${A C}_{ω}$ and applies to $C^{1}$ fields on bounded $C^{1}$ Euclidean domains. It is used in the proof of [F2]'s general residual clause; it is not applied to the periodic representative cell in this proof.

[F4]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ is countable choice for a sequence of nonempty sets. This exact assumption is carried through [F2] and the assigned dependency closure; no full Axiom of Choice is added.

Proof

1.1givenA1F1algebra

The base field $U_{b} = (S y, 0, 0)$ has $\nabla \cdot U_{b} = 0$ , $(U_{b} \cdot \nabla) U_{b} = 0$ , and $Δ U_{b} = 0$ , so with constant pressure it solves the stationary equation in [F1]. Its wall values are $0$ and $S h (1, 0, 0)$ . Subtracting this equation from [A1] and using $w = u - U_{b}$ gives $w_{t} + (U_{b} \cdot \nabla) w + (w \cdot \nabla) U_{b} + (w \cdot \nabla) w = - \nabla q + ν Δ w$ , $\nabla \cdot w = 0$ , and $w = 0$ at both walls, where $q = p / ρ_{0}$ up to an additive constant.

1.2givenA1algebra

Pointwise, $2 ∣ w_{1} w_{2} ∣ \leq w_{1}^{2} + w_{2}^{2} \leq ∣ w ∣^{2}$ . Integrating yields $∣ S ∣ ∣ \int_{Q} w_{1} w_{2} ∣ \leq \frac{∣ S ∣}{2} E (t)$ .

2.1givenA1step 1.1algebra

Put $E (t) = \int_{Q} ∣ w ∣^{2}$ . Multiply the perturbation equation in step 1.1 by $w$ and integrate over one cell. The $U_{b}$ transport integral is $\frac{1}{2} \int_{Q} \nabla \cdot (U_{b} ∣ w ∣^{2}) = 0$ : $\nabla \cdot U_{b} = 0$ , its normal component vanishes at the $y$ walls, and paired periodic-face terms cancel. The nonlinear transport integral is $\frac{1}{2} \int_{Q} \nabla \cdot (w ∣ w ∣^{2}) = 0$ because $\nabla \cdot w = 0$ , $w = 0$ at the walls, and periodic-face terms cancel. The pressure integral is $- \int_{Q} \nabla \cdot (q w) + \int_{Q} q \nabla \cdot w = 0$ for the same wall and periodic reasons. Componentwise integration by parts gives $\int_{Q} w \cdot Δ w = - ∥ \nabla w ∥_{L^{2} (Q)}^{2}$ , with zero wall term and canceling periodic faces. Since $(w \cdot \nabla) U_{b} = S w_{2} (1, 0, 0)$ , the remaining identity is $\frac{1}{2} E^{'} (t) = - ν ∥ \nabla w ∥_{L^{2} (Q)}^{2} - S \int_{Q} w_{1} w_{2}$ . These integrations use the one-dimensional FTC on one periodic cell, not the bounded $C^{1}$ Euclidean divergence theorem.

3.1givenA1F2F3F4step 2.1step 1.2algebra

From step 2.1 and step 1.2, $E^{'} (t) \leq - 2 ν ∥ \nabla w ∥_{L^{2} (Q)}^{2} + ∣ S ∣ E (t)$ . Apply [F2] with the carried exact ${A C}_{ω}$ premise [F4] to get $E^{'} (t) \leq - (2 ν / h^{2} - ∣ S ∣) E (t)$ .

4.1givenA1step 3.1algebra∎

Set $c = 2 ν / h^{2} - ∣ S ∣ > 0$ by the stated strict condition. Then $\frac{d}{d t} (e^{c t} E (t)) = e^{c t} (E^{'} (t) + c E (t)) \leq 0$ by step 3.1. Integrating from $0$ to $t$ gives $E (t) \leq e^{- c t} E (0)$ for every $t \in [0, T]$ , which is the claimed conditional decay.

Conditional nonlinear Couette energy stability

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources