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Conditional nonlinear Couette energy stability — formal worked field

Example

Assume ${A C}_{ω}$ . Define $T_{2 π} = R / (2 π Z)$ and let $Q = T_{2 π} \times (0, 1) \times T_{2 π}$ , with Lebesgue integration over one fundamental cell. Take $ρ_{0} = μ = ν = 1$ , $S = 1$ , base field $U_{b} = (y, 0, 0)$ , and pressure $p = 0$ . For any $a \in R$ , set $u (x, y, z, t) = (y + a e^{- π^{2} t} \sin (π y), 0, 0), t \geq 0.$ This field is smooth, periodic in $x, z$ , divergence-free, and has wall values $0$ at $y = 0$ and $(1, 0, 0)$ at $y = 1$ . Its perturbation $w = u - U_{b}$ satisfies $∥ w (t) ∥_{L^{2} (Q)}^{2} = e^{- 2 π^{2} t} ∥ w (0) ∥_{L^{2} (Q)}^{2} \leq e^{- t} ∥ w (0) ∥_{L^{2} (Q)}^{2}$ . The sufficient parameter condition in the supplier is $∣ S ∣ h^{2} / ν = 1 < 2$ ; this example verifies that conditional bound for one exact field.

Facts & Assumptions

Given: The periodic channel, $ρ_{0} = μ = ν = 1$ , $S = 1$ , $h = 1$ , $p = 0$ , the field $u$ , and ${A C}_{ω}$ as stated.

[A1]

On $Q = T_{2 π} \times (0, 1) \times T_{2 π}$ , $ρ_{0} = μ = ν = 1$ , $u = (y + a e^{- π^{2} t} \sin (π y), 0, 0)$ for $t \geq 0$ , with periodic $x, z$ , wall values $u ∣_{y = 0} = 0$ , $u ∣_{y = 1} = (1, 0, 0)$ , and pressure $p = 0$ .

[F1]

Conditional nonlinear Couette energy stability: for an already-existing smooth incompressible Navier-Stokes solution on a periodic channel of height $h$ with Couette shear $S$ , if $∣ S ∣ h^{2} / ν < 2$ and the theorem's stated dependencies hold, its perturbation energy satisfies $E (t) \leq e^{- (2 ν / h^{2} - ∣ S ∣) t} E (0)$ on the assumed smooth interval.

Verification

1.1givenA1algebra

The only nonzero component of $u$ is independent of $x$ , so $\nabla \cdot u = 0$ and $(u \cdot \nabla) u = 0$ . Direct differentiation gives $u_{t} = - a π^{2} e^{- π^{2} t} \sin (π y) (1, 0, 0)$ and $Δ u = - a π^{2} e^{- π^{2} t} \sin (π y) (1, 0, 0)$ ; hence $u_{t} + (u \cdot \nabla) u + ρ_{0}^{- 1} \nabla p - ν Δ u = 0$ . Since $\sin (0) = \sin (π) = 0$ , the stated wall values hold.

2.1givenA1step 1.1algebra

The perturbation is $w = a e^{- π^{2} t} \sin (π y) (1, 0, 0)$ . Integrating $\sin^{2} (π y)$ over $0 < y < 1$ and over both periods gives $E (t) = ∥ w (t) ∥_{L^{2} (Q)}^{2} = 2 π^{2} a^{2} e^{- 2 π^{2} t} = e^{- 2 π^{2} t} E (0)$ .

3.1givenA1F1step 2.1algebra∎

Here $∣ S ∣ h^{2} / ν = 1$ and $2 ν / h^{2} - ∣ S ∣ = 1$ . Applying [F1] gives $E (t) \leq e^{- t} E (0)$ , while step 2.1 verifies the sharper exact decay because $2 π^{2} \geq 1$ . The estimate is conditional on the stipulated supplier and assumptions.

Depends on

Conditional nonlinear Couette energy stability

Conditional nonlinear Couette energy stability — formal worked field

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels