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Controlled periodic smooth inviscid limit

Statement

Assume Countable Choice. Let $d \in {1, 2, 3}$ , let $s \geq 4$ be an integer, and let $u_{0} : Q_{d} \to R^{d}$ be smooth, real, and divergence-free, where $Q_{d} = R^{d} / (2 π Z)^{d}$ has normalized Lebesgue measure and divergence-free means $\sum_{j = 1}^{d} \partial_{j} (u_{0})_{j} = 0$ . The local incompressible theorem supplies, on a common interval $[0, T]$ independent of $0 \leq ν \leq 1$ , smooth solutions $u^{ν}$ of the periodic Navier–Stokes equation and $v = u^{0}$ of periodic Euler with the same initial data. For every $t \in [0, T]$ , $∥ u^{ν} (t) - v (t) ∥_{L^{2}} \leq ν \int_{0}^{t} \exp ⁣ (\int_{s}^{t} ∥ \nabla v (τ) ∥_{L^{\infty}} d τ) ∥ Δ v (s) ∥_{L^{2}} d s .$ Consequently the $L^{2}$ error is $O (ν)$ uniformly on $[0, T]$ as $ν ↓ 0$ . For each integer $m \geq s$ , the same local construction supplies a uniform $H^{m}$ bound for this family, independent of $ν$ ; therefore the $H^{s - 1}$ error is $O (ν^{1 / s})$ . The result is confined to a fixed periodic domain and this common local smooth interval; it asserts no wall inviscid limit or boundary-layer estimate.

Statement references

Smooth incompressible and symmetric-system stability

Kuksin and Shirikyan, Mathematics of Two-Dimensional Turbulence, Chapter 2, §2.1.5

Tong, Fluid Mechanics, Chapter 2, §3.1.2

Facts & Assumptions

Given: Countable Choice, $d, s, u_{0}$ as in the Statement, and the common family of smooth solutions on $[0, T]$ .

[A1]

Countable Choice: every countable family of nonempty sets has a choice function (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Smooth incompressible and symmetric-system stability: for two smooth periodic incompressible solutions with viscosities $ν_{1}, ν_{2} \geq 0$ , the $L^{2}$ difference is bounded by the initial difference and the residual $(ν_{1} - ν_{2}) Δ u^{ν_{2}}$ , with exponential factor determined by $∥ \nabla u^{ν_{2}} ∥_{\infty}$ . The local existence interval and every fixed-order Sobolev bound are uniform in viscosity for fixed smooth data.

[F2]

Periodic fluid Fourier Sobolev spaces: $∥ w ∥_{H^{r}}^{2} = \sum_{k \in Z^{d}} (1 + ∣ k ∣^{2})^{r} ∣ \hat{w} (k) ∣^{2}$ on the normalized torus, componentwise.

Proof

1.1givenA1F1

Apply [F1] with $ν_{1} = ν$ , $ν_{2} = 0$ , $u^{ν_{1}} = u^{ν}$ , and $u^{ν_{2}} = v$ . The initial difference is zero and the residual norm is $ν ∥ Δ v (s) ∥_{2}$ , so [F1] gives the displayed estimate.

2.1givenstep 1.1algebra

Smoothness of $v$ on the compact cylinder implies $L_{*} = \sup_{0 \leq t \leq T} ∥ \nabla v (t) ∥_{\infty} < \infty$ and $D_{*} = \sup_{0 \leq t \leq T} ∥ Δ v (t) ∥_{2} < \infty$ . Hence the integral in step 1.1 is at most $T e^{T L_{*}} D_{*}$ . Multiplying by $ν$ proves the uniform $O (ν)$ estimate.

3.1givenF1F2step 2.1algebra∎

By [F1], there is $M_{s} < \infty$ bounding $∥ u^{ν} (t) ∥_{H^{s}}$ for all $0 \leq ν \leq 1$ and $t \in [0, T]$ , and also bounding $∥ v (t) ∥_{H^{s}}$ . Thus $∥ u^{ν} (t) - v (t) ∥_{H^{s}} \leq 2 M_{s}$ . Weighted Cauchy–Schwarz applied to the coefficient sums in [F2] gives $∥ w ∥_{H^{s - 1}} \leq ∥ w ∥_{2}^{1 / s} ∥ w ∥_{H^{s}}^{(s - 1) / s}$ : write $w_{k}^{2 (s - 1)} ∣ \hat{w} ∣^{2} = (∣ \hat{w} ∣^{2})^{1 / s} (w_{k}^{2 s} ∣ \hat{w} ∣^{2})^{(s - 1) / s}$ and apply Holder with exponents $s$ and $s / (s - 1)$ . Apply this to $w = u^{ν} - v$ and step 2.1 to obtain $\sup_{t \in [0, T]} ∥ u^{ν} (t) - v (t) ∥_{H^{s - 1}} \leq (2 M_{s})^{(s - 1) / s} (T e^{T L_{*}} D_{*})^{1 / s} ν^{1 / s}$ .

Controlled periodic smooth inviscid limit

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources