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Smooth vorticity identities and two-dimensional norms

Statement

Assume Countable Choice. Let $d \in {2, 3}$ , let $s \geq 4$ be an integer, let $ν \geq 0$ , and let $u : [0, T] \times Q_{d} \to R^{d}$ be a smooth real solution of $u_{t} + (u \cdot \nabla) u = - \nabla π + ν Δ u$ , $\sum_{j} \partial_{j} u_{j} = 0$ , where $Q_{d} = R^{d} / (2 π Z)^{d}$ has normalized measure and $π$ is a smooth periodic scalar. For $d = 3$ , define $ω = \nabla \times u$ by $ω_{i} = \sum_{j, k = 1}^{3} ϵ_{i j k} \partial_{j} u_{k}$ , where $ϵ_{123} = 1$ and $ϵ_{i j k}$ is totally antisymmetric; define $(a \times b)_{i} = \sum_{j, k = 1}^{3} ϵ_{i j k} a_{j} b_{k}$ . Then $ω_{t} + u \cdot \nabla ω = (ω \cdot \nabla) u + ν Δ ω$ . For $d = 2$ , define the scalar $ω = \partial_{1} u_{2} - \partial_{2} u_{1}$ ; then $ω_{t} + u \cdot \nabla ω = ν Δ ω$ and $\frac{1}{2} \frac{d}{d t} ∥ ω (t) ∥_{L^{2}}^{2} = - ν ∥ \nabla ω (t) ∥_{L^{2}}^{2} .$ For every $ν \geq 0$ the scalar maximum bound $∥ ω (t) ∥_{L^{\infty}} \leq ∥ ω (0) ∥_{L^{\infty}}$ holds. These are exact identities and bounds on the given solution lifespan and assert no global existence or regularity result.

Statement references

Local smooth periodic Euler and Navier–Stokes

Kuksin and Shirikyan, Mathematics of Two-Dimensional Turbulence, Chapter 2, §2.1.6

Hou, lecture notes, equations (3.1)–(3.4)

Facts & Assumptions

Given: Countable Choice, $d, s, ν$ as in the Statement, and the smooth periodic solution 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]

Local smooth periodic Euler and Navier–Stokes: under Countable Choice, smooth divergence-free data on the normalized periodic torus give a unique local smooth solution of the stated incompressible equation for each fixed $ν \geq 0$ .

Proof

1.1givenA1F1algebra

In dimension three, set $u \times ω$ using the usual Levi-Civita component formula. Since $div u = 0$ , the identity $u \cdot \nabla u = \nabla (∣ u ∣^{2} / 2) - u \times ω$ and the component expansion $\nabla \times (u \times ω) = u div ω - ω div u + (ω \cdot \nabla) u - (u \cdot \nabla) ω$ give $\nabla \times (u \cdot \nabla u) = (u \cdot \nabla) ω - (ω \cdot \nabla) u$ . Taking the curl of the momentum equation annihilates the pressure gradient and commutes with $\partial_{t}$ and $Δ$ , yielding the stated 3D equation.

2.1givenstep 1.1algebra

In dimension two, regard $u = (u_{1}, u_{2}, 0)$ and $ω = (0, 0, \partial_{1} u_{2} - \partial_{2} u_{1})$ . The 3D curl identity from step 1.1 applies to these fields, and $(ω \cdot \nabla) u = 0$ because $u$ is independent of the third coordinate. Its third component is therefore the stated scalar transport-diffusion equation.

3.1givenstep 2.1algebra

Multiply the scalar equation in step 2.1 by $ω$ and integrate over the periodic torus. The transport integral is $\frac{1}{2} \int u \cdot \nabla (ω^{2}) d μ = 0$ by periodicity and $div u = 0$ ; integration by parts gives $\int ω Δ ω d μ = - ∥ \nabla ω ∥_{2}^{2}$ . This proves the enstrophy identity.

4.1givenstep 2.1algebra∎

Set $M = ∥ ω (0) ∥_{\infty}$ . For $ε > 0$ , the function $g (t, x) = ω (t, x) - M - ε t$ is nonpositive at $t = 0$ . If it has a positive maximum on the compact cylinder $[0, T] \times Q_{2}$ , it occurs at some $t_{*} > 0$ ; there $\nabla g = 0$ , $Δ g \leq 0$ , and the time derivative from the left is nonnegative. But the equation in step 2.1 gives $g_{t} + u \cdot \nabla g = ν Δ g - ε < 0$ , a contradiction. Apply the same argument to $- ω$ , then let $ε ↓ 0$ to obtain $∥ ω (t) ∥_{\infty} \leq M$ .

Smooth vorticity identities and two-dimensional norms

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources