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Inviscid dynamical structures near Couette flow

Zhiwu Lin, Chongchun Zeng

TL;DR

The paper investigates nonlinear inviscid damping near Couette flow in the 2D Euler equations on a channel, revealing a sharp regularity-dependent dichotomy. In $H^{s}$ with $s<\frac{3}{2}$, it constructs nontrivial steady flows with cat's eyes arbitrarily close to Couette, showing nonlinear inviscid damping cannot hold in these spaces; in $H^{s}$ with $s>\frac{3}{2}$, there are no nontrivial traveling waves, suggesting simpler long-time dynamics. It also proves linear damping holds for all initial data in $L^{2}$ with precise decay rates depending on Sobolev regularity, and discusses the minimal regularity required for nonlinear inviscid damping (indicative of a threshold near $H^{\frac{5}{2}}$). Overall, the results demonstrate a truly nonlinear separation of dynamics across the critical exponent $s=\tfrac{3}{2}$, with rich invariant structures below this threshold and rigidity above it.

Abstract

Consider inviscid fluids in a channel {-1<y<1}. For the Couette flow v_0=(y,0), the vertical velocity of solutions to the linearized Euler equation at v_0 decays in time. At the nonlinear level, such inviscid damping has not been proved. First, we show that in any (vorticity) H^{s}(s<(3/2)) neighborhood of Couette flow, there exist non-parallel steady flows with arbitrary minimal horizontal period. This implies that nonlinear inviscid damping is not true in any (vorticity) H^{s}(s<(3/2)) neighborhood of Couette flow and for any horizontal period. Indeed, the long time behavior in such neighborhoods are very rich, including nontrivial steady flows, stable and unstable manifolds of nearby unstable shears. Second, in the (vorticity) H^{s}(s>(3/2)) neighborhood of Couette, we show that there exist no non-parallel steadily travelling flows v(x-ct,y), and no unstable shears. This suggests that the long time dynamics in H^{s}(s>(3/2)) neighborhoods of Couette might be much simpler. Such contrasting dynamics in H^{s} spaces with the critical power s=(3/2) is a truly nonlinear phenomena, since the linear inviscid damping near Couette is true for any initial vorticity in L^2.

Inviscid dynamical structures near Couette flow

TL;DR

The paper investigates nonlinear inviscid damping near Couette flow in the 2D Euler equations on a channel, revealing a sharp regularity-dependent dichotomy. In with , it constructs nontrivial steady flows with cat's eyes arbitrarily close to Couette, showing nonlinear inviscid damping cannot hold in these spaces; in with , there are no nontrivial traveling waves, suggesting simpler long-time dynamics. It also proves linear damping holds for all initial data in with precise decay rates depending on Sobolev regularity, and discusses the minimal regularity required for nonlinear inviscid damping (indicative of a threshold near ). Overall, the results demonstrate a truly nonlinear separation of dynamics across the critical exponent , with rich invariant structures below this threshold and rigidity above it.

Abstract

Consider inviscid fluids in a channel {-1<y<1}. For the Couette flow v_0=(y,0), the vertical velocity of solutions to the linearized Euler equation at v_0 decays in time. At the nonlinear level, such inviscid damping has not been proved. First, we show that in any (vorticity) H^{s}(s<(3/2)) neighborhood of Couette flow, there exist non-parallel steady flows with arbitrary minimal horizontal period. This implies that nonlinear inviscid damping is not true in any (vorticity) H^{s}(s<(3/2)) neighborhood of Couette flow and for any horizontal period. Indeed, the long time behavior in such neighborhoods are very rich, including nontrivial steady flows, stable and unstable manifolds of nearby unstable shears. Second, in the (vorticity) H^{s}(s>(3/2)) neighborhood of Couette, we show that there exist no non-parallel steadily travelling flows v(x-ct,y), and no unstable shears. This suggests that the long time dynamics in H^{s}(s>(3/2)) neighborhoods of Couette might be much simpler. Such contrasting dynamics in H^{s} spaces with the critical power s=(3/2) is a truly nonlinear phenomena, since the linear inviscid damping near Couette is true for any initial vorticity in L^2.

Paper Structure

This paper contains 4 sections, 10 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Existence of Cat's-eyes in H$^{s}\ \left( s<\frac{3}{2}\right)$
  3. Non-existence of traveling waves in H$^{s}\left( s>\frac{3}{2}\right)$
  4. Linear decay problem

Key Result

Theorem 1

Fixed any $T>0$ and $0\leq s<\frac{3}{2},\ $then for any $\varepsilon>0$, there exists a steady solution $\left( u_{\varepsilon }\left( x,y\right) ,v_{\varepsilon}\left( x,y\right) \right)$ to Euler equation (Euler) with (incompre)-(Euler-bc) such that $\left( u_{\varepsilon}\left( x,y\right) and $v_{\varepsilon}\left( x,y\right)$ is not identically zero.

Theorems & Definitions (12)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Remark 1
  • Lemma 4
  • Lemma 5
  • ...and 2 more