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Illposedness of incompressible fluids in supercritical Sobolev spaces

Xiaoyutao Luo

Abstract

We prove that the 3D Euler and Navier-Stokes equations are strongly illposed in supercritical Sobolev spaces. In the inviscid case, for any $0 < s < \frac{5}{2} $, we construct a $C^\infty_c$ initial velocity field with arbitrarily small $H^{s}$ norm for which the unique local-in-time smooth solution of the 3D Euler equation develops large $\dot{H}^{s}$ norm inflation almost instantaneously. In the viscous case, the same $\dot{H}^{s}$ norm inflation occurs in the 3D Navier-Stokes equation for $0< s < \frac{1}{2} $, where $s = \frac{1}{2}$ is scaling critical for this equation.

Illposedness of incompressible fluids in supercritical Sobolev spaces

Abstract

We prove that the 3D Euler and Navier-Stokes equations are strongly illposed in supercritical Sobolev spaces. In the inviscid case, for any , we construct a initial velocity field with arbitrarily small norm for which the unique local-in-time smooth solution of the 3D Euler equation develops large norm inflation almost instantaneously. In the viscous case, the same norm inflation occurs in the 3D Navier-Stokes equation for , where is scaling critical for this equation.
Paper Structure (23 sections, 10 theorems, 104 equations, 1 figure)

This paper contains 23 sections, 10 theorems, 104 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background: the Euler case
  3. Background: the Navier-Stokes case
  4. Main ideas
  5. Outline of construction
  6. Further comments
  7. Preliminaries
  8. Notations
  9. Tools
  10. The approximate solution
  11. Auxiliary coordinates
  12. The setup
  13. Basic estimates
  14. The approximate solution
  15. Perturbation analysis
  16. ...and 8 more sections

Key Result

Theorem 1.1

The 3D Euler equation eq:euler is strongly illposed in $H^{s} (\mathbb{R}^3)$ for any $0< s < \frac{5}{2}$ in the following sense. Let $0< s < \frac{5}{2}$. For any $\epsilon>0$, there exists a vector field $u_0 \in C_c^\infty(\mathbb{R}^3)$ such that all of the following holds.

Figures (1)

  • Figure 1: Schematic of the shifted polar coordinate $(\rho , \varphi)$ in $rz$-plane. The scale of the profiles is $\mu^{-1}$ that is much smaller than $\nu^{-1}$, the distance to the origin.

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Lemma 2.2
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4: Approximation
  • ...and 9 more