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Rigidity of shear flows of the Euler equations in the plane

Riccardo Tione

Abstract

In this paper we show that steady states $u$ of the pressureless Euler equation which belong to $L^3_{loc}(\mathbb{R}^2,\mathbb{R}^2)$ are shear flows. This is achieved by combining results of degenerate Monge-Ampère-type equations with the theory of two dimensional transport equations. We also show that the problem of rigidity and flexibility for the associated differential inclusion is rigid for sequences equibounded in $L^{4+}$ and flexible for sequences equibounded in $L^{4-}$, thus displaying a gap in the rigidity exponent between the exact and the approximate problem.

Rigidity of shear flows of the Euler equations in the plane

Abstract

In this paper we show that steady states of the pressureless Euler equation which belong to are shear flows. This is achieved by combining results of degenerate Monge-Ampère-type equations with the theory of two dimensional transport equations. We also show that the problem of rigidity and flexibility for the associated differential inclusion is rigid for sequences equibounded in and flexible for sequences equibounded in , thus displaying a gap in the rigidity exponent between the exact and the approximate problem.
Paper Structure (17 sections, 16 theorems, 131 equations)

This paper contains 17 sections, 16 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Exact solutions
  3. Approximate solutions
  4. Structure of the paper
  5. Notation
  6. Proof of Theorem \ref{['t:b']}
  7. Proof of Theorem \ref{['t:b']}\ref{['st:1']}: reduction to the global case
  8. Preliminaries and proof of Theorem \ref{['t:b']}\ref{['st:2']}
  9. Proof of Theorem \ref{['t:b']}\ref{['st:3']}: unconstrained rigidity for $L^p$ vector fields
  10. Proof of Theorem \ref{['t:c']}: rigidity for non-vanishing vector fields
  11. Young measures and proof of Theorem \ref{['t:d']}
  12. Young measures
  13. Proof of Theorem \ref{['t:d']}: approximate rigidity
  14. Staircase laminates and proof of Theorem \ref{['t:e']}
  15. Laminates of finite order and staircase laminates
  16. ...and 2 more sections

Key Result

Theorem A

Let $\Omega \subset \mathbb R^2$ be a domain such that $(\overline{\Omega})^c$ has finitely many connected components and $\partial \Omega = \partial((\overline{\Omega})^c)$. Then:

Theorems & Definitions (26)

  • Theorem A: Rigidity for exact solutions
  • Theorem B: Rigidity for non-vanishing exact solutions
  • Theorem C: Approximate rigidity for $p > 4$
  • Theorem D: Approximate flexibility for $p < 4$
  • Remark 1
  • Theorem 2.1
  • Theorem 2.2
  • proof : Proof of Theorem \ref{['t:b']}\ref{['st:2']}
  • Lemma 1
  • proof
  • ...and 16 more