Rigidity of shear flows of the Euler equations in the plane

TL;DR

This paper proves that certain steady solutions of the pressureless Euler equations in the plane must be shear flows, and explores the rigidity and flexibility properties of related differential inclusions depending on function space bounds.

Contribution

It establishes a rigidity result for steady states in the pressureless Euler equations and analyzes the gap between exact and approximate solutions based on integrability conditions.

Findings

Steady states in $L^3_{loc}$ are shear flows.

Rigidity holds for sequences bounded in $L^{4+}$.

Flexibility occurs for sequences bounded in $L^{4-}$.

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-Amp\`ere-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.