The Euler Equations in planar nonsmooth convex domains

TL;DR

This paper proves the existence and uniqueness of weak solutions to the Euler equations in convex planar domains with corners, using new BMO regularity estimates, extending previous results to broader vorticity integrability ranges.

Contribution

It introduces a novel BMO regularity estimate for the Dirichlet problem on domains with corners, enabling existence and uniqueness results for Euler equations in nonsmooth convex domains.

Findings

Existence of weak solutions with L^p vorticity for 4/3<= p <= 2.

Extension of results to all 2<p< in rectangular domains.

Uniqueness of solutions with bounded initial vorticity.

Abstract

We consider the Euler system set on a bounded convex planar domain, endowed with impermeability boundary conditions. This system is a model for the barotropic mode of the Primitive Equations on a rectangular domain. We show the existence of weak solutions with L^p vorticity for 4/3<= p <= 2, extending and enriching a previous result of Taylor. In the physically interesting case of a rectangular domain, a similar result holds for all 2<p<\infty as well. Moreover, we show the uniqueness of solutions with bounded initial vorticity. The main tool is a new BMO regularity estimate for the Dirichlet problem on domains with corners.