The 2D Euler-Boussinesq equations in planar polygonal domains with Yudovich's type data

TL;DR

This paper proves the well-posedness of the 2D Euler-Boussinesq equations with Yudovich's data in polygonal domains, extending previous results by leveraging recent elliptic regularity theorems for nonsmooth domains.

Contribution

It establishes well-posedness for the 2D Euler-Boussinesq equations in polygonal domains with Yudovich's data, addressing a question from 2011 and utilizing recent elliptic regularity results.

Findings

Well-posedness of 2D Euler-Boussinesq equations in polygonal domains.

Extension of Yudovich's theory to nonsmooth domains.

Application of recent elliptic regularity results to fluid dynamics.

Abstract

We address the well-posedness of the 2D (Euler)-Boussinesq equations with zero viscosity and positive diffusivity in the polygonal-like domains with Yudovich's type data, which gives a positive answer to part of the questions raised in 2011 [Lai-Pan-Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal. 199 (2011), no. 3, 739-760]. Our analysis on the the polygonal-like domains essentially relies on the recent elliptic regularity results for such domains proved in 2013 [Bardos-Plinio-Temam, The Euler equations in planar nonsmooth convex domains, J. Math. Anal. Appl. 407 (2013), no. 1, 69-89.] and [Plinio-Temam, Grisvard's shift theorem near $l^\infty$ and yudovich theory on polygonal domains, arXiv:1310.5444]