Conditional regularity for the compressible Navier-Stokes equations with potential temperature transport

TL;DR

This paper investigates conditions under which solutions to the compressible Navier-Stokes equations with potential temperature transport remain regular, establishing local existence, uniqueness, and a blow-up criterion based on density and velocity norms.

Contribution

It provides new results on local well-posedness and a blow-up criterion for the compressible Navier-Stokes equations with temperature transport in bounded domains.

Findings

Existence and uniqueness of local strong solutions.

A blow-up criterion based on $L^ abla$-norms of density and velocity.

Applicability to 2D and 3D bounded domains.

Abstract

We study conditional regularity for the compressible Navier-Stokes equations with potential temperature transport in a bounded domain $\Omega\subset\mathbb{R}^d$, $d\in\{2,3\}$, with no-slip boundary conditions. We first prove the existence and uniqueness of local-in-time strong solutions. Further, we prove a blow-up criterion for the strong solution in terms of $L^\infty$-norms for the density and the velocity.