Global weak solutions to the two-dimensional Navier-Stokes equations of compressible heat-conducting flows with symmetric data and forces

TL;DR

This paper establishes the global existence of weak solutions for two-dimensional compressible heat-conducting Navier-Stokes equations with large, symmetric initial data and forces, using approximation and Orlicz space techniques.

Contribution

It introduces new uniform integrability estimates and a limiting process to prove global weak solutions for symmetric data in 2D compressible heat flows.

Findings

Proved global existence of weak solutions for symmetric initial data.

Derived new regularity and integrability results for approximate solutions.

Established the solution's validity in the entire space-time domain.

Abstract

We prove the global existence of weak solutions to the Navier-Stokes equations of compressible heat-conducting fluids in two spatial dimensions with initial data and external forces which are large and spherically symmetric. The solutions will be obtained as the limit of the approximate solutions in an annular domain. We first derive a number of regularity results on the approximate physical quantities in the "fluid region", as well as the new uniform integrability of the velocity and temperature in the entire space-time domain by exploiting the theory of the Orlicz spaces. By virtue of these a priori estimates we then argue in a manner similar to that in [Arch. Rational Mech. Anal. 173 (2004), 297-343] to pass to the limit and show that the limiting functions are indeed a weak solution which satisfies the mass and momentum equations in the entire space-time domain in the sense of distributions, and the energy equation in any compact subset of the "fluid region".