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

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".