Global strong solution to the two dimensional nonhomogeneous incompressible heat conducting Navier-Stokes flows with vacuum

Abstract

In this paper, we prove the unique global strong solution for the two dimensional nonhomogeneous incompressible heat conducting Navier-Stokes flows when the initial density can contain vacuum states, as long as the initial data satisfies some compatibility condition. Furthermore, our main result improves all the previous results where the initial density is strictly positive. The main ingredient of the proof is to use some critical Sobolev inequality of logarithmic type, which were originally due to Brezis-Gallouet in [ 3 ] and Brezis-Wainger in [ 4 ], some regularity properties of Stokes system and some delicate energy estimates for nonhomogeneous incompressible heat conducting flows.