# American Institute of Mathematical Sciences

November  2020, 25(11): 4317-4333. doi: 10.3934/dcdsb.2020099

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

 School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

Received  August 2011 Published  November 2020 Early access  April 2020

Fund Project: The author is partially supported by NSFC grant 11801460

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.

Citation: Yongfu Wang. Global strong solution to the two dimensional nonhomogeneous incompressible heat conducting Navier-Stokes flows with vacuum. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4317-4333. doi: 10.3934/dcdsb.2020099
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