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.

中文翻译：

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二维非均匀不可压缩导热Navier-Stokes流动的全局强解

在本文中，我们证明了当初始密度可以包含真空状态时，只要初始数据满足一定的相容性条件，二维非均匀不可压缩导热Navier-Stokes流的唯一全局强解。此外，我们的主要结果改进了所有以前的结果，其中初始密度严格为正。证明的主要内容是使用对数类型的一些关键Sobolev不等式，这些不等式最初是由[[Brezis-Gallouet]3]和Brezis-Wainger在[4]，斯托克斯系统的一些规律性，以及非均匀不可压缩导热流的一些精细的能量估计。