Global existence and large time behavior of strong solutions for nonhomogeneous heat conducting Navier–Stokes equations with large initial data and vacuum

We are concerned with an initial boundary value problem of two-dimensional nonhomogeneous heat conducting Navier–Stokes equations in bounded domains. Applying delicate energy estimates and Desjardins’ interpolation inequality, we derive the global existence and uniqueness of strong solutions. Furthermore, we also show large-time decay rates of the solution. Note that the initial data can be arbitrarily large and the initial density allows vacuum states.

Keywords

nonhomogeneous heat conducting Navier–Stokes equations, global strong solution, large-time behavior, vacuum

2010 Mathematics Subject Classification

76D03, 76D05

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This research was partially supported by National Natural Science Foundation of China (Nos. 11901474, 12071359), Exceptional Young Talents Project of Chongqing Talent (No. cstc2021ycjh-bgzxm0153), and the Innovation Support Program for Chongqing Overseas Returnees (No. cx2020082).

Received 9 January 2021

Received revised 29 September 2021

Accepted 9 November 2021

Published 26 May 2022