Global strong solution and exponential decay to the 3D incompressible Bénard system with density-dependent viscosity and vacuum

In this paper, we study the Cauchy problem of the incompressible Bénard system with density-dependent viscosity on the whole three-dimensional space. We first construct a key priori exponential estimates by the energy method, and then we prove that there is a unique global strong solution for the 3D Cauchy problem under the assumption that initial energy is suitably small. In particular, it is not required to be smallness condition for the initial density which contains vacuum and even has compact support. Finally, we obtain the exponential decay rates for the gradients of velocity, temperature field and pressure.

Keywords

Bénard system, global well-posedness, density-dependent, exponential decay rates, vacuum

2010 Mathematics Subject Classification

Primary 37-xx, 70-xx, 76Xxx, 92-xx. Secondary 34-xx, 35-xx, 80-xx, 82-xx.

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Received 12 January 2023

Published 17 May 2023