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

In this paper, we study the Cauchy problem of the incompressible Be ' nard 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.