Global Existence and Exponential Stability for 3D Nonhomogeneous Micropolar Fluid System With Density‐Dependent Viscosities and Vacuum

Abstract

ABSTRACT We consider the global well‐posedness and large‐time asymptotic behavior of strong solutions to the Cauchy problem of three‐dimensional (3D) nonhomogeneous incompressible micropolar fluids with density‐dependent viscosities and vacuum. More precisely, based on the energy method, some key a priori exponential decay‐in‐time rates of strong solutions are obtained in the whole space, provided that the initial velocity and initial micro‐rotational velocity are suitably small in some homogeneous Sobolev space which may be optimal compared with the case of homogeneous micropolar fluid equations. Note that this result is obtained without any compatibility and smallness conditions on the initial density.