Strong solutions for the Cauchy problem to three-dimensional nonhomogeneous incompressible micropolar fluid equations with vacuum

Abstract

This paper is concerned with a Cauchy problem for the three-dimensional (3D) nonhomogeneous incompressible micropolar fluid equations in the whole space. We first establish a weak Serrin-type blowup criterion for the strong solutions. It is shown that for the Cauchy problem of the 3D nonhomogeneous micropolar equations, the strong solution exists globally if the velocity satisfies the weak Serrin's condition. In particular, this criterion is independent of the micro-rotational velocity. Then as an immediate application, we prove that the Cauchy problem of micropolar fluid equations has a unique global strong solution, provided that the kinematic viscosity is sufficiently large, or the upper bound of initial density or initial kinetic energy is small enough.