Logarithmically improved blow-up criteria for the 3D nonhomogeneous incompressible Navier-Stokes equations with vacuum

Abstract

This article is devoted to the study of the nonhomogeneous incompressible Navier-Stokes equations in space dimension three. By making use of the "weakly nonlinear" energy estimate approach introduced by Lei and Zhou in [16], we establish two logarithmically improved blow-up criteria of the strong or smooth solutions subject to vacuum for the 3D nonhomogeneous incompressible Navier-Stokes equations in the whole space R^3. This results extend recent regularity criterion obtained by Kim (2006) [13]