Blow up criteria for three-dimensional incompressible Navier-Stokes-Landau-Lifshitz system in the whole space

Abstract

In this paper, we investigate the blow up criteria for the local smooth solutions to the three-dimensional incompressible Navier-Stokes-Landau-Lifshitz system via the components of the velocity field u or velocity gradient ∇u and the gradient orientation field ∇d. More precisely, the first result is the Serrin-type blow up criterion with the components of u and ∇d. The remaining is relevant to ∇u and ∇d corresponding to the Besov space with negative index. As corollaries, the blow up criteria in both of the Besov space with negative index and BMO space with reference to u and ∇d are obtained.