Weak Serrin-type blowup criterion for three-dimensional nonhomogeneous viscous incompressible heat conducting flows

Abstract

In this paper, we establish a weak Serrin’s blowup criterion for the three-dimensional (3D) nonhomogeneous viscous incompressible heat conducting flows. It is shown that for the Cauchy problem of 3D incompressible heat conducting equations, the strong solution or smooth solution exists globally if the velocity satisfies the weak Serrin’s conditions, namely, ‖ρu‖Ls(0,T,Lwr)<∞, where 2s+3r≤1, 3<r≤∞ and Lwr⫌Lr denotes the weak Lr-space. This blowup criterion is analogous to the 3D incompressible Navier–Stokes equations, in particular, it is independent of the temperature field. Additionally, the initial vacuum states are allowed.