Serrin-Type Blowup Criterion for Viscous, Compressible, and Heat Conducting Navier-Stokes and Magnetohydrodynamic Flows

摘要

This paper establishes a blowup criterion for the three-dimensional viscous,compressible, and heat conducting magnetohydrodynamic (MHD) flows. It is essentially shown that for the Cauchy problem and theinitial-boundary-value one of the three-dimensional compressible MHD flows withinitial density allowed to vanish, the strong or smooth solution existsglobally if the density is bounded from above and the velocity satisfies theSerrin's condition. Therefore, if the Serrin norm of the velocity remainsbounded, it is not possible for other kinds of singularities (such as vacuumstates vanish or vacuum appears in the non-vacuum region or even mildersingularities) to form before the density becomes unbounded. This criterion isanalogous to the well-known Serrin's blowup criterion for the three-dimensionalincompressible Navier-Stokes equations, in particular, it is independent of thetemperature and magnetic field and is just the same as that of the barotropiccompressible Navier-Stokes equations. As a direct application, it is shown that the same result also holds for thestrong or smooth solutions to the three-dimensional full compressibleNavier-Stokes system describing the motion of a viscous, compressible, and heatconducting fluid.