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

TL;DR

This paper extends Serrin's blowup criterion to three-dimensional viscous, compressible, heat-conducting MHD flows, showing that bounded velocity in the Serrin norm prevents certain singularities, ensuring global existence of solutions.

Contribution

It establishes a Serrin-type blowup criterion for 3D compressible MHD flows, independent of temperature and magnetic fields, applicable to full compressible Navier-Stokes systems.

Findings

Bounded Serrin norm of velocity prevents vacuum formation.

Global solutions exist if density remains bounded and velocity satisfies Serrin's condition.

The criterion applies to both MHD flows and full compressible Navier-Stokes equations.

Abstract

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 the initial-boundary-value one of the three-dimensional compressible MHD flows with initial density allowed to vanish, the strong or smooth solution exists globally if the density is bounded from above and the velocity satisfies the Serrin's condition. Therefore, if the Serrin norm of the velocity remains bounded, it is not possible for other kinds of singularities (such as vacuum states vanish or vacuum appears in the non-vacuum region or even milder singularities) to form before the density becomes unbounded. This criterion is analogous to the well-known Serrin's blowup criterion for the three-dimensional incompressible Navier-Stokes equations, in particular, it is independent of the temperature and magnetic field and is just the same as that of the barotropic compressible Navier-Stokes equations. As a direct application, it is shown that the same result also holds for the strong or smooth solutions to the three-dimensional full compressible Navier-Stokes system describing the motion of a viscous, compressible, and heat conducting fluid.