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

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.

25 pages

References in corpus (2)

• Global solutions to the three-dimensional full compressible magnetohydrodynamic flows
• Blow up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equations

Cited by in corpus (8)

• Global small solutions of heat conductive compressible Navier-Stokes equations with vacuum: smallness on scaling invariant quantity
• On Formation of Singularity of Spherically Symmetric Nonbarotropic Flows
• Global well-posedness to three-dimensional full compressible magnetohydrodynamic equations with vacuum
• On classical solutions to the Cauchy problem of the 2D compressible non-resistive MHD equations with vacuum
• Long-time behavior for three dimensional compressible viscous and heat-conductive gases
• Refined blow up criteria for the full compressible Navier-Stokes equations involving temperature
• Entropy-bounded solutions to the 3D compressible heat-conducting magnetohydrodynamic equations with vacuum at infinity
• Frobenius stable pluricanonical systems on threefolds of general type in positive characteristic