Weak Serrin-type blowup criterion for the 3D full compressible Navier-Stokes equations

TL;DR

This paper establishes a new blowup criterion for the 3D full compressible Navier-Stokes equations, showing that bounded temperature or velocity in the weak Serrin sense prevents certain singularities, extending previous results.

Contribution

It extends Serrin-type blowup criteria for the 3D compressible Navier-Stokes equations to include boundary conditions and removes technical assumptions for isentropic cases.

Findings

Global existence if density is bounded and temperature or velocity satisfies weak Serrin condition.

Prevents formation of vacuum or milder singularities before density blows up.

Extends previous blowup criteria results in mathematical fluid dynamics.

Abstract

We investigate weak Serrin-type blowup criterion of the three-dimensional full compressible Navier-Stokes equations for the Cauchy problem, Dirichlet problem and Navier-slip boundary condition. It is shown that the strong or smooth solution exists globally if the density is bounded from above, and either the absolute temperature or velocity satisfies the weak Serrin's condition. Therefore, if the weak Serrin norm of the absolute temperature or 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. In particular, this criterion extends those Serrin-type blowup criterion results in (Math. Ann. 390 (2024): 1201-1248; Arch. Ration. Mech. Anal. 207(2013): 303-316). Furthermore, as a by-product, for the isentropic compressible Navier-Stokes equations, we succeed in removing the technical assumption $\rho_0\in L^1$ in (J. Lond. Math. Soc. (2) 102(2020): 125--142). The initial data can be arbitrarily large and allow to contain vacuum states here.