Weak Serrin‐type criterion for the three‐dimensional viscous compressible Navier–Stokes system.

In this paper, we establish a weak Serrin‐type blowup criterion for the Cauchy problem of the three‐dimensional (3D) compressible barotropic Navier–Stokes equations in the whole space. It shows that the strong or smooth solution exists globally if the velocity satisfies the weak Serrin's condition and the L∞(0,T;Lα0)‐norm of the density is bounded, where α0 is positive constant. Therefore, if the weak 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. Furthermore, the initial data can be arbitrarily large and contain vacuum states. The proof is based on the new a priori estimates for 3D compressible Navier–Stokes equations. In particular, this extends the corresponding Huang et al.'s results (SIAM J. Math. Anal. 43 (2011) 1872–1886). [ABSTRACT FROM AUTHOR]