Non-uniqueness for the hypo-viscous compressible Navier-Stokes equations

Abstract

We study the Cauchy problem for the isentropic hypo-viscous compressible Navier-Stokes equations (CNS) under general pressure laws in all dimensions $d\geq 2$. For all hypo-viscosities $(-\Delta)^\alpha$ with $\alpha\in (0,1)$, we prove that there exist infinitely many weak solutions with the same initial data. This provides the first non-uniqueness result of weak solutions to viscous compressible fluid. Our proof features new constructions of building blocks for both the density and momentum, which respect the compressible structure. It also applies to the compressible Euler equations and the hypo-viscous incompressible Navier-Stokes equations (INS). In particular, in view of the Lady\v{z}enskaja-Prodi-Serrin criteria, the obtained non-uniqueness of $L^2_tC_x$ weak solutions to the hypo-viscous INS is sharp, and reveals that $\alpha =1$ is the sharp viscosity threshold for the well-posedness in $L^2_tC_x$. Furthermore, we prove that the H\"older continuous weak solutions to the compressible Euler equations may be obtained as a strong vanishing viscosity limit of a sequence of weak solutions to the hypo-viscous CNS.