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 [Formula: see text]. For all hypo-viscosities [Formula: see text] with [Formula: see text], 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. The construction of compressible viscous fluid models exhibits increased complexity compared to their inviscid and incompressible counterparts, primarily due to the relative rigidity of the pressure and the presence of viscosity. In order to overcome the difficulties caused by the relative rigidity of the pressure and by the viscosity, our proof provides first applications of a temporally intermittent but spatially homogeneous building block scheme and a mild density perturbation. It also applies to the compressible Euler equations and the hypo-viscous incompressible Navier–Stokes equations (INS). In particular, in view of the Ladyženskaja–Prodi–Serrin criteria, the obtained non-uniqueness of [Formula: see text] weak solutions to the hypo-viscous INS is sharp and reveals that [Formula: see text] is the sharp viscosity threshold for the well-posedness in [Formula: see text]. Furthermore, we prove that the Hölder 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.