Compressible Navier‐Stokes equations with ripped density

Abstract

AbstractWe are concerned with the Cauchy problem for the two‐dimensional compressible Navier‐Stokes equations  supplemented with general H1 initial velocity and bounded initial density not necessarily strictly positive: it may be the characteristic function of any set, for instance. In the perfect gas case, we establish global‐in‐time existence and uniqueness, provided the volume (bulk) viscosity coefficient is large enough. For more general pressure laws (like e.g., with ), we still get global existence, but uniqueness remains an open question. As a by‐product of our results, we give a rigorous justification of the convergence to the inhomogeneous incompressible Navier‐Stokes equations when the bulk viscosity tends to infinity. In the three‐dimensional case, similar results are proved for short time without restriction on the viscosity, and for large time if the initial velocity field is small enough.