Compressible Navier-Stokes equations with ripped density

Abstract

We 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., $P={\rho }^{\gamma }$ with $\gamma >1$), 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.