Compressible Navier‐Stokes equations with ripped density

We are concerned with the Cauchy problem for two-dimensional compressible Navier-Stokes equations supplemented general H1 initial velocity and bounded density not necessarily strictly positive: it may be characteristic function of any set, instance. In perfect gas case, we establish global-in-time existence uniqueness, provided volume (bulk) viscosity coefficient is large enough. For more pressure laws (like e.g., P = ρ γ $P=\rho ^\gamma$ > 1 $\gamma >1$ ), still get global existence, but uniqueness remains an open question. As a by-product our results, give rigorous justification convergence to inhomogeneous incompressible when bulk tends infinity. three-dimensional similar results proved short time without restriction on viscosity, if field small