Global-in-time well-posedness of the compressible Navier–Stokes equations with striated density

Abstract

We first show local-in-time well-posedness of the compressible Navier–Stokes equations, assuming striated regularity while no other smoothness or smallness conditions on the initial density. With these local-in-time solutions served as blocks, for less regular initial data where the vacuum is permitted, the global-in-time well-posedness follows from the energy estimates and the propagated striated regularity of the density function, if the bulk viscosity coefficient is large enough in the two-dimensional case. The global-in-time well-posedness holds also true in the three-dimensional case, provided with large bulk viscosity coefficient together with small initial energy. This solves the density-patch problem in the exterior domain for the compressible model with $W^{2,p}$-interfaces. Finally, the singular incompressible limit toward the inhomogeneous incompressible model when the bulk viscosity coefficient tends to infinity is obtained.