Zero Mach number limit in critical spaces for compressible Navier–Stokes equations

Abstract

Abstract We are concerned with the existence and uniqueness of local or global solutions for slightly compressible viscous fluids in the whole space. In [6] and [7], we proved local and global well-posedness results for initial data in critical spaces very close to the one used by H. Fujita and T. Kato for incompressible flows (see [14]). In the present paper, we address the question of convergence to the incompressible model (for ill-prepared initial data) when the Mach number goes to zero. When the initial data are small in a critical space, we get global existence and convergence. For large initial data and a bit of additional regularity, the slightly compressible solution is shown to exist as long as the corresponding incompressible solution does. As a corollary, we get global existence (and uniqueness) for slightly compressible two-dimensional fluids.