The Zero-Mach Limit of Compressible Flows

Abstract:

We prove that compressible Navier-Stokes flows in two and three space dimensions converge to incompressible Navier-Stokes flows in the limit as the Mach number tends to zero. No smallness restrictions are imposed on the external force, the initial velocity, or the time interval. We assume instead that the incompressible flow exists and is reasonably smooth on a given time interval, and prove that compressible flows with compatible initial data converge uniformly on that time interval. Our analysis shows that the essential mechanism in this process is a hyperbolic effect which becomes stronger with smaller Mach number and which ultimately drives the density to a constant.