Numerical Investigation of Cavitation in Multidimensional Compressible Flows

The compressible Navier-Stokes equations for an ideal polytropic gas are considered in ${\Cal R}^n ,n = 2,3$. The question of possible vacuum formation, an open theoretical problem, is investigated numerically using highly accurate computational methods. The flow is assumed to be symmetric about the origin with a purely radial velocity field. The numerical results indicate that there are weak solutions to the Navier-Stokes system in two and three space dimensions, which display formation of vacuum when the initial data are discontinuous and sufficiently large. The initial density is constant, while the initial velocity field is symmetric, points radially away from the origin, and belongs to $H_{loc}^B $ for all s < n/2. In addition, in the one-dimensional case, the numerical solutions are in agreement with known theoretical results.