A viscous approximation for a multidimensional unsteady Euler flow: Existence theorem for potential flow

We study a nonlinear system of partial differential equationsthat is a viscous approximation for a multidimensional unsteadyEuler potential flow governed by the conservationof mass and the Bernoulli law.The system consists of a transport equation for the density andthe viscous nonhomogeneous Hamilton-Jacobi equation for thevelocity potential.We establish the existence and regularity of global solutionsfor the nonlinear system with arbitrarily large periodicinitial data.We also prove that the density in our global solutions has apositive lower bound, that is, our solutions always stay awayfrom the vacuum,as long as the initial density has a positive lower bound.