Approximate solutions to the initial value problem for some compressible   flows in presence of shocks and void regions
  For the natural initial conditions $L^1$ in the density field (more generally a positive bounded Radon measure) and $L^\infty$ in the velocity field we obtain global approximate solutions to the Cauchy problem for the 3-D systems of isothermal and isentropic gases, the 2-D shallow water equations and the 3-D system of collisionnal self-gravitating gases. We obtain a sequence of functions which are differentiable in time and continuous in space and tend to satisfy the equations in the sense of distributions in the space variables and in the strong sense in the time variable. The method of construction relies on the study of a specific family of two ODEs in a classical Banach space (one for the continuity equation and one for the Euler equation). Standard convergent numerical methods for the solution of these ODEs can be used to provide concrete approximate solutions. It has been checked in numerous cases in which the solutions of systems of fluid dynamics are known that our constuction always gives back the known solutions. It is also proved it gives the classical analytic solutions in the domain of application of the Cauchy-Kovalevska theorem. 
Mathilde Colombeau
math.AP