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For the natural initial conditions $L^1$ in the density field (more generallya positive bounded Radon measure) and $L^\infty$ in the velocity field weobtain global approximate solutions to the Cauchy problem for the 3-D systemsof isothermal and isentropic gases, the 2-D shallow water equations and the 3-Dsystem of collisionnal self-gravitating gases. We obtain a sequence offunctions which are differentiable in time and continuous in space and tend tosatisfy the equations in the sense of distributions in the space variables andin the strong sense in the time variable. The method of construction relies onthe study of a specific family of two ODEs in a classical Banach space (one forthe continuity equation and one for the Euler equation). Standard convergentnumerical methods for the solution of these ODEs can be used to provideconcrete approximate solutions. It has been checked in numerous cases in whichthe solutions of systems of fluid dynamics are known that our constuctionalways gives back the known solutions. It is also proved it gives the classicalanalytic solutions in the domain of application of the Cauchy-Kovalevskatheorem.

Approximate solutions to the initial value problem for some compressible flows in presence of shocks and void regions