Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

1
The categories of this paper are: cs
21
The categories of this paper are: adap-org
18
The categories of this paper are: econ
14
The categories of this paper are: math-ph
15
The categories of this paper are: nucl-ex
10
The categories of this paper are: q-bio
4
The categories of this paper are: cond-mat
33
The categories of this paper are: plasm-ph
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
5
The categories of this paper are: gr-qc
28
The categories of this paper are: alg-geom
31
The categories of this paper are: ao-sci
36
The categories of this paper are: cmp-lg