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

TL;DR

This paper develops a method to construct global approximate solutions for complex fluid dynamics systems with shocks and voids, using ODE-based techniques and standard numerical methods, validated against known solutions.

Contribution

It introduces a novel approach using two ODEs in Banach space to approximate solutions for compressible flows with shocks and voids, applicable to multiple fluid systems.

Findings

Method produces solutions matching known fluid dynamics solutions.

Approach yields classical analytic solutions where applicable.

Constructs solutions with initial conditions in Radon measures.

Abstract

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.