Approximate solutions to the initial value problem for some compressible flows

M Colombeau

doi:10.1007/s00033-015-0542-x

Abstract

For the natural initial conditions L 1 in the density field (more generally a positive bounded Radon measure) and L ∞ in the velocity field, we obtain global approximate solutions to the Cauchy problem for the 3-D systems of isothermal and isentropic gases and the 2-D shallow water equations. 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 construction always gives back the known solutions. It is also proved that it gives the classical analytic solutions in the domain of application of the Cauchy–Kovalevskaya theorem.

Highlights

We construct sequences of approximate solutions with arbitrary accuracy for the 1-D, 2-D and 3-D compressible isothermal gases and isentropic gases in the presence of shocks and void regions; this applies to the shallow water equations with same proof
The need for the search of mathematical solutions to the initial value problem for compressible fluids in several dimensions when shocks show up has been recently pointed out by Lax and Serre [25,31]; our results provide sequences of approximate solutions with full mathematical proofs that tend to satisfy the equations
We have been motivated by the fact our approximate solutions are weak asymptotic solutions as considered by Albeverio et al [1,2,3], Danilov et al [15,16,17,18], Shelkovich et al [30,32,33], as an extension of the Maslov–Whitham asymptotic analysis

Summary

Introduction

We construct sequences of approximate solutions with arbitrary accuracy for the 1-D, 2-D and 3-D compressible isothermal gases and isentropic gases in the presence of shocks and void regions; this applies to the shallow water equations with same proof. 3) The ODEs for fixed One notices that the Lipschitz coefficients of the functions F, G (35) involved in the ODEs (always for fixed ) are uniformly bounded on any region of the Banach space (C(T)) for which there exist α > 0 (small) and A > 0 (large) such that for all x ∈ T one has ρ(x) ≥ α and ρ(x) ≤ A and |u(x)| ≤ A This remark implies ( to the case of Lipschitz ODEs in finite dimensional spaces) that the solution of the ODEs ceases to exist at a finite time T only when there is a sequence (tn), tn < T on which one at least of these three conditions is not satisfied uniformly in n when n → ∞. We have observed numerically that the results are independent of the arbitrariness in a choice of these parameters provided they satisfy requirements such as those in Theorem 1

Convergence to the analytic solution

Sequence of approximate solutions to the system of isentropic gas equations

Sequence of approximate solutions to the shallow water equations

Conclusion