Systems of conservation laws in higher space dimensions

TL;DR

This paper addresses the longstanding paradox between theoretical and computational results for weak solutions of higher-dimensional conservation laws, emphasizing the role of boundary-value problems and the challenges posed by non-hyperbolic systems.

Contribution

It clarifies the well-posedness issues of boundary-value problems for conservation laws in higher dimensions, especially in non-hyperbolic cases, and highlights the importance of fluid flow models for computational studies.

Findings

Successful computations relate to boundary-value problems that are only weakly well-posed.

Prescribed boundary data may not ensure local uniqueness in non-hyperbolic systems.

Fluid flow models based on Euler system reductions are promising for computational exploration.

Abstract

The existing paradox between theory and computational experiment for weak solutions of systems of conservation laws in higher space dimensions is arguably resolved. Apparently successful computations are identified with underlying boundary-value problems which are well-posed only in a weakened sense. In the absence of hyperbolicity in particular, prescribed boundary data sufficient to determine an a priori bound for an entropy weak solution need not suffice to imply local uniqueness thereof. In this context, fluid flow models based on stationary or self-similar reductions of Euler systems are distinguished as particularly attractive for computational investigation.