Constrained Systems of Conservation Laws: A Geometric Theory

TL;DR

This paper develops a geometric framework for constrained systems of conservation laws, proving local and global existence results for initial value problems in one dimension, extending classical theories to more general, physically relevant systems.

Contribution

It introduces a geometric approach to constrained conservation laws, generalizing classical theories and extending Lax's and Glimm's methods to systems with manifold-valued states.

Findings

Proved local well-posedness of the Riemann problem.

Established global existence for small total variation initial data.

Extended classical methods to systems with non-invertible accumulation functions.

Abstract

We address the Riemann and Cauchy problems for systems of $n$ conservation laws in $m$ unknowns which are subject to $m-n$ constraints ($m\geq n$). Such constrained systems generalize systems of conservation laws in standard form to include various examples of conservation laws in Physics and Engineering beyond gas dynamics, e.g., multi-phase flow in porous media. We prove local well-posedness of the Riemann problem and global existence of the Cauchy problem for initial data with sufficiently small total variation, in one spatial dimension. The key to our existence theory is to generalize the $m\times n$ systems of constrained conservation laws to $n\times n$ systems of conservation laws with states taking values in an $n$-dimensional manifold and to extend Lax's theory for local existence as well as Glimm's random choice method to our geometric framework. Our resulting existence theory allows for the accumulation function to be non-invertible across hypersurfaces.