Constrained systems of conservation laws: a geometric theory

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.

Keywords

shock waves, hyperbolic conservation laws, Glimm scheme, Riemann problem, relaxation systems, curved state space

2010 Mathematics Subject Classification

35L65

Full Text (PDF format)

M.R. is currently supported by FCT/Portugal through (GPSEinstein) PTDC/MAT-ANA/1275/2014 and UID/MAT/04459/2013. The major part of this work was done while M.R. was Post-Doctorate at IMPA in Rio de Janeiro, funded through CAPES-Brazil.

Received 2 November 2016

Accepted 27 October 2017

Published 20 April 2018