Constrained systems of conservation laws : A geometric theory

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Original languageEnglish
Pages (from-to)407–444
Number of pages37
Journal / PublicationMethods and Applications of Analysis
Issue number4
Publication statusPublished - Dec 2017
Externally publishedYes


We address the Riemann and Cauchy problems for systems of n conservation laws in m unknowns which are subject to − n constraints (mn). 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 × n systems of constrained conservation laws to n × 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.

Research Area(s)

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