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

Constrained Systems of Conservation Laws: A Geometric Theory