Divergence‐Measure Fields and Hyperbolic Conservation Laws

Abstract

. We analyze a class of $L^\infty$ vector fields, called divergence‐measure fields. We establish the Gauss‐Green formula, the normal traces over subsets of Lipschitz boundaries, and the product rule for this class of $L^\infty$ fields. Then we apply this theory to analyze $L^\infty$ entropy solutions of initial‐boundary‐value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data. The examples of conservation laws include multidimensional scalar equations, the system of nonlinear elasticity, and a class of $m\X m$ systems with affine characteristic hypersurfaces. The analysis in $L^\infty$ also extends to $L^p$.