, Volume 175, Issue 2, pp 245-267
Date: 03 Dec 2004

Divergence-Measure Fields, Sets of Finite Perimeter, and Conservation Laws

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Abstract.

Divergence-measure fields in L over sets of finite perimeter are analyzed. A notion of normal traces over boundaries of sets of finite perimeter is introduced, and the Gauss-Green formula over sets of finite perimeter is established for divergence-measure fields in L . The normal trace introduced here over a class of surfaces of finite perimeter is shown to be the weak-star limit of the normal traces introduced in Chen & Frid [6] over the Lipschitz deformation surfaces, which implies their consistency. As a corollary, an extension theorem of divergence-measure fields in L over sets of finite perimeter is also established. Then we apply the theory to the initial-boundary value problem of nonlinear hyperbolic conservation laws over sets of finite perimeter.

Communicated by C. M. Dafermos
Acknowledgement We thank WILLIAM P. ZIEMER for helpful discussions, especially on the proof of Propostion 7. GUI-QIANG CHEN’s research was supported in part by the National Science Foundation Grants.