GaussGreen theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws
Abstract
We analyze a class of weakly differentiable vector fields (\FF \colon \rn \to \rn) with the property that (\FF\in L^{\infty}) and (÷\FF) is a Radon measure. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergencemeasure field $\FF$ over the boundary of an arbitrary set of finite perimeter, which ensures the validity of the GaussGreen theorem. To achieve this, we establish a fundamental approximation theorem which states that, given a Radon measure $\mu$ that is absolutely continuous with respect to $\mathcal{H}^{N1}$ on $\rn$, any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measuretheoretic interior of the set with respect to the measure $\\mu\$. With this approximation theorem, we derive the normal trace of $\FF$ on the boundary of any set of finite perimeter, (E), as the limit of the normal traces of $\FF$ on the boundaries of the approximate sets with smooth boundary, so that the GaussGreen theorem for $\FF$ holds on (E). With these results, we analyze the Cauchy fluxes that are bounded by a Radon measure over any oriented surface (i.e. an $(N1)$dimensional surface that is a part of the boundary of a set of finite perimeter) and thereby develop a general mathematical formulation of the physical principle of balance law through the Cauchy flux. Finally, we apply this framework to the derivation of systems of balance laws with measurevalued source terms from the formulation of balance law. This framework also allows the recovery of Cauchy entropy fluxes through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws.
 Publication:

arXiv eprints
 Pub Date:
 September 2007
 arXiv:
 arXiv:0709.3673
 Bibcode:
 2007arXiv0709.3673C
 Keywords:

 Mathematics  Analysis of PDEs;
 28C05;
 26B20;
 28A05;
 26B12;
 35L65;
 35L50 (Primary);
 28A75;
 28A25;
 26B05;
 26B30;
 26B40 (Secondary)
 EPrint:
 43 pages