Gauss‐Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws

By Gui-Qiang G. Chen, William P. Ziemer, Monica Torres. 2007

132 citations recorded by the source corpus.

We analyze a class of weakly differentiable vector fields F : ℝn → ℝn with the property that F ∈ L∞ and div F is a (signed) Radon measure. These fields are called bounded divergence‐measure fields. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence‐measure field F over the boundary of an arbitrary set of finite perimeter that ensures the validity of the Gauss‐Green theorem. To achieve this, we first establish a fundamental approximation theorem which states that, given a (signed) Radon measure μ that is absolutely continuous with respect to ℋ︁N − 1 on ℝN, any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure‐theoretic interior of the set with respect to the measure ||μ||, the total variation measure. We employ this approximation theorem to derive the normal trace of F on the boundary of any set of finite perimeter E as the limit of the normal traces of F on the boundaries of the approximate sets with smooth boundary so that the Gauss‐Green theorem for F holds on E. With these results, we analyze the Cauchy flux that is bounded by a nonnegative Radon measure over any oriented surface (i.e., an (N − 1)‐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 the balance law through the Cauchy flux. Finally, we apply this framework to the derivation of systems of balance laws with measure‐valued source terms from the formulation of the balance law. This framework also allows the recovery of Cauchy entropy flux through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws. © 2008 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/a167e8d498cccb32cfd0e8811b7a693534c34e80

Citations in the corpus

• Functional Versions of Lp-Affine Surface Area and Entropy Inequalities (2014, unknown): “…becomes which can be rewritten as Taking the logarithm and using Jensen’s inequality (recall that µ is assumed to be a probability measure) we obtain We will need some version of the Gauss-Green (or Stokes) formula and refer to [14] as a general reference and for recent results on this subject.”