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

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

2007-09-24

arxiv.org/abs/0709.3673v1

Mathematics

Analysis of PDEs

43 pages

Scientific paper

We analyze a class of weakly differentiable vector fields (\FF \colon \rn \to \rn) with the property that (\FF\in L^{\infty}) and (\div \FF) is a Radon measure. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence-measure field $\FF$ over the boundary of an arbitrary set of finite perimeter, which ensures the validity of the Gauss-Green 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 $\mathcal{H}^{N-1}$ 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 $\|\mu\|$. 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 Gauss-Green 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 $(N-1)$-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 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 balance law. This framework also allows the recovery of Cauchy entropy fluxes through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws.

Affiliated with

Also associated with

No associations

Rating

Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-276445