Communications in Mathematical Sciences

Volume 13 (2015)

Number 2

A simple derivation of BV bounds for inhomogeneous relaxation systems

Pages: 577 – 586

(Fast Communication)

DOI: http://dx.doi.org/10.4310/CMS.2015.v13.n2.a17

Authors

Benoît Perthame (Laboratoire Jacques-Louis Lions, UPMC Univ Paris 6, Sorbonne Universités, Paris, France)

Nicolas Seguin (Laboratoire Jacques-Louis Lions, UPMC Univ Paris 6, Sorbonne Universités, Paris, France)

Magali Tournus (Laboratoire Jacques-Louis Lions, UPMC Univ Paris 6, Sorbonne Universités, Paris, France; and Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)

Abstract

We consider relaxation systems of transport equations with heterogeneous source terms and with boundary conditions, whose limits are scalar conservation laws. Classical bounds, and in particular BV estimates, fail in this context. They are the simplest and the most standard way to prove compactness and convergence.

We provide a novel, simple method to obtain partial BV regularity and strong compactness in this framework. The standard notion of entropy is not convenient either and we also indicate another, closely related, notion. We give two examples motivated by renal flows which consist of $2 \times 2$ and $3 \times 3$ relaxation systems with $2$-velocities, but the method is more general.

Keywords

hyperbolic relaxation, spatial heterogeneity, entropy condition, boundary conditions, strong compactness

2010 Mathematics Subject Classification

35B40, 35L03, 35L60, 35Q92

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