Well-Posedness and Singular Limit of a Semilinear Hyperbolic Relaxation System with a Two-Scale Discontinuous Relaxation Rate

Frédéric Coquel, Shi Jin, Jian Guo Liu, Li Wang

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

Nonlinear hyperbolic systems with relaxations may encounter different scales of relaxation time, which is a prototype multiscale phenomenon that arises in many applications. In such a problem the relaxation time is of O(1) in part of the domain and very small in the remaining domain in which the solution can be approximated by the zero relaxation limit which can be solved numerically much more efficiently. For the Jin–Xin relaxation system in such a two-scale setting, we establish its wellposedness and singular limit as the (smaller) relaxation time goes to zero. The limit is a multiscale coupling problem which couples the original Jin–Xin system on the domain when the relaxation time is O(1) with its relaxation limit in the other domain through interface conditions which can be derived by matched interface layer analysis.As a result, we also establish the well-posedness and regularity (such as boundedness in sup norm with bounded total variation and L1-contraction) of the coupling problem, thus providing a rigorous mathematical foundation, in the general nonlinear setting, to the multiscale domain decomposition method for this two-scale problem originally proposed in Jin et al. in Math. Comp. 82, 749–779, 2013.

Original languageEnglish (US)
Pages (from-to)1051-1084
Number of pages34
JournalArchive For Rational Mechanics And Analysis
Volume214
Issue number3
DOIs
StatePublished - Oct 17 2014
Externally publishedYes

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