L1 stability estimates for n x n conservation laws

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review

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Original languageEnglish
Pages (from-to)1-22
Journal / PublicationArchive for Rational Mechanics and Analysis
Volume149
Issue number1
Publication statusPublished - 22 Oct 1999

Abstract

Let ut + f(u)x = 0 be a strictly hyperbolic n x n system of conservation laws, each characteristic field being linearly degenerate or genuinely nonlinear. In this paper we explicitly define a functional Φ = Φ(u, v), equivalent to the L1 distance, which is "almost decreasing" i.e., Φ(u(t), v(t)) - Φ(u(s), v(s)) ≦ O (ε) · (t - s) for all t > s ≧ 0, for every pair of ε-approximate solutions u, v with small total variation, generated by a wave front tracking algorithm. The small parameter ε here controls the errors in the wave speeds, the maximum size of rarefaction fronts and the total strength of all non-physical waves in u and in v. From the above estimate, it follows that front-tracking approximations converge to a unique limit solution, depending Lipschitz continuously on the initial data, in the L1 norm. This provides a new proof of the existence of the standard Riemann semigroup generated by a n x n system of conservation laws.

Citation Format(s)

L1 stability estimates for n x n conservation laws. / BRESSAN, Alberto; LIU, Tai-Ping; YANG, Tong.
In: Archive for Rational Mechanics and Analysis, Vol. 149, No. 1, 22.10.1999, p. 1-22.

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review