L1 stability estimates for n x n conservation laws

Alberto Bressan, Tai Ping Liu, Tong Yang

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157 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1-22
Number of pages22
JournalArchive for Rational Mechanics and Analysis
Issue number1
StatePublished - Oct 22 1999

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering


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