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Archive for Rational Mechanics and Analysis

, Volume 149, Issue 1, pp 1–22 | Cite as

L1 Stability Estimates for n×n Conservation Laws

  • Alberto Bressan
  • Tai-Ping Liu
  • Tong Yang

Abstract

. Let \(u_t+f(u)_x=0\) be a strictly hyperbolic \(n\times n\) system of conservation laws, each characteristic field being linearly degenerate or genuinely nonlinear. In this paper we explicitly define a functional \(\Phi=\Phi(u,v)\), equivalent to the \(\L^1\) distance, which is “almost decreasing” i.e., \( \Phi\big( u(t),~v(t)\big)-\Phi\big( u(s),~v(s)\big)\leq \O(\ve)\cdot (t-s)\quad\hbox{for all}~~t>s\geq 0,\) 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 \({\vec L}^1\) norm. This provides a new proof of the existence of the standard Riemann semigroup generated by a n×n system of conservation laws.

Keywords

Wave Speed Hyperbolic System Tracking Algorithm Stability Estimate Characteristic Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Alberto Bressan
    • 1
  • Tai-Ping Liu
    • 2
  • Tong Yang
    • 3
  1. 1.S.I.S.S.A., Trieste, ItalyIT
  2. 2.Department of Mathematics, Stanford University, Stanford, California 94305-2125, USAUS
  3. 3.Department of Mathematics, City University of Hong KongHK

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