Communications in Mathematical Physics

, Volume 106, Issue 3, pp 481–484

BV estimates fail for most quasilinear hyperbolic systems in dimensions greater than one

  • Jeffrey Rauch
Article

Abstract

We show that for most non-scalar systems of conservation laws in dimension greater than one, one does not have BV estimates of the form
$$\begin{gathered} \parallel \overline V u(\overline t )\parallel _{T.V.} \leqq F(\parallel \overline V u(0)\parallel _{T.V.} ), \hfill \\ F \in C(\mathbb{R}),F(0) = 0,F Lipshitzean at 0, \hfill \\ \end{gathered} $$
even for smooth solutions close to constants. Analogous estimates forLp norms
$$\parallel u(\overline t ) - \overline u \parallel _{L^p } \leqq F(\parallel u(0) - \overline u \parallel _{L^p } ),p \ne 2$$
withF as above are also false. In one dimension such estimates are the backbone of the existing theory.

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Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Jeffrey Rauch
    • 1
    • 2
  1. 1.University of MichiganAnn ArborUSA
  2. 2.Centre de Mathématiques AppliquéesEcole PolytechniquePalaiseauFrance

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