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
    • University of Michigan
    • Centre de Mathématiques AppliquéesEcole Polytechnique

DOI: 10.1007/BF01207258

Cite this article as:
Rauch, J. Commun.Math. Phys. (1986) 106: 481. doi:10.1007/BF01207258


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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© Springer-Verlag 1986