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


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Brenner, Ph.: The Cauchy problem for the symmetric hyperbolic systems inL p. Math. Scand.19, 27–37 (1966)Google Scholar
  2. 2.
    Brenner, Ph.: The Cauchy problem for systems inL p andL p,α. Ark. Mat.11, 75–101 (1973)Google Scholar
  3. 3.
    Courant, R.: Methods of mathematical physics, Vol. II. New York: Interscience 1966Google Scholar
  4. 4.
    Lax, P.D.: Nonlinear hyperbolic systems. Stanford University Lecture Notes 1970Google Scholar
  5. 5.
    Majda, A.: Compressible fluid flow and systems of conservation laws in several space variables. Appl. Math. Series No. 53. Berlin, Heidelberg, New York: Springer 1984Google Scholar
  6. 6.
    Taylor, M.: Pseudodifferential operators. Princeton, NJ: Princeton University Press 1981Google Scholar
  7. 7.
    Temple, B.: PreprintGoogle Scholar

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

Personalised recommendations