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
even for smooth solutions close to constants. Analogous estimates forL p norms
withF as above are also false. In one dimension such estimates are the backbone of the existing theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Brenner, Ph.: The Cauchy problem for the symmetric hyperbolic systems inL p . Math. Scand.19, 27–37 (1966)
Brenner, Ph.: The Cauchy problem for systems inL p andL p,α . Ark. Mat.11, 75–101 (1973)
Courant, R.: Methods of mathematical physics, Vol. II. New York: Interscience 1966
Lax, P.D.: Nonlinear hyperbolic systems. Stanford University Lecture Notes 1970
Majda, A.: Compressible fluid flow and systems of conservation laws in several space variables. Appl. Math. Series No. 53. Berlin, Heidelberg, New York: Springer 1984
Taylor, M.: Pseudodifferential operators. Princeton, NJ: Princeton University Press 1981
Temple, B.: Preprint
Author information
Authors and Affiliations
Additional information
Communicated by L. Nirenberg
Research partially supported by the National Science Foundation under grant MCS-8301061.
Rights and permissions
About this article
Cite this article
Rauch, J. BV estimates fail for most quasilinear hyperbolic systems in dimensions greater than one. Commun.Math. Phys. 106, 481–484 (1986). https://doi.org/10.1007/BF01207258
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01207258