Archive for Rational Mechanics and Analysis

, Volume 179, Issue 1, pp 31–54 | Cite as

Blowup with Small BV Data in Hyperbolic Conservation Laws

Article

Abstract.

We construct weak solutions of 3×3 conservation laws which blow up in finite time. The system is strictly hyperbolic at every state in the solution, and the data can be chosen to have arbitrarily small total variation. This is thus an example where Glimm's existence theorem fails to apply, and it implies the necessity of uniform hyperbolicity in Glimm's theorem. Because our system is very simple, we can carry out explicit calculations and understand the global geometry of wave curves.

Keywords

Neural Network Complex System Total Variation Weak Solution Nonlinear Dynamics 

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References

  1. 1.
    Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18, 697–715 (1965)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Kuczma, M., Choczewski, B., Ger, R.: Iterative functional equations. Encyclopedia of Mathematics, Cambridge University Press, 1990Google Scholar
  3. 3.
    Lax, P.D.: Hyperbolic systems of conservation laws, II. Comm. Pure Appl. Math. 10, 537–566 (1957)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Smoller, J.: Shock waves and reaction-diffusion equations. Springer-Verlag, New York, 1982Google Scholar
  5. 5.
    Young, R.: Sup-norm stability for Glimm's scheme. Comm. Pure Appl. Math. 46, 903–948 (1993)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Young, R.: Exact solutions to degenerate conservation laws. SIAM J. Math. Anal. 30, 537–558 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Young, R.: Blowup in hyperbolic conservation laws. Contemp. Math. 327, 379–387 (2003)CrossRefGoogle Scholar
  8. 8.
    Young, R.: Blowup of solutions and boundary instabilities in nonlinear hyperbolic equations. Comm. Math. Sci. 2, 269–292 (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of Mathematics & StatisticsUniversity of MassachusettsAmherstU.S.A.

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