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A “forward-in-time” quadratic potential for systems of conservation laws

  • Stefano ModenaEmail author
Article
  • 47 Downloads
Part of the following topical collections:
  1. Hyperbolic PDEs, Fluids, Transport and Applications: Dedicated to Alberto Bressan for his 60th birthday

Abstract

A quadratic interaction potential \(t \mapsto \Upsilon (t)\) for hyperbolic systems of conservation laws is constructed, whose value \(\Upsilon (\bar{t})\) at time \(\bar{t}\) depends only on the present and the future profiles of the solution and not on the past ones. Such potential is used to bound the change of the speed of the waves at each interaction.

Keywords

Conservation laws Interaction estimates Quadratic potential 

Mathematics Subject Classification

35L65 

References

  1. 1.
    Bianchini, S.: Interaction estimates and Glimm functional for general hyperbolic systems. DCDS-A 9(1), 133–166 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bianchini, S., Bressan, A.: Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Math. 161, 223–342 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bianchini, S., Modena, S.: On a quadratic functional for scalar conservation laws. J. Hyperbolic Differ. Equ. 11(2), 355–435 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bianchini, S., Modena, S.: Quadratic interaction functional for systems of conservation laws: a case study. Bull. Inst. Math. Acad. Sin. (New Ser.) 9(3), 487–546 (2014)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bianchini, S., Modena, S.: Quadratic interaction functional for general systems of conservation laws. Commun. Math. Phys. 338, 1075–1152 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bressan, A.: Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem. Oxford University Press, Oxford (2000)zbMATHGoogle Scholar
  7. 7.
    Dafermos, C.: Hyberbolic Conservation Laws in Continuum Physics. Springer, Berlin (2009)Google Scholar
  8. 8.
    Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 18, 697–715 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Liu, T.-P.: The deterministic version of the Glimm scheme. Commun. Math. Phys. 57, 135–148 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Liu, T.-P.: Admissible solutions of hyperbolic conservation laws. Mem. Am. Math. Soc. 30(240), (1981). doi: 10.1090/memo/0240
  11. 11.
    Modena, S.: Quadratic interaction estimate for hyperbolic conservation laws, an overview. Contemp. Math. Fundam. Dir. 59, 148–172 (2016)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität LeipzigLeipzigGermany

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