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To the Theory of Generalized Entropy Solutions of the Cauchy Problem for a First Order Quasilinear Equation in the Class of Locally Integrable Functions

In: Hyperbolic Problems: Theory, Numerics, Applications

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  • E.Yu. Panov

    (Novgorod State University)

Abstract

We construct the theory of locally int egrable generalized entropy solutions (g.e.s.) of the Cauchy problem for a first order nonhomogeneous quasilinear equat ion in t he case when the flux is only cont inuous and satisfies the linear growth condit ion. We prove existence of maximal and minimal g.e.s., deduce sufficient conditions for uniqueness of g.e.s. Variants of comparison principle are proved, estimates of L p -norms of g.e.s, with resp ect to spatial variables are found. Uniqueness of g.e.s. is established in the case when input fun ctions are periodic in space variabl es. It is also shown that the linear growth restriction for the flux vector seems to be necessary for well-posedn ess of the Cau chy problem in classes of unbounded g.e.s.

Suggested Citation

  • E.Yu. Panov, 2003. "To the Theory of Generalized Entropy Solutions of the Cauchy Problem for a First Order Quasilinear Equation in the Class of Locally Integrable Functions," Springer Books, in: Thomas Y. Hou & Eitan Tadmor (ed.), Hyperbolic Problems: Theory, Numerics, Applications, pages 789-796, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-55711-8_74
    DOI: 10.1007/978-3-642-55711-8_74
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