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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

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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.

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Authors and Affiliations

  1. Novgorod State University, Velikiy Novgorod, Russia

    E.Yu. Panov

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  1. E.Yu. Panov
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Editor information

Editors and Affiliations

  1. Department of Applied and Computational Mathematics 217-50, California Institute of Technology, Pasadena, CA, 91125, USA

    Thomas Y. Hou

  2. Center for Scientific Computation and Mathematical Modeling (CSCAMM) Department of Mathematics, University of Maryland, College Park, MD, 20742-3289, USA

    Eitan Tadmor

  3. Institute for Physical Science & Technology (IPST), University of Maryland, College Park, MD, 20742-3289, USA

    Eitan Tadmor

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Panov, E. (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. In: Hou, T.Y., Tadmor, E. (eds) Hyperbolic Problems: Theory, Numerics, Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55711-8_74

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  • DOI: https://doi.org/10.1007/978-3-642-55711-8_74

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-62929-7

  • Online ISBN: 978-3-642-55711-8

  • eBook Packages: Springer Book Archive

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