Advertisement

Journal of Mathematical Sciences

, Volume 244, Issue 2, pp 183–197 | Cite as

Nonuniqueness of Unbounded Solutions of the Cauchy Problem for Scalar Conservation Laws

  • A. Yu. GoritskyEmail author
  • L. V. Gargyants
Article
  • 7 Downloads

Abstract

This article is aimed at studying the Cauchy problem for a first-order quasi-linear equation with a flow function of power type and unbounded initial data of power or exponential type. It is known that the Cauchy problem in the class of locally bounded functions may have several solutions. We describe all entropy solutions of this problem, which can be represented in a special form. It is shown that after the first discontinuity line (shock wave), these solutions eventually exhibit the same behavior, and their nonuniqueness actually amounts to the choice of the first shock wave.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. N. Kruzhkov, “Generalized solutions of the Cauchy problem for nonlinear first order equation,” Dokl. AN SSSR, 187, No. 1, 29–32 (1969).zbMATHGoogle Scholar
  2. 2.
    S. N. Kruzhkov, “Quasilinear first-order equations with many independent variables,” Mat. Sb., 81, No. 2, 228–255 (1970).MathSciNetzbMATHGoogle Scholar
  3. 3.
    A. Yu. Goritsky, S. N. Kruzhkov, and G. A. Chechkin, First-Order Partial Differential Equations: A Textbook [in Russian], The Center of Appl. Stud. at the Faculty of Mech. and Math. of Moscow State Univ. (1999).Google Scholar
  4. 4.
    O. A. Oleinik, “On the Cauchy problem for nonlinear equations in a class of discontinuous functions,” Dokl. AN SSSR, 95, No. 3, 451–454 (1954).MathSciNetGoogle Scholar
  5. 5.
    E. Yu. Panov, “Well-posedness classes of locally bounded generalized entropy solutions of the Cauchy problem for quasilinear first-order equations,” Fundam. Prikl. Mat., 12, No. 5, 175–188 (2006).Google Scholar
  6. 6.
    A. Yu. Goritsky, “Construction of an unbounded entropy solution of the Cauchy problem with countably many shock waves,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 3–6 (1999).Google Scholar
  7. 7.
    A. Yu. Goritsky and E. Yu. Panov, “Example of nonuniqueness of entropy solutions in the class of locally bounded functions,” Russ. J. Math. Phys., 6, No. 4, 492–494 (1999).MathSciNetzbMATHGoogle Scholar
  8. 8.
    A. Yu. Goritsky and E. Yu. Panov, “On locally bounded generalized entropy solutions of the Cauchy problem for a first-order quasilinear equation,” Tr. Mat. Inst. Steklova, 236, No. 5, 120–133 (2002).MathSciNetzbMATHGoogle Scholar
  9. 9.
    L. V. Gargyants, “Locally bounded solutions of onedimensional conservation laws,” Differ. Uravn., 52, No. 4, 481–489 (2016).MathSciNetGoogle Scholar
  10. 10.
    L. V. Gargyants, “On locally bounded solutions of the Cauchy problem for a quasilinear first-order equation with power-type flux function,” Math. Notes, 104, No. 1-2, 210–217 (2018).MathSciNetCrossRefGoogle Scholar
  11. 11.
    L. V. Gargyants, “Example of nonexistence of a positive generalized entropy solution of a Cauchy problem with unbounded positive initial data,” Russ. J. Math. Phys., 24, No. 3, 412–414 (2017).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia

Personalised recommendations