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Trudy Mat. Inst. Steklova, 2002, Volume 236, Pages 120–133 (Mi tm282)  

This article is cited in 9 scientific papers (total in 9 papers)

Locally Bounded Generalized Entropy Solutions to the Cauchy Problem for a First-Order Quasilinear Equation

A. Yu. Goritskiia, E. Yu. Panovb

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Novgorod State University after Yaroslav the Wise

Abstract: Generalized entropy solutions for a first-order quasilinear partial differential equation are studied. It is shown that the Cauchy problem for this equation is ill-posed in the class of locally bounded functions. The examples of nonexistence and nonuniqueness of solutions are constructed. Moreover, a uniqueness theorem, which holds for solutions integrable with respect to the spatial variable, is proved.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2002, 236, 110–123

Bibliographic databases:
UDC: 517.95
Received in December 2000

Citation: A. Yu. Goritskii, E. Yu. Panov, “Locally Bounded Generalized Entropy Solutions to the Cauchy Problem for a First-Order Quasilinear Equation”, Differential equations and dynamical systems, Collected papers. Dedicated to the 80th anniversary of academician Evgenii Frolovich Mishchenko, Trudy Mat. Inst. Steklova, 236, Nauka, MAIK Nauka/Inteperiodika, M., 2002, 120–133; Proc. Steklov Inst. Math., 236 (2002), 110–123

Citation in format AMSBIB
\Bibitem{GorPan02}
\by A.~Yu.~Goritskii, E.~Yu.~Panov
\paper Locally Bounded Generalized Entropy Solutions to the Cauchy Problem for a~First-Order Quasilinear Equation
\inbook Differential equations and dynamical systems
\bookinfo Collected papers. Dedicated to the 80th anniversary of academician Evgenii Frolovich Mishchenko
\serial Trudy Mat. Inst. Steklova
\yr 2002
\vol 236
\pages 120--133
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\mathnet{http://mi.mathnet.ru/tm282}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1931012}
\zmath{https://zbmath.org/?q=an:1021.35063}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2002
\vol 236
\pages 110--123


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Panov E.Y., “To the theory of generalized entropy solutions of the Cauchy problem for a first order quasilinear equation in the class of locally integrable functions”, Hyperbolic Problems: Theory, Numerics, Applications, 2003, 789–796  crossref  mathscinet  zmath  isi
    2. E. Yu. Panov, “On well-posedness classes of locally bounded generalized entropy solutions of the Cauchy problem for quasilinear first-order equations”, J. Math. Sci., 150:6 (2008), 2578–2587  mathnet  crossref  mathscinet  zmath  elib
    3. M. V. Korobkov, E. Yu. Panov, “Isentropic solutions of quasilinear equations of the first order”, Sb. Math., 197:5 (2006), 727–752  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. Lysukho P.V., Panov E.Yu., “Existence and Uniqueness of Unbounded Entropy Solutions of the Cauchy Problem for First-Order Quasilinear Conservation Laws”, Differ Equ, 47:1 (2011), 102–110  crossref  mathscinet  zmath  isi  elib  elib  scopus
    5. E. Yu. Panov, “Renormalized entropy solutions of the Cauchy problem for a first-order inhomogeneous quasilinear equation”, Sb. Math., 204:10 (2013), 1480–1515  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. Gargyants L.V., “Locally bounded solutions of one-dimensional conservation laws”, Differ. Equ., 52:4 (2016), 458–466  crossref  mathscinet  zmath  isi  elib  scopus
    7. Gargyants L.V., “Example of Nonexistence of a Positive Generalized Entropy Solution of a Cauchy Problem With Unbounded Positive Initial Data”, Russ. J. Math. Phys., 24:3 (2017), 412–414  crossref  mathscinet  zmath  isi  scopus
    8. A. Yu. Goritsky, L. V. Gargyants, “Nonuniqueness of unbounded solutions of the Cauchy problem for scalar conservation laws”, J. Math. Sci. (N. Y.), 244:2 (2020), 183–197  mathnet  crossref  elib
    9. L. V. Gargyants, A. Yu. Goritsky, E. Yu. Panov, “Constructing unbounded discontinuous solutions of scalar conservation laws using the Legendre transform”, Sb. Math., 212:4 (2021), 475–489  mathnet  crossref  crossref  isi  elib
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