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Izv. RAN. Ser. Mat., 1999, Volume 63, Issue 1, Pages 133–184 (Mi izv232)  

This article is cited in 5 scientific papers (total in 5 papers)

A non-local theory of generalized entropy solutions of the Cauchy problem for a class of hyperbolic systems of conservation laws

E. Yu. Panov

Novgorod State University after Yaroslav the Wise

Abstract: We consider a hyperbolic system of conservation laws on the space of symmetric second-order matrices. The right-hand side of this system contains the functional calculus operator $\tilde f(U)$generated in the general case only by a continuous scalar function $f(u)$. For these systems we define and describe the set of singular entropies, introduce the concept of generalized entropy solutions of the corresponding Cauchy problem, and investigate the properties of generalized entropy solutions. We define the class of strong generalized entropy solutions, in which the Cauchy problem has precisely one solution. We suggest a condition on the initial data under which any generalized entropy solution is strong, which implies its uniqueness. Under this condition we establish that the “vanishing viscosity” method converges. An example shows that in the general case there can be more than one generalized entropy solution.

DOI: https://doi.org/10.4213/im232

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English version:
Izvestiya: Mathematics, 1999, 63:1, 129–179

Bibliographic databases:

MSC: 35K45, 35K55, 35L45, 35L65
Received: 03.07.1997

Citation: E. Yu. Panov, “A non-local theory of generalized entropy solutions of the Cauchy problem for a class of hyperbolic systems of conservation laws”, Izv. RAN. Ser. Mat., 63:1 (1999), 133–184; Izv. Math., 63:1 (1999), 129–179

Citation in format AMSBIB
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\by E.~Yu.~Panov
\paper A~non-local theory of generalized entropy solutions of the Cauchy problem for a~class of hyperbolic systems of conservation laws
\jour Izv. RAN. Ser. Mat.
\yr 1999
\vol 63
\issue 1
\pages 133--184
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\crossref{https://doi.org/10.4213/im232}
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\zmath{https://zbmath.org/?q=an:0940.35137}
\transl
\jour Izv. Math.
\yr 1999
\vol 63
\issue 1
\pages 129--179
\crossref{https://doi.org/10.1070/im1999v063n01ABEH000232}
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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. E. Yu. Panov, “On the theory of generalized entropy solutions of the Cauchy problem for a class of non-strictly hyperbolic systems of conservation laws”, Sb. Math., 191:1 (2000), 121–150  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. E. Yu. Panov, “On the symmetrizability of hyperbolic matrix spaces”, St. Petersburg Math. J., 20:3 (2009), 465–471  mathnet  crossref  mathscinet  zmath  isi  elib
    3. Panov, EY, “On infinite-dimensional Keyfitz-Kranzer systems of conservation laws”, Differential Equations, 45:2 (2009), 274  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    4. Crippa G., Spinolo L.V., “An Overview on Some Results Concerning the Transport Equation and its Applications To Conservation Laws”, Communications on Pure and Applied Analysis, 9:5 (2010), 1283–1293  crossref  mathscinet  zmath  isi  scopus
    5. N.H. Risebro, F. Weber, “A note on front tracking for the Keyfitz–Kranzer system”, Journal of Mathematical Analysis and Applications, 2013  crossref  mathscinet  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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