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

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

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
\mathnet{http://mi.mathnet.ru/izv232}
\crossref{https://doi.org/10.4213/im232}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1701842}
\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}

• http://mi.mathnet.ru/eng/izv232
• https://doi.org/10.4213/im232
• http://mi.mathnet.ru/eng/izv/v63/i1/p133

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. E. Yu. Panov, “On the symmetrizability of hyperbolic matrix spaces”, St. Petersburg Math. J., 20:3 (2009), 465–471
3. Panov, EY, “On infinite-dimensional Keyfitz-Kranzer systems of conservation laws”, Differential Equations, 45:2 (2009), 274
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
5. N.H. Risebro, F. Weber, “A note on front tracking for the Keyfitz–Kranzer system”, Journal of Mathematical Analysis and Applications, 2013
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