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 Izv. RAN. Ser. Mat., 2002, Volume 66, Issue 6, Pages 91–136 (Mi izv411)

On generalized entropy solutions of the Cauchy problem for a first-order quasilinear equation in the class of locally summable functions

E. Yu. Panov

Novgorod State University after Yaroslav the Wise

Abstract: We construct a theory of locally summable generalized entropy solutions (g.e. solutions) of the Cauchy problem for a first-order non-homogeneous quasilinear equation with continuous flux vector satisfying a linear restriction on its growth. We prove the existence of greatest and least g.e. solutions, suggest sufficient conditions for uniqueness of g.e. solutions, prove several versions of the comparison principle, and obtain estimates for the $L^p$-norms of solution with respect to the space variables. We establish the uniqueness of g.e. solutions in the case when the input data are periodic functions of the space variables.

DOI: https://doi.org/10.4213/im411

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English version:
Izvestiya: Mathematics, 2002, 66:6, 1171–1218

Bibliographic databases:

UDC: 517.95
MSC: 35K45, 35K55, 35L45, 35L65

Citation: E. Yu. Panov, “On generalized entropy solutions of the Cauchy problem for a first-order quasilinear equation in the class of locally summable functions”, Izv. RAN. Ser. Mat., 66:6 (2002), 91–136; Izv. Math., 66:6 (2002), 1171–1218

Citation in format AMSBIB
\Bibitem{Pan02} \by E.~Yu.~Panov \paper On generalized entropy solutions of the Cauchy problem for a~first-order quasilinear equation in the class of locally summable functions \jour Izv. RAN. Ser. Mat. \yr 2002 \vol 66 \issue 6 \pages 91--136 \mathnet{http://mi.mathnet.ru/izv411} \crossref{https://doi.org/10.4213/im411} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1970354} \zmath{https://zbmath.org/?q=an:1071.35528} \transl \jour Izv. Math. \yr 2002 \vol 66 \issue 6 \pages 1171--1218 \crossref{https://doi.org/10.1070/IM2002v066n06ABEH000411} 

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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. Panov E.Yu., “to the Theory of Entropy Sub-Solutions of Degenerate Nonlinear Parabolic Equations”, Math. Meth. Appl. Sci.
2. Panov E.Yu., “Existence of strong traces for generalized solutions of multidimensional scalar conservation laws”, J. Hyperbolic Differ. Equ., 2:4 (2005), 885–908
3. 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
4. M. V. Korobkov, E. Yu. Panov, “Isentropic solutions of quasilinear equations of the first order”, Sb. Math., 197:5 (2006), 727–752
5. Panov E.Yu., “Existence of strong traces for quasi-solutions of multidimensional conservation laws”, J. Hyperbolic Differ. Equ., 4:4 (2007), 729–770
6. Panov E.Yu., “On infinite-dimensional Keyfitz-Kranzer systems of conservation laws”, Differ. Equ., 45:2 (2009), 274–278
7. Panov E.Yu., “Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux”, Arch. Ration. Mech. Anal., 195:2 (2010), 643–673
8. Andreianov B., Bendahmane M., Karlsen K.H., “Discrete Duality Finite Volume Schemes for Doubly Nonlinear Degenerate Hyperbolic-Parabolic Equations”, Journal of Hyperbolic Differential Equations, 7:1 (2010), 1–67
9. P. V. Lysuho, E. Yu. Panov, “Renormalized entropy solutions to the Cauchy problem for first order quasilinear conservation laws in the class of periodic functions”, J Math Sci, 2011
10. Panov E.Yu., “On the Dirichlet Problem for First Order Quasilinear Equations on a Manifold”, Trans Amer Math Soc, 363:5 (2011), 2393–2446
11. 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
12. Dmitry Golovaty, Truyen Nguyen, “On the existence, uniqueness and stability of entropy solutions to scalar conservation laws”, Journal of Differential Equations, 2012
13. E. Yu. Panov, “Renormalized entropy solutions of the Cauchy problem for a first-order inhomogeneous quasilinear equation”, Sb. Math., 204:10 (2013), 1480–1515
14. Abreu E., Colombeau M., Panov E., “Weak asymptotic methods for scalar equations and systems”, J. Math. Anal. Appl., 444:2 (2016), 1203–1232
15. Panov E.Yu., “On the Cauchy problem for scalar conservation laws in the class of Besicovitch almost periodic functions: Global well-posedness and decay property”, J. Hyberbolic Differ. Equ., 13:3 (2016), 633–659
16. Andreianov B., Gazibo M.K., “Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws”, Netw. Heterog. Media, 11:2, SI (2016), 203–222
17. E. Yu. Panov, “The Cauchy problem for a first-order quasilinear equation in the class of Besicovitch almost periodic functions”, Sb. Math., 208:8 (2017), 1207–1224
18. Cao G., Xiang W., Yang X., “Global Structure of Admissible Solutions of Multi-Dimensional Non-Homogeneous Scalar Conservation Law With Riemann-Type Data”, J. Differ. Equ., 263:2 (2017), 1055–1078
19. E. Yu. Panov, “K teorii entropiinykh reshenii nelineinykh vyrozhdayuschikhsya parabolicheskikh uravnenii”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 66, no. 2, Rossiiskii universitet druzhby narodov, M., 2020, 292–313
20. Panov E.Y., “On Decay of Entropy Solutions to Multidimensional Conservation Laws”, SIAM J. Math. Anal., 52:2 (2020), 1310–1317
21. Panov E.Yu., “On Some Properties of Entropy Solutions of Degenerate Non-Linear Anisotropic Parabolic Equations”, J. Differ. Equ., 275 (2021), 139–166
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