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

This article is cited in 21 scientific papers (total in 21 papers)

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
Received: 27.06.2001

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.  crossref  isi
    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  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  mathscinet  zmath  elib
    4. 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
    5. Panov E.Yu., “Existence of strong traces for quasi-solutions of multidimensional conservation laws”, J. Hyperbolic Differ. Equ., 4:4 (2007), 729–770  crossref  mathscinet  zmath  isi  elib  scopus
    6. Panov E.Yu., “On infinite-dimensional Keyfitz-Kranzer systems of conservation laws”, Differ. Equ., 45:2 (2009), 274–278  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    12. Dmitry Golovaty, Truyen Nguyen, “On the existence, uniqueness and stability of entropy solutions to scalar conservation laws”, Journal of Differential Equations, 2012  crossref  mathscinet  isi  scopus
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    14. Abreu E., Colombeau M., Panov E., “Weak asymptotic methods for scalar equations and systems”, J. Math. Anal. Appl., 444:2 (2016), 1203–1232  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    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  crossref  mathscinet  zmath  isi  scopus
    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  mathnet  crossref
    20. Panov E.Y., “On Decay of Entropy Solutions to Multidimensional Conservation Laws”, SIAM J. Math. Anal., 52:2 (2020), 1310–1317  crossref  mathscinet  isi
    21. Panov E.Yu., “On Some Properties of Entropy Solutions of Degenerate Non-Linear Anisotropic Parabolic Equations”, J. Differ. Equ., 275 (2021), 139–166  crossref  mathscinet  isi
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