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Izv. RAN. Ser. Mat., 1996, Volume 60, Issue 2, Pages 107–148 (Mi izv73)  

This article is cited in 22 scientific papers (total in 22 papers)

On measure-valued solutions of the Cauchy problem for a first-order quasilinear equation

E. Yu. Panov

Novgorod State University after Yaroslav the Wise

Abstract: Measure-valued solutions of the Cauchy problem are considered for a first-order quasilinear equation with only continuous flow functions. A measure-valued analogue of the maximum principle (in Lebesgue spaces) is proved. Conditions are found under which a measure-valued solution is an ordinary function. Uniqueness questions are studied. The class of “strong” measure-valued solutions is distinguished and the existence and uniqueness (under natural restrictions) of a strong measure-valued solution is proved. Questions of the convergence of sequences of measure-valued solutions are studied.

DOI: https://doi.org/10.4213/im73

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English version:
Izvestiya: Mathematics, 1996, 60:2, 335–377

Bibliographic databases:

UDC: 517.95
MSC: Primary 35L65, 35D05, 35D10; Secondary 49M30, 35F25
Received: 04.04.1995

Citation: E. Yu. Panov, “On measure-valued solutions of the Cauchy problem for a first-order quasilinear equation”, Izv. RAN. Ser. Mat., 60:2 (1996), 107–148; Izv. Math., 60:2 (1996), 335–377

Citation in format AMSBIB
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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, “A class of systems of quasilinear conservation laws”, Sb. Math., 188:5 (1997), 725–751  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. E. Yu. Panov, “An approximation scheme for measure-valued solutions of a first-order quasilinear equation”, Sb. Math., 188:1 (1997), 87–113  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. 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. Math., 63:1 (1999), 129–179  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. 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
    5. A. Yu. Goritskii, E. Yu. Panov, “Locally Bounded Generalized Entropy Solutions to the Cauchy Problem for a First-Order Quasilinear Equation”, Proc. Steklov Inst. Math., 236 (2002), 110–123  mathnet  mathscinet  zmath
    6. 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. Math., 66:6 (2002), 1171–1218  mathnet  crossref  crossref  mathscinet  zmath
    7. E. Yu. Panov, “O statisticheskikh resheniyakh zadachi Koshi dlya kvazilineinogo uravneniya pervogo poryadka”, Matem. modelirovanie, 14:3 (2002), 17–26  mathnet  mathscinet  zmath
    8. E. Yu. Panov, “Maximum and minimum generalized entropy solutions to the Cauchy problem for a first-order quasilinear equation”, Sb. Math., 193:5 (2002), 727–743  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. 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
    10. 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
    11. Panov, E, “ON WEAK COMPLETENESS OF THE SET OF ENTROPY SOLUTIONS TO A SCALAR CONSERVATION LAW”, SIAM Journal on Mathematical Analysis, 41:1 (2009), 26  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    12. 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  scopus
    13. CAROLINE BAUZET, GUY VALLET, PETRA WITTBOLD, “THE CAUCHY PROBLEM FOR CONSERVATION LAWS WITH A MULTIPLICATIVE STOCHASTIC PERTURBATION”, J. Hyper. Differential Equations, 09:04 (2012), 661  crossref  mathscinet  isi  scopus  scopus
    14. Caroline Bauzet, “Time-splitting approximation of the Cauchy problem for a stochastic conservation law”, Mathematics and Computers in Simulation, 2014  crossref  mathscinet  scopus  scopus
    15. Biswas I.H., Karlsen K.H., Majee A.K., “Conservation Laws Driven By Levy White Noise”, J. Hyberbolic Differ. Equ., 12:3 (2015), 581–654  crossref  mathscinet  zmath  isi  scopus  scopus
    16. Bauzet C., Charrier J., Gallouet T., “Convergence of flux-splitting finite volume schemes for hyperbolic scalar conservation laws with a multiplicative stochastic perturbation”, Math. Comput., 85:302 (2016), 2777–2813  crossref  mathscinet  zmath  isi  scopus  scopus
    17. 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
    18. Karlsen K.H., Storrosten E.B., “On stochastic conservation laws and Malliavin calculus”, J. Funct. Anal., 272:2 (2017), 421–497  crossref  mathscinet  zmath  isi  scopus
    19. Bauzet C., Charrier J., Gallouet T., “Numerical approximation of stochastic conservation laws on bounded domains”, ESAIM-Math. Model. Numer. Anal.-Model. Math. Anal. Numer., 51:1 (2017), 225–278  crossref  mathscinet  zmath  isi  scopus
    20. Karlsen K.H., Storrosten E.B., “Analysis of a Splitting Method For Stochastic Balance Laws”, IMA J. Numer. Anal., 38:1 (2018), 1–56  crossref  mathscinet  isi  scopus  scopus
    21. Lv G., Wu J.-L., “Heterogeneous Stochastic Scalar Conservation Laws With Non-Homogeneous Dirichlet Boundary Conditions”, J. Hyberbolic Differ. Equ., 15:2 (2018), 291–328  crossref  mathscinet  zmath  isi  scopus  scopus
    22. Majee A.K., “Convergence of a Flux-Splitting Finite Volume Scheme For Conservation Laws Driven By Levy Noise”, Appl. Math. Comput., 338 (2018), 676–697  crossref  mathscinet  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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