RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Izv. RAN. Ser. Mat.: Year: Volume: Issue: Page: Find

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

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

Full text: PDF file (2791 kB)
References: PDF file   HTML file

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

Bibliographic databases:

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

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
\Bibitem{Pan96} \by E.~Yu.~Panov \paper On measure-valued solutions of the Cauchy problem for a~first-order quasilinear equation \jour Izv. RAN. Ser. Mat. \yr 1996 \vol 60 \issue 2 \pages 107--148 \mathnet{http://mi.mathnet.ru/izv73} \crossref{https://doi.org/10.4213/im73} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1399420} \zmath{https://zbmath.org/?q=an:0882.35075} \transl \jour Izv. Math. \yr 1996 \vol 60 \issue 2 \pages 335--377 \crossref{https://doi.org/10.1070/IM1996v060n02ABEH000073} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1996VL85500005} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33747008573} 

• http://mi.mathnet.ru/eng/izv73
• https://doi.org/10.4213/im73
• http://mi.mathnet.ru/eng/izv/v60/i2/p107

 SHARE:

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, “A class of systems of quasilinear conservation laws”, Sb. Math., 188:5 (1997), 725–751
2. E. Yu. Panov, “An approximation scheme for measure-valued solutions of a first-order quasilinear equation”, Sb. Math., 188:1 (1997), 87–113
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
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
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
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
7. E. Yu. Panov, “O statisticheskikh resheniyakh zadachi Koshi dlya kvazilineinogo uravneniya pervogo poryadka”, Matem. modelirovanie, 14:3 (2002), 17–26
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
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
10. M. V. Korobkov, E. Yu. Panov, “Isentropic solutions of quasilinear equations of the first order”, Sb. Math., 197:5 (2006), 727–752
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
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
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
14. Caroline Bauzet, “Time-splitting approximation of the Cauchy problem for a stochastic conservation law”, Mathematics and Computers in Simulation, 2014
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
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
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
18. Karlsen K.H., Storrosten E.B., “On stochastic conservation laws and Malliavin calculus”, J. Funct. Anal., 272:2 (2017), 421–497
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
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
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
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
•  Number of views: This page: 371 Full text: 135 References: 75 First page: 1