Trudy Matematicheskogo Instituta imeni V.A. Steklova
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Guidelines for authors License agreement Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Trudy Mat. Inst. Steklova: Year: Volume: Issue: Page: Find

 Personal entry: Login: Password: Save password Enter Forgotten password? Register

 Trudy Mat. Inst. Steklova, 2002, Volume 236, Pages 120–133 (Mi tm282)

This article is cited in 9 scientific papers (total in 9 papers)

Locally Bounded Generalized Entropy Solutions to the Cauchy Problem for a First-Order Quasilinear Equation

A. Yu. Goritskiia, E. Yu. Panovb

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Novgorod State University after Yaroslav the Wise

Abstract: Generalized entropy solutions for a first-order quasilinear partial differential equation are studied. It is shown that the Cauchy problem for this equation is ill-posed in the class of locally bounded functions. The examples of nonexistence and nonuniqueness of solutions are constructed. Moreover, a uniqueness theorem, which holds for solutions integrable with respect to the spatial variable, is proved.

Full text: PDF file (230 kB)
References: PDF file   HTML file

English version:
Proceedings of the Steklov Institute of Mathematics, 2002, 236, 110–123

Bibliographic databases:
UDC: 517.95
Received in December 2000

Citation: A. Yu. Goritskii, E. Yu. Panov, “Locally Bounded Generalized Entropy Solutions to the Cauchy Problem for a First-Order Quasilinear Equation”, Differential equations and dynamical systems, Collected papers. Dedicated to the 80th anniversary of academician Evgenii Frolovich Mishchenko, Trudy Mat. Inst. Steklova, 236, Nauka, MAIK «Nauka/Inteperiodika», M., 2002, 120–133; Proc. Steklov Inst. Math., 236 (2002), 110–123

Citation in format AMSBIB
\Bibitem{GorPan02} \by A.~Yu.~Goritskii, E.~Yu.~Panov \paper Locally Bounded Generalized Entropy Solutions to the Cauchy Problem for a~First-Order Quasilinear Equation \inbook Differential equations and dynamical systems \bookinfo Collected papers. Dedicated to the 80th anniversary of academician Evgenii Frolovich Mishchenko \serial Trudy Mat. Inst. Steklova \yr 2002 \vol 236 \pages 120--133 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm282} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1931012} \zmath{https://zbmath.org/?q=an:1021.35063} \transl \jour Proc. Steklov Inst. Math. \yr 2002 \vol 236 \pages 110--123 

Linking options:
• http://mi.mathnet.ru/eng/tm282
• http://mi.mathnet.ru/eng/tm/v236/p120

 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. 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
2. 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
3. M. V. Korobkov, E. Yu. Panov, “Isentropic solutions of quasilinear equations of the first order”, Sb. Math., 197:5 (2006), 727–752
4. 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
5. E. Yu. Panov, “Renormalized entropy solutions of the Cauchy problem for a first-order inhomogeneous quasilinear equation”, Sb. Math., 204:10 (2013), 1480–1515
6. Gargyants L.V., “Locally bounded solutions of one-dimensional conservation laws”, Differ. Equ., 52:4 (2016), 458–466
7. Gargyants L.V., “Example of Nonexistence of a Positive Generalized Entropy Solution of a Cauchy Problem With Unbounded Positive Initial Data”, Russ. J. Math. Phys., 24:3 (2017), 412–414
8. A. Yu. Goritsky, L. V. Gargyants, “Nonuniqueness of unbounded solutions of the Cauchy problem for scalar conservation laws”, J. Math. Sci. (N. Y.), 244:2 (2020), 183–197
9. L. V. Gargyants, A. Yu. Goritsky, E. Yu. Panov, “Constructing unbounded discontinuous solutions of scalar conservation laws using the Legendre transform”, Sb. Math., 212:4 (2021), 475–489
•  Number of views: This page: 351 Full text: 124 References: 36

 Contact us: math-net2021_12 [at] mi-ras ru Terms of Use Registration to the website Logotypes © Steklov Mathematical Institute RAS, 2021