Abstract:
The existence of the maximum and minimum generalized entropy solutions of the Cauchy problem for a first-order quasilinear equation is proved in the general case of a flux vector that is merely continuous, when the uniqueness property of a generalized entropy solution does not necessarily hold. Some useful applications are presented. In particular, the uniqueness of the generalized entropy solution is established for input data that are periodic with respect to n−1n−1 linearly independent space vectors (nn is the number of space variables).
Citation:
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
\Bibitem{Pan02}
\by E.~Yu.~Panov
\paper Maximum and minimum generalized entropy solutions to the~Cauchy problem
for a~first-order quasilinear equation
\jour Sb. Math.
\yr 2002
\vol 193
\issue 5
\pages 727--743
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Linking options:
https://www.mathnet.ru/eng/sm653
https://doi.org/10.1070/SM2002v193n05ABEH000653
https://www.mathnet.ru/eng/sm/v193/i5/p95
This publication is cited in the following 15 articles:
E. Yu. Panov, “On the Theory of Entropy Sub- And Supersolutions of Nonlinear Degenerate Parabolic Equations”, J Math Sci, 2024
E. Yu. Panov, “K teorii entropiinykh sub- i superreshenii nelineinykh vyrozhdayuschikhsya parabolicheskikh uravnenii”, SMFN, 69, no. 2, Rossiiskii universitet druzhby narodov, M., 2023, 306–331
Evgeny Yu. Panov, “On Entropy Solutions of Scalar Conservation Laws with Discontinuous Flux”, Arch Rational Mech Anal, 247:5 (2023)
E. Yu. Panov, “On the Theory of Entropy Solutions of Nonlinear Degenerate Parabolic Equations”, J Math Sci, 265:6 (2022), 988
Evgeny Yu. Panov, “On decay of entropy solutions to multidimensional conservation laws in the case of perturbed periodic initial data”, J. Hyper. Differential Equations, 19:01 (2022), 141
Panov E.Yu., “On Some Properties of Entropy Solutions of Degenerate Non-Linear Anisotropic Parabolic Equations”, J. Differ. Equ., 275 (2021), 139–166
Panov E.Yu., “On Decay of Entropy Solutions to Nonlinear Degenerate Parabolic Equation With Almost Periodic Initial Data”, Lobachevskii J. Math., 42:5, SI (2021), 974–988
Panov E.Yu., “To the Theory of Entropy Sub-Solutions of Degenerate Nonlinear Parabolic Equations”, Math. Meth. Appl. Sci., 43:16 (2020), 9387–9404
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
Panov E.Y., “On Decay of Entropy Solutions to Multidimensional Conservation Laws”, SIAM J. Math. Anal., 52:2 (2020), 1310–1317
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
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
Panov E.Yu., “On the Dirichlet Problem for First Order Quasilinear Equations on a Manifold”, Trans Amer Math Soc, 363:5 (2011), 2393–2446
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
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
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