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 Trudy Mat. Inst. Steklova, 2018, Volume 301, Pages 225–240 (Mi tm3911)

On the variational approach to systems of quasilinear conservation laws

Yu. G. Rykov

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047 Russia

Abstract: The paper contains results concerning the development of a new approach to the proof of existence theorems for generalized solutions to systems of quasilinear conservation laws. This approach is based on reducing the search for a generalized solution to analyzing extremal properties of a certain set of functionals and is referred to as a variational approach. The definition of a generalized solution can be naturally reformulated in terms of the existence of critical points for a set of functionals, which is convenient within the approach proposed. The variational representation of generalized solutions, which was earlier known for Hopf-type equations, is generalized to systems of quasilinear conservation laws. The extremal properties of the functionals corresponding to systems of conservation laws are described within the variational approach, and a strategy for proving the existence theorem is outlined. In conclusion, it is shown that the variational approach can be generalized to the two-dimensional case.

 Funding Agency Grant Number Russian Science Foundation 14-21-00025 This work is supported by the Russian Science Foundation under grant 14-21-00025.

DOI: https://doi.org/10.1134/S0371968518020176

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English version:
Proceedings of the Steklov Institute of Mathematics, 2018, 301, 213–227

Bibliographic databases:

UDC: 517.956

Citation: Yu. G. Rykov, “On the variational approach to systems of quasilinear conservation laws”, Complex analysis, mathematical physics, and applications, Collected papers, Trudy Mat. Inst. Steklova, 301, MAIK Nauka/Interperiodica, Moscow, 2018, 225–240; Proc. Steklov Inst. Math., 301 (2018), 213–227

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3911
• https://doi.org/10.1134/S0371968518020176
• http://mi.mathnet.ru/eng/tm/v301/p225

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This publication is cited in the following articles:
1. A. I. Aptekarev, Yu. G. Rykov, “Variational principle for multidimensional conservation laws and pressureless media”, Russian Math. Surveys, 74:6 (2019), 1117–1119
2. Yu. G. Rykov, “Extremal properties of the functionals connected with the systems of conservation laws”, Math. Montisnigri, 46 (2019), 21–30
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