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.
\Bibitem{Ryk18}
\by Yu.~G.~Rykov
\paper On the variational approach to systems of quasilinear conservation laws
\inbook Complex analysis, mathematical physics, and applications
\bookinfo Collected papers
\serial Trudy Mat. Inst. Steklova
\yr 2018
\vol 301
\pages 225--240
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm3911}
\crossref{https://doi.org/10.1134/S0371968518020176}
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\elib{https://elibrary.ru/item.asp?id=35246353}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2018
\vol 301
\pages 213--227
\crossref{https://doi.org/10.1134/S008154381804017X}
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Linking options:
https://www.mathnet.ru/eng/tm3911
https://doi.org/10.1134/S0371968518020176
https://www.mathnet.ru/eng/tm/v301/p225
This publication is cited in the following 5 articles:
Yu. G. Rykov, “Variational Formulation of the Problem of Finding Generalized Solutions for Quasilinear Hyperbolic Systems of Conservation Laws”, Math. Notes, 110:6 (2021), 972–975
Yu. G. Rykov, “On the systems of conservation laws and on a new way to construct for them neural networks algorithms”, Lobachevskii J. Math., 42:11, SI (2021), 2645–2653
A. I. Aptekarev, Yu. G. Rykov, “Variational principle for multidimensional conservation laws and pressureless media”, Russian Math. Surveys, 74:6 (2019), 1117–1119
Yu. G. Rykov, “Extremal properties of the functionals connected with the systems of conservation laws”, Math. Montisnigri, 46 (2019), 21–30
Yuri Germanovich Rykov, “Multi-D variational principle and pressureless gas dynamics”, KIAM Prepr., 2018, no. 80, 1
Citation in format AMSBIB
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