Geodesic
On the variational approach to systems of quasilinear conservation laws
Informatics and Automation, Complex analysis, mathematical physics, and applications, Tome 301 (2018), pp. 225-240

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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. 
@article{TRSPY_2018_301_a16,
     author = {Yu. G. Rykov},
     title = {On the variational approach to systems of quasilinear conservation laws},
     journal = {Informatics and Automation},
     pages = {225--240},
     publisher = {mathdoc},
     volume = {301},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2018_301_a16/}
}
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Yu. G. Rykov. On the variational approach to systems of quasilinear conservation laws. Informatics and Automation, Complex analysis, mathematical physics, and applications, Tome 301 (2018), pp. 225-240. http://geodesic.mathdoc.fr/item/TRSPY_2018_301_a16/