Geodesic
On the variational approach to systems of quasilinear conservation laws
Trudy Matematicheskogo Instituta imeni V.A. Steklova, 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. 
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Yu. G. Rykov. On the variational approach to systems of quasilinear conservation laws. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Complex analysis, mathematical physics, and applications, Tome 301 (2018), pp. 225-240. http://geodesic.mathdoc.fr/item/TM_2018_301_a16/

[1] Aptekarev A. I., Rykov Yu. G., “On the variational representation of solutions to some quasilinear equations and systems of hyperbolic type on the basis of potential theory”, Russ. J. Math. Phys., 13:1 (2006), 4–12 | DOI | MR | Zbl

[2] Bressan A., Hyperbolic systems of conservation laws: The one-dimensional Cauchy problem, Oxford Univ. Press, Oxford, 2000 | MR | Zbl

[3] DiPerna R. J., “Measure-valued solutions to conservation laws”, Arch. Ration. Mech. Anal., 88:3 (1985), 223–270 | DOI | MR | Zbl

[4] E W., Rykov Yu. G., Sinai Ya. G., “Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics”, Commun. Math. Phys., 177:2 (1996), 349–380 | DOI | MR | Zbl

[5] Fjordholm U. S., Mishra S., Tadmor E., “On the computation of measure-valued solutions”, Acta numer., 25 (2016), 567–679 | DOI | MR | Zbl

[6] Glimm J., “Solutions in the large for nonlinear hyperbolic systems of equations”, Commun. Pure Appl. Math., 18:4 (1965), 697–715 | DOI | MR | Zbl

[7] S. K. Godunov, “A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics”, Mat. Sb., 47(89):3 (1959), 271–306 | MR | Zbl

[8] Hopf E., “The partial differential equation $u_t+uu_x=\mu u_{xx}$”, Commun. Pure Appl. Math., 3 (1950), 201–230 | DOI | MR | Zbl

[9] Keyfitz B. L., “Singular shocks, retrospective and prospective”, Confluentes math., 3:3 (2011), 445–470 | DOI | MR | Zbl

[10] Keyfitz B. L., Kranzer H. C., “A viscosity approximation to a system of conservation laws with no classical Riemann solution”, Nonlinear hyperbolic problems, Proc. Workshop (Bordeaux, 1988), Lect. Notes Math., 1402, eds. C. Carasso et al., Springer, Berlin, 1989, 185–197 | DOI | MR

[11] S. N. Kruzhkov, “First order quasilinear equations in several independent variables”, Math. USSR-Sb., 10:2 (1970), 217–243 | DOI | MR | Zbl | Zbl

[12] Lax P. D., “Hyperbolic systems of conservation laws. II”, Commun. Pure Appl. Math., 10:4 (1957), 537–566 | DOI | MR | Zbl

[13] Lax P. D., Hyperbolic partial differential equations, Courant Lect. Notes Math., 14, Amer. Math. Soc., Providence, RI, 2006 | DOI | MR | Zbl

[14] Miroshnikov A., Young R., “Weak$^*$ solutions. I: A new perspective on solutions to systems of conservation laws”, Methods Appl. Anal., 24:3 (2017), 351–382 ; arXiv: 1511.02579[math.AP] | MR

[15] O. A. Oleinik, “The Cauchy problem for nonlinear differential equations of the first order with discontinuous initial conditions”, Tr. Mosk. Mat. Obshch., 5, 1956, 433–454 | MR | Zbl

[16] Yu. G. Rykov, A variational representation of generalized solutions to quasilinear hyperbolic systems and possible algorithms for hybrid supercomputers, Preprint No. 62, Keldysh Inst. Appl. Math., Moscow, 2011

[17] Yu. G. Rykov, 2D pressureless gas dynamics and variational principle, Preprint No. 94, Keldysh Inst. Appl. Math., Moscow, 2016

[18] Yu. G. Rykov, O. B. Feodoritova, “Systems of quasilinear conservation laws and algorithmization of variational principles”, Comput. Math. Math. Phys., 55:9 (2015), 1554–1566 | DOI | DOI | MR | Zbl

[19] Sever M., Distribution solutions of nonlinear systems of conservation laws, Mem. Amer. Math. Soc., 190, Amer. Math. Soc., Providence, RI, 2007 | MR

[20] Tadmor E., Variational formulation of entropy solutions for nonlinear conservation laws: Lecture, Joint Math. Meet. Available at , Baltimore, MD, Jan. 2014 http://www.cscamm.umd.edu/tadmor/

[21] A. I. Vol'pert, “The spaces $BV$ and quasilinear equations”, Math. USSR-Sb., 2:2 (1967), 225–267 | DOI | MR | Zbl