On the variational approach to systems of quasilinear conservation laws

The theory of quasi-linear systems of conservation laws in the modern version began to develop in the second half of the last century. However, despite a number of impressive achievements, a fairly complete theory, including the multidimensional case, was built for only one conservation law. In the case of systems, fairly general results are obtained for only one spatial variable and, as a rule, under the assumption that the change of at least unknown functions is small. By expanding the concept of solutions, it was possible to find a proof of sufficiently general theorems of the existence of generalized solutions to systems of two conservation laws (one spatial variable), however, the developed technique as a whole cannot be extended even to systems of three conservation laws with one spatial variable. Accordingly, there is an assumption that the main methodologies used, namely, the low viscosity method and the method of constructing approximate solutions are insufficient. The paper offers an alternative view on the nature of quasi-linear conservation laws based on a variational representation for generalized solutions of quasi-linear systems of conservation laws. Two such representations are discussed: 1) based on the generalization of known results (starting with the works of E. Hopf) on the variational representation of solutions for one equation; 2) based on the representation of generalized solutions as functionals on the trajectory space.