CONDIONALLY CORRECT INITIAL VALUES PROBLEM FOR THE SYSTEM OF QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER

Abstract

In the theory of systems of quasilinear partial differential equations of the first order the main questions are the solvability of initial values problem and justification of the approximate methods.In this paper a class of systems of quasilinear partial differential equations of the first order is considered, for which the concept of a generalized solution is introduced. The approach of introducing a generalized solution of conditionally correct problem related to the classical Cauchy problem is also proposed. In the case of smooth initial functions, the generalized solution at least locally coincides with the classical solution, and in the general case, such a solution loses its smoothness and turns into a singular function.There is a class of problems that have a unique solution. Estimates of the convergence speed in approximate methods of solving such problems have been obtained. Approximate methods in this approach can be arbitrary in nature, for example, finite-difference methods, smoothing method, or viscosity method. A description of constructions for building an infinitenumber of solutions for the same class of such problems is also given.