ON THE REGULARIZATION OF EQUATIONS OF THE MECHANICS OF MIXTURES OF VISCOUS COMPRESSIBLE FLUIDS

Abstract

Mathematical models of multi-velocity continua, through which the motion of multicomponent mixtures are described, represent a rather extensive area of modern mechanics and mathematics. Mathematical results (statements of problems, theorems on the existence and uniqueness, properties of solutions, etc.) for such models are rather modest in comparison with the results for classical single-phase media. The present paper aims to fill this gap in some extent and is devoted to investigating the global correctness of the boundary value problem for a nonlinear system of differential equations, which is some regularity of the mathematical model of nonstationary spatial flows of a mixture of viscous compressible fluids. Construction of the solution of the problem considered in this article is a key step for the mathematical analysis of the initial model of the mixture, since it allows to obtain globally defined solutions of the latter by means of a limiting transition and, in addition, the proposed algorithm for constructing solutions to the regularized problem is practical. This algorithm is based on the finite-dimensional approximation procedure for an infinite-dimensional problem, and therefore a mathematically grounded algorithm for the numerical solution of the boundary value problem of the motion of a mixture of viscous compressible fluids in a region bounded by solid walls can be constructed on this basis. The local in time solvability of finite- dimensional problems is proved by applying the principle of contracting mappings and the local solution can be extended to an arbitrary time interval with the help of a priori estimates.