On the existence of strong solutions to regularized equations of viscous compressible fluid mixtures

Mathematical results (statements of problems, theorems on the existence and uniqueness, properties of solutions, etc.) on models of multi-velocity continuums by which the motion of multicomponent mixtures is described are rather modest in comparison with results on classical one-component media. This paper aims to fill this gap to some extent and is devoted to the study of the global correctness of the initial boundary value problem for equations of unsteady spatial motions of a mixture of viscous compressible fluids.
This paper is the first part of an extensive research that studies regularization of a mathematical model for unsteady spatial flows of a viscous compressible fluids mixture. The construction of the solution of the regularized problem is the key step for the mathematical analysis of the original mixture model because it allows to obtain globally defined solutions of the latter by passing to the limit. In addition, the proposed algorithm for constructing solutions to the regularized problem is constructive. This algorithm is based on the procedure of finitedimensional approximation of the infinite-dimensional problem and the result is a mathematically well-grounded algorithm for the numerical solution of the boundary value problem of the motion of a viscous compressible fluids mixture in the region bounded by solid walls. The local time solvability of finite-dimensional problems is proved by applying the contraction mapping principle. With the help of a priori estimates, the possibility to extend the local solution for an arbitrary period of time, as well as the possibility of a passage to the limit to an infinitedimensional problem is established. Finally, we obtain the result on the existence and uniqueness of a globally defined strong solution to the regularized problem.