Domain shape influence on the solution of the problem about the flow of a mixture of compressible viscous fluids around an obstacle

In this paper, it is studied how the solution of the boundary value problem for the motion equations of a mixture of compressible viscous fluids depends on the shape of the domain. Such a problem arises in connection with the problem of searching for the optimum shape of the obstacle which is flown around by a stream of the mixture. The solution is reduced to studying the dependence of solutions of a nonlinear system of compound-type partial differential equations on the matrix setting the deformation of the domain. Properties of coefficients of the linear system obtained for a difference of two possible solutions (corresponding to two different matrices) allow one to construct only its very weak solutions. Therefore, there appears the necessity to consider the conjugate problem and to construct its solutions (weak and strong ones). The basic results of the work are estimations allowing one to assert that the mapping associating the solution of the abovementioned boundary value problem to the matrix is a Lipschitz mapping. In particular, this implies the uniqueness of the solution of the inhomogeneous boundary value problem for the initial system of equations. On the basis of the obtained results, differentiability of the functional reflecting the drag force of the streamlined obstacle can be established, as well as an explicit formula representing the derivative of the functional.