Abstract
In this work, we consider the problem of the existence of solutions to the initial–boundary value problem for a diffusion model of inhomogeneous fluids.For theoretical justification, the methods of functional analysis and compactness techniques are applied, resulting in a priori estimates and energy inequalities. The formulation of the problem takes into account the compatibility of the initial and boundary conditions, as well as the boundedness and smoothness of the given coefficients. This system of differential equations describes the motion of a two-component fluid with components of different densities. A similar model of an inhomogeneous fluid medium was first obtained in [13] under the assumption of a small diffusion coefficient and low mass concentration of the admixture. A distinctive feature of these equations is that the unknown functions v and ρ (the velocity and density of the mixture) enter the system through higher-order derivatives and in a nonlinear manner. This circumstance, when studying boundary value problems, forces one to impose certain restrictions on the model parameters. However, these restrictions do not contradict the assumption of a small diffusion coefficient adopted in the derivation of the equations. It should also be noted that when λ=0 the obtained system of equations reduces to the classical model of an inhomogeneous viscous incompressible fluid (the Navier–Stokes equations).