Existence and Uniqueness Theorem for a Weak Solution to the Initial-Boundary Value Problem for the Kelvin–Voigt Mathematical Model of the Motion of a Mixture of Fluids

Source abstract

Equations describing three-dimensional unsteady motions of Kelvin–Voigt mixture of viscous incompressible liquids are considered. The theorem of existence and uniqueness of a weak solution of an initial boundary value problem corresponding to flows in a limited domain is proved. To prove the existence of a solution, an approximation problem is considered, its solvability and a priori estimates independent of the approximation parameter are established. After that, a transition is made to the limit on the approximation parameter and it is shown that the solutions of the approximation problem weakly converge to the solution of the original problem. The uniqueness of the solution is established using the Gronwall inequality.