Solvability of the Initial-Boundary Value Problem for the Kelvin–voigt Fluid Motion Model With Variable Density

Abstract

Summary. In the paper the solvability of the initial-boundary value problem for the Kelvin–Voigt fluid motion model with variable density is investigated. First, using the Laplace transform, from the rheological relation for the Kelvin–Voigt fluid motion model and the fluid motion equation in the Cauchy form, a system of equations that describes the motion of the Kelvin–Voigt model with variable density is obtained. For the resulting system of equations, an initial-boundary value problem is posed, a definition of its weak solution is given, and its existence is proved. The proof is carried out on the basis of an approximation-topological approach to the study of fluid dynamic problems. Namely, the problem approximating the original one is considered and its solvability is proved on the basis of one version of the Leray-Schauder theorem. Then, on the basis of a priori estimates, it is proved that from the sequence of solutions of the approximation problem it is possible to extract a subsequence that weakly converges to the solutions of the original problem.