Solvability of the initial-boundary value problem for the Kelvin–Voigt fluid motion model with variable density

The solvability of the initial-boundary value problem for the Kelvin–Voigt fluid motion model with a 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, we derive a system of equations that describes the fluid motion in the Kelvin–Voigt model with a variable density. 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 based on an approximation-topological approach to the study of fluid dynamic problems. Namely, the original problem is approximated by another one, whose solvability is proved using a 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 solution of the original problem.