Solvability of an Initial–Boundary Value Problem for the Modified Kelvin–Voigt Model with Memory along Fluid Motion Trajectories

The paper deals with proving the weak solvability of an initial–boundary value problem forthe modified Kelvin–Voigt model taking into account memory along the trajectories of motion offluid particles. To this end, we consider an approximation problem whose solvability is establishedwith the use of the Leray–Schauder fixed point theorem. Then, based on a priori estimates, weshow that the sequence of solutions of the approximation problem has a subsequence that weaklyconverges to the solution of the original problem as the approximation parameter tends to zero.