Solvability of initial-boundary value problem for the modified Kelvin–Voigt model with memory along trajectories of fluid motion

The work is devoted to proving the solvability in the weak sense of the initial-boundary value problem for the modified Kelvin–Voigt model taking into account memory along the trajectories of fluid particles motion. For this, an approximation problem is considered for which solvability is established based on the Leray–Schauder fixed point theorem. Then, based on a priori estimates, it is shown that from a sequence of solutions to the approximation problem, one can extract a subsequence that weakly converges to the solution of the original problem as the approximation parameter tends to zero.