Abstract <p>The paper establishes the solvability in the weak sense of the initial-boundary value problem for the second-order Kelvin–Voigt model with smoothed Jaumann time derivative taking into account the memory of fluid motion. For the proof, a problem approximating the original one is considered, and its solvability is established based on a priori estimates of solutions and the Leray–Schauder degree theory. After that, the limit transition is carried out as the approximation parameter tends to zero, and it is shown that the solutions to the approximation problem weakly converge to the solution to the original problem.</p>