Weak solvability of the initial-boundary value problem for a finite-order model of the inhomogeneous incompressible Kelvin-Voigt fluid without a positive lower bound on the initial condition of fluid density

In this article we investigate the weak solvability of the initial-boundary value problem for an inhomogeneous incompressible Kelvin-Voigt fluid motion model of an arbitrary finite-order in 2D and 3D cases. We do not assume that the initial condition for density is separable from zero. That is, only the non-negativity condition for the density is supposed. To prove the existence theorem, we consider some approximation problem and establish its solvability by using the Leray-Schauder theorem. Then, on the basis of a priori estimates, we go to the limit and show that the solutions of the approximation problem weakly converge to the solution of the original problem as the approximation parameter tends to zero.