Unique Strong Solvability of the Initial Boundary Value Problem for Inhomogeneous Incompressible Kelvin–Voigt Fluid Model

The existence and uniqueness of a strong solution for an inhomogeneous incompressible Kelvin–Voigt fluid motion model is proved. It is not assumed that the initial value of the fluid density is separated from zero. To prove the existence of a solution, an approximation problem is considered, its solvability is proved, and strong a priori estimates independent of the approximation parameter are established for its solutions. After that, passing to the limit as the approximation parameter tends to zero, we show that the solutions of the approximation problem converge to a strong solution of the original problem as the approximation parameter tends to zero. The uniqueness of the solution is established using the Gronwall–Bellman inequality.