UNIQUE STRONG SOLVABILITY OF THE INITIAL BOUNDARY VALUE PROBLEM FOR THE INHOMOGENEOUS INCOMPRESSIBLE KELVIN–VOIGT FLUID MODEL

The paper proves the existence and uniqueness theorem of a strong solution for a inhomogeneous incompressible Kelvin-Voigt fluid motion model. It is not assumed that the initial condition for the fluid density is separated from zero. To prove the existence of a solution, an approximation problem is considered, its solvability and strong a priori estimates for its solutions, independent of the approximation parameter, are established. After that, a passage to the limit is carried out as the approximation parameter tends to zero and it is shown that solutions to 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.