3D unsteady Navier-Stokes problem with memory and subdifferential boundary condition

We consider a mathematical model which describes the motion of a 3D unsteady fluid flow governed by Navier-Stokes system, and subjected to mixed boundary conditions with a given velocity on one part of the boundary and nonlinear slip conditions with a memory term reminiscent of Coulomb’s friction law on the other part. We establish first some regularity properties and estimates for a simplified model. Then we prove the existence of a solution to our problem by using a successive approximation technique and compactness arguments based on Helly’s theorem for the velocity field.