ON A MODEL OF THE MOTION OF A VISCOELASTIC FLUID WITH WALL SLIP
This paper is concerned with a mathematical model describing unsteady flows of an incompressible viscoelastic fluid of the Oldroyd kind in a three-dimensional bounded domain with solid impermeable boundaries. In the constitutive law, we use the total derivative. An important feature of the problem under consideration is that Navier’s slip law (the free-slip condition) at the solid surface is used instead of the classical no-slip condition. This law gives a relation between the slip velocity and the force acting on the fluid from the interphase boundary of liquid-solid. For the corresponding evolution problem we introduce the concept of generalized solutions in a class of suitable function spaces. The paper discusses an algorithm for finding solutions by the Faedo–Galerkin method. Our main result is an analogue of the Sather–Serrin theorem on the uniqueness, regularity and energy estimates of global generalized solutions.