Existence and Decay of Polymeric Flows

We study the existence and decay of solutions to kinetic models of incompressible polymeric flow. We consider dumbbell type models in the case when the drag term is co-rotational and weak solutions are constructed via a Leray-type approximation. We analyze the decay when the space of elongations is bounded, and the spatial domain of the polymer is either a bounded domain Ω ⊂ Rn, n = 2, 3 or it is the whole space R3. The decay is first established for the L2 norm of the probability density function ψ and then this decay is used to obtain L2-decay of the velocity field u. Consideration is also given to solutions where the probability density function is radial in the admissible elongation vectors q. In this case, the velocity field u becomes a solution to the incompressible Navier–Stokes equations, and thus decay follows from known results for the Navier–Stokes equations.