Decay of polymer equations and Poincare estiamtes

We study the decay and existence of solutions to some equations modeling polymeric flow. We consider the case when the drag term is corotational. We analyse the decay when when the space of elongations is bounded, and the spatial domain of the polymer is either a bounded domain $\Omega \subset \rn,\;\;n=2,3$ or the domain is the whole space $\rn,\;\;n=2,3$. The decay is first established for the probability density $\psi$ and then this decay is used to obtain decay of the velocity $u$. Consideration also is given to solutions where the probability density is radial in the admissible elongation vectors $q$. In this case the velocity $u$, will become a solution to Navier-Stokes equation, and thus decay follows from known results for the Navier-Stokes equations.

Some questions in relation to Poincaré type inequalities, and fluid equations in general, will be discussed