Abstract: Understanding the long-time behavior of fluid flows by the mathematical analysis of Navier-Stokes equations is an important and challenging problem. For the case of periodic boundary conditions and potential body forces, we survey the theory of asymptotic expansions for the solutions as time tends to infinity, and its associated normal form. By an explicit change of variable which is the formal inverse of the normalization map, this normal form was proved to be of Poincaré-Dulac type in a suitable infinite dimensional topological vector space. We then extend the asymptotic expansion result to the case of non-potential forces which decay exponentially in time. We emphasize the Gevrey-norm techniques which yield a concise and direct proof, and significantly simplify the conditions on the forces.
Title: On the theory of asymptotic expansions and normal form for Navier-Stokes equations