The optimal control design of flows described by the Navier-Stokes equations has been and still is an active area of research. For problems with an infinite time horizon, i.e. stabilization problems, methods are often based on Riccati equations arising for the linearized system. The resulting feedback laws can be interpreted as an approximation of the value function to the lowest order. Higher order approximations are tied to differentiability properties of the value function. As presented in the talk, in an appropriate functional analytic setup, the value function can be shown to be smooth. This yields a way of computing higher order feedback laws without solving the Hamilton-Jacobi-Bellman equation. Theoretical and numerical challenges such as error estimates for the feedback controls as well as a computational realization are discussed.