Kinetic and fluid equations with stiff potentials confining to a domain

In this talk, we consider two models involving extremely strong potentials , the overdamped Langevin and kinetic Fokker-Planck equations.
I will present some results on the long time behavior and both on the hydrodynamic limit and on the limit when the potential becomes stiff (confining the fluid to a domain and letting boundary conditions appear). 
This work is motivated by the kinetic derivation of immiscible fluid equations and can be viewed as a very first step. 
This is an ongoing work with Frédéric Hérau and Hélène Mathis.