Large time concentrations for solutions to kinetic equations with energy dissipation

We consider the solutions to a kinetic equation which kinetic energy converges to zero fast enough. We prove that they concentrate near the speed zero and converge towards a measure which is a product of a measure on the spacial coordinates and a Dirac mass on the speed coordinates. The difficult point here is that the full solution converges since we do not know any characterisation of the limit problem for the spatial density. We give two results of this kind, depending on the regularity of the solution, and on the assumptions. Finally we present an example of equation which describes the interactions of particles in a flow and where these theorems apply.