We shall discuss the problem of the 'trend to equilibrium' for a degenerate kinetic linear Fokker-Planck equation. The linear equation is assumed to be degenerate on a subregion of non-zero Lebesgue measure in the physical space (i.e., the equation is just a transport equation with a Hamiltonian structure in the subregion). We shall give necessary and sufficient geometric condition on the region of degeneracy which guarantees the exponential decay of the semigroup generated by the degenerate kinetic equation towards a global Maxwellian equilibrium in a weighted Hilbert space. The approach is strongly influenced by C. Villani's strategy of 'Hypocoercivity' from Kinetic theory and the 'Bardos-Lebeau-Rauch' geometric condition from Control theory. This is a joint work with Frederic Herau and Clement Mouhot.
- PDE CDT Lunchtime Seminar