2012 Young Researchers Workshop:
Kinetic Description of Multiscale Phenomena

A Vlasov Equation with Dirac Potential Used in Fusion Plasmas

Claude Bardos

ENS-Cachan

This talk is devoted to the equation:
In plasmas physics it corresponds to a Vlasov equation with a Dirac potent. It is also know as an approximation for water waves with the name of Benney equation. Following a joint work with A. Nouri well-posedness of the Cauchy problem is analyzed for a singular Vlasov equation governing the evolution of the ionic distribution function of a quasineutral fusion plasma. The Penrose criterium is adapted to the linearized problem around a time and space homogeneous distribution function showing (due to the singularity) more drastic dierences between stable and unstable situations. This pathology appears on the full non linear problem, well-posed locally in time with analytic initial data, but generally ill-posed in the Hadamard sense. Eventually with a very dierent class of solutions, mono-kinetic, which constrains the structure of the density distribution, the problem becomes locally in time well-posed. And in this last case with WKB asymptotic (cf Grenier) or inverse scattering (Jin Levermore McLaughlin and Zakharov) a remarkable link can be made between this equation and the non linear Schrordinger equation.