About the Vlasov Dirac equation

This is a report on a ongoing work with Anne Nouri from Marseille and Thierry Paul from Polytechnique Palaiseau.

I dub the equation $${\partial }_{t}f+v\cdot {\partial }_{x}f-{\partial }_x(\int f(x,w,t)dw){\partial }_{v}f=0$$ Vlasov Dirac because it is a Vlasov type equation where the standard potential has been replaced by the Dirac mass. This turns out to be an important problem for different reasons.

1 It is genuine model for numerical simulation of plasmas.
2 It exhibit some singular stabilities $\mathrm{\setminus }$ unstabilities properties which can be more easily analyzed than the standard instabilities (Landau Damping and so on..) for the original Vlasov equation.
3 It is at the cross road between mean field derivations and WKB or Wigner asymptotics.