Consider the Vlasov--Poisson--Landau system with Coulomb potential in the
weakly collisional regime on a 3-torus, i.e. βtβF(t,x,v)+viββxiββF(t,x,v)+Eiβ(t,x)βviββF(t,x,v)=Ξ½Q(F,F)(t,x,v),E(t,x)=βΞβ1(β«R3βF(t,x,v)dvβ(2Ο)31ββ«T3ββ«R3βF(t,x,v)dvdx),β with Ξ½βͺ1. We
prove that for Ο΅>0 sufficiently small (but independent of Ξ½),
initial data which are O(ϡν1/3)-Sobolev space perturbations from
the global Maxwellians lead to global-in-time solutions which converge to the
global Maxwellians as tββ. The solutions exhibit uniform-in-Ξ½
Landau damping and enhanced dissipation.
Our main result is analogous to an earlier result of Bedrossian for the
Vlasov--Poisson--Fokker--Planck equation with the same threshold. However,
unlike in the Fokker--Planck case, the linear operator cannot be inverted
explicitly due to the complexity of the Landau collision operator. For this
reason, we develop an energy-based framework, which combines Guo's weighted
energy method with the hypocoercive energy method and the commuting vector
field method. The proof also relies on pointwise resolvent estimates for the
linearized density equation.Comment: 78 Pages. Comments welcome