Landau damping and the long-time collisionless limit of the Vlasov-Poisson-Landau Equation

Abstract

In this paper, we study the Vlasov-Poisson-Landau Equations on $\mathbb{T}^3\times \mathbb{R}^3$ with small collision frequency $\nu\ll 1$. We prove that for $\nu$-independent perturbations of the global Maxwellians in Gevrey-$2_-$, solutions display uniform-in-$\nu$ Landau damping and enhanced dissipation. Moreover, the collisionless limit holds, that is, as $\nu\to 0_+$ for $0<t\ll \nu^{-\frac{1}{3}}$, the $\nu > 0$ solutions converge uniformly (and in much stronger norms) to the solution of the Vlasov-Poisson equation with the same initial data. To our knowledge, this work is hence the first justification that the collisionless prediction matches those of collisional plasmas in the nonlinear equations. The interaction between Landau damping and collisions requires several new ideas: (1) an infinite-regularity commuting vector field method, merged with Guo's weighted energy methods for the Landau operator and hypocoercivity to extract the enhanced dissipation; (2) A novel nearly-physical side treatment of the collisionless Vlasov echoes; (3) A new set of decomposition methods to treat the effects of the nonlinear collisions in the Volterra equation for the density (i.e., the ``collisional echoes'') (4) A new quasi-linearization method for treating the effect of the slowly evolving homogeneous modes over long times. As a side result, we also prove Landau damping and enhanced dissipation of $O(\epsilon\nu^{1/3})$ Sobolev-space perturbations of homogeneous distributions that are only $O(\epsilon)$ perturbations of global Maxwellians, generalizing the recent results of Chaturvedi, Luk, and Nguyen. As another side result, our methods also provide a nearly-completely physical-side proof of Mouhot and Villani's theorem in the full range of Gevrey-$3_-$.