Large-Box Transition in Nonlinear Landau Damping for the Vlasov--Yukawa Equation

Abstract

We study nonlinear Landau damping for the Vlasov--Yukawa equation in the large-box regime, as a periodic box of length $2πL$ expands to the whole space. First, for $d\ge 3$, we prove global stability and scattering in an $L$-dependent Gevrey class, with estimates uniform for $L\ge L_0$. The density obeys a two-regime upper bound: the whole-space decay rate $\langle t\rangle^{-d}$ before the critical time scale $T_{\rm disp}(L)\sim L$, and a periodic phase-mixing bound in the rescaled time $t/L$ thereafter. Second, to quantify the nonlinear resonance effect, we identify an echo-Volterra operator governing the nonlinear density memory, and provide sharp estimates for its non-collinear and collinear parts. The non-collinear time-frequency interactions remain uniformly bounded, whereas the collinear interactions do not become comparable to the background until the much later time scale $T_{\rm col}(L)\sim L^{d-1}$. Third, we investigate the stability and convergence of Sobolev initial data. For data of size $\varepsilon$, polynomial Sobolev smallness persists up to the time scale $\varepsilon^{-1} T_{\rm col}(L)$. Within the shorter time scale $T_{\rm disp}(L)\sim L$, compatible periodic solutions converge locally, as $L\to\infty$, to a global solution of the whole-space Vlasov--Yukawa equation.