Asymptotic Stability of Equilibria for Screened Vlasov–Poisson Systems via Pointwise Dispersive Estimates

Abstract

We revisit the proof of Landau damping near stable homogenous equilibria of Vlasov–Poisson systems with screened interactions in the whole space $\mathbb {R}^d$ (for $d\ge 3$) that was first established by Bedrossian, Masmoudi and Mouhot in [5]. Our proof follows a Lagrangian approach and relies on precise pointwise in time dispersive estimates in the physical space for the linearized problem that should be of independent interest. This allows to cut down the smoothness of the initial data required in [5] (roughly, we only need Lipschitz regularity). Moreover, the time decay estimates we prove are essentially sharp, being the same as those for free transport, up to a logarithmic correction.

Appendix A

We recall (1.8) for the Littlewood–Paley decomposition in $\mathbb {R}^n$ . In the paper, we use it (for $n=d$ and for $n= d+1$). Let us state the classical Bernstein Lemma.

Lemma A.1

For every $p \in [1, + \infty ]$ and any multi-index $\alpha$, there exist $c>0$, $C>0$ such that for every $u \in L^p$, we have Bernstein’s inequalities:

$$\begin{aligned} c2^{ |\alpha | q}\Vert u_{q}\Vert _{L^p} \le \Vert \partial ^\alpha (u_q )\Vert _{L^p} \le C 2^{ |\alpha | q}\Vert u_{q} \Vert _{L^p}, \quad \forall q \in \mathbb {Z}. \end{aligned}$$

(A.1)

We refer for example to [1, Chapter 2, Lemma 2.1] for the proof. As an application, we get

Lemma A.2

Let $P_{1}$ and $P_{2}$ be homogeneous polynomials of degree 1 and 2. For all $p \in [1,+\infty ]$, for all $u \in L^p$, for all $\ell \in \mathbb {N}$,

$$\begin{aligned} \Vert P_{1}(D) ( 1 - \Delta )^{-1} u \Vert _{L^p}&\lesssim \Vert u \Vert _{L^p}, \end{aligned}$$

(A.2)

$$\begin{aligned} \Vert P_{2} (D) ( 1 - \Delta )^{-1} u_\ell \Vert _{L^p}&\lesssim 2^{\ell \delta } \Vert u_{\ell } \Vert _{L^p}, \end{aligned}$$

(A.3)

where $u_\ell$ is defined as in (1.8) and $\delta \in (0,1)$ is arbitrarily small.

Proof

By using the homogeneous Littlewood–Paley decomposition and the Bernstein inequalities, we get

$$\begin{aligned} \Vert P_{1}(D) ( 1 - \Delta )^{-1} u \Vert _{L^p} \lesssim \sum _{q \in \mathbb {Z}} { 2 ^{q} \over 1 + 2^{2 q} } \Vert u_{q}\Vert _{L^p} \lesssim \Vert u \Vert _{L^p} (\sum _{q \le 0} 2^q + \sum _{q \ge 0} 2^{-q}). \end{aligned}$$

For the second estimate, we write

$$\begin{aligned} \begin{aligned} \Vert P_{2}(D) ( 1 - \Delta )^{-1} w \Vert _{L^p}&\lesssim \sum _{q \in \mathbb {Z}} { 2 ^{2q} \over 1 + 2^{2 q} } \Vert w_{q}\Vert _{L^p}\\&\lesssim \Vert w\Vert _{L^p} \sum _{q < 0} 2^{2q} + \sup _{q \ge 0} 2^{q \delta } \Vert w_{q}\Vert _{L^p} \sum _{q \ge 0} 2^{-q \delta } \end{aligned} \end{aligned}$$

and apply it for $w= u_\ell$, which ends the proof. $\square$