On the Linearized Vlasov–Poisson System on the Whole Space Around Stable Homogeneous Equilibria

Abstract

We study the linearized Vlasov–Poisson system around suitably stable homogeneous equilibria on ${\mathbb {R}}^d\times {\mathbb {R}}^d$ (for any $d \ge 1$) and establish dispersive $L^\infty$ decay estimates in the physical space.

Appendix: Radial Decreasing Equilibria Satisfy the Stability Assumption (H2)

In this section we shall prove

Proposition 8.1

Let $\mu$ satisfy (1.2) and (1.4). If $\mu (v) = F\left( { |v|^2 \over 2} \right)$ with $F'(s) <0$, $\forall s \ge 0$, then (H2) is verified.

Proof

We study the function $\tau \mapsto 1-m_{KE}(i \tau , \eta )$ for $\eta \in {\mathbb {S}}^{d-1}.$ By using (4.9), we get that $m_{KE}(i\tau , \eta ) \rightarrow 0$ when $| \tau |$ tends to $+\infty$, so that it suffices to study $\tau \mapsto 1-m_{KE}(i \tau , \eta )$ for bounded $\tau$. We have for $\gamma >0$, $|\eta |= 1$,

$$\begin{aligned} m_{KE}(z, \eta )= & {} - \int _{0}^{+ \infty } e^{-(\gamma + i \tau )s}\, i \eta \cdot \sum _{k,l} \eta _k \eta _l {\mathcal {F}}_v( v_k v_l \nabla _v \mu )(\eta s) \, d s, \\= & {} - i \int _{0}^{+ \infty } \int _{{\mathbb {R}}^d} e^{ -( \gamma + i \tau + i \eta \cdot v) t} (\eta \cdot v)^3 F'\left( {|v|^2 \over 2}\right) \, dv dt. \end{aligned}$$

We then write $v= u \eta + w$ with $w \in \eta ^{\perp }= H_{\eta }$ so that

$$\begin{aligned} m_{KE}(z, \eta )= - i \int _{0}^{+ \infty } \int _{{\mathbb {R}}} e^{ -( \gamma + i \tau + i u) t} u^3 \Phi '\left( { u^2 \over 2} \right) \, du dt \end{aligned}$$

where

$$\begin{aligned} \Phi (s)= \int _{H_{\eta }} F \left( s + {|w|^2 \over 2} \right) \, dw. \end{aligned}$$

(8.1)

This yields

$$\begin{aligned} m_{KE}(z, \eta ) = - \int _{{\mathbb {R}}} { \tau + u \over \gamma ^2 + (\tau + u)^2} u^3 \Phi '\left( { u^2 \over 2} \right) \, du - i \gamma \int _{{\mathbb {R}}} { u^3 \over \gamma ^2 + (\tau + u)^2} \Phi '\left( { u^2 \over 2} \right) \, du. \end{aligned}$$

Taking the limit $\gamma \rightarrow 0$ (following e.g. [17, Proof of Prop. 2.1]), we get that

$$\begin{aligned} m_{KE}(i\tau , \eta ) = - \text{ p.v. } \int _{{\mathbb {R}}} {u^3\Phi '\left( { u^2 \over 2} \right) \over \tau + u} \, du - i\pi \tau ^3 \Phi '\left( { \tau ^2 \over 2} \right) . \end{aligned}$$

We then observe that for bounded $\tau$ the imaginary part vanishes only for $\tau =0$ and in this case the real part is equal to

$$\begin{aligned} - \int _{{\mathbb {R}}} u^2 \Phi ' \left( {u^2 \over 2}\right) \, du= \int _{{\mathbb {R}}} \int _{{\mathbb {R}}^{d-1}} F\left( {u^2 + |w|^2 \over 2}\right) \, dw du= \int _{{\mathbb {R}}^d} \mu dv= 1. \end{aligned}$$

Therefore $1 - m_{KE}(i\tau , \eta )$ vanishes only for $\tau = 0$.

Let us now compute $\partial _{z}m_{KE}(0, \eta )$ and $\partial _{z}^2 m_{KE}(0, \eta )$. Following the same lines, we first get that

$$\begin{aligned} \partial _{z}m_{KE}(z, \eta )= & {} i \int _{0}^{+ \infty } \int _{{\mathbb {R}}} t e^{ -( \gamma + i \tau + i u) t} u^3 \Phi '\left( { u^2 \over 2} \right) \, du dt \\= & {} \int _{0}^{ + \infty } \int _{{\mathbb {R}}}e^{ -( \gamma + i \tau + i u) t} \partial _{u}\left( u^3 \Phi '\left( { u^2 \over 2} \right) \right) \, du dt \end{aligned}$$

and therefore

$$\begin{aligned} \partial _{z}m_{KE}(z, \eta )= & {} - i \int _{{\mathbb {R}}} { \tau + u \over \gamma ^2 + (\tau + u)^2} \partial _{u}\left( u^3 \Phi '\left( { u^2 \over 2} \right) \right) du \\&+ \gamma \int _{{\mathbb {R}}} { 1 \over \gamma ^2 + (\tau + u)^2} \partial _{u}\left( u^3 \Phi '\left( { u^2 \over 2} \right) \right) \, du. \end{aligned}$$

Taking the limit $\gamma \rightarrow 0$ as before, we get

$$\begin{aligned} \partial _{z}m_{KE}(0, \eta ) = - i \int _{{\mathbb {R}}} { 1 \over u } \partial _{u}\left( u^3 \Phi '\left( { u^2 \over 2} \right) \right) \, du. \end{aligned}$$

By integrating by parts as before, this yields

$$\begin{aligned} \partial _{z}m_{KE}(0, \eta ) = i \int _{{\mathbb {R}}} u \Phi '\left( { u^2 \over 2} \right) \, du = 0 . \end{aligned}$$

Finally, for $\partial _{z}^2m_{KE}(0, \eta )$, we have

$$\begin{aligned} \partial _{z}^2m_{KE}(z, \eta )= & {} - \int _{0}^{ + \infty } \int _{{\mathbb {R}}}e^{ -( \gamma + i \tau + i u) t} t \partial _{u}\left( u^3 \Phi ' \left( { u^2 \over 2} \right) \right) \, du dt \\= & {} i \int _{0}^{ + \infty } \int _{{\mathbb {R}}}e^{ -( \gamma + i \tau + i u) t} t \partial _{u}^2\left( u^3 \Phi '\left( { u^2 \over 2} \right) \right) \, du dt. \end{aligned}$$

This yields as before

$$\begin{aligned} \partial _{z}^2m_{KE}(0, \eta )= & {} - \int _{{\mathbb {R}}} { 1 \over u} \partial _{u}^2\left( u^3 \Phi ' \left( { u^2 \over 2} \right) \right) \, du= \int _{{\mathbb {R}}} { 1 \over u^2} \partial _{u}\left( u^3 \Phi ' \left( { u^2 \over 2} \right) \right) du\\= & {} 2 \int _{{\mathbb {R}}} \Phi ' \left( { u^2 \over 2} \right) du. \end{aligned}$$

By using the definition (8.1), we thus get that

$$\begin{aligned} \partial _{z}^2m_{KE}(0, \eta ) = 2 \int _{{\mathbb {R}}^d} F'\left( { |v|^2 \over 2} \right) dv \ne 0 \end{aligned}$$

and the proof of the proposition is complete. $\quad \square$