Modified Scattering for Small Data Solutions to the Vlasov

Abstract

We prove that for any global solution to the Vlasov–Maxwell system arising from compactly supported data, and such that the electromagnetic field decays fast enough, the distribution function exhibits a modified scattering dynamic. In particular, our result applies to small data solutions constructed by Glassey and Strauss.

Keywords

relativistic Vlasov–Maxwell system asymptotic properties modified scattering small data solutions compactly supported data

1. Introduction and Main Results

1.1. General Context

The relativistic Vlasov–Maxwell system models a collisionless plasma, it can be written as

\begin{aligned} \sqrt{m_{α}^{2} + | v |^{2}} \partial_{t} f_{α} + v \cdot \nabla_{x} f_{α} + e_{α} (\sqrt{m_{α}^{2} + | v |^{2}} E + v \times B) \cdot \nabla_{v} f_{α} = 0, 1 \leq α \leq N, \end{aligned}

(RVM)

\begin{aligned} \partial_{t} E = \nabla \times B - 4 π j, & \nabla \cdot E = 4 π ρ, \\ \partial_{t} B = - \nabla \times E, & \nabla \cdot B = 0, \end{aligned}

where

ρ, j

are the charge and current density of the plasma defined by

\begin{aligned} ρ = \sum_{1 \leq α \leq N} e_{α} \int_{R_{v}^{3}} f_{α} d v, j = \sum_{1 \leq α \leq N} e_{α} \int_{R_{v}^{3}} \hat{v_{α}} f_{α} d v . \end{aligned}

Here we consider the multi-species case

N \geq 2

, where

f_{α} (t, x, v)

is the density function of a species

α

with mass

m_{α} > 0

and charge

e_{α} \neq 0

. Here,

t \geq 0

denotes the time,

x \in R_{x}^{3}

the particle position and

v \in R_{v}^{3}

the particle momentum. Moreover,

(E, B) (t, x)

denotes the electromagnetic field of the plasma. For a particle of mass

1

and momentum

v \in R_{v}^{3}

, we will denote its energy by

v^{0} := ⟨ v ⟩ = \sqrt{1 + | v |^{2}}

and its relativistic speed as

\begin{aligned} \hat{v} := \frac{v}{v^{0}}, v \in R_{v}^{3} . \end{aligned}

(1.1)

In addition, we denote

v_{α} (v) := v / m_{α}

. We will simply write

v_{α}

since there is no risk of confusion. Then

\begin{aligned} \hat{v_{α}} := \frac{v}{\sqrt{m_{α}^{2} + | v |^{2}}} = \frac{v}{v_{α}^{0}}, v_{α}^{0} := \sqrt{m_{α}^{2} + | v |^{2}} . \end{aligned}

(1.2)

Note that we have

v_{α}^{0} = m_{α} ⟨ v_{α} ⟩

. Finally, the initial data

f_{α 0} = f_{α} (0, \cdot)

and

(E, B) (0, \cdot) = (E_{0}, B_{0})

also satisfy, in the electrically neutral setting, the constraint equations

\begin{aligned} \nabla \cdot E_{0} = 4 π \sum_{1 \leq α \leq N} e_{α} \int_{R_{v}^{3}} f_{α 0} d v, \nabla \cdot B_{0} = 0, \sum_{1 \leq α \leq N} e_{α} \int_{R_{x}^{3} \times R_{v}^{3}} f_{α 0} d v d x = 0. \end{aligned}

(1.3)

In 3D, the global existence problem for the classical solutions to (RVM) is still open, though various continuation criteria have been proved (see, for instance, Luk & Strain, 2016).

The case of small data solutions was first studied by Glassey and Schaeffer (1988) and Glassey and Strauss (1987). They showed that the solutions to (RVM) arising from small and compactly supported data are global in time. The compact support assumption on the momentum variable $v$ was later removed by Schaeffer (2004). More recently, without any compact support hypothesis, (Bigorgne, 2020; Wei & Yang, 2021) established propagation of regularity for the small data solution to (RVM) and (Wang, 2022) relaxed the smallness assumption on the electromagnetic field. Finally, a modified scattering dynamic was derived for the distribution function; (see Bigorgne, 2025; Pankavich & Ben-Artzi, 2024), along with a scattering map (Bigorgne, 2023b).

Similar results have been obtained for the Vlasov–Poisson equation, for instance modified scattering has been proved for small data (Choi & Kwon, 2016; Flynn et al., 2023; Ionescu et al., 2021; Pankavich, 2022) (see also Bigorgne & Ruiz, 2024; Schlue & Taylor, 2024 for more refinements). It was also shown that, in the single species case and for a non-trivial distribution function, linear scattering cannot occur (Choi & Ha, 2011). Note that modified scattering also holds in the context of stability of a point charge (Pausader & Widmayer, 2021; Pausader et al., 2024), or with the external potential $- | x |^{2} / 2$ (Bigorgne et al., 2025).

In this paper, we provide a short proof of modified scattering for the distribution functions $f_{α}$ . Compared with Pankavich and Ben-Artzi (2024), who also worked on solutions constructed by Glassey and Strauss (1987), our approach does not require to assume more regularity on the data than in Glassey and Strauss (1987).

1.2. Main Result

We assume the following properties and derive the results in this context.

Hypothesis 1.1
Assume $(f_{α}, E, B)$ is a $C^{1}$ global solution to (RVM) with initial data $(f_{α 0}, E_{0}, B_{0})$ and satisfying the following properties.
There exists $k > 0$ such that $f_{α 0}$ are non-negative $C^{1}$ functions with support in ${(x, v) | | x | \leq k, | v | \leq k}$ . Moreover, $E_{0}, B_{0}$ are $C^{1}$ with support in ${x | | x | \leq k}$ and satisfy the constraint (1.3).

There exists $C_{0} > 0$ such that, for all $(t, x) \in R_{+} \times R_{x}^{3}$ ,
$\begin{aligned} | (E, B) (t, x) | & \leq \frac{C_{0}}{(t + | x | + 2 k) (t - | x | + 2 k)}, \end{aligned}$
(1.4)

$\begin{aligned} | \nabla_{x} (E, B) (t, x) | & \leq \frac{C_{0} \log (t + | x | + 2 k)}{(t + | x | + 2 k) (t - | x | + 2 k)^{2}} . \end{aligned}$
(1.5)

Remark 1.2
In the following, we will write $a ≲ b$ when there exists $C > 0$ , independent of $t$ and depending on $(x, v)$ only through $k$ , such that $a \leq C b$ . However, $C$ will usually depend on $(m_{α}, e_{α})_{1 \leq α \leq N}$ and the initial data.
Remark 1.3
Since $(f_{α}, E, B)$ is a solution to (RVM) with compactly supported data, it follows that
$\begin{aligned} supp (E, B) (t, \cdot) \subset {x \in R_{x}^{3} | | x | \leq t + k} . \end{aligned}$
Hence the right hand sides of (1.4) and (1.5) are always bounded.
Remark 1.4
It is important to note that, according to Glassey and Schaeffer (1988, Theorem 1) and Glassey and Strauss (1987), for small compactly supported data or nearly neutral data, the unique associated classical solution to (RVM) satisfies Hypothesis 1.1. This proves that, when the data are compactly supported, Theorem 1.5 holds for small or nearly neutral data. Finally, our result applies to a subclass of the solutions constructed by Rein (1990) as well, those arising from compactly supported data.

We now state that for any $(f_{α}, E, B)$ verifying the above hypothesis, the distribution functions $f_{α}$ satisfy a modified scattering dynamic. Moreover, we are able to prove that the asymptotic limits $f_{α \infty}$ are compactly supported.
Theorem 1.5
Let $(f_{α}, E, B)$ be a solution to (RVM) satisfying Hypothesis 1.1. Then, every $f_{α}$ verifies modified scattering. More precisely there exists ${\tilde{f}}_{α \infty} \in C_{c}^{0} (R_{x}^{3} \times R_{v}^{3})$ and $E, B \in C^{0} (R_{v}^{3}, R_{v}^{3})$ such that, for any $α$ and all $t > 0, (x, v) \in R_{x}^{3} \times R_{v}^{3}$ verifying $t v_{α}^{0} - \log (t) (e_{α} / v_{α}^{0}) E (v_{α}) \cdot \hat{v_{α}} \geq 0$ ,
$\begin{aligned} | f_{α} (t v_{α}^{0} - \log (t) \frac{e_{α}}{v_{α}^{0}} E (v_{α}) \cdot \hat{v_{α}}, x + t v - \log (t) \frac{e_{α}}{v_{α}^{0}} (E (v_{α}) + \hat{v_{α}} \times B (v_{α})), v) - {\tilde{f}}_{α \infty} (x, v) | ≲ \frac{\log^{6} (2 + t)}{2 + t} . \end{aligned}$

Remark 1.6
Unlike Bigorgne (2025) and Pankavich and Ben-Artzi (2024) we modify the characteristics of the operator $v_{α}^{0} \partial_{t} + v \cdot \nabla_{x}$ instead of $\partial_{t} + \hat{v_{α}} \cdot \nabla_{x}$ . This is more consistent with the Lorentz invariance of (RVM) that we will exploit in a forthcoming article (Breton, In preparation) (see Remark 1.7). (To observe the Lorentz invariance of the Vlasov–Maxwell system, one has to write the Vlasov equation as in (RVM) rather than in (2.3) below). However, taking
$\begin{aligned} {\tilde{C}}_{v} := \hat{v} (E (v) \cdot \hat{v}) - (E (v) + \hat{v} \times B (v)), \end{aligned}$
we obtain from Theorem 1.5 the same statement as (Bigorgne, 2025; Pankavich & Ben-Artzi, 2024)
$\begin{aligned} | f_{α} (t, x + t \hat{v_{α}} + \log (t) \frac{e_{α}}{v_{α}^{0}} {\tilde{C}}_{v_{α}}, v) - f_{α \infty} (x, v) | ≲ \frac{\log^{6} (2 + t)}{2 + t}, f_{α \infty} (x, v) = {\tilde{f}}_{α \infty} (x + \frac{e_{α}}{v_{α}^{0}} \log (v_{α}^{0}) {\tilde{C}}_{v_{α}}, v) . \end{aligned}$
In fact, these two formulations can be derived from each other and it will be more convenient to prove the latter one.
Remark 1.7
In a forthcoming article (Breton, In preparation), we will show that linear scattering, that is $g_{α} (t, \cdot) \to_{t \to \infty}^{L_{x, v}^{1}} g_{α \infty}$ , is a non-generic phenomenon. More precisely, the initial data leading to linear scattering lie in a codimension 1 submanifold.
1.3. Ideas of the Proof

We detail here the arguments used to prove Theorem 1.5. Let $(f_{α}, E, B)$ be a solution to (RVM) that satisfies Hypothesis 1.1. For the sake of presentation, here we assume that $m_{α} = 1$ so that $v = v_{α}$ . We begin by composing $f_{α}$ with the linear flow to consider $g_{α} (t, x, v) = f_{α} (t, x + t \hat{v}, v)$ . We then derive two key properties

\begin{aligned} supp g_{α} (t, \cdot) \subset {(x, v) | | x | ≲ \log (2 + t), | v | \leq β}, \end{aligned}

\begin{aligned} | \nabla_{x} g_{α} (t, \cdot) | ≲ 1, | \nabla_{v} g_{α} (t, \cdot) | ≲ \log^{2} (2 + t) . \end{aligned}

Moreover, the spatial support of

f_{α} (t, \cdot)

is included in

{| x | \leq γ t + k}

, where

γ < 1

. Consequently, the Lorentz force

L (t, x, v) := E (t, x) + \hat{v} \times B (t, x)

satisfies, on the support of

f_{α}

\begin{aligned} | L (t, x, v) | ≲ (2 + t)^{- 2} . \end{aligned}

(1.6)

The first idea is to look for linear scattering, in which case we would have that

g_{α} (t, \cdot)

converges as

t \to \infty

. For this matter, we compute

\begin{aligned} \partial_{t} g_{α} (t, x, v) = & - e_{α} (E (t, x + t \hat{v}) + \hat{v} \times B (t, x + t \hat{v})) \cdot (\nabla_{v} f_{α}) (t, x + t \hat{v}, v) \\ = & e_{α} \frac{t}{v^{0}} [L (t, x + t \hat{v}, v) - (L (t, x + t \hat{v}, v) \cdot \hat{v}) \hat{v}] \cdot \nabla_{x} g_{α} (t, x, v) + O (\log^{2} (2 + t) (2 + t)^{- 2}) . \end{aligned}

With our estimates, we can merely control the first term by $t^{- 1}$ , preventing us from concluding that $g_{α} (t, \cdot)$ converges as $t \to + \infty$ . Note that this is consistent with Remark 1.7. However, by further investigating the asymptotic behavior of $(E, B)$ , we can still expect to prove that $f_{α}$ converges along modifications of the linear characteristics. To achieve this, we are led to determine the leading order contribution of the source terms $ρ$ and $j$ in the Maxwell equations. For the linearized system, the asymptotic behavior of $ρ$ and $j$ is governed by the spatial average $\int f_{α} d x$ , which, in this setting, is a conserved quantity. To this end, we begin by proving the existence of the asymptotic charge $Q_{\infty}^{α}$ such that $\int f_{α} d x$ converges to $Q_{\infty}^{α}$ . Now, let $Q_{\infty} (v) = \sum_{α} e_{α} Q_{\infty}^{α} (v)$ . This allows us to consider the asymptotic charge and the asymptotic current densities

\begin{aligned} ρ^{a s} (t, x) := \frac{1}{t^{3}} [⟨ \cdot ⟩^{5} Q_{\infty}] (\overset{ˇ}{x / t}) 1_{| x | < t}, j^{a s} (t, x) := \frac{x}{t^{4}} [⟨ \cdot ⟩^{5} Q_{\infty}] (\overset{ˇ}{x / t}) 1_{| x | < t}, \end{aligned}

(1.7)

where

u \mapsto \overset{ˇ}{u}

is the inverse of

u \mapsto \hat{u}

. These densities satisfy

\begin{aligned} | ρ (t, x) - ρ^{a s} (t, x) | + | j (t, x) - j^{a s} (t, x) | ≲ \frac{\log^{6} (2 + t)}{2 + t} . \end{aligned}

The previous arguments allow us to define $E, B \in C^{0} (R_{v}^{3}, R_{v}^{3})$ , which verify

\begin{aligned} E (t, x + t \hat{v}) = \frac{1}{t^{2}} E (v) + O (\frac{\log^{6} (2 + t)}{(2 + t)^{3}}), B (t, x + t \hat{v}) = \frac{1}{t^{2}} B (v) + O (\frac{\log^{6} (2 + t)}{(2 + t)^{3}}) . \end{aligned}

(1.8)

Remark 1.8

Although it is not immediately apparent in Sections 3.1–3.2, it turns out that $t^{- 2} E (\overset{ˇ}{x / t})$ and $t^{- 2} B (\overset{ˇ}{x / t})$ can be interpreted as follows. Let $E^{a s}$ and $B^{a s}$ be the solutions to

\begin{aligned} ◻ E^{a s} = - \nabla_{x} ρ^{a s} - \partial_{t} j^{a s}, ◻ B^{a s} = \nabla_{x} \times j^{a s}, \end{aligned}

with trivial data at

t_{0} > 0

. Then for

t \geq T (t_{0})

large enough, we have for all

| x | \leq γ t + k

\begin{aligned} E^{a s} (t, x) = t^{- 2} E (\overset{ˇ}{\frac{x}{t}}), B^{a s} (t, x) = t^{- 2} B (\overset{ˇ}{\frac{x}{t}}) . \end{aligned}

Moreover, for

t

large enough, we have

t^{- 2} E (\overset{ˇ}{x / t}) = E_{T}^{a s}

in the Glassey–Strauss decomposition of the electromagnetic field

(E^{a s}, B^{a s})

, associated to the singular distribution function

f_{α}^{a s} (t, x, v) = δ (x - t \hat{v}) Q_{\infty}^{α} (v)

through Proposition 3.1. We refer to Bigorgne (2023b, Section 5) for more information.

To conclude, it remains to prove the modified scattering statement for the density functions $f_{α}$ . Using the above estimates for the fields, we derive

\begin{aligned} \partial_{t} (f_{α} (t, x + t \hat{v}, v)) = e_{α} [- (\frac{L (v)}{t v^{0}} \cdot \hat{v}) \hat{v} + \frac{L (v)}{t v^{0}}] \cdot \nabla_{x} f (t, x + t \hat{v}, v) + O (\frac{\log^{6} (2 + t)}{(2 + t)^{2}}), \end{aligned}

(1.9)

where

L (v) = E (v) + \hat{v} \times B (v)

. We finally introduce the correction

\begin{aligned} {\tilde{C}}_{v} := \hat{v} (E (v) \cdot \hat{v}) - (E (v) + \hat{v} \times B (v)), \end{aligned}

(1.10)

which, once multiplied by

(1 / v^{0}) \log (t)

, measures how much the characteristics of the Vlasov operator deviate from the linear ones. It allows us to obtain the modified scattering statement and that the asymptotic state

f_{α \infty}

is compactly supported.

1.4. Structure of the Paper

Section 2 contains several statements that are needed in our proof of Theorem 1.5. We first introduce the function $g_{α}$ by composing $f_{α}$ with the linear flow. Then we compute the support of $f_{α}$ and two key properties on $g_{α}$ and its first order derivatives. We end this section by studying the asymptotic properties of $\int_{R_{x}^{3}} f_{α} d x$ and $\int_{R_{v}^{3}} f_{α} d v$ . Finally, in Section 3, we further investigate the asymptotic behavior of $(E, B)$ and show that the densities $f_{α}$ exhibit a modified scattering dynamic. Then, we prove the compactness of the support of ${\tilde{f}}_{α \infty}$ (see Proposition 3.11), concluding the proof of Theorem 1.5.

2. Preliminary Results

In the following sections, we consider $(f_{α}, E, B)$ a solution to (RVM) that satisfies Hypothesis 1.1. We begin this section by giving a lemma about the inverse of $v \mapsto \hat{v}$ .

Lemma 2.1
We define on ${u \in R^{3}, | u | < 1}$ the map $\overset{ˇ}{}$ by
$\begin{aligned} u \mapsto \overset{ˇ}{u} := \frac{u}{\sqrt{1 - | u |^{2}}} . \end{aligned}$
(2.1)
In particular
$\begin{aligned} \forall | u | < 1, \forall v \in R_{v}^{3}, \hat{\overset{ˇ}{u}} = u, \overset{ˇ}{\hat{v}} = v . \end{aligned}$
Finally, the Jacobian determinant of $v \in R^{3} \mapsto \hat{v}$ is $⟨ v ⟩^{- 5}$ .

Let us begin by introducing, to simplify the notations,
$\begin{aligned} f^{α} (t, x, v) = f_{α} (t, x, m_{α} v) . \end{aligned}$
(2.2)
Notice here that the support of ${f^{α}}_{0}$ is now included in ${(x, v) | | x | \leq k, | v | \leq k_{α}}$ with $k_{α} = k / m_{α}$ . Moreover, $f^{α}$ satisfies the following equation
$\begin{aligned} \partial_{t} f^{α} + \hat{v} \cdot \nabla_{x} f^{α} + \frac{e_{α}}{m_{α}} (E + \hat{v} \times B) \cdot \nabla_{v} f^{α} = 0. \end{aligned}$
(2.3)
From this we can introduce
$\begin{aligned} g^{α} (t, x, v) := f^{α} (t, x + t \hat{v}, v), g_{α} (t, x, v) := f_{α} (t, x + t \hat{v_{α}}, v) . \end{aligned}$
Notice that this notation is consistent since $g^{α} (t, x, v) = g_{α} (t, x, m_{α} v)$ . Moreover, the derivatives of $g^{α}$ can be expressed as the following
$\begin{aligned} \nabla_{x} g^{α} (t, x, v) & = (\nabla_{x} f^{α}) (t, x + t \hat{v}, v), \end{aligned}$
(2.4)

$\begin{aligned} \nabla_{v} g^{α} (t, x, v) & = \frac{t}{v^{0}} [(\nabla_{x} f^{α}) (t, x + t \hat{v}, v) - \hat{v} ((\nabla_{x} f^{α}) (t, x + t \hat{v}, v) \cdot \hat{v})] + (\nabla_{v} f^{α}) (t, x + t \hat{v}, v) . \end{aligned}$
(2.5)
This means that for $L (t, x, v) := E (t, x) + \hat{v} \times B (t, x)$ , $g^{α}$ will satisfy the following equation
$\begin{aligned} \partial_{t} g^{α} (t, x, v) + \frac{t}{v^{0}} \frac{e_{α}}{m_{α}} [\hat{v} (L (t, x + t \hat{v}, v) \cdot \hat{v}) - L (t, x + t \hat{v}, v)] \cdot \nabla_{x} g^{α} (t, x, v) + \frac{e_{α}}{m_{α}} L (t, x + t \hat{v}, v) \cdot \nabla_{v} g^{α} (t, x, v) = 0. \end{aligned}$
(2.6)
2.1. Characteristics

We begin by introducing the characteristics of $g^{α}$ . Let $X (s) = X (s, t, x, v), V (s) = V (s, t, x, v)$ be defined by the ODE

\begin{aligned} {\begin{cases} \dot{X} (s) & = \frac{s}{V^{0} (s)} \frac{e_{α}}{m_{α}} [\hat{V} (s) (L (s, X (s) + s \hat{V} (s), V (s)) \cdot \hat{V} (s)) - L (s, X (s) + s \hat{V} (s), V (s))], \\ \dot{V} (s) & = \frac{e_{α}}{m_{α}} L (s, X (s) + s \hat{V} (s), V (s)) \end{cases} \end{aligned}

(2.7)

Remark 2.2

One can easily switch between the characteristics of $f^{α}$ and $g^{α}$ . In fact, taking $X (s) := X (s) + s \hat{V} (s), V (s) := V (s)$ , we derive the following ODE

\begin{aligned} \dot{X} (s) & = \hat{V} (s), \\ \dot{V} (s) & = \frac{e_{α}}{m_{α}} L (s, X (s), V (s)) . \end{aligned}

Meaning that

(X, V)

are the characteristics of

f^{α}

starting from

X (t) = x + t \hat{v}, V (t) = v

We will now show that the support of $f^{α} (t, x, \cdot)$ is bounded, uniformly in $(t, x)$ . From this we derive that the space support of $f (t, \cdot)$ is bounded by $\hat{β_{α}} t + k$ . Here, contrary to Glassey and Strauss (1987), we do not require smallness of the initial data to prove the result.

Lemma 2.3

There exists a constant $β > 0$ such that, for all $t \geq 0$ and any $α$

\begin{aligned} supp (f^{α} (t, \cdot)) \subset {| x | \leq \hat{β_{α}} t + k, | v | \leq β_{α}}, \end{aligned}

(2.8)

where

β_{α} := β / m_{α}

. Moreover, if

g^{α} (t, x, v) \neq 0

then

\forall s \geq 0

\begin{aligned} s - | X (s) + s \hat{V} (s) | + 2 k \geq k + s (1 - {\hat{β}}_{\max}), \end{aligned}

where

β_{\max} := sup_{1 \leq α \leq N} β_{α}

. In particular, for

(x, v)

in the support of

g^{α} (t, \cdot)

, we have

\begin{aligned} t - | x + t \hat{v} | + 2 k \geq t (1 - {\hat{β}}_{\max}) . \end{aligned}

Proof.

The proof follows Wei and Yang (2021, Lemma 2.1). Let $(t, x, v)$ such that $f^{α} (t, x + \hat{t v}, v) \neq 0$ . This implies ${f^{α}}_{0} (X (0), V (0)) \neq 0$ and thus we are working with $| X (0) | \leq k, | V (0) | \leq k_{α}$ . We begin by introducing

\begin{aligned} U (t) = sup {| V (s) | | 0 \leq s \leq t} . \end{aligned}

Using the ODE satisfied by

X (s) = X (s) + s \hat{V} (s)

, one finds

\begin{aligned} s - | X (s) + s \hat{V} (s) | + 2 k \geq s (1 - \hat{U} (t)) + k \geq s (1 - \hat{U} (t)) . \end{aligned}

Here we used that

λ \in R_{+} \mapsto (λ / ⟨ λ ⟩)

is increasing. Now consider

t_{1}, t_{2} \in [0, t]

. Using (1.4) we derive,

\begin{aligned} | V (t_{1}) - V (t_{2}) | & \leq \int_{0}^{t} \frac{C}{(s - | X (s) + s \hat{V} (s) | + 2 k) (s + | X (s) + s \hat{V} (s) | + 2 k)} d s \\ \leq \int_{0}^{k / (1 - \hat{U} (t))} \frac{C}{(s + 2 k) k} d s + \int_{k / (1 - \hat{U} (t))}^{+ \infty} \frac{C}{(s (1 - \hat{U} (t)) + k) (s + 2 k)} d s . \end{aligned}

Computing the last two integrals we derive

\begin{aligned} | V (t_{1}) - V (t_{2}) | \leq \frac{C}{k} [\log (\frac{3}{2}) - \log (1 - \hat{U} (t)) + \frac{1}{2} ϕ_{0} (\frac{1}{2} - \hat{U} (t))], \end{aligned}

where

\begin{aligned} ϕ_{0} (z) = \frac{\ln (1 + z)}{z}, ϕ (0) = 1, - 1 < z < + \infty . \end{aligned}

From this we follow Wei and Yang (2021, Lemma 2.1) and find a constant

\tilde{C}

independent of

t

such that

\begin{aligned} | V (t) | \leq 2 | V (0) | + \tilde{C} \leq 2 k_{α} + \tilde{C} . \end{aligned}

Hence, there exists

β > 0

such that, for

β_{α} = β / m_{α}

\begin{aligned} | v | \leq β_{α} . \end{aligned}

It remains to prove the estimate on the support of

f^{α} (t, \cdot, v)

. Since

| v | \leq β_{α}

and

g^{α}

is constant along its characteristics, we know that

| V (s) | \leq β_{α}

. So we derive directly

\begin{aligned} | X (s) + s \hat{V} (s) | \leq | X (0) | + \hat{β_{α}} s \leq k + \hat{β_{α}} s . \end{aligned}

Implying directly

| x + t \hat{v} | \leq k + \hat{β_{α}} t

. This gives the inclusion for the support of

f^{α}

as well as the other two inequalities.

Remark 2.4

One can easily go back to $f_{α}$ to find its support. In fact, we have

\begin{aligned} supp f_{α} (t, \cdot) \subset {(x, v) \in R_{x}^{3} \times R_{v}^{3} | | x | \leq {\hat{β}}_{\max} t + k, | v | \leq β} . \end{aligned}

Moreover, since

(f_{α}, E, B)

is a solution to (RVM) with compactly supported initial data,

\begin{aligned} supp (E, B) (t, \cdot) \subset {x \in R_{x}^{3} | | x | \leq t + k} . \end{aligned}

2.2. Properties of

g^{α}

With these properties for the characteristics of $g^{α}$ in mind, we now want to estimate the support of $g^{α}$ and control its derivatives.

Proposition 2.5
There exists a constant $C > 0$ such that, for all $t \geq 0$ and any $α$ ,
$\begin{aligned} supp (g^{α} (t, \cdot)) \subset {(x, v) \in R_{x}^{3} \times R_{v}^{3} | | x | \leq C \log (2 + t), | v | \leq β_{α}} . \end{aligned}$
(2.9)
Proof.
With the previous notations, we have, thanks to (1.4),
$\begin{aligned} | \dot{X} (s) | ≲ \frac{s}{(s + | X (s) + s \hat{V} (s) | + 2 k) (s - | X (s) + s \hat{V} (s) | + 2 k)} . \end{aligned}$
Now, thanks to Proposition 2.3, we have $| \dot{X} (s) | ≲ 1 / (s + 2)$ , which implies
$\begin{aligned} | X (s) | ≲ k + \log (2 + s) ≲ \log (2 + s) . \end{aligned}$

We now estimate $g^{α}$ and its first order derivatives.
Proposition 2.6
Consider $(t, x, v)$ with $t \geq 0$ . We have the following estimates, for any $1 \leq α \leq N$ ,
$\begin{aligned} | g^{α} (t, x, v) | \leq ‖ {f^{α}}_{0} ‖_{\infty}, \end{aligned}$
(2.10)

$\begin{aligned} | \nabla_{x} g^{α} (t, x, v) | ≲ 1, \end{aligned}$
(2.11)

$\begin{aligned} | \nabla_{v} g^{α} (t, x, v) | ≲ \log^{2} (2 + t) . \end{aligned}$
(2.12)
Proof.
The first estimate is immediate by using the characteristics. Then, recall (2.6) and let $L$ be the associated operator such that $L g = 0$ . We have
$\begin{aligned} L (\partial_{x_{i}} g^{α}) = & - \frac{t}{v^{0}} \frac{e_{α}}{m_{α}} \partial_{x_{i}} [\hat{v} (L (t, x + t \hat{v}, v) \cdot \hat{v}) - L (t, x + t \hat{v}, v)] \cdot \nabla_{x} g^{α} - \frac{e_{α}}{m_{α}} \partial_{x_{i}} [L (t, x + t \hat{v}, v)] \cdot \nabla_{v} g^{α}, \\ L (\partial_{v_{i}} g^{α}) = & - t \frac{e_{α}}{m_{α}} \partial_{v_{i}} [\frac{1}{v^{0}} (\hat{v} (L (t, x + t \hat{v}, v) \cdot \hat{v}) - L (t, x + t \hat{v}, v))] \cdot \nabla_{x} g^{α} - \frac{e_{α}}{m_{α}} \partial_{v_{i}} [L (t, x + t \hat{v}, v)] \cdot \nabla_{v} g^{α} . \end{aligned}$
Now let us consider $X (s) = X (s, t, x, v), V (s) = V (s, t, x, v)$ the characteristics of $g^{α}$ . Recall Lemma 2.3, so for $g^{α} (t, x, v) \neq 0$ , we have
$\begin{aligned} s - | X (s) + s \hat{V} (s) | + 2 k \geq s (1 - {\hat{β}}_{\max}) + k . \end{aligned}$
This implies the following estimates
$\begin{aligned} | (E, B) (s, X (s) + s \hat{V} (s)) | ≲ \frac{1}{(2 + s)^{2}}, | \nabla_{x} (E, B) (s, X (s) + s \hat{V} (s)) | ≲ \frac{\log (2 + s)}{(2 + s)^{3}} . \end{aligned}$
We can now use the equation satisfied by $\partial_{x_{i}} g^{α}$ and $\partial_{v_{i}} g^{α}$ to derive, thanks to Duhamel’s principle,
$\begin{aligned} | (\nabla_{x} g^{α}) (τ, X (τ), V (τ)) | & ≲ ‖ f_{0} ‖_{C^{1}} + \int_{0}^{τ} \frac{\log (2 + s)}{(2 + s)^{2}} | \nabla_{x} g^{α} | + \frac{\log (2 + s)}{(2 + s)^{3}} | \nabla_{v} g^{α} | d s, \\ | (\nabla_{v} g^{α}) (τ, X (τ), V (τ)) | & ≲ ‖ f_{0} ‖_{C^{1}} + \int_{0}^{τ} \frac{\log (2 + s)}{(2 + s)} | \nabla_{x} g^{α} | + \frac{\log (2 + s)}{(2 + s)^{2}} | \nabla_{v} g^{α} | d s, \end{aligned}$
where in the integral $\nabla_{x} g^{α}, \nabla_{v} g^{α}$ are evaluated at $(s, X (s), V (s))$ . Since $s \mapsto ((\log (2 + s)) / ((2 + s)^{2}))$ is integrable, by Grönwall’s inequality we have for all $τ \geq 0$
$\begin{aligned} | (\nabla_{x} g^{α}) (τ, X (τ), V (τ)) | & ≲ ‖ f_{0} ‖_{C^{1}} + \int_{0}^{τ} \frac{\log (2 + s)}{(2 + s)^{3}} | \nabla_{v} g^{α} | d s, \end{aligned}$
(2.13)

$\begin{aligned} | (\nabla_{v} g^{α}) (τ, X (τ), V (τ)) | & ≲ ‖ f_{0} ‖_{C^{1}} + \int_{0}^{τ} \frac{\log (2 + s)}{(2 + s)} | \nabla_{x} g^{α} | d s . \end{aligned}$
(2.14)
We now insert (2.13) in (2.14) to derive,
$\begin{aligned} | (\nabla_{v} g^{α}) (τ, X (τ), V (τ)) | & ≲ ‖ f_{0} ‖_{C^{1}} (1 + \int_{0}^{τ} \frac{\log (2 + s)}{(2 + s)} d s) + \int_{0}^{τ} \frac{\log (2 + s)}{(2 + s)} (\int_{0}^{s} \frac{\log (2 + u)}{(2 + u)^{3}} | \nabla_{v} g^{α} | d u) d s \\ ≲ \log^{2} (2 + τ) + \log^{2} (2 + τ) \int_{0}^{τ} \frac{\log (2 + u)}{(2 + u)^{3}} | \nabla_{v} g^{α} | d u . \end{aligned}$
We now apply Gronwall’s inequality to $G (s) = | \nabla_{v} g^{α} (s, X (s), V (s)) | \log^{- 2} (2 + s)$ . Since $s \mapsto ((\log^{3} (2 + s)) / ((2 + s)^{3}))$ is integrable we derive
$\begin{aligned} \frac{1}{\log^{2} (2 + τ)} | (\nabla_{v} g^{α}) (τ, X (τ), V (τ)) | = G (τ) ≲ 1, \end{aligned}$
and the estimate on $\nabla_{v} g^{α}$ follows. Inserting the estimate on $\nabla_{v} g^{α}$ in (2.13) we derive the other estimate. Finally, taking $τ = t$ , we derive the result.
Remark 2.7
From (2.4)–(2.5) and the above proposition, one can easily derive estimates on the derivatives of $f^{α}$ . Moreover, the derivatives of $f_{α}$ (resp. $g_{α}$ ) satisfy the same estimates as those satisfied by $f^{α}$ (resp. $g^{α}$ ).
2.3. Convergence of the Spatial Average

We focus on the spatial average of $g^{α}$ since this quantity governs the asymptotic behavior of the source terms in the Maxwell equations.

Proposition 2.8
For any $α$ , there exists $Q_{\infty}^{α} \in C_{c}^{0} (R_{v}^{3})$ such that, for all $t \geq 0$ and $v \in R_{v}^{3}$ ,
$\begin{aligned} | \int_{R_{x}^{3}} g^{α} (t, x, v) d x - Q_{\infty}^{α} (v) | ≲ \frac{\log^{5} (2 + t)}{2 + t} . \end{aligned}$
(2.15)
Proof.
We begin by integrating (2.6) over $R_{x}^{3}$ to derive
$\begin{aligned} \partial_{t} \int_{R_{x}^{3}} g^{α} (t, x, v) d x = & - \frac{t}{v^{0}} \frac{e_{α}}{m_{α}} \int_{R_{x}^{3}} [\hat{v} (L (t, x + t \hat{v}, v) \cdot \hat{v}) - L (t, x + t \hat{v}, v)] \cdot \nabla_{x} g^{α} (t, x, v) d x \\ - \frac{e_{α}}{m_{α}} \int_{R_{x}^{3}} L (t, x + t \hat{v}, v) \cdot \nabla_{v} g^{α} (t, x, v) d x . \end{aligned}$
The second term of the right-hand side can directly be dealt with. Indeed, thanks to (1.4), Lemma 2.3 and Propositions 2.5–2.6, we have
$\begin{aligned} | \frac{e_{α}}{m_{α}} \int_{R_{x}^{3}} L (t, x + t \hat{v}, v) \cdot \nabla_{v} g^{α} (t, x, v) d x | ≲ \int_{| x | ≲ \log (2 + t)} \frac{\log^{2} (2 + t)}{(2 + t)^{2}} d x ≲ \frac{\log^{5} (2 + t)}{(2 + t)^{2}} . \end{aligned}$
It remains to study the first term. By integration by parts, we derive
$\begin{aligned} t \int_{R_{x}^{3}} [\hat{v} (L (t, x + t \hat{v}, v) \cdot \hat{v}) - L (t, x + t \hat{v}, v)] \cdot \nabla_{x} g^{α} (t, x, v) d x & = t \int_{R_{x}^{3}} (\nabla_{x} \cdot L) (t, x + t \hat{v}, v) g^{α} (t, x, v) d x \\ - t \int_{R_{x}^{3}} \sum_{i = 1}^{3} {\hat{v}}^{i} ((\partial_{x_{i}} L) (t, x + t \hat{v}, v) \cdot \hat{v}) g^{α} (t, x, v) d x . \end{aligned}$
Again from (1.5), Lemma 2.3 and Propositions 2.5–2.6, we obtain
$\begin{aligned} t | \int_{R_{x}^{3}} [\hat{v} (L (t, x + t \hat{v}, v) \cdot \hat{v}) - L (t, x + t \hat{v}, v)] \cdot \nabla_{x} g^{α} (t, x, v) d x | & ≲ t \int_{R_{x}^{3}} | \nabla_{x} (E, B) (t, x + t \hat{v}) | | g^{α} (t, x, v) | d x \\ ≲ \int_{| x | ≲ \log (2 + t)} \frac{\log (2 + t)}{(2 + t)^{2}} d x \\ ≲ \frac{\log^{4} (2 + t)}{(2 + t)^{2}} . \end{aligned}$

Finally, combining these two estimates, we obtain
$\begin{aligned} | \partial_{t} \int_{R_{x}^{3}} g^{α} (t, x, v) d x | ≲ \frac{\log^{5} (2 + t)}{(2 + t)^{2}} . \end{aligned}$
Now, since $\partial_{t} \int_{R_{x}^{3}} g^{α} (t, x, v) d x$ is integrable in $t$ , this proves the existence of the limit $Q_{\infty}^{α}$ . Moreover $g^{α} (t, x, \cdot)$ has its support in ${v \in R_{v}^{3}, | v | \leq β_{α}}$ so we already know that $supp (Q_{\infty}^{α}) \subset {v \in R_{v}^{3}, | v | \leq β_{α}}$ .

Finally, since $\int_{R_{x}^{3}} f (t, x, \cdot) d x$ is continuous and converges uniformly towards $Q_{\infty}^{α}$ we know that $Q_{\infty}^{α}$ is also continuous.
2.4. Link Between the Particle Density and the Asymptotic Charge

In Glassey and Strauss (1987) they prove that the velocity average decays like $(1 + t)^{- 3}$ . Here we provide the asymptotic expansion of $\int f_{α} d v$ . The following proposition justifies, for $h (v) = 1$ and $h (v) = \hat{v_{α}}$ , the asymptotic expansion of the charge and current densities $(ρ, j)$ stated in the outline of the proof. Recall, in particular, that $\hat{v_{α}} (m_{α} v) = \hat{v}$ .

Proposition 2.9
Let $h \in C^{1} (R_{v}^{3})$ . Then for any $α$ , all $t > 0$ and all $| x | < t$ we have
$\begin{aligned} | t^{3} \int_{R_{v}^{3}} h (v) f_{α} (t, x, v) d v - m_{α}^{3} [⟨ \cdot ⟩^{5} h (m_{α} \cdot) Q_{\infty}^{α}] (\overset{ˇ}{\frac{x}{t}}) | ≲ \frac{\log^{6} (2 + t)}{2 + t} sup_{| v | \leq β} (| h (v) | + | \nabla_{v} h (v) |) . \end{aligned}$
(2.16)
Proof.
Since $f_{α} (t, \cdot)$ and $Q_{\infty}^{α}$ are continuous and compactly supported, it is enough to prove the estimate for $t \geq 1$ . We begin by the change of variable $w = v_{α}$ so that
$\begin{aligned} \int_{R_{v}^{3}} h (v) f_{α} (t, x, v) d v = m_{α}^{3} \int_{R_{v}^{3}} h (m_{α} v) f^{α} (t, x, v) d v . \end{aligned}$
Applying Proposition 2.8 to $v = \overset{ˇ}{x} / t$ , in view of the support of $g^{α}$ and $Q_{\infty}^{α}$ , we derive
$\begin{aligned} | \int_{R_{y}^{3}} [⟨ \cdot ⟩^{5} h g^{α} (t, y, \cdot)] (\overset{ˇ}{\frac{x}{t}}) d y - [⟨ \cdot ⟩^{5} h Q_{\infty}^{α}] (\overset{ˇ}{\frac{x}{t}}) | ≲ \frac{\log^{5} (2 + t)}{2 + t} sup_{| v | \leq β_{α}} | h (v) | . \end{aligned}$
(2.17)
This leaves us with proving that, for any $h \in C^{1} (R_{v}^{3})$ ,
$\begin{aligned} | t^{3} \int_{R_{v}^{3}} h (v) f^{α} (t, x, v) d v - \int_{R_{y}^{3}} [⟨ \cdot ⟩^{5} h g^{α} (t, y, \cdot)] (\overset{ˇ}{\frac{x}{t}}) d y | ≲ \frac{\log^{6} (2 + t)}{2 + t} sup_{| v | \leq β_{α}} (| h (v) | + | \nabla_{v} h (v) |) . \end{aligned}$
(2.18)
First, use Lemma 2.1 and the change of variables $y = x - t \hat{v}$ to derive
$\begin{aligned} t^{3} \int_{R_{v}^{3}} h (v) f^{α} (t, x, v) d v = t^{3} \int_{| v | \leq β_{α}} h (v) g^{α} (t, x - t \hat{v}, v) d v = \int_{| x - y | \leq \hat{β_{α}} t} [⟨ \cdot ⟩^{5} h g^{α} (t, y, \cdot)] (\overset{ˇ}{\frac{x - y}{t}}) d y . \end{aligned}$
Here we observe the correct function evaluated in $\overset{ˇ}{(x - y) / t}$ instead of $\overset{ˇ}{x} / t$ . We force the desired term to appear by writing
$\begin{aligned} \int_{| x - y | < t} [⟨ \cdot ⟩^{5} h g^{α} (t, y, \cdot)] (\overset{ˇ}{\frac{x - y}{t}}) d y = & \int_{R_{y}^{3}} [⟨ \cdot ⟩^{5} h g^{α} (t, y, \cdot)] (\overset{ˇ}{\frac{x}{t}}) d y \\ + \int_{| x - y | < t} [⟨ \cdot ⟩^{5} h g^{α} (t, y, \cdot)] (\overset{ˇ}{\frac{x - y}{t}}) d y - \int_{| x - y | < t} [⟨ \cdot ⟩^{5} h g^{α} (t, y, \cdot)] (\overset{ˇ}{\frac{x}{t}}) d y \\ - \int_{| x - y | \geq t} [⟨ \cdot ⟩^{5} h g^{α} (t, y, \cdot)] (\overset{ˇ}{\frac{x}{t}}) d y \\ = & I_{0} + I_{1} + I_{2} . \end{aligned}$
We now show that $I_{1}$ and $I_{2}$ are both $O (\log^{6} (2 + t) (2 + t)^{- 1})$ .

Estimate of $I_{1}$ . For a fixed $y$ we want to apply the mean value theorem to $G : v \mapsto [⟨ \cdot ⟩^{5} h g^{α} (t, y, \cdot)] (\overset{ˇ}{v})$ . Since $| \nabla_{v} \overset{ˇ}{v} | ≲ ⟨ \overset{ˇ}{v} ⟩^{3}$ , by differentiating $G$ we get
$\begin{aligned} | \nabla_{v} G (v) | & ≲ ⟨ \overset{ˇ}{v} ⟩^{8} | \nabla_{v} h (\overset{ˇ}{v}) | | g^{α} (t, y, \overset{ˇ}{v}) | + ⟨ \overset{ˇ}{v} ⟩^{7} | h (\overset{ˇ}{v}) | | g^{α} (t, y, \overset{ˇ}{v}) | + ⟨ \overset{ˇ}{v} ⟩^{8} | h (\overset{ˇ}{v}) | | \nabla_{v} g^{α} (t, y, \overset{ˇ}{v}) | \\ ≲ sup_{| u | \leq β_{α}} ⟨ u ⟩^{8} (| g^{α} (t, y, u) | + | \nabla_{v} g^{α} (t, y, u) |) (| h (u) | + | \nabla_{v} h (u) |) \\ ≲ \log^{2} (2 + t) sup_{| u | \leq β_{α}} (| h (u) | + | \nabla_{v} h (u) |), \end{aligned}$
by Proposition 2.6. Now by the mean value theorem and using the support of $g^{α}$ , we get,
$\begin{aligned} | I_{1} | ≲ sup_{| v | \leq β_{α}} (| h (v) | + | \nabla_{v} h (v) |) \int_{| y | ≲ \log (2 + t)} \frac{| y |}{t} \log^{2} (2 + t) d y ≲ \frac{\log^{6} (2 + t)}{2 + t} sup_{| v | \leq β_{α}} (| h (v) | + | \nabla_{v} h (v) |) . \end{aligned}$
(2.19)
Estimate of $I_{2}$ . Recall that $| x | < t$ , this allows us to get, for $v = \overset{ˇ}{x} / t$ and $| y - x | \geq t$ ,
$\begin{aligned} 1 = ⟨ v ⟩^{2} (1 - {| \frac{x}{t} |}^{2}) \leq \frac{| y | (t + | x |) ⟨ v ⟩^{2}}{t^{2}} \leq 2 \frac{| y | ⟨ v ⟩^{2}}{t}, \end{aligned}$
implying that
$\begin{aligned} | I_{2} | \leq \frac{2}{t} \int_{| x - y | \geq t} | y | [⟨ \cdot ⟩^{7} h g^{α} (t, y, \cdot)] (\overset{ˇ}{\frac{x}{t}}) d y ≲ sup_{| v | \leq β_{α}} | h (v) | \frac{2}{t} \int_{| y | ≲ \log (t)} | y | d y ≲ \frac{\log^{4} (2 + t)}{2 + t} sup_{| v | \leq β_{α}} | h (v) | . \end{aligned}$
Combining these estimates, we get (2.18). With (2.17), this implies the result.
3. Modified Scattering

3.1. Estimations of the Fields

Let us start by recalling the Glassey–Strauss decomposition.

Proposition 3.1
Let $t \geq 0$ and $x \in R_{x}^{3}$ . The following decomposition of the field holds.
$\begin{aligned} E (t, x) = E_{T} (t, x) + E_{S} (t, x) + E_{data} (t, x), \end{aligned}$
(3.1)
where
$\begin{aligned} E_{T} (t, x) = - \sum_{1 \leq α \leq N} e_{α} \int_{| x - y | \leq t} \int_{R_{v}^{3}} \frac{(ω + \hat{v_{α}}) (1 - | \hat{v_{α}} |^{2})}{(1 + \hat{v_{α}} \cdot ω)^{2}} f_{α} (t - | x - y |, y, v) d v \frac{d y}{| y - x |^{2}}, \end{aligned}$
(3.2)

$\begin{aligned} E_{S} (t, x) = \sum_{1 \leq α \leq N} e_{α}^{2} \int_{| y - x | \leq t} \int_{R_{v}^{3}} \frac{ω + \hat{v_{α}}}{1 + \hat{v_{α}} \cdot ω} (E + \hat{v_{α}} \cdot B) (t - | x - y |, y) \cdot \nabla_{v} f_{α} (t - | x - y |, y, v) d v \frac{d y}{| x - y |}, \end{aligned}$
(3.3)

$\begin{aligned} E_{data} (t, x) = E (t, x) - \sum_{1 \leq α \leq N} \frac{e_{α}}{t} \int_{| y - x | = t} \int_{R_{v}^{3}} \frac{ω - (\hat{v_{α}} \cdot ω) \hat{v_{α}}}{1 + \hat{v_{α}} \cdot ω} f_{α 0} (y, v) d v d S_{y}, \end{aligned}$
(3.4)
with
$\begin{aligned} E (t, x) = \frac{1}{4 π t^{2}} \int_{| y - x | = t} [E_{0} (y) + ((y - x) \cdot \nabla) E_{0} (y) + t \nabla \times B_{0} (y)] d S_{y} - \sum_{1 \leq α \leq N} \frac{e_{α}}{4 π t} \int_{| y - x | = t} \int_{R_{v}^{3}} \hat{v_{α}} f_{α 0} (y, v) d v d S_{y}, \end{aligned}$
and $ω = \frac{x - y}{| x - y |}$ .
Remark 3.2
The same decomposition holds for $B (t, x) = B_{T} (t, x) + B_{S} (t, x) + B_{data} (t, x)$ . The expression of $B_{T}$ and $B_{S}$ is obtained by replacing $ω + \hat{v_{α}}$ by $ω \times \hat{v_{α}}$ in $E_{T}$ and $E_{S}$ . Moreover, the expression of $B_{data}$ only depends on the initial data, so the estimates follow similarly in the next propositions. In the following, we restrict ourselves to the study of $E$ as the analysis of $B$ is similar.

As stated in the introduction, we want to identify the part of $E = E_{data} + E_{S} + E_{T}$ that decays like $t^{- 2}$ for $| x | \leq γ t$ , with $γ < 1$ . We start by showing that $E_{data}$ and $E_{S}$ decay at least as $t^{- 3}$ . For this, we have to improve the estimate obtained by Glassey and Strauss (1987) for $E_{S}$ .
Proposition 3.3
For all $(t, x) \in R_{+} \times R_{x}^{3}$ , we have the following estimate
$\begin{aligned} | (E_{data}, B_{data}) (t, x) | ≲ ⟨ t ⟩^{- 1} 1_{| t - | x | | \leq k} . \end{aligned}$
(3.5)
Proof.
First, recall the expression of $E_{data}$ from (3.4). Using the support of $f_{α 0}, E_{0}, B_{0}$ every term of $E_{data}$ is bounded by
$\begin{aligned} C (\frac{1}{t} + \frac{1}{t^{2}}) \int_{\begin{matrix} | y - x | = t \\ | y | \leq k \end{matrix}} d S_{y} 1_{| t - | x | | < k}, \end{aligned}$
which implies the result.
Proposition 3.4
For all $(t, x) \in R_{+} \times R_{x}^{3}$ , we have the following estimate
$\begin{aligned} | (E_{S}, B_{S}) (t, x) | ≲ (t + | x | + 2 k)^{- 1} (t - | x | + 2 k)^{- 2} . \end{aligned}$
(3.6)
Proof.
Here we slightly refine the analysis performed for $E_{S}$ in Glassey and Strauss (1987, Lemma 6) by exploiting the support of $f_{α}$ . First recall (3.3) and $\nabla_{v} \cdot (E + \hat{v} \times B) = 0$ . By integration by parts we obtain
$\begin{aligned} E_{S} (t, x) = - \sum_{1 \leq α \leq N} e_{α}^{2} \int_{| y - x | \leq t} \int_{R_{v}^{3}} \nabla_{v} [\frac{ω + \hat{v_{α}}}{1 + \hat{v_{α}} \cdot ω}] \cdot (E + \hat{v_{α}} \times B) (t - | x - y |, y) f_{α} (t - | x - y |, y, v) d v \frac{d y}{| x - y |} . \end{aligned}$
Now, recall that the kernel where $ω$ appears is bounded on the support of $f_{α}$ . Then by Proposition 2.9 we have
$\begin{aligned} \int_{R_{v}^{3}} | (E + \hat{v_{α}} \times B) (t - | x - y |, y) | f_{α} (t - | x - y |, y, v) d v & ≲ \frac{1_{| y | \leq {\hat{β}}_{\max} (t - | y - x |) + k}}{(t - | x - y | + | y | + 2 k)^{4} (t - | x - y | - | y | + 2 k)} \\ ≲ (t - | x - y | + | y | + 2 k)^{- 5}, \end{aligned}$
by using the support of $f_{α}$ . This implies
$\begin{aligned} | E_{S} (t, x) | ≲ \int_{| y - x | \leq t} \frac{1}{(t - | y - x | + | y | + 2 k)^{5}} \frac{d y}{| x - y |} . \end{aligned}$
Now, by Glassey and Strauss (1987, Lemma 7), we have
$\begin{aligned} I := \int_{| y - x | \leq t} \frac{1}{(t - | y - x | + | y | + 2 k)^{5}} \frac{d y}{| x - y |} = \frac{1}{r} \int_{0}^{t} \int_{a}^{b} \frac{λ d λ d τ}{(τ + λ + 2 k)^{5}} \leq \frac{1}{r} \int_{0}^{t} \int_{a}^{b} \frac{d λ d τ}{(τ + λ + 2 k)^{4}}, \end{aligned}$
where $τ = t - | x - y |, λ = | y |$ and $r = | x |$ . Moreover, the bounds of the integral are $a = | r - t + τ |$ and $b = r + t - τ$ . We first write, as $b - a \leq b - (t - r - τ) = 2 r$ and $τ + b = t + r$ ,
$\begin{aligned} I ≲ \frac{1}{r} \int_{0}^{t} \frac{(b - a) (2 τ + b + a + 4 k)^{2}}{(τ + a + 2 k)^{3} (τ + b + 2 k)^{3}} d τ ≲ \frac{1}{r} \int_{0}^{t} \frac{(b - a) d τ}{(τ + a + 2 k)^{3} (τ + b + 2 k)} ≲ \frac{1}{t + r + 2 k} \int_{0}^{t} \frac{d τ}{(τ + a + 2 k)^{3}} . \end{aligned}$
If $t \leq | x | < t + k$ , then $I ≲ (t + | x | + 2 k)^{- 1} ≲ (t + | x | + 2 k)^{- 1} (t - | x | + 2 k)^{- 2}$ . Otherwise, $| x | < t$ and we can split $I$ in two parts
$\begin{aligned} I ≲ \frac{1}{t + r + 2 k} \int_{0}^{t - r} \frac{d τ}{(t - r + 2 k)^{3}} + \frac{1}{t + r + 2 k} \int_{t - r}^{t} \frac{d τ}{(τ + 2 k)^{3}} ≲ \frac{1}{(t + r + 2 k) (t - r + 2 k)^{2}} . \end{aligned}$

3.2. Asymptotic Expansion of the Fields

Having proved the estimate on $(E_{S}, E_{data})$ , it remains to study $E_{T}$ . The goal of this subsection is to find a form of asymptotic expansion for $E_{T}$ .

Proposition 3.5
Let $v \in R_{v}^{3}$ . Consider
$\begin{aligned} E_{α} (v) := - \int_{\begin{matrix} | y | \leq 1 \\ | y + \hat{v} | < 1 - | y | \end{matrix}} [⟨ \cdot ⟩^{5} W (\frac{y}{| y |}, \cdot) Q_{\infty}^{α}] (\frac{\overset{ˇ}{y + \hat{v}}}{1 - | y |}) \frac{1}{(1 - | y |)^{3}} \frac{d y}{| y |^{2}}, \end{aligned}$
(3.7)

$\begin{aligned} B_{α} (v) := - \int_{\begin{matrix} | y | \leq 1 \\ | y + \hat{v} | < 1 - | y | \end{matrix}} [⟨ \cdot ⟩^{5} W (\frac{y}{| y |}, \cdot) Q_{\infty}^{α}] (\frac{\overset{ˇ}{y + \hat{v}}}{1 - | y |}) \frac{1}{(1 - | y |)^{3}} \frac{d y}{| y |^{2}}, \end{aligned}$
(3.8)
with $W (ω, v) = \frac{(ω + \hat{v})}{⟨ v ⟩^{2} (1 + \hat{v} \cdot ω)^{2}}$ and $W (ω, v) = \frac{(ω \times \hat{v})}{⟨ v ⟩^{2} (1 + \hat{v} \cdot ω)^{2}}$ . Then, $E_{α}, B_{α} \in C^{0} (R_{v}^{3})$ .
Proof.
Using the support of $Q_{\infty}^{α}$ we know that we integrate over ${y | | y + \hat{v} | \leq \hat{β_{α}} (1 - | y |)}$ so $| y | \leq (| \hat{v} | + \hat{β_{α}}) / (1 + \hat{β_{α}}) < 1$ and thus $1 - | y | \geq 1 - (| \hat{v} | + \hat{β_{α}}) / (1 + \hat{β_{α}}) > 0$ , implying that the integral is well defined. The continuity follows as $Q_{\infty}^{α} \in C^{0} (R^{3})$ .

We begin by performing the change of variables $z = (y - x) / t$ , so that
$\begin{aligned} E_{T} (t, x) = \sum_{1 \leq α \leq N} e_{α} m_{α}^{3} E_{α, T}, \end{aligned}$
where
$\begin{aligned} E_{α, T} (t, x) := - \frac{1}{t^{2}} \int_{| y | \leq 1} \int_{R_{v}^{3}} t^{3} W (\frac{y}{| y |}, v) f^{α} (t (1 - | y |), t y + x, v) d v \frac{d y}{| y |^{2}} . \end{aligned}$

Proposition 3.6
Let $0 < γ < 1$ . Then, there exists $T (γ) \geq 0$ such that for all $t \geq T (γ)$ and all $| x | \leq γ t$
$\begin{aligned} | t^{2} E_{α, T} (t, x) - E_{α} (\overset{ˇ}{\frac{x}{t}}) | + | t^{2} B_{α, T} (t, x) - B_{α} (\overset{ˇ}{\frac{x}{t}}) | ≲ \frac{\log^{6} (2 + t)}{2 + t} I (γ), \end{aligned}$
(3.9)
where $I (γ)$ is a constant depending on $γ$ .
Proof.
We begin by showing that for $t$ large enough (depending on $γ$ ) and $| x | \leq γ t$ we have
$\begin{aligned} E_{α, T} (t, x) = - \frac{1}{t^{2}} \int_{\begin{matrix} | y | \leq 1 \\ | y + \frac{x}{t} | < 1 - | y | \end{matrix}} \int_{R_{v}^{3}} t^{3} W (\frac{y}{| y |}, v) f^{α} (t (1 - | y |), t y + x, v) d v \frac{d y}{| y |^{2}} . \end{aligned}$
Write $t^{'} = t (1 - | y |), x^{'} = t (y + (x / t))$ . On the support of $f^{α}$ we know that $| x^{'} | \leq \hat{β_{α}} t^{'} + k \leq {\hat{β}}_{\max} t^{'} + k$ . Thus, for $t^{'} > k / (1 - {\hat{β}}_{\max})$ , we have $t^{'} > | x^{'} |$ . Now, on the support of $f^{α}$ again we have
$\begin{aligned} t^{'} = t - | x^{'} - x | \geq (1 - γ) t - {\hat{β}}_{\max} t^{'} - k, \end{aligned}$
leaving us with $t^{'} \geq (t (1 - γ)) / (1 + {\hat{β}}_{\max}) - (k / (1 + {\hat{β}}_{\max}))$ . Finally, for
$\begin{aligned} t \geq T (γ) := \frac{2 k}{(1 - γ) (1 + {\hat{β}}_{\max})} + 1, \end{aligned}$
we derive $t^{'} > k / (1 - {\hat{β}}_{\max})$ and thus $t^{'} > | x^{'} |$ .

Consider now
$\begin{aligned} E_{T} (y, t) := t^{3} (1 - | y |)^{3} \int_{R_{v}^{3}} W (\frac{y}{| y |}, v) f^{α} (t (1 - | y |), t y + x, v) d v - [⟨ \cdot ⟩^{5} W (\frac{y}{| y |}, \cdot) Q_{\infty}^{α}] (\overset{ˇ}{\frac{y + \frac{x}{t}}{1 - | y |}}), \end{aligned}$
(3.10)
so that
$\begin{aligned} t^{2} E_{α, T} (t, x) - E_{α} (\overset{ˇ}{\frac{x}{t}}) = - \int_{\begin{matrix} | y | \leq 1 \\ | y + \frac{x}{t} | < 1 - | y | \end{matrix}} E_{T} (y, t) \frac{1}{(1 - | y |)^{3}} \frac{d y}{| y |^{2}} . \end{aligned}$
(3.11)
Recall (3.10). Using the support of $f^{α}$ and $Q_{\infty}^{α}$ , $E_{T} (t, y)$ vanishes for $| y | > (γ + {\hat{β}}_{\max}) / (1 + {\hat{β}}_{\max}) + (k / ((1 + {\hat{β}}_{\max}) t))$ . Thus, for $t \geq T (γ) \geq (2 k / ((1 - γ) (1 + {\hat{β}}_{\max})))$ we have, on the support of $E_{T} (t, \cdot)$ ,
$\begin{aligned} 1 - | y | \geq \frac{1}{2} (\frac{1 - γ}{1 + {\hat{β}}_{\max}}) =: K (γ) > 0. \end{aligned}$

We now apply Proposition 2.9 with $h (v) = W ((y / | y |), v)$ . Since $W \in C^{1} (S^{2} \times R^{3})$ we derive
$\begin{aligned} \int_{\begin{matrix} | y | \leq 1 \\ | y + \frac{x}{t} | < 1 - | y | \end{matrix}} | E_{T} (y, t) | \frac{1}{(1 - | y |)^{3}} \frac{d y}{| y |^{2}} ≲ K (γ)^{- 4} \int_{\begin{matrix} | y | \leq 1 \\ | y + \frac{x}{t} | < 1 - | y | \end{matrix}} \frac{\log^{6} (2 + t (1 - | y |))}{2 + t} \frac{d y}{| y |^{2}} ≲ \frac{\log^{6} (2 + t)}{2 + t} K (γ)^{- 4}, \end{aligned}$
which concludes the proof.

Before giving the final estimate, we define the asymptotic fields $E$ and $B$ by
$\begin{aligned} E := \sum_{1 \leq α \leq N} e_{α} m_{α}^{3} E_{α}, B := \sum_{1 \leq α \leq N} e_{α} m_{α}^{3} B_{α} . \end{aligned}$

Corollary 3.7
Let $0 < γ < 1, t \geq T (γ)$ and $| x | \leq γ t$ . We have the following estimate
$\begin{aligned} | t^{2} E (t, x) - E (\overset{ˇ}{\frac{x}{t}}) | + | t^{2} B (t, x) - B (\overset{ˇ}{\frac{x}{t}}) | ≲ \frac{\log^{6} (2 + t)}{2 + t} C (γ) . \end{aligned}$
(3.12)
where $C (γ)$ is a constant depending only on $γ$ and $k$ .
Proof.
This follows from the decomposition $E = E_{S} + E_{data} + \sum_{α} E_{α, T}$ and Propositions 3.3–3.4 and 3.6.
3.3. Proof of the Modified Scattering Theorem

In this subsection, we finish the proof of Theorem 1.5. First, let us detail two preliminary results.

Proposition 3.8

For any $α$ , all $t \geq 0, | v | \leq β_{α}$ and all $x$ in the support of $g^{α} (t, \cdot)$ , i.e. $| x | \leq C \log (2 + t)$ , we have

\begin{aligned} | t^{2} E (t, x + t \hat{v}) - E (v) | + | t^{2} B (t, x + t \hat{v}) - B (v) | ≲ \frac{\log^{6} (2 + t)}{2 + t} . \end{aligned}

(3.13)

Proof.

Using Corollary 3.7 we already know that for $t \geq T ({\hat{β}}_{\max})$ , since $| \hat{v} | \leq {\hat{β}}_{\max}$

\begin{aligned} | t^{2} E (t, t \hat{v}) - E (v) | ≲ \frac{\log^{6} (2 + t)}{2 + t} C ({\hat{β}}_{\max}) ≲ \frac{\log^{6} (2 + t)}{2 + t}, \end{aligned}

where here we omit the constant

C ({\hat{β}}_{\max})

since it only depends on

k

and

C_{0}

. Now we only need to prove that

\begin{aligned} | t^{2} E (t, x + t \hat{v}) - t^{2} E (t, t \hat{v}) | ≲ \frac{\log^{2} (2 + t)}{2 + t} . \end{aligned}

(3.14)

We will prove the above inequality using the mean value theorem. For this we need to consider

y = λ (x + t \hat{v}) + (1 - λ) t \hat{v} \in [t \hat{v}, x + t \hat{v}]

with

λ \in [0, 1]

, so that

y = λ x + t \hat{v}

. Using (1.5) we obtain

\begin{aligned} | \nabla_{x} E (t, y) | ≲ \frac{\log (t + | y | + 2 k)}{(t + | y | + 2 k) (t - | y | + 2 k)^{2}} . \end{aligned}

Consider

t \geq T_{1}

large enough so we have

((C \log (2 + t)) / t) \leq (1 - {\hat{β}}_{\max}) / 2

. This implies

| y | \leq ((1 + {\hat{β}}_{\max}) / 2) t

and then

t - | y | + 2 k ≳ 2 + t

as well as

\log (t + | y | + 2 k) ≲ \log (2 + t)

. This grants us

\begin{aligned} | \nabla_{x} E (t, y) | ≲ \frac{\log (2 + t)}{(2 + t)^{3}} . \end{aligned}

Now, the mean value theorem and

| x | ≲ \log (2 + t)

allow us to derive (3.14) and obtain the result for

t \geq T_{1}

. It also holds on the compact interval of time

[0, T_{1}]

since

E

and

E

are bounded.

Now, recall (2.6). Thanks to the estimate (1.4) and Propositions 2.6 and 3.8, we have

\begin{aligned} \partial_{t} g^{α} (t, x, v) = & - \frac{t}{v^{0}} \frac{e_{α}}{m_{α}} [\hat{v} (L (t, x + t \hat{v}, v) \cdot \hat{v}) - L (t, x + t \hat{v}, v)] \cdot \nabla_{x} g^{α} (t, x, v) + O (\frac{\log^{2} (2 + t)}{(2 + t)^{2}}) \\ = & - \frac{1}{t v^{0}} \frac{e_{α}}{m_{α}} [\hat{v} (L (v) \cdot \hat{v}) - L (v)] \cdot \nabla_{x} g^{α} (t, x, v) + O (\frac{\log^{6} (2 + t)}{(2 + t)^{2}}), \end{aligned}

(3.15)

where

L (v) := E (v) + \hat{v} \times B (v)

is the asymptotic Lorentz force. We observe that the first term on the right-hand side is of order

t^{- 1}

, which is not integrable. We thus consider corrections to the linear characteristics to cancel it. We define the following corrections

\begin{aligned} D_{v} := - \hat{v} \cdot L (v), C_{v} := - L (v) . \end{aligned}

(3.16)

Now, let us consider

\begin{aligned} h^{α} (t, x, v) & := f^{α} (t, x + t \hat{v} + \frac{e_{α}}{m_{α} v^{0}} \log (t) (C_{v} - \hat{v} D_{v}), v) = g^{α} (t, x + \frac{e_{α}}{m_{α} v^{0}} \log (t) (C_{v} - \hat{v} D_{v}), v), \end{aligned}

(3.17)

\begin{aligned} h_{α} (t, x, v) & := f_{α} (t, x + t \hat{v_{α}} + \frac{e_{α}}{v_{α}^{0}} \log (t) (C_{v_{α}} - \hat{v_{α}} D_{v_{α}}), v) = g_{α} (t, x + \frac{e_{α}}{v_{α}^{0}} \log (t) (C_{v_{α}} - \hat{v_{α}} D_{v_{α}}), v) . \end{aligned}

(3.18)

Proposition 3.9

For any $α$ , all $t \geq 0$ and all $(x, v) \in R_{x}^{3} \times R_{v}^{3}$ , we have

\begin{aligned} | f_{α} (t, x + t \hat{v_{α}} + \frac{e_{α}}{v_{α}^{0}} \log (t) (C_{v_{α}} - \hat{v_{α}} D_{v_{α}}), v) - f_{α \infty} (x, v) | ≲ \frac{\log^{6} (2 + t)}{(2 + t)} . \end{aligned}

(3.19)

Proof.

We directly have, thanks to the above equation (3.15) on $\partial_{t} g^{α}$ ,

\begin{aligned} \partial_{t} h^{α} (t, x, v) & = \frac{1}{t} \frac{e_{α}}{m_{α} v^{0}} (C_{v} - \hat{v} D_{v}) \cdot (\nabla_{x} g^{α}) (t, x + \frac{e_{α}}{m_{α} v^{0}} \log (t) (C_{v} - \hat{v} D_{v}), v) + (\partial_{t} g^{α}) (t, x + \frac{e_{α}}{m_{α} v^{0}} \log (t) (C_{v} - \hat{v} D_{v}), v) \\ = \frac{1}{t} \frac{e_{α}}{m_{α} v^{0}} [(C_{v} - \hat{v} D_{v}) - (\hat{v} (L (v) \cdot \hat{v}) - L (v))] \cdot \nabla_{x} g^{α} + O (\frac{\log^{6} (2 + t)}{(2 + t)^{2}}) \\ = O (\frac{\log^{6} (2 + t)}{(2 + t)^{2}}) . \end{aligned}

Consequently, there exists

{f^{α}}_{\infty} \in C^{0} (R_{x}^{3} \times R_{v}^{3})

such that

\begin{aligned} | f^{α} (t, x + t \hat{v} + \frac{e_{α}}{m_{α} v^{0}} \log (t) (C_{v} - \hat{v} D_{v}), v) - {f^{α}}_{\infty} (x, v) | ≲ \frac{\log^{6} (2 + t)}{(2 + t)} . \end{aligned}

(3.20)

This directly implies the result, with

f_{α \infty} (x, v) := {f^{α}}_{\infty} (x, v_{α})

, where we recall

v_{α}^{0} = \sqrt{m_{α}^{2} + | v |^{2}}

and

v_{α} = v / m_{α}

It remains to prove the estimate of Theorem 1.5. For this, write ${\tilde{C}}_{v} := C_{v} - \hat{v} D_{v}$ and let

\begin{aligned} h_{α} (t, x, v) & = f_{α} (t, x + t \hat{v_{α}} + \frac{e_{α}}{v_{α}^{0}} \log (t) {\tilde{C}}_{v_{α}}, v), \\ {\tilde{h}}_{α} (t, x, v) & := f_{α} (t v_{α}^{0} + \frac{e_{α}}{v_{α}^{0}} \log (t) D_{v_{α}}, x + t v + \frac{e_{α}}{v_{α}^{0}} \log (t) C_{v_{α}}, v) . \end{aligned}

Remark 3.10

Notice that for ${\tilde{h}}_{α}$ to be well-defined, one needs to have $t v_{α}^{0} + (e_{α} / v_{α}^{0}) \log (t) D_{v_{α}} \geq 0$ . Since $| v | \leq β$ , this holds whenever $t$ is large enough.

We already know that

\begin{aligned} | h_{α} (t v_{α}^{0}, x, v) - f_{α \infty} (x, v) | ≲ \frac{\log^{6} (2 + t)}{2 + t} . \end{aligned}

Moreover,

\begin{aligned} {\tilde{h}}_{α} (t, x, v) & = f_{α} (t v_{α}^{0} + \frac{e_{α}}{v_{α}^{0}} \log (t) D_{v_{α}}, x + (t v_{α}^{0} + \frac{e_{α}}{v_{α}^{0}} \log (t) D_{v_{α}}) \hat{v_{α}} + \frac{e_{α}}{v_{α}^{0}} \log (t) {\tilde{C}}_{v_{α}}, v) \\ = g_{α} (t v_{α}^{0} + \frac{e_{α}}{v_{α}^{0}} \log (t) D_{v_{α}}, x + \frac{e_{α}}{v_{α}^{0}} \log (t) {\tilde{C}}_{v_{α}}, v) . \end{aligned}

Now we know that, thanks to (1.4) and Proposition 2.6,

| \partial_{t} g_{α} (t, \cdot) | ≲ 1 / (2 + t)

. Hence, by the mean value theorem, we derive, for

t

large enough,

\begin{aligned} {\tilde{h}}_{α} (t, x, v) = g_{α} (t v_{α}^{0}, x + \frac{e_{α}}{v_{α}^{0}} \log (t) {\tilde{C}}_{v_{α}}, v) + O (\frac{\log (2 + t)}{2 + t}) . \end{aligned}

Finally, using the expression of

h_{α}

, we obtain

\begin{aligned} {\tilde{h}}_{α} (t, x, v) & = g_{α} (t v_{α}^{0}, x - \frac{e_{α}}{v_{α}^{0}} \log (v_{α}^{0}) {\tilde{C}}_{v_{α}} + \frac{e_{α}}{v_{α}^{0}} \log (t v_{α}^{0}) {\tilde{C}}_{v_{α}}, v) + O (\frac{\log (2 + t)}{2 + t}) \\ = h_{α} (t v_{α}^{0}, x - \frac{e_{α}}{v_{α}^{0}} \log (v_{α}^{0}) {\tilde{C}}_{v_{α}}, v) + O (\frac{\log (2 + t)}{2 + t}) \\ = f_{α \infty} (x - \frac{e_{α}}{v_{α}^{0}} \log (v_{α}^{0}) {\tilde{C}}_{v_{α}}, v) + O (\frac{\log^{6} (2 + t)}{2 + t}) . \end{aligned}

So we derive the estimate of Theorem 1.5 by taking

{\tilde{f}}_{α \infty} (x, v) = f_{α \infty} (x - (e_{α} / v_{α}^{0}) \log (v_{α}^{0}) {\tilde{C}}_{v_{α}}, v)

. However, to prove Theorem 1.5, it remains to show that

{\tilde{f}}_{α \infty}

has a compact support.

Proposition 3.11

There exists $T > 0$ and a constant $C > 0$ such that, for all $t \geq T$

\begin{aligned} supp h^{α} (t, \cdot) \subset {(x, v) \in R_{x}^{3} \times R_{v}^{3}, | x | \leq C, | v | \leq C}, \end{aligned}

(3.21)

i.e.

h^{α} (t, \cdot)

is compactly supported, uniformly in

t

Proof.

Recall the expression of $h^{α}$

\begin{aligned} h^{α} (t, x, v) = f^{α} (t, x + t \hat{v} + \frac{e_{α}}{m_{α} v^{0}} \log (t) (C_{v} - \hat{v} D_{v}), v) = g^{α} (t, x + \frac{e_{α}}{m_{α} v^{0}} \log (t) (C_{v} - \hat{v} D_{v}), v), \end{aligned}

and assume

g^{α} (t, x + (e_{α} / (m_{α} v^{0})) \log (t) (C_{v} - \hat{v} D_{v}), v) \neq 0

. Consider the characteristics

\begin{aligned} X (s) = X (s, t, x + \frac{e_{α}}{m_{α} v^{0}} \log (t) (C_{v} - \hat{v} D_{v}), v), V (s) = V (s, t, x + \frac{e_{α}}{m_{α} v^{0}} \log (t) (C_{v} - \hat{v} D_{v}), v) . \end{aligned}

Now recall the ODE (2.7) satisfied by

(X, V)

. We have

| \dot{V} (s) | ≲ | L (s, X (s) + s \hat{V} (s), V (s)) | ≲ (s + 2)^{- 2}

, according to Lemma 2.3 and (1.4). Thus, there exists

v_{\infty} \in R_{v}^{3}

such that

\begin{aligned} | V (s) - v_{\infty} | ≲ \frac{1}{2 + s} . \end{aligned}

Now, using (1.4)–(1.5), the mean value theorem and the above equation, we derive

\begin{aligned} \dot{X} (s) & = \frac{s}{V^{0} (s)} \frac{e_{α}}{m_{α}} [\hat{V} (s) (L (s, X (s) + s \hat{V} (s), V (s)) \cdot \hat{V} (s)) - L (s, X (s) + s \hat{V} (s), V (s))] \\ = \frac{s}{v_{\infty}^{0}} \frac{e_{α}}{m_{α}} [\hat{v_{\infty}} (L (s, X (s) + s \hat{v_{\infty}}, v_{\infty}) \cdot \hat{v_{\infty}}) - L (s, X (s) + s \hat{v_{\infty}}, v_{\infty})] + O (\frac{\log (2 + s)}{(2 + s)^{2}}) . \end{aligned}

This grants us, by applying Proposition 3.8,

\begin{aligned} \dot{X} (s) & = \frac{1}{s v_{\infty}^{0}} \frac{e_{α}}{m_{α}} [\hat{v_{\infty}} (L (v_{\infty}) \cdot \hat{v_{\infty}}) - L (v_{\infty})] + O (\frac{\log^{6} (2 + s)}{(2 + s)^{2}}) \\ = \frac{1}{s} \frac{e_{α}}{m_{α} v_{\infty}^{0}} {\tilde{C}}_{v_{\infty}} + O (\frac{\log^{6} (2 + s)}{(2 + s)^{2}}) . \end{aligned}

By integrating we derive, for

t \geq 1

\begin{aligned} | X (t) - \frac{e_{α}}{m_{α} v_{\infty}^{0}} \log (t) {\tilde{C}}_{v_{\infty}} | \leq X (1) + C . \end{aligned}

Finally, one can notice that since

| V (s) - v_{\infty} | ≲ (2 + s)^{- 1}

and

| V (s) | \leq β_{α}

, we have

| v_{\infty} | \leq β_{α}

. Then, it yields

\begin{aligned} | E (v_{\infty}) - E (V (t)) | & \leq | E (v_{\infty}) - t^{2} E (t, t \hat{v_{\infty}}) | + | E (V (t)) - t^{2} E (t, t \hat{V} (t)) | + t^{2} | E (t, t \hat{v_{\infty}}) - E (t, t \hat{V} (t)) | \\ ≲ \frac{\log^{6} (2 + t)}{2 + t} . \end{aligned}

By deriving the same estimate for

B

, we obtain for

t

large enough

\begin{aligned} \log (t) | {\tilde{C}}_{v_{\infty}} - {\tilde{C}}_{V (t)} | ≲ 1. \end{aligned}

So finally,

\begin{aligned} | X (t) - \frac{e_{α}}{m_{α} V^{0} (t)} \log (t) {\tilde{C}}_{V (t)} | ≲ 1, \end{aligned}

that is, since

V (t) = v

and

X (t) = x + \log (t) (e_{α} / m_{α} v^{0}) {\tilde{C}}_{v}

we have

\begin{aligned} | x | ≲ 1. \end{aligned}

Corollary 3.12

The functions $h_{α}$ and ${\tilde{h}}_{α}$ are compactly supported. That is $\exists M > 0$ , such that $\forall t > 0$ ,

\begin{aligned} supp h_{α} (t, \cdot) \subset {(x, v) \in R_{x}^{3} \times R_{v}^{3}, | x | \leq M, | v | \leq M}, \end{aligned}

\begin{aligned} supp {\tilde{h}}_{α} (t, \cdot) \subset {(x, v) \in R_{x}^{3} \times R_{v}^{3}, | x | \leq M, | v | \leq M} . \end{aligned}

Proof.

Since $h_{α} (t, x, v) = h^{α} (t, x, v_{α})$ , the first result is immediate. Now, notice that ${\tilde{h}}_{α} (t, x, v) = h_{α} (s, y, v)$ with

\begin{aligned} s = t v_{α}^{0} + \log (t) \frac{e_{α}}{v_{α}^{0}} D_{v_{α}}, y = x + \frac{e_{α}}{v_{α}^{0}} (\log (t) - \log (s)) {\tilde{C}}_{v_{α}} . \end{aligned}

Assume

{\tilde{h}}_{α} (t, x, v) \neq 0

, then, by Proposition 3.11,

| y | \leq C

and

| x | ≲ C + | \log (t) - \log (s) |

. Moreover,

$| \log (t) - \log (s) | \to_{t \to + \infty}^{\log} (v_{α}^{0})$ so $t \mapsto | \log (t) - \log (s) |$ is bounded, uniformly in $v$ on the support of $f^{α}$ , by $κ$ . Then $| x | ≲ C + κ$ .

Corollary 3.13

The functions ${f^{α}}_{\infty}, f_{α \infty}$ and ${\tilde{f}}_{α \infty}$ are all compactly supported.

Footnotes

Funding

The author received no financial support for the research, authorship and/or publication of this article.

Declaration of Conflicting Interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

ORCID iD

Emile Breton

References

Bigorgne

(2020). Sharp asymptotic behavior of solutions of the 3D Vlasov–Maxwell system with small data. Communications in Mathematical Physics, 376(2), 893–992.

Bigorgne

(2023b). Scattering map for the Vlasov–Maxwell system around source-free electromagnetic fields. arXiv: 2312.12214.

Bigorgne

(2025). Global existence and modified scattering for the solutions to the Vlasov-Maxwell system with a small distribution function. Analysis & PDE, 18(3), 629–714.

Bigorgne

Ruiz

R. V.

(2024). Late-time asymptotics of small data solutions for the Vlasov–Poisson system. arXiv: 2404.05812 [math.AP].

Bigorgne

Velozo Ruiz

(2025). Modified scattering of small data solutions for the Vlasov–Poisson system with a repulsive potential. SIAM Journal on Mathematical Analysis, 57(1), 714–752.

Breton

(2025). A note on the non

l^{1}

-asymptotic completeness of the Vlasov–Maxwell system. arXiv: 2509.04025.

Choi

S. H.

S. Y.

(2011). Asymptotic behavior of the nonlinear vlasov equation with a self-consistent force. SIAM Journal on Mathematical Analysis, 43(5), 2050–2077.

Choi

S. H.

Kwon

(2016). Modified scattering for the Vlasov–Poisson system. Nonlinearity, 29(9), 2755.

Flynn

Ouyang

Pausader

Widmayer

(2023). Scattering map for the Vlasov–Poisson system. Peking Mathematical Journal, 6(2), 365–392.

10.

Glassey

R. T.

Schaeffer

J. W.

(1988). Global existence for the relativistic Vlasov–Maxwell system with nearly neutral initial data. Communications in Mathematical Physics, 119(3), 353–384.

11.

Glassey

R. T.

Strauss

W. A.

(1987). Absence of shocks in an initially dilute collisionless plasma. Communications in Mathematical Physics, 113(2), 191–208.

12.

Ionescu

A. D.

Pausader

Wang

Widmayer

(2021). On the asymptotic behavior of solutions to the Vlasov–Poisson system. International Mathematics Research Notices, 2022(12), 8865–8889.

13.

Luk

Strain

R. M.

(2016). Strichartz estimates and moment bounds for the relativistic Vlasov–Maxwell system. Archive for Rational Mechanics and Analysis, 219(1), 445–552.

14.

Pankavich

(2022). Asymptotic dynamics of dispersive, collisionless plasmas. Communications in Mathematical Physics, 391(2), 455–493.

15.

Pankavich

Ben-Artzi

(2024). Modified scattering of solutions to the relativistic Vlasov–Maxwell system inside the light cone. arXiv: 2306.11725 [math.AP].

16.

Pausader

Widmayer

(2021). Stability of a point charge for the Vlasov–Poisson system: The radial case. Communications in Mathematical Physics, 385(3), 1741–1769.

17.

Pausader

Widmayer

Yang

(2024). Stability of a point charge for the repulsive Vlasov–Poisson system. Journal of the European Mathematical Society, Published online first.

18.

Rein

(1990). Generic global solutions of the relativistic Vlasov–Maxwell system of plasma physics. Communications in Mathematical Physics, 135(1), 41–78.

19.

Schaeffer

(2004). A small data theorem for collisionless plasma that includes high velocity particles. Indiana University Mathematics Journal, 53(1), 1–34.

20.

Schlue

Taylor

(2024). Inverse modified scattering and polyhomogeneous expansions for the Vlasov–Poisson system. arXiv: 2404.15885 [math.AP].

21.

Wang

(2022). Propagation of regularity and long time behavior of the 3D massive relativistic transport equation II: Vlasov–Maxwell system. Communications in Mathematical Physics, 389(2), 715–812.

22.

Wei

Yang

(2021). On the 3D relativistic Vlasov–Maxwell system with large Maxwell field. Communications in Mathematical Physics, 383(3), 2275–2307.

Modified Scattering for Small Data Solutions to the Vlasov–Maxwell System: A Short Proof

Abstract

Keywords

1. Introduction and Main Results

1.1. General Context

2. Preliminary Results

3.1. Estimations of the Fields

Footnotes

Funding

Declaration of Conflicting Interests

ORCID iD

References