Sharp Asymptotics for the Solutions of the Three-Dimensional Massless Vlasov–Maxwell System with Small Data

Abstract

This paper is concerned with the asymptotic properties of the small data solutions to the massless Vlasov–Maxwell system in 3d. We use vector field methods to derive almost optimal decay estimates in null directions for the electromagnetic field, the particle density and their derivatives. No compact support assumption in x or v is required on the initial data, and the decay in v is in particular initially optimal. Consistently with Proposition 8.1 of Bigorgne (Asymptotic properties of small data solutions of the Vlasov–Maxwell system in high dimensions. arXiv:1712.09698, 2017), the Vlasov field is supposed to vanish initially for small velocities. In order to deal with the slow decay rate of the solutions near the light cone and to prove that the velocity support of the particle density remains bounded away from 0, we make crucial use of the null properties of the system.

1 Introduction

This article is part of a series of works concerning the asymptotic behavior of small data solutions to the Vlasov–Maxwell equations. The system is a classical model for collisionless plasma and is given, for K species of particles, by¹

$$\begin{aligned}&\sqrt{m_k^2+|v|^2}\partial _t f_k+v^i \partial _i f_k +e_k\left( \sqrt{m_k^2+|v|^2} {F_{0}}^j+ v^{q} {F_{q}}^j \right) \partial _{v^j} f_k = 0, \\&\nabla ^{\mu } F_{\mu \nu } = \sum _{k=1}^K e_k \int _{v \in {\mathbb {R}}^3} \frac{v_{\nu }}{\sqrt{m_k^2+|v|^2}} f_k \mathrm{d}v, \\&\nabla ^{\mu } {}^* F_{\mu \nu } = 0, \end{aligned}$$

where

• $m_k \ge 0$ is the mass of the particles of the species k and $e_k \ne 0$ is their charge.

• The function $f_k(t,x,v)$ is the particle density of the species k, where $(t,x,v) \in {\mathbb {R}}_+ \times {\mathbb {R}}^3 \times \left( {\mathbb {R}}^3 {\setminus } \{0 \} \right)$ if $m_k=0$ and $(t,x,v) \in {\mathbb {R}}_+ \times {\mathbb {R}}^3 \times {\mathbb {R}}^3$ otherwise.

• The 2-form F(t, x), with $(t,x) \in {\mathbb {R}}_+ \times {\mathbb {R}}^3$, is the electromagnetic field and ${}^* F(t,x)$ is its Hodge dual.

In [2], we studied the massless Vlasov–Maxwell system in high dimensions ($n \ge 4$) and we proved that if the particle densities initially vanish for small velocities and if certain weighted $L^1$ and $L^2$ norms of the initial data are small enough, then the unique classical solution to the system exists globally in time. Moreover, as the smallness assumption only concerns $L^1$ and $L^2$ norms, no compact support assumption in x or v was required. We also obtained optimal pointwise decay estimates on the velocity averages of $f_k$ and their derivatives as well as improved decay estimates on the null components of the electromagnetic field and its derivatives. In the same article, we also proved that there exist smooth initial data such that the particle densities do not vanish for small velocities and for which (1)–(3) does not admit a local classical solution.²

Similar results for the massive Vlasov–Maxwell system in high dimensions are also obtained in [2]. A main difference however is that $f_k$ does not have to be supported away from $v=0$. The 3d massive case requires a refinement of our method and will be treated in [3]. We will also study the solutions of (1)–(3) in the exterior of a light cone. The strong decay satisfied by $f_k$ in such a region will allow us to lower the initial decay hypothesis on the electromagnetic field and to obtain asymptotics on the solutions in a simpler way than for the whole spacetime. This will be done in [4].

In this paper, we study the asymptotic properties of the small data solutions to the three-dimensional massless Vlasov–Maxwell, so that $m_k=0$. We start with optimal decay in v on the particle densities in the sense that we merely suppose $f_k(0,x,\cdot )$ to be integrable in v, which is a necessary condition for the source term of the Maxwell equations to be well defined. In massive Vlasov systems, powers of |v| are often lost in order to gain time decay or to exploit null properties.³ Our assumptions will force us to better understand the null structure of the equations. In fact, one of the goals of this article is to describe in full details the null structure of the system, which appears to be fundamental for proving integrability and controlling the velocity support of the particle density.

In view of their physical meaning, the functions $f_k$ are usually supposed to be nonnegative. However, as their signs play no role in this paper and since we will consider neutral plasmas, we suppose for simplicity that $K=1$ and we do not restrict the values of $f_1$ to ${\mathbb {R}}_+$. We also normalize the charge $e_1$ to 1 and we denote $f_1$ by f. The system can then be rewritten as

$$\begin{aligned} |v|\partial _t f+v^i \partial _i f +\left( |v|{F_{0}}^j+ v^{q} {F_{q}}^j \right) \partial _{v^j} f= & {} 0, \end{aligned}$$

(1)

$$\begin{aligned} \nabla ^{\mu } F_{\mu \nu }= & {} J(f)_{\nu } = \int _{v \in {\mathbb {R}}^3} \frac{v_{\nu }}{|v|} f \mathrm{d}v, \end{aligned}$$

(2)

$$\begin{aligned} \nabla ^{\mu } {}^* F_{\mu \nu }= & {} 0. \end{aligned}$$

(3)

Note that we can recover the more common form of the Vlasov–Maxwell system using the relations

$$\begin{aligned} E^i=F_{0i} \quad \text {and} \quad B^i=-{}^* F_{0i}, \end{aligned}$$

so that Eqs. (1)–(3) can be rewritten as

$$\begin{aligned}&|v| \partial _t f+v^i \partial _i f + (|v|E+v \times B) \cdot \nabla _v f = 0,&\\&\nabla \cdot E = \int _{v \in {\mathbb {R}}^3}f\mathrm{d}v, \quad \partial _t E^j = (\nabla \times B)^j -\int _{v \in {\mathbb {R}}^3} \frac{v^j}{|v|}f\mathrm{d}v,&\\&\nabla \cdot B = 0, \quad \partial _t B = - \nabla \times E.&\end{aligned}$$

We choose to work with a neutral plasma to simplify the proof but the case of a nonzero total charge will be covered in [3, 4].

1.1 Previous Results on Small Data Solutions for the Massive Vlasov–Maxwell System

Global existence for small data in dimension 3 was first established by Glassey and Strauss [10] under a compact support assumption (in space and in velocity). In [9], a similar result is obtained for the nearly neutral case. The compact support assumption in v is removed in [12] but the data still have to be compactly supported in space. Note that none of these results contain estimates on $\partial _{\mu _1}\ldots \partial _{\mu _k} \int _v f \mathrm{d}v$ and the optimal decay rate on $\int _v f \mathrm{d}v$ is not obtained by the method of [12]. They all proved decay estimates on the electromagnetic field up to first-order derivatives.

In [2], we used vector field methods, developed in [5] for the electromagnetic field and [8] for the Vlasov field, in order to remove all compact support assumptions for the dimensions $n \ge 4$. We then derived (almost) optimal decay on the solutions of the system and their derivatives, and we described precisely the behavior of the null components of F.

Recently, Wang proved in [15] a similar result for the 3d case. Using both vector field method and Fourier analysis, he replaced the compact support assumption by strong polynomial decay hypotheses in (x, v) on f and obtained optimal pointwise decay estimates on $\int _v f \mathrm{d}v$ and its derivatives.

1.2 Previous Works on Vlasov Systems Using Vector Field Methods

Properties of small data solutions of other Vlasov systems were obtained recently using vector field methods, first on the Vlasov–Nordström system, in [7, 8], and the Vlasov–Poisson system (see [13]). Vector field methods led to a proof of the stability of the Minkowski spacetime for the Einstein–Vlasov system, obtained independently by [6, 11].

Note that vector field methods can also be used to derive integrated decay for solutions to the massless Vlasov equation on curved background such as slowly rotating Kerr spacetime (see [1]).

1.3 Statement of the Main Result

The following theorem is the main result of this paper. For the notations not yet defined, see Sect. 2.

Theorem 1.1

Let $N \ge 10$, $\epsilon >0$ and $(f^0,F^0)$ an initial data set for the Vlasov–Maxwell equations (1)–(3) satisfying the smallness assumption⁴

$$\begin{aligned}&\sum _{ |\beta |+|\kappa | \le N+3} \int _{x \in {\mathbb {R}}^3} \int _{v \in {\mathbb {R}}^3} (1+|x|)^{|\beta |+2}(1+|v|)^{|\kappa |} \left| \partial _x^{\beta } \partial _v^{\kappa } f^0 \right| \mathrm{d}v \mathrm{d}x \le \varepsilon \\&\qquad \sum _{ |\gamma | \le N+2} \int _{x \in {\mathbb {R}}^3} (1+|x|)^{2|\gamma |+2} \left| \nabla _{\partial ^{\gamma }_x} F^0 \right| ^2 \mathrm{d}x \le \epsilon , \end{aligned}$$

the neutral hypothesis

$$\begin{aligned} \int _{x \in {\mathbb {R}}^3} \int _{v \in {\mathbb {R}}^3} f^0 \mathrm{d}v \mathrm{d}x =0 \end{aligned}$$

(4)

and the support assumption

$$\begin{aligned} \forall \, 0 < |v| \le 3, \quad f^0(\cdot ,v)=0. \end{aligned}$$

There exists $C>0$ and $\epsilon _0>0$ such that if $0 \le \epsilon \le \epsilon _0$, then the unique classical solution (f, F) of the system which satisfies $f(t=0)=f^0$ and $F(t=0)=F^0$ is a global solution and verifies the following estimates.

• Energy bound for the electromagnetic field F: $\forall$ $t \in {\mathbb {R}}_+$,

  $$\begin{aligned}&\sum _{\begin{array}{c} Z^{\gamma } \in {\mathbb {K}}^{|\gamma |} \\ \, |\gamma | \le N \end{array}} \int _{ {\mathbb {R}}^3} \tau _+^2 \left( \left| \alpha \left( {\mathcal {L}}_{ Z^{\gamma }}(F) \right) \right| ^2 + \left| \rho \left( {\mathcal {L}}_{ Z^{\gamma }}(F) \right) \right| ^2 + \left| \sigma \left( {\mathcal {L}}_{ Z^{\gamma }}(F) \right) \right| ^2 \right) dx\\&\quad +\,\int _{ {\mathbb {R}}^3}\tau _-^2 \left| {\underline{\alpha }} \left( {\mathcal {L}}_{ Z^{\gamma }}(F) \right) \right| ^2 \mathrm{d}x \le C\epsilon \log ^4(3+t). \end{aligned}$$

• Sharp pointwise decay estimates for the null components of ${\mathcal {L}}_{Z^{\gamma }}(F)$: $\forall$ $|\gamma | \le N-2$, $(t,x) \in {\mathbb {R}}_+ \times {\mathbb {R}}^3$,

  $$\begin{aligned} \left| \rho \left( {\mathcal {L}}_{Z^{\gamma }}(F) \right) \right| (t,x)\lesssim & {} \sqrt{\epsilon } \frac{\log ^2 (3+t)}{\tau _+^{2} \tau _-^{\frac{1}{2}}}, \qquad \left| \alpha \left( {\mathcal {L}}_{Z^{\gamma }}(F) \right) \right| (t,x) \; \lesssim \; \sqrt{\epsilon } \frac{\log ^2 (3+t)}{\tau _+^{\frac{5}{2}} }, \\ \left| \sigma \left( {\mathcal {L}}_{Z^{\gamma }}(F) \right) \right| (t,x)\lesssim & {} \sqrt{\epsilon } \frac{\log ^2 (3+t)}{\tau _+^{2} \tau _-^{\frac{1}{2}}} ,\qquad \left| {\underline{\alpha }} \left( {\mathcal {L}}_{Z^{\gamma }}(F) \right) \right| (t,x) \; \lesssim \; \sqrt{\epsilon } \frac{\log ^{\frac{5}{2}} (1+\tau _-)}{\tau _+ \tau _-^{\frac{3}{2}}}. \end{aligned}$$

• Energy bound for the particle density: $\forall$ $t \in {\mathbb {R}}_+$,

  $$\begin{aligned} \sum _{\begin{array}{c} \, {\widehat{Z}}^{\beta } \in \widehat{{\mathbb {P}}}_0^{|\beta |} \\ \, |\beta | \le N \end{array}} \sum _{z \in {\mathbf {k}}_0} \int _{x \in {\mathbb {R}}^3} \int _{v \in {\mathbb {R}}^3} \left| z {\widehat{Z}}^{\beta } f \right| (t,x,v) \mathrm{d}v \mathrm{d}x \le C\epsilon \log (3+t). \end{aligned}$$

• Vanishing property for small velocities:

  $$\begin{aligned} \forall \, (t,x,v) \in {\mathbb {R}}_+ \times {\mathbb {R}}^3 \times \left( {\mathbb {R}}^3 {\setminus } \{ 0 \} \right) , \; |v| \le 1 \, \Rightarrow \, f(t,x,v) =0. \end{aligned}$$

• Sharp pointwise decay estimates for the velocity averages of ${\widehat{Z}}^{\beta } f$: $\forall$ $|\beta | \le N-5$, $z \in {\mathbf {k}}_0$,

  $$\begin{aligned} \forall \, (t,x) \in {\mathbb {R}}_+ \times {\mathbb {R}}^3, \quad \int _{ v \in {\mathbb {R}}^3} \left| z^2 {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \; \lesssim \; \frac{\epsilon }{\tau _+^2 \tau _-}. \end{aligned}$$

Remark 1.2

One can prove a similar result if $f^0$ vanishes for the velocities v such that $|v| \le R$, with $R >0$ ($\epsilon _0$ would then also depend on R).

Remark 1.3

We say that $(f^0,F^0)$ is an initial data set for the Vlasov–Maxwell system if the function $f^0{:}\,{\mathbb {R}}^3_x \times \left( {\mathbb {R}}^3_v {\setminus } \{ 0 \} \right) \rightarrow {\mathbb {R}}$ and $F^0$ are both sufficiently regular and satisfy the constraint equations

$$\begin{aligned} \nabla ^i \left( F^0 \right) _{i0} =- \int _{v \in {\mathbb {R}}^3} f^0 \mathrm{d}v \quad \text {and} \quad \nabla ^i \left( {}^* F^0 \right) _{i0} =0. \end{aligned}$$

Remark 1.4

The neutral hypothesis (4) is a necessary condition for $\int _{{\mathbb {R}}^3} (1+r)^2|F|^2 \mathrm{d}x$ to be finite. This means that, for a sufficiently regular solution to the Vlasov–Maxwell system (f, F), the total electromagnetic charge

$$\begin{aligned} Q(t) := \lim _{r \rightarrow + \infty } \int _{{\mathbb {S}}_{t,r}} \frac{x^i}{r} F_{0i} \mathrm{d} {\mathbb {S}}_{t,r} = \int _{x \in {\mathbb {R}}^3} \int _{v \in {\mathbb {R}}^3} f \mathrm{d}v \mathrm{d}x, \end{aligned}$$

which is a conserved quantity in t, vanishes. More precisely, if $Q(0) \ne 0$, then

$$\begin{aligned} \int _{{\mathbb {R}}^3} r \left| \rho \left( F^0 \right) \right| ^2 \mathrm{d}x = +\infty , \quad \text {where} \;\; \rho \left( F^0 \right) := \frac{x^i}{r} \left( F^0 \right) _{i0}. \end{aligned}$$

We prove in Appendix C that the derivatives of F are automatically chargeless, whether or not Q vanishes.

1.4 Strategy of the Proof and Main Difficulties

The proof of Theorem 1.1 is based on energy and vector field methods and essentially relies on bounding sufficiently well the spacetime integrals of the commuted equations. The solutions of the massless Vlasov equation enjoy improved decay estimates in the null directions. More precisely, one can already see that with the following estimate (see Lemma 2.11 and Proposition 3.6), for g a solution to the free transport equation $|v| \partial _t g+v^i \partial _{i} g=0$,

$$\begin{aligned}&\forall \, (t,x) \in {\mathbb {R}}_+ \times {\mathbb {R}}^3, \quad \int _{v \in {\mathbb {R}}^3} \left| \frac{v^L}{|v|} \right| ^p \left| \frac{v^A}{|v|} \right| ^k \left| \frac{v^{{\underline{L}}}}{|v|} \right| ^q |g|(t,x,v) \mathrm{d}v \nonumber \\&\quad \lesssim \sum _{|\beta | \le 3} \frac{\Vert (1+r)^{|\beta |+p+k+q} \partial ^{\beta }_x g \Vert _{L^1_{x,v}}(t=0)}{(1+t+r)^{2+k+q}(1+|t-r|)^{1+p}}. \end{aligned}$$

(5)

This strong decay is a key element of our proof. Without it, we would have to consider modifications of the commutation vector fields of the free transport operator as in [6, 7, 11, 13] for, respectively, the Vlasov–Nordström, the Vlasov–Poisson and the Einstein–Vlasov systems. As the particles are massless, the characteristics of the transport equation and those of the Maxwell equations have the same velocity.⁵ The consequence is that, in a product such as ${\mathcal {L}}_{Z^{\gamma }}(F)\cdot {\widehat{Z}}^{\beta }f$, we cannot transform a $|t-r|$ decay in a $t+r$ one as it is done, in view of support consideration, for the massive case with compactly supported initial data. We are then led to carefully study the null structure of the equations, and in particular of the nonlinearities such as

$$\begin{aligned} v^{\mu } {{\mathcal {L}}_{Z^{\gamma }}(F)_{\mu }}^{ i} \partial _{v^i} {\widehat{Z}}^{\beta } f, \end{aligned}$$

(6)

with Z a Killing vector field and ${\widehat{Z}}$ its complete lift.⁶ The problem is that, for g a solution to $|v|\partial _t+v^{i} \partial _{i} g=0$, $\partial _v g$ essentially behaves as $(1+t+r) \partial _{t,x} g$ and the electromagnetic field, as a solution of a wave equation, only decays with a rate of $(1+t+r)^{-1}$ in the $t+r$ direction. However, from [5], we know that certain null components of the Maxwell field are expected to behave better than others. The same is true for the null components of the velocity vector v as it is suggested by (5). Moreover, we also know from [2] that $v^{{\underline{L}}}$ allows us to take advantage of the $t-r$ decay as it permits to estimate spacetime integrals by using a null foliation. Finally, the radial component of $(0,\partial _{v^1} {\widehat{Z}}^{\beta } f,\partial _{v^2} {\widehat{Z}}^{\beta } f,\partial _{v^3} {\widehat{Z}}^{\beta } f)$ costs a power of $t-r$ instead of $t+r$. The null structure of (6) is then studied in Lemma 4.1, and we can observe that each term contains at least one good component.

Another problem, specific to massless particles, arises from small velocities. We already observed in Section 8 of [2] that the velocity part V of the characteristics of

$$\begin{aligned} \partial _t +\frac{v^i}{|v|} \partial _i f+\left( F_{0i} +\frac{v^j}{|v|} F_{ji} \right) \partial _{v^i} f=0 \end{aligned}$$

(7)

can reach 0 in finite time. The consequence is that if f does not initially vanish for small velocities, the Vlasov–Maxwell system could not admit a local classical solution. This issue is reflected in the energy estimates through, schematically,

$$\begin{aligned} \left\| {\widehat{Z}} f \right\| _{L^1_{x,v}} (t) \; \le \; 2\left\| {\widehat{Z}} f \right\| _{L^1_{x,v}} (0)+\int _{0}^t \int _{x \in {\mathbb {R}}^3} \int _{v \in {\mathbb {R}}^3} |\psi (t,x,v)| \frac{|{\widehat{Z}} f|}{|v|} \mathrm{d}v \mathrm{d}x \mathrm{d}s, \end{aligned}$$

where $\psi$ is a homogeneous function of degree 0 in v. One cannot hope to close such an estimate using say the Grönwall inequality due to the factor of $\frac{1}{|v|}$ appearing in the error term on the right-hand side. In [2], we take advantage of the strong decay rate of the electromagnetic field, given by the high dimensions, to prove that the velocity support of f remains bounded away from 0 if initially true. The slow decay of F in dimension 3 forces us to exploit the null structure of the equations satisfied by the characteristics of (7) in order to recover this result. The strong decay rate satisfied by the radial component of the electric field $\rho (F)$ plays a fundamental role here. We point out that this difficulty is not present in the Einstein–Vlasov system as the Vlasov equation can be written, for a metric g and defined in terms of the cotangent variables, as

$$\begin{aligned} v_{\mu } g^{\mu \nu } \partial _{x^{\nu }} f-\frac{1}{2} v_{\mu } v_{\nu } \partial _i g^{\mu \nu } \partial _{v_i} f=0. \end{aligned}$$

One can observe that the homogeneity in v of the nonlinearity of the Vlasov equation is the same than the one of $|v| \partial _t+v^i \partial _i$, so that the velocity part of the characteristics cannot reach 0 in finite time. Local existence, for the massless Einstein–Vlasov system and for sufficiently regular initial data, has been proven by Svedberg [14].

1.5 Structure of the Paper

Section 2 presents the notations used in this article, basic results on the electromagnetic field and its null decomposition. The commutation vector fields are introduced in Sect. 2.4, and the source terms of the commuted equations are described in Sect. 2.5. Section  2.6 contains fundamental properties on the null components of the velocity vector. In Sect. 3, we introduce the norms used to study the Vlasov–Maxwell system and we present energy estimates in order to control them. We then exploit these energy norms to obtain pointwise decay estimates on both fields through Klainerman–Sobolev-type inequalities. Lemma 4.1, proved in Sect. 4, is of fundamental importance in this work since it depicts the null structure of the nonlinearities of the transport equations. In Sect. 5, we set up the bootstrap assumptions, discuss their immediate consequences and describe the main steps of the proof of Theorem 1.1. Sections 6–8 concern, respectively, the improvement of the bounds on the distribution function, the proof of $L^2$ estimates for the velocity averages of its higher-order derivatives and the improvement of the estimates on the electromagnetic field energies. In Appendix A, we prove that the Vlasov field vanishes for small velocities. In Appendix B we expose how to bound the energy norms of f and F in terms of weighted $L^1$ and $L^2$ norms of the initial data. We prove in Appendix C that the derivatives of F, for (f, F) a sufficiently regular solution to the Vlasov–Maxwell system, are automatically chargeless. Finally, Appendix D contains the proof of certain results concerning the null decomposition of the electromagnetic field.

2 Notations and Preliminaries

2.1 Basic Notations

In this paper we work on the $3+1$-dimensional Minkowski spacetime $({\mathbb {R}}^{3+1},\eta )$. We will use two sets of coordinates, the Cartesian $(t,x^1,x^2,x^3)$, in which $\eta =\mathrm{diag}(-1,1,1,1)$, and null coordinates $({\underline{u}},u,\omega _1,\omega _2)$, where

$$\begin{aligned} {\underline{u}}=t+r, \quad u=t-r \end{aligned}$$

and $(\omega _1,\omega _2)$ are spherical variables, which are spherical coordinates on the spheres $(t,r)=\mathrm{constant}$. These coordinates are defined globally on ${\mathbb {R}}^{3+1}$ apart from the usual degeneration of spherical coordinates and at $r=0$. We will also use the following classical weights,

$$\begin{aligned} \tau _+:= \sqrt{1+{\underline{u}}^2} \quad \text {and} \quad \tau _-:= \sqrt{1+u^2}. \end{aligned}$$

We denote by $(e_1,e_2)$ an orthonormal basis on the spheres and by (respectively, ) the intrinsic covariant differentiation (respectively, divergence operator) on the spheres $(t,r)=\mathrm{constant}$. Capital Latin indices (such as A or B) will always correspond to spherical variables. The null derivatives are defined by

$$\begin{aligned} L=\partial _t+\partial _r \quad \text {and} \quad {\underline{L}}=\partial _t-\partial _r, \end{aligned}$$

so that

$$\begin{aligned} L({\underline{u}})=2, \;\; L(u)=0, \quad {\underline{L}}( {\underline{u}})=0 \;\; \text {and} \;\; {\underline{L}}(u)=2. \end{aligned}$$

The velocity vector $(v^{\mu })_{0 \le \mu \le 3}$ is parameterized by $(v^i)_{1 \le i \le 3}$ and $v^0=|v|$ since we study massless particles. We introduce T, the operator defined, for all sufficiently regular function $f{:}\,[0,T[ \times {\mathbb {R}}^3_x \times \left( {\mathbb {R}}^3_v {\setminus } \{ 0 \} \right)$, by

$$\begin{aligned} T {:}\, f \mapsto v^{\mu } \partial _{\mu } f. \end{aligned}$$

We will use the notation $\nabla _v g := (0,\partial _{v^1}g, \partial _{v^2}g,\partial _{v^3}g)$ so that (1) can be rewritten

$$\begin{aligned} T_F(f) := v^{\mu } \partial _{\mu } f +F \left( v, \nabla _v f \right) =0. \end{aligned}$$

Remark 2.1

As we study massless particles, the functions considered in this paper will not be defined for $v=0$. However, for simplicity and since $\{v=0\}$ has Lebesgue measure 0, we will consider integrals over ${\mathbb {R}}^3_v$. Moreover, the distribution function f will be supported away from $v=0$ during the proof of Theorem 1.1.

We will use the notation $D_1 \lesssim D_2$ for an inequality such as $D_1 \le C D_2$, where $C>0$ is a positive constant independent of the solutions but which could depend on $N \in {\mathbb {N}}$, the maximal order of commutation. Finally, we will raise and lower indices using the Minkowski metric $\eta$. For instance, $v_{\mu } = v^{\nu } \eta _{\nu \mu }$ so that $v_0=-v^0$ and $v_i=v^i$ for all $1 \le i \le 3$.

2.2 The Problem of the Small Velocities

For technical reasons, we will use all along this paper a fixed cutoff function $\chi$ such that $\chi =1$ on $[1,+\infty [$ and $\chi =0$ on $]-\infty , \frac{1}{2}]$. We introduce the operator

$$\begin{aligned} T_F^{\chi } {:}\, g \mapsto v^{\mu } \partial _{\mu } g + \chi (|v|) F \left( v, \nabla g \right) . \end{aligned}$$

(8)

As mentioned earlier, we proved in Section 8 of [2] that because of the small velocities, there exist initial data sets for which the Vlasov–Maxwell system does not admit a local classical solution. The main idea of the proof consists in studying characteristics such that their velocity part reaches 0 in finite time. This is why we suppose in Theorem 1.1 that the Vlasov field vanishes initially for small velocities and one step of the proof will be to verify that this property remains true for all $t \in {\mathbb {R}}_+$. To circumvent difficulties related to characteristics reaching $v=0$, we will rather first define (f, F) as the solution to (2)–(3) and $T_F^{\chi }(f)=0$. Notice that none of the characteristics of the operator $T^{\chi }_F$ reaches $v=0$. Indeed, if (X, V) is one of them, we have

$$\begin{aligned} \frac{\mathrm{d} V^j}{\mathrm{d}t}(s)= \chi \left( |V|(s) \right) \frac{V^{\mu }(s)}{|V|(s)} {F_{\mu }}^j(s,X(s)). \end{aligned}$$

Consequently, if $|V(s)| < \frac{1}{2}$, then $V(t)=V(s)$ for all $t \ge s$. The goal will then to prove that if $f(0,\cdot ,\cdot )$ vanishes for all $|v| \le 3$, so does $f(t,\cdot ,\cdot )$ for all $|v| \le 1$, implying that $T_F(f)=0$ and that (f, F) is a solution to the Vlasov–Maxwell system (1)–(3).

2.3 Basic Tools for the Study of the Electromagnetic Field

As we describe the electromagnetic field in geometric form, it will be represented throughout this article by a 2-form. Let F be a 2-form defined on $[0,T[ \times {\mathbb {R}}^3_x$. Its null decomposition $(\alpha (F), {\underline{\alpha }}(F), \rho (F), \sigma (F))$, introduced by [5], is defined by

$$\begin{aligned} \alpha _A(F) = F_{AL}, \quad {\underline{\alpha }}_A(F)= F_{A {\underline{L}}}, \quad \rho (F)= \frac{1}{2} F_{L {\underline{L}} } \quad \text {and} \quad \sigma (F) =F_{12}. \end{aligned}$$

The Hodge dual ${}^* F$ of F is the 2-form given by

$$\begin{aligned} {}^* F_{\mu \nu } = \frac{1}{2} F^{\lambda \sigma } \varepsilon _{ \lambda \sigma \mu \nu }, \end{aligned}$$

where $\varepsilon _{ \lambda \sigma \mu \nu }$ are the components of the Levi–Civita symbol, and its energy–momentum tensor is

$$\begin{aligned} T[F]_{\mu \nu } := F_{\mu \beta } {F_{\nu }}^{\beta }- \frac{1}{4}\eta _{\mu \nu } F_{\rho \sigma } F^{\rho \sigma }. \end{aligned}$$

Note that $T[F]_{\mu \nu }$ is symmetric and traceless, i.e., $T[F]_{\mu \nu }=T[F]_{\nu \mu }$ and ${T[F]_{\mu }}^{\mu }=0$. This last point is specific to the dimension 3 and engenders additional difficulties in the analysis of the Maxwell equations in high dimensions (see Section 3.3.2 of [2] for more details). We have an alternative form of the Maxwell equations.

Lemma 2.2

Let G be a 2-form and J be a 1-form both sufficiently regular. Then,

$$\begin{aligned} \left\{ \begin{aligned} \nabla ^{\mu } G_{\mu \nu }&= J_{\nu } \\ \nabla ^{\mu } {}^* G_{\mu \nu }&= 0 \end{aligned} \right.&\,\, \Leftrightarrow&\left\{ \begin{aligned} \nabla _{[ \lambda } G_{\mu \nu ]}&=0 \\ \nabla _{[ \lambda } {}^* G_{\mu \nu ]}&= \varepsilon _{ \lambda \mu \nu \kappa } J^{\kappa }, \end{aligned} \right. \end{aligned}$$

where $\nabla _{[ \lambda } H_{\mu \nu ]}:= \nabla _{ \lambda } H_{\mu \nu }+\nabla _{ \mu } H_{ \nu \lambda }+\nabla _{ \nu } H_{ \lambda \mu }$.

Proof

Consider for instance $\nabla ^{\mu } G_{\mu 1} =J_1$ and $\nabla ^i {}^* G_{i0} =0$. As

$$\begin{aligned} G^{01}= & {} {}^* G_{23} \varepsilon _{ 01 23} = {}^* G_{23}, \quad G^{21} = {}^* G_{03} \varepsilon _{ 21 03} = {}^* G_{30}, \quad G^{31} = {}^* G_{02} \varepsilon _{ 31 02} = G_{02}, \\ {}^* G_{10}= & {} G_{23} \varepsilon _{ 2 3 1 0} = -G_{23}, \quad {}^* G_{20} = G_{31} \varepsilon _{ 3 1 2 0} = -G_{31}, {}^* G_{30} = G_{12}\varepsilon _{ 1 2 3 0} \; = -\; G_{12}, \end{aligned}$$

we have

$$\begin{aligned}&\nabla ^{\mu } G_{\mu 1} =J_1 \Leftrightarrow \nabla _0 {}^* G_{23}+\nabla _2 {}^* G_{30}+\nabla _3 {}^* G_{02}=J_1, \\&\quad \nabla ^i {}^* G_{i0} =0 \Leftrightarrow \nabla _1 G_{23}+\nabla _2 G_{31}+\nabla _3 G_{12} =0. \end{aligned}$$

The equivalence of the two systems can be obtained by similar computations. $\square$

We can then compute the divergence of the energy–momentum tensor of an electromagnetic field.

Corollary 2.3

Let G and J be as in the previous lemma. Then, we have $\nabla ^{\mu } T[G]_{\mu \nu }=G_{\nu \lambda } J^{\lambda }$.

Proof

Using the previous lemma, we have

$$\begin{aligned} G_{\mu \rho } \nabla ^{\mu } {G_{\nu }}^{\rho }= & {} G^{\mu \rho } \nabla _{\mu } G_{\nu \rho } = \frac{1}{2} G^{\mu \rho } (\nabla _{\mu } G_{\nu \rho }-\nabla _{\rho } G_{\nu \mu }) \\= & {} \frac{1}{2} G^{\mu \rho } \nabla _{\nu } G_{\mu \rho } = \frac{1}{4} \nabla _{\nu } (G^{\mu \rho } G_{\mu \rho }). \end{aligned}$$

Hence,

$$\begin{aligned} \nabla ^{\mu } T[G]_{\mu \nu } = \nabla ^{\mu } (G_{\mu \rho }){G_{\nu }}^{\rho }+\frac{1}{4} \nabla _{\nu } (G^{\mu \rho } G_{\mu \rho })-\frac{1}{4}\eta _{\mu \nu } \nabla ^{\mu } (G^{\sigma \rho } G_{\sigma \rho }) = G_{\nu \rho } J^{\rho }. \end{aligned}$$

$\square$

Finally, the null components of the energy–momentum tensor of a 2-form G are given by

$$\begin{aligned} T[G]_{L L}=|\alpha (G)|^2, \quad T[G]_{{\underline{L}} \, {\underline{L}}}=|{\underline{\alpha }}(G)|^2 \quad \text {and} \quad T[G]_{L {\underline{L}}}=|\rho (G)|^2+|\sigma (G)|^2.\nonumber \\ \end{aligned}$$

(9)

2.4 The Vector Fields of the Poincaré Group and Their Complete Lifts

We present in this section the commutation vector fields for the Maxwell equations and those for the relativistic transport operator. Let ${\mathbb {P}}$ be the generators of the Poincaré algebra, i.e., the set containing

$$\begin{aligned}&\text {the translations}^7 \quad \partial _{\mu }, \; 0 \le \mu \le 3,&\\&\text {the rotations} \qquad \Omega _{ij}=x^i\partial _{j}-x^j \partial _i, \; 1 \le i < j \le 3,&\\&\text {the hyperbolic rotations} \quad \Omega _{0k}=t\partial _{k}+x^k \partial _t, \; 1 \le k \le 3. \end{aligned}$$

⁷ We also consider ${\mathbb {K}}:= {\mathbb {P}}\cup \{ S \}$, where $S=x^{\mu } \partial _{\mu }$ is the scaling vector field and ${\mathbb {O}}:= \{ \Omega _{12}, \, \Omega _{13}, \, \Omega _{23} \}$, the set of the rotational vector fields. The vector fields of ${\mathbb {K}}$ are well known for commuting with the wave and the Maxwell equations (see Proposition 2.8 below). However, to commute the operator $T=v^{\mu } \partial _{\mu }$, one should consider, as in [8], the complete lifts of the vector fields of ${\mathbb {P}}$.

Definition 2.4

Let V be a vector field of the form $V^{\beta } \partial _{\beta }$. Then, the complete lift ${\widehat{V}}$ of V is defined by

$$\begin{aligned} {\widehat{V}}=V^{\beta } \partial _{\beta }+v^{\gamma } \frac{\partial V^i}{\partial x^{\gamma }} \partial _{v^i}. \end{aligned}$$

We then have ${\widehat{\partial }}_{\mu }=\partial _{\mu }$ for all $0 \le \mu \le 3$,

$$\begin{aligned}&{\widehat{\Omega }}_{ij}=x^i \partial _j-x^j \partial _i+v^i \partial _{v^j}-v^j \partial _{v^i}, \; \text {for} \; 1 \le i < j \le 3, \quad \text {and} \\&\quad {\widehat{\Omega }}_{0k} = t\partial _k+x^k \partial _t+v^0 \partial _{v^k}, \; \text {for} \; 1 \le k \le 3. \end{aligned}$$

One can check that $[T,{\widehat{Z}}]=0$ for all $Z \in {\mathbb {P}}$. As we also have $[T,S]=T$, we consider

$$\begin{aligned} \widehat{{\mathbb {P}}}_0:= \{ {\widehat{Z}} \, / \, Z \in {\mathbb {P}}\} \cup \{ S \} \end{aligned}$$

and we will, for simplicity, denote by ${\widehat{Z}}$ an arbitrary vector field of $\widehat{{\mathbb {P}}}_0$, even if S is not a complete lift. These vector fields and the averaging in v almost commute in the following sense.

Lemma 2.5

Let $f {:}\, [0,T[ \times {\mathbb {R}}^3_x \times \left( {\mathbb {R}}^3_v {\setminus } \{ 0 \} \right) \rightarrow {\mathbb {R}}$ be a sufficiently regular function. We have, almost everywhere,

$$\begin{aligned} \forall \, Z \in {\mathbb {K}}, \qquad \left| Z\left( \int _{v \in {\mathbb {R}}^3 } |f| \mathrm{d}v \right) \right| \le \int _{v \in {\mathbb {R}}^3} |f| \mathrm{d}v+ \sum _{ {\widehat{Z}} \in \widehat{{\mathbb {P}}}_0} \int _{v \in {\mathbb {R}}^3 } | {\widehat{Z}} f | \mathrm{d}v. \end{aligned}$$

Proof

Let us consider, for instance, the case where $Z=\Omega _{12}=x^1 \partial _2-x^2 \partial _1$. Then, integrating by parts in v, we have almost everywhere

$$\begin{aligned} \left| \Omega _{12}\left( \int _{v \in {\mathbb {R}}^3 } |f| \mathrm{d}v \right) \right|= & {} \left| \int _{v \in {\mathbb {R}}^3 } {\widehat{\Omega }}_{12} \left( |f| \right) \mathrm{d}v -\int _{v \in {\mathbb {R}}^3 } \Big ( v^1 \partial _{v^2} \left( |f| \right) -v^2 \partial _{v^1} \left( |f| \right) \Big ) \mathrm{d}v \right| \\= & {} \left| \int _{v \in {\mathbb {R}}^3 } \frac{ f}{| f|} {\widehat{\Omega }}_{12} \left( f \right) \mathrm{d}v +0 \right| \; \le \; \int _{v \in {\mathbb {R}}^3 } \left| {\widehat{\Omega }}_{12} \left( f \right) \right| \mathrm{d}v. \end{aligned}$$

$\square$

The vector space generated by each of the sets defined in this section is an algebra.

Lemma 2.6

Let ${\mathbb {L}}$ be either $\widehat{{\mathbb {P}}}_0$, ${\mathbb {K}}$, ${\mathbb {P}}$ or ${\mathbb {O}}$. Then for all $(Z_1,Z_2) \in {\mathbb {L}}^2$, $[Z_1,Z_2]$ is a linear combination of vector fields of ${\mathbb {L}}$.

We consider an ordering on each of the sets ${\mathbb {O}}$, ${\mathbb {P}}$, ${\mathbb {K}}$ and $\widehat{{\mathbb {P}}}_0$. We take orderings such that, if ${\mathbb {P}}= \{ Z^i / \; 1 \le i \le |{\mathbb {P}}| \}$, then ${\mathbb {K}}= \{ Z^i / \; 1 \le i \le |{\mathbb {K}}| \}$, with $Z^{|{\mathbb {K}}|}=S$, and

$$\begin{aligned} \widehat{{\mathbb {P}}}_0= \left\{ {\widehat{Z}}^i / \; 1 \le i \le |\widehat{{\mathbb {P}}}_0| \right\} , \; \text {with} \; \left( {\widehat{Z}}^i \right) _{ 1 \le i \le |{\mathbb {P}}|}=\left( \widehat{Z^i} \right) _{ 1 \le i \le |{\mathbb {P}}|} \; \text {and} \; {\widehat{Z}}^{|\widehat{{\mathbb {P}}}_0|}=S. \end{aligned}$$

If ${\mathbb {L}}$ denotes ${\mathbb {O}}$, ${\mathbb {P}}$, ${\mathbb {K}}$ or $\widehat{{\mathbb {P}}}_0$, and $\beta \in \{1, \ldots , |{\mathbb {L}}| \}^q$, with $q \in {\mathbb {N}}^*$, we will denote the differential operator $\Gamma ^{\beta _1}\ldots \Gamma ^{\beta _r} \in {\mathbb {L}}^{|\beta |}$ by $\Gamma ^{\beta }$. For a vector field Y, we will denote by ${\mathcal {L}}_Y$ the Lie derivative with respect to Y and if $Z^{\gamma } \in {\mathbb {K}}^{q}$, we will write ${\mathcal {L}}_{Z^{\gamma }}$ for ${\mathcal {L}}_{Z^{\gamma _1}}\ldots {\mathcal {L}}_{Z^{\gamma _q}}$.

Let us recall, by the following classical result, that the derivatives tangential to the cone behave better than others.

Lemma 2.7

The following relations hold,

$$\begin{aligned} (t-r){\underline{L}}=S-\frac{x^i}{r}\Omega _{0i}, \quad (t+r)L=S+\frac{x^i}{r}\Omega _{0i} \quad \text {and} \quad re_A=\sum _{1 \le i < j \le 3} C^{i,j}_A \Omega _{ij}, \end{aligned}$$

where the $C^{i,j}_A$ are uniformly bounded and depend only on spherical variables. We also have

$$\begin{aligned} (t-r)\partial _t =\frac{t}{t+r}S-\frac{x^i}{t+r}\Omega _{0i} \quad \text {and} \quad (t-r) \partial _i = \frac{t}{t+r} \Omega _{0i}- \frac{x^i}{t+r}S- \frac{x^j}{t+r} \Omega _{ij}. \end{aligned}$$

Finally, we introduce the vector field

$$\begin{aligned} {\overline{K}}_0:= \frac{1}{2}\tau _+^2 L+ \frac{1}{2}\tau _-^2 {\underline{L}}, \end{aligned}$$

which will be used as a multiplier.

2.5 Commutation of the Vlasov–Maxwell System

Let us start by proving the following result. For convenience, we extend the Kronecker symbol to vector fields, i.e., $\delta _{X, Y}=1$ if $X=Y$ and $\delta _{X, Y}=0$ otherwise.

Lemma 2.8

Let G be a 2-form and g a function, both sufficiently regular. For all ${\widehat{Z}} \in \widehat{{\mathbb {P}}}_0$,

$$\begin{aligned} {\widehat{Z}} \left( G \left( v, \nabla _v g \right) \right) = {\mathcal {L}}_Z(G) \left( v, \nabla _v g \right) +G \left( v, \nabla _v {\widehat{Z}} g \right) -2 \delta _{{\widehat{Z}},S} G \left( v, \nabla _v g \right) . \end{aligned}$$

If $\nabla ^{\mu } G_{\mu \nu } = J(g)_{\nu }$ and $\nabla ^{\mu } {}^* G_{\mu \nu } = 0$, then

$$\begin{aligned} \forall \, Z \in {\mathbb {K}}, \quad \nabla ^{\mu } {\mathcal {L}}_Z(G)_{\mu \nu } = J({\widehat{Z}}g)_{\nu }+3 \delta _{Z,S} J(g)_{\nu } \quad \text {and} \quad \nabla ^{\mu } {}^* {\mathcal {L}}_Z(G)_{\mu \nu } = 0. \end{aligned}$$

Proof

Let ${\widehat{Z}} \in \widehat{{\mathbb {P}}}_0$ and define $Z_v:={\widehat{Z}}-Z$. Then,

$$\begin{aligned} {\widehat{Z}} \left( G \left( v, \nabla _v g \right) \right)= & {} {\mathcal {L}}_Z(G) \left( v, \nabla _v g \right) +G \left( [Z,v], \nabla _v g \right) +G \left( v,[Z,\nabla _v g] \right) \\+ & {} G \left( Z_v(v), \nabla _v g\right) +\,G \left( v, Z_v \left( \nabla _v g \right) \right) . \end{aligned}$$

Note now that

• $S_v=0$ and $[S,v]=-v$,

• $[Z,v]=-Z_v(v)$ if $Z \in {\mathbb {P}}$.

The first identity is then implied by

• $[\partial , \nabla _v g]=\nabla _v \partial (g)$ and $[S, \nabla _v g ]= \nabla _v S(g)-\nabla _v g$.

• $[Z, \nabla _v g]+Z_v \left( \nabla _v g \right) = \nabla _v {\widehat{Z}}(g)$, if $Z \in {\mathbb {O}}$.

• $[Z, \nabla _v g]+Z_v \left( \nabla _v g \right) = \nabla _v {\widehat{Z}}(g)-\frac{v}{v^0}\partial _{v^i} g$ and $G(v,v)=0$, if $Z = \Omega _{0i}$.

Recall now that if⁸$Z \in {\mathbb {K}}$,

$$\begin{aligned} \nabla ^{\mu } {\mathcal {L}}_{Z}(G)_{\mu \nu } = {\mathcal {L}}_{Z} (J(g))_{\nu } +2\delta _{Z,S} J(g)_{\nu } \quad \text {and} \quad \nabla ^{\mu } {}^* {\mathcal {L}}_{Z}(G)_{\mu \nu } = 0. \end{aligned}$$

One then only have to notice that

$$\begin{aligned} {\mathcal {L}}_{S} (J(g))=J(S g)+J(g) \quad \text {and} \quad \forall \, Z \in {\mathbb {P}}, \quad {\mathcal {L}}_{Z} (J(g)) = J( {\widehat{Z}} g). \end{aligned}$$

This follows from ${\mathcal {L}}_{Z} (J(g))_{\nu } = Z (J(g)_{\nu })+ \partial _{\nu } \left( Z^{\lambda } \right) J(g)_{\lambda }$ and integration by parts in v. For instance,

$$\begin{aligned} {\mathcal {L}}_{\Omega _{12}} (J(g))_{\nu }= & {} \int _{v \in {\mathbb {R}}^3} \frac{v_{\nu }}{v^0} \left( {\widehat{\Omega }}_{12}-v^1 \partial _{v^2}+v^2 \partial _{v^1} \right) g \mathrm{d}v\\&+\delta _{1,\nu } \int _{v \in {\mathbb {R}}^3} \frac{v_2}{v^0} g \mathrm{d}v -\delta _{2, \nu } \int _{v \in {\mathbb {R}}^3} \frac{v_1}{v^0} g \mathrm{d}v \\= & {} J \left( {\widehat{\Omega }}_{12} g \right) +\delta _{2,\nu } \int _{v \in {\mathbb {R}}^3} \frac{v^1}{v^0} g \mathrm{d}v -\delta _{1, \nu } \int _{v \in {\mathbb {R}}^3} \frac{v^1}{v^0} g \mathrm{d}v\\&\qquad +\delta _{1,\nu } \int _{v \in {\mathbb {R}}^3} \frac{v_2}{v^0} g \mathrm{d}v -\,\delta _{2, \nu } \int _{v \in {\mathbb {R}}^3} \frac{v_1}{v^0} g \mathrm{d}v. \end{aligned}$$

$\square$

Iterating Lemma 2.8, we can describe the form of the source terms of the commuted Vlasov–Maxwell equations.

Proposition 2.9

Let (f, F) be a sufficiently regular solution to the Vlasov–Maxwell system (1)–(3) and $Z^{\beta } \in {\mathbb {K}}^{|\beta |}$. There exist integers $n^{\beta }_{\gamma , \kappa }$ and $m^{\beta }_{\xi }$ such that

$$\begin{aligned} T_F \left( {\widehat{Z}}^{\beta } f \right)= & {} \sum _{\begin{array}{c} |\gamma |+|\kappa | \le |\beta | \\ \, |\kappa | \le |\beta |-1 \end{array}} n^{\beta }_{\gamma , \kappa } {\mathcal {L}}_{Z^{\gamma }}(F) \left( v, \nabla _v {\widehat{Z}}^{\kappa } (f) \right) , \\ \nabla ^{\mu } {\mathcal {L}}_{Z^{\beta }}(F)_{\mu \nu }= & {} \sum _{|\xi | \le |\beta |} m^{\beta }_{\xi } J \left( {\widehat{Z}}^{\xi } f \right) _{\nu }, \\ \nabla ^{\mu } {}^* {\mathcal {L}}_{Z^{\beta }}(F)_{\mu \nu }= & {} 0. \end{aligned}$$

The main observation is that the structure of the nonlinearity $F(v,\nabla _v f)$ is conserved after commutation, which is important since if the source terms of the Vlasov equation behaved as $v^0 |F| |\partial _v f|$, we would not be able to close the energy estimates for the Vlasov field. The other conserved structure is J(f), which is also crucial since a source term behaving as $\int _v |f|\mathrm{d}v$ would prevent us to close the energy estimates for the electromagnetic field.

2.6 Weights Preserved by the Flow and Null Components of the Velocity Vector

We designate the null components of the velocity vector by $(v^L,v^{{\underline{L}}},v^{e_1},v^{e_2})$, so that

$$\begin{aligned} v=v^L L+ v^{{\underline{L}}} {\underline{L}}+v^{e_A}e_A, \quad v^L=\frac{v^0+v^r}{2} \quad \text {and} \quad v^{{\underline{L}}}=\frac{v^0-v^r}{2}. \end{aligned}$$

For simplicity we will write $v^A$ instead of $v^{e_A}$. We introduce, as in [8], the following set of weights

$$\begin{aligned} {\mathbf {k}}_0 := \left\{ \frac{v^{\mu }}{v^0} \, \Big / \, 0 \le \mu \le 3 \right\} \cup \left\{ x^{\mu }\frac{v^{\nu }}{v^0}-x^{\nu }\frac{v^{\mu }}{v^0} \, \Big / \, \mu \ne \nu \right\} \cup \left\{ x^{\mu } \frac{v_{\mu }}{v^0} \right\} \end{aligned}$$

and we will denote $x^{\mu }\frac{v^{\nu }}{v^0}-x^{\nu }\frac{v^{\mu }}{v^0}$ by $z_{\mu \nu }$ and $x^{\mu } \frac{v_{\mu }}{v^0}$ by $s_0$. They are preserved by the flow of T and by the action of $\widehat{{\mathbb {P}}}_0$. More precisely, we have the following result.

Lemma 2.10

Let $z \in {\mathbf {k}}_0$ and ${\widehat{Z}} \in \widehat{{\mathbb {P}}}_0$. Then,

$$\begin{aligned} T(z)=0, \quad {\widehat{Z}}(v^0z) \in v^0 {\mathbf {k}}_0 \cup \{ 0 \} \quad \text {and} \quad \left| {\widehat{Z}} (z) \right| \le \sum _{w \in {\mathbf {k}}_0} |w|. \end{aligned}$$

Proof

The first property ensues from straightforward computations. For the second one, let us consider for instance $tv^1-x^1v^0$, $x^1v^2-x^2v^1$, ${\widehat{\Omega }}_{12}$ and ${\widehat{\Omega }}_{02}$. We have

$$\begin{aligned} {\widehat{\Omega }}_{12}(tv^1-x^1v^0)= & {} -tv^2-x^2v^0, \qquad {\widehat{\Omega }}_{12}(x^1v^2-x^2v^1 ) = 0,\\ {\widehat{\Omega }}_{02}(tv^1-x^1v^0)= & {} x^2v^1-x^1v^2 \quad \text {and} \quad {\widehat{\Omega }}_{02}(x^1v^2-x^2v^1 ) = x^1v^0-tv^1. \end{aligned}$$

The other cases are similar, and the third property follows directly from the second one. $\square$

The following inequalities, which should be compared to those of Lemma 2.7, suggest how we will use these weights.

Lemma 2.11

We have,

$$\begin{aligned} \frac{v^L}{v^0} \lesssim \frac{1}{\tau _-} \sum _{z \in {\mathbf {k}}_0} |z|, \quad \frac{v^{{\underline{L}}}}{v^0}+\frac{|v^A|}{v^0} \lesssim \frac{1}{\tau _+} \sum _{z \in {\mathbf {k}}_0} |z| \quad \text {and} \quad |v^A| \lesssim \sqrt{v^0v^{{\underline{L}}}}. \end{aligned}$$

Proof

Note first that

$$\begin{aligned} 2(t-r)\frac{v^L}{v^0} = -s_0-\frac{x^i}{r}z_{0i}, \quad 2(t+r)\frac{v^{{\underline{L}}}}{v^0} = -s_0+\frac{x^i}{r}z_{0i} \quad \text {and} \quad rv_A = v^0 C_A^{i,j} z_{ij}, \end{aligned}$$

where $C^{i,j}_A$ are bounded functions depending only on the spherical variables such as $re_A= C^{i,j}_A \Omega _{i,j}$. This gives the first two estimates. For the last one, use also that $4r^2v^Lv^{{\underline{L}}} =\sum _{k < l} |v^0z_{kl}|^2$, which comes from

$$\begin{aligned} 4r^2v^Lv^{{\underline{L}}}= & {} (rv^0)^2-\left( x^i v_i \right) ^2 = \sum _{i=1}^3(r^2 -|x^i|^2)|v_i|^2-2\sum _{1 \le k< l \le 3} x^kx^lv_kv_l, \\ \sum _{1 \le k< l \le 3} |v^0 z_{kl}|^2= & {} \sum _{1 \le k< l \le 3} |x^k|^2 |v_l|^2+|x^l|^2 |v_k|^2-2x^kx^lv_kv_l\\= & {} \sum _{i=1}^3 \sum _{j \ne i} |x^j|^2 |v_i|^2 -2\sum _{1 \le k < l \le n} x^kx^lv_kv_l. \end{aligned}$$

$\square$

Remark 2.12

There are certain differences to the massive case, where $v^0 = \sqrt{m^2+|v|^2}$ and $m >0$.

• The inequality $1 \lesssim v^0 v^{{\underline{L}}}$ does not hold.

• As $x^{i} v_{i}-tv^0$ does not commute with the massive relativistic transport operator, we rather consider the set of weights ${\mathbf {k}}_1 := {\mathbf {k}}_0{\setminus } \{s_0 \}$ in this context. Then, the estimate $\tau _-v^L+\tau _+ v^{{\underline{L}}} \lesssim v^0 \sum _{z \in {\mathbf {k}}_1} |z|$ is merely satisfied in the exterior of the light cone.

2.7 Various Subsets of Minkowski Spacetime

We introduce here several subsets of Minkowski space depending on $t \in {\mathbb {R}}_+$, $r \in {\mathbb {R}}_+$, $u \in {\mathbb {R}}$. Let ${\mathbb {S}}_{t,r}$, $\Sigma _t$, $C_u(t)$ and $V_u(t)$, be the sets defined as

$$\begin{aligned} {\mathbb {S}}_{t,r}:= & {} \{ (s,y) \in {\mathbb {R}}_+ \times {\mathbb {R}}^3 \, / \, (s,|y|)=(t,r) \}, \; C_u(t)\\&= \{(s,y) \in {\mathbb {R}}_+ \times {\mathbb {R}}^3 / \, \, s \le t, \, s-|y|=u \}, \\ \Sigma _t:= & {} \{ (s,y) \in {\mathbb {R}}_+ \times {\mathbb {R}}^3 \, / \, s=t \}, \qquad V_u(t)\\&= \{ (s,y) \in {\mathbb {R}}_+ \times {\mathbb {R}}^3 / \, s \le t, \, s-|y| \le u \}. \end{aligned}$$

The volume form on $C_u(t)$ is given by $\mathrm{d}C_u(t)=\sqrt{2}^{-1}r^{2}\mathrm{d}{\underline{u}}\mathrm{d} {\mathbb {S}}^{2}$, where $\mathrm{d} {\mathbb {S}}^{2}$ is the standard metric on the two-dimensional unit sphere. In view of applying the divergence theorem, we also introduce

$$\begin{aligned} \Sigma _t^u = \{ (s,y) \in {\mathbb {R}}_+ \times {\mathbb {R}}^3 \, / \, s=t, \, |y| \ge s-u \}. \end{aligned}$$

We also introduce a dyadic partition of ${\mathbb {R}}_+$ by considering the sequence $(t_i)_{i \in {\mathbb {N}}}$ and the functions $(T_i(t))_{i \in {\mathbb {N}}}$ defined by

$$\begin{aligned} t_0=0, \quad t_i = 2^i \quad \text {if} \quad i \ge 1, \quad \text {and} \quad T_{i}(t)= t \mathbb {1}_{t \le t_i}(t)+t_i \mathbb {1}_{t > t_i}(t). \end{aligned}$$

We then define the truncated cones $C^i_u(t)$ adapted to this partition by

$$\begin{aligned}&C_u^i(t) := \left\{ (s,y) \in {\mathbb {R}}_+ \times {\mathbb {R}}^3 \; / \; t_i \le s \le T_{i+1}(t), \; s-|y| =u \right\} \\&\quad = \left\{ (s,y) \in C_u(t) \; / \; t_i \le s \le T_{i+1}(t) \right\} . \end{aligned}$$

The following lemma will be used several times during this paper. It depicts that we can foliate $[0,t] \times {\mathbb {R}}^3$ by $(\Sigma _s)_{0 \le s \le t}$, $(C_u(T))_{u \le t}$ or $(C^i_u(T))_{u \le t, i \in {\mathbb {N}}}$.

Lemma 2.13

Let $t>0$ and $g \in L^1([0,t] \times {\mathbb {R}}^3)$. Then

$$\begin{aligned} \int _0^t \int _{\Sigma _s} g \mathrm{d}x \mathrm{d}s = \int _{u=-\infty }^t \int _{C_u(t)} g \mathrm{d}C_u(t) \frac{\mathrm{d}u}{\sqrt{2}} = \sum _{i=0}^{+ \infty } \int _{u=-\infty }^t \int _{C^i_u(t)} g \mathrm{d}C^i_u(t) \frac{\mathrm{d}u}{\sqrt{2}}. \end{aligned}$$

Note that the sum over i is in fact finite. The second foliation is useful to take advantage of decay in the $t-r$ direction since $\Vert \tau _-^{-1} \Vert _{L^{\infty }(C_u(t))} = \tau _-^{-1}$, whereas $\Vert \tau _-^{-1} \Vert _{L^{\infty }(\Sigma _s)} = 1$. The last foliation will be used to take advantage of time decay on $C_u(t)$ as we merely have $\Vert \tau _+^{-1}\Vert _{L^{\infty }(C_u(t))} = \tau _-^{-1}$, whereas $\Vert \tau _+^{-1}\Vert _{L^{\infty }(C^i_u(t))} \le (1+t_i)^{-1} \le 3(1+t_{i+1})^{-1}$.

3 Energy and Pointwise Decay Estimates

In this section, we recall classical energy estimates for both the electromagnetic field and the Vlasov field and how to obtain pointwise decay estimates from them.

3.1 Energy Estimates

For the Vlasov field, we will use the following energy estimate.

Proposition 3.1

Let $H {:}\, [0,T[ \times {\mathbb {R}}^3_x \times \left( {\mathbb {R}}^3_v {\setminus } \{ 0 \} \right) \rightarrow {\mathbb {R}}$ and $g_0 {:}\, {\mathbb {R}}^3_x \times \left( {\mathbb {R}}^3_v {\setminus } \{ 0 \} \right) \rightarrow {\mathbb {R}}$ be two sufficiently regular functions and F a sufficiently regular 2-form. Then, g, the unique classical solution of

$$\begin{aligned} T_F(g)= & {} H \\ g(0,\cdot ,\cdot )= & {} g_0, \end{aligned}$$

satisfies, for all $t \in [0,T[$, the following estimates,

$$\begin{aligned}&\left\| \int _{v \in {\mathbb {R}}^3} |g|\mathrm{d}v \right\| _{L^1(\Sigma _t)}+ \sup _{u \le t} \left\| \int _{v \in {\mathbb {R}}^3} \frac{v^{{\underline{L}}}}{v^0} |g| \mathrm{d}v \right\| _{L^1(C_u(t))} \\\le & {} \; 2 \left\| \int _{v \in {\mathbb {R}}^3} |g_0| \mathrm{d}v \right\| _{L^1(\Sigma _0)} +\,2\int _0^t \int _{\Sigma _s} \int _{v \in {\mathbb {R}}^3}|H| \frac{\mathrm{d}v}{v^0} \mathrm{d}x\mathrm{d}s. \end{aligned}$$

Proof

Note first that as $T(|g|)=\frac{g}{|g|}H-\frac{g}{|g|} F ( v, \nabla _v g)$ and since F is a 2-form, integration by parts in v gives us

$$\begin{aligned}&\partial _{\mu } \int _v |g| \frac{v^{\mu }}{v^0}\mathrm{d}v = \int _v \left( \frac{g}{|g|}\frac{H}{v^0}-\frac{g}{|g|} F\left( \frac{v}{v^0}, \nabla _v g \right) \right) \mathrm{d}v \\&\quad = \int _v \left( \frac{g}{|g|}\frac{H}{v^0}- \frac{v^{j}v^i}{(v^0)^3}F_{j i}|g| \right) \mathrm{d}v= \int _v \frac{g}{|g|}H\frac{\mathrm{d}v}{v^0}. \end{aligned}$$

Hence, the divergence theorem applied to $\int _v |g| \frac{v^{\mu }}{v^0}\mathrm{d}v$ in the regions $[0,t] \times {\mathbb {R}}^3_x$ and $V_u(t)$, for all $u \le t$, gives

$$\begin{aligned} \int _{\Sigma _t} \int _v |g| \mathrm{d}v \mathrm{d}x \le \int _{\Sigma _0} \int _v |g| \mathrm{d}v \mathrm{d}x+\int _0^t \int _{\Sigma _s} \left| \int _v \frac{g}{|g|}H \frac{\mathrm{d}v}{v^0} \right| \mathrm{d}x \mathrm{d}s \end{aligned}$$

and

$$\begin{aligned}&\int _{\Sigma ^u_t} \int _v |g| \mathrm{d}v\mathrm{d}x+\sqrt{2} \int _{C_u(t)} \int _v \frac{v^{{\underline{L}}}}{v^0} |g| \mathrm{d}v \mathrm{d}C_u(t) \\&\le \int _{\Sigma ^u_0} \int _v |g| \mathrm{d}v\mathrm{d}x+\int _0^t \int _{\Sigma ^u_s} \left| \int _v \frac{g}{|g|}H \frac{\mathrm{d}v}{v^0} \right| \mathrm{d}x \mathrm{d}s, \end{aligned}$$

which implies the result. $\square$

We then define, for $(Q,q) \in {\mathbb {N}}^2$,

$$\begin{aligned} {\mathbb {E}}[g](t):= & {} \left\| \int _{v \in {\mathbb {R}}^3} |g|\mathrm{d}v \right\| _{L^1(\Sigma _t)}+ \sup _{u \le t} \left\| \int _{v \in {\mathbb {R}}^3} \frac{v^{{\underline{L}}}}{v^0} |g| \mathrm{d}v \right\| _{L^1(C_u(t))}, \end{aligned}$$

(10)

$$\begin{aligned} {\mathbb {E}}^{q}_{Q}[g](t):= & {} \sum _{ \begin{array}{c} {\widehat{Z}}^{\beta } \in \widehat{{\mathbb {P}}}_0^{|\beta |} \\ \, |\beta | \le Q \end{array}} \sum _{z \in {\mathbf {k}}_0} {\mathbb {E}}\left[ z^q {\widehat{Z}}^{\beta } g \right] (t). \end{aligned}$$

(11)

We now introduce the energy norms, related to the electromagnetic field, used in this paper. We consider, for the remaining of this section, G a sufficiently regular 2-form defined on $[0,T[ \times {\mathbb {R}}^3$ and we denote by $(\alpha ,{\underline{\alpha }},\rho ,\sigma )$ its null decomposition. We moreover suppose that G satisfies

$$\begin{aligned} \nabla ^{\mu } G_{\mu \nu }= & {} J_{\nu } \\ \nabla ^{\mu } {}^* G_{\mu \nu }= & {} 0, \end{aligned}$$

with J be a sufficiently regular 1-form defined on $[0,T[ \times {\mathbb {R}}^3$.

Definition 3.2

Let $k \in {\mathbb {N}}$. We define, for $t \in [0,T[$,

$$\begin{aligned} {\mathcal {E}}^{{\overline{K}}_0}[G](t):= & {} 4\int _{\Sigma _t} T[G]_{0 \nu } {\overline{K}}_0^{\nu } \mathrm{d}x+ 2 \sup _{u \le t } \int _{C_u(t)} T[G]_{L \nu }{\overline{K}}^{\nu }_0\mathrm{d}C_u(t) \\= & {} \int _{\Sigma _t} \tau _+^2|\alpha |^2+\tau _-^2|{\underline{\alpha }}|^2+(\tau _+^2+\tau _-^2)(|\rho |^2+|\sigma |^2)\mathrm{d}x\\&+\sup _{u \le t}\int _{C_u(t)} \tau _+^2|\alpha |^2+\tau _-^2(|\rho |^2+|\sigma |^2)\mathrm{d} C_u(t), \\ {\mathcal {E}}^{\partial _t,k}[G](t)= & {} \int _{\Sigma _t} \tau _-^{2} \log ^{-k}(1+\tau _-) \left( |\alpha |^2+|{\underline{\alpha }}|^2+2|\rho |^2+2|\sigma |^2 \right) \mathrm{d}x. \end{aligned}$$

For $N \in {\mathbb {N}}^*$, we also introduce

$$\begin{aligned} {\mathcal {E}}_N[G] := \sum _{\begin{array}{c} Z^{\gamma } \in {\mathbb {K}}^{|\gamma |} \\ \, |\gamma | \le N \end{array}} {\mathcal {E}}^{{\overline{K}}_0}[{\mathcal {L}}_{Z^{\gamma }}(G)] \quad \text {and} \quad {\mathcal {E}}^{k}_N[G] := \sum _{\begin{array}{c} Z^{\gamma } \in {\mathbb {K}}^{|\gamma |} \\ \, |\gamma | \le N \end{array}} {\mathcal {E}}^{\partial _t,k}[{\mathcal {L}}_{Z^{\gamma }}(G)]. \end{aligned}$$

Remark 3.3

During the proof of Theorem 1.1, we will have a small growth on ${\mathcal {E}}_N[G]$ and not on ${\mathcal {E}}^{k}_N[G]$. The second energy norm will then permit us to obtain the optimal decay rate in the $t+r$ direction on ${\underline{\alpha }}$, which will be crucial for closing the energy estimates for the Vlasov field.

The following energy estimates hold.

Proposition 3.4

We have, with ${\overline{C}} >0$ a constant depending on k, for all $t \in [0,T[$,

$$\begin{aligned} {\mathcal {E}}^{{\overline{K}}_0}[G](t)\le & {} 2 {\mathcal {E}}^{{\overline{K}}_0}[G](0)+8 \int _0^t \int _{\Sigma _s} |G_{\mu \nu } J^{\mu } {\overline{K}}_0^{\nu }| \mathrm{d}x \mathrm{d}s, \\ {\mathcal {E}}^{\partial _t, k}[G](t)\le & {} {\overline{C}} {\mathcal {E}}^{{\overline{K}}_0}[G](0)+{\overline{C}} \int _0^t \int _{\Sigma _s} \frac{\tau _-^2}{\log ^k(1+\tau _-)}|G_{\mu 0} J^{\mu }| \mathrm{d}x \mathrm{d}s\\&+{\overline{C}}\sup _{0 \le s \le t} \left( \log ^{1-k}(2+s) {\mathcal {E}}^{{\overline{K}}_0}[G](s) \right) . \end{aligned}$$

Proof

Denoting T[G] by T and using Corollary 2.3, we have, as $\nabla ^{\mu } {\overline{K}}_0^{\nu }+\nabla ^{\nu } {\overline{K}}_0^{\mu }=4t\eta ^{\mu \nu }$ and ${T_{\mu }}^{\mu }=0$,

$$\begin{aligned} \nabla ^{\mu } \left( T_{\mu \nu } {\overline{K}}_0^{\nu } \right)= & {} \nabla ^{\mu } T_{\mu \nu }{\overline{K}}_0^{\nu }+T_{\mu \nu } \nabla ^{\mu } {\overline{K}}_0^{\nu } = G_{\nu \lambda } J^{\lambda } {\overline{K}}_0^{\nu } +\frac{1}{2} T_{\mu \nu } \left( \nabla ^{\mu } {\overline{K}}_0^{\nu }+\nabla ^{\nu } {\overline{K}}_0^{\mu } \right) \\= & {} G_{\nu \lambda } J^{\lambda } {\overline{K}}_0^{\nu }. \end{aligned}$$

Applying the divergence theorem in $[0,t] \times {\mathbb {R}}^3$ and in $V_{u}(t)$, for all $u \le t$, yields

$$\begin{aligned} \int _{\Sigma _t} T_{0 \nu } {\overline{K}}_0^{\nu } \mathrm{d}x = \int _{\Sigma _0} T_{0 \nu } {\overline{K}}_0^{\nu } \mathrm{d}x-\int _0^t \int _{\Sigma _s } G_{\mu \nu } J^{\mu } {\overline{K}}_0^{\nu } \mathrm{d}x \mathrm{d}s \end{aligned}$$

(12)

and

$$\begin{aligned}&\int _{\Sigma _t^u} T_{0 \nu } {\overline{K}}_0^{\nu } \mathrm{d}x + \frac{1}{\sqrt{2}} \int _{C_{u}(t)} T_{L \nu } {\overline{K}}_0^{\nu }\mathrm{d}C_{u}(t)\nonumber \\&\quad = \int _{\Sigma ^u_0} T_{0 \nu } {\overline{K}}_0^{\nu } \mathrm{d}x-\int _0^t \int _{\Sigma ^u_s } G_{\mu \nu } J^{\mu } {\overline{K}}_0^{\nu } \mathrm{d}x \mathrm{d}s. \end{aligned}$$

(13)

Notice, using (9) and $2 {\overline{K}}_0=\tau _+^2L+\tau _-^2 {\underline{L}}$, that

$$\begin{aligned}&4T_{0 \nu } {\overline{K}}_0^{\nu } = \tau _+^2 |\alpha |+\tau _-^2|{\underline{\alpha }}|+(\tau _+^2+\tau _-^2)(|\rho |+|\sigma |), \\&\quad 2 T_{L \nu } {\overline{K}}_0^{\nu } = \tau _+^2 |\alpha |^2+\tau _-^2|\rho |^2+\tau _-^2|\sigma |^2. \end{aligned}$$

It then only remains, to obtain the first estimate, to take the supremum over all $u \le t$ in (13) and to combine it with (12). For the other one, note first using Corollary 2.3 and (9) that

$$\begin{aligned}&\left| \nabla ^{\mu } \left( \tau _-^2 \log ^{-k}(1 + \tau _-) T_{\mu 0} \right) \right| \\&\quad = \left| \tau _-^2 \log ^{-k}(1 + \tau _-) \nabla ^{\mu } T_{\mu 0}-\frac{1}{2}{\underline{L}} \left( \tau _-^2 \log ^{-k}(1 + \tau _-) \right) T_{L 0} \right| \\&\quad = \left| \tau _-^2 \log ^{-k}(1 + \tau _-) \nabla ^{\mu } T_{\mu 0} -u \log ^{-k}(1 + \tau _-) \left( 2-\frac{k \tau _-}{1 + \tau _-} \log ^{-1}(1 + \tau _-) \right) T_{L 0} \right| \\&\quad \lesssim \tau _-^2 \log ^{-k}(1+\tau _-) \left| G_{0 \lambda } J^{\lambda } \right| +\frac{\tau _+^2}{\tau _+ \log ^{k+1}(1 + \tau _+)} \left( \left| \alpha \right| ^2+\left| \rho \right| ^2+\left| \sigma \right| ^2 \right) . \end{aligned}$$

Consequently, applying the divergence theorem in $[0,t] \times {\mathbb {R}}^3$, we obtain

$$\begin{aligned}&\int _{\Sigma _t} \frac{\tau _-^2}{ \log ^{k}(1+\tau _-)} T_{00}\mathrm{d}x \lesssim \int _{\Sigma _0} (1+r)^2 T_{00}\mathrm{d}x+\int _0^t \int _{\Sigma _s } \frac{\tau _-^2}{\log ^{k}(1+\tau _-)} \left| G_{0 \nu } J^{\nu } \right| \mathrm{d}x \mathrm{d}s\\&\quad +\,\int _0^t \frac{{\mathcal {E}}^{{\overline{K}}_0}[G](s)}{(1+s) \log ^{k+1}(2+s)} \mathrm{d}s. \end{aligned}$$

The result then follows from $4T_{00} = |\alpha |^2+|{\underline{\alpha }}|^2+2|\rho |^2+2|\sigma |^2$ and $\int _0^{+\infty } (1+s)^{-1} \log ^{-2}(2+s) \mathrm{d}s < + \infty$. $\square$

3.2 Decay Estimates

3.2.1 Decay Estimates for Velocity Averages

We prove in this subsection an $L^{\infty }-L^1$ and an $L^2-L^1$ Klainerman–Sobolev inequality for velocity averages. The $L^{\infty }-L^1$ one was originally proved in [8] (see Theorem 6), and we propose here a shorter proof. Let us start with the following lemma.

Lemma 3.5

Let $g {:}\, {\mathbb {S}}^2 \times \left( {\mathbb {R}}^3_v {\setminus } \{ 0 \} \right) \rightarrow {\mathbb {R}}$ a sufficiently regular function. Then, with $\Omega ^{\beta } \in {\mathbb {O}}^{|\beta |}$,

$$\begin{aligned} \left\| \int _{v \in {\mathbb {R}}^3} |g| \mathrm{d}v \right\| _{L^{\infty }({\mathbb {S}}^2)}\lesssim & {} \sum _{ |\beta | \le 2 } \left\| \int _{v \in {\mathbb {R}}^3} \left| {\widehat{\Omega }}^{\beta } g \right| \mathrm{d}v \right\| _{L^1({\mathbb {S}}^2)},\\ \left\| \int _{v \in {\mathbb {R}}^3} |g| \mathrm{d}v\right\| _{L^{2}({\mathbb {S}}^2)}\lesssim & {} \sum _{ |\beta | \le 1 } \left\| \int _{v \in {\mathbb {R}}^3} \left| {\widehat{\Omega }}^{\beta } g \right| \mathrm{d}v \right\| _{L^1({\mathbb {S}}^2)}. \end{aligned}$$

Proof

Let $\omega \in {\mathbb {S}}^2$ and $(\theta , \varphi )\in ] 0,\pi [\times ]0, 2\pi [$ be a system of spherical coordinates such that $\omega _1:= \theta (\omega )= \frac{\pi }{2}$ and $\omega _2:=\varphi (\omega )=\pi$. Using a one-dimensional Sobolev inequality, that $|\partial _{\theta } u| \lesssim \sum _{\Omega \in {\mathbb {O}}} |\Omega u |$ and Lemma 2.5, we have,

$$\begin{aligned} \int _v |g|(\omega _1,\omega _2,v) \mathrm {d}v \,&\lesssim \, \int _{|\theta - \frac{\pi }{2}|\le \frac{\pi }{4} } \left( \left| \int _v |g|(\theta ,\omega _2,v) \mathrm {d}v \right| +\left| \partial _{\theta } \int _v |g|(\theta ,\omega _2,v) \mathrm {d}v \right| \right) \mathrm {d} \theta \\&\lesssim \, \sum _{ \begin{array}{c} \Omega ^{\kappa } \in {\mathbb {O}}^{|\kappa |} \\ \, |\kappa | \le 1 \end{array}} \int _{|\theta - \frac{\pi }{2}|\le \frac{\pi }{4} } \int _v | {\widehat{\Omega }}^{\kappa } g|(\theta ,\omega _2,v) \mathrm {d}v \sin ( \theta ) \mathrm {d} \theta . \end{aligned}$$

We obtain similarly that

$$\begin{aligned} \int _{|\theta - \frac{\pi }{2}|\le \frac{\pi }{4} } \int _v | {\widehat{\Omega }}^{\kappa } g|(\theta ,\omega _2,v) \mathrm {d}v \sin ( \theta ) \mathrm {d} \theta \,\lesssim & {} \, \sum _{ \begin{array}{c} \Omega ^{\gamma } \in {\mathbb {O}}^{|\gamma |} \\ \, |\gamma | \le 1 \end{array}} \int _{\varphi =0}^{2 \pi } \int _{|\theta - \frac{\pi }{2}|\le \frac{\pi }{4}} \int _v | {\widehat{\Omega }}^{\gamma } {\widehat{\Omega }}^{\kappa } g|(\theta ,\varphi ,v) \mathrm {d}v \sin ( \theta ) \mathrm {d} \theta \mathrm {d} \varphi \\ {}\lesssim & {} \, \sum _{ \begin{array}{c} \Omega ^{\beta } \in {\mathbb {O}}^{|\beta |} \\ \, |\beta | \le 2 \end{array}} \left\| \int _{v} \left| {\widehat{\Omega }}^{\beta } g \right| \mathrm {d}v \right\| _{L^1({\mathbb {S}}^2)}, \end{aligned}$$

which implies the first inequality. For the other one, by a standard $L^2-L^1$ Sobolev inequality, one has

$$\begin{aligned}&\left\| \int _{v \in {\mathbb {R}}^3} |g| \mathrm{d}v \right\| _{L^{2}({\mathbb {S}}^2)}{\lesssim } \left\| \int _{v \in {\mathbb {R}}^3} |g| \mathrm{d}v \right\| _{L^{1}({\mathbb {S}}^2)}{+}\left\| \partial _{\theta } \int _{v \in {\mathbb {R}}^3} |g| \mathrm{d}v \right\| _{L^1({\mathbb {S}}^2)}{+}\left\| \partial _{\varphi } \int _{v \in {\mathbb {R}}^3} |g| \mathrm{d}v \right\| _{L^1({\mathbb {S}}^2)}. \end{aligned}$$

It then remains to apply Lemma 2.5 again. $\square$

Proposition 3.6

Let $f{:}\,[0,T[ \times {\mathbb {R}}^3 \times \left( {\mathbb {R}}^3_v {\setminus } \{ 0 \} \right)$ be a sufficiently regular function, $z \in {\mathbf {k}}_0$ and $j \in {\mathbb {N}}$. Then $\forall \, (t,x) \in [0,T[ \times {\mathbb {R}}^3$

$$\begin{aligned}&\int _{v \in {\mathbb {R}}^3} |z^jf|(t,x,v) \mathrm{d}v \\&\quad \lesssim \; \frac{(j+1)^3}{\tau _+^2 \tau _-} \sum _{\begin{array}{c} {\widehat{Z}}^{\beta } \in \widehat{{\mathbb {P}}}_0^{|\beta |} \\ \, |\beta | \le 3 \end{array}} \sum _{w \in {\mathbf {k}}_0} \left\| \int _{v \in {\mathbb {R}}^3} \left| w^j {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \right\| _{ L^1(\Sigma _t)}. \end{aligned}$$

Proof

Note first that if $j \ge 1$, the inequality follows from the case $j=0$ as, using Lemma 2.10,

$$\begin{aligned} \left| {\widehat{Z}}^{\beta } \left( z^j f \right) \right| \lesssim j^{|\beta |} \sum _{w \in {\mathbf {k}}_0} \sum _{|\kappa | \le |\beta |} \left| w^j {\widehat{Z}}^{\kappa } f \right| . \end{aligned}$$

Suppose now that $j=0$ and consider $(t,x)=(t,|x| \omega ) \in [0,T[ \times {\mathbb {R}}^3$.

• If $1+t \le 2|x|$, one has, using Lemmas 2.5 and 2.7,

  $$\begin{aligned}&|x|^2 \tau _- \int _v |f|(t,|x|\omega ,v) \mathrm{d}v \\&= -|x|^2 \int _{r=|x|}^{+ \infty } \partial _r \left( \tau _- \int _v |f|(t,r\omega ,v) \mathrm{d}v \right) \mathrm{d}r\\&\quad \lesssim |x|^2 \sum _{Z \in {\mathbb {K}}} \int _{r=|x|}^{+ \infty } \left( \int _v |f|(t,r\omega ,v) \mathrm{d}v + \left| Z \left( \int _v |f|(t,r\omega ,v) \mathrm{d}v \right) \right| \right) \mathrm{d}r \\&\quad \lesssim \sum _{ {\widehat{Z}} \in \widehat{{\mathbb {P}}}_0} \int _{r=|x|}^{+ \infty } \left( \int _v |f|(t,r\omega ,v) \mathrm{d}v+ \int _v |{\widehat{Z}} f|(t,r\omega ,v) \mathrm{d}v \right) r^2 \mathrm{d}r. \end{aligned}$$

  It then remains to apply Lemma 3.5 and to remark that $\tau _+ \lesssim r$ in the region considered.

• Otherwise $1+t \ge 2|x|$, so that, with $\tau :=1+t$,

  $$\begin{aligned} \forall \, |y| \le \frac{1}{4}, \quad \tau \le 10(1+|t-|x+\tau y||). \end{aligned}$$

  Thus, for all sufficiently regular functions h, $1 \le i \le 3$ and almost all $|y| \le \frac{1}{4}$, we have, using Lemmas 2.7 and then 2.5,

  $$\begin{aligned} \left| \partial _{y^i} \left( \int _v |h|(t,x+ \tau y,v) \mathrm{d}v \right) \right|= & {} \left| \tau \int _v \Big ( \partial _i |h| \Big ) (t,x+ \tau y,v) \mathrm{d}v \right| \nonumber \\\lesssim & {} \left| (1+|t-|x+ \tau y|| ) \int _v \Big ( \partial _i |h| \Big ) (t,x+\tau y,v) \mathrm{d}v \right| \nonumber \\\lesssim & {} \sum _{Z \in {\mathbb {K}}} \left| \int _v \Big ( Z |h| \Big )(t,x+ \tau y,v) \mathrm{d}v \right| \nonumber \\\lesssim & {} \sum _{\begin{array}{c} {\widehat{Z}}^{\kappa } \in \widehat{{\mathbb {P}}}_0^{|\kappa |} \\ \, |\kappa | \le 1 \end{array}} \int _v \left| {\widehat{Z}}^{\kappa } h \right| (t,x+\tau y,v) \mathrm{d}v. \end{aligned}$$

  (14)

  Hence, using alternatively three times a one-dimensional Sobolev inequality and then (14), we get,

  $$\begin{aligned}&\int _v |f|(t,x,v) \mathrm{d}v\\&\quad \lesssim \sum _{n=0}^1 \int _{|y^1| \le \frac{1}{4 \sqrt{3}}} \left| \left( \partial _{y^1} \right) ^n \left( \int _v |f|(t,x^1+\tau y^1,x^2,x^3,v) \mathrm{d}v \right) \right| \mathrm{d}y^1 \\&\quad \lesssim \sum _{|\kappa | \le 1} \int _{|y^1| \le \frac{1}{4 \sqrt{3}}} \int _v |{\widehat{Z}}^{\kappa } f|(t,x^1+\tau y^1,x^2,x^3,v) \mathrm{d}v \mathrm{d}y^1 \\&\lesssim \sum _{n=0}^1 \sum _{ |\kappa | \le 1 } \int _{|y^1| \le \frac{1}{4 \sqrt{3}}} \int _{|y^2| \le \frac{1}{4 \sqrt{3}}} \left| \left( \partial _{y^2} \right) ^n \left( \int _v |{\widehat{Z}}^{\kappa } f|(t,x +\tau (y^1,y^2,0),v) \mathrm{d}v \right) \right| \mathrm{d}y^2\mathrm{d}y^1 \\&\lesssim \sum _{|\kappa | \le 2} \int _{|y^1| \le \frac{1}{4 \sqrt{3}}} \int _{|y^2| \le \frac{1}{4 \sqrt{3}}} \int _v |{\widehat{Z}}^{\kappa } f|(t,x+ \tau (y^1,y^2,0),v) \mathrm{d}v \mathrm{d}y^2 \mathrm{d}y^1 \\&\lesssim \sum _{n=0}^1\sum _{ |\kappa | \le 2 } \int _{|y| \le \frac{1}{4}} \left| \left( \partial _{y^3} \right) ^n \left( \int _v |{\widehat{Z}}^{\kappa } f|(t,x+\tau y,v) \mathrm{d}v \right) \right| \mathrm{d}y \\&\lesssim \sum _{|\kappa | \le 3} \int _{|y| \le \frac{1}{4}} \int _v |{\widehat{Z}}^{\kappa } f|(t,x+\tau y,v) \mathrm{d}v \mathrm{d}y. \end{aligned}$$

  The result then follows from the change of variables $z=\tau y$ and that $\tau _- \le \tau _+ \lesssim \tau$ in the region studied.

$\square$

We now turn on the $L^2-L^1$ Klainerman–Sobolev inequality.

Proposition 3.7

Let $f{:}\,[0,T[ \times {\mathbb {R}}^3 \times \left( {\mathbb {R}}^3_v {\setminus } \{ 0 \} \right)$ be a sufficiently regular function, $z \in {\mathbf {k}}_0$ and $j \in {\mathbb {N}}$. Then,

$$\begin{aligned} \forall \, t \in [0,T[, \, \left\| \tau _+ \tau _-^{\frac{1}{2}} \int _{v \in {\mathbb {R}}^3} |z^jf| \mathrm{d}v \right\| _{L^2(\Sigma _t)} \lesssim (j+1)^2 \sum _{\begin{array}{c} {\widehat{Z}}^{\beta } \in \widehat{{\mathbb {P}}}_0^{|\beta |} \\ \, |\beta | \le 2 \end{array}} \sum _{w \in {\mathbf {k}}_0} \left\| \int _{v \in {\mathbb {R}}^3} \left| w^j {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \right\| _{ L^1(\Sigma _t)}. \end{aligned}$$

Proof

As previously, we can restrict the proof to the case $j=0$. We introduce $\delta = \frac{1}{4}$ for convenience and we suppose first that $t \ge 1$. The idea is classical and consists in splitting $\Sigma _t$ into the three domains, $|x| \le \frac{t}{2}$, $|x| \ge \frac{3}{2}t$ and $\frac{1}{2}t \le |x| \le \frac{3}{2}t$.

$\bullet$ Step 1, the interior region. Applying a local two-dimensional $L^2-L^1$ Sobolev inequality to the function $x \mapsto \int _v |f|(t,tx,v) \mathrm{d}v$, we get

$$\begin{aligned}&\int _{|x| \le \frac{1}{2}} \left| \int _v |f|(t,tx,v)\mathrm{d}v \right| ^2 \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}x_3 \\&\qquad \lesssim \sum _{q=0}^1 \int _{|x_3| \le \frac{1}{2}} \left| \int _{x_1^2+x_2^2 \le \frac{1}{4}-x_3^2+\delta ^2} \int _v \left( (t \partial _{x_1,x_2})^q |f| \right) (t,tx,v)\mathrm{d}v \mathrm{d}x_1 \mathrm{d} x_2 \right| ^2 \mathrm{d}x_3. \end{aligned}$$

As $t-|tx| \ge \frac{1}{4}t$ on the domain of integration since $|x| \le \frac{1}{2}+\delta \le \frac{3}{4}$, Lemmas 2.5 and 2.7 give us

$$\begin{aligned}&\int _{|x| \le \frac{1}{2}} \left| \int _v |f|(t,tx,v)\mathrm{d}v \right| ^2 \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}x_3 \\&\qquad \lesssim \sum _{|\beta | \le 1} \int _{|x_3| \le \frac{1}{2}} \left| \int _{x_1^2+x_2^2 \le \frac{1}{4}-x_3^2+\delta ^2} \int _v \left| {\widehat{Z}}^{\beta } f \right| (t,tx,v)\mathrm{d}v \mathrm{d}x_1 \mathrm{d} x_2 \right| ^2 \mathrm{d} x_3. \end{aligned}$$

Now, one can obtain similarly, using a one-dimensional $L^2-L^1$ Sobolev inequality in the variable $x_3$, that

$$\begin{aligned}&\int _{|x| \le \frac{1}{2}} \left| \int _v |f|(t,tx,v)\mathrm{d}v \right| ^2 \mathrm{d}x_1 \mathrm{d}x_2 \mathrm{d}x_3 \\&\qquad \lesssim \sum _{|\beta | \le 2} \left| \int _{|x_3| \le \frac{1}{2}+\delta } \int _{x_1^2+x_2^2 \le \frac{1}{4}-x_3^2+\delta ^2} \int _v \left| {\widehat{Z}}^{\beta } f \right| (t,tx,v)\mathrm{d}v \mathrm{d}x_1 \mathrm{d} x_2 \mathrm{d} x_3 \right| ^2. \end{aligned}$$

Since $\tau _+^2 \tau _- \lesssim t^3$ if $|x| \le \frac{1}{2}t$, we finally obtain, by the change of variables $y=tx$,

$$\begin{aligned}&\left\| \tau _+ \tau _-^{\frac{1}{2}} \int _{v \in {\mathbb {R}}^3} |f|(t,y,v) \mathrm{d}v \right\| _{L^2(|y| \le \frac{1}{2}t)} \lesssim t^3 \left\| \int _{v \in {\mathbb {R}}^3} |f|(t,tx,v) \mathrm{d}v \right\| _{L^2(|x| \le \frac{1}{2})} \nonumber \\&\qquad \lesssim \sum _{|\beta | \le 2} \left\| \int _v \left| {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \right\| _{L^1(\Sigma _t)}. \end{aligned}$$

(15)

$\bullet$ Step 2, the exterior region. Let us introduce, for $i \in {\mathbb {N}}$, the following sets⁹

$$\begin{aligned} X_i:= & {} \left\{ y \times \Sigma _t \, / \, 3 t \times 2^{i-1} \le |y|< 3t \times 2^i \right\} , \, \text {and} \\ Y_i:= & {} \left\{ y \times \Sigma _t \, / \, 5 t \times 2^{i} \le 4 |y| < 13t \times 2^{i} \right\} . \end{aligned}$$

In the domain considered here, where $|x| \ge \frac{3}{2}t$, we have $\tau _+ \lesssim |x|$ but we cannot follow exactly what we have done for the interior region as we cannot view |x| as a parameter. However, as for $i \in {\mathbb {N}}$, $2^i t \sim \tau _+$ on $X_i$ and

$$\begin{aligned} \forall \, \frac{3}{2}-\delta \le |x| \le 3+\delta , \, |2^i t x|-t \ge \left( 2^i\frac{5}{4}-1 \right) t \ge \frac{1}{4} \times 2^i t, \end{aligned}$$

we can apply similar operations to $x \mapsto \int _v |f|(t,2^itx,v) \mathrm{d}v$ as to $x \mapsto \int _v |f|(t,tx,v) \mathrm{d}v$ previously and obtain

$$\begin{aligned}&\int _{\frac{3}{2} \le |x| \le 3} \left| \int _v |f|(t,2^i tx,v)\mathrm{d}v \right| ^2 \mathrm{d}x \\&\qquad \lesssim \sum _{|\beta | \le 2} \left| \int _{ |x_3| \le 3+\delta } \int _{\frac{9}{4}-x_3^2-\delta ^2 \le x_1^2+x_2^2 \le 9-x_3^2+\delta ^2} \int _v \left| {\widehat{Z}}^{\beta } g \right| (t,2^itx,v)\mathrm{d}v \mathrm{d}x \right| ^2 \\&\qquad \lesssim \sum _{|\beta | \le 2} \left| \int _{\frac{5}{4} \le |x| \le \frac{13}{4}} \int _v \left| {\widehat{Z}}^{\beta } g \right| (t,2^itx,v)\mathrm{d}v \mathrm{d}x \right| ^2. \end{aligned}$$

As $\tau _+^2 \tau _- \lesssim 2^{3i} t^3$ on $X_i$, we finally obtain by the change of variables $y=2^i t x$,

$$\begin{aligned} \left\| \tau _+ \tau _-^{\frac{1}{2}} \int _{v \in {\mathbb {R}}^3} |f| \mathrm{d}v \right\| _{L^2(X_i)}\lesssim & {} 2^{3i} t^3 \left\| \int _{v \in {\mathbb {R}}^3} |f|(t,2^itx,v) \mathrm{d}v \right\| _{L^2( \frac{3}{2} \le |x| \le 3)} \\\lesssim & {} \sum _{|\beta | \le 2} \left\| \int _v \left| {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \right\| _{L^1(Y_i)}. \end{aligned}$$

As $Y_i \cap Y_j = \varnothing$ if $|i-j| \ge 2$, $X_i \cap X_j = \varnothing$ if $i \ne j$ and since $\{ y \in \Sigma _t \, / \, |y| \ge \frac{3}{2}t \}=\cup _{i=0}^{+\infty } X_i$, we get

$$\begin{aligned} \left\| \tau _+ \tau _-^{\frac{1}{2}} \int _{v \in {\mathbb {R}}^3} |f|(t,y,v) \mathrm{d}v \right\| _{L^2(|y| \ge \frac{3}{2}t)} \; \lesssim \; \sum _{|\beta | \le 2} \left\| \int _v \left| {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \right\| _{L^1(\Sigma _t)}. \end{aligned}$$

(16)

$\bullet$ Step 3, the remaining domain. We now focus on the region $\frac{1}{2}t \le |x| \le \frac{3}{2}t$. We will obtain the $\tau _+$ integrated decay with the rotational vector fields through Sobolev inequalities on the spheres. To obtain the $\sqrt{\tau _-}$ decay, note first that $|u| \le \frac{1}{2}t$ in this region (recall that $u=t-|x|$). The idea to capture the decay in u will then be to divide the domain in the disjoint union of the sets

$$\begin{aligned} V_i := \{ y \in \Sigma _t \, / \, 2^{-i-1}t < |t-|y|| \le 2^{-i} t \}, \quad i \in {\mathbb {N}}^*. \end{aligned}$$

Let $\omega \in {\mathbb {S}}^2$. Applying a $L^2-L^1$ Sobolev inequality to $g{:}\, s \mapsto \int _v|f|(t,t(1-2^{-i}s)\omega ,v)\mathrm{d}v$, we obtain

$$\begin{aligned}&\int _{ \frac{1}{2} \le |s| \le 1} \left| \int _v |f|(t,t(1-2^{-i}s)\omega ,v) \mathrm{d}v \right| ^2 \mathrm{d}s \\&\qquad \lesssim \; \sum _{j=0}^1 \left| \int _{ \frac{1}{4} \le |s| \le \frac{5}{4} } \left| \int _v \left( t2^{-i} \partial _r|f| \right) ^j(t,t(1-2^{-i}s)\omega ,v) \mathrm{d}v \right| \mathrm{d}s \right| ^2. \end{aligned}$$

Since $\frac{1}{4}2^{-i}t \le |t-|t-2^{-i}ts||$ for all $i \in {\mathbb {N}}^*$ and $\frac{1}{4} \le |s| \le \frac{5}{4}$, we get, using Lemmas 2.5 and 2.7 that

$$\begin{aligned}&\int _{ \frac{1}{2} \le |s| \le 1} \left| \int _v |f|(t,t(1-2^{-i}s)\omega ,v) \mathrm{d}v \right| ^2 \mathrm{d}s \\&\qquad \lesssim \; \sum _{|\kappa | \le 1} \left| \int _{ \frac{1}{4} \le |s| \le \frac{5}{4} } \left| \int _v | {\widehat{Z}}^{\kappa } f |(t,t(1-2^{-i}s)\omega ,v) \mathrm{d}v \right| \mathrm{d}s \right| ^2. \end{aligned}$$

The change of variables $r=t(1-2^{-i}s)$ gives

$$\begin{aligned}&t2^{-i} \int _{ 2^{-i-1}t \le |t-r| \le 2^{-i}t} \left| \int _v |f|(t,r\omega ,v) \mathrm{d}v \right| ^2 \mathrm{d}r \\&\quad \lesssim \; \sum _{|\kappa | \le 1} \left| \int _{ 2^{-i-2}t \le |t-r| \le 5 \times 2^{-i-2}t } \left| \int _v | {\widehat{Z}}^{\kappa } f |(t,r\omega ,v) \mathrm{d}v \right| \mathrm{d}r \right| ^2. \end{aligned}$$

As previously with the domains $X_i$ and $Y_i$, we take the sum over $i \in {\mathbb {N}}^*$ and we get

$$\begin{aligned}&\int _{ \frac{1}{2}t \le r \le \frac{3}{2}t} |t-r| \left| \int _v |f|(t,r\omega ,v) \mathrm{d}v \right| ^2 \mathrm{d}r \\&\qquad \lesssim \; \sum _{|\kappa | \le 1} \left| \int _{ \frac{3}{8}t \le r \le \frac{13}{8}t } \left| \int _v | {\widehat{Z}}^{\kappa } f|(t,r\omega ,v) \mathrm{d}v \right| \mathrm{d}r \right| ^2. \end{aligned}$$

By simpler operations, one can also obtain that

$$\begin{aligned}&\int _{ t-\frac{1}{2} \le r \le t+\frac{1}{2}} \left| \int _v |f|(t,r\omega ,v) \mathrm{d}v \right| ^2 \mathrm{d}r \\&\qquad \lesssim \; \sum _{|\kappa | \le 1} \left| \int _{ t-\frac{5}{8} \le r \le t+\frac{5}{8} } \left| \int _v | {\widehat{Z}}^{\kappa } f|(t,r\omega ,v) \mathrm{d}v \right| \mathrm{d}r \right| ^2. \end{aligned}$$

Integrating each side of these inequalities over ${\mathbb {S}}^2$ and applying Proposition 3.5 to the right-hand sides, we get

$$\begin{aligned}&\int _{ \frac{1}{2}t \le r \le \frac{3}{2}t} \int _{\omega \in {\mathbb {S}}^2} \tau _- \left| \int _v |f|(t,r\omega ,v \mathrm{d}v \right| ^2 \mathrm{d}{\mathbb {S}}^2 \mathrm{d}r \\&\qquad \lesssim \; \sum _{|\kappa | \le 2} \left| \int _{ \frac{3}{8}t \le r \le \frac{13}{8}t } \int _{\omega \in {\mathbb {S}}^2} \left| \int _v | {\widehat{Z}}^{\kappa } f|(t,r\omega ,v) \mathrm{d}v \right| \mathrm{d} {\mathbb {S}}^2 \mathrm{d}r \right| ^2. \end{aligned}$$

Finally, multiply both sides of the inequality by $t^2$ and use $\tau _+ \lesssim t \le 2r$ on the domain of integration in order to obtain

$$\begin{aligned}&\int _{ \frac{1}{2}t \le r \le \frac{3}{2}t} \int _{\omega \in {\mathbb {S}}^2} \tau _+^2 \tau _- \left| \int _v |f|(t,r\omega ,v \mathrm{d}v \right| ^2 \mathrm{d} {\mathbb {S}}^2 r^2 \mathrm{d}r \nonumber \\&\qquad \lesssim \sum _{|\kappa | \le 2} \left| \int _{ \frac{3}{8}t \le r \le \frac{13}{8}t } \int _{\omega \in {\mathbb {S}}^2} \left| \int _v | {\widehat{Z}}^{\kappa } f|(t,r\omega ,v) \mathrm{d}v \right| \mathrm{d}{\mathbb {S}}^2 r^2 \mathrm{d}r \right| ^2. \end{aligned}$$

(17)

The result then follows from (15), (16) and (17). The case $t \le 1$ can be treated similarly, repeating the arguments of Steps 1 and 2 since in that case $\tau _+^2 \tau _- \lesssim (1+r)^3$ and

$$\begin{aligned} \Sigma _t = \{ y \in \Sigma _t \, / \, |y| \le 2^{-1} \} \cup \left( \cup _{i=0}^{+ \infty } \{ y \in \Sigma _t \, / \, 2^{i-1} \le |y| < 2^i \} \right) .of the null component \end{aligned}$$

$\square$

3.2.2 Pointwise Decay Estimates for the Electromagnetic Field

In this section, we follow mostly [5]. We first present certain identities and inequalities between quantities linked to the null decomposition of a 2-form (see Sect. 2.3 for its definition), then we recall Sobolev inequalities and, finally, we prove the desired pointwise decay estimates for the electromagnetic field.

For the remaining part of this section, we consider G a 2-form and J a 1-form, both sufficiently regular and defined on $[0,T[ \times {\mathbb {R}}^3$, such that

$$\begin{aligned} \nabla ^{\mu } G_{\mu \nu }= & {} J_{\nu }, \\ \nabla ^{\mu } {}^* G_{ \mu \nu }= & {} 0. \end{aligned}$$

Aside from Lemma 3.10 and the estimate on $\alpha (G)$ in Proposition 3.13, all the results of this subsection apply to a general 2-form.

$\bullet$ Preparatory results

To lighten the presentation, we prove the three upcoming lemmas in Appendix D.

Lemma 3.8

Let $\Omega \in {\mathbb {O}}$. Then, the operators ${\mathcal {L}}_{\Omega }$ and $\nabla _{\partial _r}$ commute with the null decomposition of G as well as with each other, i.e., denoting by $\zeta$ any of the null component $\alpha$, ${\underline{\alpha }}$, $\rho$ or $\sigma$,

$$\begin{aligned}{}[{\mathcal {L}}_{\Omega },\quad \nabla _{\partial _r}] G=0, \quad {\mathcal {L}}_{\Omega }(\zeta (G))= \zeta ( {\mathcal {L}}_{\Omega }(G) ) \quad \text {and} \quad \nabla _{\partial _r}(\zeta (G))= \zeta ( \nabla _{\partial _r}(G) ). \end{aligned}$$

Similar results hold for ${\mathcal {L}}_{\Omega }$ and $\nabla _{\partial _t}$, $\nabla _L$ or $\nabla _{{\underline{L}}}$. For instance, $\nabla _{L}(\zeta (G))= \zeta ( \nabla _{L}(G) )$.

We now give a more precise version of Lemma 3.3 of [5].

Lemma 3.9

Denoting by $\zeta$ any of the null component $\alpha$, ${\underline{\alpha }}$, $\rho$ or $\sigma$, we have

The following equation will be useful in order to obtain a strong decay estimate on $\alpha (G)$.

Lemma 3.10

Denoting by $(\alpha , {\underline{\alpha }}, \rho , \sigma )$ the null decomposition of G, we have

(18)

The following result will allow us to treat part of the interior of the light cone.

Lemma 3.11

Let U be a smooth tensor field defined on $[0,T[ \times {\mathbb {R}}^3$. Then,

$$\begin{aligned}&\forall \, t \in [0,T[, \quad \sup _{|x| \le 1+\frac{t}{2}} |U(t,x)| \\&\quad \lesssim \frac{1}{(1+t)^{\frac{5}{2}}} \sum _{|\gamma | \le 2} \Vert \tau _- {\mathcal {L}}_{Z^{\gamma }}(U)(t,y) \Vert _{L^2 \left( |y| \le 2+\frac{3}{4}t \right) }. \end{aligned}$$

Proof

As $|{\mathcal {L}}_{Z^{\gamma }}(U)| \lesssim \sum _{|\beta | \le |\gamma |} \sum _{\mu , \nu } | Z^{\beta } (U_{\mu \nu })|$, it suffices to prove the result for each component of the tensor and we can assume that U is a scalar function. Let $(t,x) \in [0,T[ \times {\mathbb {R}}^3$ such that $|x| \le 1+ \frac{1}{2}t$. Apply a standard $L^2$ Sobolev inequality to $V{:}\,y \mapsto U(t,x+\frac{1+t}{4}y)$ and then make a change of variables to get

$$\begin{aligned} |U(t,x)|=|V(0)|\lesssim & {} \sum _{|\beta | \le 2} \Vert \partial _x^{\beta } V \Vert _{L^2_y(|y| \le 1)} \\\lesssim & {} \left( \frac{1+t}{4} \right) ^{-\frac{3}{2}} \sum _{|\beta | \le 2} \left( \frac{1+t}{4} \right) ^{|\beta |} \Vert \partial _x^{\beta } U(t,\cdot ) \Vert _{L^2_y(|y-x| \le \frac{1+t}{4})}. \end{aligned}$$

Observe now that $|y-x| \le \frac{1+t}{4}$ implies $|y| \le 2+\frac{3}{4}t$ and that $1+t \lesssim \tau _-$ on that domain. Consequently, using Lemma 2.7 and that $[Z, \partial ]$, for $Z \in {\mathbb {K}}$, is either 0 or a translation, we have

$$\begin{aligned}&( 1+t )^{|\beta |+1} \Vert \partial _x^{\beta } U(t,\cdot ) \Vert _{L^2_y(|y-x| \le \frac{1+t}{4})} \\&\quad \lesssim \; \Vert \tau _-^{|\beta |+1} \partial _x^{\beta } U(t,\cdot ) \Vert _{L^2_y(|y| \le 2+\frac{3}{4}t)} \; \lesssim \; \sum _{|\gamma | \le |\beta |} \Vert \tau _- Z^{\gamma } U(t,\cdot ) \Vert _{L^2_y(|y| \le 2+\frac{3}{4}t)}. \end{aligned}$$

$\square$

We refer to Lemma 2.3 of [5] for a proof of the following two Sobolev inequalities, which will permit us to deal with the remaining region.

Lemma 3.12

Let U be a sufficiently regular tensor field defined on ${\mathbb {R}}^3$ and denote $\sum _{|\beta | \le k} \left| {\mathcal {L}}_{\Omega ^{\beta }}(U) \right| ^2$, where $\Omega ^{\beta } \in {\mathbb {O}}^{|\beta |}$, by $|U|^2_{{\mathbb {O}},k}$. There exists a uniform constant $C>0$, independent of U, such that

$$\begin{aligned} \forall \, t \in {\mathbb {R}}_+, \; \forall \, |x| \ge \frac{1}{2}t+1, \quad |U(x)|\le & {} \frac{C}{|x|\tau _-^{\frac{1}{2}}} \left( \int _{|y| \ge \frac{1}{2}t+1} |U(y)|^2_{{\mathbb {O}},2}+\tau _-^2|\nabla _{\partial _r} U(y) |^2_{{\mathbb {O}},1} \mathrm{d}y \right) ^{\frac{1}{2}}, \\ \forall \, x \ne 0, \quad |U(x)|\le & {} \frac{C}{|x|^{\frac{3}{2}}} \left( \int _{|y| \ge |x|} |U(y)|^2_{{\mathbb {O}},2}+|y|^2|\nabla _{\partial _r} U(y) |^2_{{\mathbb {O}},1} \mathrm{d}y \right) ^{\frac{1}{2}}. \end{aligned}$$

$\bullet$ Decay estimates for G We are now ready to prove the pointwise decay estimates on the electromagnetic field.

Proposition 3.13

Let $k \in {\mathbb {N}}^*$. Then, we have for all $(t,x) \in [0,T[ \times {\mathbb {R}}^3$,

$$\begin{aligned} |\rho |(t,x)+ |\sigma |(t,x)\lesssim & {} \frac{ \sqrt{{\mathcal {E}}_2[G](t)}}{\tau _+^{2}\tau _-^{\frac{1}{2}}}, \qquad |{\underline{\alpha }}|(t,x) \; \lesssim \; \sqrt{{\mathcal {E}}^{k}_2[G](t)} \frac{ \log ^{\frac{k}{2}}(1+\tau _-) }{\tau _+ \tau _-^{\frac{3}{2}}}, \\ |\alpha |(t,x)\lesssim & {} \frac{\sqrt{ {\mathcal {E}}_2[G](t) }+\sum _{|\kappa | \le 1} \Vert r^2 {\mathcal {L}}_{Z^{\kappa }}(J)_A\Vert _{L^2(\Sigma _t)}}{\tau _+^{\frac{5}{2}}}. \end{aligned}$$

Proof

We fix for this proof $(t,x) \in [0,T[ \times {\mathbb {R}}^3$. If $|x| \le 1+\frac{1}{2}t$, the result follows from Proposition 3.11. We then suppose $|x| \ge 1+\frac{t}{2}$. During this proof, $\Omega ^{\beta }$ will always denote a combination of rotational vector fields, i.e., $\Omega ^{\beta } \in {\mathbb {O}}^{|\beta |}$. Let $\zeta$ be either $\alpha$, $\rho$ or $\sigma$. As $\nabla _{\partial _r}$ and ${\mathcal {L}}_{\Omega }$ commute with the null decomposition (see Lemma 3.8), Lemma 3.12 gives us

$$\begin{aligned} r^4 \tau _- |\zeta |^2\lesssim & {} \int _{ |y| \ge \frac{t}{2}+1} |r \zeta |^2_{{\mathbb {O}},2}+\tau _-^2|\nabla _{\partial _r} (r \zeta ) |_{{\mathbb {O}},1}^2 \mathrm{d}y \\\lesssim & {} \sum _{ |\beta | \le 1 } \sum _{|\gamma | \le 2} \int _{ |y| \ge \frac{t}{2}+1} r^2| \zeta ( {\mathcal {L}}_{Z^{\gamma }} (G) |^2+\tau _-^2r^2| \zeta ( {\mathcal {L}}_{\Omega ^{\beta }} (\nabla _{\partial _r} G)) |^2 \mathrm{d}y. \end{aligned}$$

As $\nabla _{\partial _r}$ commute with ${\mathcal {L}}_{\Omega }$ as well as with the null decomposition (see Lemma 3.8), we have, using $2\partial _r= L-{\underline{L}}$ and Lemma 3.9,

$$\begin{aligned} | \zeta ( {\mathcal {L}}_{\Omega } (\nabla _{\partial _r} G)) |+| \zeta ( \nabla _{\partial _r} G) |\lesssim & {} | \nabla _L \zeta ( {\mathcal {L}}_{\Omega } (G) |\nonumber \\&\quad +| \nabla _{{\underline{L}}} \zeta ( {\mathcal {L}}_{\Omega } (G) |+| \nabla _L \zeta ( G) |+| \nabla _{{\underline{L}}} \zeta ( G) | \nonumber \\\lesssim & {} \frac{1}{\tau _-}\sum _{ |\gamma | \le 2} | \zeta ( {\mathcal {L}}_{Z^{\gamma }} (G) |. \end{aligned}$$

(19)

Since $\tau _+ \lesssim r \le \tau _+$ in the region considered, we finally obtain

$$\begin{aligned} \tau _+^4 \tau _- |\zeta |^2 \; \lesssim \; \sum _{|\gamma | \le 2} \int _{ |y| \ge \frac{t}{2}+1} \tau _+^2| \zeta ( {\mathcal {L}}_{Z^{\gamma }} (G) |^2 \mathrm{d}x \lesssim \; {\mathcal {E}}_2[G](t). \end{aligned}$$

We improve now the estimate on $\alpha$. As $\nabla ^{\mu } {\mathcal {L}}_{\Omega }$ $(G)_{\mu \nu } = {\mathcal {L}}_{\Omega }(J)_{\nu }$ and $\nabla ^{\mu } {}^* {\mathcal {L}}_{\Omega } (G)_{\mu \nu } = 0$ for all $\Omega \in {\mathbb {O}}$, we have according to (18) that

Consequently, we get using Lemma 3.9 that for all $\Omega \in {\mathbb {O}}$,

$$\begin{aligned} | \alpha ( \nabla _{\partial _r} G) |+| \alpha ({\mathcal {L}}_{\Omega } (\nabla _{\partial _r} G)) |\lesssim & {} \left| J_A \right| +\left| {\mathcal {L}}_{\Omega } (J)_A \right| \nonumber \\&\quad +\,\frac{1}{r}\sum _{ |\gamma | \le 2} | \alpha ( {\mathcal {L}}_{Z^{\gamma }} (G) |+| \rho ( {\mathcal {L}}_{Z^{\gamma }} (G) |+| \sigma ( {\mathcal {L}}_{Z^{\gamma }} (G) |.\nonumber \\ \end{aligned}$$

(20)

Applying the second inequality of Lemma 3.12 and using this time (20) instead of (19), we get

$$\begin{aligned} \tau _+^5 |\alpha |^2 \;\lesssim & {} |x|^5 |\alpha |^2 \lesssim \; \int _{ |y| \ge |x|} |r \alpha |^2_{{\mathbb {O}},2}+r^2|\nabla _{\partial _r} ( r \alpha ) |_{{\mathbb {O}},1}^2 \mathrm{d}y\\\lesssim & {} {\mathcal {E}}_2[G](t)+\sum _{|\kappa | \le 1} \Vert r^2 {\mathcal {L}}_{Z^{\kappa }}(J)_A\Vert ^2_{L^2(\Sigma _t)}. \end{aligned}$$

Applying the first inequality of Lemma 3.12 to $\tau _- \log ^{-\frac{k}{2}}(1+\tau _-) {\underline{\alpha }}$ and using the same arguments as previously, one has

$$\begin{aligned} \frac{r^2\tau _-^3}{\log ^k(1+\tau _-)} |{\underline{\alpha }} |^2\lesssim & {} \int _{ |y| \ge \frac{t}{2}+1} \left| \frac{\tau _-}{\log ^{\frac{k}{2}}(1+\tau _-)} {\underline{\alpha }} \right| ^2_{{\mathbb {O}},2}+\tau _-^2 \left| \nabla _{\partial _r} \left( \frac{\tau _-}{\log ^{\frac{k}{2}}(1+\tau _-)} {\underline{\alpha }} \right) \right| _{{\mathbb {O}},1}^2 \mathrm{d}y \\\lesssim & {} {\mathcal {E}}^{k}_2[G](t). \end{aligned}$$

$\square$

4 The Null Structure of the Nonlinearity $\mathcal {L}_{Z^{\gamma }}(F) \left( v, \nabla _v \hat{Z}^{\beta } f \right)$

In order to take advantage of the null structure of the Vlasov equation, we will expand quantities such as ${\mathcal {L}}_{Z^{\gamma }}(F) \left( v,\nabla _v g \right)$, with g a regular function, in null coordinates. We then use the following lemma.

Lemma 4.1

Let G be a sufficiently regular 2-form, $(\alpha , {\underline{\alpha }}, \rho , \sigma )$ its null components and g a sufficiently regular function. Then,

$$\begin{aligned} \left| G \left( v, \nabla _v g \right) \right| \; \lesssim \; \left( \tau _- |\rho |+\tau _+|\alpha |+\tau _+\frac{|v^A|}{v^0}|\sigma | +\tau _-\frac{|v^A|}{v^0} |{\underline{\alpha }}| +\tau _+\frac{v^{{\underline{L}}}}{v^0} |{\underline{\alpha }}| \right) \sum _{{\widehat{Z}} \in \widehat{{\mathbb {P}}}_0} \left| {\widehat{Z}} g \right| . \end{aligned}$$

Proof

Expanding $G(v, \nabla _v g )$ with null components, we obtain

$$\begin{aligned} G(v, \nabla _v g )= & {} 2 \rho \left( v^L \left( \nabla _v g \right) ^{{\underline{L}}}-v^{{\underline{L}}} \left( \nabla _v g \right) ^L \right) +v^B \varepsilon _{BA} \sigma \left( \nabla _v g \right) ^A-v^L \alpha _A \left( \nabla _v g \right) ^A\nonumber \\&+v^A \alpha _A \left( \nabla _v g \right) ^L \nonumber \\&-v^{{\underline{L}}} {\underline{\alpha }}_A \left( \nabla _v g \right) ^A+v^A {\underline{\alpha }}_A \left( \nabla _v g \right) ^{{\underline{L}}}. \end{aligned}$$

(21)

We bound the angular components of $\nabla _v g$ by merely using $v^0 \partial _{v^i} = {\widehat{\Omega }}_{0i}-t \partial _i-x^i \partial _t$. The radial component¹⁰ has a better behavior since

$$\begin{aligned} v^0\left( \nabla _v g \right) ^r = \frac{x^i}{r} v^0\partial _{v^i} g=\frac{x^i}{r} {\widehat{\Omega }}_{0i}g-Sg+(t-r) {\underline{L}} g. \end{aligned}$$

(22)

$\square$

Remark 4.2

Let us explain how this lemma reflects the null structure of the system. For this, we write $D_1 \prec D_2$ if $D_2$ is expected to behave better than $D_1$. Recall that we have the following hierarchies between the null components of G, v and $\nabla _v g$.

• ${\underline{\alpha }} \prec \rho \sim \sigma \prec \alpha$,

• $v^L \prec v^A \prec v^{{\underline{L}}}$,

• $\left( \nabla _v g \right) ^A \prec \left( \nabla _v g \right) ^r$.

We can then notice that ${\underline{\alpha }}$ is hit by $v^{{\underline{L}}}$ or $v^A \left( \nabla _v g \right) ^r$, $\rho$ by $\left( \nabla _v g \right) ^r$ and $\sigma$ by $v^A$.

5 Bootstrap Assumptions and Strategy of the Proof

Let $N \ge 10$ and $(f^0,F^0)$ be an initial data set satisfying the assumptions of Theorem 1.1. Then, by a local well-posedness argument, there exists a unique maximal solution (f, F) arising from these data to the system¹¹

$$\begin{aligned} T^{\chi }_F(f)= & {} 0, \\ \nabla ^{\mu } F_{\mu \nu }= & {} J(f)_{\nu }, \\ \nabla ^{\mu } {}^* F_{\mu \nu }= & {} 0. \end{aligned}$$

Applying Proposition B.1 and considering possibly $\epsilon _1=C_1 \epsilon$, with $C_1$ a constant depending only on N, we can suppose without loss of generality that ${\mathbb {E}}_N^2[f](0) \le \epsilon$ and ${\mathcal {E}}_N[F](0) \le \epsilon$. Let $T^* >0$ such that $[0,T^*[$ is the maximal domain of (f, F) and $T \in ]0,T^*[$ be the largest time such that¹², for all $t \in [0,T]$,

$$\begin{aligned} {\mathbb {E}}^{2}_{N-2}[f](t)\le & {} 4 \epsilon , \end{aligned}$$

(23)

$$\begin{aligned} {\mathbb {E}}^{1}_N[f](t)\le & {} 4 \epsilon \log (3+t), \end{aligned}$$

(24)

$$\begin{aligned} \sum _{|\beta | = N-1} \left\| r^2 \int _v \frac{v^A}{v^0} {\widehat{Z}}^{\beta } f \mathrm{d}v \right\| _{L^2(\Sigma _t)}\le & {} \sqrt{\epsilon } \log (3+t), \end{aligned}$$

(25)

$$\begin{aligned} {\mathcal {E}}_{N}[F](t)\le & {} 4 \epsilon \log ^4 (3+t), \end{aligned}$$

(26)

$$\begin{aligned} {\mathcal {E}}^{5}_N[F](t)\le & {} 2 {\underline{C}} \epsilon , \end{aligned}$$

(27)

where ${\underline{C}}>0$ is a positive constant which will be specified later. The third bootstrap assumption is here for convenience, we could avoid it but it would complicate the proof. Before presenting our strategy, let us write the immediate consequences of these bootstrap assumptions. Using the Klainerman–Sobolev inequality of Proposition 3.6 and the bootstrap assumption (23), one has

$$\begin{aligned} \forall \, (t,x) \in [0,T[ \times {\mathbb {R}}^3, \;\; z \in {\mathbf {k}}_0, \;\; |\beta | \le N-5, \quad \int _v \left| z^2 {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \lesssim \frac{\epsilon }{\tau _+^{2} \tau _-}. \end{aligned}$$

(28)

Applying the Klainerman–Sobolev inequality of Proposition 3.7, Lemma 2.11 and using (23) and (24), we get

$$\begin{aligned}&\forall \, t \in [0,T[, \quad \sum _{\begin{array}{c} |\beta | \le N-4 \\ \; z \in {\mathbf {k}}_0 \end{array}} \left\| \tau _+ \sqrt{\tau _-} \int _v \left| z^2 {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \right\| _{L^2(\Sigma _t)} \lesssim \epsilon , \nonumber \\&\sum _{|\beta | \le N-2} \left\| r^2 \int _v \frac{v^A}{v^0} {\widehat{Z}}^{\beta } f \mathrm{d}v \right\| _{L^2(\Sigma _t)} \lesssim \epsilon \log (3+t). \end{aligned}$$

(29)

By Proposition 3.13, commutation formula of Proposition 2.9, the bootstrap assumptions (25), (26), (27) and the estimate (29)–(29), we obtain that, for all $(t,x) \in [0,T[ \times {\mathbb {R}}^3$ and $|\gamma | \le N-2$,

$$\begin{aligned} \left| \rho \left( {\mathcal {L}}_{Z^{\gamma }}(F) \right) \right| (t,x)\lesssim & {} \sqrt{\epsilon }\frac{\log ^2(3+t)}{\tau _+^{2} \tau _-^{\frac{1}{2}}}, \qquad \left| \alpha \left( {\mathcal {L}}_{Z^{\gamma }}(F) \right) \right| (t,x) \; \lesssim \; \sqrt{\epsilon }\frac{\log ^2(3+t)}{\tau _+^{\frac{5}{2}} },\nonumber \\ \end{aligned}$$

(30)

$$\begin{aligned} \left| \sigma \left( {\mathcal {L}}_{Z^{\gamma }}(F) \right) \right| (t,x)\lesssim & {} \sqrt{\epsilon }\frac{\log ^2(3+t)}{\tau _+^{2} \tau _-^{\frac{1}{2}}}, \qquad \left| {\underline{\alpha }} \left( {\mathcal {L}}_{Z^{\gamma }}(F) \right) \right| (t,x) \; \lesssim \; \sqrt{\epsilon }\frac{\log ^{\frac{5}{2}}(1+\tau _-)}{\tau _+ \tau _-^{\frac{3}{2}}}. \nonumber \\ \end{aligned}$$

(31)

Applying Proposition A.1, one obtains that f vanishes for small velocities, i.e.,

$$\begin{aligned} \forall \, t \in [0,T[, \; x \in {\mathbb {R}}^3, \; 0 < |v| \le 1, \; f(t,x,v)=0. \end{aligned}$$

(32)

In view of the support of $\chi$, we then obtain that $T_F(f)=0$ on [0, T[, so that (f, F) is the unique classical solution to the Vlasov–Maxwell system (1)–(3) on [0, T[. The remainder of the article is devoted to the improvement of the bootstrap assumptions (23)–(27), which will imply Theorem 1.1 as it will prove that $T=T^*$ and then $T^* = + \infty$. The proof is divided in three parts.

• 1. First, we improve the bootstrap assumptions (23) and (24) by using Proposition 3.1. To bound the spacetime integrals arising from this energy estimate, we make crucial use of the null structure of the nonlinearity ${\mathcal {L}}_{Z^{\gamma }}(F)(v,\nabla _v {\widehat{Z}}^{\beta } f )$ as well as (30), (31) and (28).

• 2. Then, in of view of improving (25)–(27), the next step consists in proving $L^2$ estimates on quantities such as $\int _v | z {\widehat{Z}}^{\beta } f | \mathrm{d}v$. To treat the higher-order derivatives, we rewrite all transport equations as an inhomogeneous system of Vlasov equations. To handle the homogenous part, we take advantage of the smallness assumption on the $N+3$ derivatives of f at $t=0$, (30) and (31). The inhomogeneous part G will be schematically decomposed as a product KY, with $\int _v |Y| \mathrm{d}v$ a decaying function and $|K|^2Y$ an integrable function in (x, v).

• 3. Finally, we improve the bounds on the energy norms of the electromagnetic field through Proposition 3.4. The null structure of the source terms of the Maxwell equations is fundamental for us here.

6 Improvement of the Energy Bound on the Particle Density

The purpose of this section is to improve the bootstrap assumptions (23) and (24). Note first that

$$\begin{aligned} \forall \, z \in {\mathbf {k}}_0, \; q \in \{1,2 \}, \; |\beta | \le N, \;\; T_F(z^q {\widehat{Z}}^{\beta } f )= & {} T_F(z^q) {\widehat{Z}}^{\beta } f + z^qT_F( {\widehat{Z}}^{\beta } f )\\= & {} q z^{q-1} F(v,\nabla _v z ) {\widehat{Z}}^{\beta } f + z^q T_F( {\widehat{Z}}^{\beta } f ) \end{aligned}$$

and recall from (32) that $T_F(f)=0$ on [0, T[. Thus, in view of the energy estimate of Proposition 3.1, ${\mathbb {E}}^{2}_{N-2}[f](0) \le \epsilon$ and commutation formula of Proposition 2.9, ${\mathbb {E}}^{2}_{N-2}[f] \le 3 \epsilon$ on [0, T[ ensues, if $\epsilon$ is small enough, from the following proposition.

Proposition 6.1

Let $z \in {\mathbf {k}}_0$, $|\zeta | \le N-2$, $|\gamma | \le N-2$ and $|\xi | \le N-3$. Then,

$$\begin{aligned} I_{\zeta }^{z,2} \;:= & {} \; \int _0^t \int _{\Sigma _s} \int _v \left| z F \left( v,\nabla _v z \right) {\widehat{Z}}^{\zeta } f \right| \frac{\mathrm{d}v}{v^0} \mathrm{d}x \mathrm{d}s \; \lesssim \; \epsilon ^{\frac{3}{2}}, \\ K^{z,2}_{\gamma , \xi } \;:= & {} \; \int _0^t \int _{\Sigma _s} \int _v \left| z^2 {\mathcal {L}}_{Z^{\gamma }}(F) \left( v,\nabla _v {\widehat{Z}}^{\xi } f \right) \right| \frac{\mathrm{d}v}{v^0} \mathrm{d}x \mathrm{d}s \; \lesssim \; \epsilon ^{\frac{3}{2}}. \end{aligned}$$

Similarly, the following result implies, if $\epsilon$ is small enough, that ${\mathbb {E}}^1_N[f](t) \le 3 \epsilon \log (3+t)$ for all $t \in [0,T[$.

Proposition 6.2

Let $z \in {\mathbf {k}}_0$, $|\zeta | \le N$, $\gamma$ and $\xi$ such that $|\gamma |+|\xi | \le N$ and $|\xi | \le N-1$. We have,

$$\begin{aligned} I^{z,1}_{\zeta }:= & {} \int _0^t \int _{\Sigma _s} \int _v \left| F \left( v,\nabla _v z \right) {\widehat{Z}}^{\zeta } f \right| \frac{\mathrm{d}v}{v^0} \mathrm{d}x \mathrm{d}s \; \lesssim \; \epsilon ^{\frac{3}{2}} \log (3+t), \\ K^{z,1}_{\gamma , \xi }:= & {} \int _0^t \int _{\Sigma _s} \int _v \left| z {\mathcal {L}}_{Z^{\gamma }}(F) \left( v,\nabla _v {\widehat{Z}}^{\xi } f \right) \right| \frac{\mathrm{d}v}{v^0} \mathrm{d}x \mathrm{d}s \; \lesssim \; \epsilon ^{\frac{3}{2}} \log (3+t). \end{aligned}$$

The proofs are based on the analysis, through Lemma 4.1, of quantities such as ${\mathcal {L}}_{Z^{\gamma }}(F) \left( v, \nabla _v {\widehat{Z}}^{\beta } f \right)$. We then prove the following preparatory lemma.

Lemma 6.3

Let $|\gamma | \le N-2$ and $h{:}\,[0,T[ \times {\mathbb {R}}^3_x \times ({\mathbb {R}}^3_v {\setminus } \{ 0\} )$ be a sufficiently regular function. Then,

$$\begin{aligned} \left| {\mathcal {L}}_{Z^{\gamma }}(F) \left( v, \nabla _v h \right) \right|\lesssim & {} \left( \frac{\sqrt{\epsilon }}{\tau _+^{\frac{5}{4}}}+\frac{ \sqrt{\epsilon } v^{{\underline{L}}}}{ \tau _-^{\frac{5}{4}}v^0} \right) \sum _{{\widehat{Z}} \in \widehat{{\mathbb {P}}}_0} \left| {\widehat{Z}} h \right| , \\ \left| F \left( v,\nabla _v z \right) \right|\lesssim & {} \left( \frac{\sqrt{\epsilon }}{\tau _+^{\frac{5}{4}}}+\frac{ \sqrt{\epsilon } v^{{\underline{L}}}}{ \tau _-^{\frac{5}{4}}v^0} \right) \sum _{w \in {\mathbf {k}}_0} |w|. \end{aligned}$$

Proof

Let $(\alpha , {\underline{\alpha }}, \rho , \sigma )$ be the null decomposition of ${\mathcal {L}}_{Z^{\gamma }}(F)$. Using Lemma 4.1, we have

$$\begin{aligned} \left| {\mathcal {L}}_{Z^{\gamma }}(F) \left( v, \nabla _v h \right) \right| \, \lesssim \, \sum _{{\widehat{Z}} \in \widehat{{\mathbb {P}}}_0} \left( \tau _- \left| \rho \right| +\tau _+\left| \alpha \right| + \tau _+ \frac{|v^A|}{v^0} \left| \sigma \right| + \tau _- \frac{|v^A|}{v^0} \left| {\underline{\alpha }} \right| + \tau _+ \frac{v^{{\underline{L}}}}{v^0} \left| {\underline{\alpha }} \right| \right) \left| {\widehat{Z}} h \right| . \end{aligned}$$

According to the pointwise estimates (30), (31) and the inequality $|v^A| \lesssim \sqrt{v^0 v^{{\underline{L}}}}$ (see Lemma 2.11), one has

$$\begin{aligned} \tau _- \left| \rho \right| +\tau _+\left| \alpha \right|\lesssim & {} \frac{\sqrt{\epsilon }}{\tau _+^{\frac{5}{4}}}, \quad \tau _+\frac{v^{{\underline{L}}}}{v^0}|{\underline{\alpha }}| \lesssim \frac{\sqrt{\epsilon }v^{{\underline{L}}}}{\tau _-^{\frac{5}{4}}v^0}, \quad \frac{|v^A|}{v^0} (\tau _+ |\sigma |+\tau _- |{\underline{\alpha }} |) \lesssim \frac{\sqrt{\epsilon } \sqrt{ v^{{\underline{L}}}}}{\tau _+^{\frac{3}{4}} \tau _-^{\frac{1}{2}}\sqrt{v^0}} \nonumber \\\lesssim & {} \frac{\sqrt{\epsilon }}{\tau _+^{\frac{5}{4}} }+\frac{ \sqrt{\epsilon } v^{{\underline{L}}}}{ \tau _-^{\frac{5}{4}}v^0}, \end{aligned}$$

(33)

which implies the first inequality. The second one follows directly since, by Lemma 2.10, $\sum _{{\widehat{Z}} \in \widehat{{\mathbb {P}}}_0} | {\widehat{Z}}( z) | \lesssim \sum _{w \in {\mathbf {k}}_0} |w|$. $\square$

The remainder of the section is devoted to the proof of Propositions 6.1 and 6.2.

6.1 Proof of Proposition 6.1

Let $z \in {\mathbf {k}}_0$, $|\zeta | \le N-2$, $|\gamma | \le N-2$ and $|\xi | \le N-3$. Using successively Lemma 6.3, that $1 \le v^0$ on the support of f [see (32)], that $[0,t] \times {\mathbb {R}}^3$ can be foliated by $(C_u(t))_{u \le t}$ (see Lemma 2.13) and ${\mathbb {E}}^2_{N-2}[f] \le 4 \epsilon$, which comes from the bootstrap assumption (23), we have,

$$\begin{aligned} I^{z,2}_{\zeta }+K^{z,2}_{\gamma , \xi }\lesssim & {} \sum _{|\beta | \le N-2} \sum _{w \in {\mathbf {k}}_0} \int _0^t \int _{\Sigma _s} \int _v \frac{\sqrt{\epsilon }}{\tau _+^{\frac{5}{4}}} \left| w^2 {\widehat{Z}}^{\beta } f \right| \frac{\mathrm{d}v}{v^0} \mathrm{d}x \mathrm{d}s\\&+\,\int _0^t \int _{\Sigma _s} \int _v \frac{\sqrt{\epsilon }}{\tau _-^{\frac{5}{4}}} \frac{v^{{\underline{L}}}}{v^0}\left| w^2 {\widehat{Z}}^{\beta } f \right| \frac{\mathrm{d}v}{v^0} \mathrm{d}x \mathrm{d}s \\\lesssim & {} \sum _{\begin{array}{c} |\beta | \le N-2 \\ \, w \in {\mathbf {k}}_0 \end{array}} \int _0^t \frac{\sqrt{\epsilon }}{(1+s)^{\frac{5}{4}}} \int _{\Sigma _s} \int _v \left| w^2 {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \mathrm{d}x \mathrm{d}s\\&+\,\int _{u=-\infty }^t \int _{C_u(t)} \int _v \frac{\sqrt{\epsilon }}{\tau _-^{\frac{5}{4}}} \frac{v^{{\underline{L}}}}{v^0}\left| w^2 {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \mathrm{d}C_u(t) \mathrm{d}u \\\lesssim & {} \sqrt{\epsilon } \sum _{|\beta | \le N-2} \sum _{w \in {\mathbf {k}}_0} \int _0^t \frac{{\mathbb {E}}[ w^2 {\widehat{Z}}^{\beta } f ](s)}{(1+s)^{\frac{5}{4}}} \mathrm{d}s\\&+\int _{u=-\infty }^t \frac{1}{\tau _-^{\frac{5}{4}}} \int _{C_u(t)} \int _v \frac{v^{{\underline{L}}}}{v^0}\left| w^2 {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \mathrm{d}C_u(t) \mathrm{d}u \\\lesssim & {} \sqrt{\epsilon } \int _0^t \frac{{\mathbb {E}}^2_{N-2}[f](s)}{(1+s)^{\frac{5}{4}}} \mathrm{d}s+\sqrt{\epsilon }\int _{u=-\infty }^t \frac{1}{\tau _-^{\frac{5}{4}}} {\mathbb {E}}^2_{N-2}[f](t) \mathrm{d}u \\\lesssim & {} \epsilon ^{\frac{3}{2}} \int _0^{+ \infty } \frac{\mathrm{d}s}{(1+s)^{\frac{5}{4}}}+\epsilon ^{\frac{3}{2}}\int _{u=-\infty }^{+\infty } \frac{\mathrm{d}u}{\tau _-^{\frac{5}{4}}}. \end{aligned}$$

We then deduce that $I^{z,2}_{\zeta }+K^{z,2}_{\gamma , \xi } \lesssim \epsilon ^{\frac{3}{2}}$, which concludes the proof of Proposition 6.1.

6.2 Proof of Proposition 6.2

We fix $z \in {\mathbf {k}}_0$, $\zeta$, $\gamma$ and $\xi$ satisfying $|\zeta | \le N$, $|\gamma |+|\xi | \le N$ and $|\xi | \le N-1$. Suppose first that $|\gamma | \le N-2$. Hence, following the computations of Subsection 6.1 and using that ${\mathbb {E}}^1_N[f](t) \le 4 \log (3+t)$ by the bootstrap assumption (24), we get

$$\begin{aligned} I^{z,1}_{\zeta }+K^{z,1}_{\gamma , \xi }\lesssim & {} \sqrt{\epsilon } \sum _{|\beta | \le N} \sum _{w \in {\mathbf {k}}_0} \int _0^t \frac{{\mathbb {E}}[ w {\widehat{Z}}^{\beta } f ](s)}{(1+s)^{\frac{5}{4}}}\mathrm{d}s\\&+\int _{u=-\infty }^t \frac{1}{\tau _-^{\frac{5}{4}}} \int _{C_u(t)} \int _v \frac{v^{{\underline{L}}}}{v^0}\left| w {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \mathrm{d}C_u(t) \mathrm{d}u \\\lesssim & {} \sqrt{\epsilon } \int _0^t \frac{{\mathbb {E}}^1_{N}[f](s)}{(1+s)^{\frac{5}{4}}} \mathrm{d}s+\sqrt{\epsilon }\int _{u=-\infty }^t \frac{1}{\tau _-^{\frac{5}{4}}} {\mathbb {E}}^1_{N}[f](t) \mathrm{d}u \\\lesssim & {} \epsilon ^{\frac{3}{2}} \int _0^{+ \infty } \frac{\log (3+s)}{(1+s)^{\frac{5}{4}}} \mathrm{d}s+\epsilon ^{\frac{3}{2}}\log (3+t) \int _{u=-\infty }^{+\infty } \frac{\mathrm{d}u}{\tau _-^{\frac{5}{4}}} \; \lesssim \; \epsilon ^{\frac{3}{2}} \log (3+t). \end{aligned}$$

We now consider the cases where $|\gamma | \ge N-1$, so that $|\xi | \le 1$. Let us denote the null decomposition of ${\mathcal {L}}_{Z^{\gamma }}(F)$ by $(\alpha , {\underline{\alpha }}, \rho , \sigma )$. Using Lemma 4.1 and that $1 \le v^0$ on the support of f, we are led to bound, for all $|\beta |=|\kappa |+1 \le 2$, the following integrals,

$$\begin{aligned} I_F:= & {} \int _0^t \int _{\Sigma _s}\int _v \left( \tau _- \left| \rho \right| +\tau _+\left| \alpha \right| +\tau _+|\sigma | \frac{|v^{A}|}{v^0} \right) \left| z {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \mathrm{d}x \mathrm{d}s, \\ I_{{\underline{\alpha }}}:= & {} \int _0^t \int _{\Sigma _s} \int _v \left| {\underline{\alpha }} \right| \frac{\tau _-|v^A| + \tau _+v^{{\underline{L}}}}{v^0}\left| z {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \mathrm{d}x \mathrm{d}s. \end{aligned}$$

Using $\tau _+ v^{{\underline{L}}}+\tau _+|v^A| \lesssim v^0 \sum _{w \in {\mathbf {k}}_0}|w|$ (which comes from Lemmas 2.11), the Cauchy–Schwarz inequality, the bootstrap assumption (27) and the estimate (29), we get

$$\begin{aligned} I_{{\underline{\alpha }}}\lesssim & {} \sum _{w \in {\mathbf {k}}_0} \int _{0}^{t} \left\| |{\underline{\alpha }}| \right\| _{L^2(\Sigma _s)} \left\| \int _v \left| (w^2+z^2) {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \right\| _{L^2(\Sigma _s)} \mathrm{d}s \\\lesssim & {} \int _{0}^{t} \sqrt{{\mathcal {E}}^{5}_N[F](s)} \frac{\mathrm{d}s}{1+s} \; \lesssim \; \epsilon ^{\frac{3}{2}} \log (3+t). \end{aligned}$$

For $I_F$, in order to apply Lemma 2.13, notice first that we have by the estimate (28), for $u \le t$ and $i \in {\mathbb {N}}$,

$$\begin{aligned} \left\| \int _v \left| w^2 {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \right\| ^2_{L^2(C^i_u(t))}\lesssim & {} \int _{C_u^i(t)} \frac{\epsilon ^2}{\tau _+^4 \tau _-^2} \mathrm{d} C_u^i(t) \lesssim \frac{\epsilon ^2}{\tau _-^{\frac{9}{4}}(1+t_i)^{\frac{1}{4}}} \int _{{\underline{u}}=2t_i-u}^{2t_{i+1}-u} \frac{r^2}{\tau _+^{\frac{7}{2}}} \mathrm{d} {\underline{u}} \\\lesssim & {} \frac{\epsilon ^2}{\tau _-^{\frac{9}{4}}(1+2^i)^{\frac{1}{4}}}. \end{aligned}$$

Hence, using $\tau _+|v^A| \lesssim v^0 \sum _{w \in {\mathbf {k}}_0} |w|$, the Cauchy–Schwarz inequality and the bootstrap assumption (26), we obtain

$$\begin{aligned} I_{F}= & {} \sum _{w \in {\mathbf {k}}_0} \sum _{i=0}^{+\infty } \int _{u=-\infty }^t \int _{C^i_u(t)} \left( \tau _- \left| \rho \right| +\tau _+\left| \alpha \right| +|\sigma | \right) \int _v \left| w^2 {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \mathrm{d}C_u(t)^i \mathrm{d}u \\\lesssim & {} \sum _{w \in {\mathbf {k}}_0} \sum _{i=0}^{+\infty } \int _{u=-\infty }^t \left\| \tau _- \left| \rho \right| +\tau _+\left| \alpha \right| +|\sigma | \right\| _{L^2(C_u^i(t))} \left\| \int _v \left| w^2 {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \right\| _{L^2(C^i_u(t))} \mathrm{d}u \\\lesssim & {} \sum _{i=0}^{+\infty } \int _{u=-\infty }^t \sqrt{{\mathcal {E}}_N[F]( T_{i+1}(t))} \frac{\epsilon }{\tau _-^{\frac{9}{8}}(1+2^i)^{\frac{1}{8}}} \mathrm{d}u \\\lesssim & {} \; \epsilon ^{\frac{3}{2}} \sum _{i=0}^{+\infty } \frac{\log ^2(3+2^{i+1})}{(1+2^i)^{\frac{1}{8}}} \int _{u=-\infty }^{+\infty } \frac{\mathrm{d}u}{\tau _-^{\frac{9}{8}}} \; \lesssim \; \epsilon ^{\frac{3}{2}}. \end{aligned}$$

This concludes the proof and the improvement of the bootstrap assumptions (23) and (24).

7 $L^2$ Estimates on the Velocity Averages of the Vlasov Field

In view of the energy estimate of Proposition 3.4, we have to prove $L^2_x$ estimates on quantities such as $\int _v | z {\widehat{Z}}^{\beta } f | \mathrm{d}v$, for $|\beta | \le N$. If $|\beta | \le N-2$, we can use a Klainerman–Sobolev inequality to obtain a sufficient decay rate (see Proposition 7.9 below). The main part of this section then consists in deriving such estimates for $|\beta | \ge N-1$. For this purpose, we follow the strategy used in [8] (Section 4.5.7) and adapted in [2] for the Vlasov–Maxwell system. Contrary to [2], we will have to keep more of the null structure of the system. This will force us to add a new hierarchy on the functions studied here. Let us first rewrite the system and then we will explain how we will proceed. Let $I_1$ and $I_2$ be the following ordered sets,

$$\begin{aligned} I_1:= & {} \{ \beta \; \text {multi-index} \, / \, N-5 \le |\beta | \le N \} = \{ \beta _{1,1},\ldots ,\beta _{1,|I_1|} \}, \\ I_2:= & {} \{ \beta \; \text {multi-index} \, / \, |\beta | \le N-5 \} =\{ \beta _{2,1},\ldots ,\beta _{2,|I_2|} \}. \end{aligned}$$

Remark 7.1

Contrary to [2], we have $I_1 \cap I_2 \ne \varnothing$.

We also consider, for $N-5 \le k \le N$, $I^k_1 := \{ \beta \in I_1 \, / \, |\beta | =k \}$, and two vector-valued fields R and W of respective length $|I_1|$ and $|I_2|$ such that

$$\begin{aligned} R_i= {\widehat{Z}}^{\beta _{1,i}}f \quad \text {and} \quad W_i = {\widehat{Z}}^{\beta _{2,i}}f. \end{aligned}$$

We will sometimes abusively write $i \in I^k_1$ instead of $\beta _{1,i} \in I^k_1$. Let us denote by ${\mathbb {V}}$ the module over the ring $C^0 \left( [0,T[ \times {\mathbb {R}}^3_x \times \left( {\mathbb {R}}^3_v {\setminus } \{0 \} \right) \right)$ generated by $( \partial _{v^l})_{1 \le l \le 3}$. We now rewrite the Vlasov equations satisfied by R and W.

Lemma 7.2

There exist three matrix-valued functions $A {:}\, [0,T[ \times {\mathbb {R}}^3 \times \left( {\mathbb {R}}^3_v {\setminus } \{0 \} \right) \rightarrow {\mathfrak {M}}_{|I_1|}({\mathbb {V}})$, $D {:}\, [0,T[ \times {\mathbb {R}}^3 \times \left( {\mathbb {R}}^3_v {\setminus } \{0 \} \right) \rightarrow {\mathfrak {M}}_{|I_2|}({\mathbb {V}})$ and $B {:}\, [0,T[ \times {\mathbb {R}}^3 \times \left( {\mathbb {R}}^3_v {\setminus } \{0 \} \right) \rightarrow {\mathfrak {M}}_{|I_1|,|I_2|}({\mathbb {V}})$ such that

$$\begin{aligned} T_F(R)+AR=B W \quad \text {and} \quad T_F(W)=DW. \end{aligned}$$

Moreover, if $1 \le i \le |I_1|$, A and B are such that $T_F(R_i)$ is a linear combination of

$$\begin{aligned}&{\mathcal {L}}_{Z^{\gamma }}(F) \left( v, \nabla _v R_j \right) , \quad \text {with} \quad |\beta _{1,j}| < |\beta _{1,i}| \quad \text {and} \quad |\gamma | + |\beta _{1,j}| \le |\beta _{1,i}|,&\\&{\mathcal {L}}_{Z^{\xi }}(F) \left( v, \nabla _v W_j \right) , \quad \text {with} \quad |\beta _{2,j}| \le N-6 \quad \text {and} \quad |\xi | \le N.&\end{aligned}$$

Similarly, if $1 \le i \le I_2$, D is such that $T_F(W_i)$ is a linear combination of

$$\begin{aligned}&{\mathcal {L}}_{Z^{\gamma }}(F) \left( v, \nabla _v W_j \right) , \quad \text {with} \quad |\beta _{2,j}| \le N-6 \quad \text {and} \quad |\gamma | \le N-5.&\end{aligned}$$

Note also, using (28), that

$$\begin{aligned}&\forall \, (t,x) \in [0,T[ \times {\mathbb {R}}^3, \;\; z \in {\mathbf {k}}_0, \;\; 1 \le q \le |I_2|, \quad \int _v |z^2W_q| \mathrm{d}v \lesssim \frac{\epsilon }{\tau _+^{2}\tau _-}.&\end{aligned}$$

Remark 7.3

Notice that if $\beta _{1,i} \in I^{N-5}_1$, then $A_i^q=0$ for all $1 \le q \le |I_1|$. Note also that if $p \ge 1$ and $\beta _{1,i} \in I^{N-5+p}$, we have $|\gamma | \le p$.

Proof

One only has to apply the commutation formula of Proposition 2.9 to ${\widehat{Z}}^{\beta _{1,i}} f$ or ${\widehat{Z}}^{\beta _{2,i}} f$ and to replace each quantity such as ${\widehat{Z}}^{\kappa } f$, for $|\kappa | \ne N-5$, by the corresponding component of R or W. If $|\kappa | = N-5$, we replace it by the corresponding component of R. $\square$

The goal is to obtain an $L^2$-estimate on R. For this, let us split it in $R:=H+G$, where

$$\begin{aligned} \left\{ \begin{array}{ll} T^{\chi }_F(H)+ AH=0 \;, \; H(0,\cdot ,\cdot )=R(0,\cdot ,\cdot ),\\ T^{\chi }_F(G)+ AG= BW \;, \; G(0,\cdot ,\cdot )=0 \end{array} \right. \end{aligned}$$

and then prove $L^2$ estimates on the velocity averages of H and G. To do it, we will schematically establish that $G=KW$, with K a matrix such that ${\mathbb {E}}[KKW]$ do not growth too fast, and then use the pointwise decay estimates on $\int _v |z^2W|\mathrm{d}v$ to obtain the expected decay rate on $\Vert \int _v |G| \mathrm{d}v \Vert _{L^2_x}$. For $\Vert \int _v |H| \mathrm{d}v \Vert _{L^2_x}$, we will make crucial use of Klainerman–Sobolev inequalities so that we will need to commute the transport equation satisfied by H and prove $L^1$-bounds such as we did in the proof of Proposition 6.1. Contrary to what we did in [2], we keep the v derivatives in order to take advantage of the good behavior of radial component of $\nabla _v g$. This is why we put the derivatives of order $N-5$ in both R and W.

Remark 7.4

If we proceed as in [2], we would not be able to use the estimate $\left( \nabla _v g \right) ^r \sim \tau _- {\widehat{Z}}g$ and an analogous result to Lemma 4.1 would give the term $\tau _+|{\underline{\alpha }}| \frac{|v^A|}{v^0}|{\widehat{Z}} g|$. In our case (the three-dimensional one), a lack of decay in the $t+r$ direction prevents us to deal with it.

7.1 The Homogeneous System

In order to obtain $L^{\infty }$, and then $L^2$, estimates on $\int _v |H| \mathrm{d}v$, we will have to commute at least three times the transport equation satisfied by each component of H. However, if $\beta _{1,i} \in I^k_1$, with $k \ge N-4$, we need to control the $L^1$ norm of ${\widehat{Z}}^{\kappa } H_j$, with $|\kappa |=4$ and $j \in I^{k-1}_1$, to bound $\Vert {\widehat{Z}}^{\xi } H_i \Vert _{L^1_{x,v}}$, with $|\xi | = 3$. We then consider the following energy norm

$$\begin{aligned} {\mathbb {E}}_H = \sum _{z \in {\mathbf {k}}_0} \, \sum _{k=0}^5 \, \sum _{ |\beta | \le 3+k} \, \sum _{i \in I^{N-k}_1} \, {\mathbb {E}}[z^2{\widehat{Z}}^{\beta } H_i]. \end{aligned}$$

We have the following commutation formula.

Lemma 7.5

Let $0 \le k \le 5$, $|\beta | \le 3+k$ and $i \in I^{N-k}_1$. Then, if H vanishes for all $|v| \le 1$, $T_F({\widehat{Z}}^{\beta } H_i)$ can be written as a linear combination of terms of the form

$$\begin{aligned}&{\mathcal {L}}_{Z^{\gamma }}(F) \left( v, \nabla _v {\widehat{Z}}^{\xi } H_j \right) , \quad \text {with} \quad |\gamma | \le 8 \le N-2, \quad |\xi | \le |\beta |, \quad |\beta _{1,j}| \le |\beta _{1,i}|, \\&\quad |\xi |+|\beta _{1,j}| < |\beta |+|\beta _{1,i}|. \end{aligned}$$

Proof

If H vanishes for all $|v| \le 1$, we have $T_F(H)+AH=0$. Hence, according to Proposition 2.9, the source terms which arise from the commutator $[T_F,{\widehat{Z}}^{\beta }]$ are such as those described in this lemma, with $j=i$. The other ones come from ${\widehat{Z}}^{\beta } \left( T_F \left( H_i \right) \right)$ (use Lemma 7.2, Remark 7.3 and Lemma 2.8 to check that they are of the researched form). $\square$

As $H(0,\cdot ,\cdot )=R(0,\cdot ,\cdot )$, it then follows that $H(0,\cdot v)=0$ for all $|v| \le 3$ and, applying Proposition¹³B.2, that there exists $C_H>0$ such that ${\mathbb {E}}_H(0) \le C_H \epsilon$. Consequently, using Corollary A.5 and following the proof of Proposition 6.1, one can prove that, for $\epsilon$ small enough,

$$\begin{aligned}&\forall \, t \in [0,T[, \quad {\mathbb {E}}_H (t) \le 3C_H\\&\epsilon , \forall \, (t,x) \in [0,T[ \times {\mathbb {R}}^3, \, 0 < |v|\\&\quad \le 1, \quad H(t,x,v)=0. \end{aligned}$$

By Proposition 3.6, we then obtain, for $0 \le k \le 5\, \mathrm{and}\, z \,{\in }\, {k}_0$,

$$\begin{aligned} \forall \, (t,x) \in [0,T[ \times {\mathbb {R}}^3, \;\; z \,{\in }\, {k}_0, \;\; 1 \le j \le |I^{N-k}_1|, \;\; |\beta | \le k, \quad \int _v |z^2{\widehat{Z}}^{\beta } H_j| \mathrm{d}v \lesssim \frac{\epsilon }{\tau _+^2 \tau _-}.\nonumber \\ \end{aligned}$$

(34)

Remark 7.6

Proceeding as in Subsection 17.2 of [6], we could avoid any hypothesis on the higher-order derivatives of $F^0$.

7.2 The Inhomogeneous System

Start by noticing that G vanishes for all $|v| \le 1$ since $G=R-H$. We then deduce from $\chi (|v|) =1$ for all $|v| \ge 1$ that G satisfies $T_F(G)+AG=BW$. To derive an $L^2$ estimate on G, we cannot commute the transport equation because B contains top order derivatives of the electromagnetic field. Instead, we follow the methodology of [8] (see Section 4.5.7). We kept the v derivatives of G in the matrix A so that we could use the null structure in a better way. In order to obtain $L^1$-bounds on quantities introduced below, we now need to rewrite these v derivatives. This is the purpose of the following lemma.

Lemma 7.7

There exists $p \ge 1$, a vector-valued field Y of length p, which vanishes for $|v| \le 1$, and three matrix-valued functions ${\overline{A}} {:}\, [0,T[ \times {\mathbb {R}}^3 \times \left( {\mathbb {R}}^3_v {\setminus } \{0 \} \right) \rightarrow {\mathfrak {M}}_{|I_1|}({\mathbb {R}})$, ${\overline{B}} {:}\, [0,T[ \times {\mathbb {R}}^3 \times \left( {\mathbb {R}}^3_v {\setminus } \{0 \} \right) \rightarrow {\mathfrak {M}}_{|I_1|,p}({\mathbb {R}})$, ${\overline{D}} {:}\, [0,T[ \times {\mathbb {R}}^3 \times \left( {\mathbb {R}}^3_v {\setminus } \{0 \} \right) \rightarrow {\mathfrak {M}}_{p}({\mathbb {R}})$ such that

$$\begin{aligned} T_F(G)+{\overline{A}}G= {\overline{B}} Y, \quad T_F(Y)= {\overline{D}} Y \quad \text {and} \quad \sum _{z \in {\mathbf {k}}_0} \int _v |z^2Y| \mathrm{d}v \lesssim \frac{\epsilon }{\tau _+^2 \tau _-}. \end{aligned}$$

Moreover, ${\overline{A}}$ and ${\overline{B}}$ are such that, if $i \in \llbracket 1, |I_1| \rrbracket$, $T_F(G_i)$ can be bounded by a linear combination of terms of the form,

$$\begin{aligned}&\left( \tau _-\left| \rho \left( {\mathcal {L}}_{Z^{\gamma }}(F) \right) \right| +\tau _+\left| \alpha \left( {\mathcal {L}}_{Z^{\gamma }}(F) \right) \right| \right) |G_j|, \\&\quad \left( \tau _+ \frac{|v^A|}{v^0} \left| \sigma \left( {\mathcal {L}}_{Z^{\gamma }}(F) \right) \right| + \frac{\tau _- |v^A|+ \tau _+ v^{{\underline{L}}}}{v^0} \left| {\underline{\alpha }} \left( {\mathcal {L}}_{Z^{\gamma }}(F) \right) \right| \right) |G_j|, \\&\quad \left( \tau _-\left| \rho \left( {\mathcal {L}}_{Z^{\xi }}(F) \right) \right| +\tau _+\left| \alpha \left( {\mathcal {L}}_{Z^{\xi }}(F) \right) \right| +\left| \sigma \left( {\mathcal {L}}_{Z^{\xi }}(F) \right) \right| +\left| {\underline{\alpha }} \left( {\mathcal {L}}_{Z^{\xi }}(F) \right) \right| \right) |z Y_q|, \end{aligned}$$

where $j \in \llbracket 1, |I_1| \rrbracket$, $|\gamma | \le 5$, $q \in \llbracket 1, p \rrbracket$, $|\xi | \le N$ and $z \in V$. Similarly, ${\overline{D}}$ is such that, if $i \in \llbracket 1, p \rrbracket$, $T_F(Y_i)$ can be bounded by a linear combination of terms of the form,

$$\begin{aligned}&\left( \tau _-\left| \rho \left( {\mathcal {L}}_{Z^{\gamma }}(F) \right) \right| +\tau _+\left| \alpha \left( {\mathcal {L}}_{Z^{\gamma }}(F) \right) \right| \right) |Y_j|, \\&\quad \left( \tau _+ \frac{|v^A|}{v^0} \left| \sigma \left( {\mathcal {L}}_{Z^{\gamma }}(F) \right) \right| + \frac{\tau _- |v^A|+ \tau _+ v^{{\underline{L}}}}{v^0} \left| {\underline{\alpha }} \left( {\mathcal {L}}_{Z^{\gamma }}(F) \right) \right| \right) |Y_j|, \end{aligned}$$

where $j \in \llbracket 1, p \rrbracket$ and $|\gamma | \le N-5$.

Proof

The strategy of the proof is the following. If $\partial _{v^k} G_j$ appears in $T_F(G)+AG=BW$, then, by Lemma 7.2, $j \in I^k_1$, with $N-5 \le k \le N-1$. We then transform it with $v^0 \partial _{v^k} = {\widehat{\Omega }}_{0k}-x^k \partial _t-t \partial _k$ and express it, with controllable error terms, as a combination of $(G_l)_{l \in I^{k+1}_1}$. The other manipulations are similar to those made in Sect. 6 when we applied Lemma 4.1. Let us denote, for $j \in I_1 {\setminus } I_1^N$ and ${\widehat{Z}} \in \widehat{{\mathbb {P}}}_0$, by $j_{{\widehat{Z}}}$ the index such that $R_{j_{{\widehat{Z}}}}= {\widehat{Z}} {\widehat{Z}}^{\beta _{1,j}} f = {\widehat{Z}} R_j$. Hence, by (34) and since $R=H+G$, we have, for all $j \in I_1 {\setminus } I^N_1$, and for any $(z,\hat{Z})\in k_0\times \hat{P}_0$

$$\begin{aligned}&\forall \, (t,x) \in [0,T[ \times {\mathbb {R}}^3, \; \quad \int _v |z|^2|G_{j_{{\widehat{Z}}}}-{\widehat{Z}} G_j|\mathrm{d}v\nonumber \\&\quad = \int _v |z|^2|H_{j_{{\widehat{Z}}}}-{\widehat{Z}} H_j|\mathrm{d}v \lesssim \frac{\epsilon }{\tau _+^2 \tau _-}. \end{aligned}$$

(35)

Let $p^0 := |I_2|+|I_1 {\setminus } I^N_1|$ and $Y^0$ a vector-valued field¹⁴ of length $p^0$ containing each component of W and each $G_{j_{{\widehat{Z}}}}-{\widehat{Z}} G_j$, for $j \in I_1 {\setminus } I^N_1$. We order the components of $Y^0$ such as $Y^0_{j_{{\widehat{Z}}}}=G_{j_{{\widehat{Z}}}}-{\widehat{Z}} G_j$. In view of (35) and Lemma 7.2, $\int _v |z^2Y^0| \mathrm{d}v$ satisfies the desired pointwise decay estimate on $\int _v |z^2Y| \mathrm{d}v$. We now fix $i \in I_1$. Applying Lemma 7.2, one can see that $T_F(G_i)$ can be written as a linear combination of the following terms.

• Those coming from BW,

  $$\begin{aligned} {\mathcal {L}}_{Z^{\xi }}(F) \left( v, \nabla _v W_j \right) , \quad \text {with} \quad |\beta _{2,j}| \le N-6 \quad \text {and} \quad |\xi | \le N, \end{aligned}$$

  leading, by Lemma 4.1 and $\tau _+v^{{\underline{L}}}+\tau _+ |v^A| \lesssim v^0 \sum _{w \in {\mathbf {k}}_0} |w|$ (see Lemma 2.11), to the announced terms involving Y.

• Those coming from AW,

  $$\begin{aligned} {\mathcal {L}}_{Z^{\gamma }}(F) \left( v, \nabla _v G_j \right) , \quad \text {with} \quad |\beta _{1,j}| < |\beta _{1,i}| \quad \text {and} \quad |\gamma |+|\beta _{1,j}| \le |\beta _{1,i}|. \end{aligned}$$

  Then, expand ${\mathcal {L}}_{Z^{\gamma }}(F) \left( v, \nabla _v G_j \right)$ in null components using formula (21). We now rewrite the angular components of $\nabla _v G_j$ using $v^0 \partial _{v^k}= {\widehat{\Omega }}_{0k}-x^k \partial _t-t \partial _k$, so that

  $$\begin{aligned} v^0\partial _{v^k} G_j = G_{j_{{\widehat{\Omega }}_{0k}}}-x^k G_{j_{\partial _t}}-t G_{j_{\partial _k}}-Y^0_{j_{{\widehat{\Omega }}_{0k}}} +x^k Y^0_{j_{\partial _t}}+t Y^0_{j_{\partial _k}}. \end{aligned}$$

  For the radial component, use (22) to obtain

  $$\begin{aligned}&v^0\left( \nabla _v G_j \right) ^r = \frac{x^q}{r} \left( G_{j_{{\widehat{\Omega }}_{0q}}}-Y^0_{j_{{\widehat{\Omega }}_{0q}}} \right) \\&\quad -\,G_{j_S}+Y^0_{j_S}+(t-r) \left( G_{j_{\partial _t}}-Y^0_{j_{\partial _t}}-\frac{x^q}{r} G_{j_{\partial _q}}+\frac{x^q}{r} Y^0_{j_{\partial _q}} \right) . \end{aligned}$$

This concludes the construction of ${\overline{A}}$, ${\overline{B}}$. To obtain an equation for $Y^0$, we will see that we need to consider a bigger vector than $Y^0$. Let $i \in \llbracket 1, p^0 \rrbracket$. If $Y^0_i=W_q$, with $q \in I_2$, we can build the line i of ${\overline{D}}$ using Lemmas 7.2 and 4.1. Otherwise, $Y^0_i = {\widehat{Z}} H_j-H_{j_{{\widehat{Z}}}}$ and by Lemma 7.5 we see that functions such as $\partial _v {\widehat{Z}} H_r$, with $|\beta _{1,r}| < |\beta _{1,j}|$, appear in certain source terms of $T_F(Y^0_i)$. We then consider the vector-valued field Y containing $Y^0$ and all the quantities ${\widehat{Z}}^{\kappa } H_j$ such as $\beta _{1,j} \in I^{N-5+k}_1$ and $|\kappa |+k \le 5$. It remains to use (34) and Lemmas 4.1, 7.5. $\square$

Consider now K satisfying $T^{\chi }_{ F}(K)+\chi {\overline{A}}K+\chi K {\overline{D}}= \chi {\overline{B}}$ and $K(0,\cdot ,\cdot )=0$. Hence, $KY=G$ since they both initially vanish and $T_F(KY)+{\overline{A}}KY={\overline{B}}Y$ in view of the velocity support of Y. The goal now is to control the energy

$$\begin{aligned} {\mathbb {E}}_G := \sum _{i=0}^{|I_1|} \sum _{j=0}^p \sum _{q=0}^p {\mathbb {E}}\left[ \left| K_i^j \right| ^2 Y_q \right] . \end{aligned}$$

We will then be naturally led to use that

$$\begin{aligned} T_F\left( |K^j_i|^2 Y_q\right) = |K^j_i |^2 {\overline{D}}^r_q Y_r-2\left( {\overline{A}}^r_i K^j_r +K^r_i {\overline{D}}^j_r \right) K^j_i Y_q+2 {\overline{B}}^j_iK^j_iY_q. \end{aligned}$$

(36)

Proposition 7.8

If $\epsilon$ is small enough, we have ${\mathbb {E}}_G(t) \lesssim \epsilon \log ^2 (3+t)$ for all $t \in [0,T[$.

Proof

Let $T_0 \in [0,T[$ the largest time such that ${\mathbb {E}}_G(t) \lesssim \epsilon \log ^2 (3+t)$ for all $t \in [0,T_0[$. By continuity, $T_0 >0$. The remaining of the proof consists in improving this bootstrap assumption, which would imply the result. The computations will be similar as those of the proof of Proposition 6.2. Let $i \in \llbracket 1, |I_1| \rrbracket$ and $(j,q) \in \llbracket 1, p \rrbracket ^2$. According to the energy estimate of Proposition 3.1 and (36), it suffices to prove that

$$\begin{aligned} I_{{\overline{A}},{\overline{D}}}= & {} \int _0^t \int _{\Sigma _s} \int _v \left| |K^j_i |^2 {\overline{D}}^r_q Y_r-2\left( {\overline{A}}^r_i K^j_r +K^r_i {\overline{D}}^j_r \right) K^j_i Y_q \right| \frac{\mathrm{d}v}{v^0}\mathrm{d}x \mathrm{d}s \nonumber \\\lesssim & {} \; \epsilon ^{\frac{3}{2}} \log ^2 (3+t), \end{aligned}$$

(37)

$$\begin{aligned} I_{{\overline{B}}}= & {} \int _0^t \int _{\Sigma _s} \int _v \left| {\overline{B}}^j_iK^j_iY_q \right| \frac{\mathrm{d}v}{v^0}\mathrm{d}x \mathrm{d}s \; \lesssim \; \epsilon ^{\frac{3}{2}} \log ^2 (3+t). \end{aligned}$$

(38)

According to Proposition 7.7 and (33), one has, using ${\mathbb {E}}_G (t) \lesssim \epsilon \log ^2 (3+t)$ and $1 \le v^0$ on the support of Y,

$$\begin{aligned} I_{{\underline{A}}, {\overline{D}}}\lesssim & {} \int _0^t \int _{\Sigma _s} \int _v \left( \frac{\sqrt{\epsilon }}{\tau _+^{\frac{5}{4}}}+\frac{\sqrt{\epsilon } v^{{\underline{L}}}}{ \tau _-^{\frac{5}{4}} v^0} \right) |K|^2 |Y| \mathrm{d}v \mathrm{d}x \mathrm{d}s \, \lesssim \, \sqrt{\epsilon } \int _0^t \frac{{\mathbb {E}}_G(s)}{(1+s)^{\frac{5}{4}}} \mathrm{d}s \\&+\,\sqrt{\epsilon } \int _{u=-\infty }^t \frac{{\mathbb {E}}_G(t)}{\tau _-^{\frac{5}{4}}} \mathrm{d}u \, \lesssim \, \epsilon ^{\frac{3}{2}}\log ^2 (3+t). \end{aligned}$$

We now turn on (38), where the electromagnetic field is differentiated too many times to be estimated pointwise. According to Proposition 7.7, $1 \le v^0$ on the support of Y and using the Cauchy–Schwarz inequality in v, we can bound $\int _v | {\overline{B}}^j_iK^j_iY_q | \frac{\mathrm{d}v}{v^0}$ by a linear combination of terms of the form

$$\begin{aligned}&\bullet \, \Phi _F^{\xi } := \left| \int _v |z^2Y| \mathrm{d}v \int _v | K|^2 |Y| \mathrm{d}v \right| ^{\frac{1}{2}}\left( \tau _-\left| \rho _{\xi } \right| +\tau _+\left| \alpha _{\xi } \right| +\left| \sigma _{\xi } \right| \right) , \\&\quad \bullet \, \Phi _{{\underline{\alpha }}}^{\xi }:= \left| \int _v |z^2Y| \mathrm{d}v \int _v | K|^2 |Y| \mathrm{d}v \right| ^{\frac{1}{2}} \left| {\underline{\alpha }}_{\xi } \right| , \end{aligned}$$

where $|\xi | \le N$ and $(\alpha _{\xi },{\underline{\alpha }}_{\xi }, \rho _{\xi }, \sigma _{\xi })$ is the null decomposition of ${\mathcal {L}}_{Z^{\xi }}(F)$. Now, fix $|\xi | \le N$. Using the Cauchy–Schwarz inequality in x, the bootstrap assumption (27) and ${\mathbb {E}}_G(t) \le \epsilon \log ^2(3+t)$, we have

$$\begin{aligned} \int _0^t \int _{\Sigma _s} \Phi ^{\xi }_{{\underline{\alpha }}} \mathrm{d}x \mathrm{d}s\lesssim & {} \int _0^t \Vert {\underline{\alpha }}_{\xi } \Vert _{L^2(\Sigma _s)} \left\| \left| \int _v |z^2Y| \mathrm{d}v \int _v | K|^2 |Y| \mathrm{d}v \right| ^{\frac{1}{2}} \right\| _{L^2(\Sigma _s)} \mathrm{d}s \\\lesssim & {} \int _0^t \sqrt{{\mathcal {E}}^5_N[F](s)} \left\| \int _v |z^2Y| \mathrm{d}v \right\| _{L^{\infty }(\Sigma _s)}^{\frac{1}{2}} \left\| \int _v | K|^2 |Y| \mathrm{d}v \right\| _{L^1(\Sigma _s)}^{\frac{1}{2}} \mathrm{d}s \\\lesssim & {} \epsilon ^{\frac{3}{2}} \int _0^t \frac{\sqrt{{\mathbb {E}}_G(s)}}{1+s} \mathrm{d}s \; \lesssim \; \epsilon ^{\frac{3}{2}} \log ^2(3+t). \end{aligned}$$

By the inequality $2ab \le a^2+b^2$ and $\tau _+^2 \tau _- \int _v |z^2 Y| \mathrm{d}v \lesssim \epsilon$, one has

$$\begin{aligned} \Phi _F^{\xi } \le \frac{\epsilon }{\tau _+^{\frac{5}{4}}} \int _v | K|^2 |Y| \mathrm{d}v + \frac{\epsilon }{ \tau _+^{\frac{3}{4}} \tau _-} \left( \tau _-\left| \rho _{\xi } \right| +\tau _+\left| \alpha _{\xi } \right| +\left| \sigma _{\xi } \right| \right) ^2, \end{aligned}$$

so that, by Lemma 2.13 and the bootstrap assumption (26)

$$\begin{aligned}&\int _0^t \int _{\Sigma _s} \Phi ^{\xi }_F \mathrm{d}x \mathrm{d}s\\&\quad \lesssim \epsilon \int _0^t \frac{{\mathbb {E}}_G(s)}{(1+s)^{\frac{5}{4}}} \mathrm{d}s+\sum _{i =0}^{+\infty } \int _{u=-\infty }^t \frac{\epsilon }{\tau _-^{\frac{5}{4}} } \int _{C_u^i(t)} \frac{1}{\sqrt{\tau _+}} \left( \tau _-\left| \rho _{\xi } \right| +\tau _+\left| \alpha _{\xi } \right| +\left| \sigma _{\xi } \right| \right) ^2 \mathrm{d}C_u^i(t) \mathrm{d}u \\&\quad \lesssim \epsilon ^{\frac{3}{2}}+ \epsilon \sum _{i =0}^{+\infty } \int _{-\infty }^t \frac{{\mathcal {E}}_N[F](T_{i+1}(t))}{ \tau _-^{\frac{5}{4}} \sqrt{1+t_i}} \mathrm{d}u \; \lesssim \; \epsilon ^{\frac{3}{2}}+ \epsilon ^2 \sum _{i =0}^{+\infty } \frac{\log ^4(1+2^{i+1})}{\sqrt{1+2^i}} \int _{-\infty }^{+\infty } \frac{\mathrm{d}u}{ \tau _-^{\frac{5}{4}} } \; \lesssim \; \epsilon ^{\frac{3}{2}}. \end{aligned}$$

This concludes the improvement of the bootstrap assumption on ${\mathbb {E}}_G$ and then the proof. $\square$

7.3 The $L^2$ Estimates

In order to improve the bound on the electromagnetic field energy, we will use the following estimates.

Proposition 7.9

Let $z \in {\mathbf {k}}_0$ and $|\beta | \le N$. Then,

$$\begin{aligned} \forall \, t \in [0,T[, \quad \left\| \tau _+ \sqrt{\tau _-} \int _v \left| z {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \right\| _{L^2(\Sigma _t)} \lesssim \epsilon \log (3+t). \end{aligned}$$

The logarithmical growth can be removed for $|\beta | \le N-4$.

Proof

The cases $|\beta | \le N-4$ ensue from (29). Suppose now that $|\beta | \ge N-3$, so that there exists $j \in \llbracket 1, |I_1| \rrbracket$ such that ${\widehat{Z}}^{\beta } f = H_j+G_j$. It then suffices to prove that both $H_j$ and $G_j$ satisfy such $L^2$-estimates. For $H_j$, one only has to use ${\mathbb {E}}_H \le 3\epsilon$ on [0, T[ and the Klainerman–Sobolev inequality of Proposition 3.7. For $G_j$, recall that $G_j = K_j^q Y_q$ and use $\int _v |z^2 Y| \mathrm{d}v \lesssim \epsilon \tau _+^{-2}$, which comes from Proposition 7.7, and the Cauchy–Schwarz inequality in v in order to obtain

$$\begin{aligned} \left\| \int _v \left| z G_j \right| \mathrm{d}v \right\| _{L^2(\Sigma _t)}= & {} \left\| \int _v |z| \left| K_j^q Y_q \right| \mathrm{d}v \right\| _{L^2(\Sigma _t)} \\\lesssim & {} \left\| \int _v \left| zY \right| \mathrm{d}v \right\| ^{\frac{1}{2}}_{L^{\infty }(\Sigma _t)}\left\| \int _v \left| K_j^q \right| ^2 \left| Y_q \right| \mathrm{d}v \right\| ^{\frac{1}{2}}_{L^1(\Sigma _t)} \lesssim \frac{\sqrt{\epsilon }}{1+t} \sqrt{{\mathbb {E}}_G(t)}. \end{aligned}$$

It then remains to use Proposition 7.8, which gives ${\mathbb {E}}_G(t) \lesssim \epsilon \log ^2(3+t)$. $\square$

Combining this Proposition with the inequality $r |v^A| \lesssim v^0 \sum _{w \in {\mathbf {k}}_0} |w|$ (see Lemma 2.11), one can then improve the bootstrap assumption (25) if $\epsilon$ is small enough.

8 The Energy Bounds of the Electromagnetic Field

The last part of the proof consists in improving the bootstrap assumptions (26) and (27). According to the energy estimate of Proposition 3.4, commutation formula of Proposition 2.9 and ${\mathcal {E}}_N[F](0) \le \epsilon$, ${\mathcal {E}}_N[F](t) \le 3 \epsilon \log ^{4}(3+t)$ and ${\mathcal {E}}^{5}[F](t) \le {\underline{C}} \epsilon$ for all $t \in [0,T[$ follow, if $\epsilon$ is small enough and ${\underline{C}}$ chosen large enough, from

$$\begin{aligned} \sum _{|\gamma | \le N} \sum _{|\beta | \le N} \int _0^t \int _{\Sigma _s} \left| {\overline{K}}_0^{\mu } {\mathcal {L}}_{Z^{\gamma }}(F)_{\mu \nu } \int _v \frac{v^{\nu }}{v^0} {\widehat{Z}}^{\beta } f \mathrm{d}v \right| \mathrm{d}x \mathrm{d}s\lesssim & {} \epsilon ^{\frac{3}{2}} \log ^4 (3+t), \\ \sum _{|\gamma | \le N} \sum _{|\beta | \le N} \int _0^t \int _{\Sigma _s} \frac{\tau _-^2}{\log ^5(1+\tau _-)} \left| {\mathcal {L}}_{Z^{\gamma }}(F)_{0 \nu } \int _v \frac{v^{\nu }}{v^0} {\widehat{Z}}^{\beta } f \mathrm{d}v \right| \mathrm{d}x \mathrm{d}s\lesssim & {} \epsilon ^{\frac{3}{2}}. \end{aligned}$$

Fix $|\beta | \le N$, $|\gamma | \le N$, denote by $(\alpha , {\underline{\alpha }}, \rho , \sigma )$ the null decomposition of ${\mathcal {L}}_{Z^{\gamma }}(F)$ and recall that ${\overline{K}}_0^L=\frac{1}{2} \tau _+^2$ and ${\overline{K}}_0^{{\underline{L}}}=\frac{1}{2} \tau _-^2$. Expanding ${\overline{K}}_0^{\mu } {\mathcal {L}}_{Z^{\gamma }}(F)_{\mu \nu } J( {\widehat{Z}}^{\beta } f )^{\nu }$ and ${\mathcal {L}}_{Z^{\gamma }}(F)_{0 \nu } J( {\widehat{Z}}^{\beta } f )^{\nu }$ in null coordinates, we can observe that it suffices to prove that

$$\begin{aligned} I:= & {} \int _0^t \int _{\Sigma _s} \int _v \left( \tau _+^2 |\rho | \frac{v^{{\underline{L}}}}{v^0}+\tau _+^2 |\alpha | \frac{|v^A|}{v^0} +\tau _-^2 |\rho |\frac{v^L}{v^0}+\tau _-^2|{\underline{\alpha }}|\frac{|v^{A}|}{v^0} \right) \left| {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \mathrm{d}x \mathrm{d}s \\\lesssim & {} \epsilon ^{\frac{3}{2}} \log ^4 (3+t), \\ I_0:= & {} \int _0^t \int _{\Sigma _s} \int _v \frac{\tau _-^2}{\log ^5(1+\tau _-)} \left( |\rho | + |\alpha |+ |{\underline{\alpha }}|\frac{|v^{A}|}{v^0} \right) \left| {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \mathrm{d}x \mathrm{d}s \; \lesssim \; \epsilon ^{\frac{3}{2}}. \end{aligned}$$

Using the Cauchy–Schwarz inequality in x, $\tau _+v^{{\underline{L}}}+\tau _+|v^A|+\tau _- v^L \lesssim v^0 \sum _{w \in {\mathbf {k}}_0} |w|$ (see Lemmas 2.11), the bootstrap assumption (26) and Proposition 7.9, we have

$$\begin{aligned} I\lesssim & {} \sum _{w \in {\mathbf {k}}_0} \int _{0}^t \left\| \tau _+ |\rho | +\tau _+ |\alpha |+\tau _- {\underline{\alpha }} \right\| _{L^2(\Sigma _s)} \left\| \int _v \left| w {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \right\| _{L^2(\Sigma _s)}\mathrm{d}s \\\lesssim & {} \epsilon \int _{0}^t \sqrt{{\mathcal {E}}_N[F](s)} \frac{ \log (3+s)}{1+s} \mathrm{d}s \; \lesssim \; \epsilon ^{\frac{3}{2}} \int _{0}^t \frac{\log ^3(3+s)}{1+s}\mathrm{d}s \; \lesssim \; \epsilon ^{\frac{3}{2}} \log ^4(3+t). \end{aligned}$$

Similarly, using $\tau _+|v^A|+\tau _- v^0 \lesssim v^0 \sum _{w \in {\mathbf {k}}_0} |w|$, we obtain

$$\begin{aligned} I_0\lesssim & {} \sum _{w \in {\mathbf {k}}_0} \int _{0}^t \left\| \tau _+ |\rho | +\tau _+ |\alpha |+\tau _- {\underline{\alpha }} \right\| _{L^2(\Sigma _s)} \left\| \frac{\tau _-}{\tau _+ \log ^5(1+\tau _-)}\int _v \left| w {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \right\| _{L^2(\Sigma _s)}\mathrm{d}s \\\lesssim & {} \int _{0}^t \sqrt{{\mathcal {E}}_N[F](s)} \left\| \frac{\tau _+ \sqrt{\tau _-}}{\tau _+^{\frac{3}{2}}} \int _v \left| w {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \right\| _{L^2(\Sigma _s)} \mathrm{d}s \; \lesssim \; \epsilon ^{\frac{3}{2}} \int _{0}^t \frac{\log (3+s)}{(1+s)^{\frac{3}{2}}} \mathrm{d}s \; \lesssim \; \epsilon ^{\frac{3}{2}}. \end{aligned}$$

These two estimates allow us to improve the bootstrap assumptions (26) and (27) if $\epsilon$ is small enough, which concludes the proof.

Appendices

The Vlasov Field Vanishes for Small Velocities

Let F be a smooth 2-form defined on $[0,T[ \times {\mathbb {R}}^3$ which satisfies

$$\begin{aligned} \forall \, (t,x) \in [0,T[ \times {\mathbb {R}}^3, \quad |F|(t,x) \, \lesssim \, \frac{\sqrt{\epsilon }}{\tau _+ \tau _-} \quad \text {and} \quad |\rho (F)|(t,x) \, \lesssim \, \frac{\sqrt{\epsilon }}{\tau _+^{\frac{3}{2}}}. \end{aligned}$$

(39)

The aim of this section is to prove the following result.

Proposition A.1

Let f be a classical solution to $T^{\chi }_F(f)=0$ such that $f(0,\cdot ,v)=0$ for all $|v| \le 3$. Then if $\epsilon$ is small enough, we have

$$\begin{aligned} \forall \, (t,x,v) \in [0,T[ \times {\mathbb {R}}^3 \times ({\mathbb {R}}^3 {\setminus } \{ 0 \} ), \quad |v| \le 1 \; \Rightarrow \; f(t,x,v)=0. \end{aligned}$$

The proof is based on the study of the characteristics of the system. As $f_0(\cdot ,v)=0$ for all $|v| \le 3$, we consider $(x,v) \in {\mathbb {R}}^3 \times {\mathbb {R}}^3$ such that $|v| \ge 3$ and (X, V) the characteristic of the operator $T^{\chi }_F$ such that $(X(0),V(0))=(x,v)$. Our goal is to prove $\inf _{[0,T[} |V| \ge 1$, which would imply Proposition A.1. Then, suppose that |V| reaches the value 1 and define

$$\begin{aligned} t_1 := \inf \{ s \in [0,T[ \, / \, |V(s)|=1 \}, \quad t_0 := \sup \{ s \in [0,t_1] \, / \, |V(s)| = 2 \}. \end{aligned}$$

As V is continuous, $t_0$ and $t_1$ are well defined. In view of the support of $\chi$, (X, V) satisfies the following system of ODEs on $[t_0,t_1]$. For all $1\le i \le 3$,

$$\begin{aligned} \frac{\mathrm{d}X^i}{\mathrm{d}s}(s)=\frac{V^i(s)}{|V(s)|} \quad \text {and}\quad \frac{\mathrm{d}V^i}{\mathrm{d}s}(s) =F_{0i}(s,X(s))+\frac{V^j(s)}{|V(s)|} F_{ji}(s,X(s)). \end{aligned}$$

(40)

We then deduce, since F is a 2-form, that $\frac{d\left( |V|^2 \right) }{\mathrm{d}s}=2V^i F_{0i}$, which implies

$$\begin{aligned} \forall \, t_0 \le t \le t_1, \quad |V(t)|^2 \ge |V(t_0)|^2-2\int _{t_0}^t \left| V^i(s) F_{0i}(s,X(s)) \right| \mathrm{d}s. \end{aligned}$$

(41)

Before presenting the strategy of the proof, let us introduce certain subsets of $[t_0,t_1]$ and two constants. Note that if $\epsilon$ is small enough, we can suppose that $22 \le t_0 < t_1$. We can then introduce two constants $\delta >0$ and $K>0$ independent of $\epsilon$ and satisfying

$$\begin{aligned} 4\delta \le 1+\delta \le K < \frac{\pi }{4 \sqrt{2}} t_0^{\frac{1}{4}} \quad \text {and} \quad 2^{-\frac{5}{2}} K^2-\delta > 2 \delta . \end{aligned}$$

(42)

We also consider, for $Q >0$, the following subsets of $[t_0,t_1]$,

$$\begin{aligned} {\mathcal {A}}_Q := \{ s \in [t_0,t_1] \, / \, |s-|X(s)|| \ge Q s^{\frac{1}{4}} \} \quad \text {and} \quad \overline{{\mathcal {A}}}_Q := [t_0,t_1] {\setminus } {\mathcal {A}}_Q. \end{aligned}$$

Then, using (41) and $\sup _{[t_0,t_1]} |V| \le 2$, we have for all $t \in [t_0,t_1]$,

$$\begin{aligned} |V(t)|^2\ge & {} 4-4\int _{{\mathcal {A}}_{\delta }} \left| \frac{V^i(s)}{|V(s)|} F_{0i}(s,X(s)) \right| \mathrm{d}s -4 \int _{\overline{{\mathcal {A}}}_{\delta }} \left| \frac{V^i(s)}{|V(s)|}-\frac{X^i(s)}{s} \right| \left| F_{0i}(s,X(s)) \right| \mathrm{d}s \\&-4 \int _{\overline{{\mathcal {A}}}_{\delta }} \frac{|X(s)|}{s} \left| \frac{X^i(s)}{|X(s)|} F_{0i}(s,X(s)) \right| \mathrm{d}s. \end{aligned}$$

The result would ensue if we could bound the three integrals on the right-hand side of the last inequality by $C \sqrt{\epsilon }$, with $C>0$ a constant independent of T and (x, v). Indeed, we would then obtain, for $\epsilon < (2C)^{-2}$,

$$\begin{aligned} \forall \, t \in [t_0,t_1], \quad |V(t)| \ge \sqrt{2}, \end{aligned}$$

which would contradict $|V(t_1)| = 1$. We can easily bound two of these integrals, using either the strong decay rate of F away from the light cone or the strong decay rate satisfied by the null component $\rho$ and that $\frac{|X(s)|}{s}$ is bounded near the light cone. More precisely, using (39), the definition of ${\mathcal {A}}_{\delta }$ and $t_0 \ge 1$, one has

$$\begin{aligned} \int _{{\mathcal {A}}_{\delta }} \left| \frac{V^i(s)}{|V(s)|} F_{0i}(s,X(s)) \right| \mathrm{d}s\lesssim & {} \int _0^{+\infty } \frac{\sqrt{\epsilon } \mathrm{d}s}{(1+s)(1+\delta s^{\frac{1}{4}})} \; \lesssim \; \sqrt{\epsilon }, \\ \int _{\overline{{\mathcal {A}}}_{\delta }} \frac{|X(s)|}{s} \left| \frac{X^i(s)}{|X(s)|} F_{0i}(s,X(s)) \right| \mathrm{d}s\lesssim & {} \int _{\overline{{\mathcal {A}}}_{\delta }}(1+\delta ) \left| \rho (F)(s,X(s)) \right| \mathrm{d}s \\\lesssim & {} \int _0^{+ \infty } \frac{\sqrt{\epsilon }}{(1+s)^{\frac{3}{2}}}\mathrm{d}s \; \lesssim \; \sqrt{\epsilon }. \end{aligned}$$

For the last integral, observe, in view of (39), that

$$\begin{aligned} I_1:= & {} \int _{\overline{{\mathcal {A}}}_{\delta }} \left| \frac{V^i(s)}{|V(s)|}-\frac{X^i(s)}{s} \right| \left| F_{0i}(s,X(s)) \right| \mathrm{d}s \nonumber \\\lesssim & {} \sqrt{\epsilon } \int _{\overline{{\mathcal {A}}}_{\delta }} \left| \frac{V(s)}{|V(s)|}-\frac{X(s)}{s} \right| \frac{1}{1+|s-|X(s)||} \frac{\mathrm{d}s}{1+s}. \end{aligned}$$

(43)

As $|s-|X(s)||$ is small on $\overline{{\mathcal {A}}}_{\delta }$, the goal is to obtain enough decay from $\frac{V(s)}{|V(s)|}-\frac{X(s)}{s}$. The rough idea behind the following computations is the following. As $|s-X(s)|$ is small, then, by (40), $s \sim |X(s)| \sim \left| \int _0^s \frac{V(\tau )}{|V(\tau )|} \mathrm{d} \tau \right|$ and we almost have equality in the triangular inequality

$$\begin{aligned} \left| \int _0^s \frac{V(\tau )}{|V(\tau )|} \mathrm{d} \tau \right| \le \int _0^s \left| \frac{V(\tau )}{|V(\tau )|} \right| \mathrm{d} \tau = s. \end{aligned}$$

$\frac{V}{|V|}$ then almost keeps a constant direction $\mathbf {u}$, so that

$$\begin{aligned} \frac{X(s)}{s} \sim \frac{1}{s} \int _0^s \frac{V(\tau )}{|V(\tau )|} \mathrm{d} \tau \sim \mathbf {u} \sim \frac{V}{|V|}. \end{aligned}$$

In order to bound (43), let us introduce, for $Q > 0$ and $\delta _0>0$, the following subsets of $[t_0,t_1]$,

$$\begin{aligned}&{\mathcal {B}}_Q := \left\{ s \in [t_0,t_1] \, / \, \left| \frac{V(s)}{|V(s)|}-\frac{X(s)}{s} \right| > \frac{Q}{ s^{\frac{1}{4}} } \right\} , \\&\quad \overline{{\mathcal {B}}_Q}:= [t_0,t_1] {\setminus } {\mathcal {B}}_Q \quad \text {and} \\&\quad {\mathcal {C}}^{\delta _0}_Q:= \overline{{\mathcal {A}}}_{\delta _0} \cap {\mathcal {B}}_Q. \end{aligned}$$

In view of the definition of ${\mathcal {C}}^{\delta }_{4K}$ and (43), $I_1 \lesssim \sqrt{\epsilon }$ would ensue if we prove

$$\begin{aligned} I \, := \, \int _{{\mathcal {C}}^{\delta }_{4K}} \left| \frac{V^i(s)}{|V(s)|}-\frac{X^i(s)}{s} \right| \left| F_{0i}(s,X(s)) \right| \mathrm{d}s \, \lesssim \, \sqrt{\epsilon }. \end{aligned}$$

(44)

From now, we suppose that ${\mathcal {C}}^{\delta }_{4K} \ne \varnothing$ as otherwise, $I=0$. We start by the following two results.

Lemma A.2

Let $s \in {\mathcal {C}}^{\delta }_{4K}$. Then, $[s,\min (t_1,s+s^{\frac{3}{4}})] \subset {\mathcal {B}}_{2K}$.

Proof

Let $t \in [s,\min (t_1,s+s^{\frac{3}{4}})]$. As $s \in {\mathcal {B}}_{4K}$ and by the triangle inequality, one has

$$\begin{aligned} \frac{4K}{s^{\frac{1}{4}}} \le \left| \frac{V^i(s)}{|V(s)|}-\frac{X^i(s)}{s} \right| \le \left| \frac{V^i(t)}{|V(t)|}-\frac{X^i(t)}{t} \right| +\left| \frac{V^i(s)}{|V(s)|}-\frac{V^i(t)}{|V(t)|} \right| +\left| \frac{X^i(s)}{s}-\frac{X^i(t)}{t} \right| . \end{aligned}$$

By the mean value theorem applied to the function $\frac{V}{|V|}$ and using the estimate (39), we have

$$\begin{aligned} \left| \frac{V^i(s)}{|V(s)|}-\frac{V^i(t)}{|V(t)|} \right| \, \le \, C_1 \sup _{\tau \in [s,t[} |F(\tau ,X(\tau ))| |t-s| \, \le \, \frac{C_0\sqrt{\epsilon }}{1+s}|t-s| \, \le \, \frac{C_0\sqrt{\epsilon }}{s}|t-s|. \end{aligned}$$

Using $|X(s)| \le s+\delta s^{\frac{1}{4}}$ and $|X(t)-X(s)| \le |t-s|$, which results from (40), we have

$$\begin{aligned} \left| \frac{X^i(s)}{s}-\frac{X^i(t)}{t} \right| \, \le \, \frac{(t-s)|X^i(s)|+s|X^i(t)-X^i(s)|}{ts} \, \le \, \frac{2+\delta }{t}|t-s| \; \le \; \frac{ 2+\delta }{s}|t-s|. \end{aligned}$$

Thus, as $s \le t \le s+s^{\frac{3}{4}}$ and $2K \ge 2+2 \delta$ (see (42)), it follows, for $\epsilon$ small enough,

$$\begin{aligned} \frac{4K}{s^{\frac{1}{4}}}-\frac{C_0 \sqrt{\epsilon }+2+\delta }{s}|t-s| \ge \frac{4K-C_0 \sqrt{\epsilon }-2-\delta }{s^{\frac{1}{4}}} > \frac{2K}{t^{\frac{1}{4}}}, \quad \text {so that} \quad t \in {\mathcal {B}}_{2K}. \end{aligned}$$

$\square$

Lemma A.3

Suppose that $s \in {\mathcal {C}}^{\delta }_{4K}$ and let $t_*(s)$ be equal to $\inf \{ t \in [s,t_1] \, / \, t \notin {\mathcal {C}}_{2K}^{2 \delta } \}$ if it is well defined and $t_*(s)=t_1$ otherwise. Then,

$$\begin{aligned} \forall \, t \in [s,t_*(s)], \quad t-|X(t)| \ge s-|X(s)|+K^2\sqrt{t}-K^2\sqrt{s}. \end{aligned}$$

Proof

Let $g{:}\,t \mapsto t-|X(t)|$, so that $g'(t)= 1-\langle \frac{V(t)}{|V(t)|}, \frac{X(t)}{|X(t)|} \rangle$. Let us estimate $\theta \in [0, \pi [$, the angle between V and X. For $t \in [s,t_*(s)[$, we have

• $\left| \frac{V(t)}{|V(t)|}-\frac{X(t)}{|X(t)|} \right| \le |\theta (t) |$ since $\frac{V}{|V|}$ and $\frac{X}{|X|}$ are unit vectors.

• $\frac{2K}{t^{\frac{1}{4}}} \le \left| \frac{V(t)}{|V(t)|}-\frac{X(t)}{t} \right|$ as $t \in B_{2K}$ and $\left| \frac{|X(t)|}{t}-1 \right| \le \frac{2\delta }{t^{\frac{3}{4}}}$ since $t \in {\overline{A}}_{2\delta }$.

We then obtain, using $4 \delta \le K$, the trivial fact $\left| \frac{X(t)}{t}-\frac{X(t)}{|X(t)|} \right| =\left| \frac{|X(t)|}{t}-1 \right|$ and the triangle inequality, that

$$\begin{aligned} \frac{ \sqrt{2} K}{t^{\frac{1}{4}}} \,\le & {} \, \frac{2K}{t^{\frac{1}{4}}}-\frac{2\delta }{t^{\frac{3}{4}}} \, \le \, \left| \frac{V(t)}{|V(t)|}-\frac{X(t)}{t} \right| -\left| \frac{X(t)}{t}-\frac{X(t)}{|X(t)|} \right| \nonumber \\\le & {} \, \left| \frac{V(t)}{|V(t)|}-\frac{X(t)}{|X(t)|} \right| \, \le \, |\theta (t) |. \end{aligned}$$

Consequently, as $1-\frac{1}{4}\phi ^2 \ge \cos \phi$ for $\phi \in [0, \frac{\pi }{4}]$ and since $\sqrt{2} K \le \frac{\pi }{4}t_0^{\frac{1}{4}}$, we obtain

$$\begin{aligned} g'(t)= & {} 1-\cos \theta (t) \ge 1-\cos \left( \frac{\sqrt{2} K}{t^{\frac{1}{4}}} \right) \\\ge & {} \frac{K^2}{ 2\sqrt{t}} \quad \text {and then} \quad g(t) \ge g(0)+K^2(\sqrt{t}-\sqrt{s}). \end{aligned}$$

$\square$

The strategy now is to prove that ${\mathcal {C}}^{\delta }_{4K}$ is composed of pieces sufficiently well separated for (44) to hold. It is then convenient to consider a dyadic partition of $[t_0,t_1]$, which leads us to introduce, for all $i \in {\mathbb {N}}$,

$$\begin{aligned} {\mathcal {C}}^i_{4K} := {\mathcal {C}}^{\delta }_{4K} \cap [2^i,2^{i+1}[ \quad \text {and, if} \; {\mathcal {C}}^i_{4K} \ne \varnothing , \quad s^i = \inf {\mathcal {C}}^i_{4K} \quad \text {and} \quad t_*^i = t_*(s^i). \end{aligned}$$

Corollary A.4

Let $i \in {\mathbb {N}}$ such that ${\mathcal {C}}^i_{4K} \ne \varnothing$. Then, ${\mathcal {C}}^i_{4K} \subset [s^i, \min (t_*^i,2^{i+1})]$. Moreover, if $|X(s^i)|-s^i >0$, $t \mapsto t-|X(t)|$ is positive on $[2^{i+2},t_1]$.

Proof

Let $i \in {\mathbb {N}}$ and suppose that ${\mathcal {C}}^i_{4K} \ne \varnothing$. We assume moreover that $t_*^i < t_1$ since there is nothing to prove otherwise and we introduce $T^i=\min (t_1,2^{i+2})$. We will use several times that $s-|X(s)| > 0$, for $s \in [t_0,t_1]$, implies that $t \mapsto t-|X(t)|$ is positive on $[s,t_1]$ since it is an increasing function. As $t_*^i \in {\mathcal {A}}_{2 \delta } \cup \overline{{\mathcal {B}}}_{2K}$ by definition, we have two cases to study.

• If $t_*^i \in \overline{{\mathcal {B}}}_{2K}$, then, by Lemma A.2, $t_*^i \ge \tau ^i := s^i+|s^i|^{\frac{3}{4}}$. Hence, using Lemma A.3, we get

  $$\begin{aligned} \tau ^i -|X(\tau ^i)|\ge & {} s^i -|X(s^i)|+ K^2 \sqrt{\tau ^i}-K^2 \sqrt{s^i} \\\ge & {} -\delta \left| s^i \right| ^{\frac{1}{4}}+K^2 \sqrt{s^i} \left( \sqrt{1+|s^i|^{-\frac{1}{4}}}-1 \right) . \end{aligned}$$

  Since $4\sqrt{1+h}-4 \ge h$ for all $h \in [0,1]$ and $t \mapsto t -|X(t)|$ increases, we obtain, for all $t \in [\tau ^i,T^i]$,

  $$\begin{aligned}&t-|X(t)| \ge \tau ^i -|X(\tau ^i)| \ge -\delta 2^{\frac{i+2}{4}}+ K^2 \sqrt{s^i} \frac{1}{4}|s^i|^{-\frac{1}{4}} \ge (2^{-\frac{10}{4}}K^2-\delta )2^{\frac{i+2}{4}} \nonumber \\&\quad \ge 2 \delta \left| T^i \right| ^{\frac{1}{4}} \ge 2 \delta t^{\frac{1}{4}}. \end{aligned}$$

  (45)

  so that¹⁵$[\tau ^i,T_i] \subset {\mathcal {A}}_{2 \delta }$. As $C^i_{4K} \subset {\overline{A}}_{\delta }$ and ${\mathcal {A}}_{2 \delta } \cap {\overline{A}}_{\delta } = \varnothing$, we obtain $C^i_{4K} \subset [s^i, \tau ^i] \subset [s^i, t_*^i]$. Observe also that $\tau ^i-|X(\tau ^i)| >0$, which implies that $t \mapsto t-|X(t)|$ is positive on $[\tau ^i,t_1] \subset [2^{i+2},t_1]$.

• Otherwise $t_*^i \in {\mathcal {A}}_{2 \delta }$. If $t_*^i \ge T^i$, we have $\tau ^i \le 2^{i+2} \le t_*^i$ so that $C^i_{4K} \subset [s^i, 2^{i+1}] \subset [s^i, t_*^i]$. The positivity of $t \mapsto t-|X(t)|$ on $[2^{i+2},t_1]$ then follows from (45). Otherwise, as $|t_*^i-|X(t_*^i)|| \ge 2 \delta |t_*^i|^{\frac{1}{4}}$ and $t \mapsto t -|X(t)|$ increases, we necessarily have $t_*^i-|X(t_*^i)| \ge 0$ since $s^i-|X(s^i)| \ge - \delta |s^i|^{\frac{1}{4}}$. Hence, using again that $t \mapsto t -|X(t)|$ increases, it yields

  $$\begin{aligned} \forall \, t \in [t_*^i,T^i], \quad t-|X(t)| \ge 2 \delta \left| t_*^i \right| ^{\frac{1}{4}}> \delta t^{\frac{1}{4}}, \quad \text {as} \quad 2 \delta \left| t_*^i \right| ^{\frac{1}{4}} \ge 2 \delta 2^{\frac{i}{4}} > \delta 2^{\frac{i+2}{4}}. \end{aligned}$$

  We can then conclude that $[t_*^i,T^i] \subset {\mathcal {A}}_{\delta }$, implying that $C^i_{4K} \subset [s^i,t_*^i]$ since $A_{\delta } \cap C^i_{4K} = \varnothing$. We also proved that $t \mapsto t-|X(t)|$ on $[2^{i+2},t_1]$ as $t_*^i-|X(t_*^i)| \ge 0$ and $t_*^i \le 2^{i+2}$.

$\square$

We are now able to bound I. Let ${\mathcal {D}} := \{ i \in {\mathbb {N}} \, / \, C^i_{4K} \ne \varnothing \}$. Suppose first that $t \ge |X(t)|$ for all $t \in {\mathcal {C}}_{4K}^{\delta }$. Then, using (39), Lemma A.3 and Corollary A.4,

$$\begin{aligned} I\lesssim & {} \sum _{i \in {\mathcal {D}} } \int _{{\mathcal {C}}^i_{4K}} \frac{\sqrt{\epsilon }}{(1+s)(1+|s-|X(s)||)} \mathrm{d}s \; \lesssim \; \sum _{i \in {\mathcal {D}} } \int _{s^i}^{2^{i+1}} \frac{\sqrt{\epsilon }}{(1+s)(1+K^2\sqrt{s}-K^2\sqrt{s^i})} \mathrm{d}s \\\lesssim & {} \sum _{i \in {\mathcal {D}} } \sqrt{\epsilon } \int _{s^i}^{2^{i+1}} \frac{1+\sqrt{s}+\sqrt{s^i}}{(1+s)(1+s-s^i)} \mathrm{d}s \; \lesssim \; \sum _{i =0 }^{+ \infty } \frac{\sqrt{\epsilon }}{2^{\frac{i}{2}}} \int _{2^i}^{2^{i+1}} \frac{1}{(1+s-2^i)} \mathrm{d}s \\\lesssim & {} \sqrt{\epsilon } \sum _{i =0 }^{+ \infty } \frac{\log (1+2^i)}{2^{\frac{i}{2}}} \; \lesssim \; \sqrt{\epsilon }. \end{aligned}$$

Otherwise, with $p= \min {\mathcal {D}}$ and according to Corollary A.4, we have $t \ge |X(t)|$ for all $t \ge 2^{p +2}$ and the result then follows from

$$\begin{aligned} \sqrt{\epsilon } \int _{{\mathcal {C}}^{p}_{4K} \cup {\mathcal {C}}^{p+1}_{4K}} \frac{\mathrm{d}s}{1+s} \le \sqrt{\epsilon } \int _{2^p}^{2^{p +2}} \frac{\mathrm{d}s}{1+s} \le \sqrt{\epsilon } \log (4). \end{aligned}$$

In order to apply this result in Sect. 7.1, we need to adapt it to an echeloned system of transport equations.

Corollary A.5

Let $k \in {\mathbb {N}}^*$ and, for all $1 \le j < i \le k$ and $1 \le q \le 3$, let $A^{q,j}_i$ be a sufficiently regular matrix-valued function defined on $[0,T[ \times {\mathbb {R}}^3 \times \left( {\mathbb {R}}^3 {\setminus } \{ 0 \} \right)$. Consider $g=(g_1,\ldots ,g_k)$, where each $g_i$ is a vector-valued field, a classical solution on [0, T[ to the system

$$\begin{aligned} T^{\chi }_F(g_1)=0, \;\; T_F^{\chi }(g_i)=A^{q,j}_i \partial _{v^q} g_{j} \, 2 \le i \le k. \end{aligned}$$

If $g(0,\cdot ,v)=0$ for all $|v| \le 3$, then $g(t,\cdot ,v)=0$ for all $|v| \le 1$.

Proof

Let $(t,x,v) \in [0,T[ \times {\mathbb {R}}^3 \times \left( {\mathbb {R}}^3 {\setminus } \{ 0 \} \right)$ such that $|v| < 1$. We denote by $(X_s,V_s)$ the value in s of the characteristic of the operator $T_F^{\chi }$ which was equal to (x, v) in $s=t$. By Duhamel’s formula, we have

$$\begin{aligned} g_1(t,x,v)= & {} g_1(s,X_s,V_s) = g_1(0,X_0,V_0), \nonumber \\ g_i(t,x,v)= & {} g_i(0,X_0,V_0)+\int _0^t A^{q,j}_i(s,X_s,V_s) \partial _{v^q} g_j(s,X_s,V_s) \mathrm{d}s\nonumber \\&\quad \text {for all } \,2 \le i \le k. \end{aligned}$$

(46)

According to the proof of Proposition A.1,

$$\begin{aligned} |V_0| < 3, \; \text {so that} \; \forall \; 1 \le i \le k, \; g_i(0,X_0,V_0)=0 \end{aligned}$$

(47)

since otherwise we would have $|V_t|=|v| \ge 1$. Fix now $s \in [0,t]$ and consider $w \in {\mathbb {R}}^3$ such that $|w|< \frac{1}{2}|v|$. We denote by $(X_{\tau }^{w,s}, V_{\tau }^{w,s})$ the value in $\tau$ of the characteristic of $T^{\chi }_F$ which was equal to $(X_s,V_s+w)$ in $\tau =s$. Then,

$$\begin{aligned} g_1(s,X_s,V_s+w)=g_1(0,X_0^{w,s},V_0^{w,s}). \end{aligned}$$

By continuous dependence on the initial condition of the solutions to

$$\begin{aligned}&\frac{\mathrm{d}X^i}{\mathrm{d}s}(s)=\frac{V^i(s)}{|V(s)|}, \quad \frac{\mathrm{d}V^i}{\mathrm{d}s}(s)=\chi (|V(s)|)F_{0i}(s,X(s))\\&\qquad +\,\frac{V^j(s)}{|V(s)|} \chi (|V(s)|) F_{ji}(s,X(s)), \quad 1 \le i \le 3, \end{aligned}$$

and since $|V_0| < 3$, there exists ${\overline{\delta }} >0$ (depending on (t, s, x, v)) such that $|V_0^{w,s}| <3$ for all $|w| < {\overline{\delta }}$. Hence,

$$\begin{aligned} \forall \, |w| < {\overline{\delta }}, \, g_1(s,X_s,V_s+w)=0, \, \text {so that} \, \forall \, 1 \le q \le 3, \quad \partial _{v^q} g_1 (s,X_s,V_s)=0.\nonumber \\ \end{aligned}$$

(48)

Repeating the argument, one can obtain

$$\begin{aligned} \forall \, |w| \le {\overline{\delta }}, \;\; \forall \, \tau \in [0,s], \;\; \exists {\underline{\delta }} > 0, \;\; \forall \, |{\underline{w}}| \le {\underline{\delta }}, \quad g_1(\tau ,X_{\tau }^{w,s},V_{\tau }^{w,s}+{\underline{w}})=0, \end{aligned}$$

which implies

$$\begin{aligned} \forall \, |w| \le {\overline{\delta }}, \;\; \forall \, \tau \in [0,s], \, \forall \, 1 \le q \le 3, \quad \partial _{v^q} g_1 (\tau ,X_{\tau }^{w,s},V_{\tau }^{w,s})=0. \end{aligned}$$

(49)

Combining (46), (47) and (48), we get $g_2(t,x,v)=0$. Using (49) and $|V_0^{w,s}| <3$ for all $|w| < {\overline{\delta }}$, it yields, in view of the support of $g_2(0,\cdot ,\cdot )$,

$$\begin{aligned}&\forall \, |w| < {\overline{\delta }}, \quad g_2(s,X_s,V_s+w)=g_2(0,X_0^{w,s},V_0^{w,s})\\&\quad +\,\int _0^s A^{q,j}_i( \tau , X_{\tau }^{w,s}, V_{\tau }^{w,s}) \partial _{v^q} g_1 ( \tau , X_{\tau }^{w,s}, V_{\tau }^{w,s}) \mathrm{d} \tau =0. \end{aligned}$$

We then deduce that $\partial _{v^q} g_2(s,X_s,V_s)=0$ for all $q \in \llbracket 1,3 \rrbracket$ and $s \in [0,t]$, so that, by (46), $g_3(t,x,v)=0$. We then proved the desired result if $k=3$. The general case can be treated similarly by an induction. $\square$

Bounding the Initial Norms

We consider in this section $(f^0,F^0)$ satisfying the hypotheses of Theorem 1.1 and (f, F) the unique classical solution of (1)–(3) arising from these data.

Proposition B.1

There exists a constant $C_1$, depending only on N, such that

$$\begin{aligned} {\mathcal {E}}_N[F](0) \le C_1 \epsilon \; \text {and} \; {\mathbb {E}}^2_{N+3}[f](0) \le C_1 \epsilon . \end{aligned}$$

The proof is a corollary of the following proposition and that, for all ${\widehat{Z}}^{\beta } \in \widehat{{\mathbb {P}}}_0^{|\beta |}$ and $Z \in {\mathbb {K}}^{|\gamma |}$,

$$\begin{aligned} |{\widehat{Z}}^{\beta } f |\lesssim & {} \sum _{|\alpha _2|+|\alpha _1|+p \le |\beta |} \tau _+^{|\alpha _1|+p} |v|^{|\alpha _2|} | \partial _t^p \partial _x^{\alpha _1}\partial _{v}^{\alpha _2} f |, \quad \left| {\mathcal {L}}_{Z^{\gamma }}(F) \right| \\\lesssim & {} \sum _{|\kappa |+q \le |\gamma |} \tau _+^{|\kappa |+q} | \nabla _{\partial _t}^q \nabla _x^{\kappa } F|. \end{aligned}$$

Proposition B.2

We have, for all $|\alpha _2|+|\alpha _1|+p \le N+3$ and $|\kappa |+q \le N+2$,

$$\begin{aligned}&\left\| (1+|x|)^{|\alpha _1|+p+2} |v|^{|\alpha _2|} \partial _t^p \partial _x^{\alpha _1}\partial _{v}^{\alpha _2} f \right\| _{L^1_v L^1(\Sigma _0)} \lesssim \epsilon , \; \\&\quad \left\| (1+|x|)^{|\kappa |+q+1} \nabla ^q_{\partial _t} \nabla _x^{\kappa } F \right\| ^2_{L^2(\Sigma _0)} \lesssim \epsilon .\nonumber \end{aligned}$$

(50)

Proof

Note that $\tau _+^2=1+|x|^2$ on $\Sigma _0$. The proof consists of an induction on $\max (p,q)$. The result holds for $\max (p,q)=0$ in view of the hypotheses on $(f^0,F^0)$. Let $r \in \llbracket 1, N+2 \rrbracket$ and suppose that (50)–(51) is satisfied for all $p \le r$ and all $q \le r$. If $r < N+2$, fix $|\kappa | \le N+2-(r+1)$ and notice that, using (2) and Lemma 2.2,

$$\begin{aligned} \partial _t F_{0i} = \partial ^j F_{ji}+\int _{v \in {\mathbb {R}}^3} \frac{v^i}{v^0} f \mathrm{d}v \quad \text {and} \quad \partial _t F_{ij}=\partial _i F_{0j}+\partial _j F_{i0}. \end{aligned}$$

Then, by a standard $L^2_x-L^1_x$ Sobolev inequality and using the induction hypothesis twice, we get

$$\begin{aligned} \left\| \tau _+^{|\kappa |+r+2} \nabla ^{r+1}_{\partial _t} \nabla _x^{\kappa } F \right\| ^2_{L^2(\Sigma _0)}\le & {} 2\left\| \tau _+^{|\kappa |+r+2} \nabla _x \nabla ^r_{\partial _t} \nabla _x^{\kappa } F \right\| ^2_{L^2(\Sigma _0)}\\&+\left\| \tau _+^{|\kappa |+r+2} \int _{v} \partial ^r_t \partial _x^{\kappa } f \mathrm{d}v \right\| ^2_{L^2(\Sigma _0)} \\\lesssim & {} \epsilon + \sum _{|\beta | \le 2} \left\| \tau _+^{|\kappa |+r+2} \int _{v} \partial _x^{\beta } \partial ^r_t \partial _x^{\kappa } f \mathrm{d}v \right\| ^2_{L^1(\Sigma _0)} \; \lesssim \; \epsilon +\epsilon ^2, \end{aligned}$$

since $r+|\kappa |+|\beta | \le N+3$. We now turn on the Vlasov field and we do not suppose $r < N+2$ anymore. Start by fixing $\alpha _1$ and $\alpha _2$ such that $|\alpha _1|+|\alpha _2|+r+1 \le N+3$. Iterating the commutation formula $T_F(\partial _{\mu } f ) = - {\mathcal {L}}_{\partial _{\mu }} (F)(v, \nabla _v f)$ and using ${\mathcal {L}}_{\partial _{\mu }}= \nabla _{\partial _{\mu }}$, we get

$$\begin{aligned} \partial _t \partial _t^r \partial ^{\alpha _1}_x f=-\frac{v^i}{|v|} \partial _i \partial _t^r \partial ^{\alpha _1}_x f+\sum _{q+p=r} \sum _{\begin{array}{c} \; |\kappa |+|\beta _1| = |\alpha _1| \\ p+|\beta _1| \le r+|\alpha _1|-1 \end{array}} \frac{v^{\mu }}{|v|} \nabla ^q_{\partial _t} \nabla ^{\kappa }_x {F_{\mu }}^{ j} \partial _{v^j} \partial _t^p \partial _x^{\beta _1} f. \end{aligned}$$

Taking the $\partial ^{\alpha _2}_v$ derivative of each side and multiplying them by $\tau _+^{|\alpha _1|+r+3}|v|^{|\alpha _2|}$, one obtains

$$\begin{aligned}&\tau _+^{|\alpha _1|+r+1+2}|v|^{|\alpha _2|} |\partial _t^{r+1} \partial ^{\alpha _1}_x \partial ^{\alpha _2}_v f| \le \sum _{|\alpha _3|+n=|\alpha _2|} \tau _+^{|\alpha _1|+1+r+2} |v|^{|\alpha _2|-n}| \partial _x \partial _t^r \partial ^{\alpha _1}_x \partial ^{\alpha _3}_v f| \nonumber \\ \end{aligned}$$

(51)

$$\begin{aligned}&\quad +\, \sum _{\begin{array}{c} \, |\kappa |+|\beta _1| = |\alpha _1| \\ p+|\beta _1| \le r+|\alpha _1|-1 \end{array}} \sum _{\begin{array}{c} \; q+p=r \\ |\alpha _3|+n=|\alpha _2| \end{array}} \tau _+^{|\alpha _1|+r+3} |v|^{|\alpha _2|-n} \left| \nabla ^q_{\partial _t} \nabla ^{\kappa }_x F \partial _{v} \partial _t^p \partial _x^{\beta _1} \partial _{v}^{\alpha _3} f \right| . \end{aligned}$$

(52)

By the induction hypothesis, the $L^1_v L^1(\Sigma _0)$ norm of the terms of (51) is bounded by $\epsilon$. Consider parameters such as in the sum in (52).

$\bullet$ If $q+|\kappa | \le N$, then, using a standard $L^{\infty }-L^2$ Sobolev inequality on $\tau _+^{q+|\kappa |+1} \nabla ^q_{\partial _t} \nabla ^{\kappa }_x F$ and that $|v| \ge 3$ on the support of $f^0$, we get

$$\begin{aligned}&\tau _+^{|\alpha _1|+r+3} |v|^{|\alpha _2|-n} \left| \nabla ^q_{\partial _t} \nabla ^{\kappa }_x F \partial _{v} \partial _t^p \partial _x^{\beta _1} \partial _{v}^{\alpha _3} f \right| \\&\quad \lesssim \tau _+^{|\beta _1|+p+2} |v|^{|\alpha _3|+1} \left| \partial _t^p \partial _x^{\beta _1} \partial _v \partial _{v}^{\alpha _3} f \right| \sum _{|\gamma | \le |\kappa |+2} \left\| \tau _+^{|\kappa |+q+1} \nabla ^{q}_{\partial _t} \nabla _x^{\gamma } F \right\| _{L^2_x}. \end{aligned}$$

The $L^1_v L^1(\Sigma _0)$ norm of the left-hand side of the previous inequality is then bounded by $\epsilon ^{\frac{3}{2}}$ according to the induction hypothesis and $p+q=r$.

$\bullet$ Otherwise $|\beta _1|+p \le 2$ and, by the Cauchy–Schwarz inequality in x,

$$\begin{aligned}&\left\| \tau _+^{|\alpha _1|+r+3} |v|^{|\alpha _2|-n} \nabla ^q_{\partial _t} \nabla ^{\kappa }_x F \partial _{v} \partial _t^p \partial _x^{\beta _1} \partial _{v}^{\alpha _3} f \right\| _{L^1_{v,x}} \\&\quad \le \left\| \tau _+^{|\kappa |+q+1} \nabla ^q_{\partial _t} \nabla ^{\kappa }_x F \right\| _{L^2_{x}} \left\| \tau _+^{|\beta _1|+p+2} |v|^{|\alpha _3|} \partial _t^p \partial _x^{\beta _1} \partial _v \partial _{v}^{\alpha _3} f \right\| _{L^1_vL^2_{x}}. \end{aligned}$$

The left-hand side of the previous inequality can be bounded by $\epsilon ^{\frac{3}{2}}$. Indeed, as $|v| \ge 3$ on the support of $f^0$ and using a $L^2_x-L^1_x$ Sobolev inequality,¹⁶ it yields

$$\begin{aligned}&\left\| \tau _+^{|\beta _1|+p+2} |v|^{|\alpha _3|} \partial _t^p \partial _x^{\beta _1} \partial _v \partial _{v}^{\alpha _3} f \right\| _{L^1_vL^2(\Sigma _0)} \\&\quad \lesssim \sum _{ |\beta | \le |\beta _1|+2} \left\| \tau _+^{|\beta _1|+p+2} |v|^{|\alpha _3|+1} \partial _t^p \partial _x^{\beta } \partial _v \partial _{v}^{\alpha _3} f \right\| _{L^1_v L^1 (\Sigma _0) }. \end{aligned}$$

It remains to use the induction hypothesis twice. This concludes the induction and then the proof. $\square$

All Derivatives of F are Chargeless

The aim of this section is to prove the following result, which also applies to massive particles.

Proposition C.1

Let $N_0 \ge 2$ and (f, F) be a sufficiently regular solution to the Vlasov–Maxwell system on [0, T] such that

$$\begin{aligned} \forall \, t \in [0,T], \quad \sum _{ |\beta | \le N_0} \int _{\Sigma _t} \int _{v \in {\mathbb {R}}^3} \left| {\widehat{Z}}^{\beta } f \right| \mathrm{d}v \mathrm{d}x +\sum _{ |\gamma | \le N_0} \int _{\Sigma _t} \left| {\mathcal {L}}_{Z^{\gamma }}(F) \right| ^2 \mathrm{d}x < + \infty . \end{aligned}$$

Then, for all $1 \le |\gamma | \le N_0$, ${\mathcal {L}}_{Z^{\gamma }}(F)$ is chargeless, i.e.,

$$\begin{aligned} -\lim _{r \rightarrow + \infty } \int _{{\mathbb {S}}_{0,r}} \rho \left( {\mathcal {L}}_{Z^{\gamma }}(F) \right) \mathrm{d}{\mathbb {S}}_{0,r} = 0. \end{aligned}$$

This ensues from the commutation formula of Proposition 2.9 and the following lemma.

Lemma C.2

Fix $p \ge 1$ and let $H_0$, $H_1$, ..., $H_p$ be sufficiently regular 2-forms defined on $[0,T] \times {\mathbb {R}}^3$ and $h_0$, $h_1$, ..., $h_p$ be sufficiently regular functions defined on $[0,T] \times {\mathbb {R}}^3_x \times {\mathbb {R}}^3_v$ such that

$$\begin{aligned} \nabla ^{\mu } \left( H_0 \right) _{ \mu \nu } = J(h_0)_{\nu } \quad \text {and} \quad T(h_0)=\sum _{1 \le i \le p} H_i \left( v, \nabla _v h_i \right) . \end{aligned}$$

Suppose moreover that

$$\begin{aligned}&\forall \, t \in [0,T], \quad \sum _{\lambda =0}^p \Vert H_{\lambda } \Vert _{L^2_x} (t)\\&\quad +\,\Vert h_{\lambda } \Vert _{L^1_{x,v}} (t) + \Vert (1+r) \nabla _{t,x} h_{\lambda } \Vert _{L^1_{x,v}} (t)+ \Vert (1+|v|) \nabla _{v} h_{\lambda } \Vert _{L^1_{x,v}} (t) < + \infty . \end{aligned}$$

is finite for all $t\in [0,T]$. Then, ${\mathcal {L}}_{Z}(H_0)$ is chargeless for all $Z \in {\mathbb {K}}$.

Remark C.3

Note that in dimension $n \ne 3$, we merely have that ${\mathcal {L}}_S(H_0)$ is chargeless if and only if $H_0$ is.

Proof

To lighten the proof, we suppose that $p=1$ and we denote $(H_0,h_0)$ by (G, g) and $(h_1,H_1)$ by (H, h). Consider first $Z \in {\mathbb {P}}$. Then, by Lemma 2.8, $\nabla ^{\mu } {\mathcal {L}}_Z(G)_{\mu 0}= J({\widehat{Z}} g)_0$ so that, using the divergence theorem,

$$\begin{aligned} Q(t):= \lim _{r \rightarrow + \infty } \int _{ {\mathbb {S}}_{t,r} } \rho \left( {\mathcal {L}}_{Z}(G) \right) \mathrm{d} {\mathbb {S}}_{t,r} = - \int _{x \in {\mathbb {R}}^3} \int _{v \in {\mathbb {R}}^3} {\widehat{Z}} g \mathrm{d}v \mathrm{d}x. \end{aligned}$$

(53)

• If $Z=\Omega \in {\mathbb {O}}$ is a rotational vector field, the result does not depend of the source term of the Maxwell equations. Indeed, using Lemma 3.8 and the divergence theorem on ${\mathbb {S}}_{t,r}$, one has,

  

  As $\Omega$ is tangential to ${\mathbb {S}}_{t,r}$, we have , so that $Q(t)=0$.

• If $Z = \partial _i$ is a spatial translation, an integration by parts on the right-hand side of (53) gives the result.

• If $Z=\partial _t$, then, as $\partial _t g=-\frac{v^j}{v^0} \partial _j g+H \left( \frac{v}{v^0}, \nabla _v h \right)$, integration by parts (in x and in v) gives us, as H is a 2-form,

  $$\begin{aligned} Q(t)= & {} \int _{x \in {\mathbb {R}}^3} \int _{v \in {\mathbb {R}}^3} \frac{v^i}{v^0} \partial _i g-H \mathrm{d}v \mathrm{d}x \nonumber \\= & {} \int _{x \in {\mathbb {R}}^3} \int _{v \in {\mathbb {R}}^3} \frac{v^i v^j}{(v^0)^3} H_{ij} h \mathrm{d}v \mathrm{d}x =0. \end{aligned}$$

  (54)

• If $Z= \Omega _{0i}$ is a Lorentz boost, then an integration by parts in x on $t \partial _i g$ and in v on $v^0 \partial _{v^i} g$ gives

  $$\begin{aligned} Q(t)= & {} - \int _{x \in {\mathbb {R}}^3} \int _{v \in {\mathbb {R}}^3} t \partial _i g+x^i \partial _t g+v^0 \partial _{v^i} g \mathrm{d}v \mathrm{d}x \\= & {} \int _{x \in {\mathbb {R}}^3} \int _{v \in {\mathbb {R}}^3}\frac{v^i}{v^0} gdv \mathrm{d}x-\int _{x \in {\mathbb {R}}^3} \int _{v \in {\mathbb {R}}^3} x^i \partial _t g \mathrm{d}v \mathrm{d}x. \end{aligned}$$

  Using again $\partial _t g=-\frac{v^j}{v^0} \partial _j g+H$ and integrating by parts, we have

  $$\begin{aligned} \int _{x \in {\mathbb {R}}^3} \int _{v \in {\mathbb {R}}^3} x^i \partial _t g \mathrm{d}v \mathrm{d}x= & {} -\int _{v \in {\mathbb {R}}^3} \frac{v^j}{v^0} \int _{x \in {\mathbb {R}}^3}x^i \partial _j g \mathrm{d}x \mathrm{d}v+ \int _{x \in {\mathbb {R}}^3} x^i \int _{v \in {\mathbb {R}}^3} H \mathrm{d}v \mathrm{d}x \\= & {} \int _{x \in {\mathbb {R}}^3} \int _{v \in {\mathbb {R}}^3} \frac{v^i}{v^0}g \mathrm{d}v \mathrm{d}x-\int _{x \in {\mathbb {R}}^3} \int _{v \in {\mathbb {R}}^3} x^i\frac{v^k v^j}{(v^0)^3} H_{kj} h \mathrm{d}v \mathrm{d}x. \end{aligned}$$

  As H is a 2-form, we finally obtain that $Q(t)=0$.

For the case of the scaling vector field, note first by Lemma 2.8 that $\nabla ^{\mu } {\mathcal {L}}_S (G)_{\mu 0 } = J(Sg)_0+3J(g)_0$. Hence,

$$\begin{aligned} \lim _{r \rightarrow + \infty } \int _{ {\mathbb {S}}_{t,r} } \rho {\mathcal {L}}_{S}(G) \mathrm{d} {\mathbb {S}}_{t,r}= & {} - \int _{x \in {\mathbb {R}}^3} \int _{v \in {\mathbb {R}}^3} x^{\mu } \partial _{\mu }g+3g \mathrm{d}v \mathrm{d}x\\= & {} -t \int _{x \in {\mathbb {R}}^3} \int _{v \in {\mathbb {R}}^3} \partial _t g \mathrm{d}v \mathrm{d}x. \end{aligned}$$

Recall from (54) that the integral on the right-hand side of the last equation is equal to 0. This concludes the proof. $\square$

Proof of Lemmas 3.8, 3.9 and 3.10

Let G be a 2-form and J be a 1-form, both sufficiently regular and defined on $[0,T[ \times {\mathbb {R}}^3$, such that

$$\begin{aligned} \nabla ^{\mu } G_{\mu \nu }= & {} J_{\nu }, \\ \nabla ^{\mu } {}^* G_{ \mu \nu }= & {} 0. \end{aligned}$$

Let us successively prove Lemmas 3.8, 3.9 and 3.10.

Lemma D.1

Let $\Omega \in {\mathbb {O}}$. Then, denoting by $\zeta$ any of the null component $\alpha$, ${\underline{\alpha }}$, $\rho$ or $\sigma$,

$$\begin{aligned}{}[{\mathcal {L}}_{\Omega }, \nabla _{\partial _r}] G=0, \, {\mathcal {L}}_{\Omega }(\zeta (G))= \zeta ( {\mathcal {L}}_{\Omega }(G) ) \, \text {and} \, \nabla _{\partial _r}(\zeta (G))= \zeta ( \nabla _{\partial _r}(G) ). \end{aligned}$$

Similar results hold for ${\mathcal {L}}_{\Omega }$ and $\nabla _{\partial _t}$, $\nabla _L$ or $\nabla _{{\underline{L}}}$. For instance, $\nabla _{L}(\zeta (G))= \zeta ( \nabla _{L}(G) )$.

Proof

Let $\Omega \in {\mathbb {O}}$. The property $[{\mathcal {L}}_{\Omega }, \nabla _{\partial _r}] G=0$ follows from $[\Omega , \partial _r]=0$, straightforward computations and that, in Cartesian coordinates and for a vector field X,

$$\begin{aligned} {\mathcal {L}}_{X} (G)_{\mu \nu } = X(G_{\mu \nu } )+ \partial _{\mu } ( X^{\lambda } ) G_{\lambda \nu }+\partial _{\nu } (X^{\lambda } ) G_{\mu \lambda } \quad \text {and} \quad \nabla _X(G)_{\mu \nu } = X(G_{\mu \nu } ).\nonumber \\ \end{aligned}$$

(55)

Aside from ${\mathcal {L}}_{\Omega } (\sigma (G))= \sigma ( {\mathcal {L}}_{\Omega }(G))$, the remaining identities ensue from $[\Omega , L]=[\Omega , {\underline{L}}]=\nabla _{\partial _r} L=\nabla _{\partial _r} {\underline{L}}=\nabla _{\partial _r} e_A=0$ since, for instance,

$$\begin{aligned}&2 \nabla _{\partial _r} \rho (G) = (\nabla _{\partial _r} G) (L, {\underline{L}} )+G(\nabla _{\partial _r} L, {\underline{L}})+G(L, \nabla _{\partial _r} {\underline{L}}) \quad \text {and} \\&\quad {\mathcal {L}}_{\Omega }(\alpha )={\mathcal {L}}_{\Omega }(G)( \cdot , L)+G(\cdot , [\Omega ,L]). \end{aligned}$$

Using that $[\Omega , e_A]=C_{\Omega }^B(\omega _1,\omega _2) e_B$, where $C^B_{\Omega }$ are bounded functions on the sphere, we directly obtain $|\Omega \sigma (G) | \lesssim |\sigma ( {\mathcal {L}}_{\Omega }(G)) |+|\sigma (G)|$, which is good enough for proving the following results of this appendix. To obtain $\Omega (\sigma (G))= \sigma ( {\mathcal {L}}_{\Omega }(G))$, one can check, with straightforward computations that, for a 2-form H,

$$\begin{aligned} \rho ({}^* H ) = - \sigma (H) \, \text {and} \, {}^* {\mathcal {L}}_{\Omega } (H)= {\mathcal {L}}_{\Omega } ( {}^* H). \end{aligned}$$

It then follows

$$\begin{aligned} \Omega \sigma (G) =-\Omega \rho ( {}^* G ) = -\rho ({\mathcal {L}}_{\Omega } ( {}^* H))=-\rho ({}^* {\mathcal {L}}_{\Omega } ( G))=\sigma ({\mathcal {L}}_{\Omega } ( G))). \end{aligned}$$

For the results concerning the operator $\nabla _{\partial _t}$, use $\nabla _{\partial _t} L=\nabla _{\partial _t} {\underline{L}}=\nabla _{\partial _t} e_A=0$. Finally, for $\nabla _L$ and $\nabla _{{\underline{L}}}$, recall that $L=\partial _t+\partial _r$ and ${\underline{L}}=\partial _t-\partial _r$. $\square$

Lemma D.2

Denoting by $\zeta$ any of the null component $\alpha$, ${\underline{\alpha }}$, $\rho$ or $\sigma$, we have

Proof

Let $\zeta \in \{ \alpha , {\underline{\alpha }}, \rho , \sigma \}$. As $\nabla _{{\underline{L}}}$ commute with the null decomposition (see the previous Lemma) and $(t-r){\underline{L}}= S-\frac{x^i}{r} \Omega _{0i}$ (see Lemma 2.7), we have,

$$\begin{aligned}&(t-r) \nabla _{{\underline{L}}} \zeta \left( G \right) = \zeta \left( \nabla _{(t-r) {\underline{L}}} G \right) = \zeta \left( \nabla _S G -\frac{x^i}{r} \nabla _{\Omega _{0i}} G \right) \\&\quad = \zeta \left( {\mathcal {L}}_S (G) \right) -2 \zeta (G)-\frac{x^i}{r} \zeta \left( {\mathcal {L}}_{\Omega _{0i}}(G) \right) +\frac{x_i}{r} \zeta \left( H^i \right) , \end{aligned}$$

since ${\mathcal {L}}_S(G)=\nabla _S(G)+2G$ and where $H^i={\mathcal {L}}_{\Omega _{0i}}(G)-\nabla _{\Omega _{0i}}(G)$. We have, using (55),

$$\begin{aligned}&H^i_{\mu \nu } =0 \quad \text {if} \quad \mu , \nu \notin \{0,i \} \;\; \text {or} \;\; \mu , \nu \in \{ 0, i \}. \quad \text {If} \quad \nu \notin \{0,i \}, \quad H^i_{0\nu }=G_{i \nu } \quad \text {and} \nonumber \\&\quad H^i_{i \nu } = G_{0 \nu }. \end{aligned}$$

(56)

Consequently,

$$\begin{aligned} \frac{x_i}{r} \rho \left( H^i \right) = \frac{x_i}{r} H^i_{r 0 } = \frac{x^i}{r}\frac{x^j}{r} G_{j i} =0, \quad \text {so that} \quad | {\underline{L}} \rho ( G)| \lesssim \frac{1}{\tau _-} \sum _{|\gamma | \le 1} |\rho ({\mathcal {L}}_{Z^{\gamma }}(G)|. \end{aligned}$$

It remains to study $x_i \zeta \left( H^i \right)$ for $\zeta \in \{ \alpha , {\underline{\alpha }}, \sigma \}.$ Let us treat together the case of $\alpha$ and ${\underline{\alpha }}$ by computing $x_iH^i_{A0}$ and $x_iH^i_{Ar}$. Recall that $e_A = \sum _{1 \le k < l \le 3} C^{k,l}(\omega _1,\omega _2) \frac{\Omega _{kl}}{r}$ where $C^{k,l}$ are bounded functions on the sphere. As

$$\begin{aligned} x_i H^i \left( \frac{\Omega _{kl}}{r}, \partial _t \right)= & {} \frac{x_ix^k}{r} H^i_{l0}-\frac{x_ix^l}{r} H^i_{k0}=\frac{x^ix^k}{r} G_{li}-\frac{x^ix^l}{r} G_{ki}=rG \left( \frac{\Omega _{kl}}{r}, \partial _r \right) , \\ x_i H^i \left( \frac{\Omega _{kl}}{r}, \partial _r \right)= & {} \frac{x_ix^kx^j}{r^2} H^i_{lj}-\frac{x_ix^lx^j}{r^2} H^i_{kj} \\= & {} x^k \left( \frac{x_ix^i - (x^l)^2}{r^2} G_{l0} + \frac{x_lx^j}{r^2} G_{0j} - \frac{(x^l)^2}{r^2} G_{0l} \right) \\&- x^l \left( \frac{x_ix^i - (x^k)^2}{r^2} G_{k0} + \frac{x_kx^j}{r^2} G_{0j} - \frac{(x^k)^2}{r^2} G_{0k} \right) \\= & {} r G \left( \frac{\Omega _{kl}}{r}, \partial _t \right) , \end{aligned}$$

we obtain

$$\begin{aligned} | \nabla _{{\underline{L}}} \left( \alpha ( G) \right) _A|\lesssim & {} \frac{1}{\tau _-} \sum _{|\gamma | \le 1} |\alpha ({\mathcal {L}}_{Z^{\gamma }}(G)_A| \quad \text {and} \quad |\nabla _{{\underline{L}}} \left( {\underline{\alpha }}( G) \right) _A| \\\lesssim & {} \frac{1}{\tau _-} \sum _{|\gamma | \le 1} |{\underline{\alpha }}({\mathcal {L}}_{Z^{\gamma }}(G)_A|. \end{aligned}$$

For the remaining case, $\zeta =\sigma$, straightforward computations give

$$\begin{aligned}&x_i H^i \left( \frac{\Omega _{kl}}{r}, \frac{\Omega _{pq}}{r} \right) = \frac{x^k x^p x_i}{r^2} H^i_{lq}+\frac{x^l x^q x_i}{r^2} H^i_{kp}-\frac{x^k x^q x_i}{r^2} H^i_{lp}\\&\quad -\,\frac{x^l x^p x_i}{r^2} H^i_{kq} =0, \quad \text {so that} \quad x_i H^i_{AB}=0. \end{aligned}$$

os that $x_iH_{AB}=0$. The proof for $\nabla _{L}$ is similar as it also commutes with the null decomposition and since $(t+r)L=S+\frac{x^i}{r}\Omega _{0i}$. Finally, for the angular derivatives, use that ${\mathcal {L}}_{\Omega }$ commute with the null decomposition and that for a function u and a 1-form U tangential to the 2-spheres,

In order to prove the last inequality, recall that $|\Gamma _{Ar}^B| \lesssim r^{-1}$, where $\Gamma$ are the Christofel symbols of the Minkowski spacetime in the nonholonomic basis $(\partial _t, \partial _r,e_1,e_2)$, so that

$\square$

Lemma D.3

Denoting by $(\alpha , {\underline{\alpha }}, \rho , \sigma )$ the null decomposition of G, we have

Proof

Let us start by proving, for $B \ne A$,

(57)

Suppose for instance that $e_B=e_1$ and $e_A=e_2$. Then, as ,

It remains to notice that, as G is a 2-form and,

Recall now from Lemma 2.2 that $\nabla _{[\lambda } G_{\mu \nu ] }=0$. Taking the $(L,{\underline{L}},A)$ component of this tensorial equation, we get, as $\nabla _L {\underline{L}}=\nabla _{{\underline{L}}} L=0$ and $\nabla _{e_A} L=-\nabla _{e_A} {\underline{L}} = \frac{1}{r}e_A$,

(58)

Similarly, taking $\nu =A$ in $\nabla ^{\mu } G_{\mu \nu }= J_{\nu }$, we obtain, using (57) and since $\nabla ^L = -\frac{1}{2} \nabla _{{\underline{L}}}$ and $\nabla ^{{\underline{L}}} = -\frac{1}{2} \nabla _L$,

(59)

It remains to add half of (58)–(59). $\square$