On the Cauchy Problem for the Vlasov-Maxwell-Fokker-Planck System in Low Regularity Space

Abstract

In this study, we investigate the Cauchy problem for the Vlasov-Maxwell-Fokker-Planck system near a global Maxwellian in low regularity space. We establish the existence of global mild solutions to the system by employing the energy method, provided that the perturbative initial data is sufficiently small. Moreover, despite the absence of zeroth-order dissipation for the magnetic field, we are able to derive exponential decay estimates for solutions in higher-order regularity space. This is achieved by leveraging the higher-order dissipation properties of the magnetic field, which are deduced from the Maxwell equation.

1. Introduction

The Vlasov-Maxwell-Fokker-Planck system is a set of equations that describe the behavior of charged particles in a plasma under the influence of electromagnetic fields. It combines elements from several areas of physics and mathematics, providing a comprehensive framework for understanding plasma dynamics. The Vlasov equation describes the particle distribution, Maxwell’s equations provide the electromagnetic fields, and the Fokker-Planck equation accounts for collisional effects. This system is widely used in plasma physics, particularly in the study of fusion plasmas, space plasmas, and plasma-based technologies such as plasma propulsion and plasma processing [1,2].

This paper focus on the Cauchy problem for the Vlasov-Maxwell-Fokker-Planck system, which is delineated by the following equations [3,4]:

$$\left\{\begin{array}{c}{\mathsf{\partial }}_{t}F+v·{\nabla }_{x}F+(E+v\times B)·{\nabla }_{v}F=L_{FP}F,\hfill \\ {\mathsf{\partial }}_{t}E-{\nabla }_x\times B=-{\int }_{{\mathbb{R}}^3}vFdv,\hfill \\ {\mathsf{\partial }}_{t}B+{\nabla }_x\times E=0,\hfill \\ {\nabla }_x·E={\int }_{{\mathbb{R}}^3}Fdv-1,{\nabla }_x·B=0,\hfill \end{array}\right.$$

(1)

with initial data

$$F(0,x,v)=F_0(x,v),E(0,x)=E_0\left(x\right),B(0,x)=B_0\left(x\right),$$

(2)

and the compatibility conditions

$${\nabla }_x·E_0={\int }_{{\mathbb{R}}^3}{F}_0(x,v)dv-1,{\nabla }_x·B_0=0,$$

where the unknown non-negative function $F(t,x,v)$ describes the spatially periodic distribution of particles at time $t\ge 0$, with spatial coordinates $x=(x_1,x_2,x_3)$ defined on the three-dimensional torus ${\mathbb{T}}^3:={[0,2\pi ]}^3$ and velocity coordinates $v=(v_1,v_2,v_3)\in {\mathbb{R}}^3$.The electromagnetic field $\left[E\right(t,x),B(t,x\left)\right]$, which is self-consistently generated, is coupled to the distribution function $F(t,x,v)$ through the Maxwell equations. The Fokker-Planck operator $L_{FP}$, crucial for describing the stochastic acceleration of particles, is defined as

$$L_{FP}F={\nabla }_v·(vF+{\nabla }_{v}F).$$

The global Maxwellian equilibrium state which is a fundamental reference in our analysis is defined as

$$\mu \left(v\right)={\left(2\pi \right)}^{-{\displaystyle \frac{3}{2}}}exp(-{\displaystyle \frac{{\left|v\right|}^2}{2}}).$$

This equilibrium distribution is characterized by a Gaussian profile that reflects the thermal equilibrium of the particle system in the absence of external influences. In this manuscript, we consider the perturbation solution with the form

$$F(t,x,v)=\mu +{\mu }^{{\displaystyle \frac{1}{2}}}f(t,x,v).$$

(3)

Then the Cauchy problem (1) and (2) is reformulated as

$$\left\{\begin{array}{c}{\mathsf{\partial }}_{t}f+v·{\nabla }_{x}f+(E+v\times B)·{\nabla }_{v}f-{\displaystyle \frac{1}{2}}E·vf-E·v{\mu }^{{\displaystyle \frac{1}{2}}}=L_{FP}f,\hfill \\ {\mathsf{\partial }}_{t}E-{\nabla }_x\times B=-{\int }_{{\mathbb{R}}^3}v{\mu }^{{\displaystyle \frac{1}{2}}}fdv,\hfill \\ {\mathsf{\partial }}_{t}B+{\nabla }_x\times E=0,\hfill \\ {\nabla }_x·E={\int }_{{\mathbb{R}}^3}{\mu }^{{\displaystyle \frac{1}{2}}}fdv,{\nabla }_x·B=0,\hfill \end{array}\right.$$

(4)

with initial data

$$f(0,x,v)=f_0(x,v),E(0,x)=E_0\left(x\right),B(0,x)=B_0\left(x\right),$$

(5)

where the Fokker-Planck operator $L_{FP}$ is given by

$$L_{FP}f={\mu }^{-{\displaystyle \frac{1}{2}}}{\nabla }_v·(\mu {\nabla }_v({\mu }^{-{\displaystyle \frac{1}{2}}}f)(={\Delta }_{v}f+{\displaystyle \frac{1}{4}}{(6-|v|}^2)f.$$

(6)

Now define the velocity orthogonal projection

$$\mathbf{P}:L^2\left({\mathbb{R}}_v^3\right)\to \mathrm{Span}\left\{{\mu }^{{\displaystyle \frac{1}{2}}},v_i{\mu }^{{\displaystyle \frac{1}{2}}}(1\le i\le 3)\right\},$$

such that for any given function $f(t,x,v)\in L^2\left({\mathbb{R}}_v^3\right)$, one has that

$$\mathbf{P}f=a(t,x){\mu }^{{\displaystyle \frac{1}{2}}}+b(t,x)·v{\mu }^{{\displaystyle \frac{1}{2}}},$$

(7)

where

$$\begin{array}{cc}\hfill a=& {\int }_{{\mathbb{R}}^3}{\mu }^{{\displaystyle \frac{1}{2}}}fdv={\int }_{{\mathbb{R}}^3}{\mu }^{{\displaystyle \frac{1}{2}}}\mathbf{P}fdv,\hfill \\ \hfill b_i=& {\int }_{{\mathbb{R}}^3}{v}_i{\mu }^{{\displaystyle \frac{1}{2}}}fdv={\int }_{{\mathbb{R}}^3}{v}_i{\mu }^{{\displaystyle \frac{1}{2}}}\mathbf{P}fdv,i=1,2,3.\hfill \end{array}$$

(8)

Consequently, the solution $f(t,x,v)$ of the Vlasov-Maxwell-Fokker-Planck system (4) can be decomposed into macroscopic and microscopic components with respect to the global Maxwellian $\mu$, as introduced in reference [5]. This decomposition is expressed as

$$f(t,x,v)=\mathbf{P}f(t,x,v)+\{\mathbf{I}-\mathbf{P}\}f(t,x,v),$$

(9)

where $\mathbf{I}$ denotes the identity operator, while $\mathbf{P}f$ and $\{\mathbf{I}-\mathbf{P}\}f$ are referred to as the macroscopic and microscopic components of $f(t,x,v)$, respectively.

By integrating Equation (1)₁ with respect to the velocity variable v, we derive the local balance laws

$${\mathsf{\partial }}_t{\int }_{{\mathbb{R}}^3}Fdv+{\nabla }_x·{\int }_{{\mathbb{R}}^3}vFdv=0.$$

Then by employing Equations (3) and (8), we can deduce the macroscopic system [6]

$${\mathsf{\partial }}_{t}a+{\nabla }_x·b=0.$$

(10)

Furthermore, integrating the aforementioned identity with respect to x, we obtain the conservation law of mass

$${\int }_{{\mathbb{T}}^3}{\int }_{{\mathbb{R}}^3}{\mu }^{{\displaystyle \frac{1}{2}}}f(t,x,v)dvdx={\int }_{{\mathbb{T}}^3}{\int }_{{\mathbb{R}}^3}{\mu }^{{\displaystyle \frac{1}{2}}}f(0,x,v)dvdx.$$

(11)

Now we recall the related works of this manuscript. The Vlasov-Maxwell-Fokker-Planck Equation (1) has been extensively studied. Wollman [7] established the local existence and uniqueness of the smooth solutions to the Vlasov-Maxwell system. Diperna and Lions [2] established the stability of solutions in weak topologies and, based on this stability result, deduced the global existence of weak solutions for the Vlasov-Maxwell systems with large initial data. For the special one and one half dimensional case, the global-in-time existence and uniqueness of classical solutions for the relativistic version of the Vlasov-Maxwell-Fokker-Planck system was proved by [8,9]. For a comprehensive discussion of the limiting problems related to the Vlasov-Maxwell-Fokker-Planck system, we direct the reader to [1,10] and the references included therein.

The nonlinear energy method referred to as the macro-micro decomposition method in [5] for the Boltzmann equation has recently emerged as a powerful tool for proving classical global existence of solutions in perturbative frameworks. This approach has successfully been applied to construct global-in-time solutions for various kinetic models and also provide time decay rate estimates [11,12]. For the Vlasov-Maxwell-Boltzmann system, Guo [13] pioneered the study of global existence of solutions to the periodic initial boundary value problem near global Maxwellian equilibrium states for the hard sphere model. The global in time classical solutions and the large-time behavior for the Cauchy problem with cutoff collision kernels were studied in [14,15,16] and the non-cutoff case in [17].

As for the Vlasov-Maxwell-Fokker-Planck equation, Yang and Yu [18] obtained the global existence of classical solutions near Maxwellian based on the energy method by combining the compensating function. Additionally, the convergence rate in time of the solution is obtained. Further results on the global existence of solutions to the Vlasov-Maxwell-Fokker-Planck system near Maxwellians in the whole space, achieved through the application of the refined energy method, are presented in [4,6].

However, these results are obtained in Sobolev space $H_x^N(N\ge 2)$ which required high regularity on the initial data. Recently, Duan et al. constructed the global-in-time mild solutions to both Landau and non-cutoff Boltzmann equation by introducing a low regularity function space $L_k^1{L}_T^{\infty }{L}_v^2$ and also obtained the large-time behavior of solutions. Motivated by this method, we aim to establish the global-in-time existence of solutions to the Vlasov-Maxwell-Fokker-Planck equation in low regularity spaces.

The rest of this paper is organized as follows. Section 2 presents our main results. Section 3 lists some fundamental lemmas for subsequent use. Section 4 are devoted to deducing the desired uniform energy estimates which included the proofs of main Theorems 1 and 2. The Section 5 is conclusions.

2. Main Results

For the sake of convenience in exposition, we first introduce some several simplified notations.

• $A\lesssim B$ signifies that there exists a positive constant $C>0$ such that $A\le CB$.
• $A\sim B$ indicates that both $A\lesssim B$ and $B\lesssim A$ hold simultaneously.
• The inner product is defined as $(f,g)=f·\overline{g}$ for any complex vectors $f,g\in {\mathbb{C}}^3$.
• The notation ${(·,·)}_{L_v^2}$ represents the complex inner product over the space $L_v^2$, defined as

  $${(f,g)}_{L_v^2}={\int }_{{\mathbb{R}}_v^3}f\left(v\right)\overline{g\left(v\right)}dv,$$

with the associated $L^2$ norm given by ${\parallel f\parallel }_{L_v^2}={\left({\int }_{{\mathbb{R}}_v^3}{\left|f\right|}^{2}dv\right)}^{1/2}$.

• Let $\nu \left(v\right)=1+{\left|v\right|}^2$, and define the corresponding norm ${\parallel ·\parallel }_{\nu }$ as

  $${\parallel f\parallel }_{\nu }^2={\int }_{{\mathbb{R}}_v^3}\left({\nu \left(v\right)\left|f\right|}^2+{\left|{\nabla }_{v}f\right|}^2\right)dv.$$

  (12)
• The symbol $\mathcal{R}$ represents the real part of a complex number.
• For any complex vector $k\in {\mathbb{C}}^3$, $⟨k⟩$ is defined as $⟨k⟩=\sqrt{1+{\left|k\right|}^2}$.
• The Fourier transformation of $f(t,x,v)$ with respect to the spatial variable $x\in {\mathbb{T}}^3$ is given by

  $$\widehat{f}(t,k,v)={\int }_{{\mathbb{T}}^3}{e}^{-ik·x}f(t,x,v)dx,k\in {\mathbb{Z}}^3,$$

  and ${\parallel ·\parallel }_{L_k^1}$ is defined by

  $${\parallel f\parallel }_{L_k^1}={\int }_{{\mathbb{Z}}_k^3}\left|f\left(k\right)\right|d\Sigma \left(k\right)=\sum _{k\in {\mathbb{Z}}^3}\left|f\left(k\right)\right|,$$
  $d\Sigma \left(k\right)$ denotes the discrete measure in ${\mathbb{Z}}^3$.

Inspired by the work in [19], we employ the low regularity function space $L_k^1{L}_T^{\infty }{L}_v^2$, which is endowed with the norm

$${\parallel f\parallel }_{L_k^1{L}_T^{\infty }{L}_v^2}:={\int }_{{\mathbb{Z}}_k^3}\underset{0\le t\le T}{sup}{\parallel \widehat{f}(t,k,·)\parallel }_{L_v^2}d\Sigma \left(k\right)<\infty .$$

(13)

In order to state the results of this work, we introduce the energy functional $\mathcal{E}\left(f\right(t\left)\right)$ and the corresponding dissipation functional $\mathcal{D}\left(f\right(t\left)\right)$, defined by

$$\mathcal{E}\left(f\left(t\right)\right)\equiv {\int }_{{\mathbb{Z}}_k^3}\underset{0\le \tau \le t}{sup}\left(\parallel \widehat{f}{(\tau ,k,·)\parallel }_{L_v^2}+|\widehat{E}(\tau ,k)|+|\widehat{B}(\tau ,k)|\right)d\Sigma \left(k\right),$$

(14)

and

$$\begin{array}{cc}\hfill \mathcal{D}\left(f\right(t\left)\right)& \equiv {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\parallel \{\mathbf{I}-\mathbf{P}\}\widehat{f}(\tau ,k,·)\parallel }_{\nu }^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|(\widehat{a},\widehat{b})(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & +{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|\widehat{E}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right),\hfill \end{array}$$

(15)

respectively. To achieve exponential temporal decay, it is imperative to have a higher-order regularity of spatial variable function space $L_{k,m}^1{L}_T^{\infty }{L}_v^2$ with norm defined as

$${\parallel f\parallel }_{L_{k,m}^1{L}_T^{\infty }{L}_v^2}:={\int }_{{\mathbb{Z}}_k^3}\underset{0\le t\le T}{sup}{\parallel {⟨k⟩}^m\widehat{f}(t,k)\parallel }_{L_v^2}d\Sigma \left(k\right),$$

(16)

for any integer $m\ge 1$. Additionally, we define the energy functional ${\mathcal{E}}_m\left(f\left(t\right)\right)$ and the dissipation functional ${\mathcal{D}}_m\left(f\left(t\right)\right)$ as

$${\mathcal{E}}_m\left(f\left(t\right)\right)\equiv {\int }_{{\mathbb{Z}}_k^3}\underset{0\le \tau \le t}{sup}\left({\parallel ⟨k⟩}^m\widehat{f}{(\tau ,k,·)\parallel }_{L_v^2}{+|⟨k⟩}^m\widehat{E}{(\tau ,k)|+|⟨k⟩}^m\widehat{B}(\tau ,k)|\right)d\Sigma \left(k\right),$$

(17)

and

$$\begin{array}{cc}\hfill {\mathcal{D}}_m\left(f\left(t\right)\right)& \equiv {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\parallel {⟨k⟩}^m\{\mathbf{I}-\mathbf{P}\}\widehat{f}(\tau ,k,·)\parallel }_{\nu }^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|{⟨k⟩}^m(\widehat{a},\widehat{b})(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & +{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|{⟨k⟩}^m\widehat{E}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|{⟨k⟩}^m\widehat{B}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right),\hfill \end{array}$$

(18)

respectively.

Remark 1. 

The dissipation functional ${\mathcal{D}}_m\left(f\left(t\right)\right)(m\ge 1)$ includes the high order dissipation of the magnetic field B which is derived from the Maxwell Equation (39).

Theorem 1. 

Assume that the initial data $f_0(x,v)=f(0,x,v)$ satisfies the mass conservation law

$${\int }_{{\mathbb{T}}^3}{\int }_{{\mathbb{R}}^3}{\mu }^{{\displaystyle \frac{1}{2}}}f(t,x,v)dvdx={\int }_{{\mathbb{T}}^3}{\int }_{{\mathbb{R}}^3}{\mu }^{{\displaystyle \frac{1}{2}}}f(0,x,v)dvdx=0,$$

(19)

$B(t,x)$ is spatially periodic function, i.e., ${\int }_{{\mathbb{T}}_x^3}B(t,x)dx=0$. Then there is ${ϵ}_0>0$ such that if the initial data $F_0(x,v)=\mu +{\mu }^{{\displaystyle \frac{1}{2}}}{f}_0(x,v)\ge 0$ and

$$\begin{array}{c}\hfill \mathcal{E}\left(f_0\right)=\parallel f_0{\parallel }_{L_k^1{L}_v^2}+\parallel {\widehat{E}}_0\left(k\right){\parallel }_{L_k^1}+{\parallel {\widehat{B}}_0\left(k\right)\parallel }_{L_k^1}\le {ϵ}_0,\end{array}$$

the Cauchy problem (4) and (5) possesses a unique global mild solution $f(t,x,v)$, $t>0$, $x\in {\mathbb{T}}^3$, $v\in {\mathbb{R}}^3$. The solution satisfies the positivity condition $F(t,x,v)=\mu +{\mu }^{{\displaystyle \frac{1}{2}}}f(t,x,v)\ge 0$ as well as the uniform estimate

$$\mathcal{E}\left(f\left(t\right)\right)+\mathcal{D}\left(f\left(t\right)\right)\lesssim \mathcal{E}\left(f_0\right),$$

(20)

for any $t>0$.

Theorem 2. 

Under the assumption of Theorem 1, then there exists a positive constant ${ϵ}_0>0$ such that if the initial data $F_0(x,v)=\mu +{\mu }^{{\displaystyle \frac{1}{2}}}{f}_0(x,v)\ge 0$ and

$$\begin{array}{c}\hfill {\mathcal{E}}_1\left(f_0\right)=\parallel f_0{\parallel }_{L_k^1{L}_v^2}+\parallel ⟨k⟩{\widehat{E}}_0\left(k\right){\parallel }_{L_k^1}+{\parallel ⟨k⟩{\widehat{B}}_0\left(k\right)\parallel }_{L_k^1}\le {ϵ}_0,\end{array}$$

the Cauchy problem (4) and (5) possesses a unique global mild solution $f(t,x,v)$, $t>0$, $x\in {\mathbb{T}}^3$, $v\in {\mathbb{R}}^3$. The solution satisfies the positivity condition $F(t,x,v)=\mu +{\mu }^{{\displaystyle \frac{1}{2}}}f(t,x,v)\ge 0$ as well as the uniform estimate

$${\mathcal{E}}_1\left(f\left(t\right)\right)+{\mathcal{D}}_1\left(f\left(t\right)\right)\lesssim {\mathcal{E}}_1\left(f_0\right),$$

(21)

for any $t>0$. Furthermore, there exists a constant $\lambda >0$ such that the solution satisfies the exponential time decay estimate

$$\begin{array}{c}\hfill {\parallel f\left(t\right)\parallel }_{L_k^1{L}_v^2}+\parallel ⟨k⟩\widehat{E}{(t,k)\parallel }_{L_k^1}+{\parallel ⟨k⟩\widehat{B}(t,k)\parallel }_{L_k^1}\lesssim e^{-\lambda t}{\mathcal{E}}_1\left(f_0\right),\end{array}$$

(22)

for any $t\ge 0$.

Remark 2. 

• In contrast to the integer-order Sobolev space $H_{x,v}^N(N\ge 4)$ used in [6,18]. The regularity requirement for the initial data is significantly weaker here, as the embedding $H_x^2\hookrightarrow L_k^1$ holds.
• Owing to the lack of zeroth-order dissipation in the system, i.e.,

  $${\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|\widehat{B}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right),$$
  we are able to establish the global existence of the equation’s solution, yet we are unable to demonstrate its exponential time decay. However, we are fortunate to ascertain its higher-order dissipation, i.e.,

  $${\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|{⟨k⟩}^m\widehat{B}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right),m\ge 1,$$
  allowing us to obtain the exponential decay of the solution. For detailed calculations, please refer to the proof of Theorem 2.

3. Basic Eatimates

In this section, we present several fundamental results essential for subsequent discussions. The first lemma concerns the Fokker-Planck operator $L_{FP}$. As established in [3,20], the Fokker-Planck operator $L_{FP}$ exhibits coercivity, meaning there exists a positive constant ${\lambda }_0$ such that

$${(L_{FP}f,f)}_{L_v^2}\ge {\lambda }_0{\parallel \{\mathbf{I}-\mathbf{P}\}f\parallel }_{\nu }^2+{\left|b\right|}^2.$$

(23)

Since $L_{FP}$ is a linear operator, the following lemma holds:

Lemma 1 

([3,20]). There exists a constant ${\lambda }_0>0$ such that

$$-\mathcal{R}{(L_{FP}\widehat{f},\widehat{f})}_{L_v^2}\ge {\lambda }_0\parallel \{\mathbf{I}-\mathbf{P}\}\widehat{f}(t,k,·){\parallel }_{\nu }^2+{|\widehat{b}(t,k)|}^2.$$

(24)

The subsequent lemmas address the estimate of nonlinear terms, beginning with the estimate on $E·{\nabla }_{v}f$.

Lemma 2. 

There exists a sufficiently small constant $\eta >0$ such that

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|{\left(\widehat{E}\ast {\nabla }_v\widehat{f},\widehat{f}\right)}_{L_v^2}\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)& \lesssim \eta {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\parallel \widehat{f}(\tau ,k,·)\parallel }_{L_v^2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & +\mathcal{E}\left(f\left(t\right)\right){\int }_{{\mathbb{Z}}_l^3}{\left({\int }_0^t{\parallel {\nabla }_v\widehat{f}(\tau ,l,·)\parallel }_{L_v^2}^{2}d\tau \right)}^{1/2}d\Sigma \left(l\right)\hfill \\ \hfill & \lesssim \eta \mathcal{D}\left(f\right(t\left)\right)+\mathcal{E}\left(f\right(t\left)\right)\mathcal{D}\left(f\right(t\left)\right),\hfill \end{array}$$

(25)

and

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|{\left(\widehat{E}\ast {\nabla }_v\widehat{f},{\mu }^{{\displaystyle \frac{1}{4}}}\right)}_{L_v^2}\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)& \lesssim \mathcal{E}\left(f\left(t\right)\right){\int }_{{\mathbb{Z}}_l^3}{\left({\int }_0^t{\parallel {\nabla }_v\widehat{f}(\tau ,l,·)\parallel }_{L_v^2}^{2}d\tau \right)}^{1/2}d\Sigma \left(l\right)\hfill \\ \hfill & \lesssim \mathcal{E}\left(f\right(t\left)\right)\mathcal{D}\left(f\right(t\left)\right).\hfill \end{array}$$

(26)

Proof. 

(i): By Fubini’s theorem, we have that

$$\begin{array}{cc}\hfill \left|{\left(\widehat{E}\ast {\nabla }_v\widehat{f},\widehat{f}\right)}_{L_v^2}\right|& =\left|{\int }_{{\mathbb{R}}_v^3}\left({\int }_{{\mathbb{Z}}_l^3}\widehat{E}(\tau ,k-l){\nabla }_v\widehat{f}(\tau ,l,v)d\Sigma \left(l\right)\right)\overline{\widehat{f}}(\tau ,k,v)dv\right|\hfill \\ \hfill & =\left|{\int }_{{\mathbb{Z}}_l^3}\widehat{E}(\tau ,k-l)\left({\int }_{{\mathbb{R}}_v^3}{\nabla }_v\widehat{f}(\tau ,l,v)\overline{\widehat{f}}(\tau ,k,v)dv\right)d\Sigma \left(l\right)\right|\hfill \\ \hfill & \lesssim {\int }_{{\mathbb{Z}}_l^3}|\widehat{E}(\tau ,k-l)|\parallel {\nabla }_v\widehat{f}{(\tau ,l,·)\parallel }_{L_v^2}{\parallel \widehat{f}(\tau ,k,·)\parallel }_{L_v^2}d\Sigma \left(l\right).\hfill \end{array}$$

The following term is then bounded by applying the Cauchy-Schwarz’s inequality with respect to the time $\tau$ and Young’s inequality

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t\left|{\left(\widehat{E}\ast {\nabla }_v\widehat{f},\widehat{f}\right)}_{L_v^2}\right|d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & \lesssim {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\int }_{{\mathbb{Z}}_l^3}|\widehat{E}(\tau ,k-l)|\parallel {\nabla }_v\widehat{f}{(\tau ,l,·)\parallel }_{L_v^2}{\parallel \widehat{f}(\tau ,k,·)\parallel }_{L_v^2}d\Sigma \left(l\right)d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & \lesssim {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left({\int }_{{\mathbb{Z}}_l^3}|\widehat{E}(\tau ,k-l)|\parallel {\nabla }_v\widehat{f}{(\tau ,l,·)\parallel }_{L_v^2}d\Sigma \left(l\right)\right)}^{2}d\tau \right)}^{1/4}\hfill \\ \hfill & \times {\left({\int }_0^t{\parallel \widehat{f}(\tau ,k,·)\parallel }_{L_v^2}^{2}d\tau \right)}^{1/4}d\Sigma \left(k\right)\hfill \\ \hfill & \lesssim \eta {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\parallel \widehat{f}(\tau ,k,·)\parallel }_{L_v^2}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{4\eta }}{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left({\int }_{{\mathbb{Z}}_l^3}|\widehat{E}(\tau ,k-l)|\parallel {\nabla }_v\widehat{f}{(\tau ,l,·)\parallel }_{L_v^2}d\Sigma \left(l\right)\right)}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right),\hfill \end{array}$$

(27)

where $\eta >0$ is a sufficiently small universal constant. For the second term in the aforementioned inequality, we employ Minkowski’s inequality, that is,

$${\parallel \parallel ·\parallel }_{L_l^1}{\parallel }_{L_t^2}\le \parallel {\parallel ·\parallel }_{L_t^2}{\parallel }_{L_l^1},$$

leads to the following bound:

$$\begin{array}{cc}\hfill & {\left({\int }_0^t{\left({\int }_{{\mathbb{Z}}_l^3}|\widehat{E}(\tau ,k-l)|\parallel {\nabla }_v\widehat{f}{(\tau ,l,·)\parallel }_{L_v^2}d\Sigma \left(l\right)\right)}^{2}d\tau \right)}^{1/2}\hfill \\ \hfill \le & {\int }_{{\mathbb{Z}}_l^3}{\left({\int }_0^t|\widehat{E}{(\tau ,k-l)|}^2{\parallel {\nabla }_v\widehat{f}(\tau ,l,·)\parallel }_{L_v^2}^{2}d\tau \right)}^{1/2}d\Sigma \left(l\right).\hfill \end{array}$$

(28)

By invoking Fubini’s theorem and the translation invariance property, we have that

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left({\int }_{{\mathbb{Z}}_l^3}|\widehat{E}(\tau ,k-l)|\parallel {\nabla }_v\widehat{f}{(\tau ,l,·)\parallel }_{L_v^2}d\Sigma \left(l\right)\right)}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & \le {\int }_{{\mathbb{Z}}_k^3}{\int }_{{\mathbb{Z}}_l^3}{\left({\int }_0^t|\widehat{E}{(\tau ,k-l)|}^2{\parallel {\nabla }_v\widehat{f}(\tau ,l,·)\parallel }_{L_v^2}^{2}d\tau \right)}^{1/2}d\Sigma \left(l\right)d\Sigma \left(k\right)\hfill \\ \hfill & \le {\int }_{{\mathbb{Z}}_k^3}{\int }_{{\mathbb{Z}}_l^3}\underset{0\le \tau \le t}{sup}|\widehat{E}(\tau ,k-l)|{\left({\int }_0^t{\parallel {\nabla }_v\widehat{f}(\tau ,l,·)\parallel }_{L_v^2}^{2}d\tau \right)}^{1/2}d\Sigma \left(l\right)d\Sigma \left(k\right)\hfill \\ \hfill & ={\int }_{{\mathbb{Z}}_l^3}{\int }_{{\mathbb{Z}}_k^3}\underset{0\le \tau \le t}{sup}|\widehat{E}(\tau ,k-l)|{\left({\int }_0^t{\parallel {\nabla }_v\widehat{f}(\tau ,l,·)\parallel }_{L_v^2}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)d\Sigma \left(l\right)\hfill \\ \hfill & ={\int }_{{\mathbb{Z}}_l^3}\left({\int }_{{\mathbb{Z}}_k^3}\underset{0\le \tau \le t}{sup}|\widehat{E}(\tau ,k-l)|d\Sigma \left(k\right)\right){\left({\int }_0^t{\parallel {\nabla }_v\widehat{f}(\tau ,l,·)\parallel }_{L_v^2}^{2}d\tau \right)}^{1/2}d\Sigma \left(l\right)\hfill \\ \hfill & =\left({\int }_{{\mathbb{Z}}_k^3}\underset{0\le \tau \le t}{sup}|\widehat{E}(\tau ,k)|d\Sigma \left(k\right)\right){\int }_{{\mathbb{Z}}_l^3}{\left({\int }_0^t{\parallel {\nabla }_v\widehat{f}(\tau ,l,·)\parallel }_{L_v^2}^{2}d\tau \right)}^{1/2}d\Sigma \left(l\right).\hfill \end{array}$$

(29)

Note that (12), we have that

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\parallel {\nabla }_v\{\mathbf{I}-\mathbf{P}\}\widehat{f}(\tau ,k,·)\parallel }_{L_v^2}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & \lesssim {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\parallel \{\mathbf{I}-\mathbf{P}\}\widehat{f}(\tau ,k,·)\parallel }_{\nu }^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right).\hfill \end{array}$$

By virtue of the fact that

$$\begin{array}{c}\hfill {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\parallel {\nabla }_v\mathbf{P}\widehat{f}(\tau ,k,·)\parallel }_{L_v^2}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\lesssim {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|(\widehat{a},\widehat{b})(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right),\end{array}$$

and the definition of $\mathcal{D}\left(f\right(t\left)\right)$ given in (15), we can deduce that

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\parallel {\nabla }_v\widehat{f}(\tau ,k,·)\parallel }_{L_v^2}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)& \le {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\parallel {\nabla }_v\{\mathbf{I}-\mathbf{P}\}\widehat{f}(\tau ,k,·)\parallel }_{L_v^2}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & +{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\parallel {\nabla }_v\mathbf{P}\widehat{f}(\tau ,k,·)\parallel }_{L_v^2}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\lesssim \mathcal{D}\left(f\left(t\right)\right).\hfill \end{array}$$

(30)

Similarly, the following result is derived by using analogous steps

$${\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\parallel \widehat{f}(\tau ,k,·)\parallel }_{L_v^2}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\lesssim \mathcal{D}\left(f\left(t\right)\right).$$

(31)

Combining (27), (29), (30) and (31) with the definition of $\mathcal{E}\left(f\right(t\left)\right)$ given in (14), the desired result (25) is achieved.

(ii): Since

$$\begin{array}{cc}\hfill |({\widehat{E}\ast {\nabla }_v\widehat{f},{\mu }^{{\displaystyle \frac{1}{4}}})}_{L_v^2}|& \lesssim {\int }_{{\mathbb{Z}}_l^3}|\widehat{E}(\tau ,k-l)|\parallel {\nabla }_v\widehat{f}{(\tau ,l,·)\parallel }_{L_v^2}d\Sigma \left(l\right),\hfill \end{array}$$

combining (28), (29), (30) and (31), the desired result (26) is followed by the same process as the estimate of (i). □

The following lemma provides the estimate for the term $v\times B·{\nabla }_{v}f$.

Lemma 3. 

It holds that

$$\begin{array}{c}\hfill {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|{\left(v\times \widehat{B}\ast {\nabla }_v\widehat{f},\widehat{f}\right)}_{L_v^2}\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\lesssim \eta \mathcal{D}\left(f\left(t\right)\right)+\mathcal{E}\left(f\left(t\right)\right)\mathcal{D}\left(f\left(t\right)\right),\end{array}$$

(32)

and

$$\begin{array}{c}\hfill {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|{\left(v\times \widehat{B}\ast {\nabla }_v\widehat{f},{\mu }^{{\displaystyle \frac{1}{4}}}\right)}_{L_v^2}\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\lesssim \mathcal{E}\left(f\left(t\right)\right)\mathcal{D}\left(f\left(t\right)\right),\end{array}$$

(33)

where $\eta >0$ is a sufficiently small universal constant.

Proof. 

(i): Following a similar approach as in Lemma 2, we can derive the following bound

$$\left|{\left(v\times \widehat{B}\ast {\nabla }_v\widehat{f},\widehat{f}\right)}_{L_v^2}\right|\lesssim {\int }_{{\mathbb{Z}}_l^3}|\widehat{B}(\tau ,k-l)|\parallel {\nabla }_v\widehat{f}{(\tau ,l,·)\parallel }_{L_v^2}\parallel \left|v\right|\widehat{f}{(\tau ,k,·)\parallel }_{L_v^2}d\Sigma \left(l\right),$$

which leads to the following inequality

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t\left|{\left(v\times \widehat{B}\ast {\nabla }_v\widehat{f},\widehat{f}\right)}_{L_v^2}\right|d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & \lesssim {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\int }_{{\mathbb{Z}}_l^3}|\widehat{B}(\tau ,k-l)|\parallel {\nabla }_v\widehat{f}{(\tau ,l,·)\parallel }_{L_v^2}\parallel \left|v\right|\widehat{f}{(\tau ,k,·)\parallel }_{L_v^2}d\Sigma \left(l\right)d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & \lesssim {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left({\int }_{{\mathbb{Z}}_l^3}|\widehat{E}(\tau ,k-l)|\parallel {\nabla }_v\widehat{f}{(\tau ,l,·)\parallel }_{L_v^2}d\Sigma \left(l\right)\right)}^{2}d\tau \right)}^{1/4}\hfill \\ \hfill & \times {\left({\int }_0^t\parallel \left|v\right|\widehat{f}{(\tau ,k,·)\parallel }_{L_v^2}^{2}d\tau \right)}^{1/4}d\Sigma \left(k\right)\hfill \\ \hfill & \lesssim \eta {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t\parallel \left|v\right|\widehat{f}{(\tau ,k,·)\parallel }_{L_v^2}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{4\eta }}{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left({\int }_{{\mathbb{Z}}_l^3}|\widehat{E}(\tau ,k-l)|\parallel {\nabla }_v\widehat{f}{(\tau ,l,·)\parallel }_{L_v^2}d\Sigma \left(l\right)\right)}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & \lesssim \eta \mathcal{D}\left(f\right(t\left)\right)+\mathcal{E}\left(f\right(t\left)\right)\mathcal{D}\left(f\right(t\left)\right),\hfill \end{array}$$

(34)

the final inequality is justified by the fact that ${\left|v\right|\left|f\right|}^2\le \nu \left(v\right){\left|f\right|}^2$. Additional details are omitted as they are analogous to Lemma 2. □

The next lemma provides an estimate for the term $v·Ef$. The proof follows a similar approach to Lemma 2 and is therefore omitted for the sake of conciseness.

Lemma 4. 

It holds that

$$\begin{array}{c}\hfill {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|{\left(v·\widehat{E}\ast \widehat{f},\widehat{f}\right)}_{L_v^2}\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\lesssim \eta \mathcal{D}\left(f\left(t\right)\right)+\mathcal{E}\left(f\left(t\right)\right)\mathcal{D}\left(f\left(t\right)\right),\end{array}$$

(35)

and

$$\begin{array}{c}\hfill {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|{\left(v·\widehat{E}\ast f,{\mu }^{{\displaystyle \frac{1}{4}}}\right)}_{L_v^2}\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\lesssim \mathcal{E}\left(f\left(t\right)\right)\mathcal{D}\left(f\left(t\right)\right),\end{array}$$

(36)

where $\eta >0$ is a sufficiently small universal constant.

4. Uniform Energy Estimates

4.1. Microscopic Estimates

We begin by deriving an estimate for the microscopic dissipation of the solution f to the system (4).

Lemma 5. 

Let $f(\tau ,x,v)$ be a smooth solution to the Cauchy problem (4) on the time interval [0, t], then it holds that

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{Z}}_k^3}\underset{0\le \tau \le t}{sup}\left(\parallel \widehat{f}{(\tau ,k,·)\parallel }_{L_v^2}+|\widehat{E}(\tau ,k)|+|\widehat{B}(\tau ,k)|\right)d\Sigma \left(k\right)\hfill \\ \hfill & +{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\parallel \{\mathbf{I}-\mathbf{P}\}\widehat{f}(\tau ,k,·)\parallel }_{\nu }^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|\widehat{b}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & \lesssim \parallel {\widehat{f}}_0{\parallel }_{L_k^1{L}_v^2}+\parallel {\widehat{E}}_0{\left(k\right)\parallel }_{L_k^1}+{\parallel {\widehat{B}}_0\left(k\right)\parallel }_{L_k^1}+\eta \mathcal{D}\left(f\left(t\right)\right)+\mathcal{E}\left(f\left(t\right)\right)\mathcal{D}\left(f\left(t\right)\right),\hfill \end{array}$$

(37)

where the constant $\eta >0$ is sufficiently small.

Proof. 

Taking the Fourier transform of Equation (4)₁ with respect to $x\in {\mathbb{T}}^3$, we obtain

$$\begin{array}{cc}\hfill & {\mathsf{\partial }}_{\tau }\widehat{f}(\tau ,k,v)+iv·k\widehat{f}(\tau ,k,v)+[\widehat{E}\ast {\nabla }_v\widehat{f}](\tau ,k,v)+[v\times \widehat{B}\ast {\nabla }_v\widehat{f}](\tau ,k,v)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}[v·\widehat{E}\ast \widehat{f}](\tau ,k,v)-\widehat{E}(\tau ,k)·v{\mu }^{{\displaystyle \frac{1}{2}}}=L_{FP}\widehat{f}(\tau ,k,v),\hfill \end{array}$$

(38)

where the convolutions are taken with respect to $k\in {\mathbb{Z}}^3$:

$$\begin{array}{ccc}\hfill [\widehat{E}\ast {\nabla }_v\widehat{f}](\tau ,k,v)& =& {\int }_{{\mathbb{Z}}_l^3}\widehat{E}(\tau ,k-l){\nabla }_v\widehat{f}(\tau ,l,v)d\Sigma \left(l\right),\hfill \\ \hfill [v\times \widehat{B}\ast {\nabla }_v\widehat{f}](\tau ,k,v)& =& {\int }_{{\mathbb{Z}}_l^3}v\times \widehat{B}(\tau ,k-l){\nabla }_v\widehat{f}(\tau ,l,v)d\Sigma \left(l\right),\hfill \\ \hfill [v·\widehat{E}\ast \widehat{f}](\tau ,k,v)& =& {\int }_{{\mathbb{Z}}_l^3}v·\widehat{E}(\tau ,k-l)\widehat{f}(\tau ,l,v)d\Sigma \left(l\right).\hfill \end{array}$$

Next, taking the Fourier transform with $x\in {\mathbb{T}}^3$ for the Maxwell equations yields

$$\left\{\begin{array}{cc}\hfill & {\mathsf{\partial }}_{\tau }\widehat{E}-ik\times \widehat{B}=-\widehat{b},\hfill \\ \hfill & {\mathsf{\partial }}_{\tau }\widehat{B}+ik\times \widehat{E}=0,\hfill \\ \hfill & ik·\widehat{E}=\widehat{a},ik·\widehat{B}=0.\hfill \end{array}\right.$$

(39)

Due to (8) and (39), it follows that

$$\begin{array}{cc}\hfill {(-\widehat{E}·v{\mu }^{{\displaystyle \frac{1}{2}}},\widehat{f})}_{L_v^2}& =-\widehat{E}·\overline{\widehat{b}}=\widehat{E}({\mathsf{\partial }}_{\tau }\overline{\widehat{E}}-\overline{ik\times \widehat{B}})=\widehat{E}{\mathsf{\partial }}_{\tau }\overline{\widehat{E}}+\widehat{E}·ik\times \overline{\widehat{B}}\hfill \\ \hfill & =\widehat{E}{\mathsf{\partial }}_{\tau }\overline{\widehat{E}}-\overline{\widehat{B}}·(ik\times \widehat{E})=\widehat{E}{\mathsf{\partial }}_{\tau }\overline{\widehat{E}}+\widehat{B}{\mathsf{\partial }}_{\tau }\overline{\widehat{B}}={\displaystyle \frac{1}{2}}{\mathsf{\partial }}_{\tau }\left(\right|\widehat{E}{|}^2+|\widehat{B}{|}^2).\hfill \end{array}$$

Multiplying Equation (38) by the complex conjugate of $\widehat{f}(\tau ,k,v)$ and integrating over v, further taking the real part of this identity and integrating with respect to $\tau$, yields the following result

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}\left(\parallel \widehat{f}{(\tau ,k,·)\parallel }_{L_v^2}^2+|\widehat{E}{(\tau ,k)|}^2+{\left|\widehat{B}(\tau ,k)\right|}^2\right)-{\int }_0^{\tau }\mathcal{R}{(L_{FP}\widehat{f},\widehat{f})}_{L_v^2}d\tau \hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\left(\parallel {\widehat{f}}_0{(k,·)\parallel }_{L_v^2}^2+|{\widehat{E}}_0{\left(k\right)|}^2+{\left|{\widehat{B}}_0\left(k\right)\right|}^2\right)-{\int }_0^{\tau }\mathcal{R}{(\widehat{E}\ast {\nabla }_v\widehat{f},\widehat{f})}_{L_v^2}d\tau \hfill \\ \hfill & -{\int }_0^{\tau }\mathcal{R}{(v\times \widehat{B}\ast {\nabla }_v\widehat{f},\widehat{f})}_{L_v^2}d\tau +{\int }_0^{\tau }\mathcal{R}{({\displaystyle \frac{1}{2}}v·\widehat{E}\ast \widehat{f},\widehat{f})}_{L_v^2}d\tau ,\hfill \end{array}$$

(40)

where ${\widehat{f}}_0(k,·)=\widehat{f}(0,k,·)$, ${\widehat{E}}_0\left(k\right)=\widehat{E}(0,k)$, ${\widehat{B}}_0\left(k\right)=\widehat{B}(0,k)$. Utilizing the coercivity estimate of $L_{FP}$ (24) and taking the square root of both sides, one can obtain that

$$\begin{array}{cc}\hfill & \parallel \widehat{f}{(\tau ,k,·)\parallel }_{L_v^2}+|\widehat{E}(\tau ,k)|+|\widehat{B}(\tau ,k)|+{\left({\int }_0^{\tau }{\parallel \{\mathbf{I}-\mathbf{P}\}\widehat{f}(\tau ,k,·)\parallel }_{\nu }^{2}d\tau \right)}^{1/2}\hfill \\ \hfill & +{\left({\int }_0^{\tau }{\left|\widehat{b}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}\hfill \\ \hfill & \lesssim \parallel {\widehat{f}}_0{(k,·)\parallel }_{L_v^2}+|{\widehat{E}}_0\left(k\right)|+|{\widehat{B}}_0\left(k\right)|+{\left({\int }_0^{\tau }\left|\mathcal{R}{(\widehat{E}\ast {\nabla }_v\widehat{f},\widehat{f})}_{L_v^2}\right|d\tau \right)}^{1/2}\hfill \\ \hfill & +{\left({\int }_0^{\tau }\left|\mathcal{R}{(v\times \widehat{B}\ast {\nabla }_v\widehat{f},\widehat{f})}_{L_v^2}\right|d\tau \right)}^{1/2}+{\left({\int }_0^{\tau }\left|\mathcal{R}{({\displaystyle \frac{1}{2}}v·\widehat{E}\ast \widehat{f},\widehat{f})}_{L_v^2}\right|d\tau \right)}^{1/2}.\hfill \end{array}$$

(41)

Furthermore, by taking the supremum over $0\le \tau \le t$ on both sides of (41) and integrating the resulting inequality with respect to $d\Sigma \left(k\right)$ over ${\mathbb{Z}}^3$, one can derive that

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{Z}}_k^3}\underset{0\le \tau \le t}{sup}\left(\parallel \widehat{f}{(\tau ,k,·)\parallel }_{L_v^2}+|\widehat{E}(\tau ,k)|+|\widehat{B}(\tau ,k)|\right)d\Sigma \left(k\right)\hfill \\ \hfill & +{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\parallel \{\mathbf{I}-\mathbf{P}\}\widehat{f}(\tau ,k,·)\parallel }_{\nu }^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|\widehat{b}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & \lesssim \parallel {\widehat{f}}_0{\parallel }_{L_k^1{L}_v^2}+\parallel {\widehat{E}}_0\left(k\right)+{\parallel {\widehat{B}}_0\left(k\right)\parallel }_{L_k^1}+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t\left|{\left(\widehat{E}\ast {\nabla }_v\widehat{f},\widehat{f}\right)}_{L_v^2}\right|d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & +{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t\left|{\left(v\times \widehat{B}\ast {\nabla }_v\widehat{f},\widehat{f}\right)}_{L_v^2}\right|d\tau \right)}^{1/2}d\Sigma \left(k\right)+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t\left|{\left({\displaystyle \frac{1}{2}}v·\widehat{E}\ast \widehat{f},\widehat{f}\right)}_{L_v^2}\right|d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & \lesssim \parallel {\widehat{f}}_0{\parallel }_{L_k^1{L}_v^2}+\parallel {\widehat{E}}_0{\left(k\right)\parallel }_{L_k^1}+{\parallel {\widehat{B}}_0\left(k\right)\parallel }_{L_k^1}+\eta \mathcal{D}\left(f\left(t\right)\right)+\mathcal{E}\left(f\left(t\right)\right)\mathcal{D}\left(f\left(t\right)\right),\hfill \end{array}$$

where Lemmas 2 and 3 are used. Thus, the proof is completed. □

4.2. Macroscopic Estimates

In this section, we establish uniform estimates for the macroscopic component $\widehat{a}(\tau ,k)$ by adopting a strategy similar to that in [19], utilizing a duality argument.

Theorem 3. 

Under the hypotheses of Theorem 1, we have that

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|\widehat{a}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|\widehat{E}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & \lesssim \mathcal{E}\left(f\left(t\right)\right)+\parallel {\widehat{f}}_0{\parallel }_{L_k^1{L}_v^2}+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|\widehat{b}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & +{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\parallel \{\mathbf{I}-\mathbf{P}\}\widehat{f}(\tau ,k,·)\parallel }_{\nu }^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)+\mathcal{E}\left(f\left(t\right)\right)\mathcal{D}\left(f\left(t\right)\right).\hfill \end{array}$$

(42)

Proof. 

To derive the estimate for a, we choose a test function

$$\widehat{\Phi }(\tau ,k,v)\in C^1((0,\infty )\times {\mathbb{Z}}^3\times {\mathbb{R}}^3).$$

Applying the Fourier transform to (4), taking the $L_v^2$ inner product of the resulting equation with $\widehat{\Phi }$, and integrating the resultant over the interval $[0,t]$, we can obtain that

$$\begin{array}{cc}\hfill & {(\widehat{f},\widehat{\Phi })}_{L_v^2}{{|}_{\tau =t}-{(\widehat{f},\widehat{\Phi })}_{L_v^2}|}_{\tau =0}-{\int }_0^t{(\widehat{f},{\mathsf{\partial }}_{\tau }\widehat{\Phi })}_{L_v^2}d\tau -{\int }_0^t{(\widehat{f},v·ik\widehat{\Phi })}_{L_v^2}d\tau -{\int }_0^t{(\widehat{E},v·{\mu }^{{\displaystyle \frac{1}{2}}}\widehat{\Phi })}_{L_v^2}d\tau \hfill \\ \hfill & ={\int }_0^t{(L_{FP}\widehat{f},\widehat{\Phi })}_{L_v^2}d\tau +{\int }_0^t{(\widehat{H},\widehat{\Phi })}_{L_v^2}d\tau ,\hfill \end{array}$$

where

$$H=-(E+v\times B)·{\nabla }_{v}f+{\displaystyle \frac{1}{2}}E·vf,$$

(43)

is the nonlinear terms. The notations ${(\widehat{f},\widehat{\Phi })}_{L_v^2}{|}_{\tau =t}={(\widehat{f},\widehat{\Phi })}_{L_v^2}\left(t\right)$ and ${(\widehat{f},\widehat{\Phi })}_{L_v^2}{|}_{\tau =0}={(\widehat{f},\widehat{\Phi })}_{L_v^2}\left(0\right)$ are employed. By employing the macro-micro decomposition, we derive that

$$\begin{array}{cc}\hfill {\int }_0^t{(\mathbf{P}\widehat{f},v·ik\widehat{\Phi })}_{L_v^2}d\tau & =\underset{J_1}{\underbrace{(\widehat{f},\widehat{\Phi })\left(t\right)-{(\widehat{f},\widehat{\Phi })}_{L_v^2}\left(0\right)-{\int }_0^t{(\widehat{f},{\mathsf{\partial }}_t\widehat{\Phi })}_{L_v^2}d\tau }}\underset{J_2}{\underbrace{-{\int }_0^t{(\widehat{E},v·{\mu }^{{\displaystyle \frac{1}{2}}}\widehat{\Phi })}_{L_v^2}d\tau }}\hfill \\ \hfill & \underset{J_3}{\underbrace{-{\int }_0^t{(\{\mathbf{I}-\mathbf{P}\}\widehat{f},v·ik\widehat{\Phi })}_{L_v^2}d\tau }}\underset{J_4}{\underbrace{-{\int }_0^t{(L_{FP}\widehat{f},\widehat{\Phi })}_{L_v^2}d\tau }}\underset{J_5}{\underbrace{-{\int }_0^t{(\widehat{H},\widehat{\Phi })}_{L_v^2}d\tau }}.\hfill \end{array}$$

Thanks to (19), we deduce the conservation law for mass, namely,

$${\int }_{{\mathbb{T}}^3}a(\tau ,x)dx={\int }_{{\mathbb{T}}^3}a(0,x)dx=0,$$

which implies that $\widehat{a}(\tau ,0)=0$. We now choose the test function as

$$\widehat{\Phi }(\tau ,k,v)={\left(\right|v|}^2-10)v·ik{\widehat{\varphi }}_a(\tau ,k){\mu }^{{\displaystyle \frac{1}{2}}},$$

(44)

where ${\widehat{\varphi }}_a(\tau ,k)$ is a solution to

$${\left|k\right|}^2{\widehat{\varphi }}_a(\tau ,k)=\widehat{a}(\tau ,k).$$

Given that $\widehat{a}(\tau ,0)=0$, the function ${\widehat{\varphi }}_a(\tau ,k)$ can be formally expressed as ${\widehat{\varphi }}_a(\tau ,k)=\widehat{a}(\tau ,k)/{\left|k\right|}^2$ for any $k\in {\mathbb{Z}}^3$, with the understanding that ${\widehat{\varphi }}_a(\tau ,0)=0$. Combining this representation with the estimate for a provided in [19], we obtain that

$${\int }_0^t{(\mathbf{P}\widehat{f},v·ik\widehat{\Phi })}_{L_v^2}d\tau =5{\int }_0^t{\left|\widehat{a}(\tau ,k)\right|}^{2}d\tau ,$$

and

$$\begin{array}{cc}\hfill & |J_1|\lesssim \parallel \widehat{f}{(t,k,·)\parallel }_{L_v^2}^2+\parallel {\widehat{f}}_0{\left(k\right)\parallel }_{L_v^2}^2+{\int }_0^t|\widehat{b}{(\tau ,k)|}^{2}d\tau +{\int }_0^t{|\{\mathbf{I}-\mathbf{P}\}\widehat{f}(\tau ,k)|}_{\nu }^{2}d\tau ,\hfill \\ \hfill & |J_3|\lesssim \eta {\int }_0^t|\widehat{a}{(\tau ,k)|}^{2}d\tau +C_{\eta }{\int }_0^t{|\{\mathbf{I}-\mathbf{P}\}\widehat{f}(\tau ,k)|}_{\nu }^{2}d\tau ,\hfill \end{array}$$

where the constant $\eta >0$ is sufficiently small.

For the estimation of $J_2$, we have the following

$$\begin{array}{cc}\hfill J_2& =-{\int }_0^t{(\widehat{E},v·{\mu }^{{\displaystyle \frac{1}{2}}}\widehat{\Phi })}_{L_v^2}d\tau =-{\int }_0^t{\left(\widehat{E},v·{\left(\right|v|}^2-10)v·ik{\widehat{\varphi }}_a(\tau ,k)\mu \right)}_{L_v^2}d\tau \hfill \\ \hfill & =5{\int }_0^t(\widehat{E},ik\widehat{{\varphi }_a})d\tau =5{\int }_0^t(\widehat{E},ik{\displaystyle \frac{\widehat{a}}{{\left|k\right|}^2}})d\tau =-5{\int }_0^t(\widehat{E},\widehat{E})d\tau =-5{\int }_0^t{\left|\widehat{E}(\tau ,k)\right|}^{2}d\tau ,\hfill \end{array}$$

where we have used ${\left|k\right|}^2{\widehat{\varphi }}_a(\tau ,k)=\widehat{a}(\tau ,k)$ and $ik·\widehat{E}=\widehat{a}$ in (39).

For the estimate of $J_4$, we can get

$$\begin{array}{cc}\hfill |J_4|& =\left|{\int }_0^t{(L_{FP}\widehat{f},\widehat{\Phi })}_{L_v^2}d\tau \right|=|{\int }_0^t({\Delta }_v\widehat{f}+{\displaystyle \frac{1}{4}}{(6-|v|}^2{)\widehat{f},\widehat{\Phi })}_{L_v^2}d\tau |\hfill \\ \hfill & \le \underset{J_{4,1}}{\underbrace{\left|{\int }_0^t{({\Delta }_v\widehat{f},\widehat{\Phi })}_{L_v^2}d\tau \right|}}+\underset{J_{4,2}}{\underbrace{\left|{\int }_0^t({\displaystyle \frac{1}{4}}{(6-|v|}^2{)\widehat{f},\widehat{\Phi })}_{L_v^2}d\tau \right|}},\hfill \end{array}$$

by using (6). By applying the macro-micro decomposition from (9) and (7), we obtain that

$$\begin{array}{cc}\hfill J_{4,1}& \le \left|{\int }_0^t{({\Delta }_v\mathbf{P}\widehat{f},\widehat{\Phi })}_{L_v^2}d\tau \right|+\left|{\int }_0^t{({\Delta }_v\{\mathbf{I}-\mathbf{P}\}\widehat{f},\widehat{\Phi })}_{L_v^2}d\tau \right|\hfill \\ \hfill & \le \underset{J_{4,1}^1}{\underbrace{\left|{\int }_0^t{(\widehat{a}{\Delta }_v{\mu }^{{\displaystyle \frac{1}{2}}},\widehat{\Phi })}_{L_v^2}d\tau \right|}}+\underset{J_{4,1}^2}{\underbrace{\left|{\int }_0^t{(\widehat{b}{\Delta }_v(v{\mu }^{{\displaystyle \frac{1}{2}}}),\widehat{\Phi })}_{L_v^2}d\tau \right|}}+\underset{J_{4,1}^3}{\underbrace{\left|{\int }_0^t{({\Delta }_v\{\mathbf{I}-\mathbf{P}\}\widehat{f},\widehat{\Phi })}_{L_v^2}d\tau \right|}}.\hfill \end{array}$$

Since ${\Delta }_v{\mu }^{{\displaystyle \frac{1}{2}}}=({\displaystyle \frac{1}{4}}{v}^2-{\displaystyle \frac{3}{2}}){\mu }^{{\displaystyle \frac{1}{2}}}$ and (44), we deduce that

$$\begin{array}{cc}\hfill J_{4,1}^1& =\left|{\int }_0^t{\left(\widehat{a}{\Delta }_v{\mu }^{{\displaystyle \frac{1}{2}}}{,\left(\right|v|}^2-10)v·ik{\widehat{\varphi }}_a(\tau ,k){\mu }^{{\displaystyle \frac{1}{2}}}\right)}_{L_v^2}d\tau \right|\hfill \\ \hfill & =\left|{\int }_0^t{\left(\widehat{a}({\displaystyle \frac{1}{4}}{v}^2-{\displaystyle \frac{3}{2}}){\mu }^{{\displaystyle \frac{1}{2}}}{,\left(\right|v|}^2-10)v·ik{\widehat{\varphi }}_a(\tau ,k){\mu }^{{\displaystyle \frac{1}{2}}}\right)}_{L_v^2}d\tau \right|=0,\hfill \end{array}$$

since the integrand is an odd function with respect to v. Considering $k\in {\mathbb{Z}}^3$ and ${\left|k\right|}^2{\widehat{\varphi }}_a(\tau ,k)=\widehat{a}(\tau ,k)$, it follows that

$$\begin{array}{c}\hfill J_{4,1}^2\lesssim {\int }_0^t|\widehat{b}(\tau ,k)\left|\right|k\left|\right|{\widehat{\varphi }}_a(\tau ,k)|d\tau \lesssim \eta {\int }_0^t|\widehat{a}{(\tau ,k)|}^{2}d\tau +C_{\eta }{\int }_0^t{\left|\widehat{b}(\tau ,k)\right|}^{2}d\tau ,\end{array}$$

by applying Young’s inequality. Similarly, one can establish that

$$\begin{array}{cc}\hfill J_{4,1}^3& =\left|{\int }_0^t{(\{\mathbf{I}-\mathbf{P}\}\widehat{f},{\Delta }_v\widehat{\Phi })}_{L_v^2}d\tau \right|d\tau \hfill \\ \hfill & \lesssim \eta {\int }_0^t|\widehat{a}{(\tau ,k)|}^{2}d\tau +C_{\eta }{\int }_0^t{|\{\mathbf{I}-\mathbf{P}\}\widehat{f}(\tau ,k)|}_{\nu }^{2}d\tau .\hfill \end{array}$$

Regarding the estimate of $J_{4,2}$, we can demonstrate that

$$\begin{array}{c}\hfill J_{4,2}\lesssim \eta {\int }_0^t|\widehat{a}{(\tau ,k)|}^{2}d\tau +C_{\eta }{\int }_0^t|\widehat{b}{(\tau ,k)|}^{2}d\tau +C_{\eta }{\int }_0^t{\left|\{\mathbf{I}-\mathbf{P}\}\widehat{f}(\tau ,k)\right|}_{\nu }^{2}d\tau .\end{array}$$

For the term $J_5$, applying Young’s inequality yields

$$\begin{array}{cc}\hfill |J_5|& ={\int }_0^t{(\widehat{H}{,\left(\right|v|}^2-10)v·ik{\widehat{\varphi }}_a(\tau ,k){\mu }^{{\displaystyle \frac{1}{2}}})}_{L_v^2}d\tau \hfill \\ \hfill & \lesssim \eta {\int }_0^t|\widehat{a}{(\tau ,k)|}^{2}d\tau +C_{\eta }{\int }_0^t\left|\right(|\widehat{H}(\tau ,k)|,{\mu }^{{\displaystyle \frac{1}{4}}}{{)}_{L_v^2}|}^{2}d\tau .\hfill \end{array}$$

Collecting the above estimates, we obtain that

$$\begin{array}{cc}\hfill {\int }_0^t|\widehat{a}{(\tau ,k)|}^{2}d\tau +{\int }_0^t{|\widehat{E}(\tau ,k)|}^{2}d\tau & \lesssim \parallel \widehat{f}{(k,t)\parallel }_{L_v^2}^2+\parallel {\widehat{f}}_0{\left(k\right)\parallel }_{L_v^2}^2+{\int }_0^t{|\widehat{b}(\tau ,k)|}^{2}d\tau \hfill \\ \hfill & +{\int }_0^t|\{\mathbf{I}-\mathbf{P}\}\widehat{f}{(\tau ,k)|}_{\nu }^{2}d\tau +{\int }_0^t\left|\right(|\widehat{H}(\tau ,k)|,{\mu }^{{\displaystyle \frac{1}{4}}}{\left)\right|}^{2}d\tau .\hfill \end{array}$$

Furthermore, by (43), Lemmas 2–4, it holds that

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|\widehat{a}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|\widehat{E}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & \lesssim \mathcal{E}\left(f\left(t\right)\right)+{\int }_{{\mathbb{Z}}_k^3}{\parallel {\widehat{f}}_0\left(k\right)\parallel }_{L_v^2}d\Sigma \left(k\right)+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|\widehat{b}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & +{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\parallel \{\mathbf{I}-\mathbf{P}\}\widehat{f}(\tau ,k,·)\parallel }_{\nu }^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)+\mathcal{E}\left(f\left(t\right)\right)\mathcal{D}\left(f\left(t\right)\right),\hfill \end{array}$$

where

$${\int }_{{\mathbb{Z}}_k^3}\parallel \widehat{f}{(t,k,·)\parallel }_{L_v^2}d\Sigma \left(k\right)\le {\int }_{{\mathbb{Z}}_k^3}\underset{0\le \tau \le t}{sup}{\parallel \widehat{f}(\tau ,k,·)\parallel }_{L_v^2}d\Sigma \left(k\right)\le \mathcal{E}\left(f\left(t\right)\right),$$

is used. Thus this completes the proof the theorem. □

Remark 3. 

It is important to highlight that the symmetry of the periodic domain has been utilized $\left(\right|k|\ge 1)$ for the estimate of $J_1$,i.e.,

$$\begin{array}{cc}\hfill |(\widehat{f},\widehat{\Phi })\left(t\right)|& =|{\int }_{{\mathbb{R}}_v^3}\widehat{f}{(t,k,v)\left(\right|v|}^2-10)v·ik{\widehat{\varphi }}_a(t,k){\mu }^{{\displaystyle \frac{1}{2}}}dv|\hfill \\ \hfill & \lesssim \left|k\right||{\widehat{\varphi }}_a(t,k)\left|\right({\int }_{{\mathbb{R}}_v^3}|\widehat{f}{|}^2{dv)}^{{\displaystyle \frac{1}{2}}}\le {\displaystyle \frac{|\widehat{a}(t,k)|}{\left|k\right|}}({\int }_{{\mathbb{R}}_v^3}|\widehat{f}{{|}^{2}dv)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le {\displaystyle \frac{|\widehat{a}(t,k)|}{\left|k\right|}}\left|k\right|({\int }_{{\mathbb{R}}_v^3}|\widehat{f}{|}^2{dv)}^{{\displaystyle \frac{1}{2}}}=|\widehat{a}(t,k)\left|\right({\int }_{{\mathbb{R}}_v^3}|\widehat{f}{{|}^{2}dv)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \lesssim ({\int }_{{\mathbb{R}}_v^3}|\widehat{f}{|}^2{dv)}^{{\displaystyle \frac{1}{2}}}({\int }_{{\mathbb{R}}_v^3}|\widehat{f}{|}^2{dv)}^{{\displaystyle \frac{1}{2}}}={\parallel \widehat{f}(t,k,·)\parallel }_{L_v^2}^2.\hfill \end{array}$$

If $\left(\right|k|\le 1)$, then the following inequality is not valid

$${\displaystyle \frac{|\widehat{a}(t,k)|}{\left|k\right|}}({\int }_{{\mathbb{R}}_v^3}|\widehat{f}{|}^2{dv)}^{{\displaystyle \frac{1}{2}}}\le {\displaystyle \frac{|\widehat{a}(t,k)|}{\left|k\right|}}\left|k\right|({\int }_{{\mathbb{R}}_v^3}|\widehat{f}{{|}^{2}dv)}^{{\displaystyle \frac{1}{2}}}.$$

4.3. Proof of Theorem 1

Suppose the Cauchy problem (4) and (5) has a unique local solution $f(t,x,v)$ on the time interval $0\le t\le T$ for some $0<T<\infty$, and the solution $f(t,x,v)$ satisfies the a priori bound

$$\underset{0\le t\le T}{sup}\mathcal{E}\left(f\left(t\right)\right)\le \delta ,$$

(45)

where the constant $\delta >0$ is sufficiently small. Using a continuation argument, we can iteratively extend this solution to a global one. Therefore, it suffices to establish uniform-in-time energy estimates for $f(t,x,v)$ under the a priori assumption (45).

Lemma 6. 

There exists a constant $\delta >0$ such that if

$$\underset{0\le t\le T}{sup}\mathcal{E}\left(f\left(t\right)\right)\le \delta ,$$

for any $0<T<\infty$, then it follows that

$$\mathcal{E}\left(f\left(t\right)\right)+\mathcal{D}\left(f\left(t\right)\right)\le \mathcal{E}\left(f_0\right).$$

Proof. 

By selecting $\lambda >0$ to be sufficiently small and adding $\lambda \times$ (42) to (37), we can get that

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{Z}}_k^3}\underset{0\le \tau \le t}{sup}\left(\parallel \widehat{f}{(\tau ,k,·)\parallel }_{L_v^2}+|\widehat{E}(\tau ,k)|+|\widehat{B}(\tau ,k)|\right)d\Sigma \left(k\right)\hfill \\ \hfill & +{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\parallel \{\mathbf{I}-\mathbf{P}\}\widehat{f}(\tau ,k,·)\parallel }_{\nu }^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|\widehat{b}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & +\lambda {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|\widehat{a}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)+\lambda {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|\widehat{E}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & \lesssim \parallel {\widehat{f}}_0{\parallel }_{L_k^1{L}_v^2}+\parallel {\widehat{E}}_0{\left(k\right)\parallel }_{L_k^1}+{\parallel {\widehat{B}}_0\left(k\right)\parallel }_{L_k^1}+\eta \mathcal{D}\left(f\left(t\right)\right)+\mathcal{E}\left(f\left(t\right)\right)\mathcal{D}\left(f\left(t\right)\right).\hfill \end{array}$$

By selecting $\eta >0$ sufficiently small and utilizing the definitions of $\mathcal{E}\left(f\right(t\left)\right)$ in (14) and $\mathcal{D}\left(f\right(t\left)\right)$ in (15) yields

$$\mathcal{E}\left(f\left(t\right)\right)+\mathcal{D}\left(f\left(t\right)\right)\lesssim \mathcal{E}\left(f_0\right)+\mathcal{E}\left(f\left(t\right)\right)\mathcal{D}\left(f\left(t\right)\right).$$

By taking $\delta >0$ to be appropriately small, we ensure that

$$\mathcal{E}\left(f\left(t\right)\right)+\mathcal{D}\left(f\left(t\right)\right)\le \mathcal{E}\left(f_0\right),$$

which completes the proof of the lemma. □

The existence and uniqueness of local-in-time solution to the Cauchy problem (4) can be constructed based on the energy functional $\mathcal{E}\left(f\right(t\left)\right)$ in (14). The details which are similar to those in reference [19] are omitted for brevity.

Given the smallness assumption on $\mathcal{E}\left(f_0\right)$, combined with Lemma 6, we have that

$$\mathcal{E}\left(f\left(t\right)\right)+\mathcal{D}\left(f\left(t\right)\right)\lesssim \mathcal{E}\left(f_0\right).$$

By combining this result with the local-in-time existence, the global mild solution and its uniqueness follow directly from a standard continuity argument. This concludes the proof of global existence and the uniform estimate (20), thereby establishing Theorem 1.

4.4. Proof of Theorem 2

In order to get the exponential time decay, we require the high regularity of spatial variable, which includes high-order dissipation of B. Taking the $L_v^2$ inner product of the Fourier transform (4)₁ with ${⟨k⟩}^2\widehat{f}$, we have that

$$\begin{array}{cc}\hfill & ({\mathsf{\partial }}_t\widehat{f},{⟨k⟩}^2\widehat{f})+(iv·k\widehat{f},{⟨k⟩}^2\widehat{f})+(\widehat{E}\ast {\nabla }_v\widehat{f},{⟨k⟩}^2\widehat{f})+(v\times \widehat{B}\ast {\nabla }_v\widehat{f},{⟨k⟩}^2\widehat{f})\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}(v·\widehat{E}\ast \widehat{f},{⟨k⟩}^2\widehat{f})-(\widehat{E}·v{\mu }^{{\displaystyle \frac{1}{2}}},{⟨k⟩}^2\widehat{f})=(L_{FP}\widehat{f},{⟨k⟩}^2\widehat{f}).\hfill \end{array}$$

Using arguments similar to those that led to (37) and (42), we have that

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{Z}}_k^3}\underset{0\le \tau \le t}{sup}\left(\parallel ⟨k⟩\widehat{f}{(\tau ,k,·)\parallel }_{L_v^2}+|⟨k⟩\widehat{E}(\tau ,k)|+|⟨k⟩\widehat{B}(\tau ,k)|\right)d\Sigma \left(k\right)\hfill \\ \hfill & +{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\parallel ⟨k⟩\{\mathbf{I}-\mathbf{P}\}\widehat{f}(\tau ,k,·)\parallel }_{\nu }^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|⟨k⟩\widehat{b}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & \lesssim \parallel ⟨k⟩{\widehat{f}}_0{\parallel }_{L_k^1{L}_v^2}+\parallel ⟨k⟩{\widehat{E}}_0{\left(k\right)\parallel }_{L_k^1}+{\parallel ⟨k⟩{\widehat{B}}_0\left(k\right)\parallel }_{L_k^1}+\eta {\mathcal{D}}_1\left(f\left(t\right)\right)+{\mathcal{E}}_1\left(f\left(t\right)\right){\mathcal{D}}_1\left(f\left(t\right)\right),\hfill \end{array}$$

(46)

and

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|⟨k⟩\widehat{a}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|⟨k⟩\widehat{E}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & \lesssim {\mathcal{E}}_1\left(f\left(t\right)\right)+{\parallel ⟨k⟩{\widehat{f}}_0\parallel }_{L_k^1{L}_v^2}+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|⟨k⟩\widehat{b}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & +{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\parallel ⟨k⟩\{\mathbf{I}-\mathbf{P}\}\widehat{f}(\tau ,k,·)\parallel }_{\nu }^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)+{\mathcal{E}}_1\left(f\left(t\right)\right){\mathcal{D}}_1\left(f\left(t\right)\right),\hfill \end{array}$$

(47)

where the constant $\eta >0$ is sufficiently small. We now derive estimate for the dissipation of B from the Maxwell Equation (39). From ${\mathsf{\partial }}_{\tau }\widehat{E}-ik\times \widehat{B}=-\widehat{b}$ in (39)₁, we have that

$$\begin{array}{cc}\hfill |k\times \widehat{B}{|}^2& =(ik\times \widehat{B},ik\times \widehat{B})=(ik\times \widehat{B},{\mathsf{\partial }}_{\tau }\widehat{E}+\widehat{b})\hfill \\ \hfill & ={\mathsf{\partial }}_{\tau }(ik\times \widehat{B},\widehat{E})-(ik\times {\mathsf{\partial }}_{\tau }\widehat{B},\widehat{E})+(ik\times \widehat{B},\widehat{b})\hfill \\ \hfill & ={\mathsf{\partial }}_{\tau }(ik\times \widehat{B},\widehat{E})+{|k\times \widehat{E}|}^2+(ik\times \widehat{B},\widehat{b}),\hfill \end{array}$$

(48)

where $-(ik\times {\mathsf{\partial }}_{\tau }\widehat{B},\widehat{E})=-(k\times (k\times \widehat{E}),\widehat{E})={|k\times \widehat{E}|}^2$. Noticing that ${|k|}^2|\widehat{B}{|}^2={|k\times \widehat{B}|}^2$ due to $k·\widehat{B}=0$, we integrate Equation (48) over the interval $[0,t]$ with respect to $\tau$ to derive that

$$\begin{array}{cc}\hfill {\int }_0^t{\left|k\right|}^2{\left|\widehat{B}(\tau ,k)\right|}^{2}d\tau & \le (ik\times \widehat{B},\widehat{E})(t,k)+(ik\times \widehat{B},\widehat{E})(0,k)\hfill \\ \hfill & +{\int }_0^t{\left|k\right|}^2|\widehat{E}{(\tau ,k)|}^{2}d\tau +\eta {\int }_0^t{\left|k\right|}^2|\widehat{B}{(\tau ,k)|}^{2}d\tau +{\displaystyle \frac{1}{4\eta }}{\int }_0^t{\left|\widehat{b}(\tau ,k)\right|}^{2}d\tau \hfill \\ \hfill & \le {\left|k\right|}^2|\widehat{B}{(t,k)|}^2+|\widehat{E}{(t,k)|}^2+{\left|k\right|}^2|\widehat{B}{(0,k)|}^2+{\left|\widehat{E}(0,k)\right|}^2\hfill \\ \hfill & +{\int }_0^t{\left|k\right|}^2|\widehat{E}{(\tau ,k)|}^{2}d\tau +\eta {\int }_0^t{\left|k\right|}^2|\widehat{B}{(\tau ,k)|}^{2}d\tau +{\displaystyle \frac{1}{4\eta }}{\int }_0^t{\left|\widehat{b}(\tau ,k)\right|}^{2}d\tau ,\hfill \end{array}$$

where the Young’s inequality is applied. Consequently, we have that

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|k\right|}^2{|\widehat{B}(\tau ,k)|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)& \lesssim {\mathcal{E}}_1\left(f\left(t\right)\right)+{\mathcal{E}}_1\left(f_0\right)+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|k\right|}^2{\left|\widehat{E}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & +{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|\widehat{b}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right),\hfill \end{array}$$

(49)

for a sufficiently small constant $\eta >0$. Moreover, given that $\widehat{B}(t,k)={\int }_{{\mathbb{T}}_x^3}{e}^{-ik·x}B(t,x)dx,k\in {\mathbb{Z}}^3$ and the assumption ${\int }_{{\mathbb{T}}_x^3}B(t,x)dx=0$ from Theorem 2, we have that $\widehat{B}(t,0)=0$ and the following estiamte

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{⟨k⟩}^2{\left|\widehat{B}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)& ={\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{(1+|k|}^2\left)\right|\widehat{B}{(\tau ,k)|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & ={\int }_{\{k\in {\mathbb{Z}}^3|\left|k\right|=0\}}{\left({\int }_0^t{(1+|k|}^2\left)\right|\widehat{B}{(\tau ,k)|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & +{\int }_{\{k\in {\mathbb{Z}}^3|\left|k\right|\ge 1\}}{\left({\int }_0^t{(1+|k|}^2\left)\right|\widehat{B}{(\tau ,k)|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & \le {\int }_0^t{\left|\widehat{B}(\tau ,0)\right|}^{2}d\tau +\sqrt{2}{\int }_{\{k\in {\mathbb{Z}}^3|\left|k\right|\ge 1\}}{\left({\int }_0^t{\left|k\right|}^2{\left|\widehat{B}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & \le \sqrt{2}{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|k\right|}^2{\left|\widehat{B}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right).\hfill \end{array}$$

By combining the aforementioned inequality with the estimate of (49), one can obtain that

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{⟨k⟩}^2{\left|\widehat{B}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)& \lesssim {\mathcal{E}}_1\left(f\left(t\right)\right)+{\mathcal{E}}_1\left(f_0\right)+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|k\right|}^2{\left|\widehat{E}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & +{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{\left|\widehat{b}(\tau ,k)\right|}^{2}d\tau \right)}^{1/2}d\Sigma \left(k\right).\hfill \end{array}$$

(50)

Taking an appropriate linear combination of inequalities (46), (47) and (50), we obtain that

$${\mathcal{E}}_1\left(f\left(t\right)\right)+{\mathcal{D}}_1\left(f\left(t\right)\right)\lesssim {\mathcal{E}}_1\left(f_0\right)+{\mathcal{E}}_1\left(f\left(t\right)\right){\mathcal{D}}_1\left(f\left(t\right)\right).$$

Given the smallness assumption on ${\mathcal{E}}_1\left(f_0\right)$, we can deduce that

$${\mathcal{E}}_1\left(f\left(t\right)\right)+{\mathcal{D}}_1\left(f\left(t\right)\right)\lesssim {\mathcal{E}}_1\left(f_0\right),$$

which yields (21) in Theorem 2. Together with the local-in-time existence, the global mild solution and its uniqueness are immediately established via a standard continuity argument.

In order to derive the time decay estimates for the solution, we introduce

$$\widehat{\tilde{f}}=e^{\lambda t}\widehat{f},\widehat{\tilde{E}}=e^{\lambda t}\widehat{E},\widehat{\tilde{E}}=e^{\lambda t}\widehat{E},$$

with $\lambda >0$ to be chosen later. Given that $\widehat{f}$ and $\widehat{E},\widehat{B}$ satisfy the system (38) and (39), then $\widehat{\tilde{f}}$ and $\widehat{\tilde{E}},\widehat{\tilde{B}}$ satisfy

$$\begin{array}{c}\hfill {\mathsf{\partial }}_t\widehat{\tilde{f}}+iv·k\widehat{\tilde{f}}+e^{-\lambda t}\widehat{\tilde{E}}\ast {\nabla }_v\widehat{\tilde{f}}+e^{-\lambda t}(v\times \widehat{\tilde{B}})\ast {\nabla }_v\widehat{\tilde{f}}-{\displaystyle \frac{1}{2}}{e}^{-\lambda t}v·\widehat{\tilde{E}}\ast \widehat{\tilde{f}}-\widehat{\tilde{E}}·v{\mu }^{{\displaystyle \frac{1}{2}}}=L_{FP}\widehat{\tilde{f}}+\lambda \widehat{\tilde{f}},\end{array}$$

(51)

and

$$\left\{\begin{array}{cc}\hfill & {\mathsf{\partial }}_t\widehat{\tilde{E}}-ik\times \widehat{\tilde{B}}=-\widehat{b}+e^{-\lambda t}\widehat{\tilde{E}},\hfill \\ \hfill & {\mathsf{\partial }}_t\widehat{\tilde{B}}+ik\times \widehat{\tilde{E}}=e^{-\lambda t}\widehat{\tilde{B}},\hfill \\ \hfill & ik·\widehat{\tilde{E}}=\widehat{a},ik·\widehat{\tilde{B}}=0,\hfill \end{array}\right.$$

(52)

with initial data

$$\begin{array}{c}\hfill \widehat{\tilde{f}}(0,k,v)={\widehat{f}}_0(k,v),\widehat{\tilde{E}}={\widehat{E}}_0(k,v),\widehat{\tilde{B}}={\widehat{B}}_0(k,v).\end{array}$$

By applying the approach employed in the derivation of Lemma 6, we can derive that

$$\begin{array}{cc}\hfill {\mathcal{E}}_1\left(\tilde{f}\left(T\right)\right)+{\mathcal{D}}_1\left(\tilde{f}\left(T\right)\right)& \lesssim {\mathcal{E}}_1\left({\tilde{f}}_0\right)+\sqrt{\lambda }{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^T{\parallel ⟨k⟩\widehat{\tilde{f}}(t,k,·)\parallel }_{L_v^2}^{2}dt\right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & +\sqrt{\lambda }{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^T{|⟨k⟩\widehat{\tilde{E}}(t,k)|}^{2}dt\right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & +\sqrt{\lambda }{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^T{|⟨k⟩\widehat{\tilde{B}}(t,k)|}^{2}dt\right)}^{1/2}d\Sigma \left(k\right),\hfill \end{array}$$

for any $T>0$. Since

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^T{\parallel ⟨k⟩\widehat{\tilde{f}}(t,k,·)\parallel }_{L_v^2}^{2}dt\right)}^{1/2}d\Sigma \left(k\right)+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^T{\parallel ⟨k⟩\widehat{\tilde{E}}(t,k,·)\parallel }_{L_v^2}^{2}dt\right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & +{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^T{\parallel ⟨k⟩\widehat{\tilde{B}}(t,k,·)\parallel }_{L_v^2}^{2}dt\right)}^{1/2}d\Sigma \left(k\right)\lesssim {\mathcal{D}}_1\left(\tilde{f}\left(T\right)\right),\hfill \end{array}$$

(53)

then choosing $\lambda >0$ to be sufficiently small in (53) yields

$${\mathcal{E}}_1\left(\tilde{f}\left(T\right)\right)+{\mathcal{D}}_1\left(\tilde{f}\left(T\right)\right)\lesssim {\mathcal{E}}_1\left({\tilde{f}}_0\right).$$

(54)

From the definition of ${\mathcal{E}}_m(\tilde{f}(T))$ in (17) and the Minkowski’s inequality ${\parallel \parallel ·\parallel }_{L_k^1}{\parallel }_{L_T^{\infty }}\le {\parallel \parallel ·\parallel }_{L_T^{\infty }}{\parallel }_{L_k^1}$, one can deduce that

$${\int }_{{\mathbb{Z}}_k^3}\left(\parallel ⟨k⟩\widehat{\tilde{f}}{(t,k,·)\parallel }_{L_v^2}+|⟨k⟩\widehat{\tilde{E}}(t,k)|+|⟨k⟩\widehat{\tilde{B}}(t,k)|\right)d\Sigma \left(k\right)\lesssim {\mathcal{E}}_1\left({\tilde{f}}_0\right).$$

Recalling that $\widehat{\tilde{f}}=e^{\lambda t}\widehat{f},\widehat{\tilde{E}}=e^{\lambda t}\widehat{E},\widehat{\tilde{E}}=e^{\lambda t}\widehat{E}$, one can derive that

$${\int }_{{\mathbb{Z}}_k^3}\left(\parallel ⟨k⟩\widehat{f}{(t,k,·)\parallel }_{L_v^2}+|⟨k⟩\widehat{E}(t,k)|+|⟨k⟩\widehat{B}(t,k)|\right)d\Sigma \left(k\right)\lesssim e^{-\lambda t}{\mathcal{E}}_1\left(f_0\right),$$

which gives (22). This completes the proof of Theorem 2.

5. Conclusions

This paper discusses the Cauchy problem for the Vlasov-Maxwell-Fokker-Planck system near a global Maxwellian in low regularity space $L_k^1{L}_T^{\infty }{L}_v^2$ in periodic domain. The global-in-time existence of solutions to the system is established under the assumption that the perturbative initial data is sufficiently small. With the help of higher-order dissipation properties of the magnetic field, the exponential time decay can be also obtained. Future research will consider the global existence of solutions to the system in the whole space or in a general bounded domain with different boundary conditions.