Global Existence of Solutions to the Cauchy Problem for the Relativistic Vlasov–Maxwell–Fokker–Planck System in Low-Regularity Spaces

Abstract

This paper establishes the global-in-time existence and uniqueness of mild solutions to the relativistic Vlasov–Maxwell–Fokker–Planck (VMFP) system near a global Maxwellian equilibrium. We adopt a low-regularity functional framework, namely the mixed-norm space $L_k^1{L}_T^{\infty }{L}_p^2$ introduced for kinetic equations, which requires only integrability in the Fourier frequency variable and avoids high-order spatial differentiability. By employing a macro–micro decomposition, we derive macroscopic estimates for the hydrodynamic density and electric field, complemented by coercive estimates for the microscopic dissipation. Under a smallness assumption on the initial perturbation measured in this low-regularity norm, we derive a uniform a priori bound for the associated energy functional. This work provides the global existence result for the relativistic VMFP system in such low-regularity spaces, significantly relaxing the regularity requirements of previous classical Sobolev approaches.

1. Introduction and Main Results

1.1. The Model

The relativistic Vlasov–Maxwell–Fokker–Planck system describes the evolution of a particle distribution function coupled with self-consistent electromagnetic fields. It reads

$$\left\{\begin{array}{c} {\partial }_{t}F+\tilde{p}·{\nabla }_{x}F+(E+\tilde{p}\times B)·{\nabla }_{p}F={\nabla }_p·(pF+D{\nabla }_{p}F),\hfill \\ {\partial }_{t}E-{\nabla }_x\times B=-{\int }_{{\mathbb{R}}^3}\tilde{p}Fdp,\hfill \\ {\partial }_{t}B+{\nabla }_x\times E=0,\hfill \\ {\nabla }_x·E={\int }_{{\mathbb{R}}^3}Fdp-1,{\nabla }_x·B=0,\hfill \end{array}\right.$$

(1)

with initial data

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

(2)

and the compatibility conditions

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

(3)

where $F(t,x,p)\ge 0$ is the spatially periodic distribution function for the particles at time $t\ge 0$, with spatial coordinates $x=(x_1,x_2,x_3)\in {\mathbb{T}}^3:={[0,2\pi ]}^3$ and momentum $p=(p_1,p_2,p_3)\in {\mathbb{R}}^3$. The self-consistent electromagnetic field $\left[E\right(t,x),B(t,x\left)\right]$ is coupled with the distribution function $F(t,x,p)$ through the Maxwell equations. The particle energy is given by $p_0=⟨p⟩=\sqrt{1+{|p|}^2}$, and $\tilde{p}={\displaystyle \frac{p}{p_0}}$. Moreover,

$$D=\left(D_{ij}\right)={\displaystyle \frac{I+p\otimes p}{\sqrt{1+{|p|}^2}}},$$

is the relativistic diffusion matrix [1,2], and I denotes the identity matrix.

Kinetic equations track the time evolution of particle distribution functions defined over phase space, serving as indispensable foundational models throughout plasma physics, high-energy astrophysics, and controlled fusion research. Of particular importance is the VMFP system, which self-consistently couples the dynamical evolution of particle distributions to induced electromagnetic fields. The Fokker–Planck collision operator therein encodes diffusive scattering interactions between particles and a thermal reservoir or homogeneous background medium. Once particle velocities become comparable to the speed of light, relativistic kinematic effects cannot be neglected, motivating the analysis of the fully relativistic VMFP system. From a mathematical perspective, the system combines intricate nonlinearities and strong cross-field coupling, rendering its global-in-time well-posedness a longstanding central challenge within mathematical kinetic theory.

1.2. Related Works

Now we recall the related works of this manuscript. Research on the relativistic VMFP system is rooted in the theoretical development of Vlasov-type equations in the non-relativistic context. Foundational early work can be traced back to DiPerna and Lions [3] on the existence of global weak solutions for the Vlasov–Maxwell system. Using regularization methods and stability analysis, they first established a global weak solution theory for this model in the absence of the Fokker–Planck term. Carrillo and Labrunie [4] established the global existence of weak solutions for the non-relativistic regime, as well as the global existence of characteristic solutions under the quasi-relativistic framework.

For the Vlasov system including the Fokker–Planck diffusion term, Bostan and Goudon [5] investigated the high-electric-field limit, revealing the asymptotic behavior of the system as the electric field strength tends to infinity. Guo [6] pioneered the theory of global existence of classical solutions for the Vlasov–Maxwell–Boltzmann system near a Maxwellian equilibrium. By introducing macro–micro decomposition and nonlinear energy methods, he proved the global stability of perturbations under appropriate regularity assumptions. This work became a paradigm for much subsequent research. Carrillo-Duan-Moussa [7] further extended the study to the coupled system of the Vlasov–Fokker–Planck equation and the compressible Euler equations, proving the existence of global classical solutions near equilibrium.

As non-relativistic theory matured, researchers began to tackle the more challenging relativistic cases. Regarding the relativistic Fokker–Planck equation, Arnold and Toshpulatov [8] proved exponential time decay toward the global equilibrium within weighted $L^2$ and Sobolev space, while Alcántara and Calogero [9] showed its $L^1$ convergence to solutions of the non-relativistic counterpart in the non-relativistic limit the speed of light $c\to \infty$. Pankavich and Michalowski [10] investigated the one half dimensional relativistic VMFP and established the global existence and uniqueness of classical solutions to its Cauchy problem and uncovered a regularity enhancement effect of the distribution function with respect to the momentum variable. Lai [11] established the existence of classical solutions to the one and a half dimensional relativistic VMFP system under the assumption that the initial density decays sufficiently rapidly in the momentum variables. Authors in [12,13,14,15] established the existence of global classical solutions for the VMFP system. Yang [16] constructed global solutions to the VMFP system with relativistic transport where the diffusive matrix is an identity one and derived their optimal time decay rates via a combined framework of compensating functions and energy methods. Liu and Ma [2] investigated the Cauchy problem for the relativistic VMFP system in the whole space. By identifying a new structure in the linearized relativistic Fokker–Planck operator and establishing coercive estimates, they obtained the existence of global classical solutions and large-time decay rates for perturbed initial data with appropriate regularity and integrability. Notably, they proved that the unweighted energy decays at a polynomial rate and that the decay rate of low-order derivatives of the solution, except for the magnetic field, is faster than the optimal decay rate—a novel phenomenon unique to the relativistic Fokker–Planck model. Liu and Xiao [17] investigated the relativistic Vlasov–Maxwell–Boltzmann system for short-range interactions, while Liu and Zhao [18] analyzed the optimal large-time decay for the relativistic Landau-Maxwell system. Dong et al. [19] recently studied the local well-posedness of the relativistic Vlasov–Maxwell–Landau system with specular reflection boundary conditions, incorporating boundary effects into consideration.

However, most of these results have been obtained in high-regularity Sobolev spaces, which impose strong regularity requirements on the initial data. To establish the well-posedness of solutions in low-regularity spaces, Duan-Liu-Sakamoto-Strain [20] introduced a novel class of low-regularity function spaces, $L_k^1{L}_T^{\infty }{L}_v^2$, and proved the existence of global mild solutions for the Landau equation and the non-cutoff Boltzmann equation under low-regularity assumptions on the initial data. Their approach relies on refined energy estimates. The present paper aims to apply this low-regularity functional framework to investigate the Cauchy problem for the relativistic VMFP system. In particular, under the assumption that the initial perturbation is sufficiently small, we establish the existence and uniqueness of a global mild solution.

1.3. Perturbation

We are interested in constructing global solutions to the system (1)–(3) close to an equilibrium state, namely the relativistic global Maxwellian

$$J\left(p\right)={\displaystyle \frac{e^{-⟨p⟩}}{{\rho }_0}}={\displaystyle \frac{e^{-\sqrt{1+{|p|}^2}}}{{\rho }_0}},$$

where the background charge constant ${\rho }_0={\int }_{{\mathbb{R}}^3}{e}^{-\sqrt{1+{|p|}^2}}dp$, ensuring that ${\int }_{{\mathbb{R}}^3}J\left(p\right)dp=1$. Define the perturbation $f=f(t,x,p)$ by

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

(4)

Then, the Cauchy problem (1)–(3) is reformulated as

$$\left\{\begin{array}{c} {\partial }_{t}f+\tilde{p}·{\nabla }_{x}f+(E+\tilde{p}\times B)·{\nabla }_{p}f-{\displaystyle \frac{1}{2}}E·\tilde{p}f-E·\tilde{p}{J}^{{\displaystyle \frac{1}{2}}}+Lf=0,\hfill \\ {\partial }_{t}E-{\nabla }_x\times B=-{\int }_{{\mathbb{R}}^3}\tilde{p}{J}^{{\displaystyle \frac{1}{2}}}fdp,\hfill \\ {\partial }_{t}B+{\nabla }_x\times E=0,\hfill \\ {\nabla }_x·E={\int }_{{\mathbb{R}}^3}{J}^{{\displaystyle \frac{1}{2}}}fdp,{\nabla }_x·B=0,\hfill \end{array}\right.$$

(5)

with initial data

$$f(0,x,p)=f_0(x,p)=J^{-{\displaystyle \frac{1}{2}}}(F_0(x,p)-J),E(0,x)=E_0\left(x\right),B(0,x)=B_0\left(x\right),$$

(6)

and the compatibility conditions

$${\nabla }_x·E_0={\int }_{{\mathbb{R}}^3}{J}^{{\displaystyle \frac{1}{2}}}{f}_0(x,p)dp,{\nabla }_x·B_0=0.$$

(7)

Here, the Fokker–Planck operator L is given by

$$Lf=-J^{-{\displaystyle \frac{1}{2}}}{\nabla }_p·\left(JD{\nabla }_p\left(f{J}^{-{\displaystyle \frac{1}{2}}}\right)\right)=-{\nabla }_p·\left(D{\nabla }_{p}f\right)+{\displaystyle \frac{p^2}{4p_0}}f-{\displaystyle \frac{3}{2}}f.$$

(8)

It follows from [2,21] that the linearized operator L is non-negative and its null space is

$$\mathcal{N}=\mathrm{span}\left\{J^{{\displaystyle \frac{1}{2}}}\right\}.$$

Denote by $\mathbf{P}$ the orthogonal projection from $L^2\left({\mathbb{R}}_p^3\right)$ onto $\mathcal{N}$. For any function $f(t,x,p)\in L^2\left({\mathbb{R}}_p^3\right)$, we have

$$\mathbf{P}f=a(t,x)J^{{\displaystyle \frac{1}{2}}},$$

with

$$\begin{array}{c}\hfill a(t,x)={\int }_{{\mathbb{R}}^3}{J}^{{\displaystyle \frac{1}{2}}}f(t,x,p)dp={\int }_{{\mathbb{R}}^3}{J}^{{\displaystyle \frac{1}{2}}}\mathbf{P}f(t,x,p)dp.\end{array}$$

(9)

Thus we obtain the macro–micro decomposition

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

(10)

where $\mathbf{I}$ is the identity. $\mathbf{P}f$ and $\{\mathbf{I}-\mathbf{P}\}f$ are called the macroscopic and microscopic components, respectively.

As shown in [2], the operator L is coercive i.e., there exists a constant $\lambda >0$ such that

$${(Lf,f)}_{L_p^2}\ge \lambda {|\{\mathbf{I}-\mathbf{P}\}f|}_D^2,$$

(11)

where the norm ${|·|}_D$ is defined by

$${|f|}_D^2={\int }_{{\mathbb{R}}_p^3}\left\{{\displaystyle \sum _{ij}^3}{D}^{ij}{\partial }_{p_i}f{\partial }_{p_j}f+{\displaystyle \frac{p^2}{4p_0}}f\right\}dp\gtrsim {\int }_{{\mathbb{R}}_p^3}\left({\displaystyle \frac{|{\nabla }_p{f|}^2}{p_0}}+p_0{f}^2\right)dp.$$

(12)

1.4. Macro Structure

In this subsection, we derive the macroscopic conservation laws that follow from the relativistic VMFP system (1). These relations form the foundation for the subsequent analysis of the fluid-kinetic coupling.

Conservation of mass. We begin by integrating the first equation of (1) with respect to p. This yields

$${\partial }_t{\int }_{{\mathbb{R}}^3}Fdp+{\nabla }_x·{\int }_{{\mathbb{R}}^3}\tilde{p}Fdp+{\int }_{{\mathbb{R}}^3}(E+\tilde{p}\times B)·{\nabla }_{p}Fdp=0.$$

(13)

The last integral vanishes identically. Indeed, an integration by parts in p gives

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^3}(E+\tilde{p}\times B)·{\nabla }_{p}Fdp=-{\int }_{{\mathbb{R}}^3}F{\nabla }_p·\left(E+\tilde{p}\times B\right)dp=0,\end{array}$$

since ${\nabla }_p·(E+\tilde{p}\times B)=0$. Substituting the distribution function $F(t,x,p)=J+J^{{\displaystyle \frac{1}{2}}}f(t,x,p)$ into (13), we can get that the mass conservation law for the macroscopic density

$${\partial }_{t}a+{\nabla }_x·b=0,$$

(14)

where

$$a(t,x)={\int }_{{\mathbb{R}}^3}{J}^{{\displaystyle \frac{1}{2}}}f(t,x,p)dp,b(t,x)={\int }_{{\mathbb{R}}_p^3}\tilde{p}{J}^{{\displaystyle \frac{1}{2}}}f(t,x,p)dp.$$

Integrating (14) over the spatial torus ${\mathbb{T}}^3$ and applying the periodic boundary conditions, the divergence term vanishes, yielding the global mass conservation

$${\int }_{{\mathbb{T}}_x^3}{\int }_{{\mathbb{R}}_p^3}{J}^{{\displaystyle \frac{1}{2}}}f(t,x,p)dpdx={\int }_{{\mathbb{T}}_x^3}{\int }_{{\mathbb{R}}_p^3}{J}^{{\displaystyle \frac{1}{2}}}f(0,x,p)dpdx.$$

(15)

This identity ensures that the total perturbed mass remains constant in time, which is essential for controlling the zero-frequency mode in the Fourier analysis.

Momentum balance. We now derive the evolution equation for the momentum flux b. Multiplying the first Equation (5) by $\tilde{p}{J}^{1/2}$ and integrate over p, we have

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^3}\tilde{p}{J}^{1/2}{\partial }_{t}fdp+{\int }_{{\mathbb{R}}^3}\tilde{p}{J}^{1/2}(\tilde{p}·{\nabla }_{x}f)dp-{\int }_{{\mathbb{R}}^3}\tilde{p}{J}^{1/2}Gdp\hfill \\ \hfill & -{\int }_{{\mathbb{R}}^3}\tilde{p}E·\tilde{p}Jdp+{\int }_{{\mathbb{R}}^3}\tilde{p}{J}^{1/2}Lfdp=0,\hfill \end{array}$$

where the nonlinear term G is given by

$$G={\displaystyle \frac{1}{2}}E·\tilde{p}f-(E+\tilde{p}\times B)·{\nabla }_{p}f.$$

Now we split $f=\mathbf{P}f+\{\mathbf{I}-\mathbf{P}\}f$ with $\mathbf{P}f=a{J}^{1/2}$. Then,

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^3}\tilde{p}{J}^{1/2}(\tilde{p}·{\nabla }_{x}f)dp={\nabla }_x·{\int }_{{\mathbb{R}}^3}\tilde{p}\otimes \tilde{p}{J}^{1/2}fdp.\hfill \\ \hfill & ={\nabla }_x·a{\int }_{{\mathbb{R}}^3}\tilde{p}\otimes \tilde{p}Jdp+{\nabla }_x·{\int }_{{\mathbb{R}}^3}\tilde{p}\otimes \tilde{p}{J}^{1/2}\{\mathbf{I}-\mathbf{P}\}fdp\hfill \\ \hfill & =c_0{\nabla }_{x}a++{\nabla }_x·{\int }_{{\mathbb{R}}^3}\tilde{p}\otimes \tilde{p}{J}^{1/2}\{\mathbf{I}-\mathbf{P}\}fdp,\hfill \end{array}$$

where

$$c_0={\int }_{{\mathbb{R}}^3}{\tilde{p}}_1^{2}Jdp={\int }_{{\mathbb{R}}^3}{\tilde{p}}_2^{2}Jdp={\int }_{{\mathbb{R}}^3}{\tilde{p}}_3^{2}Jdp.$$

Moreover, a direct computation gives

$$-{\int }_{{\mathbb{R}}^3}{\tilde{p}}_i(E·\tilde{p})Jdp=-c_0{E}_i,{\int }_{{\mathbb{R}}^3}\tilde{p}{J}^{1/2}{\partial }_{t}fdp={\partial }_{t}b.$$

Collecting all the above identities and rearranging the terms, we arrive at the momentum balance equation

$$\begin{array}{c}\hfill {\partial }_{t}b+c_0{\nabla }_{x}a-c_{0}E+{\nabla }_x·{\int }_{{\mathbb{R}}^3}\tilde{p}\otimes \tilde{p}{J}^{1/2}\{\mathbf{I}-\mathbf{P}\}fdp={\int }_{{\mathbb{R}}^3}\tilde{p}{J}^{1/2}(G-Lf)dp\end{array}$$

(16)

This equation describes the evolution of the macroscopic momentum flux under the combined effects of the pressure gradient, the electric field forcing, the kinetic stress from the non-fluid modes, and the collisional and nonlinear contributions on the right-hand side. Together, (14) and (16) constitute the two fundamental macroscopic constraints that will be exploited in the following estimates.

1.5. Notation and Function Spaces

We introduce the notation and the function spaces that will be used throughout the paper. We begin with some standard notation. For two nonnegative quantities A and B, we write $A\lesssim B$ if there exists a constant $C>0$ such that $A\le CB$. $A\sim B$ means $A\lesssim B$ and $B\lesssim A$. The inner product in $L_p^2$ is denoted by

$$⟨f,g⟩={(f,g)}_{L_p^2}={\int }_{{\mathbb{R}}_p^3}f\left(p\right)\overline{g\left(p\right)}dp,$$

and the associated norm is ${|f|}_{L_p^2}={⟨f,f⟩}^{{\displaystyle \frac{1}{2}}}$. $\mathcal{R}$ stands for the real part of a complex number, which will appear in energy estimates involving Fourier modes.

Inspired by [20], we work in the low-regularity space $L_k^1{L}_T^{\infty }{L}_p^2$, which combines integrability in the Fourier frequency domain with temporal supremum norms and spatial $L^1$ regularity. Let $f(t,x,p)$ be a function defined for $t\ge 0$, $x=(x_1,x_2,x_3)\in {\mathbb{T}}^3:={[0,2\pi ]}^3$, and $p=(p_1,p_2,p_3)\in {\mathbb{R}}^3$. We define its Fourier transform with respect to the spatial variable x as

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

Here, $k=(k_1,k_2,k_3)\in {\mathbb{Z}}^3$ is the frequency vector. The use of Fourier series allows us to decompose the solution into discrete frequency modes. To simplify the notation, we give the discrete counting measure $d\Sigma \left(k\right)$ on ${\mathbb{Z}}^3$, so that integration over k is understood as summation over all lattice points

$${\int }_{{\mathbb{Z}}^3}g\left(k\right)d\Sigma \left(k\right)=\sum _{k\in {\mathbb{Z}}^3}g\left(k\right).$$

For any two functions f and g defined on the frequency lattice, their spatial convolution is given by

$$\begin{array}{ccc}\hfill (f\ast g)(t,k)& =& {\int }_{{\mathbb{Z}}^3}f(t,k-r)g(t,r)d\Sigma \left(r\right),k\in {\mathbb{Z}}^3.\hfill \end{array}$$

which corresponds to the standard convolution of Fourier coefficients. The norm of the space $L_k^1{L}_T^{\infty }{L}_p^2$ is then defined as

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

(17)

This norm is a mixed-norm space that encodes low regularity in the spatial variable (via $L^1$ summability in k) while retaining sufficient control in time and momentum to close energy estimates without requiring high spatial differentiability.

To state our main results, we need give two key functionals: the energy functional ${\mathcal{E}}_T\left(f\right)$ and the corresponding energy dissipation rate functional ${\mathcal{D}}_T\left(f\right)$:

$${\mathcal{E}}_T\left(f\right):={\int }_{{\mathbb{Z}}_k^3}\underset{0\le t\le T}{\sup }\left(|\widehat{f}{(t,k,·)|}_{L_p^2}+|\widehat{E}(t,k)|+|\widehat{B}(t,k)|\right)d\Sigma \left(k\right)$$

(18)

and

$$\begin{array}{cc}\hfill {\mathcal{D}}_T\left(f\right)& :={\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^T{|\{\mathbf{I}-\mathbf{P}\}\widehat{f}(t,k,·)|}_D^{2}dt\right)}^{1/2}d\Sigma \left(k\right)+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^T{|\widehat{a}(t,k)|}^{2}dt\right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & +{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^T{|\widehat{E}(t,k)|}^{2}dt\right)}^{1/2}d\Sigma \left(k\right)\hfill \end{array}$$

(19)

respectively. The first term in ${\mathcal{D}}_T\left(f\right)$ measures the time-integrated dissipation of the microscopic component, weighted by the norm ${|·|}_D$. The second and third terms capture the dissipation of the macroscopic density a and the electric field E, which arises from the coupling with the electromagnetic fields and the transport terms.

1.6. Main Results

We now state our main global well-posedness result for the relativistic VMFP system within the low-regularity mixed-norm space.

Theorem 1.

Let the initial data $f_0(x,p)=f(0,x,p)$ satisfy the conservation of mass

$${\int }_{{\mathbb{T}}^3}{\int }_{{\mathbb{R}}^3}{J}^{{\displaystyle \frac{1}{2}}}f(t,x,p)dpdx={\int }_{{\mathbb{T}}^3}{\int }_{{\mathbb{R}}^3}{J}^{{\displaystyle \frac{1}{2}}}f(0,x,p)dpdx=0,$$

(20)

and assume that the corresponding distribution function $F_0(x,p)=\mu +{\mu }^{{\displaystyle \frac{1}{2}}}{f}_0(x,p)\ge 0$. There exists a constant ${ϵ}_0>0$ such that if

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

then the Cauchy problem (5)–(7) admits a unique global mild solution $f(t,x,p)$, $t>0$, $x\in {\mathbb{T}}^3$, $p\in {\mathbb{R}}^3$. Moreover, the solution satisfies $F(t,x,p)=\mu +{\mu }^{{\displaystyle \frac{1}{2}}}f(t,x,p)\ge 0$ and the uniform estimate

$${\mathcal{E}}_T\left(f\right)+{\mathcal{D}}_T\left(f\right)\lesssim {ϵ}_0,$$

(21)

for any $T>0$.

1.7. The Strategy

The core strategy is to prove the global existence of mild solutions to the relativistic VMFP system within a low-regularity function space framework, by combining macro–micro decomposition with energy-dissipation closure estimates. The main steps are as follows:

• Linearization and decomposition. The distribution function is decomposed into a global Maxwellian equilibrium plus a perturbation. Using the null structure of the linearized Fokker–Planck operator, the perturbation is split into a macroscopic density (the null-space component) and a microscopic component. The former lacks intrinsic dissipation, while the latter is subject to coercive dissipation.
• Construction of energy and dissipation functionals. We define the energy functional ${\mathcal{E}}_T$, which controls the amplitudes of the perturbation and the electromagnetic fields, and the dissipation functional ${\mathcal{D}}_T$, which includes the microscopic dissipation as well as the dissipation of the macroscopic density and the electric field.
• Two-level estimate closure.

  (a)

  Macroscopic estimate: By employing the Fourier transform, the dissipation of the macroscopic density and the electric field is controlled in terms of the microscopic dissipation and the initial data.

  (b)

  Microscopic estimate: An energy estimate for the perturbed equation is performed, where the coercivity of the operator is used to extract the microscopic dissipation, and the nonlinear terms are bounded by ${\mathcal{E}}_T{\mathcal{D}}_T$.

  (c)

  The macroscopic estimate, multiplied by a small constant, is added to the microscopic estimate. Using the smallness assumption on the initial data, the higher-order nonlinear terms are absorbed, yielding a time-uniform a priori bound ${\mathcal{E}}_T+{\mathcal{D}}_T\lesssim {ϵ}_0.$

• Global continuation. Combining the local-in-time existence with a continuity argument, the solution is extended to arbitrarily long times, thereby obtaining a global mild solution.

In contrast to the approach in reference [22], the present paper adopts a different technical strategy for deriving the dissipative estimates for the macroscopic density a and the electric field E. In our proof, we directly multiply the macroscopic equations by the weight factor ${\displaystyle \frac{ik\widehat{a}}{1+{|k|}^2}}$> [23] and perform energy integration. The key trick is to invoke Gauss’s law $ik·\widehat{E}=\widehat{a}$ from the Maxwell equations, which converts the cross term into a positive definite dissipation term for the electric field, namely ${\displaystyle \frac{{|k|}^2{|\widehat{E}|}^2}{1+{|k|}^2}}$. Through this algebraic manipulation, we are able to directly and explicitly extract the dissipative structure of the macroscopic density and the electric field.

By contrast, reference [22] employs a more subtle duality approach. The authors construct a velocity-dependent test function of the form $\Phi ={(|v|}^2-10)v·ik{\widehat{\varphi }}_a{\mu }^{1/2}$. By taking the $L_v^2$-inner product of the perturbed equation with this test function, and then using integration by parts together with the self-adjointness of the Fokker–Planck operator, they successfully eliminate the interference term involving the macroscopic momentum $\widehat{b}$. This reduces the estimate for the macroscopic density a to controlling the microscopic component and the nonlinear terms. In summary, the method employed in the present paper is more direct, as it fully exploits the intrinsic structure of the Maxwell equations to rapidly derive the desired dissipative estimates.

The rest of the paper is arranged in the following order. We devote Section 2 to the derivation of essential macroscopic estimates. The existence proof for our main result is established in Section 3, and Section 4 draws the concluding remarks. The nonlinear estimates employed throughout the paper are summarized in Appendix A.

2. Macroscopic Estimates

In this section, we derive uniform estimates for the macroscopic density $\widehat{a}(t,k)$ and $\widehat{E}(t,k)$ of the solution. Taking the Fourier transform in the spatial variable x of (14) and (16) leads to

$$\left\{\begin{array}{c} {\partial }_t\widehat{a}+ik·\widehat{b}=0,\hfill \\ {\partial }_t\widehat{b}+c_{0}ik\widehat{a}-c_0\widehat{E}+ik·⟨\tilde{p}\otimes \tilde{p}{J}^{{\displaystyle \frac{1}{2}}},\{\mathbf{I}-\mathbf{P}\}\widehat{f}⟩=⟨\tilde{p}{J}^{{\displaystyle \frac{1}{2}}},\widehat{G}-L\widehat{f}⟩.\hfill \end{array}\right.$$

(22)

The Fourier-transformed Maxwell equations read

$$\left\{\begin{array}{c} {\partial }_t\widehat{E}-ik\times \widehat{B}=-\widehat{b},\hfill \\ {\partial }_t\widehat{B}+ik\times \widehat{E}=0,\hfill \\ ik·\widehat{E}=\widehat{a},ik·\widehat{B}=0.\hfill \end{array}\right.$$

(23)

Lemma 1.

Under the assumptions of Theorem 1, the following estimate holds:

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

(24)

Proof. 

Multiplying ${\left(22\right)}_2$ by ${\displaystyle \frac{ik\overline{\widehat{a}}}{1+{|k|}^2}}$ and integrating the result over $[0,T]$ leads to

$$\begin{array}{cc}\hfill & c_0{\int }_0^T{\displaystyle \frac{{|k|}^2{|\widehat{a}|}^2}{1+{|k|}^2}}dt-c_0{\int }_0^T{\displaystyle \frac{ik\overline{\widehat{a}}\widehat{E}}{1+{|k|}^2}}dt\hfill \\ \hfill & \le |{\int }_0^T{\displaystyle \frac{k{\partial }_t\widehat{b}\overline{\widehat{a}}}{1+{|k|}^2}}dt|+{\int }_0^T{\displaystyle \frac{{|k|}^2|\overline{\widehat{a}}|}{1+{|k|}^2}}|⟨\tilde{p}\otimes \tilde{p}{J}^{{\displaystyle \frac{1}{2}}},\{\mathbf{I}-\mathbf{P}\}\widehat{f}⟩|dt\hfill \\ \hfill & +{\int }_0^T{\displaystyle \frac{|k\overline{\widehat{a}}|}{1+{|k|}^2}}|⟨\tilde{p}{J}^{{\displaystyle \frac{1}{2}}},\widehat{G}⟩|dt+{\int }_0^T{\displaystyle \frac{|k\overline{\widehat{a}}|}{1+{|k|}^2}}|⟨\tilde{p}{J}^{{\displaystyle \frac{1}{2}}},L\widehat{f}⟩|dt={\displaystyle \sum _{j=1}^4}{I}_j.\hfill \end{array}$$

(25)

From ${\left(23\right)}_3$, we have

$$-c_0{\int }_0^T{\displaystyle \frac{ik\overline{\widehat{a}}\widehat{E}}{1+{|k|}^2}}dt=-c_0{\int }_0^T{\displaystyle \frac{ik(ik·\overline{\widehat{E}})\widehat{E}}{1+{|k|}^2}}dt=c_0{\int }_0^T{\displaystyle \frac{{|k|}^2{|\widehat{E}|}^2}{1+{|k|}^2}}dt.$$

Using ${\left(22\right)}_1$ and Young’s inequality, we estimate $I_1$ as

$$\begin{array}{cc}\hfill I_1& \lesssim |{\int }_0^T{\displaystyle \frac{k{\partial }_t\left(\widehat{b}\overline{\widehat{a}}\right)}{1+{|k|}^2}}dt|+|{\int }_0^T{\displaystyle \frac{k\widehat{b}{\partial }_t\overline{\widehat{a}}}{1+{|k|}^2}}dt|\hfill \\ \hfill & \lesssim {\displaystyle \frac{|k|}{1+{|k|}^2}}|\widehat{b}(T,k)\widehat{a}(T,k)+\widehat{b}(0,k)\widehat{a}(0,k)|+{\displaystyle \frac{{|k|}^2}{1+{|k|}^2}}{\int }_0^T{|\widehat{b}(t,k)|}^{2}dt\hfill \\ \hfill & \lesssim |\widehat{b}{(T,k)|}^2+|\widehat{a}{(T,k)|}^2+|\widehat{b}{(0,k)|}^2+|\widehat{a}{(0,k)|}^2+{\int }_0^T{|\widehat{b}(t,k)|}^{2}dt\hfill \\ \hfill & \lesssim |\widehat{f}{(T,k,·)|}_{L_p^2}^2+|\widehat{f}{(0,k,·)|}_{L_p^2}^2+{\int }_0^T{|\{\mathbf{I}-\mathbf{P}\}\widehat{f}(t,k,·)|}_{L_p^2}^{2}dt.\hfill \end{array}$$

where we have used ${\displaystyle \frac{|k|}{1+{|k|}^2}}\le 1,{\displaystyle \frac{{|k|}^2}{1+{|k|}^2}}\le 1$, $|\widehat{a}(t,k)|\lesssim |\widehat{f}{(t,k,·)|}_{L_p^2}$ and

$$|\widehat{b}(t,k)|\lesssim |\{\mathbf{I}-\mathbf{P}\}\widehat{f}{(t,k,·)|}_{L_p^2}\lesssim {|\widehat{f}(t,k,·)|}_{L_p^2}.$$

For $I_2$, Young’s inequality yields the following estimate

$$\begin{array}{cc}\hfill I_2& \lesssim {\int }_0^T{\displaystyle \frac{{|k|}^2|\overline{\widehat{a}}|}{1+{|k|}^2}}{|\{\mathbf{I}-\mathbf{P}\}\widehat{f}(t,k,·)|}_{L_p^2}dt\hfill \\ \hfill & \lesssim \eta {\int }_0^T{\displaystyle \frac{{|k|}^2{|\widehat{a}|}^2}{1+{|k|}^2}}dt+C_{\eta }{\int }_0^T{\displaystyle \frac{{|k|}^2}{1+{|k|}^2}}{|\{\mathbf{I}-\mathbf{P}\}\widehat{f}(t,k,·)|}_{L_p^2}^{2}dt\hfill \\ \hfill & \lesssim \eta {\int }_0^T{\displaystyle \frac{{|k|}^2{|\widehat{a}|}^2}{1+{|k|}^2}}dt+C_{\eta }{\int }_0^T{|\{\mathbf{I}-\mathbf{P}\}\widehat{f}(t,k,·)|}_{L_p^2}^{2}dt.\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}\hfill I_3& \lesssim \eta {\int }_0^T{\displaystyle \frac{{|k|}^2{|\widehat{a}|}^2}{1+{|k|}^2}}dt+C_{\eta }{\int }_0^T{\displaystyle \frac{1}{1+{|k|}^2}}|⟨\tilde{p}{J}^{{\displaystyle \frac{1}{2}}},\widehat{G}⟩{|}^{2}dt\hfill \\ \hfill & \lesssim \eta {\int }_0^T{\displaystyle \frac{{|k|}^2{|\widehat{a}|}^2}{1+{|k|}^2}}dt+C_{\eta }{\int }_0^T|⟨\tilde{p}{J}^{{\displaystyle \frac{1}{2}}},\widehat{G}⟩{|}^{2}dt\hfill \\ \hfill & \lesssim \eta {\int }_0^T{\displaystyle \frac{{|k|}^2{|\widehat{a}|}^2}{1+{|k|}^2}}dt+{\mathcal{E}}_T\left(f\right){\mathcal{D}}_T\left(f\right)dt,\hfill \end{array}$$

where we have used ${\displaystyle \frac{1}{1+{|k|}^2}}\le 1$ and the Lemma A1. Using integration by parts and Hölder’s inequality yields

$$\begin{array}{cc}\hfill |⟨\tilde{p}{J}^{{\displaystyle \frac{1}{2}}},{\nabla }_p·\left(D{\nabla }_p\{\mathbf{I}-\mathbf{P}\}\widehat{f}\right)⟩|& =|⟨{\nabla }_p\left(\tilde{p}{J}^{{\displaystyle \frac{1}{2}}}\right),D{\nabla }_p\{\mathbf{I}-\mathbf{P}\}\widehat{f}⟩|\hfill \\ \hfill & =|⟨{\nabla }_p\left(\tilde{p}{J}^{{\displaystyle \frac{1}{2}}}\right)D\sqrt{p_0},{\displaystyle \frac{1}{\sqrt{p_0}}}{\nabla }_p\{\mathbf{I}-\mathbf{P}\}\widehat{f}⟩|\hfill \\ \hfill & \lesssim |{\displaystyle \frac{1}{\sqrt{p_0}}}{\nabla }_p\{\mathbf{I}-\mathbf{P}\}\widehat{f}{(t,k,·)|}_{L_p^2},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill |⟨\tilde{p}{J}^{{\displaystyle \frac{1}{2}}},{\displaystyle \frac{p^2}{4p_0}}\{\mathbf{I}-\mathbf{P}\}\widehat{f}⟩|& =|⟨\tilde{p}{J}^{{\displaystyle \frac{1}{2}}}{\displaystyle \frac{p^2}{4p_0}},\{\mathbf{I}-\mathbf{P}\}\widehat{f}⟩|\lesssim {|\{\mathbf{I}-\mathbf{P}\}\widehat{f}(t,k,·)|}_{L_p^2},\hfill \\ \hfill |⟨\tilde{p}{J}^{{\displaystyle \frac{1}{2}}},{\displaystyle \frac{3}{2}}\{\mathbf{I}-\mathbf{P}\}\widehat{f}⟩|& \lesssim |\{\mathbf{I}-\mathbf{P}\}\widehat{f}{(t,k,·)|}_{L_p^2}.\hfill \end{array}$$

Hence, we get that

$$\begin{array}{cc}\hfill |⟨\tilde{p}{J}^{{\displaystyle \frac{1}{2}}},L\{\mathbf{I}-\mathbf{P}\}\widehat{f}⟩|& =|⟨\tilde{p}{J}^{{\displaystyle \frac{1}{2}}},-{\nabla }_p·\left(D{\nabla }_p\{\mathbf{I}-\mathbf{P}\}\widehat{f}\right)+{\displaystyle \frac{p^2}{4p_0}}\{\mathbf{I}-\mathbf{P}\}\widehat{f}-{\displaystyle \frac{3}{2}}\{\mathbf{I}-\mathbf{P}\}\widehat{f}⟩|\hfill \\ \hfill & \lesssim |{\displaystyle \frac{1}{\sqrt{p_0}}}{\nabla }_p\{\mathbf{I}-\mathbf{P}\}\widehat{f}{(t,k,·)|}_{L_p^2}+{|\{\mathbf{I}-\mathbf{P}\}\widehat{f}(t,k,·)|}_{L_p^2}.\hfill \end{array}$$

Furthermore, we can obtain that

$$\begin{array}{cc}\hfill I_4& ={\int }_0^T{\displaystyle \frac{|k\overline{\widehat{a}}|}{1+{|k|}^2}}|⟨\tilde{p}{J}^{{\displaystyle \frac{1}{2}}},L\{\mathbf{I}-\mathbf{P}\}\widehat{f}⟩|dt\hfill \\ \hfill & \lesssim \eta {\int }_0^T{\displaystyle \frac{{|k|}^2{|\widehat{a}|}^2}{1+{|k|}^2}}dt+C_{\eta }{\int }_0^T{\displaystyle \frac{1}{1+{|k|}^2}}|⟨\tilde{p}{J}^{{\displaystyle \frac{1}{2}}},L\{\mathbf{I}-\mathbf{P}\}\widehat{f}⟩{|}^{2}dt\hfill \\ \hfill & \lesssim \eta {\int }_0^T{\displaystyle \frac{{|k|}^2{|\widehat{a}|}^2}{1+{|k|}^2}}dt+{\int }_0^T{|\{\mathbf{I}-\mathbf{P}\}\widehat{f}(t,k,·)|}_D^{2}dt.\hfill \end{array}$$

By taking square roots on both sides, applying the $L_k^1$ norm to (25), and combining the above estimates for $I_1$ through $I_4$ with a sufficiently small $\eta >0$ sufficiently small, we obtain

$$\begin{array}{cc}\hfill & c_0{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^T{\displaystyle \frac{{|k|}^2{|\widehat{a}|}^2}{1+{|k|}^2}}dt\right)}^{1/2}dk+c_0{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^T{\displaystyle \frac{{|k|}^2{|\widehat{E}|}^2}{1+{|k|}^2}}dt\right)}^{1/2}dk\hfill \\ \hfill \lesssim & {\int }_{{\mathbb{Z}}_k^3}|\widehat{f}{(T,k,·)|}_{L_p^2}dk+{\int }_{{\mathbb{Z}}_k^3}{|\widehat{f}(0,k,·)|}_{L_p^2}dk+{\mathcal{E}}_T\left(f\right){\mathcal{D}}_T\left(f\right)\hfill \\ \hfill & +{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^T{|\{\mathbf{I}-\mathbf{P}\}\widehat{f}(t,k,·)|}_D^{2}dt\right)}^{1/2}dk\hfill \\ \hfill \lesssim & {\mathcal{E}}_T\left(f\right)+{\parallel {\widehat{f}}_0\parallel }_{L_k^1{L}_p^2}+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^t{|\{\mathbf{I}-\mathbf{P}\}\widehat{f}(t,k,·)|}_D^{2}dt\right)}^{1/2}d\Sigma \left(k\right)+{\mathcal{E}}_T\left(f\right){\mathcal{D}}_T\left(f\right).\hfill \end{array}$$

Since $|k|\ge 1$, it follows that ${\displaystyle \frac{1}{2}}\le {\displaystyle \frac{{|k|}^2}{1+{|k|}^2}}\le 1$. Consequently,

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

(26)

This completes the proof of the lemma. □

3. Global Existence

To establish the global-in-time existence of solutions to the Cauchy problem (5)–(7), we first derive estimates for the microscopic dissipation of f. These estimates quantify the dissipative effects encoded in the linear operator L, and will later be combined with the macroscopic estimates from Section 2 to close the energy functionals and obtain uniform a priori estimates.

Lemma 2.

Let $f(t,x,p)$ be a smooth solution to the Cauchy problem (5)–(7) on the interval [0, T], then the following estimate holds

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

(27)

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

Proof. 

Taking the Fourier transform of ${\left(5\right)}_1$ with $x\in {\mathbb{T}}^3$ yields

$$\begin{array}{cc}\hfill & {\partial }_t\widehat{f}(t,k,p)+i\tilde{p}·k\widehat{f}(t,k,p)+[\widehat{E}\ast {\nabla }_p\widehat{f}](t,k,p)+[(\tilde{p}\times \widehat{B})\ast {\nabla }_p\widehat{f}](t,k,p)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}[\tilde{p}·\widehat{E}\ast \widehat{f}](t,k,p)-\widehat{E}(t,k)·\tilde{p}{J}^{{\displaystyle \frac{1}{2}}}+L\widehat{f}(t,k,p)=0.\hfill \end{array}$$

(28)

Here, the convolutions are understood in the sense of $k\in {\mathbb{Z}}^3$:

$$\begin{array}{ccc}\hfill [\widehat{E}\ast {\nabla }_p\widehat{f}](t,k,p)& =& {\int }_{{\mathbb{Z}}_r^3}\widehat{E}(t,k-r){\nabla }_p\widehat{f}(t,r,p)d\Sigma \left(r\right),\hfill \\ \hfill [(\tilde{p}\times \widehat{B})\ast {\nabla }_p\widehat{f}](t,k,p)& =& {\int }_{{\mathbb{Z}}_r^3}\tilde{p}\times \widehat{B}(t,k-r){\nabla }_p\widehat{f}(t,r,p)d\Sigma \left(r\right),\hfill \\ \hfill [\tilde{p}·\widehat{E}\ast \widehat{f}](t,k,p)& =& {\int }_{{\mathbb{Z}}_r^3}\tilde{p}·\widehat{E}(t,k-r)\widehat{f}(t,r,p)d\Sigma \left(r\right).\hfill \end{array}$$

From (23), we have

$$\begin{array}{cc}\hfill {(-\widehat{E}·\tilde{p}{J}^{{\displaystyle \frac{1}{2}}},\widehat{f})}_{L_p^2}& =-\widehat{E}·\overline{\widehat{b}}=\widehat{E}({\partial }_t\overline{\widehat{E}}-\overline{ik\times \widehat{B}})=\widehat{E}{\partial }_t\overline{\widehat{E}}+\widehat{E}·ik\times \overline{\widehat{B}}\hfill \\ \hfill & =\widehat{E}{\partial }_t\overline{\widehat{E}}-\overline{\widehat{B}}·(ik\times \widehat{E})=\widehat{E}{\partial }_t\overline{\widehat{E}}+\widehat{B}{\partial }_t\overline{\widehat{B}}={\displaystyle \frac{1}{2}}{\partial }_t(|\widehat{E}{|}^2+|\widehat{B}{|}^2).\hfill \end{array}$$

Multiplying (28) by the complex conjugate of $\widehat{f}(t,k,p)$, integrating over p, taking the real part, and then integrating with respect to t, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}\left(|\widehat{f}{(t,k,·)|}_{L_p^2}^2+|\widehat{E}{(t,k)|}^2+{|\widehat{B}(t,k)|}^2\right)+{\int }_0^t\mathcal{R}{(L\widehat{f},\widehat{f})}_{L_p^2}d\tau \hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\left(\parallel {\widehat{f}}_0{(k,·)\parallel }_{L_p^2}^2+|{\widehat{E}}_0{\left(k\right)|}^2+{|{\widehat{B}}_0\left(k\right)|}^2\right)-{\int }_0^t\mathcal{R}{(\widehat{E}\ast {\nabla }_p\widehat{f},\widehat{f})}_{L_p^2}d\tau \hfill \\ \hfill & -{\int }_0^t\mathcal{R}{(\tilde{p}\times \widehat{B}\ast {\nabla }_p\widehat{f},\widehat{f})}_{L_p^2}d\tau +{\int }_0^t\mathcal{R}{({\displaystyle \frac{1}{2}}\tilde{p}·\widehat{E}\ast \widehat{f},\widehat{f})}_{L_p^2}d\tau ,\hfill \end{array}$$

(29)

where ${\widehat{f}}_0(k,·)=\widehat{f}(0,k,·)$, ${\widehat{E}}_0\left(k\right)=\widehat{E}(0,k)$, ${\widehat{B}}_0\left(k\right)=\widehat{B}(0,k)$. Applying the coercivity estimate of L (11) and taking the square root of both sides, we deduce

$$\begin{array}{cc}\hfill & |\widehat{f}{(t,k,·)|}_{L_p^2}+|\widehat{E}(t,k)|+|\widehat{B}(t,k)|+{\left({\int }_0^t{|\{\mathbf{I}-\mathbf{P}\}\widehat{f}(t,k,·)|}_D^{2}d\tau \right)}^{1/2}\hfill \\ \hfill & \lesssim |{\widehat{f}}_0{(k,·)|}_{L_p^2}+|{\widehat{E}}_0\left(k\right)|+|{\widehat{B}}_0\left(k\right)|+{\left({\int }_0^t\left|\mathcal{R}{(\widehat{E}\ast {\nabla }_p\widehat{f},\widehat{f})}_{L_p^2}\right|d\tau \right)}^{1/2}\hfill \\ \hfill & +{\left({\int }_0^t\left|\mathcal{R}{(\tilde{p}\times \widehat{B}\ast {\nabla }_p\widehat{f},\widehat{f})}_{L_p^2}\right|d\tau \right)}^{1/2}+{\left({\int }_0^t\left|\mathcal{R}{({\displaystyle \frac{1}{2}}\tilde{p}·\widehat{E}\ast \widehat{f},\widehat{f})}_{L_p^2}\right|d\tau \right)}^{1/2}.\hfill \end{array}$$

(30)

Taking the supremum over $0\le t\le T$ on both sides of (30) and integrating with respect to $d\Sigma \left(k\right)$ over ${\mathbb{Z}}^3$, we get

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{Z}}_k^3}\underset{0\le t\le T}{\sup }\left(|\widehat{f}{(t,k,·)|}_{L_p^2}+|\widehat{E}(t,k)|+|\widehat{B}(t,k)|\right)d\Sigma \left(k\right)\hfill \\ \hfill & +{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^T{|\{\mathbf{I}-\mathbf{P}\}\widehat{f}(t,k,·)|}_D^{2}dt\right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & \lesssim \parallel {\widehat{f}}_0{\parallel }_{L_k^1{L}_p^2}+\parallel {\widehat{E}}_0{\left(k\right)\parallel }_{L_k^1}+{\parallel {\widehat{B}}_0\left(k\right)\parallel }_{L_k^1}+{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^T\left|{\left(\widehat{E}\ast {\nabla }_p\widehat{f},\widehat{f}\right)}_{L_p^2}\right|dt\right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & +{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^T\left|{\left(\tilde{p}\times \widehat{B}\ast {\nabla }_p\widehat{f},\widehat{f}\right)}_{L_p^2}\right|dt\right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & +{\int }_{{\mathbb{Z}}_k^3}{\left({\int }_0^T\left|{\left({\displaystyle \frac{1}{2}}\tilde{p}·\widehat{E}\ast \widehat{f},\widehat{f}\right)}_{L_p^2}\right|dt\right)}^{1/2}d\Sigma \left(k\right)\hfill \\ \hfill & \lesssim \parallel {\widehat{f}}_0{\parallel }_{L_k^1{L}_p^2}+\parallel {\widehat{E}}_0{\left(k\right)\parallel }_{L_k^1}+{\parallel {\widehat{B}}_0\left(k\right)\parallel }_{L_k^1}+\eta {\mathcal{D}}_T\left(f\right)+{\mathcal{E}}_T\left(f\right){\mathcal{D}}_T\left(f\right).\hfill \end{array}$$

where Lemma A1 has been employed. This completes the proof. □

To derive uniform-in-time a priori estimates for solutions to the Cauchy problem (5) and (7), we need impose the following smallness assumption

$${\mathcal{E}}_T\left(f\right)\le \delta ,$$

(31)

where $\delta >0$ is a sufficiently small constant.

Lemma 3.

Under the assumptions of Theorem 1 and the assumption (31), it holds that

$${\mathcal{E}}_T\left(f\right)+{\mathcal{D}}_T\left(f\right)\lesssim {ϵ}_0.$$

(32)

Proof. 

Choose a constant $\kappa >0$ and add $\kappa \times$ (24) to (27). This yields

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

Choosing $\kappa >0$ sufficiently small, we absorb the terms containing $\kappa$ on the right-hand side to obtain

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

Next, selecting $\eta >0$ sufficiently small gives

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

Finally, taking $\delta$ sufficiently small (recall ${\mathcal{E}}_T\left(f\right)\le \delta$) leads to the desired result (32). □

Proof of Theorem 1.

The local-in-time existence and uniqueness of the solution to the Cauchy problem (5) can be established based on the energy functional given by ${\mathcal{E}}_T\left(f\right)$. The details which are similar to those in reference [20] are omitted for brevity. Combing this local-in-time existence, we obtain the global mild solution and uniqueness and the uniform estimate (21) immediately by the standard continuity argument with the smallness assumption on the initial data ${ϵ}_0$. □

4. Conclusions

To summarize, we establish the global existence and uniqueness of mild solutions to the relativistic VMFP system near a global Maxwellian equilibrium, provided the initial perturbation is sufficiently small. The low-regularity space $L_k^1{L}_T^{\infty }{L}_p^2$ circumvents the high-order Sobolev regularity requirements of previous works. Combining macro–micro decomposition with a delicately constructed energy-dissipation framework, we derive a time-uniform a priori estimate for the corresponding energy functional. Theoretically, we extend the low-regularity analytical machinery originally developed for the Landau and Boltzmann equations to the relativistic VMFP system, which substantially weakens the high Sobolev regularity restrictions on initial data imposed by classical theories. This generalization expands the applicable range of the aforesaid analytical framework and furnishes a rigorous mathematical justification for the global well-posedness of this fundamental kinetic model. Methodologically, the present work validates that the low-regularity energy method is applicable to intricate coupled multi-physics systems, and offers new analytical insights for future research on other intractable kinetic equations.