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Spectrum Structure and Behaviors of the Vlasov-Maxwell-Boltzmann Systems

The spectrum structures and behaviors of the Vlasov-Maxwell-Boltzmann (VMB) systems for both two species and one species are studied in this paper. The analysis shows the effect of the Lorentz force induced by the electro-magnetic field leads to some different structure of spectrum from the classical Boltzmann equation and the closely related Vlasov-Poisson-Boltzmann system. And the significant difference between the two-species VMB model and one-species VMB model are given. The structure in high frequency illustrates the hyperbolic structure of the Maxwell equation. Furthermore, the long time behaviors and the optimal convergence rates to the equilibrium of the Vlasov-Maxwell-Boltzmann systems for both two species and one species are established based on the spectrum analysis, and in particular the phenomena of the electric field dominating and magnetic field dominating are observed for the one-species Vlasov-Maxwell-Boltzmann system.


Introduction
The Vlasov-Maxwell-Boltzmann system is a fundamental model in plasma physics for the describing the time evolution of dilute charged particles, such as electrons and ions, under the influence of the self-induced Lorentz forces governed by Maxwell equations, cf. [2] for derivation and the physical background. In the literatures, there are two basic models, one is called two-species Vlasov-Maxwell-Boltzmann system that describes both the time evolution of ions and electrons where F ± = F ± (t, x, v) are number density distribution functions of charged particles, and E(t, x), B(t, x) denote the electro and magnetic fields, respectively. Here, the operator Q(·, ·) describing the binary elastic collisions is given by where The other one called one-species Valsov-Maxwell-Boltzmann system that takes account of the fact that the ion is much heavier than the electron and the electron moves faster than the ion so that the time evolution of electron can be considered under a fixed background of ion distribution where the time evolution of electrons is considered under the influence of a fixed ions background n b (x), and F = F (x, v, t) is the number density function of electrons. And the operator Q(·, ·) is defined by (1.2).
The Vlasov-Maxwell-Boltzmann system has been intensively studied and many important progress has been made in [7,6,10,11,19]. For instance, Guo [10] has first established the global existence of classical solutions in three-dimensional torus when the initial data is a small perturbation of a global Maxwellian, and Strain [19] proved the the corresponding global existence result of classical solutions in R 3 . The diffusive limit for two-species VMB system was shown in [11]. The recent important investigation of long time behavior of global solution near the global Maxwellian studied in [7,6] shed light on the complicacy of the Valsov-Maxwell-Boltzmann (VMB) system. Therein, it was shown by the method of compensated functions in [6] that the total energy of the linearized one-species VMB system decays at the rate (1 + t) − 3 8 (but the decay rate of the nonlinear system has not been obtained since the decay rate of the linear system obtained therein seems to be not enough to deal with the time evolution of the nonlinear terms) and in [7] that the total energy of nonlinear two-species VMB system decays at the rate (1 + t) − 3 4 . A natural and interesting question follows then, what is the main characters of the structures and timeasymptotical behaviors of the Valsov-Maxwell-Boltzmann system on the transport of charged particles under the influence of electromagnetic fields governed by the Maxwell equation and/or mutual interaction between charged particles. One of the methods to investigate this properties is the analysis of spectrum structures of the Valsov-Maxwell-Boltzmann system. However, in contrast to the works on Boltzmann equation [8,15,16,20,21,22] and Vlasov-Poisson-Boltzmann system [13,14], the spectrum of the linearized Valsov-Maxwell-Boltzmann system around a global equilibrium has not been given despite of its importance. The main purpose of this paper is to fill in this gap.
The main purpose of this paper is to consider the structures of the linearized systems for the above VMB systems (1.1) and (1.3) around a global equilibrium so that some specific properties influenced by the electromagnetic fields and/or the mutual interaction between charged particles are revealed. These spectrum structures are important for understanding the behavior of the solutions to these systems both locally in space and globally in time. Indeed, the main purpose in the present paper is to continue the project to study the structures and behaviors on the transport of charged particles under the influence of electric fields, electromagnetic fields, or magnetic field. As it has already been studied in [3,13,14] about the structure structures and behaviors of both one-species and two-species Vlasov-Poisson-Boltzmann systems, the influence of electric field gives rise to some complicated phenomena on the transport of charged particles. Indeed, it was shown in [3,13] that the global distribution function to the one-species Vlasov-Poisson-Boltzmann system tends to the global Maxwellian at the optimal rate (1 + t) − 1 4 in R 3 which is slower than the Boltzmann equation and is caused mainly due to the slower but optimal decay (1 + t) − 1 4 of the electric fields. On the other hand, with the influence of mutual interaction of charged particles included, the global distribution function and electric field of the two-species Vlasov-Poisson-Boltzmann system was proved in [23,14] to converge to the equilibrium at the optimal rate (1 + t) − 3 4 for the distribution functions and e −O(1)t for the electric field, where the key issue lies in the fact that the mutual interaction of charged particles leads to spectral gap.
In the present paper, we shall establish the structure of the spectrum in details for both two-species and one-species linearized VMB systems, analyze the corresponding semigroups to the linearized operators and show the optimal decay rates of global solutions to the linearized VMB systems, and finally obtain the (optimal) timeasymptotical behaviors of global solutions to the nonlinear VMB systems of both two-species and one-species types. To be more precisely, we first establish the structure of the spectrum in details for two linearized VMB systems and reveal the effect of electromagnetic fields on the distribution of spectrum of linearized operators. This effect gives rise to a completely different distribution of the spectrum of the linearized operators for both two-species and one-species VMB system. Indeed, the influence of electromagnetic fields on the transport of one-species charged particles causes the linearized VMB system admits higher order eigenvalues (spectrum) λ 6 = λ 7 = −O(1)|ξ| 4 at lower frequency 0 < |ξ| ≪ 1 besides those behaving like λ j = ji − O(1)|ξ| 2 for j = 0, ±1 density, momentum and energy related to the distribution function f are shown to decay at the different optimal rates (1 + t) − 5 4 , (1 + t) − 5 8 and (1 + t) − 3 4 respectively (refer to Theorem 2.11 for details). Furthermore, we can also study rigorously the time-asymptotical behaviors of global solutions to the nonlinear VMB system and in particular obtain the optimal time decay rate (1 + t) − 5 8 of the distribution function f , the optimal time decay rate (1 + t) − 3 8 of the magnetic fields B, and the faster time decay rate (1 + t) − 3 4 ln(1 + t) of electric field E (refer to Theorem 2.12 for details). This gives more information than those obtained by energy method in [7,6] where only the upper bound of the decay rate of total energy was obtained. Here, we should mention that the time-convergence rate (1 + t) − 3 8 of the total energy of global solution for linearized one-species VMB system in [7] indeed corresponds to the case of the magnetic field dominating phenomena but without the above analysis in details made in the present paper, and the results corresponding to the electric field dominating phenomena is never observed before. In particular, we obtain the optimal decay rates of the solution to the nonlinear one-species VMB system (refer to Theorem 2.12 for details) which was not solved in [7].
The rest of this paper will be organized as follows. In Section 2, the main results on spectrum structures and time-asymptotic behaviors of global solutions for two-species VMB and one-species VMB are stated in section 2.1 and section 2.2 respectively. In Section 3 and Section 4, the spectrum structures of the two linearized systems for both two-species and one-species charge motion will be analyzed with detailed description in low and high frequency regions. Based on this analysis on the linearized operators, the decomposition of the corresponding semigroups generated by these operators will be given in Section 5 together with the optimal convergence rates to the equilibrium in time. The optimal convergence rates of the global solution to the original nonlinear system will be studied in the last section.

Two-species VMB system
We first consider the Cauchy problem for the two-species Valsov-Maxwell-Boltzmann system (1.1) as follows (2.4) In order to study the spectrum structure of the system (2.4), it is convenient to consider the following system for F 1 = F + + F − and F 2 = F + − F − that takes care of the cancellation in the original system:  In the following, we will consider the spectrum of the operator by linearizing the system (2.5) around an equilibrium state (F * 1 , F * 2 , E * , B * ) = (M (v), 0, 0, 0) with M (v) being the normalized Maxwellian given by Set Then the system (2.5) for (F 1 , F 2 , E, B) can be written as the following system for (f 1 , f 2 , E, B): where , The linearized collision operators L and L 1 can be written as, cf. [1,24], where ν(v) is called the collision frequency, K and K 1 are self-adjoint compact operators on L 2 (R 3 v ) with real symmetric integral kernels k(v, v * ) and k 1 (v, v * ). The null space of the operator L, denoted by N 0 , is a subspace spanned by the orthonoraml basis {χ j , j = 0, 1, · · · , 4} given by ; (2.12) and the null space of the operator L 1 , denoted by N 1 , is spanned only by √ M . For later use, denote by L 2 (R 3 ) the Hilbert space of complex valued functions on R 3 with the inner product and the norm given by And let P 0 , P d be the projection operators from L 2 (R 3 v ) to the subspace N 0 , N 1 with 13) (2.14) From the Boltzmann's H-theorem, the linearized collision operators L and L 1 are non-positive, precisely, there is a constant µ > 0 such that (L 1 f, f ) ≤ −µ P r f 2 , f ∈ D(L 1 ), (2.16) where D(L) and D(L 1 ) are the domains of L and L 1 given by In addition, for the hard sphere model, ν satisfies Without the loss of generality, we choose ν(0) ≥ ν 0 ≥ µ > 0 throughout this paper.
From the system (2.6)-(2.10) for (f 1 , f 2 , E, B), we have the following decoupled linearized system for f 1 and (f 2 , E, B): The equation (2.18) is simply the linearized Boltzmann equation around a global Maxwellian so that its spectrum structure is well established since 1970s. Therefore, we only need to study the spectrum structure of the linear system (2.19)-(2.22) on (f 2 , E, B).
For convenience of notations, rewrite the linearized system for and where the operators B 0 and A 0 are operators on

25)
with Take the Fourier transform in (2.24) with respect to x to have Note that it is difficult to study the spectrum structure of the system (2.28) directly because of the constraints onÊ andB given in the second and third equations. One of the key observations in this paper is that by using the identity F = (F · y)y − y × y × F for any F ∈ R 3 and y ∈ S 2 , we can firstly solve forV = (f 2 , ω ×Ê, ω ×B) with ω = ξ/|ξ| so that by combining these two constraints, we have the full information onÛ . For this reason, we consider the following system forV : Here, for ξ = 0,B Before further discussion, we will give a remark on the eigenvalues and eigenfunctions of the original system and the reduced system (2.29). (2.32) The eigenvalues of the system (2.32) are same as those of (2.23) and (2.29), and the eigenfunctions of the system (2.32) can be obtained as linear combinations of those for (2.23) and (2.29). In fact, let λ be an eigenvalue with the corresponding eigenfunction denoted by φ of (2.23), and β be an eigenvalue with the corresponding eigenvector denoted by Ψ = (ψ, E, B) of (2.29). Then U = (φ, φ, 0, 0) is the corresponding eigenvector with the eigenvalue λ, and V = (ψ, −ψ, − iξ is the corresponding eigenvector with the eigenvalue β. By the above argument, from now on, we will focus on the system (2.24). It is interesting to find out that the spectrum structure depends on the decomposition of the asymptotics in low frequency and high frequency like the classical Boltzmann equation and the Vlasov-Poisson-Boltzmann system, but depends on the low-intermediatehigh frequencies. This is mainly due to the hyperbolic structure of the Maxwell equations, in particular the effect of the magnetic field on the velocity field. More precisely, the spectrum of linearized operator contain an eigenvalue in low frequency located in a small neighborhood of the origin, and two eigenvalues in high frequency located in two small neighborhoods centered at the points λ = ±i|ξ| respectively. Except these eigenvalues, there is a spectral gap for the intermediate frequency. Precisely, we have the following result on the spectrum structure.
There exists a constant r 1 > 0 such that the spectrum β ∈ σ(Â 1 (ξ)) ⊂ C for s = |ξ| > r 1 consists of four eigenvalues {β j (s), j = 1, 2, 3, 4} in the domain Reλ > −µ/2. In particular, the eigenvalues satisfy for two positive constants c 1 and c 2 . For any r 1 > r 0 > 0, there exists α = α(r 0 , r 1 ) > 0 such that for r 0 ≤ |ξ| ≤ r 1 , Based on the spectrum structure given in the above theorem, the semigroup generated by the linearized operator of the Vlasov-Maxwell-Boltmann system can be decomposed in three parts according to the frequency in low, high and intermediate regions so that the optimal time decay rates of the solution can be obtained for the linearized system. Note that a higher regularity on the initial data is needed because of the spectrum structure in the high frequency region. With the estimates on the semigroup, the optimal decay in time of the solution to the original nonlinear problem can also be obtained. Before stating results on nonlinear problem, let us first introduce the following notations.
so that the Sobolev spaces H N and H N w are given by equipped with the norms For q ≥ 1, denote In the following, denote by · L 2 x,v and · L 2 ξ,v the norms of the function spaces L 2 (R 3 x × R 3 v ) and L 2 (R 3 ξ × R 3 v ) respectively, and by · L 2 x , · L 2 ξ and · L 2 v the norms of the function spaces L 2 (R 3 x ), L 2 (R 3 ξ ) and L 2 (R 3 v ) respectively. For any integer m ≥ 1, denote by · H m x and · L 2 v (H m x ) the norms in the spaces H m (R 3 x ) and x )) respectively. Moreover, introduce a weighted Hilbert space L 2 ξ (R 3 v ) for ξ = 0 defined by

For any vectors
, define a weighted inner product and the corresponding norm by and another L 2 inner product and norm by For simplicity, denote x . With the above preparation, we first state the estimates on the semigroup to linearized system. Theorem 2.3. The semigroup S(t, ξ) = e tÂ1(ξ) with ξ = sω ∈ R 3 and s = |ξ| = 0 can be decomposed into that has the following properties where C 3 ξ = {y ∈ C 3 : y·ξ = 0}. Here, (λ j (s), Ψ j (s, ω)) and (β j (s), Φ j (s, ω)) are the eigenvalues and eigenvectors of the operatorÂ 1 (ξ) in low and high frequency regions with properties given in Theorem 3.12 and Theorem 3.15. And S 3 (t, ξ)U =: Then, we have the optimal time convergence rates of global solutions to linearized system (2.24).
Theorem 2.4. Let (f 2 (t), E(t), B(t)) be a solution of the system (2.24). If the initial data
Finally for the two-species system, we have the optimal time convergence rates of global solutions to the original nonlinear system (2.6)-(2.11) as follows.
Theorem 2.6. Under the conditions given in Theorem 2.5, if we further assume that there exist positive constants d 0 , d 1 > 0 and a small constant r 0 > 0 so that the Fourier transform (f 1,0 ,f 2,0 ,Ê 0 ,B 0 ) of the initial data then the global solution (f 1 , f 2 , E, B) to the system (2.6)-(2.11) satisfies for t > 0 large with two constants C 2 > C 1 > 0, and in particular for j = 0, 1, 2, 3, 4 Remark 2.7. From the above theorems, Remark 2.1 and those estimates obtained in [13,14], we justify the different time-asymptotic phenomena of charged particle transport at mesoscope level that are determined by the irrotational electric field and electromagnetic field respectively. Indeed, it was shown in [13] that the distribution function converges to the global Maxwellian at the same optimal rate (1+t) − 1 4 as the irrotational electric field for the unipolar Vlasov-Poisson-Boltzmann in R 3 x × R 3 v ; and in [14] it shows that the distribution function converges to the global Maxwellian at the optimal rate (1 + t) − 3 4 , which is slower than the exponential decay rate of the irrotational electric field for bipolar Vlasov-Poisson-Boltzmann equations in R 3 x × R 3 v . While in the appearance of electro-magnetic force, the electric field decays at the optimal rate (1 + t) − 5 4 , which is faster than those of the distribution function and the magnetic field. In particular, it is the rotational vector field part of electric field for linear system that decays at algebraic rate, while the gradient vector field part of electric field decays exponentially in time.

One-species VMB system
We now turn to the one-species Vlasov-Maxwell-Boltzmann system where the time evolution of electrons is considered under the influence of a fixed ion background n b (x). That is, consider is the number density function of electrons, and n b (x) > 0 is assumed to be a constant representing the spatially uniform density of the ionic background. Take n b = 1 without loss of generality.
The one-species VMB system (2.42) has a stationary solution (F * , E * , B * ) = (M (v), 0, 0) with M (v) being the normalized global Maxwellian defined above. Set Then the one-species VMB system (2.42) for (F, E, B) is reformulated in terms of (f, E, B) into Consider the linearized one-speices Vlasov-Maxwell-Boltzmann system: For convenience of notations, rewrite the linearized system for with the operator A 2 defined by Similarly, consider the following system forV = (f , ω ×Ê, ω ×B): Here, for ξ = 0,B Different from the two-species VMB, the spectrum of the linearized operator of one-species VMB consists of nine eigenvalues in low frequency located in small neighborhoods of three points 0, ±i, and two eigenvalues in high frequency located in two small neighborhoods centered at the points λ = ±i|ξ| respectively in complex plane. Except these eigenvalues, there is a spectral gap for the intermediate frequency. More precisely, we have Theorem 2.8. There exists a constant r 0 > 0 so that the spectrum λ ∈ σ(Â 3 (ξ)) ⊂ C for ξ = sω with |s| ≤ r 0 and ω ∈ S 2 consists of nine points {λ j (s), −1 ≤ j ≤ 7} in the domain Reλ > −µ/2. The spectrum λ j (s) are C ∞ functions of s for |s| ≤ r 0 . In particular, the eigenvalues λ j (s) have the following asymptotic expansions when |s| ≤ r 0 where a j > 0 (0 ≤ j ≤ 3) and b j > 0 (1 ≤ j ≤ 2) are constants defined in Theorem 4.9.
being held, then the unique solution (f (t), E(t), B(t)) to the system (2.52) exists globally in time and satisfies for any α, α ′ ∈ N 3 with α ′ ≤ α that

59)
where and below δ(α, α ′ ) =: ∂ α Furthermore, if we assume further that l ≥ 1 and there exist two constants d 0 , d 1 > 0 such that the Fourier , then the following optimal time decay rates holds for t > 0 being large enough and C 2 ≥ C 1 > 0 being two constants, and in particular 67) 68) 69) being held, then there exists globally in time a unique solution (f (t), E(t), B(t)) to the system (2.52) which satisfies for any α, α ′ ∈ N 3 with α ′ ≤ α that

73)
where and below δ(α, α ′ ) =: Furthermore, if we also assume that l ≥ 2 and there exist a constant d 0 > 0 such that the Fourier transform , B(t)) satisfies the following optimal time decay rates for t > 0 being large enough and C 2 ≥ C 1 > 0 being two constants, and in particular Finally, we have the time convergence rates of global solutions to the nonlinear system (2.43)-(2.47) as follows.
x ≤ δ 0 with δ 0 > 0 being small enough. Then, the unique solution (f, E, B) to the VMB system (2.43)-(2.47) exists globally in time and belongs to H N +3 Moreover, if it also holds ∇ x · E 0 = (f 0 , χ 0 ), then the global solution satisfies for k = 0, 1 that

87)
and in particular Furthermore, if there also exist a constant d 0 > 0 and a small constant r 0 > 0 so that the Fourier transform for t > 0 large enough with two constants C 2 > C 1 , and in particular Remark 2.13. Let us give an example of the initial function (f 0 , E 0 , B 0 ) which satisfies the assumptions of Theorem 2.12. For a positive constant d 0 , we define (f 0 , E 0 , B 0 ) as Remark 2.14. In the case of ∇ x · E 0 = (f 0 , √ M ), the solutions of the nonlinear VMB system have no decay rates. The main reason is that the magnetic field decay too slow. For instance, we assume that f, E, B decay at the same rates (1 + t) − 1 4 , (1 + t) − 1 4 and (1 + t) − 3 8 respectively as the linear solution, then we can obtain that the key quadratic nonlinear term decays at most at the rate (1 + t) − 1 2 . Thus, making use of the Duhamel's principle to represent the solution via the semigroup, the corresponding nonlinear term to magnetic field B generated from the nonhomogeneous source decays as Thus, the bootstrap argument breaks down and one can not expect the magnetic field to decay as (1 + t) − 3 8 in the nonlinear case.

Spectral analysis for two species case
For the study on the spectrum structure of the two-species VMB, in the following subsection, we will first investigate some properties of the operatorÂ 1 (ξ) that lead to the decription of its spectra and resolvent. And then the asymptotics of its eigenvalues and eigenfunctions in low and high frequency regions will be given in the next two subsections.

Spectrum structure
First of all, note that P d is a self-adjoint operator satisfying (P d f, P d g) = (P d f, g) = (f, P d g). Hence, (3.1) Also, note thatB 1 (ξ) is a linear operator from the space L 2 ξ (R 3 ) to itself, and for any y ∈ C 3 ξ , . Direct computation shows the dissipation of bothÂ 1 (ξ) andÂ 1 (ξ) * , namely, SinceÂ 1 (ξ) is a densely defined closed operator, it follows from Corollary 4.4 on p.15 of [18] that the operator Define a 6 × 6 matrix by Proof. First, we compute the eigenvalues of the operator B 3 (ξ). For this, consider It follows that Multiplying (3.7) by λ and using (3.8) and (3.3) give This completes the proof of the lemma.

Now denote by T a linear operator on
, and we define the corresponding norms of T by We will make use of the following decomposition associated with the operatorÂ 1 (ξ) for |ξ| > 0 where G 1 (ξ), G 2 (ξ) are defined by (3.11). For Reλ > −ν 0 and λ = ±i|ξ|, we have Let K 1 , K 4 be the operators on the space X and Y , and K 2 , K 3 be the operators Y → X and X → Y respectively. Let K be a matrix operator on X × Y defined by Then, we have Lemma 3.4. If the norms of K 1 , K 2 , K 3 and K 4 satisfy Proof. Decompose I + K into it follows that Proof. The proof of (3.22), (3.23), (3.26) and (3.27) can be found in Lemma 2.3 of [13]. We only need to prove (3.24) and (3.25). Since With Lemma 3.5, we can investigate the spectrum set of the operatorÂ 1 (ξ) in the intermediate and high frequency regions.
We now investigate the spectrum and resolvent sets ofÂ 1 (ξ) in low frequency. For this, we decompose (3.34) Lemma 3.7. Let ξ = 0 and Q(ξ) defined by (3.34). We have Proof. The proof of (3.35)-(3.37) can be found in [14]. Repeating a same argument as to the estimate (3.25), we can obtain (3.38). Hence, we omit the details for brevity.

Asymptotics in low frequency
In this subsection, we study in the low frequency region, the asymptotics of the eigenvalues and eigenvectors of the operatorÂ 1 (ξ). In terms of (2.31), the eigenvalue problemÂ Let f be the eigenfunction of (3.47), we rewrite f in the form By Lemma 3.7, (3.34) and (3.50), the microscopic part f 1 can be represented for any Reλ > −µ by Substituting (3.51) into (3.49) and (3.48), we obtain the eigenvalue problem for (C 0 , X, Y ) as By changing variable (v · ξ) → |ξ|v 1 and using the rotational invariance of the operator L 1 , we have the following transformation.
the application of the implicit function theorem implies that there exist constants r 0 , r 1 > 0 and a unique C ∞ function λ 0 (s) such that Similarly to Lemma 3.10, we can prove that there is b 1 > 0 so that This together with the continuity of D 2 (λ, 0) imply that there is a constant δ 1 > 0 such that it follows that Thus for r 0 > 0 small enough, We then conclude that the equation With Lemmas 3.10-3.11, we are able to construct the eigenvector Ψ j (s, ω) corresponding to the eigenvalue λ j in the low frequency. Indeed, we have Theorem 3.12. There exist two constants r 0 > 0 and b 2 > 0 so that the spectrum λ ∈ σ(Â 1 (ξ)) ⊂ C for ξ = sω with |s| ≤ r 0 and ω ∈ S 2 consists of two points {λ j (s), j = 1, 2} in the domain Reλ > −b 2 . The spectrum λ j (s) and the corresponding eigenvector Ψ j (s, ω) = (ψ j , X j , Y j )(s, ω) are C ∞ functions of s for |s| ≤ r 0 . In particular, the eigenvalues admit the following asymptotic expansion for |s| ≤ r 0 The eigenvectors Ψ j = (ψ j , X j , Y j ) are orthogonal to each other and satisfy , and the coefficients ψ j,n , X j,n , Y j,n are given by Here, W j (j = 1, 2) are two orthonormal vectors satisfying W j · ω = 0.
Proof. The eigenvalues λ j (s) and the eigenvectors Ψ j (s, ω) = (ψ j , X j , Y j )(s, ω), j = 1, 2, can be constructed as follows. Let b 2 = min{b 0 , b 1 } and take λ j = λ(s) to be the solution of the equation D 1 (λ, s) = 0 defined in Lemma 3.11, and choose C 0 = 0, and X j = ω × W j with W j , j = 1, 2, being two linearly independent vectors satisfying W j · ω = 0 and which satisfy (Ψ 1 (s, ω), Ψ * 2 (s, ω)) = 0. We can normalize them by taking To study the asymptotic expression of eigenvectors in the low frequency, we can take Taylor expansion of both eigenvalues and eigenvectors as Substituting the above expansions into (3.70), we obtain b j (0) = 0, b ′ j (0) = a 1 and b j (−s) = −b j (s). This and (3.69) give the expansion of Ψ j (s, ω) for j = 1, 2, stated in (3.68). And then it completes the proof of the theorem.

Asymptotics in high frequency
We now turn to study the asymptotic expansions of the eigenvalues and eigenvectors in the high frequency region. Firstly, recalling the eigenvalue problem for Reλ ≥ −ν 0 /2 and |ξ| > R 0 . Then it follows from (3.71) that Substituting (3.73) into (3.72) and using the transformation Multiplying (3.74) by λ and using (3.75) and (3.3), we obtain Similar to the proof of Lemma 3.5, we can obtain Lemma 3.13. For any δ > 0, we have As consequence, it holds that for |ξ| > R 0 We now study the equation (3.77) as follows.
In particular, there are two constants c 1 , c 2 > 0 such that Proof. For any fixed s > R 0 , we define a function of λ as . It is straightforward to verify that a solution of D(λ, s) = 0 for any fixed s > R 0 is a fixed point of G 1 j (λ). Consider an equivalent equation of (3.83) as . By (3.81), when R 1 > 0 is large enough and δ > 0 is small enough, it holds for s > R 1 and |β| ≤ δ that Thus G 2 j (β) is a contraction mapping on B δ (0) and there is a unique fixed point β j (s) of G 2 j (β). Thus λ j (s) = jis + β j (s) is the solution of D(λ, s) = 0 and |β j (s)| ≤ Cs −1/2 because |B j (β, s)| ≤ Cs −1/2 due to (3.81).
We now turn to prove (3.82). For this, we decompose β j (s), j = ±1, into Note that In the following, we will prove , we obtain for s > 1 that As for by the change variables (u 1 , u 2 , u 3 ) = ((v 1 ± 1)s, v 2 , v 3 ), we obtain for s > 1 that where C 4 > 0 is a constant. By Lemma 3.13, for any 0 < ǫ ≪ 1 there exists s > R 1 such that This and (3.88) lead to Thus, by combining with (3.86), (3.87) and (3.88), there exist constants C 5 , C 6 > 0 such that For I 2 , we have The following theorem gives the asymptotic expansions of eigenvalues and eigenvectors in the high frequency region.

Spectral analysis for one-species case
In this section, we will study the spectrum structure of the linearized system 2.54 of one-species VMB. It is interesting to find out that its structure is very different in the low frequency region from the case of two-species. And this essential difference comes from the lack of the cancellation in the one-species system.
With the help of Lemmas 4.2-4.4, we obtain the spectral and resolvent sets of the operatorÂ 3 (ξ) by applying the similar argument as Lemma 3.8.

Asymptotics in low frequency
We study the low frequency asymptotics of the eigenvalues and eigenvectors of the operatorÂ 3 (ξ) in this subsection. In terms of (2.56), the eigenvalue problemÂ 3 (4.21) By macro-micro decomposition, the eigenfunction f of (4.21) can be decomposed into f = f 0 + f 1 =: P 0 f + P 1 f. Then the first equation of (4.21) gives rise to Substituting (4.23) into (4.22), we obtain the eigenvalue problem for (f 0 , X, Y ) as where To solve the eigenvalue problem (4.24), we write f 0 ∈ N 0 as f 0 = 4 j=0 W j χ j . Taking the inner product of the first equation of (4.24) and χ j for j = 0, 1, 2, 3, 4 respectively, we have the equations about λ and (W 0 , W, W 4 , X, Y ) with W =: (W 1 , W 2 , W 3 ) and X = (X 1 , ξ for Reλ > −µ: We apply the following transform to simplify the system (4.25)-(4.27).
With the help of (4.30)-(4.33), the equations (4.25)-(4.27) can be simplified as where Multiplying (4.35) by ω i and making the summation of resulted equations with respect to i = 1, 2, 3, we have  Multiplying (4.28) by λ − s 2 R 22 , we obtain Then we multiply above equation by λ and using (4.29) and (3.3) to have Moreover, λ j (s) is an even function and satisfies In particular, λ 0 (s) is a real function.
Moreover, λ j (s) is an even function and satisfies In particular, λ 0 (s) is a real function.
And the corresponding eigenvectors Ψ j (s, ω) = (ψ j , X j , Y j )(s, ω) are defined by which satisfy the orthonormal relation: Then following the similar argument as Theorem 3.12, we can obtain the expansion of Ψ j (s, ω) in (4.55) and (4.56). Hence, we omit the detail for brevity.

Asymptotics in high frequency
The structure of the spectrum for the one-species VMB in high frequency region is similar to the two-species case so that we only sketch the key points here. Recalling the eigenvalue problem Similar to two species case, we obtain The eigenvectors Φ j = (φ j , X j , Y j ) are orthogonal to each other and satisfy (4.63) Here, W j (j = 1, 2, 3, 4) are normal vectors satisfying W j · ω = 0, W 1 · W 2 = 0, W 1 = W 3 , W 2 = W 4 .

The linearized system
In this section, we consider the Cauchy problems (2.24) and (2.52) for the linearized Vlasov-Maxwell-Boltzmann equations in two and one-species and give the optimal time decay rates of the solution based on on spectrum structures obtained in the previous sections.

Semigroup for two-species
Before giving the theorem on the semigroup, we firstly prepare some lemmas on its properties.

(5.3)
In addition, for any x = 0 and U ∈ C 6 it holds Proof. Since iB 3 (ξ) is a self-adjoint operator on C 6 satisfying (3.10), we can prove (5.3) and (5.4) by applying a similar argument as the one for Lemma 3.2 in [13].

Optimal convergence rates for two-species
Based on the decomposition of the semigroup given in the previous subsection, we now study the optimal convergence rates of the solution of the linearized system to the equilibrium.
Then e tA0 U 0 is the solution of the system (2.24). By Lemma 3.1, it holds that This means that the linear operator A 0 generates a strongly continuous contraction semigroup e tA0 on D l , and therefore, U (t) = e tA0 U 0 is a global solution to (2.24) for the linearized Vlasov-Maxwell-Boltzmann equations with initial data U 0 ∈ D l . First of all, we have the upper bounds of the time decay rates given in ) be a solution of the system (2.24). If the initial data Proof. By (5.43) and Theorem 5.4, we have for ω = ξ/|ξ| that By (5.19) and we can estimate the terms on the right hand side of (5.49)-(5.51) as follows: In the low frequency region, by (5.17), we have where T j (ξ), j = 1, 2, are the linear operators with the norm T j (ξ) being uniformly bounded for |ξ| ≤ r 0 . By (3.67) and (3.68), we have where W j , j = 1, 2, is given by (3.68). Since where η 1 > 0 denotes a generic constant that will also be used later, we obtain by (5.53)-(5.55) that In the high frequency region, by (5.18), we have it follows from (5.62)-(5.64) that In fact, the above time decay rates are optimal as shown in ) be a solution of the system (2.24). Assume that the initial data for t > 0 large enough with C 2 ≥ C 1 > 0 as two generic constants.
By (5.55), we have This and (5.73) give (5.70) for t > 0 being large enough. The proof is then completed.

Corresponding results for one-species
Corresponding to the linearized two-species VMB, the decomposition and estimation on the semigroup together with the optimal convergence rates can be obtained for the one-species VMB based on its spectrum structure. However, in the following, we can see that the estimates are very different from the case of two-species mainly due to the lack of cancellation in the one-species model. In addition, for any x > −µ and Lemma 5.9. The operator G 6 (ξ) generates a strongly continuous unitary group on N 0 × C 3 ξ × C 3 ξ for any fixed |ξ| = 0, which satisfies for t > 0 and U ∈ N 0 ∩ L 2 In addition, for any Proof. Since the operator iG 6 (ξ) is self-adjoint on N 0 × C 3 ξ × C 3 ξ with respect to the inner product (·, ·) ξ defined by (4.18), we can prove (5.84) and (5.85) by applying a similar argument as Lemma 3.2 in [13].
With the help of Lemmas 4.3-4.5 and Lemmas 5.8-5.10, we have the decomposition of the semigroup S(t, ξ) = e tÂ3(ξ) as following, the detail of the proof is omitted for brevity.
Theorem 5.11. The semigroup S(t, ξ) = e tÂ3(ξ) with ξ = sω ∈ R 3 and s = |ξ| = 0 has the following decomposition with (λ j (s), Ψ j (s, ω)) and (β j (s), Φ j (s, ω)) being the eigenvalue and eigenvector of the operatorÂ 3 (ξ) given by Theorem 4.9 and Theorem 4.10 for |ξ| ≤ r 0 and |ξ| > r 1 respectively, and S 3 (t, ξ) =: S(t, ξ) − S 1 (t, ξ) − S 2 (t, ξ) satisfies that there exists a constant κ 0 > 0 independent of ξ such that Based on the decomposition of the semigroup given in Theorem 5.11, we now study the optimal convergence rates of the solutions to the linearized system around an equilibrium. For Then e tA2 U 0 is the solution of the system (2.52). By Lemma 4.1, it holds that This implies that the linear operator A 2 generates a strongly continuous contraction semigroup e tA2 in D l , and therefore, U (t) = e tA2 U 0 is a global solution to the IVP (2.52) for the linearized one-species Vlasov-Maxwell-Boltzmann equation for U 0 ∈ D l .

The nonlinear system
In this section, we prove the large time decay rates of the solution to the Cauchy problem for Vlasov-Maxwell-Boltzmann systems with the estimates on the linearized problem obtained in Section 5.

Energy estimates for two species
We first obtain some energy estimates. Let N be a positive integer and U = (f 1 , f 2 , E, B), and for k ≥ 0. For brevity, we write E N (U ) = E N,0 (U ), H N (U ) = H N,0 (U ) and D N (U ) = D N,0 (U ) for k = 0. Firstly, by taking the inner product between χ j (j = 0, 1, 2, 3, 4) and (2.6), we obtain a compressible Euler-Maxwell type system Taking the microscopic projection P 1 on (2.6), we have where the nonlinear term G 1 is denoted by By (6.7), we can express the microscopic part P 1 f 1 as Substituting (6.9) into (6.4)-(6.6), we obtain a compressible Navier-Stokes-Maxwell type system ∂ t n 1 + div x m 1 = 0, (6.10) where the viscosity and heat conductivity coefficients κ 1 , κ 2 > 0 and the remainder terms R 1 , R 2 , R 3 , R 4 are given by By taking the inner product between √ M and (2.7), we obtain ∂ t n 2 + div x m 2 = 0. (6.13) Taking the microscopic projection P r on (2.7), we have where the nonlinear term G 2 is denoted by By (6.14), we can express the microscopic part P r f 2 as where the viscosity coefficient κ 3 > 0 and the remainder terms R 5 , R 6 are defined by The following lemma is from [4,9]. 4,9]). It holds that ,v ). Lemma 6.2 (Macroscopic dissipation). Let (n 1 , m 1 , q 1 ) and (n 2 , E, B) be the strong solutions to (6.10)-(6.12) and (6.17)-(6.19) respectively. Then, there are two constants s 0 , s 1 > 0 such that Proof. First of all, we prove (6.20). Taking the inner product between ∂ α x m 1 and ∂ α x (6.11) with |α| ≤ N − 1, we have For the second and third terms on the left hand side of (6.22), we have and x . (6.24) The first and second terms on the right hand side of (6.22) are bounded by C E N (U )D N (U ). The last term can be estimated by where we have used Lemma 6.1 to obtain Therefore, it follows from (6.22)-(6.25) that Similarly, taking the inner product between ∂ α x q 1 and ∂ α x (6.12) with |α| ≤ N − 1, we have Again, taking the inner product between ∂ α x ∇ x n 1 and ∂ α x (6.5) with |α| ≤ N − 1 gives Next, we turn to show (6.21). Taking the inner product between ∂ α x n 2 and ∂ α x (6.17) with |α| ≤ N − 1, we have Similarly, taking the inner product between ∂ α x E and ∂ α x (6.18) with |α| ≤ N − 1 gives And taking the inner product between ∂ α x ∇ x × B and ∂ α x (6.18) with |α| ≤ N − 1 gives This and the fact that ∇ x × B 2 x . In the following, we shall estimate the microscopic terms to close the energy estimate.
In order to close the energy estimate, we need to estimate the terms ∂ α x ∇ v f with |α| ≤ N − 1. For this, we rewrite (2.6) and (2.7) as Let 1 ≤ k ≤ N , and choose α, β with |β| = k and |α| + |β| ≤ N . Taking inner product between ∂ α x ∂ β v P 1 f 1 and (6.43), between ∂ α x ∂ β v P r f 2 and (6.44) respectively, and then taking summation of the resulted equations, we have d dt Then taking summation 1≤k≤N p k (6.45) with constants p k chosen by we obtain (6.35). The proof of the lemma is then completed.
Taking the inner product between E and (6.18), between f 2 and (2.7) respectively, we have Taking the summation of s 2 (6.48) + (6.49) with s 2 > 0 large enough and ǫ > 0 small enough, we have d dt Taking the inner product between (6.43) and P 1 f 1 , we have is sufficiently small, then the solution U = (f 1 , f 2 , E, B)(t, x, v) to the two-species Vlasov-Maxwell-Boltzmann system (2.6)-(2.11) satisfies x . (6.53)

Convergence rates for two-species
Based on the above energy estimates and the convergence rates of the solution to the linearized system, the convergence rates of the solution to the two-species VMB can be summarized in the following theorem. and similarly First, we consider the time decay rate of the macroscopic density, momentum and energy of f 1 . It follows from (5.78), (6.63) and (6.64) that where we have used ∇ k x G 1 (s) L 2 x,v + G 1 (s) L 2,1 ≤ C(1 + s) − 3 2 Q(t) 2 , k = 0, 1.

Corresponding results on one-species
Finally, we give the corresponding results on the one-species VMB. Let N be a positive integer and U = (f, E, B), and for k ≥ 0. For brevity, we write E 1 N (U ) = E 1 N,0 (U ), H 1 N (U ) = H 1 N,0 (U ) and D 1 N (U ) = D 1 N,0 (U ) for k = 0.