The Vlasov-Maxwell-Fokker-Planck system with relativistic transport in the whole space

Acta Mathematica Scientia 2017,37B(5):1237–1261 http://actams.wipm.ac.cn

THE VLASOV-MAXWELL-FOKKER-PLANCK SYSTEM WITH RELATIVISTIC TRANSPORT IN THE WHOLE SPACE∗

À¤)

Dongcheng YANG (

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China E-mail : [email protected] Abstract In this paper, we consider the Vlasov-Maxwell-Fokker-Planck system with relativistic transport in the whole space. The global solutions to this system near the relativistic Maxwellian are constructed and the optimal time decay rate of global solutions are also obtained by an approach by combining the compensating function and energy method. Key words

relativistic Vlasov-Maxwell-Fokker-Planck system; compensating function; global solutions; optimal time decay rate

2010 MR Subject Classification

1

35F20; 76P05; 82C40

Introduction

The dilute charged particles interacting both through collisions and the self-consistent electro-magnetic field can be described by the non-relativistic or relativistic Vlasov-MaxwellFokker-Planck system in the plasma physics (cf. [1, 2, 6, 9, 17, 24, 25, 27]). In this paper, we consider the Vlasov-Maxwell-Fokker-Planck system with relativistic transport p p p ∂t F + · ∇x F + (E + × B) · ∇p F = ∇p · (∇p F + F ), (1.1) p0 p0 p0 Z p ∂t E − ∇x × B = − F dp, (1.2) p R3 0 ∂t B + ∇x × E = 0, Z ∇x · E = (F − J)dp, R3

(1.3)

∇x · B = 0

(1.4)

with initial data F (0, x, p) = F0 (x, p), E(0, x) = E0 (x), B(0, x) = B0 (x), here F (t, x, p) is the distribution function for the particles at the t ≥ 0, with spatial coordinate x = (x1 , x2 , x3 ) ∈ R3 p and momentum p = (p1 , p2 , p3 ) ∈ R3 . The energy of a particle is given by p0 = 1 + |p|2 . The purpose of this paper is to construct global solutions and study the asymptotic behavior to the system (1.1)–(1.4) near an equilibrium state, that is, a relativistic global Maxwellian ∗ Received

June 7, 2016; revised November 22, 2016. The research of the author was supported partially by the NNSFC Grant (11371151) and the Scientific Research Foundation of Graduate School of South China Normal University.

1238 J(p) = e−

ACTA MATHEMATICA SCIENTIA

√

1+|p|2

Vol.37 Ser.B

. We normalize the background charge ρ0 > 0 so that Z √ 2 e− 1+|p| dp = ρ0 . R3

Then we define perturbation f (t, x, p) around this relativistic global Maxwellian by √ F = J + Jf. Then (1.1) yields the following equation for the perturbation f (t, x, p) p√ p · ∇x f − E · J + Lf = G (1.5) ∂t f + p0 p0 with initial data f (0, x, p) = f0 (x, p), here the linearized Fokker-Planck operator L is given by Lf = −∆p f +

|p|2 3p20 − |p|2 f − f. 4p20 2p30

(1.6)

The nonlinear term is given by p 1 p E · f − (E + × B) · ∇p f. 2 p0 p0 Accordingly, the coupled Maxwell system takes the form Z p √ ∂t E − ∇x × B = − f Jdp, R3 p0 G=

∂t B + ∇x × E = 0, Z √ ∇x · E = f Jdp,

∇x · B = 0

R3

with initial data

f (0, x, p) = f0 (x, p),

E(0, x) = E0 (x),

B(0, x) = B0 (x).

(1.7)

(1.8) (1.9) (1.10)

(1.11)

To present the results in this paper, the following notations are needed. Let multi-indices α and β be α = [α1 , α2 , α3 ], β = [β1 , β2 , β3 ], respectively. We denote ∂βα = ∂xα11 ∂xα22 ∂xα33 ∂pβ11 ∂pβ22 ∂pβ33 . If each component of α is not greater than the corresponding one of α, ¯ we use the standard α ¯ ¯ notation α ≤ α ¯ . And α < α ¯ means that α ≤ α ¯ and |α| < |α|. Cα is the usual binomial coefficient. In addition, A ∼ B means λA ≤ B ≤ λ1 A for a generic constant 0 < λ < 1. h·, ·i is used to denote the standard L2 inner product in R3p , and (·, ·) for the one in R3x × R3p . For two complex vectors a, b ∈ C3 , we use (a|b) = a · ¯b to denote the dot product, where ¯b is the ordinary complex conjugate of b. For any integer m ≥ 0, we use H m to denote the usual Hilbert spaces H m (Rnx × Rnp ) or H m (Rnx ). | · |2 denotes L2 norms in R3p , and k · k denotes L2 norms in R3x or R3x × R3p . Throughout this paper, C is used to denote a positive constant which changes from line to line. For r ≥ 1, we define the standard spatial-momentum mixed Lebesgue space Zr = L2p (Lrx ) with the norm ! 21  Z Z 2/r

kf kZr =

R3

R3

|f (x, p)|r dx

dp

,

f = f (x, p) ∈ Zr .

Corresponding to the linearized operator L, the dissipation rate is given by the following norms  1/2  1/2 p 2 p |f |D = |∇p f |22 + | f |2 , kf kD = k∇p f k2 + k f k2 . 2p0 2p0

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In the following, we use ej with 1 ≤ j ≤ 4 to denote the functions √ √ √ √ −1/2 −1/2 −1/2 −1/2 ρ0 J, ρ1 p1 J, ρ1 p2 J, ρ1 p3 J, (1.12) √ √ where ρ1 = hp1 J, p1 Ji. Notice that they are orthogonal to each other with respect to the standard inner product. It follows from (1.6) that L is non-negative and self-adjoint and the null space of L is the one dimensional space N = span{e1 }. We define P as the projection in L2 (R3p ) to the null space N with (t, x) fixed, we decompose any function f (t, x, p) as f (t, x, p) = Pf + (I − P)f, (1.13) here Pf and (I − P)f are called the macroscopic and microscopic components of the function f respectively. Since the null space is only one dimensional, we set Pf (t, x, p) = af (t, x)e1 .

(1.14)

From Lemma 2.1 in [19], we can obtain |f |2D ∼ |∇p f |22 + |f |22

(1.15)

and there exists a constant δ > 0 such that hLf, f i ≥ δ|(I − P)f |2D .

(1.16)

To prove the existence of global solutions, a key step is to devise an energy estimate so that the a priori estimate can be closed. For this, given a solution [f (t, x, p), E(t, x), B(t, x)] to the Vlasov-Maxwell-Fokker-Planck system with relativistic transport, we define an instant full energy functional and an instant high-order energy functional. The instant full energy functional EN (t) is defined by X X X k∂ α B(t)k2 . (1.17) k∂βα f (t)k2 + k∂ α E(t)k2 + EN (t) ∼ |α|+|β|≤N

|α|≤N

|α|≤N

h Similarly, the instant high-order energy functional EN (t) is defined by X X X h k∂ α B(t)k2 . EN k∂βα f (t)k2 + k∂ α E(t)k2 + (t) ∼

(1.18)

Correspondingly, the dissipation rate DN (t) satisfies X X k∂ α Pf (t)k2 . DN (t) = k∂βα (I − P)f (t)k2D +

(1.19)

|α|+|β|≤N

|α|+|β|≤N

|α|≤N

1≤|α|≤N

|α|≤N

Throughout this paper, the Sobolev index N is chosen to be larger than 4. With these preparations, the main result can be stated as follows. √ Theorem 1.1 Assume that [f0 , E0 , B0 ] satisfies (1.10). Let F0 (x, p) = J + Jf0 (x, p) ≥ 0. There exists a small constant ε > 0 such that if EN (0) ≤ ε, then system (1.5) and (1.8)–(1.10) √ have a unique global solution [f (t, x, p), E(t, x), B(t, x)] with F (t, x, p) = J + Jf (t, x, p) ≥ 0, which satisfies d EN (t) + λDN (t) ≤ 0. (1.20) dt

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Similarly, we can also obtain the following high-order energy inequality. Theorem 1.2 Assume that all the assumptions of Theorem 1.1 hold. Then there is a h functional EN (t) such that the following estimate holds d h E (t) + λDN (t) ≤ Ck∇x × B(t)k2 . (1.21) dt N The next main results are concerned with the large time behavior of the solution of system (1.5) and (1.8)–(1.10) obtained in Theorem 1.1. For any integer j, we write ǫj = Ej (0) + kf0 k2Z1 + kE0 k2L1 + kB0 k2L1 .

(1.22)

Theorem 1.3 Let [f (t, x, p), E(t, x), B(t, x)] be the solution to system (1.5) and (1.8)– (1.10) obtained in Theorem 1.1. If ǫN +2 is sufficiently small, for any t ≥ 0, then 3

EN (t) ≤ CǫN +2 (1 + t)− 2 .

(1.23)

In addition, if ǫN +5 is sufficiently small, for any t ≥ 0, then 5

h EN (t) ≤ CǫN +5 (1 + t)− 2 .

(1.24)

Remark 1.4 From the above results, we notice that electric field E(t) and macroscopic components Pf (t) decay faster than the magnetic field B(t) in L2 -norm. Exponential time decay rate of global solutions to the Vlasov-Poisson-Fokker-Planck system, which is the system in the absence of a magnetic field, was shown in [15, 19]. However time decay rate of global solutions here is only polynomial due to the existence of the magnetic field. In the rest of the introduction, we will review some related works to this paper. Global existence of the solutions to the classical Vlasov-Poisson-Fokker-Planck (VPFP) system were studied by many authors and we will not attempt to list all related references. Classical solutions to the VPFP system were obtained in the case of two dimensions in [21] and for small data in the case of three dimensions in [23]. The existence of global smooth solutions in an L1 setting posed in three dimensions was obtained in [3]. Existence and exponential time decay rate of global solutions to the classical VPFP system near Maxwellian were proved in [15] and for the relativistic VPFP system, the similar results were obtained in [19] by a different method. In addition to the above works, there were some other works (cf. [9–11, 17] and references therein). However, the Vlasov-Maxwell-Fokker-Planck (VMFP) system is more complicated because the coupled Maxwell system gives rise to some analytic difficulty. For this system, the collisionless case was investigated and the global classical solutions exist if control of the momentum support of the distribution function (cf. [11, 12] and references therein). Notice that the local well-posedness of smooth solutions was given in [26]. In this direction, [5] concerns a reduced version of the Vlasov-Maxwell system. On the other hand, for the VMFP system, the global existence of weak solutions was obtained in [8, 9] in different settings. Global existence and uniqueness of smooth solutions were proved in the relativistic version of the VMFP system in the special one and one half dimensional framework [17, 22]. For the various limiting problems on the VMFP system, we refer to [1, 2, 14, 20] and references therein. As related, note that there is some progress on the full Vlasov-Maxwell-Boltzmann system around an equilibrium in [7, 13, 18] and references therein.

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The more physical relativistic Fokker-Planck model was introduced, which contains Lorentz invariant, and also the related mathematical theory was studied in [4]. Global-in-time existence and uniqueness of classical solutions were proved in a dimensionally-reduced version of the VMFP system (cf. [24, 25] and references therein). Existence and a slow time decay of global solutions to the classical VMFP system near the steady state were proved in [27]. Chae also obtained the similar global existence in [6] as that in [27]. However the optimal time decay rate of global solution obtained in [6, 27] to the steady state remains open, although the optimal time decay rate of global solution to the Vlasov-Maxwell-Boltzmann system the steady state have been obtained in [7]. Motivated by [6, 7, 19, 27], we obtain global solution to the VMFP system with relativistic transport near the steady state and then we prove the optimal time decay rate of global solution in this paper. As for the classical system studied in [27], we first construct the compensating function of linear VMFP system with relativistic transport and then we combine this, the estimates of the electromagnetic field and basic energy analysis to obtain global existence. Note that the construction of the compensating function was based on the methods in [11, 16, 27] and the estimates of the electromagnetic field were obtained by the similar arguments as [27] such that our global solutions do not contain the time derivatives. Due to this the proof of the optimal time decay rate is more involved in that we are forced to use the estimates of the electromagnetic field and the techniques of the integral by parts repeatedly to compensate this fact. On the other hand, the key estimate (2.12) does not contain the estimates of the magnetic term and we have to use the Maxwell system to recover this term, which was motivated by [7]. Thus we obtain another crucial estimate (4.13) and then use this, the estimates of the electromagnetic field, integration by parts and some similar estimates as that in [7] to obtain the optimal time decay estimates. We believe that the optimal time decay rate of global solution to the classical VMFP system can be obtained in [6, 27] by this methods.

2

Compensating Function

In the section, we will construct the compensating function for the system (1.5) and (1.8)– (1.10). The construction follows from the work in [11, 16, 19, 27, 28] for some other kinetic equations. f be the subspace spanned by the following four moments containing the null space Let W

N and the images of N under the mappings f (p) 7→

pj p0 f (p)

(j = 1, 2, 3)

f = span{ej |j = 1, 2, 3, 4}, W

f where ej is as defined in (1.12). Let P0 be the orthogonal projection from L2 (R3p ) onto W P0 f =

4 X

k=1

hf, ek iek .

Consider the Vlasov-Maxwell-Fokker-Planck system with relativistic transport p p√ ∂t f + · ∇x f − E · J + Lf = G, p0 p0

(2.1)

and (1.8)–(1.10). Set Wk = hf, ek i, (k = 1, 2, 3, 4), and W = [W1 , W2 , W3 , W4 ]T . Then we have

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from (2.1) that ∂t W +

X

Vol.37 Ser.B

¯ =G ¯ + R, V j ∂xj W + LW

j

¯ are the symmetric matrices defined by where V j (j = 1, 2, 3) and L 4  3 X p j ¯ = {hL[el ], ek i}4 · ξ)e , e i , L , V (ξ) = V ξ = h( k l j k,l=1 p0 k,l=1 j=1 ¯ is the vector component hG, ej i, here R denotes the remaining term which contains either and G the factor (I − P0 )f or E(t, x). The following lemma follows from [19, 27] and its proof is simple as follows.

Lemma 2.1 There exist three 4×4 real constant skew-symmetric matrices Rj (j = 1, 2, 3) and positive constants c1 and c2 such that R(ω) ≡

3 X

R j ωj

j=1

satisfies 4 X

RehhR(ω)V (ω)W, W ii ≥ c1 |W1 |2 − c2

k=2

|Wk |2

for all W ∈ C4 and ω ∈ S2 as the unit sphere in R3 , here hh·, ·ii represents the inner product in C4 and Rez denotes the real part of z ∈ C. Proof

Notice that  4 p V (ω) = V ωj = h( · ω)ek , el i . p0 k,l=1 j=1 3 X

j

A direct calculation gives 

0

 ω  1 V (ω) = V ω j = c0  ω2 j=1  ω3 3 X

j

ω1

ω2

0

0

0

0

0

0

ω1

ω2

0

0

0

0

0

0

ω3

 0   0  0

for some constant c0 > 0. Let



0

 −ω  1 R(ω) = R ωj =  −ω2 j=1  −ω3 3 X

j



ω3



 0  . 0  0

(2.2)

It follows that there are some positive constants c1 and c2 such that RehhR(ω)V (ω)W, W ii ≥ c1 |W1 |2 − c2 and this completes the proof of Lemma 2.1.

4 X

k=2

|Wk |2 , 

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Next, we define a compensating function S(ω) for the solution of (2.1). For any given ω ∈ S2 , set R(ω) = {rij (ω)}4i,j=1 and let S(ω)f =

4 X

k,ℓ=1

λrkℓ (ω)hf, eℓ iek

for

λ > 0,

f ∈ L2 (R3 ).

(2.3)

Lemma 2.2 The compensating function S(ω) defined above enjoys the following properties (i) S(·) is C∞ on S2 with values in the space of bounded linear operators on L2 (R3 ), and S(−ω) = −S(ω) for all ω ∈ S2 . (ii) iS(ω) is self-adjoint on L2 (R3 ) for all ω ∈ S2 . (iii) There exist constants λ > 0 and c0 > 0 such that for all f ∈ L2 (R3 ) and ω ∈ S2 , p RehS(ω)( · ω)f, f i + hLf, f i ≥ c0 (|Pf |22 + |(I − P)f |2D ). (2.4) p0 The proof of this lemma is similar to that in [19, 27] for the compensating function by Lemma 2.1, we omit its proof here. By using the compensating function S(ω), we will derive some energy estimate on the linearized Vlasov-Maxwell-Fokker-Planck system with relativistic transport. Set ω = ξ/|ξ| and take the Fourier transform in x of (2.1) to have √ p ˆ ˆ · p J + Lfˆ = G. ∂t fˆ + i|ξ|( · ω)fˆ − E (2.5) p0 p0 By taking the inner product of (2.5) with fˆ, we have 1 ˆ2 p√ ˆ ˆ fˆi. J, f i + hLfˆ, fˆi = RehG, ∂t |f |2 − RehEˆ · 2 p0 Notice that the Fourier transform of the Maxwell system (1.8)–(1.10) gives Z  p ˆ√  ˆ ˆ f Jdp, ∂ E − iξ × B = −  t   R3 p0    ˆ + iξ × E ˆ = 0, ∂t B    Z  √   ˆ = 0. ˆ= iξ · E fˆ Jdp, iξ · B

(2.6)

(2.7)

R3

Hence it holds from (2.7) that p√ ˆ 1 ˆ 2 + |B| ˆ 2 ) + Re(iξ × B| ˆ E) ˆ − Re(iξ × E| ˆ B) ˆ RehEˆ · J, f i = − ∂t (|E| p0 2 1 ˆ 2 + |B| ˆ 2 ). = − ∂t (|E| 2 Then it follows from this and (2.6) that 1 ˆ 2 + |B| ˆ 2 ) + hLfˆ, fˆi = RehG, ˆ fˆi. ∂t (|fˆ|22 + |E| (2.8) 2 Applying −i|ξ|S(ω) to (2.5) gives   √ p ˆ · p J − i|ξ|S(ω)Lfˆ = −i|ξ|S(ω)G. ˆ (2.9) −i|ξ|S(ω)∂t fˆ + |ξ|2 S(ω) ( · ω)fˆ + i|ξ|S(ω)E p0 p0 The inner product of the above equation with fˆ yields p p√ ˆ Reh−i|ξ|S(ω)∂t fˆ, fˆi + |ξ|2 RehS(ω)( · ω)fˆ, fˆi + Rehi|ξ|S(ω)Eˆ · J, f i p0 p0

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n o ˆ fˆi . = |ξ|Re hiS(ω)Lfˆ, fˆi − hiS(ω)G,

(2.10)

Since iS(ω) is self-adjoint, the first term is just − 12 |ξ|∂t hiS(ω)fˆ, fˆi. By multiplying (1 + |ξ|2 ) to (2.8), and by adding κ times (2.10), we have    κ|ξ| (1 + |ξ|2 )  ˆ 2 2 2 ˆ ˆ ˆ ˆ ∂t |f |2 + |E| + |B| − hiS(ω)f , f i 2 2   p 2 2 2 ˆ ˆ ˆ ˆ ˆ ˆ + (1 + |ξ| − κ|ξ| )hLf , f i + κ|ξ| RehS(ω)( · ω)f , f i + hLf , f i p0 n o ˆ fˆi + κ|ξ|Re hiS(ω)Lfˆ, fˆi − hiS(ω)G, ˆ fˆi = (1 + |ξ|2 )RehG, p√ ˆ J, f i. (2.11) − κRehi|ξ|S(ω)Eˆ · p0 For the second term on the left-hand side of (2.11), when 0 < κ < 1, we have (1 + |ξ|2 − κ|ξ|2 )hLfˆ, fˆi ≥ (1 − κ)(1 + |ξ|2 ) · δ|(I − P)fˆ|2D . And by Lemma 2.2, the third term on the left hand side of (2.11) is bounded by   p κ|ξ|2 RehS(ω)( · ω)fˆ, fˆi + hLfˆ, fˆi ≥ κ|ξ|2 · c0 (|Pfˆ|22 + |(I − P)fˆ|2D ). p0

Notice that

S(ω)Lf =

4 X

k,ℓ=1

and

λrkℓ (ω)hL(I − P)f, eℓ iek

|hL(I − P)f, eℓ i| = |h(I − P)f, Leℓ i| ≤ C|(I − P)f |D , where we used the self-adjoint property of L and the exponential decay of eℓ (p) in p. By (2.3), we obtain n o ˆ fˆi ≤ Cη κ|(I − P)fˆ|2D + ηκ|ξ|2 |fˆ|22 + Cη |G| ˆ 22 , Cκ|ξ|Re hiS(ω)Lfˆ, fˆi − hiS(ω)G,

where Cη > 0 depends on η. By (2.3) and (1.12) we have hi|ξ|S(ω)Eˆ ·

4 X p√ p√ ˆ J, f i = iλ|ξ|rkℓ (ω)hEˆ · J, eℓ ihfˆ, ek i p0 p0 k,ℓ=1

=

4 X 4 X

ˆℓ−1 hPfˆ, ek i iλc1 |ξ|rkℓ (ω)E

k=1 ℓ=2 4 X 4 X

+

√ √ −1/2 ρ1 h pp10 J, p1 Ji

k=1 ℓ=2

iλc1 |ξ|rkℓ (ω)Eˆℓ−1 h(I − P)fˆ, ek i,

here c1 = > 0. For the first part in the above equation, we use (1.10), (1.12), (1.14) and the matrix R(ω) in (2.2) to obtain 4 X 4 X

k=1 ℓ=2

−1/2

ˆℓ−1 hPfˆ, ek i = λρ iλc1 |ξ|rkℓ (ω)E 0

ˆ 2. c1 |ξ · E|

Notice that rk,ℓ (ω) = 0 when 2 ≤ k ≤ 4, 2 ≤ ℓ ≤ 4. For the second part of the above equation, we have 4 X 4 X ˆℓ−1 h(I − P)fˆ, ek i = 0. iλc1 |ξ|rkℓ (ω)E k=1 ℓ=2

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For the first term on the right hand side of (2.11), we have ˆ fˆi ≤ η(1 + |ξ|2 )|fˆ|2 + Cη (1 + |ξ|2 )|G| ˆ 2. (1 + |ξ|2 )RehG, 2 2 It follows from (2.7) that ˆ 2 = ρ0 |Pfˆ|2 . |ξ · E| 2 By choosing κ, η > 0 small enough and combining the above estimates, we know from (1.13) and (1.15) that there exist constants δ1 , δ2 > 0 such that h i ˆ 2 + |B| ˆ 2 ) − κ|ξ|hiS(ω)fˆ, fˆi + δ1 (1 + |ξ|2 )|(I − P)fˆ|2 ∂t (1 + |ξ|2 )(|fˆ|22 + |E| D ˆ 2 ≤ C(1 + |ξ|2 )|G| ˆ 2. + δ2 (1 + |ξ|2 )|Pfˆ|22 + δ2 |ξ · E| 2

(2.12)

This is a main estimate obtained by the compensating function, which will be used for later energy analysis. Next we will give the estimate of the nonlinear term. Lemma 2.3 Let |α| + |β| ≤ N and G = −(E +

p p0

× B) · ∇p f + 21 E ·

p p0 f .

It holds that

k∂βα Gk2 ≤ CEN (t)DN (t). Proof

Notice that X p α−α1 1 X α1 β1 α1 ∂βα G = Cα Cβ ∂ E · ∂β1 ( )∂β−β f− Cαα1 ∂ α1 E · ∇p ∂βα−α1 f 1 2 p0 X p α−α1 − Cαα1 Cββ1 ∂β1 ( ) × ∂ α1 B · ∇p ∂β−β f. 1 p0

We consider the third term. If |α1 | ≥ N2 , one has

2 p

α−α1 2 α−α1 f |2 k∂ α1 Bk2 f

∂β1 ( ) × ∂ α1 B · ∇p ∂β−β

≤ C sup |∇p ∂β−β 1 1 p0 x∈R3 X ′ k∇p ∂βα′ f k22 k∂ α1 Bk2 ≤C |α′ |+|β ′ |≤N

≤ CEN (t)DN (t), here we used Sobolev imbedding, that is, for any h ∈ H 2 (R3 ), we have X ′ sup |h| ≤ C k∇x ∂ α hk. x∈R3

|α′ |≤1

If |α − α1 | + |β − β1 | ≥ N2 , one has

2 p

α−α1 α−α1 f k2 f

∂β1 ( ) × ∂ α1 B · ∇p ∂β−β

≤ C sup |∂ α1 B|22 k∇p ∂β−β 1 1 p0 x∈R3 X ′ α−α1 ≤C k∂ α Bk22 k∇p ∂β−β f k2 1 |α′ |≤N

≤ CEN (t)DN (t). This completes the estimate of the third term. The other terms can be treated similarly. Lemma 2.4 For system (1.5) and (1.8)–(1.10), we have the following estimates   Z
d  X k∂ α f k2 + k∂ α Ek2 + k∂ α Bk2 − κ (1 + |ξ|2 )N −1 |ξ|hiS(ω)fˆ, fˆidξ  dt R3 |α|≤N



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X

+ δ1

|α|≤N

and

 d  X dt

1≤|α|≤N

+ δ1

k(I − P)∂ α f k2D + δ2

X

kP∂ α f k2 ≤ CEN (t)DN (t)

|α|≤N

k∂ α f k2 + k∂ α Ek2 + k∂ α Bk

X

1≤|α|≤N

k(I − P)∂ α f k2D + δ2

Vol.37 Ser.B


2

X

−κ

1≤|α|≤N

Z

R3

(2.13)



(1 + |ξ|2 )N −2 |ξ|3 hiS(ω)fˆ, fˆidξ 

kP∂ α f k2 ≤ CEN (t)DN (t),

(2.14)

where κ > 0 is a small constant and EN (t) and DN (t) are defined in (1.17) and (1.19).

Proof Multiplying (2.12) by (1 + |ξ|2 )N −1 and integrating it over ξ give   Z X
d  (1 + |ξ|2 )N −1 |ξ|hiS(ω)fˆ, fˆidξ  k∂ α f k2 + k∂ α Ek2 + k∂ α Bk2 − κ dt 3 R |α|≤N X X + δ1 k(I − P)∂ α f k2D + δ2 kP∂ α f k2 |α|≤N

+ δ2

Z

R3

|α|≤N

ˆ 2 dξ ≤ C (1 + |ξ|2 )N −1 |ξ · E|

Z

R3

ˆ 22 dξ. (1 + |ξ|2 )N |G|

By (1.14) and equation (2.7), one has Z Z ˆ 2 dξ = ρ0 (1 + |ξ|2 )N −1 |ξ · E| (1 + |ξ|2 )N −1 |Pfˆ|22 dξ = ρ0 R3

R3

(2.15)

X

|α|≤N −1

kP∂ α f k2 .

(2.16)

Notice that G is as (1.7). By Lemma 2.3 and the property of Fourier transform, we have that Z X ˆ 22 dξ ≤ C (1 + |ξ|2 )N |G| k∂ α Gk2 ≤ CEN (t)DN (t). (2.17) R3

|α|≤N

By (2.16) and (2.17), (2.13) follows from (2.15). On the other hand, multiplying (2.12) by (1 + |ξ|2 )N −2 |ξ|2 and integrating it over ξ give   Z
d  X k∂ α f k2 + k∂ α Ek2 + k∂ α Bk2 − κ (1 + |ξ|2 )N −2 |ξ|3 hiS(ω)fˆ, fˆidξ  dt 3 R 1≤|α|≤N Z X X ˆ 2 dξ k(I − P)∂ α f k2D + δ2 kP∂ α f k2 + δ2 (1 + |ξ|2 )N −2 |ξ|2 |ξ · E| + δ1 ≤C

Z

1≤|α|≤N

R3

1≤|α|≤N

R3

ˆ 2 dξ. (1 + |ξ|2 )N −1 |ξ|2 |G| 2

An argument similar to the above gives (2.14), and this completes the proof of Lemma 2.4. 

3

Global Classical Solutions

In the section, we will prove global solutions to system (1.5) and (1.8)–(1.10). First, we derive the estimates of the electromagnetic field through the system (1.5) and (1.8)–(1.10) such that we can close the energy estimates. Lemma 3.1 Let β > 0. There exists C > 0 such that 1 h∂β Lf, ∂β f i ≥ |∂β f |2D − C|f |22 . 2

(3.1)

No.5

D.C. Yang: VLASOV-MAXWELL-FOKKER-PLANCK MODEL

1247

For the Fokker-Planck operator (1.6), and by the Leibniz formula, we have    2 X β  3p20 − |p|2 |p| h∂β Lf, ∂β f i = hL∂β f, ∂β f i + Cβ 1 ∂β−β1 − ∂ f, ∂ f β β 1 4p20 2p30

Proof

β1 <β

p ∂β f |22 − C|∂β f |22 ≥ +| 2p0    2 X β  3p20 − |p|2 |p| − ∂ + Cβ 1 ∂β−β1 f, ∂ f . β1 β 4p20 2p30 β1 <β   2 3p20 −|p|2 − and ∂p ( pp0 ) are bounded. For any small η > 0, one has Since ∂β−β1 |p| 2 3 4p0 2p0    2 X β  X 3p20 − |p|2 |p| 1 − ∂β1 f, ∂β f ≤ η|∂β f |22 + Cη Cβ ∂β−β1 |∂β1 f |22 . 2 3 4p0 2p0 |∇p ∂β f |22

β1 <β

β1 <β

By a standard interpolation lemma, one has X |∂β1 f |22 ≤ η|∇p ∂β f |22 + Cη |f |22 . β1 ≤β

By combining the above inequalities with (1.15), we obtain the estimate of (3.1).



Lemma 3.2 Let [f (t), E(t), B(t)] be a classical solution to (1.5) and (1.8)–(1.10) with the following norms well defined. Then there exists C > 1 such that X k∂ α Ek2 |α|≤N −1

 X √ p  ∂ α E · h∂ α (I − P)f, ∂ α E · ∇x × ∂ α B  dx Ji + p0 R3 |α|≤N −1 |α|≤N −2 X X α 2 α k∂ (I − P)f k + C k∇x ∂ Pf k2 + CEN (t)DN (t). +C

d ≤C dt

Z



X

|α|≤N

(3.2)

|α|≤N −1

√ First, the inner product of (1.5) with pp0 J gives p p√ p√ p√ Ji + h · ∇x f, Ji + hLf, Ji ρ2 E = h∂t f, p0 p0 p0 p0   p 1 p p√ + (E + × B) · ∇p f − E · f, J , p0 2 p0 p0 √ p √ p1 1 where ρ2 = h p0 J, p0 Ji. By taking the |α|th order spatial derivative on the above equation and multiplying it by ∂ α E, we have Z Z p√ p p√ ρ2 k∂ α Ek2 = ∂ α E · h∂t ∂ α f, Jidx + ∂ α E · h · ∇x ∂ α f, Jidx p0 p0 p0 R3 R3 Z p√ + ∂ α E · hL∂ α f, Jidx p0 3     ZR p p p√ α α α 1 + ∂ E · ∂ (E + × B) · ∇p f − ∂ ( E · f ), J dx p0 2 p0 p0 R3 Proof

=

4 X

Ii .

i=1

We now estimate I1 − I4 . For the term I1 , we apply (1.8) to obtain Z d p√ I1 = ∂ α E · h∂ α (I − P)f, J idx dt R3 p0

(3.3)

1248

ACTA MATHEMATICA SCIENTIA

−

Z

R3

(∇x × ∂ α B − h∂ α f,

Vol.37 Ser.B

p√ p√ Ji) · h∂ α (I − P)f, Jidx. p0 p0

(3.4)

To estimate I1 , we discuss about the following two cases because of the different estimation on the magnetic field B. When |α| ≤ N − 2, we have Z d p√ I1 ≤ ∂ α E · h∂ α (I − P)f, Jidx + ηk∇x × ∂ α Bk2 + Cη k∂ α (I − P)f k2 . (3.5) dt R3 p0

When |α| = N − 1, by applying integration by parts to the term ∇x × ∂ α B, we have Z d p√ I1 ≤ ∂ α E · h∂ α (I − P)f, Jidx + ηk∂ α Bk2 dt R3 p0 + Cη k∂ α ∇x (I − P)f k2 + Ck∂ α (I − P)f k2 .

(3.6)

Notice that in (3.5) and (3.6) the order of derivative on B(t, x) is between 1 and N −1. This is essential as kBk2 and k∂ N Bk2 do not appear in the dissipation rate of the energy functional. For |α| > 0, let α1 = |α| − 1 ≥ 0. By taking ∂ α1 on (1.8) and (1.10), we have p√ ∇x × ∂ α1 B = ∂t ∂ α1 E + h∂ α1 f, Ji, ∇x · ∂ α1 B = 0. p0 Therefore, we have from (1.8) and (1.9) that k∂ α Bk2 ≤ k∇x ∂ α1 Bk2 = k∇x × ∂ α1 Bk2 Z Z p√ α1 α1 Ji · ∇x × ∂ α1 Bdx = ∂t ∂ E · ∇x × ∂ Bdx + h∂ α1 f, p 3 3 0 R Z ZR d α1 α1 α1 = ∂ E · ∇x × ∂ Bdx + ∂ E · ∇x × (∇x × ∂ α1 E)dx dt R3 R3 Z p√ Ji · ∇x × ∂ α1 Bdx + h∂ α1 f, p0 R3 Z d ≤ ∂ α1 E · ∇x × ∂ α1 Bdx + ηk∇x × ∂ α1 Bk2 dt R3 + Ck∇x ∂ α1 Ek2 + Cη k∂ α1 (I − P)f k2 .

By taking η > 0 small enough, we have Z d α 2 k∂ Bk ≤ C ∂ α1 E · ∇x × ∂ α1 Bdx + Ck∇x ∂ α1 Ek2 + Cη k∂ α1 (I − P)f k2 , dt R3

(3.7)

since the term ηk∇x × ∂ α1 Bk2 is absorbed by the left hand side. For the term I2 , we have I2 ≤ ηk∂ α Ek2 + Cη k∇x ∂ α f k2 .

(3.8)

I3 ≤ ηk∂ α Ek2 + Cη k∂ α (I − P)f k2 .

(3.9)

For the term I3 , we have

In addition, for the term I4 , it is straightforward to show that for 0 < |α| ≤ N − 1, I4 ≤ ηk∂ α Ek2 + Cη EN (t)DN (t), where we have used Lemma 2.3. By taking η > 0 small enough and |α| = N − 1, from (3.6) to (3.10), we have Z Z d p√ d k∂ α Ek2 ≤ C ∂ α E · h∂ α (I − P)f, Jidx + C ∂ α1 E · ∇x × ∂ α1 Bdx dt R3 p0 dt R3

(3.10)

No.5

D.C. Yang: VLASOV-MAXWELL-FOKKER-PLANCK MODEL

 + Cη k∇x ∂ α f k2 + k∂ α (I − P)f k2 + k∂ α1 (I − P)f k2 + Cη EN (t)DN (t).

1249 (3.11)

On the other hand, by taking η > 0 small enough and 1 ≤ |α| ≤ N − 2, from (3.5) and (3.7)–(3.10), we have Z Z d p√ d k∂ α Ek2 ≤ C ∂ α E · h∂ α (I − P)f, ∂ α E · ∇x × ∂ α Bdx Jidx + C dt R3 p0 dt R3  (3.12) + ηk∇x ∂ α Ek2 + Cη k∇x ∂ α f k2 + k∂ α (I − P)f k2 + Cη EN (t)DN (t).

Notice that

Z

R3

h(E +

p p p√ 1 p√ × B) · ∇p f, JiE + h( E · f ), JiEdx p0 p0 2 p0 p0

≤ ηkEk2 + Cη EN (t)DN (t).

(3.13)

Choosing η > 0 small enough, taking summation of (3.11) over |α| = N − 1 and summation of (3.12) over 1 ≤ |α| ≤ N − 2, inequality (3.2) follows from (3.13) and (3.5) with |α| = 0. This completes the proof of Lemma 3.2.  Lemma 3.3 For system (1.5) and (1.8)–(1.10), we have the following estimates   X d Cα,β k∂βα f k2 + k∂βα f k2D dt 1≤|β|,|α|+|β|≤N X X k∂ α Ek2 + CEN (t)DN (t). k∂ α f k2 + C ≤C |α|≤N

(3.14)

|α|≤N −1

Proof We take α and β with |α| + |β| ≤ N and |β| ≥ 1. Applying ∂βα to (1.5) and taking the inner product with ∂βα f over R3x × R3p to obtain    
1 d α 2 p p√ J), ∂βα f + ∂βα Lf, ∂βα f k∂β f k + ∂βα ( · ∇x f ), ∂βα f − ∂βα (E · 2 dt p0 p0
α α = ∂β G, ∂β f . (3.15) We will estimate (3.15) term by term. Since ∂β ( pp0 ) is bounded, for the second term on the left-hand side of (3.15), we have   X p α 2 ∂βα ( · ∇x f ), ∂βα f ≤ ηk∂βα f k2 + Cη k∇x ∂β−β ′fk . p0 ′ 0<β ≤β

Since |β| > 0 and |α| ≤ N − 1, the third term on the left-hand side of (3.15) is bounded by   p√ α α J), ∂β f ≤ Cη k∂ α Ek2 + ηk∂βα f k2 . ∂β (E · p0 For the fourth term on the left-hand side of (3.15), by Lemma 3.1, we have 1 α 2 k∂ f k − Ck∂ α f k2 . 2 β D For the first term on the right-hand side of (3.15), we have from Lemma 2.3 that
∂βα G, ∂βα f ≤ ηk∂βα f k2 + Cη EN (t)DN (t). (∂βα Lf, ∂βα f ) ≥

By combining the above inequalities, we have d α 2 k∂ f k + k∂βα f k2D dt β

1250

ACTA MATHEMATICA SCIENTIA

≤ Cηk∂βα f k2 + C

X

(k∂ α Ek2 + k∂ α f k2 ) + Cη

|α|≤N −1

Vol.37 Ser.B

X

0<β ′ ≤β

α 2 k∇x ∂β−β ′fk .

For any 1 ≤ |β| ≤ N , by taking η > 0 small enough, the summation of the above inequality over |α| + |β| ≤ N by using (1.15) and a suitable linear combination gives (3.14). And this completes the proof of Lemma 3.3.  Next we will prove Theorem 1.1 by using the above estimates. Proof of Theorem 1.1 Combining (3.14) and (3.2) , we have from (1.14) that X X Z dn p√ κ0 Jidx Cα,β k∂βα f k2 − C ∂ α E · h∂ α (I − P)f, dt p0 3 1≤|β|,|α|+|β|≤N |α|≤N −1 R o X Z X −C ∂ α E · ∇x × ∂ α Bdx + κ0 k∂βα f k2D |α|≤N −2

≤C

X

|α|≤N

R3

1≤|β|,|α|+|β|≤N

k∂ α f k2 + CEN (t)DN (t).

(3.16)

Here κ0 > 0 small enough. By (1.13), (1.15), (2.13) and (3.16), we can obtain X X d k∂ α (I − P)f k2D + C0 δ2 kP∂ α f k2 EN (t) + C0 δ1 dt |α|≤N |α|≤N X α 2 k∂β f kD ≤ CEN (t)DN (t), + κ0

(3.17)

1≤|β|,|α|+|β|≤N

here EN (t) and DN (t) are defined as (1.17) and (1.19) and the precise expression of EN (t) is Z X
EN (t) = C0 k∂ α f k2 + k∂ α Ek2 + k∂ α Bk2 − C0 κ (1 + |ξ|2 )N −1 |ξ|hiS(ω)fˆ, fˆidξ |α|≤N

+ κ0

R3

X

1≤|β|,|α|+|β|≤N

−C

X

|α|≤N −2

Z

R3

Cα,β k∂βα f k2 − C

X

|α|≤N −1

∂ α E · ∇x × ∂ α Bdx ∼

X

|α|≤N

Z

R3

∂ α E · h∂ α (I − P)f,

p√ Jidx p0


k∂ α f k2 + k∂ α Ek2 + k∂ α Bk2 ,

here κ0 > 0 and κ > 0 are small enough, C0 >> C large enough and we used the fact that S(ω) is bounded. Thus it follows from (3.17) and (1.19) that for some λ > 0, d EN (t) + λDN (t) ≤ CEN (t)DN (t). (3.18) dt Based on the known result on the local existence of classical solution, and the a priori assumption that EN (t) ≤ 2ε with ε small enough, we have from (3.18) that d EN (t) + λDN (t) ≤ 0 dt for some positive constant λ. Thus, we have Z t EN (t) + λ DN (s)ds ≤ EN (0). 0

This implies EN (t) ≤ EN (0) ≤ ε so that the priori estimate is closed. In what follows we will prove Theorem 1.2.



No.5

D.C. Yang: VLASOV-MAXWELL-FOKKER-PLANCK MODEL

1251

Multiplying (1.5) by f and integrating over R3x × R3p to obtain   p√ 1 d 2 kf k − E · J, f + (Lf, f ) = (G, f ). (3.19) 2 dt p0

Proof of Theorem 1.2

By the Maxwell system (1.8)–(1.10), we can obtain   Z Z Z p√ p √ − E· J, f = − E· E · (−∂t E + ∇x × B)dx f Jdpdx = − p0 R3 R3 p0 R3 Z 1 d = kEk2 − E · ∇x × Bdx 2 dt R3 1 d ≥ kEk2 − ηkEk2 − Cη k∇x × Bk2 . 2 dt By (1.14) and (1.10), one has 2 2 kPf k2 = ρ−1 0 k∇x · Ek ≤ Ck∇x Ek .

By (1.16), Lemma 2.3 and the above estimates, we have from (3.19) that d {kf k2 + kEk2 } + δk(I − P)f k2D + ηkPf k2 dt X k∂ α Ek2 + CEN (t)DN (t). ≤ Cη k∇x × Bk2 + Cη

(3.20)

|α|≤N −1

For any η0 ∈ (0, η) small enough, it follows from (3.14) and (3.20) that o X dn Cα,β k∂βα f k2 + δ1 k(I − P)f k2D + η ′ kPf k2 kf k2 + kEk2 + η0 dt 1≤|β|,|α|+|β|≤N X k∂βα f k2D + η0 1≤|β|,|α|+|β|≤N

≤ Cη0

X

1≤|α|≤N

k∂ α f k2 + Cη

X

|α|≤N −1

k∂ α Ek2 + Cη k∇x × Bk2 + CEN (t)DN (t).

(3.21)

A suitable linear combination of (3.21) and (3.2) yields X dn Cα,β k∂βα f k2 kf k2 + kEk2 + η0 dt 1≤|β|,|α|+|β|≤N o X Z X Z p√ ∂ α E · ∇x × ∂ α Bdx ∂ α E · h∂ α (I − P)f, Jidx − Cη − Cη p0 3 3 |α|≤N −1 R |α|≤N −2 R X + (δ1 − Cη)k(I − P)f k2D + η ′ kPf k2 + η0 k∂βα f k2D ≤ Cη

X

1≤|α|≤N

1≤|β|,|α|+|β|≤N

α

2

2

k∂ f k + Cη k∇x × Bk + CEN (t)DN (t),

(3.22)

here η > 0 small enough. It follows from (3.22) and (2.14) that X d h EN (t) + (δ − Cη)k(I − P)f k2D + ηkPf k2 + η k∂βα f k2D dt 1≤|β|,|α|+|β|≤N X X α 2 + (δ1 − Cη) k(I − P)∂ f kD + (δ2 − Cη) kP∂ α f k2 1≤|α|≤N 2

≤ Cη k∇x × Bk + CEN (t)DN (t),

1≤|α|≤N

(3.23)

1252

ACTA MATHEMATICA SCIENTIA

where h EN (t) ∼

X

|α|≤N

(k∂ α f k2 + k∂ α Ek2 ) +

X

1≤|α|≤N

Vol.37 Ser.B

k∂ α Bk2 .

By EN (t) ≤ Cε and η > 0 small enough, we can obtain from (3.23) and (1.19) that

d h E (t) + λDN (t) ≤ Ck∇x × Bk2 . dt N This completes the proof of Theorem 1.2.

4



Time Decay for the Linearized System

In the section, we are concerned with time-decay properties of solutions to the linearized Vlasov-Maxwell-Fokker-Planck system with relativistic transport. The proof of time-decay rate is based on an approach by compensating function and energy method. For simplicity, we write Z √ p Ψ= (4.1) (I − P)f Jdp, R3 p0 U = [f, E, B],

U0 = [f0 , E0 , B0 ].

Formally, the solution to system (1.5) and (1.8)–(1.10) is denoted by U (t) = U I (t) + U II (t), U I (t) = A(t)U0 , U II (t) =

Z

0

(4.2)

U I (t) = [f I , E I , B I ],

(4.3)

t

A(t − s)[G(s), 0, 0]ds,

U II (t) = [f II , E II , B II ],

(4.4)

where A(t) is the linear solution operator for the linearized homogeneous system corresponding to system (1.5) and (1.8)–(1.10) with G = 0. In the following, we introduce the norms k · kHm and k · kZr with m ≥ 0 and r ≥ 0 given by kU k2Hm = kf k2L2p(Hxm ) + kEk2H m + kBk2H m ,

kU kZr = kf kZr + kEkLrx + kBkLrx

(4.5)

for U = [f, E, B]. With the above preparation, the main result of this section can be stated as follows. Theorem 4.1 Assume that U is defined in (4.2)–(4.4) as the solution to system (1.5) and (1.8)–(1.10). Then for any 1 ≤ r ≤ 2, ℓ ≥ 0, and m ≥ 0 be an integer, we have the following result, for any t ≥ 0, 3

1

1

m

ℓ

I − 2 ( r − 2 )− 2 k∇m kU0 kZr + C(1 + t)− 2 k∇m+ℓ U 0 k H0 , (4.6) x U kH0 ≤ C(1 + t) x Z t Z t 1 1 II 2 (1+t−s)−ℓk∇m+ℓ G(s)k2 ds, (4.7) k∇m (1+t−s)−3( r − 2 )−m kG(s)k2Zr ds+C x U k H0 ≤ C x 0

0

∇m+ℓ x

here, if ℓ is not integer, is regarded as the fractional spatial derivative in terms of the Fourier transform. ˆ and ξ × B ˆ are not included in the key estimates (2.12). However, these The terms ξ × E can be recovered from the Maxwell equations, which is motivated by [7]. In the following, we ˆ and ξ × B. ˆ will obtain the dissipation rate related to ξ × E

No.5

1253

D.C. Yang: VLASOV-MAXWELL-FOKKER-PLANCK MODEL

Lemma 4.2 For any t ≥ 0 and ξ ∈ R3 , it holds that " # ˆ E/ρ ˆ 2) ˆ2 ˆ E) ˆ C|ξ|2 Re(Ψ|2 |ξ × B| Re(−iξ × B| ∂t − +λ 2 2 2 2 (1 + |ξ| ) (1 + |ξ| ) (1 + |ξ|2 )2   |ξ|2 |ξ|2 ˆ 2+λ ˆ 2 ≤ C |(I − P)fˆ|2 + |G| ˆ2 . |ξ · E| | E| +λ 2 2 (1 + |ξ|2 )2 (1 + |ξ|2 )2 Proof system

(4.8)

ˆ 2 , recall the Fourier transform in x for the Maxwell To obtain the estimate of |ξ × B|  ˆ − iξ × B ˆ = −Ψ, ˆ  ∂t E  ˆ + iξ × E ˆ = 0, ∂t B    ˆ = ρ1/2 a ˆ = 0. iξ · E iξ · B f, 0 c

It follows that

(4.9)

ˆ E) ˆ = Re(−iξ × ∂t B| ˆ E) ˆ + Re(−iξ × B|∂ ˆ t E) ˆ ∂t Re(−iξ × B| 2 2 ˆ + Re(iξ × B| ˆ Ψ). ˆ ˆ − |ξ × B| = |ξ × E| ˆ 2 = |ξ|2 |B| ˆ 2 , we have from the above equation that Noticing from (4.9) that |ξ × B| ! ˆ E) ˆ ˆ2 ˆ2 Re(−iξ × B| |ξ|2 |B| |ξ|2 |E| ∂t + λ ≤ + C|(I − P)fˆ|22 . (1 + |ξ|2 )2 (1 + |ξ|2 )2 (1 + |ξ|2 )2 Recalling (2.5) and (4.1), we have that √ √ ˆ p Ji, ˆ = −ρ−1/2 ρ2 iξc ˆ − h p · iξ(I − P)fˆ, p Ji + h−Lfˆ + G, ∂t Ψ af + ρ2 E 0 p0 p0 p0 √ p1 √ p1 where ρ2 = h p0 J, p0 Ji. By (4.11) and (4.9), we can obtain

(4.10)

(4.11)

ˆ E/ρ ˆ 2 ) ≤ −2ρ−1 |ξ · E| ˆ 2 − (2 − η)|E| ˆ 2 + Cη (1 + |ξ|2 )|(I − P)fˆ|22 + Cη |G| ˆ 22 . Re(−∂t Ψ|2 0 By (4.11) and (4.9), we can also obtain ˆ t E/ρ ˆ 2 ) ≤ η|B| ˆ 2 + Cη (1 + |ξ|2 )|(I − P)fˆ|22 . Re(−Ψ|2∂ Thus we have from the above two estimates that ˆ E/ρ ˆ 2) − ∂t Re(Ψ|2 ˆ E/ρ ˆ 2 ) + Re(−Ψ|2∂ ˆ t E/ρ ˆ 2) = Re(−∂t Ψ|2 2 ˆ2 ˆ2 ˆ2 ˆ2 ˆ2 ≤ −2ρ−1 0 |ξ · E| − (2 − η)|E| + Cη (1 + |ξ| )|(I − P)f |2 + Cη |G|2 + η|B| .

This implies that − ∂t ≤η

ˆ E/ρ ˆ 2) |ξ|2 Re(Ψ|2 2 (1 + |ξ| )2

!

ˆ2 |ξ|2 |B| |ξ|2 + Cη 2 2 (1 + |ξ| ) 1 + |ξ|2

2 |ξ|2 ˆ 2 + (2 − η)|ξ| |E| ˆ2 |ξ · E| 2 2 2 2 (1 + |ξ| ) (1 + |ξ| )   ˆ 22 . |(I − P)fˆ|22 + |G|

+λ

(4.12)

Therefore, by a suitable linear combination of (4.12) and (4.10), then choosing η > 0 small enough, we obtain (4.8), and this completes the proof of Lemma 4.2.  Now, we will deduce the time decay of the linearized system (1.5) and (1.8)–(1.10).

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Lemma 4.3 Let U = [f, E, B] be the solution to the linearized system (1.5) and (1.8)– (1.10), then there exists a time-frequency function E(t, ξ) such that ∂t E(t, ξ) + for any t ≥ 0 and ξ ∈ R3 , where

λ|ξ|2 ˆ 22 E(t, ξ) ≤ C|G| (1 + |ξ|2 )2

ˆ 2 + |B| ˆ 2. E(t, ξ) ∼ |fˆ|22 + |E| Proof

where

(4.14)

By multiplying κ1 to (4.8) and by adding (2.12)/(1 + |ξ|2 ), we have

∂t E(t, ξ) + δ1 |(I − P)fˆ|2D + δ2 |Pfˆ|22 + δ2 + κ1 λ

(4.13)

2 ˆ 2 1 ˆ 2 + κ1 λ |ξ| |B| |ξ · E| 1 + |ξ|2 (1 + |ξ|2 )2

|ξ|2 |ξ|2 ˆ 2 + κ1 λ ˆ 2 ≤ C|G| ˆ 22 + Cκ1 (|(I − P)fˆ|22 + |G| ˆ 22 ), |ξ · E| |E| (1 + |ξ|2 )2 (1 + |ξ|2 )2 κ|ξ| ˆ 2 + |B| ˆ2− E(t, ξ) = |fˆ|22 + |E| hiS(ω)fˆ, fˆi (1 + |ξ|2 ) " # ˆ E) ˆ ˆ E/ρ ˆ 2) Re(−iξ × B| C|ξ|2 Re(Ψ|2 + κ1 − . (1 + |ξ|2 )2 (1 + |ξ|2 )2

One can fix κ > 0, and κ1 > 0 small enough such that (4.14) holds. Hence ˆ2 |ξ|2 |B| ∂t E(t, ξ) + λ|(I − P)fˆ|2D + λ|Pfˆ|22 + λ (1 + |ξ|2 )2 2 2 |ξ| |ξ| ˆ 2+λ ˆ 2 ≤ C|G| ˆ 22 . +λ |ξ · E| |E| 2 2 (1 + |ξ| ) (1 + |ξ|2 )2

This completes the proof of Lemma 4.3.



In what follows we are in position to prove Theorem 4.1 by Lemma 4.3. Proof of the Theorem 4.1 We rewrite (4.13), for any t ≥ 0 and ξ ∈ R3 , as ˆ 22 , ∂t E(t, ξ) + q(ξ)E(t, ξ) ≤ C|G| where q(ξ) = λ|ξ|2 /(1 + |ξ|2 )2 . Therefore by the Gronwall inequality we obtain Z t −q(ξ)t ˆ ξ)|2 ds. E(t, ξ) ≤ e E(0, ξ) + C e−q(ξ)(t−s) |G(s, 2

(4.15)

0

From (4.14), we can obtain Z 2 m 2 m 2 |ξ|2m E(t, ξ)dξ ∼ k∇m x f (t)k + k∇x E(t)k + k∇x B(t)k .

(4.16)

R3

First, we will prove (4.6), one can apply (4.15) with G = 0 to get Z Z Z |ξ|2m E(t, ξ)dξ ≤ |ξ|2m e−q(ξ)t E(0, ξ)dξ + |ξ|2m e−q(ξ)t E(0, ξ)dξ = I1 + I2 . R3

|ξ|≤1

For I1 , we have I1 ≤

Z

λ

|ξ|≤1

≤C

|ξ≥1

Z

2

|ξ|2m e− 4 |ξ| t E(0, ξ)dξ

|ξ|≤1

2m − λ |ξ|2 t 4

|ξ|

e

|fˆ0 |22 dξ + C

Z

|ξ|≤1

2 λ |ξ|2m e− 4 |ξ| t |Eˆ0 |2 dξ

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+C

Z

2

λ

|ξ|≤1

1255

|ξ|2m e− 4 |ξ| t |Bˆ0 |2 dξ

= I11 + I12 + I13 . From the H¨older and Hausdorff-Young inequalities, we have # p1 "Z "Z # q1 p  2 2q 2m − λ |ξ| t ˆ dξ |f0 |2 dξ I11 ≤ C |ξ| e 4 |ξ|≤1

|ξ|≤1

≤C

"Z

|ξ|≤1

p  2 λ |ξ|2m e− 4 |ξ| t dξ

−3( r1 − 12 )−m

≤ C(1 + t)

where p1 + 1q = 1 and 1r + Similarly, we have

1 2q

# p1 "Z

R3

Z

R3

 r2 # |f0 | dx dp r

kf0 k2Zr ,

= 1 for 1 ≤ p ≤ ∞ and 1 ≤ r ≤ 2. 1

1

1

1

I12 ≤ C(1 + t)−3( r − 2 )−m kE0 k2Lrx ,

I13 ≤ C(1 + t)−3( r − 2 )−m kB0 k2Lrx .

Hence

  1 1 I1 ≤ C(1 + t)−3( r − 2 )−m kf0 k2Zr + kE0 k2Lrx + kB0 k2Lrx .

The integration over |ξ| ≥ 1 is estimated as Z Z − λt I2 ≤ |ξ|2m e 4|ξ|2 E(0, ξ)dξ ≤ |ξ|≥1

≤ C(1 + t)−ℓ

Z

|ξ|≥1

|ξ|≥1

λt 1 − 4|ξ| 2 e 2ℓ |ξ| |ξ|≥1

|ξ|2m+2ℓ E(0, ξ)dξ sup

|ξ|2m+2ℓ E(0, ξ)dξ


≤ C(1 + t)−ℓ k∇m+ℓ f0 k2 + k∇m+ℓ E0 k2 + k∇m+ℓ B0 k2 . x x x

Collecting the above estimates, we have 3

1

1

m

ℓ

I − 2 ( r − 2 )− 2 kU0 kZr + C(1 + t)− 2 k∇m+ℓ U 0 k H0 . k∇m x x U kH0 ≤ C(1 + t)

Similarly, to prove (4.7), one can apply (4.15) with U0 = 0 and thus E(0, ξ) = 0 to bound E(t, ξ) as follow Z Z tZ ˆ 22 dξds. |ξ|2m E(t, ξ)dξ ≤ C |ξ|2m e−q(ξ)(t−s) |G| R3

0

R3

Then, from the H¨ older and Hausdorff-Young inequalities, the integration over |ξ| ≤ 1 is estimated as # Z t "Z Z t 1 1 2m −q(ξ)(t−s) ˆ 2 |ξ| e |G|2 dξ ds ≤ C (1 + t − s)−3( r − 2 )−m kG(s)k2Zr ds. 0

|ξ|≤1

0

The other integration over |ξ| ≥ 1 is estimated as # Z t "Z Z t 2m −q(ξ)(t−s) ˆ 2 |ξ| e |G|2 dξ ds ≤ C (1 + t − s)−ℓ k∇m+ℓ G(s)k2 ds. x 0

|ξ|≥1

0

Hence, from the above estimates, we have Z t Z t II 2 −3( 1r − 12 )−m 2 U k ≤ C (1 + t − s) G(s)k2 ds. k∇m kG(s)k ds + C (1 + t − s)−ℓ k∇m+ℓ 0 x Zr x H 0

This completes the proof of Theorem 4.1.

0



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Time Decay for the Nonlinear System

In the section, our goal is to prove the time decay of the global solution to the nonlinear system (1.5) and (1.8)–(1.10) obtained in Theorem 1.1, namely, Theorem 1.3. The global solution U = [f, E, B] to the nonlinear system (1.5) and (1.8)–(1.10) obtained in Theorem 1.1 can be written as Z t U (t) = A(t)U0 + A(t − s)[G(s), 0, 0]ds, (5.1) 0

where A is defined in (4.3). In the following, we shall first prove the time-decay estimate (1.23) for the full instant energy functional EN (t). Assume that ǫN +2 defined in (1.22) is sufficiently small, then inequality (1.20) for the N + 1 and N + 2 order also holds. Let 1 < ℓ < 2. Multiplying (1.20) by (1 + t)ℓ and taking time integration over [0, t] give Z t Z t ℓ ℓ (1 + t) EN (t) + λ (1 + s) DN (s)ds ≤ EN (0) + ℓ (1 + s)ℓ−1 EN (s)ds. (5.2) 0

0

By definitions (1.17) and (1.19) of EN (t) and DN (t) implies   X X α 2 α 2 k∂ B(t)k + DN (t) . k∂ E(t)k + EN (t) ≤ C |α|≤N

|α|≤N

Plugging this inequality into (5.2) gives Z t (1 + t)ℓ EN (t) + λ (1 + s)ℓ DN (s)ds 0 Z t X Z t (1 + s)ℓ−1 k∂ α B(s)k2 ds + Cℓ (1 + s)ℓ−1 DN (s)ds ≤ EN (0) + Cℓ |α|≤N

+ Cℓ

X Z

0

|α|≤N

From (3.2), we have X k∂ α E(t)k2 |α|≤N

d ≤ CDN +1 (t) + C dt

Z

R3

0

0

t

(1 + s)ℓ−1 k∂ α E(s)k2 ds.

 X

|α|≤N

(5.3)

p√ ∂ E · h∂ (I − P)f, Ji + p0 α

X

α

|α|≤N −1

α

0

(5.4)

(5.5)

0

By (3.7), we have X k∂ α B(t)k2 ≤ C 1≤|α|≤N



∂ E · ∇x × ∂ B dx.

It follows from (5.4) that Z t X Z t k∂ α E(s)k2 ds ≤ CEN (0) + CEN (t) + C DN +1 (s)ds. |α|≤N

α

X

|α1 |≤N −1

+C

X

d dt

|α1 |≤N −1

Z

R3

∂ α1 E · ∇x × ∂ α1 Bdx

k∇x ∂ α1 E(t)k2 + C

X

|α1 |≤N −1

k∂ α1 (I − P)f k2 .

(5.6)

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D.C. Yang: VLASOV-MAXWELL-FOKKER-PLANCK MODEL

Thus, by (5.6) and (5.5), we can obtain Z t X Z t k∂ α B(s)k2 ds ≤ CEN (0) + CEN (t) + C DN +1 (s)ds. 0

1≤|α|≤N

1257

(5.7)

0

By (5.4), we have that for any ℓ ∈ (1, 2), X Z t (1 + s)ℓ−1 k∂ α E(s)k2 ds |α|≤N

≤C

Z

0

t

(1 + s)ℓ−1 DN +1 (s)ds

0

+C(1 + s)ℓ−1

 X

Z

R3

−C(ℓ − 1) +

X

|α|≤N −1

Z

Z

|α|≤N

t

(1 + s)ℓ−2

Z

∂ α E · h∂ α (I − P)f,

R3

0

 X

|α|≤N

p√ Ji + p0

∂ α E · h∂ α (I − P)f,

X

|α|≤N −1

p√ Ji p0

 t ∂ α E · ∇x × ∂ α B dx

0

 ∂ α E · ∇x × ∂ α B dxds

t

(1 + s)ℓ−1 DN +1 (s)ds + C(1 + t)ℓ−1 EN (t) + CEN (0) X Z t X Z t k∂ α B(s)k2 ds k∂ α E(s)k2 ds + C(ℓ − 1) +C(ℓ − 1)

≤C

0

+C(ℓ − 1)

|α|≤N

0

|α|≤N

0

X Z

1≤|α|≤N

0

t

k∂ α (I − P)f k2 ds.

(5.8)

Recalling ℓ ∈ (1, 2), we have from (5.5), (5.7) and (5.8) that X Z t (1 + s)ℓ−1 k∂ α E(s)k2 ds 0

|α|≤N

≤C

Z

t

(1 + s)ℓ−1 DN +1 (s)ds + C(1 + t)ℓ−1 EN (t) + CEN (0).

0

(5.9)

Similarly, one has from (5.6), (5.8) and (5.9) that X Z t (1 + s)ℓ−1 k∂ α B(s)k2 ds 1≤|α|≤N

≤C

Z

0

0

t

(1 + s)ℓ−1 DN +1 (s)ds + C(1 + t)ℓ−1 EN (t) + CEN (0).

By (5.9) and (5.10), we have from (5.3) that Z t ℓ (1 + t) EN (t) + λ (1 + s)ℓ DN (s)ds 0 Z t ≤ CEN (0) + Cℓ (1 + s)ℓ−1 kB(s)k2 ds + Cℓ(1 + t)ℓ−1 EN (t) 0 Z t + Cℓ (1 + s)ℓ−1 DN +1 (s)ds. 0

(5.10)

(5.11)

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We use the similar estimates as (1.20) for N + 2 to obtain Z t EN +2 (t) + λ DN +2 (s)ds ≤ CEN +2 (0).

(5.12)

0

The similar arguments as (5.3) imply that Z t (1 + t)ℓ−1 EN +1 (t) + λ (1 + s)ℓ−1 DN +1 (s)ds 0

Z

t

≤ CEN +1 (0) + C(ℓ − 1) (1 + s)ℓ−2 kB(s)k2 ds + C(ℓ − 1)(1 + t)ℓ−2 EN +1 (t) 0 Z t X Z t ℓ−2 + C(ℓ − 1) (1 + s) DN +1 (s)ds + C(ℓ − 1) (1 + s)ℓ−2 k∂ α E(s)k2 ds 0

+ C(ℓ − 1)

X

1≤|α|≤N +1

Z

0

|α|≤N +1

t

0

(1 + s)ℓ−2 k∂ α B(s)k2 ds.

Recalling ℓ ∈ (1, 2), we have from (5.5), (5.7), (5.12) and the above inequality that Z t (1 + t)ℓ−1 EN +1 (t) + λ (1 + s)ℓ−1 DN +1 (s)ds 0 Z t Z t ≤ CEN +1 (0) + C(ℓ − 1) kB(s)k2 ds + CEN +1 (t) + DN +2 (s)ds 0 0 Z t ≤ CEN +2 (0) + C(ℓ − 1) kB(s)k2 ds.

(5.13)

0

Combining (5.11) and (5.13), one can obtain Z t Z t (1 + t)ℓ EN (t) + λ (1 + s)ℓ DN (s)ds ≤ CEN +2 (0) + Cℓ (1 + s)ℓ−1 kB(s)k2 ds. 0

(5.14)

0

Denote

∞ EN (t) = sup (1 + s)3/2 EN (s). 0≤s≤t

It follows from Theorem 4.1 and representation (5.1) of U (t) that kB(t)k2 ≤ CkU (t)k2H0 ≤ C(1 + t)−3/2 kU0 k2Z1 T H3/2 + C

Z

0

t

(1 + t − s)−3/2 kG(s)k2Z1 T H3/2 ds

∞ ≤ C(1 + t)−3/2 kU0 k2Z1 T H2 + C(1 + t)−3/2 EN (0)EN (t),

(5.15)

here we have the following facts that

kG(s)k2Z1 ≤ CkE(s)k2 kf (s)k2 + (kE(s)k2 + kB(s)k2 )k∇p f (s)k2

2 ∞ ≤ CEN (s) ≤ CEN (0)EN (s) ≤ C(1 + s)−3/2 EN (0)EN (s)

(5.16)

and kG(s)k2H2 ≤ C(kE(s)k2H 2 + kB(s)k2H 2 )(kf (s)k2H2 + k∇p f (s)k2H2 )

2 ∞ ≤ CEN (s) ≤ CEN (0)EN (s) ≤ C(1 + s)−3/2 EN (0)EN (s).

(5.17)

Now, fix a constant ǫ > 0 close to 0. Taking ℓ = 3/2 + ǫ in (5.14) yields Z t Z t (1 + t)3/2+ǫ EN (t) + λ (1 + s)3/2+ǫ DN (s)ds ≤ CEN +2 (0) + Cℓ (1 + s)3/2+ǫ−1 kB(s)k2 ds. 0

0

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Plugging (5.15) into the above inequality gives   ∞ (t) , (1 + t)3/2+ǫ EN (t) ≤ CEN +2 (0) + C(1 + t)ǫ kU0 k2Z1 T H2 + EN (0)EN

which, after dividing by (1 + t)ǫ , implies that for any t ≥ 0,   ∞ ∞ EN (t) ≤ C EN +2 (0) + kU0 k2Z1 T H2 + EN (0)EN (t) .

Since EN +2 (0) + kU0 k2Z1 T H2 ≤ ǫN +2 , and EN (0) ≤ ǫN +2 are sufficiently small, one has 3

EN (t) ≤ CǫN +2 (1 + t)− 2 ,

(5.18)

which gives the desired time decay estimate (1.23). Next, we will prove the time-decay estimate (1.24) for the high-order instant energy funch tional EN (t). It follows from Theorem 1.2 that d h E (t) + λDN (t) ≤ Ck∇x × B(t)k2 . dt N By (1.18) and (1.19), we notice that   X X α 2 2 h α 2 k∂ B(t)k + kE(t)k + DN (t) . EN (t) ≤ C k∂ E(t)k +

(5.19)

1≤|α|≤N

1≤|α|≤N

For any η > 0 small enough, we have that d h h E (t) + ηEN (t) + (λ − η)DN (t) ≤ C dt N

X

1≤|α|≤N


k∂ α E(t)k2 + k∂ α B(t)k2 + ηkE(t)k2 .

Then for some η > 0 small enough, we have from Lemma 3.2 that X
d eh h EN (t) + η EeN (t) ≤ C k∂ α E(t)k2 + k∂ α B(t)k2 , dt 1≤|α|≤N

h here the functional EeN (t) is defined as  Z  X X p√ h h ∂ α E · ∇x × ∂ α B dx Ji + EeN (t) = EN ∂ α E · h∂ α (I − P)f, (t) − Cη p0 R3 |α|≤N −2 |α|≤N −1 X X X h k∂ α B(t)k2 ∼ EN k∂βα f (t)k2 + k∂ α E(t)k2 + (t). (5.20) ∼ |α|+|β|≤N

1≤|α|≤N

|α|≤N

Thus we can obtain d eh h E (t) + η EeN (t) ≤ C dt N ≤C

X

1≤|α|≤N

X

1≤|α|≤N

k∂ α E(t)k2 + k∂ α B(t)k2




k∂ α f (t)k2 + k∂ α E(t)k2 + k∂ α B(t)k2

≤ Ck∇x U (t)k2HN −1 .


(5.21)

Denote h,∞ h EeN (t) = sup (1 + s)5/2 EeN (s). 0≤s≤t

If we use the imbedding theorem, we can obtain

h,∞ h kG(s)k2Z1 T HN +4 ≤ CEN +5 (s)EeN (s) ≤ C(1 + s)−5/2 EN +5 (0)EeN (s).

(5.22)

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As before, it follows from Theorem 4.1 and representation (5.2) of U (t) that Z t 2 −5/2 2 T k∇x U (t)kHN −1 ≤ C(1 + t) kU0 kZ1 HN +4 + C (1 + t − s)−5/2 kG(s)k2Z1 T HN +4 ds 0

≤ C(1 + t)−5/2 kU0 k2Z1 T HN +4 + CEN +5 (0)

≤ C(1 + t)−5/2 (kU0 k2Z1 T HN +4 +

Z

t

h (s)ds (1 + t − s)−5/2 EeN

0 h,∞ e EN +5 (0)EN (t))

h,∞ ≤ CǫN +5 (1 + t)−5/2 (1 + EeN (t)).

By this, (5.20) and (5.21), we see that Z t h h (0) + C e−η(t−s) k∇x U (s)k2HN −1 ds EeN (t) ≤ e−ηt EeN 0 Z t h,∞ h (0) + CǫN +5 (1 + EeN (t)) e−η(t−s) (1 + s)−5/2 ds ≤ e−ηt EeN 0

h,∞ (t). ≤ CǫN +5 (1 + t)−5/2 + CǫN +5 (1 + t)−5/2 EeN

It follows from this and (5.22) that

h,∞ h,∞ EeN (t) ≤ CǫN +5 + CǫN +5 EeN (t).

By this and (5.20) and (5.22), we have that

h h EN (t) ≤ C EeN (t) ≤ CǫN +5 (1 + t)−5/2 .

This completes the proof of Theorem 1.3.



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