Stability of the relativistic Vlasov–Maxwell–Boltzmann system for short range interaction

Applied Mathematics and Computation 265 (2015) 854–882

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Stability of the relativistic Vlasov–Maxwell–Boltzmann system for short range interaction Fanghua Ma a, Xuan Ma b,∗ a b

Department of Mathematics, Jinan University, Guangzhou 510632, P.R. China Department of Mathematics, Guangdong University of Education, Guangzhou 510303, China

a r t i c l e

i n f o

a b s t r a c t

Keywords: Relativistic Vlasov–Maxwell–Boltzmann system Short range interaction Stability

The Cauchy problem of the relativistic Vlasov–Maxwell–Boltzmann system for short range interaction is investigated. For perturbative initial data with suitable regularity and integrability, we prove the large time stability of solutions to the relativistic Vlasov–Maxwell–Boltzmann system, and also obtain as a byproduct the convergence rates of solutions. For the proof, a new interactive instant energy functional is introduced to capture the the macroscopic dissipation and the very weak electro-magnetic dissipation of the linearized system. A refined time–velocity weighted energy method is also applied to compensate the weaker dissipation of the linearized collision operator in the case of non-hard potential models. The results also extend the case of “hard ball” model considered by Guo and Strain (2012) to the short range interactions. © 2015 Elsevier Inc. All rights reserved.

1. Introduction 1.1. Considered system We investigate the following relativistic Vlasov–Maxwell–Boltzmann system

∂t F+ + p˜ · ∇x F+ + (E + p˜ × B ) · ∇ p F+ = Q (F+ , F+ ) + Q (F+ , F− ), ∂t F− + p˜ · ∇x F− − (E + p˜ × B ) · ∇ p F− = Q (F− , F+ ) + Q (F− , F− ).

(1.1)

The self-consistent electromagnetic field satisfies the Maxwell equations

∂t E − ∇x × B = − ∂t B + ∇x × E = 0, ∇x · E =



R3



R3

p˜ (F+ − F− ) dp,

(F+ − F− ) dp,

(1.2)

∇x · B = 0.

Here F± = F± (t, x, p) ≥ 0 stand for the number densities of irons (+ ) and electrons (− ) which have position x = (x1 , x2 , x3 ) ∈ R3 and momentum p = ( p1 , p2 , p3 ) ∈ R3 at time  t ≥ 0, and E(t, x), B(t, x) denote the electro and magnetic fields, respectively. The ˜ energy of a particle is given by p0 =  p = 1 + | p|2 . Here and in the sequel, we denote pp by p. 0

∗

Corresponding author. Tel.: +86 20 34113296. E-mail addresses: [email protected] (F. Ma), [email protected] (X. Ma).

http://dx.doi.org/10.1016/j.amc.2015.05.043 0096-3003/© 2015 Elsevier Inc. All rights reserved.

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

855

The initial data of the coupled system above is given by

F± (0, x, p) = F0,± (x, p),

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

B(0, x ) = B0 (x )

(1.3)

satisfying the compatibility conditions

∇ x · E0 =



R3

∇x · B0 = 0.

(F0,+ − F0,− ) dp,

The relativistic Boltzmann collision operator Q(·, ·) in (1.1) takes the form of

Q (F, G ) =

1 p0



R3

dq q0



R3

dq q0



R3

dp W[F ( p )G(q ) − F ( p)G(q )], p0

(1.4)

here the “transition rate” W = W( p, q| p , q ) is defined as

W = s σ (, θ )δ ( p0 + q0 − p0 − q0 )δ (3) ( p + q − p − q ), is the delta function in one variable, δ (3)

and δ of momentum and energy:

p + q = p + q,

(1.5)

is the delta function in three variables. The delta functions express the conservation

p0 + q0 = p0 + q0 .

(1.6)

The quantity s in (1.5) is the square of the energy in the “center of momentum”, p + q = 0 , and is given as s = s ( p, q ) = −( p0 + q0 , p + q ) ( p0 + q0 , p + q ) = 2( p0 q0 − p · q + 1 ) ≥ 4, where denotes the Lorentz inner product which is given by

( p0 , p) (q0 , q ) = −p0 q0 + p · q. The relative momentum ϱ in (1.5) is denoted by

 =  ( p, q ) =



( p0 − q0 , p − q ) ( p0 − q0 , p − q ) =



2( p0 q0 − p · q − 1 ) ≥ 0.

The angle θ in (1.5) is then given by

cos θ =

( p0 − q0 , p − q ) ( p0 − q0 , p − q ) . 2

(1.7)

The relativistic differential cross section σ (ϱ, θ ) depends only on the relative momentum ϱ and the deviation angle θ . In the mathematical literature of the relativistic Boltzmann equation it is hard to find precise physically relevant examples of the differential cross section. Here, we use the following short range interaction form [31]

σ ( , θ ) =

Cθ , s

(1.8)

where Cθ is a constant. Our goal in the paper is to study the stability of the Cauchy problem of (1.1)–(1.3) under the assumption (1.8). 1.2. Reformulation, weight functions and norms We now turn to the presentation of our main result. The global relativistic Maxwellian (the Jüttner solution) is given by

J ( p) =

exp(−p0 ) , 4π K2 (1 )

where K2 (· ) is the modified Bessel function K2 (z ) = We set the perturbation in a standard way

 z2 +∞ 2 (t 3 0

− 1 )3/2 e−zt dt.

F± = J + J1/2 f± . Use [·, ·] to denote the column vector in R2 . Set F = [F+ , F− ] and f = [ f+ , f− ]. Then the Cauchy problems (1.1)–(1.3) can be reformulated as

⎧ ˜ 1/2 q 1 + L f = ∂t f + p˜ · ∇x f + q 0 (E + p˜ × B ) · ∇ p f − E · pJ ⎪  ⎪ ⎪ ⎪ ⎨∂t E − ∇x × B = − ˜ 1/2 ( f+ − f− ) dp, pJ R3 ∂t B + ∇x × E = 0, ⎪ ⎪  ⎪ ⎪ ⎩∇x · E = J1/2 ( f+ − f− ) dp, ∇x · B = 0

q0 E 2

· p˜ f + ( f, f ), (1.9)

R3

with initial data

f± (0, x, p) = f0,± (x, p),

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

B(0, x ) = B0 (x )

(1.10)

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F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

satisfying the compatibility condition



∇ x · E0 =

∇x · B0 = 0.

J1/2 ( f0,+ − f0,− ) dp,

R3

(1.11)

Here, q 0 = diag(1, −1 ), q 1 = [1, −1], and the linearized collision term Lf and the nonlinear collision term  (f, f) are respectively defined by

( f, g) = [+ ( f, g),

L f = [L+ f, L− f ],

− ( f, g)],

with

L± f = −2J−1/2 Q (J1/2 f± , J ) − J−1/2 Q (J, J1/2 { f± + f∓ } ),

± ( f, g) = J−1/2 Q (J1/2 f± , J1/2 g± ) + J−1/2 Q (J1/2 f± , J1/2 g∓ ).

(1.12)

As in [21], the null space of the linearized operator L is given by





N = span [1, 0]J1/2 , [0, 1]J1/2 , [pi , pi ]J1/2 (1 ≤ i ≤ 3 ), [ p,  p]J1/2 . Let P be the orthogonal projection from L2p × L2p to N . Given f(t, x, p), one can write P as P f = [P+ f, P− f ] with

P± f = {a± (t, x ) + b(t, x ) · p + c (t, x ) p}μ1/2 ,

(1.13)

where the coefficient functions are determined by f in the way (4.3). In what follows, we introduce the weight functions and norms used throughout the paper. First of all, define



−τ

wτ ,λ = wτ ,λ (t, p) =  p

exp

λ

(1 + t )

1  p , 0 < ϑ ≤ , ϑ

(1.14)

4

where constants τ ∈ R and λ ≥ 0 are two parameters which may vary in different places. Note that the dependence of wτ , λ on parameter ϑ has been neglected without any confusion. For simplicity, we write wτ = wτ ,0 when λ = 0. For any function f(t, x, p), define

| f (x )|2τ ,λ = Denote

| f |2ν =

 R3



R3

w2τ ,λ (t, p)| f |2 dp,

f 2τ ,λ =



R3

| f (x )|2τ ,λ dx.

 p−1 | f |2 dp.

and define

| f |2ν,τ ,λ = |wτ ,λ f |2ν , f 2ν,τ ,λ =

 R3

| f |2ν,τ ,λ dx.

For simplicity, we also use the notation

| f ( p)|2L2 = | p f ( p)|2L2 and f ( p) 2L2 =



 R3

| f ( p)|2L2 dx,

where ∈ R. We also use · H N to denote the standard Sobolev norm in R3 with respect to the variables x. 1.3. Main result Let I = [I+ , I− ] with I± f = f± . To study the global existence by means of the energy method, inspired by [6], the temporal energy functionals and the corresponding dissipation rate functionals are defined by

EN, ,λ (t ) ∼

E N (t ) ∼

|α|≤N

|α|≤N

and

DN, ,λ (t ) =

|α|≤N

 α  ∂β {I − P} f (t )2

|α|+|β|− ,λ

+ (E, B ) 2H N ,

∂ α f (t ) 2 + (E, B ) 2HN ,

|α|+|β|≤N



|α|+|β|≤N

+ DN (t ) =

∂ α (a± , b, c ) 2 +

(1.15)

(1.16)

 α 

∂β {I − P} f (t )2 +

∇x ∂ α (a± , b, c ) 2 + a+ − a− 2 + E 2HN−1 + ∇x B 2HN−2 ν,|α|+|β|− ,λ

λ

(1 + t )1+ϑ

|α|≤N−1

|α|+|β|≤N

∂ α {I − P} f (t ) 2ν +

 1/2 α   p ∂β {I − P} f (t )2

|α|+|β|− ,λ

|α|≤N−1

,

∇x ∂ α (a± , b, c ) 2 + a+ − a− 2 + E 2HN−1 + ∇x B 2HN−2 ,

(1.17)

(1.18)

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

857

where the integer N ≥ 0 and ≥ 0 are parameters which may differ in different places and also satisfy − N ≥ 0. Note that E N (t ) and DN (t ) contain no velocity differentiation and no extra velocity weight, and also that the last term of DN, ,λ (t ) on the righthand side of (1.17) is the extra dissipation term induced by the time–velocity weight function w ,λ0 (t, ξ ) defined by (1.14) which will be used in this paper to compensate the weaker dissipation of the linearized collision operator for non-hard potentials. Note that such a term disappears when λ = 0. Let constants N1 ≥ 13, 1 ≥ 1 + N1 , λ0 > 0, 0 < ϑ ≤ 14 and 0 > 0 be given; the exact choice of N1 , 1 λ0 , ϑ and 0 can be seen in the later proof. In terms of those given constants, the temporal energy norm X(t) is defined by

X (t ) = sup



0≤s≤t





E N1 (s ) + (1 + s ) 2 E N1 −2 (s ) + sup (1 + s )2(1+ϑ ) ∇x (E, B )(s ) 2H 5

+ sup

0≤s≤t

3



0≤s≤t

(1 + s )

−

1+0 2

3



EN1 , 1 ,λ0 (s ) + EN1 −1, 1 ,λ0 (s ) + (1 + s ) 2 EN1 −3, 1 −1,λ0 (s ) .

(1.19)

The main result of the paper is stated as follows. Theorem 1.1. Assume σ (ϱ, θ ) satisfies (1.8), take ϑ = 14 , N1 ≥ 13, 1 ≥ 1 + N1 , 2 > 15 4 , and λ0 > 0 suitably small, and take also 0 > 0 small enough. Let f0 = [ f0,+ , f0,− ] satisfy F± (0, x, p) = J ( p) + J1/2 ( p) f0,± (x, p) ≥ 0. If

Y0 =

|α|+|β|≤N1

 α   ∂β f 0 

|α|+|β|− 1 ,λ0

+ (E0 , B0 ) H N1 ∩L1 + w− 2 f0 Z1

is sufficiently small, then there exist properly defined energy functionals EN, ,λ (t ) and E N (t ) in the definition (1.19) of X(t)-norm such that the Cauchy problems (1.9)–(1.11) admits a unique global solution (f(t, x, p), E(t, x), B(t, x)) satisfying F± (t, x, p) = J ( p) + J1/2 ( p) f± (t, x, p) ≥ 0 and

X (t )  Y02

(1.20)

for all time t ≥ 0. The relativistic kinetic systems (Landau–Maxwell system, Vlasov–Maxwell–Boltzmann system, etc.) are the central equations to describe charged particles in viewpoint of physics. Standard references which discuss relativistic kinetic theory include [2,3,14,15,24]. We also refer to [1,4,7,16–20,25–30,33–35,37–40] for several previous results in this field. We now review the mathematical results most closely related to the relativistic Vlasov–Maxwell–Boltzmann system discussed in this paper. For the non-relativistic Vlasov–Maxwell–Boltzmann system, the global existence of solutions to the periodic initial boundary value problem near the global Maxwellian was firstly investigated by Guo [21]. Then, the rate of convergence to Maxwellians with any polynomial speed in large time was shown by Guo and Strain [36]. For the Cauchy problem in the whole space, the global in time classical solutions were constructed by Strain [32]. And recently, the large-time behavior of classical solutions in the situation of both cutoff and non-cutoff potentials were studied by Duan and Strain [13] and Duan et al. [8] respectively. However, the project that proving a stability theorem for the relativistic Vlasov–Maxwell–Boltzmann system and generalizing the results in [13] and [8] to a relativistic setting was easily stopped due to a severe difficulty of lack of regularity in the momentum p variables for the relativistic Boltzmann equation. Very recently, Guo and Strain [22] proved the momentum regularity and stability of the relativistic Vlasov–Maxwell–Boltzmann system with “hard ball” condition by combing the Glassey–Strauss and center of momentum frame of the collision operator. The aim of this paper is to extend the results in [22] for the “hard ball” condition to the short range interactions. The proof of Theorem 1.1 is based on a refined energy method with the weight wτ , λ (t, ξ ) containing the time-velocity-dependent exponential factor. The main difficulty in the proof is to deal with two types of nonlinearities: one stems from the coupling term (E + p˜ × B ) · ∇ξ F due to interactions between the self-consistent Lorentz force and gas particles in the relativistic framework, and the other is induced by the nonlinear Boltzmann collision operator for the short range interactions. To overcome such difficulty, compared with the previous works [6,9–11,34,37] and [22], one of our main contributions of the paper is to introduce the exponential weight estimate into the relativistic Boltzmann operator, and the other one is that it is much harder to apply the strategy of [6,22] through designing the X(t)-norm to obtain its closed global-in-time bound. Let’s illustrate the technical parts of the proof in more details. Unlike the case of “hard ball” model studied in [22], the dissipation of the linearized Boltzmann collision operator for the physically interesting short range interaction is weaker in the sense that it is degenerate in the large-velocity domain. However, the introduction of the time-velocity-dependent weight wτ , λ (t, ξ ) to the energy norm EN, ,λ (t ) can generate the extra dissipation corresponding to the last term in the energy dissipation rate functional DN, ,λ (t ) defined by (1.17), so that the weaker dissipation of the linearized collision operator is possibly compensated. For that, it is necessary to deduce the exponentially weighted estimates on the first equation of system (1.9). Therefore, the velocitygrowth effect arising from the coupling term (E + p˜ × B ) · ∇ξ F can be controlled through the extra dissipation balanced by the time-decay of the electromagnetic field. It is worth pointing out that such a weight function wτ , λ (t, ξ ) seems also necessary to deal with the Lorentz force in the case of relativistic non-hard potential. The rest of the paper is arranged as follows. In Section 2, we introduce two typical frames of the relativistic collision operators. In Section 3, we carry out the weighted estimates on L and  , which is one of the key technical part of the paper. In Section 4, we deduce the optimal time decay rates of the linearized relativistic Vlasov–Maxwell–Boltmzann system. In Section 5, we shall show how to use our previous estimates to conclude the proof of Theorem 1.1 following the arguments from [6,8].

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F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

Notations. Throughout this paper, C denotes some generic positive (generally large) constant and κ denotes some generic positive (generally small) constant, where both C and κ may take different values in different places. A ࣠ B means that there is a generic constant C > 0 such that A ≤ CB. A ∼ B means A ࣠ B and B ࣠ A. We use L2 to denote the usual Hilbert spaces L2 = L2x,p or L2x with the norm · , and use ·,· to denote the inner product over L2x,p or L2p . We also use HN to stand for the Sobolev space. The mixed velocity-space Lebesgue space Z1 = L2p (L1x ) = L2 (R3p ; L1 (R3x )) is used. For multi-indices α = (α1 , α2 , α3 ) and

β = (β1 , β2 , β3 ), ∂βα = ∂xα ∂ pβ = ∂xα11 ∂xα22 ∂xα33 ∂ pβ11 ∂ pβ22 ∂ pβ33 . The length of α is |α| = α1 + α2 + α3 and similar for |β |. If each component of α  is not greater than that of α , we denote the condition by α  ≤ α . We also define α  < α if α  ≤ α and |α  | < |α |. Some usual notations as in [12,13] are also given as follows. For two complex vectors√ z1 , z2 ∈ C3 , (z1 |z2 ) = z1 · z¯2 denotes the dot product in the complex field C, where z¯2 is the complex conjugate of z2 . And i = −1 ∈ C is the pure imaginary unit. gˆ(k ) denotes the Fourier transform of g = g(x ) with respect to the variable x. z denotes the real part of the complex number z. 2. The relativistic collision operators In this section, we will present the reductions of the collision operator (1.4). 2.1. Center of momentum reduction One may use Lorentz transformations as described in [31,35] to reduce the delta functions in (1.4) and obtain

Q (F, G ) = =



  dq dq dp W[F ( p )G(q ) − F ( p)G(q )]   3 3 3 q q 0 R R R p0 0  dq dωv( p, q )σ (, θ )[F ( p )G(q ) − F ( p)G(q )],

1 p0  R3

(2.1)

S2

where ω = (ω1 , ω2 , ω3 ) ∈ S2 and v = v( p, q ) is the Møller velocity given by

v = v( p, q ) =



| p˜ − q˜|2 − | p˜ × q˜|2 =



√

s

2p0 q0

.

We can refer to [31] for more details about Lorentz transformations. The post collisional momentum in the expression (2.1) can be written:

    p0 + q0 ( p + q) · ω ω+ − 1 ( p + q) , √ s | p + q|2     p+q p0 + q0 ( p + q) · ω q = − ω+ − 1 ( p + q) . √ 2 s | p + q|2 p+q + p = 2 

(2.2)

The energies are then

p0 + q0  + √ ( p + q ) · ω, 2 2 s p0 + q0  − √ ( p + q ) · ω. q0 = 2 2 s

p0 =

The angle θ in the reduced expression (2.1) is defined by

cos θ =

υ · ω, |υ|

where υ ∈ R3 has a complicated expression as given in [35] but its precise form will be inessential. Now we turn to the expression given by Glassey and Strauss in [18]. 2.2. Glassey–Strauss reduction Glassey–Strauss have derived an alternative form for relativistic operator without using the center-of-momentum. We will skip their argument and write down the result as follows.

Q (F, G ) =

  dq dq dp W[F ( p )G(q ) − F ( p)G(q )]   R3 q0 R3 q0 R 3 p0  s σ ( , θ ) dq dω B( p, q, ω )[F ( p )G(q ) − F ( p)G(q )], 3 2 p0 q0 R S

1 p0 

= where the kernel is



(2.3)

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

B( p, q, ω ) =

859

( p0 + q0 )2 |ω · ( p0 q − q0 p)| . 2 [( p0 + q0 )2 − (ω · ( p + q ))2 ]

In this expression, the post collisional momentum are given as follows

p = p + a ( p, q, ω )ω, q = q − a ( p, q, ω )ω, where

2( p0 + q0 ){ω · ( p0 q − q0 p)} . ( p0 + q0 )2 − (ω · ( p + q ))2

a ( p, q, ω ) =

and the energies can be expressed as

p+q · ω, p0 + q0 p+q · ω. q0 = q0 − a ( p, q, ω ) p0 + q0

p0 = p0 + a ( p, q, ω )

These formulas clearly satisfy the collisional conservations (1.6). The angle (1.7) in (2.3) can then be reduced to

cos θ = 1 −

{ω · ( p0 q − q0 p)}2 .  ( p0 + q0 )2 − (ω · ( p + q ))2 8

2

Moreover, assuming the collisions are elastic as in (1.6), we have the invariance

ω · ( p0 q − q0 p) = ω · ( p0 q − q0 p ). Therefore, for fixed ω ∈ S2 , B( p, q, ω ) = B( p , q , ω ). The Jacobian for the transformation (p, q) → (p , q ) in these variables [17] is

p q ∂ ( p , q ) = − 0 0. ∂ ( p, q ) p0 q0

(2.4)

From [22], it follows that



 dq R3

S2

dω

s σ ( , θ )

p0 q0

B( p, q, ω )G( p, q, p , q ) =



 dq

R3

S2

dωv( p, q )σ (, θ )G( p, q, p , q ),

(2.5)

where (p , q ) on the right hand side are given by (2.2) and G( p, q, p , q ) : R3 × R3 × R3 × R3 → R is a given function. Moreover the Jacobian (2.4) effectively works for (p , q ) in (2.2) as







dp R3

dq R3

S2

dωv( p, q )σ (, θ )G( p, q, p , q ) =







dp R3

dq R3

S2

dωv( p, q )σ (, θ )G( p , q , p, q ).

3. Weighted estimates on  and L This section is devoted to deducing the key estimates on  and L with respect to the weight w ,λ0 (t, ξ ), which is one of the main techniques used for the proof of the global stability of the relativistic Vlasov–Maxwell–Boltzmann system for the short range interaction. In light of (1.8), for scalar functions g1 , g2 and h, we use the following notations





T (g1 , g2 ) = J −1/2 Q J 1/2 g1 , J 1/2 g2 , T1 (g1 , g2 ) =

 

=

R3 ×S2

R3 ×S2

v( p, q )σ (, θ )J1/2 (q )[g1 ( p )g2 (q ) − g1 ( p)g2 (q )]dqdω Cθ  1/2 √ J (q )g1 ( p )g2 (q )dqdω − p0 q0 s

 R\3 ×S2

Cθ  1/2 √ J (q )g1 ( p)g2 (q )dqdω p0 q0 s

= T1gain − T1loss ,  s σ ( , θ ) T2 (g1 , g2 ) = B( p, q, ω )J1/2 (q )[g1 ( p )g2 (q ) − g1 ( p)g2 (q )]dqdω 3 2 p0 q0 R ×S  Cθ = B( p, q, ω )J1/2 (q )[g1 ( p )g2 (q ) − g1 ( p)g2 (q )]dqdω 3 2 p 0 q0 R ×S = T2gain − T2loss , and

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F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882







  Cθ   J (q ) h( p ) J (q ) − h( p) J (q ) dqdω √ p0 q0 s    Cθ   J (q ) J ( p )h(q ) − J ( p)h(q ) dqdω √ p0 q0 s

L h = − T1 (h,

=−  −

R3 ×S2

R3 ×S2

J ) + T1 (



J, h )







ν ( p)h − Kh,

=

(3.1)

where

ν ( p)h = h( p)T1loss (1, J1/2 ) = h( p)

 R3 ×S2



Cθ  √ J (q )dqdω, p0 q0 s

(3.2)



Kh = T1gain (h, J1/2 ) + T1gain (J1/2 , h ) − T1loss (J1/2 , h )      Cθ   = J (q ) h( p ) J (q ) + J ( p )h(q ) dqdω √ R3 ×S2 p0 q0 s   Cθ   − J (q ) J ( p)h(q )dqdω √ R3 ×S2 p0 q0 s = K2 h − K1 h.

(3.3)

From (2.5), one can see that the above operators T , L , ν and K have the following expressions:

⎧ T (g1 , g2 ) = T1 (g1 , g2 ) = T2 (g1 , g2 ), ⎪ ⎪   ⎪   ⎪ ⎪ ⎪ L h = − T ( h, J ) + T ( J, h ) , 2 2 ⎨ loss 1/2 ν ( p) = T2 (1, J ), ⎪ ⎪ ⎪ ⎪ K2 h = T2gain (h, J1/2 ) + T2gain (J1/2 , h ), ⎪ ⎪ ⎩ K1 h = T2loss (J1/2 , h ).

With the above notations, recalling (2.1), (2.3) and (1.12), it is straightforward to see

± ( f, g) = Ti ( f± , g± ) + Ti ( f± , g∓ ),

i = 1 or 2,

and

L± f = 2ν ( p) f± − 2Ti gain ( f± , J1/2 ) + Ti loss (J1/2 , f+ + f− ) − Ti gain (J1/2 , f+ + f− ),

i = 1 or 2.

3.1. Weighted estimates on  In order to make the weighted estimates on (1.9), particularly on L and  , we will split the desired estimate into two different cases. Those cases correspond to the following two different integration regions

A = {| p| ≤ 1} ∪ {| p| ≥ 1, | p| ≤ 2q},

A¯ = {| p| ≥ 1, | p| ≥ 2q}.

To make our presentation more clear, we also introduce the smooth test function χ (x ) ∈ C0∞ ([0, ∞ )) satisfying



1, if x ∈ [0, 1],

χ (x ) =

0,

if x ≥ 2.

To kill the singularity occurring on p = q and p = −q, we use the splitting 1 = χA ( p, q ) + χA¯ ( p, q ) with



χA ( p, q ) = χ ( p0 ) + (1 − χ ( p0 ))χ

| p| q0



,



χA¯ ( p, q ) = (1 − χ ( p0 )) 1 − χ



| p| q0



.

We now decompose T (g1 , g2 ) = T2,A + T1,A¯ using the definitions of T1 and T2 as



T2,A =

R3 ×S2

 T1,A¯ =

R3 ×S2

Cθ B( p, q, ω )J1/2 (q )[g1 ( p )g2 (q ) − g1 ( p)g2 (q )]χA ( p, q )dqdω, p0 q0 Cθ  1/2 √ J (q )[g1 ( p )g2 (q ) − g1 ( p)g2 (q )]χA¯ ( p, q )dqdω. p0 q0 s

By these important decomposition, we will deduce the main estimate: Lemma 3.1. Assume λ0 > 0 and suitable small and ≤ 0. For any |β | ≥ 0, one has

 2         

   w ,λ ∂β T (g1 , g2 ), ∂β g3   w ,λ0 ∂β g3  w ,λ0 ∂β1 g1  wλ0 ∂β2 g2  + wλ0 ∂β1 g1  w ,λ0 ∂β2 g2  . 0 ν ν ν ν ν β1 +β2 ≤β

(3.4)

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

This lemma will follow directly from the later lemmas below. To avoid taking derivatives for the singular factor of |ω · ( pp − 0 change of variables q → u (for fixed p) as

q q0

861

)| inside B( p, q, ω ) for T2,A , we introduce the following

u = p0 q − q0 p, from which we have that

q=





 q0 u p u p+ = p · u + ( p · u )2 + p20 + u2 − , p0 p0 p0 p0

and

q0 = ( p · u ) +



( p · u )2 + p20 + u2 .

According to [22], one also have

     ∂u  p20 p20 q j pi ∂ ui ∂ ui   = p0 δi j − ,  = det = ( p q − p · q ) ≥ . 0 0 ∂qj q0 ∂q  ∂qj q0 q0

We next express T2,A as

 T2,A =

R3 ×S2





Cθ  ∂ q  |ω · u| BJ1/2 (q )[g1 ( p )g2 (q ) − g1 ( p)g2 (q )]χA ( p, q )dudω, p0 q0  ∂ u 

(3.5)

where

 B= B( p, q, ω ) =

( p0 + q0 )2 . 2 [( p0 q0 )2 − (ω · ( p + q ))2 ]

We take the p−derivatives ∂ β of (3.5) to obtain

|∂β T2,A | ≤

 



R3 ×S2

+



dudω1| p|≤2q0 XAβ0 

     ∂β1 g1 ( p ) ∂β2 g2 (q )Yββ21  

R3 ×S2

dudω1| p|≤2q0 XAβ0 (∂β1 g1 )( p)

where the sum is over β0 + β1 + β2 = β , and

 

XAβ0 = XAβ0 (u, p, ω ) = |ω · u|∂βp,q



0



   ∂β2 g2 (q )Yβ2 ,







 1  ∂ q  1/2 BJ (q )χA ( p, q ) Yβ0 .   p0 q0 ∂ u β

Also ∂β denotes the mixed partial derivatives with respect to variables p and q, Yβ2 and Yβi (i = 0, 2 ) are the terms which 0 1 result from applying the chain rule to the post-collisional momentum (p , q ) and momentum q respectively. The next step is to reverse the change of variables q → u, after the change of variables u → q: p,q

Lemma 3.2. For λ0 > 0 suitably small and ≤ 0, on the set A, we obtain

w2 ,λ0 XAβ0 ( p0 q − q0 p, p, ω ) 

J1/8 (q ) , p0

     β2  Yβ1 ( p0 q − q0 p, p, ω ), Yβ2 ( p0 q − q0 p, p)  qn0 , where n ≥ 1 is a fixed large integer which depends upon λ, and β . Proof. When 0 ≤ λ0 



exp

λ0

p0 ( 1 + t )ϑ

1 16 ,



we see that



 exp

λ0

(1 + t )

q ϑ 0

≤ J−1/8 (q ),

on set A. Therefore the proof of Lemma 3.2 directly follows from [22].  Lemma 3.3. For λ0 > 0 suitably small and ≤ 0, we have the following estimates for T2,A :

 2  w ,λ ∂β T2,A   e −p1280 0



dqdω1| p|≤2q0 J 16 (q ) 1

3 2 β1 +β2 ≤β R ×S

        

 ∂β1 g1 ( p ) ∂β2 g2 (q ) + ∂β1 g1 ( p) ∂β2 g2 (q ) .

Moreover, from that estimate one can deduce the following uniform upper bound

 2     

   w ,λ ∂β T2,A (g1 , g2 ), ∂β g3   ∂β g3  ∂β1 g1  ∂β2 g2  . 0 ν ν ν β1 +β2 ≤β

(3.6)

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F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

The proof of Lemma 3.3 is similar to that of [22], we omit the details. We are now in a position to prove the estimates for T1,A¯ . By a change of variable p − q → u and using the product ruler as well as a reverse change of variables, we have

      p,q     β  1/2  |∂β T1,A¯ | ≤ dqdω1| p|≥2q0 ∂β √ J (q )  ∂β1 g1 ( p ) ∂β2 g2 (q )Zβ2  0 1 p0 q0 s R3 ×S2      

    + dqdω1| p|≥2q0 ∂βp,q √ J1/2 (q ) ∂β1 g1 ( p) ∂β2 g2 (q ). 0



(3.7)

p0 q0 s

R3 ×S2

β

Here Zβ2 is the collection of sums of product of momentum derivatives of p and q , from (2.2), which result from the chain ruler 1

of differentiation. Again the sum is over β0 + β1 + β2 ≤ β . We then have: ¯ for any β ≥ 0, it holds that Lemma 3.4. Let ( p, q ) ∈ A,

   p,q ∂  β



√ J1/2 (q )

p0 q0 s

     √ J1/4 (q ),  p0 q0 s

(3.8)

|∂β p | + |∂β q |  qn0 .

(3.9)

Proof. (3.9) is proved in [22]. (3.8) holds due to the fact that

√ 2≤



p0    p0 . q0



We now obtain: Lemma 3.5. It holds that

 

 ∂β T1,A¯  

3 2 β1 +β2 ≤β R ×S

dqdω1| p|≥2q0



1

√ J 8 (q )

p0 q0 s

         

 ∂β1 g1 ( p ) ∂β2 g2 (q ) +  ∂β1 g1 ( p) ∂β2 g2 (q ) .

(3.10)

Moreover, for λ0 > 0 suitably small and ≤ 0, we have the following estimates for T2,A :

 2         

   w ,λ ∂β T1,A¯ (g1 , g2 ), ∂β g3   w ,λ0 ∂β g3  w ,λ0 ∂β1 g1  wλ0 ∂β2 g2  + wλ0 ∂β1 g1  w ,λ0 ∂β2 g2  . 0 ν ν ν ν ν

(3.11)

β1 +β2 ≤β

Proof. We only prove (3.11), since one can easily get (3.10) from Lemma 3.4 and (3.7). By (3.10), we have

 2   w ,λ ∂β T1,A¯ (g1 , g2 ), ∂β g3  0 



6 2 β1 +β2 ≤β R ×S

+

dpdqdω1| p|≥2q0



6 2 β1 +β2 ≤β R ×S



√ J 8 (q )w2 ,λ0 ( p)



    ∂β1 g1 ( p ) ∂β2 g2 (q )|∂β g3 ( p)|





1

p0 q0 s

dpdqdω1| p|≥2q0

√ J 8 (q )w2 ,λ0 ( p)∂β1 g1 ( p) 1

p0 q0 s

  ∂β2 g2 (q )|∂β g3 ( p)|



= I1 + I2 . For I1 , we get from Cauchy–Schwartz’s inequality and pre-post change of variables that

I1 



R6 ×S2

β1 +β2 ≤β



×

R6 ×S2



β1 +β2 ≤β

+

β1 +β2 ≤β

1 8

√ J (q ) p0 q0 s





√ J 8 ( q ) p0 q0 s 1

w2 ,λ0

( p)|∂β g3 ( p)|

12

2

1    2

2 ∂β1 g1 ( p ) ∂β2 g2 (q ) w2 ( p )w2λ0 ( p )w2λ0 (q ) + w2 (q )w2λ0 ( p )w2λ0 (q )



    2 1 dpdqdω w2 ( p)w2λ0 ( p)w2λ0 (q ) ∂β1 g1 ( p) ∂β2 g2 (q ) p0 q0 R6 ×S2

ν

β1 +β2 ≤β



dp dq dω

  w ,λ0 ∂β g3 



dpdqdω



  w ,λ0 ∂β g3 



ν

    2 1 dpdqdω w2 (q )w2λ0 ( p)w2λ0 (q ) ∂β1 g1 ( p) ∂β2 g2 (q ) 6 2 p q 0 0 R ×S

         

w ,λ0 ∂β g3  w ,λ0 ∂β1 g1  wλ0 ∂β2 g2  + wλ0 ∂β1 g1  w ,λ0 ∂β2 g2  , ν ν ν ν ν

where we have used the fact that



√

s

12

≤ 1,  p ≤  p  + q  and  p−   p − + q − , for ≤ 0.

12

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

863

The second term I2 clearly has the desired upper bound in (3.11) using the Cauchy–Schwartz inequality. This completes the proof of Lemma 3.5.  Combing Lemma 3.1, we can deduce the following weighted estimates on the nonlinear collision term with respect to the time-velocity weight w ,λ0 (t, p) which will play an important role in performing the nonlinear energy estimates in the next section. Lemma 3.6. Assume λ0 > 0 and suitable small and ≤ 0. For any β ≥ 0 and α ≥ 0, one has

         2 α  

   w ,λ ∂β T (g1 , g2 ), ∂βα g3   w ,λ0 ∂βα g3  w ,λ0 ∂ α1 g1  wλ0 ∂ α2 g2  + wλ0 ∂ α1 g1  w ,λ0 ∂ α2 g2  . β1 β2 β1 β2 0 ν ν

β1 +β2 ≤β α1 +α2 =α

ν

ν

ν

Next, we turn to control the weighted estimates on the nonlinear collision term  with respect to the time-velocity exponential weight w ,λ0 (t, p) in terms of the temporal energy functionals EN, ,λ0 (t ), E N (t ) defined by (1.15) and (1.16) and the

corresponding entropy dissipation rate DN, ,λ0 (t ), DN (t ) defined by (1.17) and (1.18). For this issue, we prove the following Lemma 3.7. For λ0 > 0 and suitable small, and assume l − 1 ≥ N ≥ 10, 0 < ϑ ≤ 1/4, |α| + |β| ≤ N. Then one has 1/2 |∂ α ± ( f, f ), ∂ α {I± − P± } f |  EN−3,l−1, λ0 (t )D N (t ),     2 1/2  w|α|+|β|−l,λ0 ∂βα ± ( f, f ), ∂βα {I± − P± } f   EN−3,l−1, λ0 (t )DN,l,λ0 (t ).

(3.12) (3.13)

Moreover, if |α | > 0, it follows that

  w2

|α|−l,λ0 ∂

α

±

 1/2 ( f, f ), ∂ α f±   EN−3,l−1, λ0 (t )DN,l,λ0 (t ).

(3.14)

Here and in the sequel, in the case when an undetermined energy functional EN, ,λ (t ) appears on the right-hand side of inequalities, it is always understood to take exactly the right-hand expression of (1.15). Proof. Noticing l − 1 ≥ N ≥ 10, by virtue of Lemma 3.6 and Sobolev’s inequality, one can prove (3.12) without any difficulty. Thus, to complete the proof of Lemma 3.7, we only prove (3.13) detailedly in the following part, since the proof of (3.14) is similar and easier. Recalling Lemma 3.6, it suffices to estimate

R

      w ,λ0 ∂βα {I± − P± } f  w ,λ0 ∂ α1 f±  wλ0 ∂ α2 f±  dx, β β 1 2 ν 3

R

      w ,λ0 ∂βα {I± − P± } f  wλ0 ∂ α1 f±  w ,λ0 ∂ α2 f±  dx. β β 1 2 ν 3



I3 =

β1 +β2 ≤β α1 +α2 =α

and



I4 =

β1 +β2 ≤β α1 +α2 =α

ν

ν

ν

ν

Without loss of generality, we only compute I3 in the following. For this, by splitting f± = P± f + {I± − P± } f, we have

I3  with

I3,1 + I3,2 + I3,3 + I3,4 , 

  |∂ α1 (a± , b, c )||∂ α2 (a± , b, c )|w|α|+|β|−l,λ0 ∂βα {I± − P± } f ν dx,        = |∂ α1 (a± , b, c )|wλ0 ∂βα22 {I± − P± } f  w|α|+|β|−l,λ0 ∂βα {I± − P± } f ν dx, ν R3        = w|α|+|β|−l,λ0 ∂βα1 {I± − P± } f  |∂ α2 (a± , b, c )|w|α|+|β|−l,λ0 ∂βα {I± − P± } f  dx,  1 ν 3 ν R           = wλ0 ∂βα22 {I± − P± } f  w|α|+|β|−l,λ0 ∂βα11 {I± − P± } f  w|α|+|β|−l,λ0 ∂βα {I± − P± } f ν dx.

I3,1 = I3,2 I3,3 I3,4

R3

ν

R3

ν

Next, we only present the estimates for I3, 1 I3, 3 , the others being similar. We divide our computations into following three cases. Case 1. |α| + |β| ≤ N/2. Noticing that α1 + β1 + α2 + β2 ≤ α + β , in this case, L∞ x −norm can be used to control both functions α α involving differentiations ∂β 1 and ∂β 2 . Thus, by applying Sobolev’s inequality, we obtain

I3,1 

|α1 |≤N/2

1

2





∇x ∂ α1 (a± , b, c ) H1 ∂ α2 (a± , b, c ) w|α|+|β|−l,λ0 ∂βα {I± − P± } f 

1/2  EN−3,l−1, λ (t )DN,l,λ0 (t ), 0

where the fact that N ≥ 10 and N/2 + 2 ≤ N − 3 has been used.

ν

864

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

Similarly, one has

      w|α|+|β|−l,λ0 ∂βα11 {I± − P± } f  ∂ α2 ∇x (a± , b, c ) H1 w|α|+|β|−l,λ0 ∂βα {I± − P± } f ν

I3,3 

ν

|α2 |≤N/2



1/2 EN−3,l−1, λ0

(t )DN,l,λ0 (t ).

Case 2. |α| + |β| ≥ N/2 and |α1 | + |β1 | ≥ |α2 | + |β2 |. In this case, |α2 | + |β2 | ≤ N/2 and |α1 | + |β1 | ≥ N/4. For I3, 1 , if |α 1 | > 0, since N/2 + 2  N − 3, we see that 1/2

∂ α1 (a± , b, c )  DN,l, λ0 (t ), 1/2

∂ α2 (a± , b, c ) L∞  ∂ α2 (a± , b, c ) H2  EN−3,l−1, λ0 (t ),

which imply that

I3,1 

0<|α1 |≤N





∂ α1 (a± , b, c )

|α2 |≤N/2

1/2 EN−3,l−1, λ0

 

∂ α2 (a± , b, c ) L∞ w|α|+|β|−l,λ0 ∂βα {I± − P± } f ν

(t )DN,l,λ0 (t ).

If |α1 | = 0, then α2 = α . We find that 1/2 1/2 α2

(a± , b, c ) L∞  ∇x (a± , b, c ) H1  DN,l, λ0 (t ), ∂ (a± , b, c )  EN−3,l−1,λ0 (t ).

Therefore

I3,1  (a± , b, c ) L∞

|α|=|α2 |≤N/2



1/2 EN−3,l−1, λ0

 

∂ α2 (a± , b, c ) w|α|+|β|−l,λ0 ∂βα {I± − P± } f ν

(t )DN,l,λ0 (t ).

As to I3, 3 , it follows form Sobolev’s inequality that

      w|α|+|β|−l,λ0 ∂βα11 {I± − P± } f  ∂ α2 (a± , b, c ) H2 w|α|+|β|−l,λ0 ∂βα {I± − P± } f ν

I3,3 

|α2 |≤N/2

1/2  EN−3,l−1, λ (t )DN,l,λ0 (t ). 0

Case 3. |α| + |β| ≥ N/2 and |α1 | + |β1 | ≤ |α2 | + |β2 |. In this case, |α1 | + |β1 | ≤ N/2 and |α2 | + |β2 | ≥ N/4. It is easy to see that the estimates on I3, 1 in this case are the same as that of Case 2, and we thus omit the details of its proof for brevity. Now we shall estimate I3, 3 carefully. Firstly, since |α| + |β| ≥ N/2 ≥ 5 and |α1 | + |β1 | ≤ |α2 | + |β2 |, we see that |α| + |β| − |α1 | − |β1 | − 2 ≥ 1. Then it follows that

|α1 |+|β1 |≤N/2 2|α1 |+2|β1 |≤|α|+|β|

      w|α|+|β|−l,λ0 ∂βα11 {I± − P± } f   ∞  ν L

|α1 |+|β1 |≤N/2 2|α1 |+2|β1 |≤|α|+|β|

      w|α|+|β|−l,λ0 ∂βα11 {I± − P± } f  

ν H2





1/2 1/2  min DN,l, λ (t ), EN−3,l−1,λ (t ) , 0

0

which yields that for α 2 > 0

I3,3 

|α1 |+|β1 |≤N/2 2|α1 |+2|β1 |≤|α|+|β|

      w|α|+|β|−l,λ0 ∂βα11 {I± − P± } f   2 ∂ α2 (a± , b, c ) ν H

  1/2 × w|α|+|β|−l,λ0 ∂βα {I± − P± } f   EN−3,l−1, λ0 (t )DN,l,λ0 (t ), ν and for α2 = 0

I3,3 

|α1 |+|β1 |≤N/2

      w|α|+|β|−l,λ0 ∂βα11 {I± − P± } f  (a± , b, c ) H2 w|α|+|β|−l,λ0 ∂βα {I± − P± } f ν ν

1/2  EN−3,l−1, λ (t )DN,l,λ0 (t ). 0

This completes the proof of Lemma 3.7.  With Lemmas 3.3 and 3.5 in hand, one can prove the following estimates. Lemma 3.8. Let ζ (p) be a smooth function that decays in p exponentially, and let |α | ≤ N, N ≥ 8. Writing

∂ α ( f, f ) =

α1 +α2 =α

(∂ α1 f, ∂ α2 f ),

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

865

one has

  1/2 1/2    (∂ α1 f, ∂ α2 f )ζ ( p) dp  E N (t )DN (t ).

Lemma 3.9. Let ≤ 0, N ≥ 8. It holds that

|α|≤N

w ∂ α T ( f, f ) + w T ( f, f ) Z1 

|α|≤N

 2   w + 12 ∂ α f  .

3.2. Weighted estimates on L In this subsection, we deduce some weighted estimates on the linearized collision operator L with respect to the time-velocity exponential weight w ,λ0 (t, p). Remembering the notations (3.1)–(3.3), we now give the basic estimates for ν and K. Lemma 3.10. Under the condition of (1.8), it holds that

ν ( p) ∼ p−1 0 .

(3.15)

Moreover, for |β | ≥ 1, we have

|∂β ν ( p)|  p−2 0 .

(3.16)

Proof. We only prove (3.16), the estimate for (3.15) is similar to that of Lemma 3.1 in [34]. Noticing that loss ν ( p) = T loss (1, J1/2 ) = T1,loss (1, J1/2 ) + T2,A (1, J1/2 ). A¯

loss (1, J 1/2 ). From (3.6), we find To prove (3.16), it suffices to estimate ∂β T loss (1, J1/2 ) and ∂β T2,A ¯

   −p0 loss ∂β T2,A (1, J1/2 )  e 128 β2 ≤β

For

∂β T1,loss (1, J1/2 ), A¯

1,A





dqdωJ 16 (q )∂β2 J1/2 (q )  p−2 0 . 1

R3 ×S2

since

   √  p  s   q|β| 0 ∂  , √  β p0 s  p20

¯ we get from (3.4) that on set A,

    1/2  ( 1, J ) ∂β T1,loss  A¯

    p   J (q )  dqdω1| p|≥2q0 ∂β  p−2 √ 0 . p0 s  q0 R3 ×S2

This completes the proof of Lemma 3.10.  With Lemma 3.10 in hand, one can see that Lemma 3.11. Let |β | ≥ 1, ∈ R and λ0 ≥ 0 suitably small. For any small η > 0, there exists a large R > 0 and Cη > 0 such that

2 

    w ,λ0 ∂β  g2 − Cη |g|22 , ∂β {ν ( p)g}, w2 ,λ0 ∂β g  w ,λ0 ∂β gν − η L ν BR

|β  |≤|β|

where BR denotes the closed ball in R3p with center zero and radius a constant R. Let us now turn to estimate K. For results in this direction, we have Lemma 3.12. Assume |β | ≥ 1, ∈ R and λ0 ≥ 0 suitably small. For any small η > 0, there exists a large R > 0 and Cη > 0 such that



∂

w2 ,λ0 β Kg1 ,



∂β g2  η

|β1 ≤β

!

|w ,λ0 ∂β1 g1 |ν + Cη |g1 |L2B

R

  w ,λ0 ∂β g2  . ν

Proof. The proof of (3.17) is similar as [22, pp.668, Proposition 8], we omit the details for brevity.  Lemma 3.13. Assume |β | ≥ 1, ∈ R and λ0 ≥ 0 suitably small. One has





w2 ,λ0 ∂β L g, ∂β g  |w ,λ0 ∂β g|2ν − C

β1 < β

For β = 0, one also has





w2 ,λ0 L g, g  |w ,λ0 g|2ν − C |w ,λ0 g|2L2 , BC

|w ,λ0 ∂β1 g|2ν − C |w ,λ0 ∂β g|2L2 . BC

(3.17)

866

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

and

L g, g  |{I − P}g|2ν . As a direct application of Lemmas 3.11–3.13, we have the following weighted coercivity estimates on the linearized collision operator L with respect to the time–velocity exponential weight w ,λ0 (t, ξ ). Lemma 3.14. Assume |β | ≥ 1, ∈ R and λ0 ≥ 0 suitably small. It holds that

|β|≤N





w2 ,λ0 ∂β Lg, ∂β g 

|β|≤N

|w ,λ0 ∂β g|2ν − C |g|2L2 , BC

where g is a vector function in R2 . 4. Linearized analysis Consider the Cauchy problem on the linearized system with a microscopic source S = S(t, x, p) = [S+ (t, x, p), S− (t, x, ξ )]:

⎧ ˜ 1/2 q 1 + L f = S, ∂t f + p˜ · ∇x f − E · pJ ⎪ ⎪ ⎪ ⎪ ⎪ ˜ 1/2 , f+ − f− , ⎨∂t E − ∇x × B = − pJ ∂t B + ∇x × E = 0, ⎪ ⎪ ⎪ ∇x · E = J1/2 , f+ − f− , ∇x · B = 0, ⎪ ⎪ ⎩ [ f, E, B]|t=0 = [ f0 , E0 , B0 ],

(4.1)

where initial data [f0 , E0 , B0 ] satisfies the compatibility condition

∇ x · E0 =



R3

∇x · B0 = 0,

J1/2 ( f0,+ − f0,− ) dp,

(4.2)

and the source term S is assumed to satisfy



R3

J1/2 (S+ − S− ) dp = 0.

To consider the solution to the Cauchy problem (4.1), for simplicity, we denote U = [ f, E, B], U0 = [ f0 , E0 , B0 ] so that one can formally write

U (t ) = A(t )U0 +



t

0

A(t − s )[S(s ), 0, 0] ds,

where A(t ) is the linear solution operator for the Cauchy problem on the linearized homogeneous system corresponding to (4.1) in the case when S = 0. We also remark that the operator A(t ) is a bounded operator in some weighted Sobolev space. 4.1. Macro structure We first note that for the relativistic Maxwellian J, we have the normalization for some integrals as follows

 C0 =  1122 = C00

R3

 pJdp, C 00 = p21 p22 Jdp,  p2

R3

Recalling (1.13), we see that





 R3

 p2 Jdp, C 11 = 

1111 C00 =



R3

p41 Jdp,  p2

bi (t, x ) =



pi J1/2 , f+ + f−

 R3

p21 Jdp,

11 C00 =

 R3

C011 =

R3

R3 J ( p)dp

p21

 p

= 1, then we introduce the notation

Jdp,

p21 Jdp.  p2



a± (t, x ) = J1/2 , f± = J1/2 , P± f − C 0 c, 1 2









=

pi J1/2 , P± f

 ,

C   C  1  pJ1/2 , f+ + f− − 12 C 0 J1/2 , f+ + f− 2 . c (t, x ) = C 00 − (C 0 )2 11

11

(4.3)

Taking velocity integrations of the first equation of (4.1) with respect to the velocity moments

J1/2 ,

pi J1/2 ,

i = 1, 2, 3,

 pJ1/2 ,

one has

   1/2    ˜ ∂t a± + C 0 c + C011 ∇x · b + ∇x · pJ , {I± − P± } f = J1/2 , S± ,

(4.4)

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

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#







867







˜ i J1/2 , {I± − P± } f = pi J1/2 , S± −L± f , ∂t C 11 bi + pi J1/2 , {I± − P± } f + ∂i C011 a± + C 11 c ∓ C011 Ei + ∇x · pp "  #     ∂t C 00 c + C 0 a± +  pJ1/2 , {I± − P± } f + C 11 ∇x · b + ∇x · pJ1/2 , {I± − P± } f =  pJ1/2 , S± −L± f .

(4.5) (4.6)

Define the high-order moment functions ( f± ) = (i j ( f± ))3×3 and ( f± ) = (1 ( f± ), 2 ( f± ), 3 ( f± )) by

i j ( f ± ) =

 p p i

j

 p





− A1 J1/2 , f± ,

i ( f± ) =

 1   − A2 pi J1/2 , f± ,  p

where A1 and A2 satisfy



C 11 C011 − A1



C0

1122 = C00 − A1 C011

and

A2 =

C011 C 11

respectively. Further taking momentum integrations of the first equation of (4.1) with respect to the above high-order moments, one has

$

∂t

 C 11 − A1  ii ({I± − P± } f ) − 0 0 {I± − P± } f,  pJ1/2 + C 

%



C 11 − A1 C 0 −

C 00 C011 − A1 C0

& ' c

 C 11 − A  ∂i bi = ii (r± + S± ) + 0 0 1 r± + S± ,  pJ1/2 , C     p 1/2 1122 ∂t i j ({I± − P± } f ) + C00 J , i = j, ∂ j bi + ∂i b j = i j (r± + S± ) + A1 ∇x · {I± − P± } f,  p     (C 11 )2 (C 11 )2 11 11 ∂t i ({I± − P± } f ) + C00 − 011 − 011 Ei = i (r± + S± ), ∂i a± ∓ C00 1111 1122 + C00 − C00



C

C

(4.7) (4.8) (4.9)

where

r = [r+ , r− ],

r± = − p˜ · ∇x {I± − P± } f − L± f.

Notice that we have used (4.6) and (4.4) to derive (4.7) and (4.8) respectively. Taking the mean value of every two equations with ± sign for (4.4)–(4.6) and noticing S = 0, we have for 1 ≤ i ≤ 3

⎧      3 ⎪ C011 C 0 1 ( C 0 )2 a+ + a− ⎪ 11 ⎪ 1 − + C − · b + ∂i i ({I − P} f · [1, 1] ) = 0, ∂ ∇ t x ⎪ 0 ⎪ 2 2 C 00 C 00 ⎪ i=1 ⎪ ⎪ a + a  ⎨ C 11 3 1 − ∂t bi + C011 ∂i + + C 11 ∂i c + ∂ j i j ({I − P} f · [1, 1] ) = 0, 2 2 2 ⎪ ⎪ j=1 ⎪     ⎪ 3 ⎪ ⎪ C 00 C 11 1 0 11 ⎪ C − c + C − ∂i i ({I − P} f · [1, 1] ) = 0, ∂ ∇x · b + ⎪ t 0 ⎩ 0 0 C

C

2

i=1

where we have also used (4.4) to derive (4.10)3 . Write









˜ 1/2 , f+ − f− , H = H (t, x ) = pJ and

H = H (t, x ) = pJ1/2 , f+ − f− . Then,

















˜ − p]J ˜ 1/2 , {I − P} f = pJ ˜ 1/2 , {I − P} f · q 1 , H = [ p, H = [p, −p]J1/2 , {I − P} f = pJ1/2 , {I − P} f · q 1 . Taking difference of two equations with ± sign for (4.4) and (4.5) and also noticing S = 0, we obtain

∂t (a+ − a− ) + ∇x · H = 0,   ∂t H + C011 ∇x (a+ − a− ) − 2C011 E = [p, −p]J1/2 , r · q 1 . In particular, for the nonlinear system (1.9), the non-homogeneous source S = [S+ (t, x, p), S− (t, x, p)] takes the form of

1 S± = ± E · p˜ f± ∓ (E + p˜ × B ) · ∇ p f± + ± ( f, f ). 2 Then, it is straightforward to compute from integration by parts that

J1/2 , S±  = 0.

(4.10)

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F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

4.2. Time decay for the linearized system For the linearized homogeneous system, repeating the argument employed in [6], one can prove Proposition 4.1. Let S = 0, and let [f, E, B] be the solution to the Cauchy problems (4.1) and (4.2) of the linearized homogeneous system. Define the velocity weight function W = W ( p) by

W ( p) =  p 2 . 1

(4.11)

Then, for ≥ 0 and α ≥ 0 with m = |α|,

    α  

+ low W ∂ f  + ∂ α (E, B )  (1 + t )−σm  ∗ f0  + (E0 , B0 ) L1x W Z1     high   + (1 + t )−j /2 W + ∗ ∇xj +1 ∂ α f0  + ∇xj +1 ∂ α (E0 , B0 ) ,

(4.12)

where

σm =

m 3 + , 4 2

ell∗low > 2σm ,

ell∗high > 0, 0 ≤ j < high . ∗

(4.13)

The following remark is concerned with Proposition 4.1. Remark 4.1. The extra (j + 1 ) − th order derivative of the initial data is required in order to deduce the time decay rate of [f, E, B]. This results essentially from the coupling of the hyperbolic Maxwell equations but not due to the technique of the approach, see [5] for the analysis of the Green’s function of the damping Euler–Maxwell system. Now, we will prove Proposition 4.1 by using as in [13] the following lemma. Lemma 4.1. (i) For any t ≥ 0, it holds that

  ˆ B] ˆ |2 + κ|(I − P ) fˆ|2ν ≤ 0. ∂t | fˆ|2 + |[E,

(4.14)

(ii) There is a time–frequency interactive functional E0int,(1 ) (t, k ) defined by

E0int,(1)



3

1 aˆ+ + aˆ− (t, k ) = i ki 2 2 1 + |k| i=1

+

3 

1 ˆi i k jb 2 1 + |k| i, j=1

  ˆ | i {I − P} f · [1, 1]

  1 | i j {I − P} fˆ · [1, 1] 2

−

3  1122

C00 1 ˆi i ki b 1111 − C 1122 1 + |k|2 C00 00 i, j=1

+

3  1122

3C00 1 ˆi i ki b 1111 − C 1122 1 + |k|2 C00 00 i=1

+

  1 |  j j {I − P} fˆ · [1, 1] 2

  1 | ii {I − P} fˆ · [1, 1] 2

 3  κ1 ˆi , ˆ b i k | c i 1 + |k|2 i=1

(4.15)

with a properly chosen constant 0 < κ 1  1 such that

∂t  E0int,(1) (t, k ) +

2   κ|k|2   2 ˆ 2   a+ + a− + |b| + |cˆ|2  {I − P} fˆ . 2 1 + |k| ν

(4.16)

(iii) There exists a constant 0 < κ 2  1 such that

       ∂t  Hˆ | i k a+ − a− ∂t  Hˆ | Eˆ ∂t  i k × Bˆ | Eˆ − − κ2 1 + |k|2 1 + |k|2 (1 + |k|2 )2  2  2 κ k · Eˆ κ  ˆ2 |k × Bˆ|2 +κ a+ − a−  + + E +κ 2 2 1 + |k| 1 + |k| (1 + |k|2 )2  2    {I − P} fˆ . ν

(4.17)

(iv) Let [f, E, B] be the solution to the Cauchy problem (4.1) with S = 0. Then there is a time-frequency interactive functional E0int (t, k ) such that

ˆ B] ˆ |2 , |E0int (t, k )|  | fˆ|22 + |[E,

(4.18)

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

and

869

  ˆ B] ˆ |2 + κ0  E0int (t, k ) + κ|{I − P} fˆ|2ν ∂t | fˆ|22 + |[E,  2  κ|k|2   2 ˆ 2 κ κ|k|2 a+ + a− + |b| + |cˆ|2 + κ a+ + − a−  + |Eˆ|2 + |Bˆ|2 ≤ 0, 2 2 1 + |k| 1 + |k| (1 + |k|2 )2

(4.19)

where κ 0 > 0 is a small constant such that

ˆ B] ˆ |2 + κ0  E0int (t, k ) ∼ | fˆ|22 + |[E, ˆ B] ˆ |2 . | fˆ|22 + |[E,

(4.20)

Proof. In that follows, we only prove assertion (ii), since the other assertions are much similar to that of [13]. From (4.8), it follows that 1122 1122 C00 (bi + ∂ii bi ) = C00

1122 ∂ j j bi + 2C00 ∂ii bi

j=i

=−

1 ∂ j ∂t i j ({I − P} f · [1, 1] ) 2 j=i

1

1122 1122 + ∂ j i j (r · [1, 1] ) + A1 ∇x · ({I − P} f · [1, 1] ) + 2C00 ∂ii bi − C00 ∂i j b j 2 j=i

j=i

1 =− ∂ j ∂t i j ({I − P} f · [1, 1] ) 2 j=i

1

1122 1122 + ∂ j i j (r · [1, 1] ) + A1 ∇x · ({I − P} f · [1, 1] ) + 3C00 ∂ii bi − C00 ∂i ∇ · b. 2

(4.21)

j=i

On the other hand, we get from (4.7) that 1122 C00 ∂i ∇ · b = −

2(

−

1122 C00 1111 1122 C00 − C00

2(

)

1122 3C00 1111 C00

−

1122 C00

)

∂i ∂t  j j ({I − P} f · [1, 1] )

j

%

C



11

0

− A1 C −

C 00 C011 − A1

&

C0

∂i ∂t c +

2(

1122 C00 1111 1122 C00 − C00

)

∂i  j j (r · [1, 1] ),

(4.22)

j

and 1122 3C00 ∂ii bi = −

−

1122 3C00

1111 − C 1122 ) 2(C00 00

2(

1122 3C00 1111 C00

−

1122 C00

∂i ∂t ii ({I − P} f · [1, 1] ) %  11 00

)

C 11 − A1 C 0 −

C0 − A 1

C

C0

& ∂i ∂t c +

2(

1122 3C00 1111 C00

1122 ) − C00

∂i ii (r · [1, 1] ).

(4.23)

Thus (4.21)–(4.23) implies that 1122

1

C00 ∂ j ∂t i j ({I − P} f · [1, 1] ) − ∂i ∂t  j j ({I − P} f · [1, 1] ) 1111 1122 ) 2 2 ( C − C 00 00 j=i j

+ =

2(

1 j=i

−

1122 3C00

1111 C00

2

2(

1122 ) − C00

1122 ∂i ∂t ii ({I − P} f · [1, 1] ) + C00 (bi + ∂ii bi )



∂ j i j (r · [1, 1] ) + A1 ∇x · ({I − P} f · [1, 1] )

1122 C00 1111 1122 C00 − C00

)

∂i  j j (r · [1, 1] ) +

j

2(

1122 3C00 1111 C00

1122 ) − C00

∂i ii (r · [1, 1] ).

In light of (4.10), by the Fourier energy estimates on (4.24) and integration by parts, we obtain

 3

1 ˆi  ∂ i k jb t 1 + |k|2 i, j=1

−

  1 | i j {I − P} fˆ · [1, 1] 2

 3 1122

C00 1 ˆi  ∂ i ki b t 1111 − C 1122 1 + |k|2 C00 00 i, j=1

  1 |  j j {I − P} fˆ · [1, 1] 2

(4.24)

870

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

+

 3 1122

3C00 1 ∂t i ki bˆ i 2 1111 1122 1 + |k| C00 − C00 i=1

+

κ|k| ˆ 2 |k| |b|  η |(aˆ+ + aˆ− , cˆ)|2 + |{I − P} fˆ|2ν , 1 + |k|2 1 + |k|2 2

  1 | ii {I − P} fˆ · [1, 1] 2

2

(4.25)

here and in the sequel, η is a positive and suitably small constant. Similarly, by (4.9) and (4.10), we have

   3

1 κ|k|2 |k|2 ˆ 2 ˆ · [1, 1] ˆ ˆ { f  ∂ i k ( + a ) |  I − P } + |aˆ+ + aˆ− |2  η |b| + |{I − P} fˆ|2ν , a t + − i i 1 + |k|2 1 + |k|2 1 + |k|2

(4.26)

  3 2 2

1 |k|2 ˆ i + κ|k| |cˆ|2  |k| ˆ b  ∂ i k | |bˆ |2 + |aˆ+ + aˆ− |2 + |{I − P} fˆ|2ν . c t i 1 + |k|2 1 + |k|2 1 + |k|2 1 + |k|2

(4.27)

i=1

and

i=1

Therefore, (4.16) follows from (4.25)–(4.27). This completes the proof of Lemma 4.1.  The proof of Proposition 4.1. With assertion (iv) of Lemma 4.1 in hand, one can obtain (4.12) by performing the similar calculations as that of Theorem 3.1 in [6], we omit the details for brevity. This completes the proof of Proposition 4.1.  5. Global a priori estimates In this section we are going to prove Theorem 1.1, the main result of the paper. The key point is to deduce the uniform-in-time a priori estimates on solutions to the relativistic Vlasov–Maxwell–Boltzmann system

⎧ ∂t f + p˜ · ∇x f − E · p˜μ1/2 q1 + L f = S, ⎪ ⎪  1/2  ⎪ ⎨ ˜ ∂t E − ∇x × B = − pJ , f+ − f− , ∂t B + ∇x × E = 0, ⎪ ⎪ ⎪ ⎩∇ · E = J1/2 , f − f , ∇ · B = 0, x + − x

(5.1)

where the nonlinear term S = [S+ , S− ] is given by

S = ( f, f ) +

1 q 0 E · p˜ f − q 0 (E + p˜ × B ) · ∇ p f. 2

(5.2)

For that, let (f, E, B) be a smooth solution to (5.1) over the time interval 0 ≤ t ≤ T with initial data (f0 , E0 , B0 ) for some 0 < T ≤ ∞, and further suppose that (f, E, B) satisfies

X (t ) ≤ δ02 ,

(5.3)

where X(t) is given in (1.19) and the constant δ 0 > 0 is sufficiently small. Here recall that X(t) also depends on parameters N1 , 1 , λ, ϑ and 0 which will be fixed in the proof, what we want to do in the following is to deduce some a priori estimates on (f(t, x, p), E(t, x), B(t, x)) based on the a priori assumption (5.3). To make the presentation easy to follow, we divide this section into several subsections and the first one is concerned with the macro dissipation of the relativistic Vlasov–Maxwell–Boltzmann system. 5.1. Macro structure and macro dissipation Now we define the macro dissipation DN,mac (t ) by

DN,mac (t ) =

|α|≤N−1

∇x ∂ α (a± , b, c ) 2 + a+ − a− 2 + E 2HN−1 + ∇x B 2HN−2 .

With the above macro structure of the system (5.1) in hand, we have Lemma 5.1. For any integer N with 8 ≤ N ≤ N1 , there is an interactive energy functional ENint (t ) such that

|ENint (t )| 



∂ α f 2 + ∂ α (E, B ) 2



(5.4)

|α|≤N

and

d int E (t ) + κ DN,mac (t )  dt N

|α|≤N

∂ α {I − P} f 2ν + E N (t )DN (t )

(5.5)

hold for 0 ≤ t ≤ T. Proof. Basing on the previous work [12] and [13], it is a quite standard process to obtain (5.5) with (5.4) being satisfied. We hence omit the details for brevity. 

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

871

5.2. Uniform spatial energy estimate In this subsection we derive the basic energy estimates on E N1 (t ) which contains only the spatial derivatives. Lemma 5.2. Let l1 − 1 ≥ N1 ≥ 12. There is an energy functional E N1 (t ) such that

d δ0 E N (t ) + κ DN1 (t )  D (t ) + E N1 (t )EN1 −3, 1 −1,λ0 (t ) dt 1 (1 + t )1+ϑ N1 , 1 ,λ0

(5.6)

holds for 0 ≤ t ≤ T. Proof. It is straightforward to establish the energy identities

1 d 2 dt



 ∂ α S, ∂ α f .

∂ α f 2 + ∂ α (E, B ) 2 + L∂ α f, ∂ α f  =

|α|≤N1

|α|≤N1

(5.7)

|α|≤N1

Moreover, from Lemma 5.1 as well as (5.3),

d int EN1 (t ) + κ DN1 ,mac (t )  dt

|α|≤N1

∂ α {I − P} f 2ν + δ02 DN1 (t ).

(5.8)

Then, since δ 0 > 0 can be small enough, the proper linear combination of (5.7) and (5.8) implies that there is an energy functional E N1 (t ) satisfying (1.15) such that

d E N (t ) + κ DN1 (t )  IN(11 ) (t ), dt 1 where

IN(11 ) (t ) = S, f  +

(5.9)

∂ α S, ∂ α f .

1≤|α|≤N1

Finally, we claim that

IN(11 ) (t )  EN1/2−3,l 1

λ

1 −1, 0

(t )DN1 (t ) +

δ0 1/2 1/2 D (t ) + E N1 (t )EN1/2 (t )DN1 (t ). 1 −3, 1 −1,λ0 (1 + t )1+ϑ N1 , 1 ,λ0

(5.10)

Therefore, the desired estimate (5.6) follows from plugging (5.10) into (5.9) and applying (5.3) and the Cauchy–Schwarz inequality to the first and third terms on the right-hand side of (5.10), respectively. Thus to complete the proof of Lemma 5.2, we only need to prove (5.10). To this end, we first consider the estimate of IN(1 ) (t ) corresponding to  (f, f) in the nonlinear term S. Using Lemma 3.7, it 1

directly follows that it is bounded up to a generic constant by EN1/2−3,l 1

hence,

∇x (E, B ) H5 ≤

1 −1,λ0

(t )DN1 (t ). Recall from the definition of X(t), and

X 1/2 (t ) δ0 ≤ . (1 + t )1+ϑ (1 + t )1+ϑ

For the zero-order term related to the electromagnetic field, it holds that

1 1 q 0 E · p˜ f − q 0 (E + p˜ × B ) · ∇ p f, f  = q 0 E · p˜ f, f  2  2 

R3 ×R3

|E |(|P f |2 + |{I − P} f |2 ) dxdp

 E · (a± , b, c ) L∞ (a± , b, c ) + E L∞ 1/2

 E N1 (t )DN1 (t ) +

δ0 D (t ), (1 + t )1+ϑ N1 , 1 ,λ0

 R3 ×R3

|{I − P} f |2 dxdp

where we have used 1 ≥ 1. For the ∂ α derivative term related to (E, B) with 1 ≤ |α | ≤ N1 , we write

∂ α (E · p˜ f ), ∂ α f  = =

α1 ≤α

α1 ≤α

Cαα1

 α  ∂ 1 E · p˜∂ α−α1 f, ∂ α f

Cαα1

 α  α α  Cα1 ∂ 1 E · p˜∂ α −α1 {I − P} f, ∂ α f . ∂ 1 E · p˜∂ α−α1 P f, ∂ α f + α1 ≤α

By Sobolev’s inequality, one can easily prove that the first term on the right-hand side of (5.11) is bounded by 1/2

C E N1 (t )DN1 (t ).

(5.11)

872

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

Now we turn to estimate the second term on the right hand side of (5.11). Notice that

l1 − 1 ≥ N1 , then if |α 1 | ≤ 4, it follows that

∂ α1 E · p˜∂ α−α1 {I − P} f, ∂ α f  

|α1 |≤4, α1 ≤α

|α1 |≤4



    1  1   

∂ α1 E L∞  p 2 ∂ α−α1 {I − P} f  ·  p− 2 ∂ α f 

δ0 D (t ). (1 + t )1+ϑ N1 , 1 ,λ0

On the other hand, for |α 1 | ≥ 5 and N1 ≥ 10, we also see that

|α − α1 | + 2 ≤ N1 − 3,  p 2 ≤ w|α−α1 |+3−l1 ( p). 1

With the above observation, we obtain

∂ α1 E · p˜∂ α−α1 {I − P} f, ∂ α f  

|α1 |≥5, α1 ≤α

|α1 |≥5, α1 ≤α

    1  1   

∂ α1 E · sup  p 2 ∂ α−α1 {I − P} f   p− 2 ∂ α f  2 x

1/2

 E N1 (t )EN1/2−3, 1

Lp

1/2

λ

1 −1, 0

(t )DN1 (t ),

where the Sobolev inequality g L∞ ≤ C ∇x g H 1 for any function g = g(x ) ∈ H 2 has been used. Combing both cases, one can see that

∂ α (E · p˜ f ), ∂ α f  

|α|≤N1

δ0 1/2 1/2 D (t ) + E N1 (t )EN1/2 (t )DN1 (t ). 1 −3, 1 −1,λ0 (1 + t )1+ϑ N1 , 1 ,λ0

Finally, in a similar way as before, it holds that

∂ α {( p˜ × B ) · ∇ p f }, ∂ α f  = 

0<α1 ≤α



Cαα1 ( p˜ × ∂ α1 B ) · ∇ p ∂ α −α1 f, ∂ α f



δ0 1/2 1/2 D (t ) + E N1 (t )EN1/2 (t )DN1 (t ). 1 −3, 1 −1,λ0 (1 + t )1+ϑ N1 , 1 ,λ0

Therefore, the claimed inequality (5.10) follows by collecting all the estimates and the proof of Lemma 5.2 is complete.  Remark 5.1. We would like to point out here that l1 − 1 ≥ N1 is needed in order to control the worst term

E · p˜∂ α f, ∂ α f .

|α|=N1

And |α 1 | ≥ 5 cannot be improved if one intends to control the term

 

 

sup  p 2 ∂ α −α1 f  1

x

by

EN1/2−3, −1,λ (t ). 1 1 0

L2p

This implies that derivatives of the electromagnetic field (E, B) which enjoys the explicit time-decay rate can

be up to order six.

5.3. The highest-order energy estimate with weight In this subsection we turn to the weighted energy estimates on EN1 , 1 ,λ0 (t ). As pointed out in [6], due to the regularityloss property of the whole system, two difficulties naturally come out, that is, the weighted highest-order energy functional EN1 , 1 ,λ0 (t ) can only be expected to increase in time and it is also a problem to obtain the weighted estimate on derivatives of the ˜ 1/2 . To overcome the first difficulty, we shall refine in the following lemma the nonlinear highest order N1 for the linear term E · pJ estimates in order to make use of the time-decay property of the lower-order energy functional EN1 −3, 1 −1,λ0 (t ), and postpone ˜ 1/2 to Lemma 5.4 in terms of the trick firstly introduced in [23]. the estimate on E · pJ Lemma 5.3. There is an energy functional EN1 , 1 ,λ0 (t ) with λ0 > 0, ϑ =

d (t ) + κ DN1 , 1 ,λ0 (t )  E dt N1 , 1 ,λ0

|α|=N1

1 4

 α  ˜ 1/2 , w2|α|− 1 ,λ0 ∂ α f ∂ E · pJ

and l1 − 1 ≥ N1 ≥ 12 such that (5.12)

for 0 ≤ t ≤ T. Proof. Starting from the first equation of (5.1), the energy estimate on ∂ α f with 1 ≤ |α | ≤ N1 weighted by the time-velocity dependent function w|α|− 1 ,λ0 = w|α|− 1 ,λ0 (t, p) gives

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

873

 ϑλ0  ξ 1/2 ∂ α f 2 1+ ϑ |α|− 1 ,λ0 (1 + t ) 1≤|α|≤N1 1≤|α|≤N1



   ˜ 1/2 , w2|α|− 1 ,λ0 ∂ α f . = ∂ α S, w2|α|− 1 ,λ0 ∂ α f + ∂ α E · pJ

1 d 2 dt

∂ α f 2|α|− 1 ,λ0 +

1≤|α|≤N1





L∂ α f, w2|α|− 1 ,λ0 ∂ α f +

(5.13)

1≤|α|≤N1

Similarly, from the first equation of (5.1), one has the weighted energy estimate on {I − P} f

 1 d ϑλ   p1/2 {I − P} f 2

{I − P} f 2− 1 ,λ0 + κ I − P f 2ν,− 1 ,λ0 + 1+ ϑ − 1 ,λ0 2 dt (1 + t ) 



 S, w2− 1 ,λ0 {I − P} f + DN1 (t )E N1 (t ) + DN1 (t ).

(5.14)

and the weighted energy estimate on {I − P}∂βα f with |α| + |β| ≤ N1 and |β | ≥ 1

1 d 2 dt

|β|≥1 |α|+|β|≤N1

 α   ∂β {I − P } f  2

|α|+|β|− 1 ,λ0

+κ

|β|≥1 |α|+|β|≤N1



|α|≤N1



|β|≥1 |α|+|β|≤N1

+

ν,|α|+|β|− 1 ,λ0

∂ α {I − P} f 2ν,|α|− 1 ,λ +

+

   α  ∂β { I − P } f  2

|β|≥1 |α|+|β|≤N1

+

λ

(1 + t )1+ϑ

 1/2 α   p ∂β {I − P} f 2

|α|+|β|− 1 ,λ0





∇x ∂ α (a± , b, c ) 2 + ∂ α E 2



|α|≤N1 −1

 p   i ∂βα−+eβi1 {I − P} f, w2|α|+|β|− 1 ,λ0 ∂βα {I − P} f ∂β 1  p

 α  ∂β S, w2|α|+|β|− 1 ,λ0 ∂βα {I − P} f + E N1 (t )DN1 (t ),

(5.15)

where |β1 | = 1 and ei denotes the multi-index with the ith element unit and the rest ones zeros. To be continued, we need to deal with the term



|β|≥1 |α|+|β|≤N1

∂β 1

 p   i ∂βα−+eβi1 {I − P} f, w2|α|+|β|− 1 ,λ0 ∂βα {I − P} f  p

carefully. Since

   pi   ∂β1  p    p−1 , |β |=1

1

we get from Cauchy–Schwartz’s inequality that



|β|≥1 |α|+|β|≤N1

∂β 1

 p   i ∂βα−+eβi1 {I − P} f, w2|α|+|β|− 1 ,λ0 ∂βα {I − P} f  p

 

 

 ∂βα−+eβi {I − P} f  1

ν,|α +ei |+|β −β1 |−l1 ,λ0

 2    Cη ∂βα−+eβi {I − P} f  1

 α   ∂β {I − P } f  2

ν,|α +ei |+|β −β1 |−l1 ,λ0

ν,|β|−l1 ,λ0



2

+ η∂βα {I − P} f 

ν,|α|+|β|−l1 ,λ0

.

(5.16)

Then, the proper linear combination of (5.7), (5.8), (5.13)–(5.16) implies that there is an energy functional EN1 , 1 ,λ0 (t ) satisfying (1.15) such that

d E (t ) + κ DN1 , 1 ,λ0 (t )  IN(21 , ) 1 ,λ0 (t ) + dt N1 , 1 ,λ0

|α|=N1

where

 α  ˜ 1/2 , w2|α|− 1 ,λ0 ∂ α f , ∂ E · pJ

(5.17)

874

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

) IN(2, (t ) = S, f  + 1 1 ,λ0

+

N1

1≤|α|≤N1

    ∂ α S, w2|α|− 1 ,λ0 ∂ α f ∂ α S, ∂ α f  + S, w2|α|− 1 ,λ0 {I − P} f + 1≤|α|≤N1

(

Cm

|β|=m |α|+|β|≤N1

m=1

(

)* A

+

 α  ∂β S, w2|α|+|β|− 1 ,λ0 ∂βα {I − P} f . )*

(5.18)

+

B

We now claim that ) IN(2, (t )  EN1/2 (t )DN1 , 1 ,λ0 (t ) + E N1 (t )DN1 , 1 ,λ0 (t ) + 1 1 ,λ0 1 −3, 1 −1,λ0 1/2

1

λ

(1 + t )1+ϑ ∇x (E, B ) H5 DN1 , 1 ,λ0 (t ).

(5.19)

By the assumption X (t ) ≤ δ02 , one gets that ) IN(2, (t )  δ0 DN1 , 1 ,λ0 (t ). 1 1 ,λ0

(5.20)

Then putting (5.20) into (5.17), we see that the desired estimate (5.12) follows. Thus to complete the proof of Lemma 5.3, we only need to prove (5.19). To this end, for brevity, we only present the estimate of A and B on the right-hand side of (5.18) since the estimate on other terms is simpler or follows in the completely same way. Take α with 1 ≤ |α | ≤ N1 . For the inner product term related to ∂ α  (f, f), by using (3.14) in Lemma 3.7, it follows that

 α  (t )DN1 , 1 ,λ0 (t ). ∂ ( f, f ), w2|α|− 1 ,λ0 ∂ α f  EN1/2 1 −3, 1 −1,λ0

Next, for the term E · p˜ f in S, one has

 α  α α  Cα1 ∂ 1 E · p˜∂ α −α1 f, w2|α|− 1 ,λ0 ∂ α f ∂ (E · p˜ f ), w2|α|− 1 ,λ0 ∂ α f = 



∂ α1 E L∞

|α1 |≤2

+

|α1 |≥3 or α1 =α



1

λ0

α1 ≤α

 R3 ×R3

w2|α|− 1 ,λ0



 |∂ α−α1 f |2 + |∂ α f |2 dxdp

    1  1   

∂ α1 E · sup  p 2 w|α|− 1 ,λ0 ∂ α−α1 f  2  p− 2 w|α|− 1 ,λ0 ∂ α f  x

Lp

(1 + t )1+ϑ ∇x E H3 DN1 , 1 ,λ0 (t ) + E N1 (t )DN1 , 1 ,λ0 (t ), 1/2

where we have used the Sobolev inequality g L∞  ∇ g H 1 . For the term (E + p˜ × B ) · ∇ p f in S, the difference point is that it contains the velocity derivative of order one. Our goal is to prove

 α  1 1/2 ∂ [(E + p˜ × B ) · ∇ p f ], w2|α|− 1 ,λ0 ∂ α f  E N1 (t )DN1 , 1 ,λ0 (t ) + (1 + t )1+ϑ ∇x (E, B ) H4 DN1 , 1 ,λ0 (t ). λ0

(5.21)

In fact, one can deduce that

 α  ∂ [(E + p˜ × B ) · ∇ p f ], w2|α|− 1 ,λ0 ∂ α f   1 = (E + p˜ × B ) · ∇ p w2|α|− 1 ,λ0 , − |∂ α f |2 2

  + Cαα1 (∂ α1 E + p˜ × ∂ α1 B ) · ∇ p ∂ α −α1 f, w2|α|− 1 ,λ0 ∂ α f . 0<α1 ≤α

Here, it is straightforward to see that the first term on the right is bounded in a rough way by

C E L∞  

1

λ0 1

λ0

 R3 ×R3



  p−1 + (1 + t )−ϑ w2|α|− 1 ,λ0 |∂ α f |2 dxdp

(1 + t )1+ϑ ∇x E H1



R3 ×R3

λ0 w2 |∂ α f |2 dxdp (1 + t )1+ϑ |α|− 1 ,λ0

(1 + t )1+ϑ ∇x E H1 DN1 , 1 ,λ0 (t ).

We split the second term on the right-hand side of (5.22) into

(5.22)

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882



0<α1 ≤α

Cαα1 (∂ α1 E + p˜ × ∂ α1 B ) · ∇ p ∂ α −α1 f, w2|α|− 1 ,λ0 ∂ α f

0<α1 ≤α

Cαα1 (∂ α1 E + p˜ × ∂ α1 B ) · ∇ p ∂ α −α1 P f, w2|α|− 1 ,λ0 ∂ α f





+





=

0<α1 ≤α

875



Cαα1 (∂ α1 E + p˜ × ∂ α1 B ) · ∇ p ∂ α −α1 {I − P} f, w2|α|− 1 ,λ0 ∂ α f .

(5.23)

In a similar way to estimate (5.11), we only compute the second term on the right-hand side of (5.23). For doing this, when |α 1 | ≤ 4, it is bounded by

C ∂ α1 (E, B ) L∞





R3 ×R3

 ∇x ∂ α1 (E, B ) H 1 

1

λ0





w2|α|− 1 ,λ0 |∇ p ∂ α −α1 {I − P} f |2 + w2|α|− 1 ,λ0 |∂ α f |2 dxdp

R3 ×R3

    w1+|α−α1 |− 1 ,λ0 ∇ p ∂ α−α1 {I − P} f 2 + w|α|− 1 ,λ0 ∂ α f 2 dxdp

(1 + t )1+ϑ ∇x (E, B ) H5 DN1 , 1 ,λ0 (t ),

where we have used the fact that |α − α1 | + 1 ≤ |α|. When |α 1 | ≥ 5, one can see that

 p 2 w|α|− 1 ,λ0 (t, ξ )   p− 2 w3+|α−α1 |− 1 ,λ0 (t, ξ ), 1

1

which implies that the second term on the right-hand side of (5.23) can be dominated by

 

     − 12   p w|α|− 1 ,λ0 ∂ α f   2 x Lp 

 1/2  − 12  α −α1 +α   E N1 (t ) {I − P} f DN1/2 (t )  p w1+|α−α1 +α |− 1 ,λ0 ∇ p ∂ 1 , 1 ,λ0

C ∂ α1 (E, B ) · sup  p 2 w|α|− 1 ,λ0 ∇ p ∂ α −α1 {I − P} f  1

|α  |≤2



1/2 E N1

(t )DN1 , 1 ,λ0 (t ).

Therefore (5.21) is true. Collecting all the above estimates, we see that 1/2 1 −3, 1 −1,λ0

A  EN

1/2

(t )DN1 , 1 ,λ0 (t ) + E N1 (t )DN1 , 1 ,λ0 (t ) +

1

λ0

(1 + t )1+ϑ ∇x (E, B ) H5 DN1 , 1 ,λ0 (t ).

As to B, letting |α| + |β| ≤ N1 with |β | ≥ 1, applying (3.13) in Lemma 3.7, we obtain

 α  (t )DN1 , 1 ,λ0 (t ). ∂β ( f, f ), w2|α|+|β|− 1 ,λ0 ∂βα {I − P} f  EN1/2 1 −3, 1 −1,λ0

Next, for the term E · p˜ f in S, we only consider the estimates by

 α  ∂β (E · p˜{I − P} f ), w2|α|+|β|− 1 ,λ0 ∂βα {I − P} f  

α ,β = Cα ,β ∂ α1 E · ∂β1 p˜∂βα−−βα1 {I − P} f, w2|α|+|β|− 1 ,λ0 ∂βα {I − P} f 1 1 1 α1 ≤α ,|β1 |≤1





∂ α1 E L∞

|α1 |≤4

+

|α1 |≥5 or α1 =α



1

λ0

 R3 ×R3

w2|α|+|β|− 1 ,λ0



 |∂βα−−βα11 {I − P} f |2 + ∂βα {I − P} f |2 dxdp

    1   1  

∂ α1 E · sup  p 2 w|α|+|β|− 1 ,λ0 ∂βα−−βα11 {I − P} f  2  p− 2 w|α|+|β|− 1 ,λ0 ∂βα {I − P} x

Lp

(1 + t )1+ϑ ∇x (E, B ) H5 DN1 , 1 ,λ0 (t ) + E N1 (t )DN1 , 1 ,λ0 (t ). 1/2

For the term (E + p˜ × B ) · ∇ p f in S, the difference point is that it contains the velocity derivative of order one and the growth of ξ . For brevity, we only estimate the following term

  α ∂β [ p˜ × B · ∇ p {I − P} f ], w2|α|+|β|− 1 ,λ0 ∂βα {I − P} f  

α ,β = Cα ,β (∂β1 p˜ × ∂ α1 B ) · ∇ p ∂βα−−βα1 {I − P} f, w2|α|+|β|− 1 ,λ0 ∂βα {I − P} f . 1 1 1 0<α1 +β1

When |α 1 | ≤ 4, (5.24) is bounded by

(5.24)

876

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

C ∂ α1 B L∞  ∇x 

1

λ0



 R3 ×R3

∂ α1 B

H 1

(1 + t )





1+ϑ



w2|α|+|β|− 1 ,λ0 |∇ p ∂βα−−βα1 {I − P} f |2 + w2|α|+|β|− 1 ,λ0 |∂βα {I − P} f |2 dxdp

R3 ×R3

1

 |w|α−α1 |+|β −β1 |+1− 1 ,λ0 ∇ p ∂βα−−βα11 {I − P} f |2 + |w|α|+|β|− 1 ,λ0 ∂βα {I − P} f |2 dxdp

∇x (E, B ) H5 DN1 , 1 ,λ0 (t ).

When |α 1 | ≥ 5, (5.24) is dominated by

 

      − 12  p w|α|+|β|− 1 ,λ0 ∂βα {I − P} f   1 x L2p 

  1/2  12   E N1 (t ) (t )  p w1+|α−α1 +α |+|β −β1 |− 1 ,λ0 ∇ p ∂βα−−βα11 +α {I − P} f DN1/2 1 , 1 ,λ0

C ∂ α1 B · sup  p 2 w|α|+|β|− 1 ,λ0 |∇ξ ∂βα−−βα1 {I − P} f | 1

|α  |≤2



1/2 E N1

(t )DN1 , 1 ,λ0 (t ).

Thus (5.19) holds true for B. This proves the desired inequality (5.19) and completes the proof of Lemma 5.3  Now we give a remark to explain the choice of our new weight w|α|+|β|− 1 ,λ0 . Remark 5.2. In fact, the weight w|α|+|β|− 1 ,λ0 is designed to treat the following three delicate terms

|α1 |=1

( p˜ × ∂ α1 B ) · ∇ξ ∂βα−α1 {I − P} f,

|β1 |=1 |α|+|β|≤N1

∂β 1

(5.25)

 p  i ∂ α+ei {I − P} f,  p β −β1

(5.26)

∂ α1 E · p˜∂ α−α1 f.

(5.27)

|α1 |≥5

More precisely, to control (5.27), we need the weight depends on |α |, if so, (5.26) can be dominated only when the weight function depends on |β |. And the exponential part of the weight w|α|+|β|− 1 ,λ0 is required to absorb the nonnegative growth of p˜ θ

in (5.25). It is easily to see that the exponential weight ep can be relaxed to the case e p for any 0 < θ ≤ 1.

At this point, we are ready to obtain the closed estimate on the first portion of the time-weighted energy norm X(t) in the following Lemma 5.4. Assume l1 − 1 ≥ N1 ≥ 13. It holds that



sup 0≤s≤t

1+0 2

E N1 (s ) + (1 + s )−



EN1 , 1 ,λ0 (s ) +



t 0

DN1 (s ) ds +

 0

t

( 1 + s )−

1+0 2

DN1 , 1 ,λ0 (s ) ds  Y02 + X 2 (t )

(5.28)

for 0 ≤ t ≤ T. Proof. In fact, the time integration of (5.6) gives

E N1 (t ) +



t

0

DN1 (s ) ds  Y02 + δ0



t 0

(1 + s )−1−ϑ DN1 , 1 ,λ0 (s ) ds +

 0

t

E N1 (s )EN1 −3, 1 −1,λ0 (s ) ds.

(5.29)

Furthermore, from multiplying (5.6) by (1 + t )−0 and then taking the time integration, it follows that

(1 + t )−0 E N1 (t ) +  Y02 + δ0



t 0



t

(1 + s )−0 DN1 (s ) ds +

0



t 0

(1 + s )−1−ϑ −0 DN1 , 1 ,λ0 (s ) ds +

(1 + s )−1−0 E N1 (s ) ds



t

0

(1 + s )−0 E N1 (s )EN1 −3, 1 −1,λ0 (s ) ds.

(5.30)

Combining (5.29) and (5.30) gives

E N1 (t ) +



t

0

 Y02 + δ0

DN1 (s ) ds + 

t 0



t 0

(1 + s )−1−0 E N1 (s ) ds

(1 + s )−1−ϑ DN1 , 1 ,λ0 (s ) ds +

 Y02 + X 2 (t ) + δ0

 0

t

 0

t

E N1 (s )EN1 −3, 1 −1,λ0 (s ) ds

(1 + s )−1−ϑ DN1 , 1 ,λ0 (s ) ds,

where to obtain the second inequality, we have used

(5.31)

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882



877



3

E N1 (s ) + (1 + s ) 2 EN1 −3, 1 −1,λ0 (s ) ≤ X (t ).

sup 0≤s≤t

From (5.12), multiplying it by (1 + t )−(1+0 )/2 and taking the time integration yields

( 1 + t )−

1+0 2

 Y02 +

EN1 , 1 ,λ0 (t ) +



t

|α|=N1 0



t

0

( 1 + s )−

( 1 + s )−

1+0 2

1+0 2



DN1 , 1 ,λ0 (s ) ds +

 α  ˜ 1/2 , w2|α|− 1 ,λ0 ∂ α f . ∂ E · pJ

t 0

( 1 + s )−

3+0 2

EN1 , 1 ,λ0 (s ) ds

(5.32) (5.32)

By the Cauchy–Schwarz inequality, the right-hand second term of (5.32) is bounded up to a generic constant by



t



|α|=N1 0

(1 + s )

−1−0

2 



1  

∂ α E 2 +  p− 2 ∂ α f 

 ds 

0

t

(1 + s )−1−0 E N1 (s ) ds +



t 0

DN1 (s ) ds.

Then, in terms of the above estimates, taking the sum of (5.31) and (5.32) and using the fact that δ 0 > 0 is small enough, we arrive at

E N1 (t ) + (1 + t )−  +

t 0

( 1 + s )−

1+0 2

3+0 2

EN1 , 1 ,λ0 (t ) +



t 0

DN1 (s ) ds +



0

t

( 1 + s )−

1+0 2

DN1 , 1 ,λ0 (s ) ds +



0

t

(1 + s )−1−0 E N1 (s ) ds

EN1 , 1 ,λ0 (s ) ds  Y02 + X 2 (t ).

(5.33)

Therefore, (5.28) follows, and then this completes the proof of Lemma 5.4.  5.4. Decay of electromagnetic fields and macro components In this step, we will use directly the Duhamel’s principle to obtain the time-decay of the electromagnetic field (E, B) and the macro components (a± , b, c) up to order six in terms of the time-decay of the weighted high-order energy function EN1 −3, 1 −1,λ0 (t ) which follows from the boundedness of X(t). The Duhamel principle together with Proposition 4.1 imply Lemma 5.5. Suppose l1 − 1 ≥ N1 ≥ 13. It holds that



sup 0≤s≤t

 5 3 (1 + s ) 2 ∇x (E, B ) 2H5 + (1 + s ) 2 (a± , b, c, E, B ) 2  Y02 + X 2 (t )

(5.34)

for 0 ≤ t ≤ T. Proof. Recall the mild form

U (t ) = A(t )U0 +



t 0

A(t − s )[S(s ), 0, 0] ds,

(5.35)

which denotes the solutions to the Cauchy problem on system (5.1) with initial data U0 = ( f0 , E0 , B0 ), where the nonlinear term S is given by (5.2). The linearized analysis for the homogeneous system in Proposition 4.1 implies

∇x PE,B {A(t )U0 } H5

     low 3  (1 + t ) W f0  + (E0 , B0 ) L1x Z1   7 

 7  high    − 54 + (1 + t ) W 3 ∇x2 ∂ α f0  + ∇x2 ∂ α (E0 , B0 ) , − 54

1≤|α|≤6

where PE, B means the projection along the electro and magnetic components in the solution (f, E, B), W = W ( p) is defined by high

, 3 (4.11), and constants low 3

low > 15/2, 3 high

and also low , 3 3

high 3

are chosen to satisfy

> 5/2,

are sufficiently close to 15/2 and 5/2 respectively. By interpolation of derivatives,

   5  low 

∇x PE,B {A(t )U0 } H5  (1 + t )− 4 W 3 f0  + (E0 , B0 ) L1x Z1  

 high 5   + ( 1 + t )− 4 W 3 ∂ α f0  + ∂ α (E0 , B0 ) . 3≤|α|≤10

Applying this time-decay property to the mild form (5.35) gives

878

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

  5  low  (1 + t − s )− 4 W 3 S(s ) ds Z1 0  t  high 

5  3 α  ( 1 + t − s )− 4 + W ∂ S(s ) ds,

∇x (E, B ) H5  (1 + t )− 4 Y0 + 5



0

t

(5.36)

3≤|α|≤10

where we have used the definition (1.1) for Y0 . Noticing W ( p) = w−1/2 ( p) and By applying Lemma 3.9, it is straightforward to obtain

  

  low   high  W 3 S(t ) + W 3 ∂ α S(t )  EN1 −3, 1 −1,λ0 (t ). Z1

(5.37)

3≤|α|≤10

Here, we have used the choice of N1 − 3 ≥ 10, 1 − 1 ≥ N1 . Recall X(t) norm, and hence 3

EN1 −3, 1 −1,λ0 (s ) ≤ (1 + s )− 2 X (t ),

0 ≤ s ≤ t.

Plugging these estimates into (5.36), the further computations yield



sup 0≤s≤t

 5 (1 + s ) 2 ∇x (E, B ) 2H5  Y02 + X 2 (t ).

(5.38)

Moreover, to obtain the time-decay of (a± , b, c, E, B) , we use the linearized time-decay property

    low  4

Pf {A(t )U0 } + PE,B {A(t )U0 }  (1 + t ) W f0  + (E0 , B0 ) L1x Z1   high 5   5    − 34 

4 + (1 + t ) W ∇x2 f0  + ∇x2 (E0 , B0 ) , − 34

high

where Pf means the projection along the f-component in the solution (f, E, B), and constants low , 4 4 ,

low 4

high

4

> 3/2 and also

(E, B ) H5 in (5.38), one has



sup 0≤s≤t

low , 4

high

4

are chosen to satisfy

are sufficiently close to 3/2. Therefore, in the completely same way for estimating ∇x

 3 (1 + s ) 2 (a± , b, c, E, B ) 2  Y02 + X 2 (t ).

(5.39)

Thus, combining (5.38) and (5.39) gives the desired estimate (5.34). This then completes the proof of Lemma 5.5.  Remark 5.3. Notice that in the proof of (5.37), the inequality N1 − 3 ≥ 10 was used, which then yields to require N1 ≥ 13 in Theorem 1.1. 5.5. The compensating energy estimate with weight In this subsection we obtain the uniform-in-time boundedness of the energy functional EN1 −1, 1 ,λ0 (t ). Notice that this is consistent with the estimation on the linearized system. The main observation in the nonlinear analysis is that those remaining terms in the energy inequalities are time-space integrable. Lemma 5.6. Assume l1 − 1 ≥ N1 ≥ 13. It holds that

sup EN1 −1, 1 ,λ0 (s ) +

0≤s≤t



t 0

DN1 −1, 1 ,λ0 (s ) ds  Y02 + X 2 (t )

(5.40)

for 0 ≤ t ≤ T. Proof. Similarly for obtaining (5.17), one has

d ) E (t ) + κ DN1 −1, 1 ,λ0 (t )  IN(21 −1, (t ) + 1 ,λ0 dt N1 −1, 1 ,λ0

|α|=N1 −1

 α  ˜ 1/2 , w2|α|− 1 ,λ0 ∂ α f , ∂ E · pJ

(5.41)

where IN(2 )−1, ,λ (t ) is defined by (5.18) with N1 replaced by N1 − 1. Following the same way as in the proof of (5.19) one can 1 1 0 obtain that ) IN(2−1, (t )  (1 + t )1−ϑ EN1/2 (t )DN1 −1, 1 ,λ0 (t ) + 1 1 ,λ0 1 −3, 1 −1,λ0

1

λ0

(1 + t )1+ϑ ∇x (E, B ) H5 DN1 −1, 1 ,λ0 (t )

1/2

+ E N1 −1 (t )DN1 −1, 1 ,λ0 (t ).

(5.42)

Further using



sup 0≤s≤t



E N1 −1 (s ) + EN1 −1, 1 ,λ0 (s ) + (1 + s )2(1+ϑ ) ∇x (E, B ) 2H 5 + (1 + s ) 2 EN1 −3, 1 −1,λ0 (s ) ≤ X (t ) ≤ δ02 , 3

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

879

it follows that ) IN(2−1, (t )  δ0 DN1 −1, 1 ,λ0 (t ) + (1 + t )− 4 X (t )DN1/2 (t ). 1 1 ,λ0 1 −1, 1 ,λ0 3

(5.43)

From the Cauchy–Schwarz inequality, the right-hand second term of (5.41) is estimated by

|α|=N1 −1

 α  ˜ 1/2 , w2|α|− 1 ,λ0 ∂ α f  ∂ E · pJ

   2 1 1   η p− 2 ∂ α f  + ∂ α E 2 η |α|=N −1

1



1

ηDN1 −1, 1 ,λ0 (t ) + DN1 (t ) η

(5.44)

for any η > 0. Then, by applying again the Cauchy–Schwarz inequality with η to the right-hand second term of (5.43), plugging the resultant estimate together with (5.44) into (5.41), and choosing η > 0 small enough, one has

d 3 E (t ) + κ DN1 −1, 1 ,λ0 (t )  DN1 (t ) + (1 + t )− 2 X 2 (t ). dt N1 −1, 1 ,λ0

(5.45)

Recall that from (5.33),



t 0

DN1 (s ) ds  Y02 + X 2 (t ).

Therefore, (5.40) follows by the time integration of (5.45). This completes the proof of Lemma 5.6.  5.6. Decay of the lower order energy To obtain the closed estimate on the energy norm X(t), it remains to obtain the time-decay of the lower-order energy functional EN1 −3, 1 −1,λ0 (t ) and E N1 −2 (t ) through the time-weighted estimate as well as the iterative trick as in [6]. Notice that smoothness-loss and velocity–weight-loss in EN1 −3, 1 −1,λ0 (t ) result from the regularity-loss of the electromagnetic field and the degeneration of collisional kernels for soft potentials. Here we emphasize that although the proof of the following lemma looks similar to that in [6], the full details will be provided since most of subscripts in the energy functional EN, ,λ (t ) take the completely different form, and one has to carefully check the validity of all the estimates. Lemma 5.7. It holds that



sup 0≤s≤t

 3 (1 + s ) 2 [EN1 −3, 1 −1,λ0 (s ) + E N1 −2 (s )]  Y02 + X 2 (t )

(5.46)

for 0 ≤ t ≤ T. Proof. First recall from Lemmas 5.4 and 5.6

E N1 (t ) + EN1 −1, 1 ,λ0 (t ) +



0

t

DN1 (s ) + DN1 −1, 1 ,λ0 (s ) ds  Y02 + X 2 (t ).

(5.47)

To obtain the time-decay of EN1 −3, 1 −1,λ0 (t ) and E N1 −2 (t ), we will make the time-weighted estimate. For brevity of presentation we write (2 ) JN, , λ (t ) = 0

|α|=N

 α  ˜ 1/2 , w2|α|− ,λ0 ∂ α f . ∂ E · pJ

From the proof of Lemmas 5.2 and 5.3, cf. (5.6) and (5.17), one has the Lyapunov inequalities

⎧ d ) ⎪ (t ), ⎨ E N1 −1 (t ) + κ DN1 −1 (t )  IN(11 −1 dt

⎪ (2 ) (2 ) ⎩dE N1 −2, 1 − 12 ,λ0 (t ) + κ DN1 −2, 1 − 12 ,λ0 (t )  IN −2, − 1 ,λ (t ) + JN −2, − 1 ,λ (t ). dt

1

1

2

0

1

1

2

(5.48)

0

Those terms on the right can be estimated as follows. Similar to (5.10), it holds that ) IN(11 −1 (t )  EN1/2−3, 1

Here, noticing that EN1/2−3, 1

λ

1 −1, 0

(t )DN1 −1 (t ) +

1 −1,λ0

δ0 1/2 1/2 D (t ) + E N1 −1 (t )EN1/2 (t )DN1 −1 (t ). 1 −3, 1 −1,λ0 (1 + t )1+ϑ N1 −1, 1 ,λ0

(t ) ≤ X 1/2 (t ) ≤ δ0 is small enough for the first term on the right and applying the Cauchy–

Schwarz inequality to the third term on the right, it then follows from the first equation of (5.48) that

d δ0 E N −1 (t ) + κ DN1 −1 (t )  D (t ) + E N1 −1 (t )EN1 −3, 1 −1,λ0 (t ). dt 1 (1 + t )1+ϑ N1 −1, 1 ,λ0 Moreover, similar to (5.42), it holds that

(5.49)

880

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

) IN(2−2, 1

(t )  EN1/2 (t )DN1 −2, 1 − 12 ,λ0 (t ) + 1 −3, 1 −1,λ0

λ

1 1− 2 , 0

1

λ0

(1 + t )1+ϑ ∇x (E, B ) H5 DN1 −2, 1 − 12 ,λ0 (t )

1/2

+ E N1 −2 (t )DN1 −2, 1 − 1 ,λ0 (t ), 2

which by using X (t ) ≤ δ02 for the first two terms on the right and the Cauchy–Schwarz inequality for the last term, further implies ) IN(2−2, 1

(t )  δ0 DN1 −2, 1 − 12 ,λ0 (t ).

λ

1 1− 2 , 0

(5.50)

Again from the Cauchy–Schwarz inequality with η > 0, ) JN(2−2, 1

λ

1 1− 2 , 0

(t ) 

   2 1   1 η p 2 ∂ α f  + ∂ α E 2 . η |α|=N −2

(5.51)

1

Then, by plugging (5.50) and (5.51) into the second equation of (5.48), taking the sum of the resultant inequality multiplied by a proper small constant κ 1 > 0 and another inequality (5.49), and using smallness of δ 0 > 0 and η > 0, one has







d E N1 −1 (t ) + κ1 EN1 −2, 1 − 1 ,λ0 (t ) + κ DN1 −1 (t ) + κ1 DN1 −2, 1 − 1 ,λ0 (t ) 2 2 dt



δ0 D (t ) + E N1 −1 (t )EN1 −3, 1 −1,λ0 (t ). (1 + t )1+ϑ N1 −1, 1 ,λ0



(5.52)

Further from multiplying it by (1 + t ) 2 + with  > 0 fixed small enough and taking the time integration, it follows 1

  t   1 1 (1 + t ) 2 + E N1 −1 (t ) + EN1 −2, 1 − 12 ,λ0 (t ) + (1 + s ) 2 + DN1 −1 (s ) + DN1 −2, 1 − 12 ,λ0 (s ) ds 

0

t



+

− 12 −ϑ +



t

(1 + s ) DN1 −1, 1 ,λ0 (s ) ds + (1 + s ) 2 + E N1 −1 (s )EN1 −3, 1 −1,λ0 (s ) ds 0 0  t   1 + (1 + s )− 2 + E N1 −1 (s ) + EN1 −2, 1 − 12 ,λ0 (s ) ds. Y02

1

(5.53)

0

Here, since  > 0 is small enough, the second term on the right is bounded by Y02 + X 2 (t ) directly by (5.47), the third term on the right is bounded by

  1 δ02 sup (1 + s ) 2 + E N1 −1 (s ) , 0≤s≤t

due to the fact that



sup

 3 (1 + s ) 2 EN1 −3, 1 − γ +2s ,λ0 (s ) ≤ X (t ) ≤ δ02 , γ

0≤s≤t

and the fourth term on the right is bounded by Y02 + X 2 (t ) by noticing

E N1 −1 (t ) + EN1 −2, 1 − 1 ,λ0 (t )  DN1 (t ) + DN1 −1, 1 ,λ0 (t ) + (a± , b, c, B ) 2 , 2

and further using (5.47) as well as Lemma 5.5. Hence, we arrive from (5.53) at



sup 0≤s≤t

  1 (1 + s ) 2 + E N1 −1 (s ) + EN1 −2, 1 − 12 ,λ0 (s )



+

t 0

  1 (1 + s ) 2 + DN1 −1 (s ) + DN1 −2, 1 − 12 ,λ0 (s ) ds  Y02 + X 2 (t ).

In a similar way to obtain (5.52), starting with the Lyapunov inequalities

⎧ d ) ⎪ ⎨ E N1 −2 (t ) + κ DN1 −2 (t )  IN(11 −2 (t ), dt

⎪ (2 ) (2 ) ⎩dE N1 −3, 1 −1,λ0 (t ) + κ DN1 −3, 1 −1,λ0 (t  IN1 −3, 1 −1,λ0 (t ) + JN1 −3, 1 −1,λ0 (t ), dt

one can prove





d E N1 −2 (t ) + κ2 EN1 −3, 1 −1,λ0 (t ) + κ DN1 −2 (t ) + κ2 DN1 −3, 1 −1,λ0 (t ) dt

δ0 D (t ) + E N1 −2 (t )EN1 −3, 1 −1,λ0 (t ) 1 (1 + t )1+ϑ N1 −2, 1 − 2 ,λ0 for a properly chosen constant κ 2 > 0. 

(5.54)

F. Ma, X. Ma / Applied Mathematics and Computation 265 (2015) 854–882

881

Further multiplying it by (1 + t ) 2 + and taking the time integration gives 3



t

3 3 (1 + t ) 2 + E N1 −2 (t ) + EN1 −3, 1 −1,λ0 (t ) + (1 + s ) 2 + DN1 −2 (s ) + DN1 −3, 1 −1,λ0 (s ) ds 0

 Y02



+

 +

t 0

t 0

δ0 (1 + s )

1 2+

 −ϑ D

λ

N1 −2, 1 − 12 , 0

(s ) ds +



t 0

(1 + s ) 2 + E N1 −2 (s )EN1 −3, 1 −1,λ0 (s ) ds 3



1 (1 + s ) 2 + E N1 −2 (s ) + EN1 −3, 1 −1,λ0 (s ) ds.

(5.55)

Here, notice again that  > 0 is a fixed constant small enough. Then, the second term on the right is bounded by Y02 + X 2 (t ) by (5.54), the third term on the right is bounded by X2 (t) due to



sup 0≤s≤t

 3 (1 + s ) 2 E N1 −2 (s ) + EN1 −3, 1 −1,λ0 (s ) ≤ X (t ),

and as before, the fourth term on the right is bounded by



C (1 + t ) Y02 + X 2 (t )



by noticing

E N1 −2 (t ) + EN1 −3, 1 −1,λ0 (t )  DN1 −1 (t ) + DN1 −2, 1 − 1 ,λ0 (t ) + (a± , b, c, B ) 2 , 2

and further using (5.54) as well as Lemma 5.5. Therefore, the desired inequality (5.46) follows by putting these estimates into (5.55). This then completes the proof of Lemma 5.7.  5.7. Global existence We are now in a position to complete the Proof of Theorem 1.1. Recall X(t)-norm (1.19). From Lemmas 5.4– 5.7, it follows that

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