Optimal large-time decay of the relativistic Landau–Maxwell system

Available online at www.sciencedirect.com

ScienceDirect J. Differential Equations 256 (2014) 832–857 www.elsevier.com/locate/jde

Optimal large-time decay of the relativistic Landau–Maxwell system Shuangqian Liu a,∗ , Huijiang Zhao b a Department of Mathematics, Jinan University, Guangdong, PR China b School of Mathematics and Statistics, Wuhan University, PR China

Received 26 May 2013 Available online 23 October 2013

Abstract The Cauchy problem of the relativistic Landau–Maxwell system in R3 is investigated. For perturbative initial data with suitable regularity and integrability, we obtain the optimal large-time decay rates of the relativistic Landau–Maxwell system. For the proof, a new interactive instant energy functional is introduced to capture the macroscopic dissipation and the very weak electromagnetic dissipation of the linearized system. The iterative method is applied to handle the time-decay rates of the full instant energy functional because of the regularity-loss property of the electromagnetic field. © 2013 Elsevier Inc. All rights reserved. MSC: primary 76X05; secondary 82D10 Keywords: Optimal time-decay rates; Relativistic Landau–Maxwell system; Regularity-loss

Contents 1.

2.

Introduction . . . . . . . . . . . . . . . . . . . 1.1. The Cauchy problem . . . . . . . . . 1.2. Reformulation and norms . . . . . 1.3. Main result . . . . . . . . . . . . . . . 1.4. Previous results and our approach Linearized analysis . . . . . . . . . . . . . . . 2.1. Macro structure . . . . . . . . . . . .

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* Corresponding author.

E-mail addresses: [email protected] (S. Liu), [email protected] (H. Zhao). 0022-0396/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jde.2013.10.004

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2.2. Time decay for the linearized system . . . . . . . . . . Optimal time-decay rates . . . . . . . . . . . . . . . . . . . . . . . 3.1. Macro dissipation . . . . . . . . . . . . . . . . . . . . . . . h (t) . . . . . 3.2. Lyapunov inequalities for EN (t) and EN 3.3. Decay of the lower-order instant energy functional 3.4. Decay of the higher-order instant energy functional Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.

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1. Introduction 1.1. The Cauchy problem We consider the following relativistic Landau–Maxwell system ∂t F+ + p˜ · ∇x F+ + (E + p˜ × B) · ∇p F+ = C(F+ , F+ ) + C(F+ , F− ), ∂t F− + p˜ · ∇x F− − (E + p˜ × B) · ∇p F− = C(F− , F+ ) + C(F− , F− ).

(1.1)

The self-consistent electromagnetic field satisfies the relativistic Maxwell equations  ∂t E − ∇x × B = −

p(F ˜ + − F− ) dp,

R3

 ∇x · E =

∂t B + ∇x × E = 0, (F+ − F− ) dp,

∇x · B = 0.

(1.2)

R3

Here F± = F± (t, x, p)  0 stand for the number densities of ions (+) 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 fields, respectively. The energy of a particle is  the electro and magnetic p 2 given by p0 = p = 1 + |p| , and p˜ = p0 . 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 · E 0 =

(F0,+ − F0,− ) dp,

∇x · B0 = 0.

R3

We define relativistic four vectors as P = (p0 , p) and Q = (q0 , q), and we use  to denote the Lorentz inner product, which is given by

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S. Liu, H. Zhao / J. Differential Equations 256 (2014) 832–857

P  Q = p0 q0 − p · q. The relativistic Landau collision operator C(·,·) in (1.1) takes the form of  E(F, G) = ∇p ·

  Φ(P , Q) ∇p F (p)G(q) − F (p)∇q G(q) dq,

R3

here the relativistic collision kernel is given by the 3 × 3 non-negative matrix Φ(P , Q) =

Ξ (P , Q) Π(P , Q), p0 q 0

and  −3/2 Ξ (P , Q) = (P  Q)2 (P  Q)2 − 1 ,   2 Π(P , Q) = (P  Q) − 1 I3 − (p − q) ⊗ (p − q) + {P  Q − 1}(p ⊗ q + q ⊗ p). It is well known that 3  i=1





3 pj qj pi qi Φ ij (P , Q) − Φ ij (P , Q) − = = 0, p0 q 0 p0 q 0 j =1

and 

Φ ij (P , Q)ωi ωj > 0,

if ω = d

i,j

p q − p0 q 0

∀d ∈ R.

This property leads to the five relativistic collision invariants [1, pi (1  i  3), p0 ], i.e.    1 1 p C(F, G)(p) + p C(G, F )(p) dp = 0. p0 p 0 R3 Notice that all the physical parameters, such as the particle masses, the light speed, and all other involving constants, have been chosen to be unit for simplicity of presentation and also without loss of generality. 1.2. Reformulation and norms The global relativistic Maxwellian (the Jüttner solution) is given by J (p) =

exp(−p) , 4πK2 (1)

where K2 (·) is the modified Bessel function K2 (z) =

 z2 +∞ 2 (t 3 0

− 1)3/2 e−zt dt.

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We set the perturbation in a standard way F± = J + J 1/2 f± . Use [·,·] to denote the column vector in R2 . Set F = [F+ , F− ] and f = [f+ , f− ]. Then the Cauchy problem (1.1), (1.2), (1.3) can be reformulated as ⎧ m0 ⎪ ˜ 1/2 m1 + Lf = E · pf ˜ + Γ (f, f ), ∂t f + p˜ · ∇x f + m0 (E + p˜ × B) · ∇p f − E · pJ ⎪ ⎪ ⎪ 2  ⎪ ⎪ ⎪ ⎪ ⎪ ˜ 1/2 (f+ − f− ) dp, ⎪ ⎨ ∂t E − ∇x × B = − pJ (1.4) R3 ⎪ ⎪ ∂t B + ∇x× E = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ∇x · E = J 1/2 (f+ − f− ) dp, ∇x · B = 0 ⎪ ⎪ ⎪ ⎩ R3

with initial data f± (0, x, p) = f0,± (x, p),

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

satisfying the compatibility condition  ∇x · E0 = J 1/2 (f0,+ − f0,− ) dp,

B(0, x) = B0 (x)

∇x · B0 = 0.

(1.5)

(1.6)

R3

Here, m0 = diag(1, −1), m1 = [1, −1], and the linearized collision term Lf and the nonlinear collision term Γ (f, f ) are respectively defined by Lf = [L+ f, L− f ],

  Γ (f, g) = Γ+ (f, g), Γ− (f, g) ,

with     L± f = −2J −1/2 E J 1/2 f± , J − J −1/2 E J, J 1/2 {f± + f∓ } ,     Γ± (f, g) = J −1/2 E J 1/2 f± , J 1/2 g± + J −1/2 E J 1/2 f± , J 1/2 g∓ .

(1.7) (1.8)

As in [29], the null space of the linearized operator L is given by     N = span [1, 0]J 1/2 , [0, 1]J 1/2 , [pi , pi ]J 1/2 (1  i  3), p, p J 1/2 . Let P be the orthogonal projection from L2p × L2p to N . Given f (t, x, p), one can write P as Pf = [P+ f, P− f ] with   P± f = a± (t, x) + b(t, x) · p + c(t, x)p J 1/2 ,

(1.9)

where the coefficient functions are determined by f in the way (2.3). In what follows, we introduce the weight functions and norms used throughout the paper. First of all, define

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 σ (p) = ij

Φ ij (P , Q)J (q) dq. R3

With this, for any scalar function f (t, x, p), the weighted norms for dissipation are defined by   |f |2σ =

2σ ij ∂pi f ∂pj f + R3



f 2σ

=

 1 pi p j 2 f dp, 2 p02



 1 pi pj 2 f dp dx. 2σ ∂pi f ∂pj f + 2 p02 ij

R3 ×R3

As to a vector function g = [g1 , g2 ], we also use the corresponding L2 -norms   |g|2σ =

σ ij ∂pi g1 ∂pj g1 + σ ij ∂pi g2 ∂pj g2 + R3

 1 pi pj 2 1 pi pj 2 g + g dp, 4 p02 1 4 p02 2



g 2σ

=

|g|σ dx. R3

It has been proved that [29] |f |2σ ∼ |f |22 + |∇p f |22 . 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 obtain the global existence and the optimal time decay of the system (1.4), motivated by [7,10], the temporal energy functionals and the corresponding dissipation rate functionals are defined by EN (t) ∼

   ∂ α (a± , b, c) + |α|N

ENh (t) ∼

 |α|+|β|N

   ∂ α (a± , b, c)2 +



    α ∂ {I − P}f (t) + (E, B)2 N , β H

β

|α|+|β|N

1|α|N

   ∂ α (E, B)2 ,

  α ∂ {I − P}f  +

(1.10)

1|α|N

(1.11) and 

DN (t) =

|α|+|β|N

 α  ∂ {I − P}f (t)2 + β σ

+ a+ − a− where the integer N  4.

2

+ E 2H N−1

   ∇x ∂ α (a± , b, c)2 |α|N−1

+ ∇x B 2H N−2 ,

(1.12)

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The main result of the paper is stated as follows. Theorem 1.1. Assume N  4. Let f0 = [f0,+ , f0,− ] satisfy F± (0, x, p) = J (p) + J 1/2 (p) × f0,± (x, p)  0. There exist EN (t) and ENh (t) such that if EN (0) is sufficiently small, then the Cauchy problem (1.4)–(1.6) admits a unique global solution [f (t, x, p), E(t, x), B(t, x)] satisfying F± (t, x, p) = J (p) + J 1/2 (p)f± (t, x, p)  0 and d EN (t) + κEN (t)  0, dt  2 d h EN (t) + κDN (t)  ∇x (a± , b, c) + ∇x × B 2 , dt for all time t  0. Moreover, let Y0 =



 α    ∂ f0  + (E0 , B0 ) β

|α|+|β|N

H N ∩L1

+ f0 Z1

(1.13)

small enough, if N  6, it holds that EN−2 (t)  (1 + t)−3/2 Y02 ,

(1.14)

h EN−5 (t)  (1 + t)−5/2 Y02 .

(1.15)

and if N  9, it holds that

Remark 1.1. As in [29,33], N  4 is required to get the global existence of the system (1.4)–(1.6), however, to obtain the decay rates in (1.14) and (1.15), N must be relaxed to N  6 and N  9 respectively. Since our main concern in this paper is to show the optimal large-time decay rates of the relativistic Maxwell–Landau system near relativistic Maxwellians through the perturbation method, the problem on determining the critical value of N is beyond the scope of this manuscript. Remark 1.2. The decay rates in (1.14) and (1.15) are optimal in the sense that they are coincide with these rates of the solutions of the linearized relativistic Landau–Maxwell system (2.1). 1.4. Previous results and our approach In the perturbative context, there have been extensive investigations on the global classical solvability and the decay rate of convergence for the kinetic equations in the last decade, see the first result by Ukai [32] and also [1–3,5,6,8,9,11,14,15,21–24,26,28,30] and references therein. In plasma physics, the relativistic Landau–Maxwell system is the most fundamental and complete model for describing the dynamics of a dilute collisional plasma in which particles interact through Coulombic collisions through their self-consistent electromagnetic field. There is a brief history of the mathematical theory of the Landau–Maxwell (relativistic or non-relativistic) system. In 1994, Zhan [35] constructed the local existence of weak solution of the non-relativistic

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S. Liu, H. Zhao / J. Differential Equations 256 (2014) 832–857

Landau–Maxwell equation with one specie of particles. In 2000, Lemou [20] analyzed the spectral properties and the dispersion relations of the linearized relativistic Landau equation with no electromagnetic field. Lemou’s work is a first natural step when one looks for solutions near relativistic equilibrium and their large-time behaviors [12,13,33,34]. We shall point out that Strain and Guo [29] obtained the first global in time classical solutions of the relativistic Landau– Maxwell system on torus. Although the Vlasov–Maxwell–Boltzmann system is not an accurate model to describe the completely ionized particles, it is a fundamental model for dynamics of weak ionized particles, interested readers may refer to the textbook [19, Chapter 6]. In what follows, we mention some mathematical results related to the 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 [16]. Then, the rate of convergence to Maxwellians with any polynomial speed in large time was shown by Guo and Strain [17,27] for the Vlasov–Maxwell–Boltzmann system on the periodic box in both the relativistic and classical situations. For the Cauchy problem in the whole space, the global in time classical solutions were constructed by Strain [25]. And recently, the large-time behavior of classical solutions to the Vlasov–Maxwell–Boltzmann system for both cutoff and non-cutoff potentials were studied by Duan and Strain [10] and Duan, Liu, Yang and Zhao [7] respectively. In this paper, we intend to study the optimal large-time decay rates of the relativistic Landau– Maxwell system in the whole space. Our proof is based on the Fourier analysis of the linearized relativistic Landau–Maxwell system and a refined nonlinear energy method. For the linearized system, compared with [10,31], a more complicated interactive instant energy functional E0int,(1) (t, k) defined by (2.17) is introduced to capture the zeroth order macro dissipation. We particularly apply two quantities H and H to derive the dissipation of (a+ − a− ), which cannot be encountered in the similar issues in [10,31]. To obtain the large-time decay rates of the nonlinear system, an iterative argument is employed to treat the regularity-loss phenomenon [18] in the energy dissipation rate due to the degenerately dissipative property from the relativistic Maxwell system. The rest of the paper is arranged as follows. In Section 2, we carry out the analysis of the linearized relativistic Landau–Maxwell system and get the optimal time-decay rates of the solutions. In Section 3, we shall prove series of lemmas to obtain the decay rates of the instant energy functionals so as to conclude the proof of Theorem 1.1. 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 | · |2 to denote the L2 norm in R3p and · to denote the L2 norm in R3p × R3x or R3x , and use ·,· to denote the inner product over L2x,p or L2p . The mixed velocity-space Lebesgue space Z1 = L2p (L1x ) = L2 (R3p ; L1 (R3x )) is used. β

α

β

β

β

For multi-indices α = (α1 , α2 , α3 ) and β = (β1 , β2 , β3 ), ∂βα = ∂xα ∂p = ∂xα11 ∂xα22 ∂x33 ∂p11 ∂p22 ∂p33 . The length of α is |α| = α1 + α2 + α3 and similar for |β|. 2. 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, p)]:

S. Liu, H. Zhao / J. Differential Equations 256 (2014) 832–857

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⎧ ˜ 1/2 m1 + Lf = S, ∂t f + p˜ · ∇x f − E · pJ ⎪ ⎪ ⎪ ⎪ ˜ 1/2 , f+ − f− , ⎨ ∂t E − ∇x × B = − pJ ∂t B + ∇x× E = 0,  ⎪ ⎪ 1/2 ⎪ ∇ , ∇x · B = 0, · E = J , f − f ⎪ x + − ⎩ [f, E, B]|t=0 = [f0 , E0 , B0 ],

(2.1)

where initial data [f0 , E0 , B0 ] satisfies the compatibility condition  ∇x · E0 = J 1/2 (f0,+ − f0,− ) dp, ∇x · B0 = 0,

(2.2)

R3

and the source term S is assumed to satisfy  J 1/2 (S+ − S− ) dp = 0. R3

To consider the solution to the Cauchy problem (2.1), for simplicity, we denote U = [f, E, B], U0 = [f0 , E0 , B0 ] so that one can formally write t

  A(t − s) S(s), 0, 0 ds,

U (t) = A(t)U0 + 0

where A(t) is the linear solution operator for the Cauchy problem on the linearized homogeneous system corresponding to (2.1) in the case when S = 0. 2.1. Macro structure Before continuing our investigation of the  system (2.1), we first note that for the relativistic Maxwellian J , we have the normalization R3 J (p) dp = 1, then we introduce the notation for some integrals as follows 



C0 =

pJ dp, R3

C 00 =

 p2 J dp,

C 11 =

R3



1122 = C00 R3

p12 p22 J p2

dp,

 1111 C00 = R3

 C011 =

p12 J dp, R3

p14 J p2

dp,

 11 C00 = R3

R3

p12 J p2

p12 J dp, p dp.

Recalling (1.9), we see that     a± (t, x) = J 1/2 , f± = J 1/2 , P± f − C 0 c, bi (t, x) = c(t, x) =

1 1/2 , f + 2 pi J C 11

1 1/2 , f + 2 pJ

+ f− 

=

pi J 1/2 , P± f  , C 11

+ f−  − 12 C 0 J 1/2 , f+ + f−  . C 00 − (C 0 )2

(2.3)

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S. Liu, H. Zhao / J. Differential Equations 256 (2014) 832–857

Taking velocity integrations of the first equation of (2.1) with respect to the velocity moments J 1/2 ,

pi J 1/2 ,

i = 1, 2, 3,

pJ 1/2 ,

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

(2.4)

       ∂t C 11 bi + pi J 1/2 , {I± − P± }f + ∂i C011 a± + C 11 c ∓ C011 Ei + ∇x · pp ˜ i J 1/2 , {I± − P± }f   = pi J 1/2 , S± −L± f , (2.5)      ∂t C 00 c + C 0 a± + pJ 1/2 , {I± − P± }f + C 11 ∇x · b + ∇x · pJ 1/2 , {I± − P± }f   = pJ 1/2 , S± −L± f . (2.6) Define the high-order moment functions Θ(f± ) = (Θij (f± ))3×3 and Λ(f± ) = (Λ1 (f± ), Λ2 (f± ), Λ3 (f± )) by  Θij (f± ) =

 p i pj − A1 J 1/2 , f± , p

 Λi (f± ) =

 1 − A2 pi J 1/2 , f± , p

(2.7)

where A1 and A2 satisfy C 11 (C011 − A1 ) 1122 = C00 − A1 C011 C0 and A2 =

C011 C 11

respectively. Further taking momentum integrations of the first equation of (2.1) with respect to the above high-order moments one has     C 11 − A1  {I± − P± }f, pJ 1/2 ∂t Θii {I± − P± }f − 0 0 C

   1111 C 00 (C011 − A1 ) 1122 + C 11 − A1 C 0 − ∂i bi − C00 c + C00 0 C = Θii (r± + S± ) +

 C011 − A1  r± + S± , pJ 1/2 , 0 C

  1122 ∂t Θij {I± − P± }f + C00 (∂j bi + ∂i bj )   p 1/2 = Θij (r± + S± ) + A1 ∇x · {I± − P± }f, J , p

(2.8)

i = j,

(2.9)

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841





(C011 )2 (C011 )2 11 11 ∂t Λi {I± − P± }f + C00 − 11 ∂i a± ∓ C00 − 11 Ei C C 



= Λi (r± + S± ),

(2.10)

where r = [r+ , r− ],

r± = −p˜ · ∇x {I± − P± }f − L± f.

Notice that we have used (2.6) and (2.4) to derive (2.8) and (2.9) respectively. Taking the mean value of every two equations with ± sign for (2.4), (2.5), (2.6) and noticing S = 0,





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

3 11   1 a + + a− C ⎪ 11 11 ⎪ ⎪ ∂j Θij {I − P}f · [1, 1] = 0, ∂t bi + C0 ∂i + C ∂i c + ⎪ ⎪ 2 2 2 ⎪ ⎪ j =1 ⎪ ⎪ ⎪



⎪ 3 00 11 ⎪   C 1 ⎪ 11 ⎪ C0 − C ⎪ ∂i Λi {I − P}f · [1, 1] = 0, ∂t c + C0 − 0 ∇x · b + ⎩ 2 C0 C

(2.11)

i=1

for 1  i  3. Write   1/2 H = H (t, x) = pJ ˜ , f+ − f− , and   H = H(t, x) = pJ 1/2 , f+ − f− . Then,     1/2 H = [p, ˜ −p]J ˜ 1/2 , {I − P}f = pJ ˜ , {I − P}f · m1 ,     H = [p, −p]J 1/2 , {I − P}f = pJ 1/2 , {I − P}f · m1 . Taking difference of two equations with ± sign for (2.4) and (2.5) and also noticing S = 0, ∂t (a+ − a− ) + ∇x · H = 0,   ∂t H + C011 ∇x (a+ − a− ) − 2C011 E = [p, −p]J 1/2 , r · [1, −1] .

(2.12) (2.13)

In particular, for the nonlinear system (1.4), the non-homogeneous source S = [S+ (t, x, p), S− (t, x, p)] takes the form of

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1 S± = ± E · pf ˜ ± ∓ (E + p˜ × B) · ∇p f± + Γ± (f, f ). 2

(2.14)

Then, it is straightforward to compute from integration by parts that 

 J 1/2 , S± = 0.

2.2. Time decay for the linearized system For the linearized homogeneous system, one can prove Theorem 2.1. Let S = 0, and assume [f, E, B] be the solution to the Cauchy problem (2.1), (2.2) of the linearized homogeneous system. Then, for   0 and α  0 with m = |α|,   α   α ∂ f  + ∂ (E, B)        (1 + t)−σm f0 Z1 + (E0 , B0 )L1 + (1 + t)−/2 ∇x ∂ α f0  x    α   + ∇x ∂ (E0 , B0 ) ,

(2.15)

where σm =

3 m + . 4 2

The following remarks are concerned with Theorem 2.1. Remark 2.1. The extra -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 relativistic Maxwell equations but not due to the technique of the approach, see [4] for the analysis of the Green’s function of the damping Euler–Maxwell system. Remark 2.2. Here, if  is not integer, ∇x is regarded as the fractional spatial derivative in terms of the Fourier transform. Now, we will prove Theorem 2.1 by using as in [10] the following lemma. Some usual notations as in [9,10] are 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 ˆ denotes the Fourier transform of g = g(x) with i = −1 ∈ C is the pure imaginary unit. g(k) respect to the variable x. z denotes the real part of the complex number z. Lemma 2.1. (i) For any t  0, it holds that       ˆ B] ˆ 2 + κ (I − P)fˆ2  0. ∂t |fˆ|2 + [E, σ

(2.16)

S. Liu, H. Zhao / J. Differential Equations 256 (2014) 832–857 int,(1)

(ii) There is a time–frequency interactive functional E0 E0int,(1) (t, k) =

843

(t, k) defined by

3   aˆ + + aˆ −   1 ˆ iki  Λi {I − P}f · [1, 1] 2 1 + |k|2 i=1

3  1 ikj bˆi + 1 + |k|2 i,j =1

1    ˆ  Θij {I − P}f · [1, 1] 2

3 1122  C00 1 iki bˆi − 1111 − C 1122 1 + |k|2 C00 00 i,j =1

+

3 1122  3C00 1 iki bˆi 1111 − C 1122 1 + |k|2 C00 00 i=1

+

κ1 1 + |k|2

1    ˆ  Θjj {I − P}f · [1, 1] 2

1    ˆ {I − P} f · [1, 1] Θ  ii 2

3 

(iki cˆ | bˆi ),

(2.17)

i=1

with a properly chosen constant 0 < κ1  1 such that int,(1)

∂t E0

(t, k) +

2   κ|k|2  ˆ 2 + |c|  |a+ + a− |2 + |b| ˆ 2  {I − P}fˆσ . 2 1 + |k|

(2.18)

(iii) There exists a constant 0 < κ2  1 such that ˆ ˆ  ∂t (Hˆ | ik(a+ − a− )) ∂t (Hˆ | E) ∂t (ik × Bˆ | E) − − κ 2 1 + |k|2 1 + |k|2 (1 + |k|2 )2  + κ|a+ − a− |2 +  2  {I − P}fˆσ .

ˆ 2 ˆ 2 κ|k · E| κ ˆ 2 + κ |k × B| + | E| 1 + |k|2 1 + |k|2 (1 + |k|2 )2 (2.19)

(iv) Let [f, E, B] be the solution to the Cauchy problem (2.1) with S = 0. Then there is a time– frequency interactive functional E0int (t, k) such that     int E (t, k)  |fˆ|2 + [E, ˆ B] ˆ 2 , 2 0

(2.20)

and       ˆ B] ˆ 2 + κ0 E int (t, k) + κ {I − P}fˆ2 ∂t |fˆ|22 + [E, 0 σ +

 κ|k|2  ˆ 2 + |c|   |a+ + a− |2 + |b| ˆ 2 + κ|a+ − a− |2 2 1 + |k|

+

2 κ ˆ 2 + κ|k| ˆ 2 | E| |B| 1 + |k|2 (1 + |k|2 )2

 0,

(2.21)

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S. Liu, H. Zhao / J. Differential Equations 256 (2014) 832–857

where κ0 > 0 is a small constant such that     ˆ B] ˆ 2 + κ0 E int (t, k) ∼ |fˆ|22 + [E, ˆ B] ˆ 2 . |fˆ|22 + [E, 0

(2.22)

Proof. In what follows, we only prove assertion (ii), since the other assertions are much similar to that of [10]. From (2.9), it follows that 1122 1122 C00 (bi + ∂ii bi ) = C00



1122 ∂jj bi + 2C00 ∂ii bi

j =i

=−

1 2

j =i

+

  ∂j ∂t Θij {I − P}f · [1, 1]

1 2

j =i

     ∂j Θij r · [1, 1] + A1 ∇x · Λ {I − P}f · [1, 1]

1122 1122 ∂ii bi − C00 + 2C00



∂ij bj

j =i

=−

1 2

j =i

+

  ∂j ∂t Θij {I − P}f · [1, 1]

1 2

j =i

     ∂j Θij r · [1, 1] + A1 ∇x · Λ {I − P}f · [1, 1]

1122 1122 ∂ii bi − C00 ∂i ∇ · b. + 3C00

(2.23)

On the other hand, we get from (2.8) that 1122 C00 ∂i ∇ · b = −

1122 C00 1122 ) − C00

1111 2(C00

− +



  ∂i ∂t Θjj {I − P}f · [1, 1]

j

1122 3C00 C 11 1111 − C 1122 ) 2(C00 00 1122 C00 1111 2(C00

1122 ) − C00



− A1 C 0 −

C 00 (C011 − A1 ) ∂i ∂t c C0

  ∂i Θjj r · [1, 1] ,

(2.24)

j

and 1122 3C00 ∂ii bi = −

1122 3C00 1111 2(C00

− +

∂∂Θ 1122 ) i t ii − C00





{I − P}f · [1, 1]



C 00 (C011 − A1 ) 11 0 C − A C − ∂i ∂t c 1 1111 − C 1122 ) C0 2(C00 00 1122 3C00

1122 3C00 1111 2(C00

∂Θ 1122 ) i ii − C00



 r · [1, 1] .

(2.25)

S. Liu, H. Zhao / J. Differential Equations 256 (2014) 832–857

845

Thus (2.23), (2.24) and (2.25) imply that 1 j =i

2

  ∂j ∂t Θij {I − P}f · [1, 1] − +

=

j =i

−

1111 2(C00

1122 ) − C00

1122  3C00 ∂i ∂t Θii {I − P}f 1111 1122 2(C00 − C00 )

1 2



1122 C00

  ∂i ∂t Θjj {I − P}f · [1, 1]

j

 1122 · [1, 1] + C00 (bi + ∂ii bi )

     ∂j Θij r · [1, 1] + A1 ∇x · Λ {I − P}f · [1, 1] 

1122 C00 1111 2(C00

1122 ) − C00

  ∂i Θjj r · [1, 1] +

j

1122 3C00 1111 2(C00

∂Θ 1122 ) i ii − C00



 r · [1, 1] . (2.26)

In light of (2.11), by the Fourier energy estimates on (2.26) and integration by parts, we obtain 3  1 ∂t ikj bˆi 1 + |k|2 i,j =1

1    ˆ  Θij {I − P}f · [1, 1] 2

3 1122  C00 1 ∂t iki bˆi − 1111 − C 1122 1 + |k|2 C00 00 i,j =1

3 1122  3C00 1 + ∂t iki bˆi 1111 − C 1122 1 + |k|2 C00 00 i=1

η

|k|2 1 + |k|2

1    ˆ  Θjj {I − P}f · [1, 1] 2

1   κ|k|2 ˆ 2  ˆ |b|  Θii {I − P}f · [1, 1] + 2 1 + |k|2

 2  2 (aˆ + + aˆ − , c) ˆ  + {I − P}fˆ , σ

(2.27)

here and in the sequel, η is a positive and suitably small constant. Similarly, by (2.10) and (2.11), we have 3 2      1  Λi {I − P}fˆ · [1, 1] + κ|k| |aˆ + + aˆ − |2 ik ∂ ( a ˆ + a ˆ ) t i + − 1 + |k|2 1 + |k|2 i=1

η

2 |k|2 ˆ 2  |b| + {I − P}fˆσ , 2 1 + |k|

(2.28)

and 3  1 κ|k|2 ∂t (iki cˆ | bˆi ) + |c| ˆ2 2 1 + |k| 1 + |k|2 i=1



 2 |k|2 ˆ 2 |k|2 | b| + |aˆ + + aˆ − |2 + {I − P}fˆσ . 2 2 1 + |k| 1 + |k|

(2.29)

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S. Liu, H. Zhao / J. Differential Equations 256 (2014) 832–857

Therefore, (2.18) follows from (2.27), (2.28) and (2.29). This completes the proof of Lemma 2.1. 2 Proof of Theorem 2.1. With assertion (iv) of Lemma 2.1 in hand, one can obtain (2.15) by performing the similar calculations as that of Theorem 3.1 in [5], we omit the details for brevity. This completes the proof of Theorem 2.1. 2 3. Optimal time-decay rates In this section we are going to prove Theorem 1.1, the main result of the paper. 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 Landau–Maxwell system. 3.1. Macro dissipation Lemma 3.1. Let [f, E, B]√be the solution to the Cauchy problem (2.1) with the non-homogeneous terms S± satisfying S± , J  = 0. Let N  4 be an integer. Then, there are two time–frequency interaction functionals ENint (t) and ENint,h (t) satisfying      int    E (t)  ∂ α f 2 + ∂ α [E, B]2 , N  int,h  E  N (t) 

|α|N

(3.1)

|α|N

         ∂ α Pf 2 + ∂ α {I − P}f 2 + ∂ α E 2 + |α|N

1|α|N

|α|N

   ∂ α B 2 , 1|α|N

(3.2) such that for any t  0, ∂t ENint (t) + κ

     ∂ α ∇x (a± , b, c)2 + κ (a+ − a− )2 |α|K−1

   ∂ α E 2 + κ

+κ

|α|N−1

   ∂ α {I − P}f 2 +  σ |α|N



 α 2 ∂ B 

1|α|N −1

    ∂ α S, ζ (p) 2 ,

(3.3)

|α|N−1

and 

∂t ENint,h (t) + κ

 α  ∂ ∇x (a± , b, c)2 + κ a+ − a− 2

H1

1|α|N−1

   ∂ α E 2 + κ

+κ

|α|N−1

   ∂ α {I − P}f 2 +  σ |α|N

where ζ (p) is defined as in Lemma A.2.



 α 2 ∂ B 

1|α|N −1

    ∂ α S, ζ (p) 2 ,

|α|N−1

(3.4)

S. Liu, H. Zhao / J. Differential Equations 256 (2014) 832–857

847

Proof. Take α with |α|  N − 1. Denote int,(α)

EN

(t) =

3  



  ∂ α a+ + ∂ α a− , Λi {I − P}∂ α f · [1, 1] 2

∇x

i=1

3  

+

i,j =1

−

+



  1 ∂j ∂ bi , Θij {I − P}∂ α f · [1, 1] 2 α

1122 C00

3  

1111 − C 1122 C00 00

i,j =1

1122 3C00

3  

1111 − C 1122 C00 00

i=1

+ κ1

3 



  1 ∂i ∂ bi , Θjj {I − P}∂ α f · [1, 1] 2 α



  1 ∂i ∂ bi , Θii {I − P}∂ α f · [1, 1] 2 αˆ



 ∂i ∂ α c, ∂ α bi .

i=1

One can use the same argument as for obtaining (2.18) to deduce   2 d int,(α) (t) + κ ∂ α ∇x (a+ + a− ), b, c  EN dt       ∂ α (I − P)f 2 +  ∂ α S, ζ (p) 2 .  |α  ||α|+1

σ

(3.5)

And performing the similar calculations as for obtaining (2.19), one has  d α ∂ H, ∇x ∂ α (a+ − a− ) + κ dt 

 |α  ||α|+1



  α ∂ (a+ − a− )2

|α||α  ||α|+1

 α    ∂ (I − P)f 2 +  ∂ α S, ζ (p) 2 . σ

(3.6)

To include the√zero-order term E in the dissipation, we go back to (2.13). When S = 0 but satisfies S± , J  = 0, it holds that   1 1 p 1/2 ∂t H + ∇x (a+ − a− ) − 2E = 11 J , (r + S) · [1, −1] . C011 C0 p

(3.7)

From applying ∂ α (|α|  N − 1) to (3.7) and then taking the inner product with ∂ α E over R3 × R3 , we have  2  2 2∂ α E  + ∂ α (a+ − a− )  

    α   α p 1/2 α α  ∂t ∂ H, ∂ E +  ∂ (S + r) · [1, −1], J , ∂ E  p

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S. Liu, H. Zhao / J. Differential Equations 256 (2014) 832–857



 2    d α ∂ H, ∂ α E − ∂ α H, ∂t ∂ α E + η∂ α E  dt     α 2  α 2 C     ∂ (I − P)f σ + ∂ S, ζ (p) + , η 

(3.8)

|α ||α|+1

where in the last inequality we have used integrations by parts in the t -variable and the Cauchy– Schwarz inequality with η. Notice that from the second equation of (2.1),     − ∂ α H, ∂t ∂ α E = ∂ α H, ∂ α H − ∂ α ∇x × B     ∂ α (I − P)f 2 + η  σ |α  ||α|+1

    ∂ α ∇x × B 2 .

(3.9)

|α  |N−2

Remark 3.1. Note that when |α| = N − 1, to avoid the N -th order derivative of B, one must distribute one derivative of B to H by integration by parts. And this argument is coincide with the linearized analysis (2.19). Therefore, (3.8) together with (3.9) imply −

2   d α ∂ H, ∂ α E + ∂ α E   dt

 |α  ||α|+1

 α  ∂ (I − P)f 2 + η σ

    ∂ α ∇x × B 2 |α  |N−2

 2  2 + ∂ α ∇x (a+ − a− ) +  ∂ α S, ζ (p)  .

(3.10)

Next, let |α|  N − 2; it follows from the second and third equations of (2.1) that −

2  2  2   d α ∂ ∇x × B, ∂ α E + ∂ α ∇x × B   ∂ α (I − P)f σ + ∂ α ∇x E  . dt

(3.11)

Now, we set ENint,h (t) =



int,(α)

EN

1|α|N−1

− κ3





(t) +



0|α|N−1



 ∂ α H, ∂ α E − κ4

0|α|N−1

 ∂ α H, ∇x ∂ α (a+ − a− ) 



 ∂ α ∇x × B, ∂ α E ,

0|α|N−2

for a suitably chosen constant 0 < κ4  κ3  1. It is straightforward to see that (3.2) holds true. Moreover by taking η in (3.10) small enough and then taking the sum of (3.5), (3.6), (3.10) × κ3 and (3.11) × κ4 , we see that (3.4) is valid. In a similar way, basing on the obtained estimates, ENint (t) can be constructed to satisfy both (3.1) and (3.3). This completes the proof of Lemma 3.1. 2 3.2. Lyapunov inequalities for EN (t) and ENh (t) In this section, our aim is to deduce the uniform-in-time a priori estimates on solutions to the relativistic Landau–Maxwell system (2.1) and (2.14).

S. Liu, H. Zhao / J. Differential Equations 256 (2014) 832–857

849

For that, let (f, E, B) be a smooth solution to (2.1) 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 EN (t)  ε0 , where EN (t) is given in (1.10) and the constant ε0 > 0 is sufficiently small. Lemma 3.2. There exist energy functionals EN (t) and ENh (t) with N  4 such that d EN (t) + κDN (t)  0, dt

(3.12)

 2 d h EN (t) + κDN (t)  ∇x (a± , b, c) + ∇x × B 2 , dt

(3.13)

and

for 0  t  T . Proof. We only prove (3.13), since the proof for (3.12) is similar and easier. For this, we divide our computations into following two steps. Step 1. Uniform spatial energy estimates. Starting from the first equation of (2.1), the energy estimate on ∂ α f with 1  |α|  N gives d dt

   ∂ α [f, E, B]2 + 1|α|N



 

 2 κ ∂ α {I − P}f σ 

1|α|N

 ∂ α S, ∂ α f .

(3.14)

1|α|N

Similarly, by the first equation of (2.1), one has the energy estimate on {I − P}f     d  {I − P}f 2 + E 2 + κ {I − P}f 2 σ dt  α  2  E, ∇x × B + ∇x Pf + ∂ S, {I − P}f  η E 2 +

  C

∇x × B 2 + ∇x Pf 2 + ∂ α S, {I − P}f . η

(3.15)

Applying now Lemmas A.1 and A.2 and Sobolev’s inequality, we see that 

 ∂ α S, {I − P}f +

 

 1/2 ∂ α S, ∂ α f  EN (t)DN (t),

(3.16)

1|α|N

and     ∂ α S, ζ (p) 2  EN (t)DN (t). |α|N−1

Combining (3.4), (3.14), (3.15), (3.16) and (3.17), we arrive at

(3.17)

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S. Liu, H. Zhao / J. Differential Equations 256 (2014) 832–857

 ∂t κ5 ENint,h (t) + 

+κ

      ∂ α [f, E, B]2 + {I − P}f 2 + E 2 1|α|N

 α  ∂ ∇x (a± , b, c)2 + κ a+ − a− 2

H1

1|α|N−1

   ∂ α E 2 + κ

+κ

|α|N−1



   α 2 2 ∂ B  + κ ∂ α {I − P}f σ |α|N

1|α|N−1

1/2  ∇x Pf 2 + ∇x × B 2 + EN (t)DN (t) + EN (t)DN (t),

(3.18)

for 0 < κ5  1. Step 2. Energy estimates with mixed derivatives. For the energy estimate on ∂βα {I − P}f with |α| + |β|  N and |β|  1, one has N d  Cm dt m=1





  α ∂ {I − P}f 2 + κ

  α ∂ {I − P}f 2

β

|β|=m |α|+|β|N

β

|β|1 |α|+|β|N

 DN (t) + EN (t)DN (t) +

N 



Cm

m=1



σ

 ∂βα S, ∂βα {I − P}f ,

(3.19)

|β|=m |α|+|β|N

where DN (t) is defined as DN (t) =

   ∂ α (a± , b, c)2 + 1|α|N

+



   ∂ α E 2 + a+ − a− 2 |α|N−1

H1

    α 2 ∂ B  + ∂ α {I − P}f 2 . σ

1|α|N−1

|α|N

Then, a proper linear combination of (3.18) and (3.19) implies that there is an energy functional ENh (t) satisfying (1.11) such that d h E (t) + κDN (t) dt N 1/2

 ∇x Pf 2 + ∇x × B 2 + EN (t)DN (t) + EN (t)DN (t),

(3.20)

where Lemma A.1 and Sobolev’s inequality are used to obtain the bound 

 1/2 ∂βα S, ∂βα {I − P}f  EN (t)DN (t).

(3.13) follows from (3.20) since EN (t) is small enough uniformly in 0  t  T . This completes the proof of Lemma 3.2. 2

S. Liu, H. Zhao / J. Differential Equations 256 (2014) 832–857

851

3.3. Decay of the lower-order instant energy functional In this subsection, our purpose is to obtain the time decay of the lower-order energy functional EN−2 (t) through the Duhamel principle as well as the iterative trick as in [5]. Notice that smoothness-loss in EN−2 (t) results from the regularity-loss of the electromagnetic field. More precisely, letting X(t) = sup (1 + s)3/2 EN−2 (s), 0st

we intend to prove that X(t)  Y02 .

(3.21)

We now consider the solution to the Cauchy problem (1.4) where initial data [f0 , E0 , B0 ] satisfies the compatibility condition (1.5). The Duhamel principle together with Theorem 2.1 implies Lemma 3.3. Suppose N  6. It holds that 2 2   5 3 sup (1 + s) 2 ∇x (E, B)H N−6 + (1 + s) 2 (a± , b, c, E, B)  Y02 + X 2 (t). (3.22)

0st

Proof. Recall the mild form t U (t) = A(t)U0 +

  A(t − s) S(s), 0, 0 ds,

(3.23)

0

which denotes the solutions to the Cauchy problem on system (2.1) with initial data U0 = (f0 , E0 , B0 ), where the nonlinear term S is given by (2.14). The linearized analysis for the homogeneous system in Theorem 2.1 implies    ∇x PE,B A(t)U0  − 54

 (1 + t)

H N−6

  

f0 Z1 + (E0 , B0 )L1



− 54

+ (1 + t)



x

  52 α   52 α ∇x ∂ f0  + ∇x ∂ (E0 , B0 ) ,

(3.24)

1|α|N−5

where PE,B means the projection along the electro and magnetic components in the solution (f, E, B). By interpolation of derivatives,    ∇x PE,B A(t)U0 

H N−6

   5  (1 + t)− 4 f0 Z1 + (E0 , B0 )L1 x      5 ∂ α f0  + ∂ α (E0 , B0 ) . + (1 + t)− 4 3|α|N−2

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S. Liu, H. Zhao / J. Differential Equations 256 (2014) 832–857

Applying this time-decay property to the mild form (3.23) gives   ∇x (E, B)

H N−6

− 54

 (1 + t)

t Y0 +

 5 (1 + t − s)− 4 S(s)Z ds 1

0

t +



5

(1 + t − s)− 4

 α  ∂ S(s) ds,

(3.25)

3|α|N−2

0

where we have used the definition (1.13) for Y0 . By applying Lemma A.3, it is straightforward to obtain   S(t)

Z1

+



 α  ∂ S(t)  EN−2 (t).

(3.26)

3|α|N−2

Here, we have used the choice of N − 2  4. Recall X(t) norm, and hence 3

EN−2 (s)  (1 + s)− 2 X(t),

0  s  t.

Plugging these estimates into (3.25), the further computations yield 2   5 sup (1 + s) 2 ∇x (E, B)H N−6  Y02 + X 2 (t).

(3.27)

0st

Moreover, to obtain the time decay of (a± , b, c, E, B) , we use the linearized time-decay property           Pf A(t)U0  + PE,B A(t)U0   (1 + t)− 34 f0 Z + (E0 , B0 ) 1 1 L x

− 34

+ (1 + t)

  32   32 ∇x f0  + ∇x (E0 , B0 ) ,

where Pf means the projection along the f -component in the solution (f, E, B). Therefore, in the completely same way for estimating ∇x (E, B) H N−6 in (3.27), one has 2   3 sup (1 + s) 2 (a± , b, c, E, B)  Y02 + X 2 (t).

(3.28)

0st

Thus, combining (3.27) and (3.28) gives the desired estimate (3.22). This then completes the proof of Lemma 3.3. 2 Remark 3.2. The choice of the norm ∇x (E, B) H N−6 is due to the fact that the (N − 5)-th order derivative comes up in (3.24) can guarantee (3.26) holds true.

S. Liu, H. Zhao / J. Differential Equations 256 (2014) 832–857

853

In what follows, we consider the point-wise time-decay estimates on EN−2 in terms of

(a± , b, c, B) . Proof of (3.21). First recall from Lemma 3.2 d EN−1 (t) + κDN−1 (t)  0. dt 1

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

(1 + t)

1 2 +

t EN−1 (t) + κ

1

(1 + s) 2 + DN−1 (s) ds

0

t  Y02

+

1

(1 + s)− 2 + EN−1 (s) ds.

(3.29)

0

Here, since  > 0 is small enough, the second term on the right is bounded by Y02 + X 2 (t) by noticing  2 EN−1 (t)  DN (t) + (a± , b, c, B) . Hence, we arrive from (3.29) and (3.28) at   1 sup (1 + s) 2 + EN−1 (s) + 0st

t

1

(1 + s) 2 + DN−1 (s) ds  Y02 + X 2 (t).

(3.30)

0

Likewise multiplying d EN−2 (t) + κDN−2 (t)  0 dt

(3.31)

3

by (1 + t) 2 + and taking the time integration gives (1 + t)

3 2 +

t EN−2 (t) + κ

3

(1 + s) 2 + DN−2 (s) ds

0

t  Y02 +

1

(1 + s) 2 + EN−2 (s) ds.

(3.32)

0

Here, notice again that  > 0 is a fixed constant and small enough. Then, the second term on the right is dominated by

854

S. Liu, H. Zhao / J. Differential Equations 256 (2014) 832–857

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

(3.33)

by noticing  2 EN−2 (t)  DN−1 (t) + (a± , b, c, B) . Therefore, letting  → 0, (3.30), (3.32) and (3.33) yield that   3 sup (1 + s) 2 EN−2 (s)  Y02 + X 2 (t),

0st

2

which implies that (3.21) is true. In addition, one can see that (1.14) follows from (3.21). 3.4. Decay of the higher-order instant energy functional

h In this subsection, we shall prove the time-decay estimates on EN−5 (t). Since N  9, it follows from Lemma 3.2 and the definitions (1.11) and (1.12) that

 2 d h h (t)  ∇x (a± , b, c) + ∇x × B 2 + EN −5 (t) + κEN−5 dt 



 α  ∂ U (t)2 ,

   ∂ α [E, B]2 |α|=N −5

(3.34)

1|α|N−5

where U (t) = [f, E, B](t). On the other hand, following the proof of (3.27) and applying Lemma 3.3 and (3.21), one can see that 

 α  ∂ U (t)2  (1 + t)−5/2 Y 2 . 0

1|α|N−5

Using (3.34), this implies from the Gronwall inequality that (1.15) holds true for any t  0. Remark 3.3. Notice that in the proof of (3.31) and (3.34), the inequalities N − 2  4 and N − 5  4 are used, which then yields to require N  6 and N  9 respectively in Theorem 1.1. Acknowledgments SQL was supported by grants from the National Natural Science Foundation of China under contracts 11101188 and 11271160. HJZ was supported by a grant from the National Natural Science Foundation of China under contract 10925103. This work was also partially supported by a grant from the National Natural Science Foundation of China under contract 11261160485 and the Fundamental Research Funds for the Central Universities.

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855

Appendix A In this appendix, we collect some known basic estimates needed in the proof of the main result of the paper. We first give the following notations. For scalar functions g1 , g2 and h, we define   T (g1 , g2 ) = J −1/2 C J 1/2 g1 , J 1/2 g2 , and      L h = − T h, J 1/2 + T J 1/2 , h . With the above notations, recalling (1.7) and (1.8), it is straightforward to see Γ± (f, g) = T (f± , g± ) + T (f± , g∓ ), and     L± f = −2T f± , J 1/2 − T J 1/2 , f+ + f− . We now list the following propositions which have been proved in [29, p. 291, Lemma 7] and [29, p. 293, Theorem 4]. Proposition A.1. Assume |β|  1. One has ∂β L g, ∂β g  |∂β g|2σ − C



|∂β1 g|2σ − C|g|22 .

β1 <β

Proposition A.2. It holds that  2 Lg, g  {I − P}g σ , where g is a vector function in R2 . Moreover for |β|  1, it holds that 

∂β Lg, ∂β g 

|β|N



|∂β g|2σ − C|g|2σ .

|β|N

Proposition A.3. For any β  0 and α  0, one has  α   ∂ T (g1 , g2 ), ∂ α g3   β



β

β1 +β2 β α1 +α2 =α

 α   α1  α2      ∂ g3  ∂ g1 ∂ g2  + ∂ α1 g1  ∂ α2 g2  . β β1 β2 β1 σ σ σ β2

With Proposition A.3 in hand, as in [25, p. 557, Lemma 7] and [7, p. 176, Lemma 2.4], one can prove

856

S. Liu, H. Zhao / J. Differential Equations 256 (2014) 832–857

Lemma A.1. Assume N  4 and |α| + |β|  4, then one has  α   ∂ Γ± (f, f ), ∂ α {I± − P± }f   E 1/2 (t)DN (t). β

β

N

Moreover, if |α| > 0, it follows that  α   ∂ Γ± (f, f ), ∂ α f±   E 1/2 (t)DN (t). N

Lemma A.2. Let ζ (p) be a smooth function that decays in p exponentially, and let |α|  N , N  4. Writing ∂ α Γ (f, f ) =



  Γ ∂ α1 f, ∂ α2 f ,

α1 +α2 =α

one has        Γ ∂ α1 f, ∂ α2 f ζ (p) dp   E 1/2 (t)D1/2 (t). N N   Lemma A.3. For N  4, it holds that      ∂ α Γ (f, f ) + Γ (f, f ) |α|N

Z1





 α 2 ∂ f  . β

|α|+|β|N

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