Optimal convergence rates of Landau equation with external forcing in the whole space

Acta Mathematica Scientia 2009,29B(4):1035–1062 http://actams.wipm.ac.cn

OPTIMAL CONVERGENCE RATES OF LANDAU EQUATION WITH EXTERNAL FORCING IN THE WHOLE SPACE∗ Dedicated to Professor Wu Wenjun on the occasion of his 90th birthday

)

Yang Tong (

Department of Mathematics, City University of Hong Kong, Hong Kong, China Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China E-mail: [email protected]


)

Yu Hongjun (

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Abstract In this paper, we combine the method of constructing the compensating function introduced by Kawashima and the standard energy method for the study on the Landau equation with external forcing. Both the global existence of solutions near the time asymptotic states which are local Maxwellians and the optimal convergence rates are obtained. The method used here has its own advantage for this kind of studies because it does not involve the spectrum analysis of the corresponding linearized operator. Key words compensating function; energy method; Landau equation; optimal convergence rates 2000 MR Subject Classification

1

35Q99; 35B40

Introduction

In this paper, we are concerned with the Landau equation under the influence of external forcing and source. That is, consider ∂t F + v · ∇x F − (∇x Φ + E) · ∇v F = Q(F, F ) + S

(1.1)

with initial condition F (0, x, v) = F0 (x, v). Here, F (t, x, v) is the distribution function of particles at time t ≥ 0, located at x = (x1 , x2 , x3 ) ∈ R3 with velocity v = (v1 , v2 , v3 ) ∈ R3 . The external force takes the form of −(∇x Φ+E) with given functions Φ = Φ(x) and E = E(t, x). In the following discussion, E(t, x) is assumed to decay in time so that the time asymptotic ∗ Received

January 30, 2009. The research of the first author was supported by Strategic Research Grant of City University of Hong Kong, 7002129, and the Changjiang Scholar Program of Chinese Educational Ministry in Shanghai Jiao Tong University. The research of the second author was supported partially by NSFC (10601018) and partially by FANEDD

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state is governed only by the potential Φ(x). Moreover, the source term S = S(t, x, v) is also assumed to be a given function. Q is the usual bilinear collision operator defined by   Q(F, G)(v) = ∇v · φ(v − v  )[F (v  )∇v G(v) − G(v)∇v F (v  )]dv  R3

=

3  i,j=1

 ∂i

R3

φij (v − v  )[F (v  )∂j G(v) − G(v)∂j F (v  )]dv  ,

(1.2)

where ∂i = ∂vi . The non-negative matrix φ is given by  vi vj  2+γ φij (v) = δij − |v| . |v|2 Here, γ is a parameter leading to the standard classification of the hard potential (γ > 0), Maxwellian molecule (γ = 0) or soft potential (γ < 0), cf. [1]. In particular, γ = −3 corresponds to the Coulomb interaction in plasma physics. The following study is restricted to the case γ ≥ −2. Throughout this paper, we consider solutions which are perturbations near a local Maxwellian. Without loss of generality, define the perturbation f (t, x, v) by F = M +M 1/2 f, where the local  2 Maxwellian is M (v) = exp − Φ(x) − |v|2 . Then equation (1.1) for the perturbation f (t, x, v) is 1 ∂t f + v · ∇x f − (∇x Φ + E) · ∇v f + v · Ef + e−Φ Lf = e−Φ/2 Γ(f, f ) + S (1.3) 2 with f (0, x, v) = f0 (x, v). Denote μ(v) = exp(−|v|2 /2), then M = μ(v)e−Φ . The linearized collision operator L is defined by Lf ≡ −μ−1/2 [Q(μ, μ1/2 f ) + Q(μ1/2 f, μ)] = −Af − Kf, where Af = μ−1/2 Q(μ, μ1/2 f ), Kf = μ−1/2 Q(μ1/2 f, μ), and the bilinear collision operator Γ(f, g) and the source term S are given by Γ(f, g) = μ−1/2 Q(μ1/2 f, μ1/2 g), S = eΦ/2 μ−1/2 S − e−Φ/2 μ1/2 v · E.

(1.4)

By H-theorem, L is non-negative and self-adjoint on L2 (R3v ) with domain D(L) = {f ∈ L2 (R3v ) Lf ∈ L2 (R3v )}. Furthermore, set ψ1 (v) = 1, ψj+1 (v) = vj , j = 1, 2, 3 and ψ5 (v) = |v|2 . Then the null space of L is the five dimensional space (j = 1, 2, 3) √ √ √ N = span{ψ1 μ, ψj+1 μ, ψ5 μ}.

(1.5)

For any fixed (t, x) and any function f (t, x, v), we define P as the projection in L2 (R3v ) to the null space N . Then decompose f (t, x, v) uniquely by f (t, x, v) = Pf + (I − P)f.

(1.6)

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Here, Pf and (I − P)f are called the macroscopic and microscopic components of the function f , respectively. In this paper, the following notations are used. Let α and β be α = [α1 , α2 , α3 ] and β = [β1 , β2 , β3 ], respectively. Denote ∂βα ≡ ∂xα11 ∂xα22 ∂xα33 ∂vβ11 ∂vβ22 ∂vβ33 . If each component of β is not greater than corresponding one of β, we use the standard notation ¯ β ≤ β. And β < β means that β ≤ β and |β| < |β|. Cββ is the usual binomial coefficient. For the study of the optimal time decay rates, the space Zq = L2 (R3v ; Lq (R3x )) is used with its norm defined by

   q2  12 f Zq = |f (x, v)|q dx dv . R3

R3

We will use ·, · to denote the standard L2 inner product in R3v , and (·, ·) for the one in R3x ×R3v . | · |2 denotes the L2 norm in R3v and · to denote the L2 norms in R3x × R3v . For the weight function in v, we use w = w(v) = (1 + |v|)γ+2 . From now on, C denotes a generic positive constant which may vary from line to line. We now come back to the Landau equation. Note that the collision frequency is given by  σ ij (v) = φij (v − v  )μ(v  )dv  . (1.7) R3

Accordingly, the following weighted L2 norms will be used:   {σ ij ∂i g∂j g + σ ij vi vj g 2 }dv, |g|2σ = 1≤i,j≤3

g 2σ =

R3

  1≤i,j≤3

R3 ×R3

{σ ij ∂i g∂j g + σ ij vi vj g 2 }dvdx.

For the energy estimates, the instant energy functional for a solution f (t, x, v), denoted by El (t) has the equivalent relation with some l ≥ 0 representing the index in the weight in v,  El (t) ∼ El (t) = wl ∂βα f (t) 2 . (1.8) |α|+|β|≤N

In addition, the corresponding dissipation functional denoted by Dl (t) satisfies   l (t) = ∂ α Pf (t) 2 + wl ∂βα (I − P)f (t) 2σ . Dl (t) ∼ D 1≤|α|≤N

(1.9)

|α|+|β|≤N

In the following discussion, we will choose the Sobolev space index N ≥ 8. Notice that from [1, 9], there exists C > 0 such that C|w1/2 g|2 ≤ |g|σ .

(1.10)

Notice that El (t) and Dl (t) do not contain the temporal derivatives and Dl (t) does not contain the zero-th order derivative of the macroscopic component in the solution. With the above preparation, the main results of this paper can be stated as follows. √ Theorem 1.1 Let F0 (x, v) = M + M f0 (x, v) ≥ 0 and γ ≥ −1. Suppose that

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(H0 ) The functions E = E(t, x), S = S(t, x, v) and f0 = f0 (x, v) satisfy E ∈ Cb0 (R+ ; H N (R3x )), S ∈ Cb0 (R+ ; H N (R3x × R3v )), f0 ∈ H N (R3x × R3v ); (H1 ) There are constants ε > 0 and l ≥ 0 such that Φ(x), E(t, x) and f0 are bounded in the sense that E˜l (0) ≤ ε, and  ∂ α Φ(x) L3 (R3 ) ≤ ε, (1.11) Φ(x) L2 (R3 ) + 1≤|α|≤N +1



2 ≤ ε. (1 + |x|)∂ α E(t, x) L∞ + |x|E(t, x) L∞ t,x t (Lx )

(1.12)

|α|≤N

Moreover E(t, x) and S(t, x) decay in time like   ∂ α E(t, x) 2 + wl−1/2 ∂βα (μ−1/2 S(t, x)) 2 ≤ ε(1 + t)−1− , |α|≤N

(1.13)

|α|+β|≤N

where  is any positive constant. Then there is a constant ε1 > 0 such that for any ε ≤ ε1 , the Cauchy problem of the Landau equation (1.3) has a unique global classical solution f (t, x, v) √ with F (t, x, v) = M + M f (t, x, v) ≥ 0 for (1.1), which satisfies  t El (t) + Dl (s)ds ≤ Cε. (1.14) 0

If we further assume l ≥ 1,  =

3 2

and

f0 Z1 + ∇x Φ L6/5 (R3 ) ≤ ε,

(H2 )

E(t) L1x + μ−1/2 S Z1 ≤ ε(1 + t)−5/4 ,

(1.15) (1.16)

we have the optimal time decay of the solution to its time asymptotic state f (t) 2 = Pf (t) 2 + (I − P)f (t) 2 ≤ Cε(1 + t)−3/2 ,   ∂ α Pf (t) 2 + wl ∂βα (I − P)f (t) 2 ≤ Cε(1 + t)−5/2 . 1≤|α|≤N

(1.17) (1.18)

|α|+|β|≤N

Note that if E(t, x) = 0 and S(t, x, v) = 0, equation (1.3) becomes ∂t f + v · ∇x f − ∇x Φ · ∇v f + e−Φ Lf = e−Φ/2 Γ(f, f )

(1.19)

with f (0, x, v) = f0 (x, v). Corresponding to Theorem 1.1, we have the following theorem about the existence and optimal convergence rate which holds for larger range of γ. √ Theorem 1.2 Let F0 (x, v) = M + M f0 (x, v) ≥ 0 and γ ≥ −2. If we assume (1.11) and E˜l (0) ≤ ε for some small constant ε > 0, then the Cauchy problem of the Landau equation √ (1.19) has a unique global classical solution f (t, x, v) with F (t, x, v) = M + M f (t, x, v) ≥ 0, which satisfies  t

El (t) +

Dl (s)ds ≤ CEl (0).

(1.20)

0

If we further assume (1.15) and l ≥ 1, we have the optimal time decay: f (t) 2 = Pf (t) 2 + (I − P)f (t) 2 ≤ Cε(1 + t)−3/2 ,

(1.21)

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∂ α Pf (t) 2 +

1≤|α|≤N



wl ∂βα (I − P)f (t) 2 ≤ Cε(1 + t)−5/2 .

1039 (1.22)

|α|+|β|≤N

Finally in the introduction, we review some related works to the study in this paper. Recently, there is some progress on the L2 energy methods for the Boltzmann equation. One was initiated by Liu-Yu [16] and developed by Liu-Yang-Yu [15] in terms of the macro-micro decomposition around the local Maxwellian defined by the solution. The other was proposed by Guo [ 9, 10, 11] about the decomposition around a global Maxwellian. On the other hand, instead of using the spectrum analysis initiated by Grad and finished by Ukai and others, Kawashima [13] introduced the compensating function method which is based on the Fourier transform for the Botlzamnn equation. With these methods, the authors in [24] study the nonlinear stability and the optimal time decay of the solutions for the relativistic Boltzmann and Landau equations near a uniform equilibrium by combining the Kawashima’s compensating function method and the macro-micro decomposition of the solution. The main observation is that the energy estimate on the macroscopic component can be obtained by improving the Kawashima’s compensating function method [13]. And the advantage is not only to obtain the global existence of the solution without time derivative, but also to obtain the optimal time decay to the equilibrium. As a further study by using the method introduced in [24], in this paper, we consider the Landau equation with external forcing. As for Landau equation, there have been intensive investigations, cf. [1, 2, 9, 12, 14, 17, 21] and references therein. More precisely, Desvillettes and Villani [2] proved the global existence and uniqueness of classical solutions for spatially homogeneous Landau equation for hard potentials and a large class of initial data. Degond and Lemou [1] studied the spectral properties and the dispersion relation of linearized Landau operator. Guo [9] constructed global classical solutions near a global Maxwellian in a periodic box. Hsiao and Yu extended Guo’s results to the whole space in [12]. It is shown in [14] that (1.19) has a unique global solution and it converges to the steady state with a rate O((1 + t)−1/4 ). We should also mention that recently Duan-Ukai-Yang-Zhao [6] introduced a method by combining the spectrum analysis and energy method for the study on the optimal time decay rate for the Boltzmann equation and systems of fluid dynamics such as Navier-Stokes equations, cf. also [5]. Some of the ideas from this work will be used here, however, we do not need to consider the spectrum of the corresponding linearized equation because the compensating function coming from the Fourier transform will show to be enough. We believe that the method used here can be applied to the study on some more complicated systems in the kinetic theories.

2

Basic Estimates

For later use, we derive some estimates on the the bilinear collision operator Γ(f, f ). Lemma 2.1 For any g1 = g1 (x, v) and g2 = g2 (x, v) with the following norms well defined, we have   Γ(g1 , g2 ) Z1 ≤ C ∂β1 g1 · w∂β2 g2 . (2.1) |β1 |≤2

Proof By (1.2), Γ(g1 , g2 ) can be written as Γ(g1 , g2 ) = μ−1/2 (v)Q(μ1/2 g1 , μ1/2 g2 )

|β2 |≤2

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−1/2

=μ

 (v)∂i

R3

Vol.29 Ser.B

  φij (v − v  )μ1/2 (v)μ1/2 (v  ) g1 (v  )∂j g2 (v) − ∂j g1 (v  )g2 (v) dv 



 v vj  j − g2 (v  )g1 (v)dv  (v)∂i φij (v − v  )μ1/2 (v)μ1/2 (v) 2 2 R3    = μ−1/2 (v)∂i φij (v − v  )μ1/2 (v  )μ1/2 (v) g1 (v  )∂j g2 (v) − ∂j g1 (v  )g2 (v) dv  3 R

  vi  = ∂i − φij (v − v  )μ1/2 (v  ) g1 (v  )∂j g2 (v) − ∂j g1 (v  )g2 (v) dv  2 R3    = ∂i φij (v − v  )μ1/2 (v  ) g1 (v  )∂j g2 (v) − ∂j g1 (v  )g2 (v) dv  R3    1 − φij (v − v  )vi μ1/2 (v  ) g1 (v  )∂j g2 (v) − ∂j g1 (v  )g2 (v) dv  , (2.2) 2 R3 −1/2

+μ

where we have used the fact that 3 

ij

φ (v − v



)(vi

i=1

− vi ) =

3 

φij (v − v  )(vj − vj ) = 0.

j=1

Since |μ1/2 (v  )| + |∂i μ1/2 (v  )| ≤ Cμ1/4 (v  ), the Cauchy-Schwartz inequality implies  φij (v − v  )∂i (μ1/2 (v  )g1 (v  ))dv  R3  ≤ φij (v − v  )[μ1/2 (v  ) + |∂i μ1/2 (v  )|][|g1 (v  )| + |∂i g1 (v  )|]dv  R3 1/2

 ≤C |φij (v − v  )|2 μ1/2 (v  )dv  [|g1 |2 + |∂i g1 |2 ] R3

≤ C(1 + |v|)γ+2 [|g1 |2 + |∂i g1 |2 ].

(2.3)

Thus,    φij (v − v  )μ1/2 (v  ) g1 (v  )∂j g2 (v) − ∂j g1 (v  )g2 (v) dv  L1x ∂i 2 R3   γ+2 ≤C ∂β1 g1 ∂β2 g2 (v) L2x (1 + |v|) |β1 |≤2

≤C



2

|β2 |≤2

∂β1 g1

|β1 |≤2



w∂β2 g2 .

|β2 |≤2

This gives the first part of (2.2). Similar argument leads to the proof of the second part of (2.2) and we skip the detail. This completes the proof of the lemma. Lemma 2.2 For any l ≥ 1, it holds that 

∂ α Γ(f, f ) 2 ≤ C El (t)E l (t),

|α|≤1

where E l (t) =

 1≤|α|≤N

∂ α Pf (t) 2 +



wl ∂βα (I − P)f (t) 2 .

|α|+|β|≤N

l (t) by the norms on the microscopic component. Note that E l (t) is different from D

(2.4)

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Proof By (2.2), we have 

∂ α Γ(f, f ) =

Cαα1 μ−1/2 (v)Q(μ1/2 ∂ α1 f, μ1/2 ∂ α2 f )

α1 +α2 =α



=



Cαα1 ∂i

R3

α1 +α2 =α

−



Cαα1

α1 +α2 =α

1 2

  φij (v − v  )μ1/2 (v  ) ∂ α1 f (v  )∂jα2 f (v) − ∂jα1 f (v  )∂ α2 f (v) dv 



R3

φij (v − v  )vi μ1/2 (v  )

  · ∂ α1 f (v  )∂jα2 f (v) − ∂jα1 f (v  )∂ α2 f (v) dv  .

(2.5)

By using (2.3), we have 

|∂ α Γ(f, f )|2 ≤ C

 

α1 +α2 =α



|∂βα11 f |2

|β1 |≤2

 |(1 + |v|)γ+2 ∂βα22 f |2 .

|β2 |≤2

Thus, 

2  |∂β1 f |2 ∞

∂ α Γ(f, f ) 2 ≤ C

|α|=1

|β1 |≤2

Lx



(1 + |v|)γ+2 ∂βα22 f 2

|α2 |=1,|β2 |≤2

2  +C (1 + |v|)γ+2 |∂β2 f |2 ∞ ≤C



|β2 |≤2



Lx

∂βα f 2 ·

|α|+|β|≤N





∂βα11 f 2

|α1 |=1,|β1 |≤2

(1 + |v|)γ+2 ∂βα f 2

1≤|α|,|α|+|β|≤N

≤ C El (t)E l (t). Similar argument leads to the proof of the case when |α| = 0. This completes the proof of the lemma. We now recall two basic estimates from [9] in the following two lemmas. Lemma 2.3 The linearized operator L has the properties that Lg, h = g, Lh, Lg, g ≥ 0. And Lg = 0 if and only if g = Pg. Furthermore, there exists a δ > 0 such that

Lg, g ≥ δ|(I − P)g|2σ .

(2.6)

Lemma 2.4 For any η > 0 and l ≥ 0, there exist constants Cη > 0 and C > 0 such that

w2l ∂β Lg, ∂β g ≥ |wl ∂β g|2σ − η



|wl ∂β1 g|2σ − Cη |g|2σ ,

(2.7)

β1 ≤β

and | w2l ∂βα Γ(f, g), ∂βα h|

 l α1 l α−α1 1 ≤C |wl ∂βα h|σ , |wl ∂βα11 f |2 |wl ∂βα−α g| + |w ∂ f | |w ∂ g| σ σ 2 β β 2 1 2

(2.8)

α1 ≤α,β1 +β2 ≤β

where |α| + |β| ≤ N . In the following, we prove one more weighted estimate on the nonlinear collision operator.

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Lemma 2.5 Let l ≥ 0 and |α| + |β| ≤ N . Then for any η > 0, there is some constant Cη > 0 such that

 2l α l (t). (2.9) w ∂β Γ(f, f ), ∂βα (I − P)f ≤ η wl ∂βα (I − P)f 2σ + Cη El (t)D Proof By using the decomposition (1.6), we have Γ(f, f ) = Γ(Pf, Pf ) + Γ(Pf, (I − P)f ) + Γ((I − P)f, Pf ) + Γ((I − P)f, (I − P)f ). By (2.8), we obtain

 2l α w ∂β Γ(Pf, Pf ), ∂βα (I − P)f   1 ≤C [|wl ∂βα11 Pf |2 |wl ∂βα−α Pf |σ 2 R3

α1 ≤α,β1 +β2 ≤β

1 +|wl ∂βα11 Pf |σ |wl ∂βα−α Pf |2 ]|wl ∂βα (I − P)f |σ dx. 2

(2.10)

Notice that there exists C > 1 such that |g|2σ ≥ C −1 |w1/2 g|2 ,

|wl ∂β Pf |2σ ≤ C|Pf |2 .

(2.11)

We only consider the first term in (2.10) because the second term can be estimated similarly. If |α1 | + |β1 | ≤ N/2, for any η > 0, we easily get  1 |wl ∂βα11 Pf |2 |wl ∂βα−α Pf |σ |wl ∂βα (I − P)f |σ dx 2 R3   1 wl ∇x ∂ α ∂βα11 Pf 2 wl ∂βα−α Pf 2σ ≤ η wl ∂βα (I − P)f 2σ + η 2 |α |≤1

≤ η wl ∂βα (I − P)f 2σ + η





∇x ∂ α ∂ α1 Pf 2 ∂ α−α1 Pf 2 ,

|α |≤1

which is bounded by the right hand side of (2.9). The discussion on the case when |α1 | + |β1 | ≥ N/2 is similar. By applying Lemma 2.4 again, we have

 2l α w ∂β Γ(Pf, (I − P)f ), ∂βα (I − P)f   1 ≤C [|wl ∂βα11 Pf |2 |wl ∂βα−α (I − P)f |σ 2 α1 ≤α,β1 +β2 ≤β

R3

1 +|wl ∂βα11 (I − P)f |2 |wl ∂βα−α Pf |σ ]|wl ∂βα (I − P)f |σ dx. 2

Again, we only consider the first term because the second term can be estimated similarly. If |α1 | + |β1 | ≤ N/2, by using the Sobolev inequality, we obtain  1 |wl ∂βα11 Pf |2 |wl ∂βα−α (I − P)f |σ |wl ∂βα (I − P)f |σ dx 2 R3   1 wl ∂ α ∂βα11 Pf 2 wl ∂βα−α (I − P)f 2σ ≤ η wl ∂βα (I − P)f 2σ + η 2 |α |≤2

≤ η wl ∂βα (I − P)f 2σ + η



|α |≤2



1 ∂ α ∂ α1 Pf 2 wl ∂βα−α (I − P)f 2σ . 2

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If |α − α1 | + |β2 | ≤ N/2, we also have  1 |wl ∂βα11 Pf |2 |wl ∂βα−α (I − P)f |σ |wl ∂βα (I − P)f |σ dx 2 R3   1 wl ∂ α ∂βα−α (I − P)f 2σ wl ∂βα11 Pf 2 ≤ η wl ∂βα (I − P)f 2σ + η 2 |α |≤2

≤ η wl ∂βα (I − P)f 2σ + η





|α |≤2

1 wl ∂ α ∂βα−α (I − P)f 2σ ∂ α1 Pf 2 . 2

Both are bounded by the right hand side of (2.9). Since the other cases can be estimated similarly, we complete the proof of this lemma.

3

Energy Estimates and Compensating Function

In this section, we will first consider the energy estimate on the microscopic component. For this, we have the following equation from (1.3) [∂t + v · ∇x + e−Φ L](I − P)f 1 = e−Φ/2 Γ(f, f ) + (∇x Φ + E) · ∇v f − v · Ef + S − [∂t + v · ∇x ]Pf. 2

(3.1)

In fact, the estimate on the microscopic component can be obtained by applying the standard energy method because of the dissipation of the linearized collision operator on the microscopic component. Lemma 3.1 Under the assumptions (H0 ) and (H1 ), for any l ≥ 0, we have d

 wl ∂βα (I − P)f 2 + wl ∂βα (I − P)f 2σ dt 1≤|β|,|α|+|β|≤N   l (t). (3.2) ≤ Cε(1 + t)−1− + C ∂ α Pf 2 + C wl ∂ α (I − P)f 2σ + C El (t)D |α|≤N

1≤|α|≤N

0) on (3.1), we obtain Proof By applying ∂βα (β =  β α Cβ 1 ∂β1 v · ∇x ∂β−β (I − P)f + e−Φ ∂βα L(I − P)f [∂t + v · ∇x ]∂βα (I − P)f + 1 +



|β1 |=1

Cαα1 ∂ α−α1 e−Φ ∂βα1 L(I

− P)f = ∂βα (e−Φ/2 Γ(f, f )) + (∇x Φ + E) · ∇v ∂βα f

α1 <α

+



Cαα1 (∂ α−α1 ∇x Φ + ∂ α−α1 E) · ∇v ∂βα1 f −

α1 <α

+∂βα S −



 α1 ≤α,β1 ≤β

∂β−β1 Cββ1 v

· ∇x ∂βα1 Pf − [∂t + v · ∇x ]∂βα Pf.

1 α1 β1 α−α1 C C ∂β v · ∂ α1 E∂β−β f 1 2 α β 1 (3.3)

β1 <β

We take the inner product of (3.3) over R3 × R3 with w2l ∂βα (I − P)f . The first term on the d left hand side is 12 dt wl ∂βα (I − P)f 2 . By H¨ older inequality, the second term on the left hand side of the inner product is bounded by  α η wl ∂βα (I − P)f 2 + Cη wl ∇x ∂β−β (I − P)f 2 . (3.4) 1 |β1 |=1

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Notice that the assumption (1.11) and the Sobolev imbedding theorem imply that  |∂ α Φ(x)| ≤ Cε. sup x∈R3

(3.5)

|α|≤N −1

Then by Lemma 2.4, for any η > 0, the third term on the left hand side in the inner product satisfies (w2l e−Φ ∂βα L(I − P)f, ∂βα (I − P)f )  ≥ wl ∂βα (I − P)f 2σ − η wl ∂βα1 (I − P)f 2σ − Cη ∂ α (I − P)f 2σ . β1 ≤β

In the following, we only consider the case when |α − α1 | = 1 for the last term on the left hand side of (3.3) because the other cases can be estimated similarly. By Lemma 2.4, we obtain 

|(w2l ∂ α−α Φe−Φ ∂β L[(∂ α1 (I − P)f ], ∂βα (I − P)f )|   ≤C |∂ α−α1 Φ| · |wl ∂βα11 (I − P)f |σ |wl ∂βα (I − P)f |σ dx |α−α1 |=1,β1 ≤β

R3

≤ Cε wl ∂βα (I − P)f 2σ + Cε

 α1 <α,β1 ≤β

wl ∂βα11 (I − P)f 2σ .

For the nonlinear collision operator, by (1.11) and (3.5), Lemma 2.5 gives l (t). |(w2l ∂βα (e−Φ/2 Γ(f, f )), ∂βα (I − P)f )| ≤ η wl ∂βα (I − P)f 2σ + Cη El (t)D For the second term on the right hand side of (3.3), recalling (H0 ), (H1 ) and |β| ≥ 1, we apply the decomposition on f as (1.6) to obtain |(w2l (∇x Φ + E) · ∇v ∂βα (I − P)f, ∂βα (I − P)f )| = |((∇x Φ + E) · ∇v w2l ∂βα (I − P)f, ∂βα (I − P)f )| ≤ C( ∇x Φ L∞ + E L∞ ) wl ∂βα (I − P)f 2 ≤ Cε wl ∂βα (I − P)f 2 ; t,x |(w2l (∇x Φ + E) · ∇v ∂βα Pf, ∂βα (I − P)f )| ≤ ( ∇x Φ L3 (R3 ) + |x|E L∞ )( wl ∇v ∇x ∂βα Pf 2 + wl ∂βα (I − P)f 2 ) t,x  ≤ Cε ∂ α Pf 2 + Cε wl ∂βα (I − P)f 2 , 1≤|α|≤N g ≤ C ∇x g and H¨ older inequality. Similarly, when where we have the Hardy inequality |x| |α − α1 | ≤ N/2, the third term on the right hand side of (3.3) satisfies

|(w2l (∂ α−α1 ∇x Φ + ∂ α−α1 E) · ∇v ∂βα1 (I − P)f, ∂βα (I − P)f )| ≤ C( ∂ α−α1 ∇x Φ L∞ + ∂ α−α1 E L∞ )( wl ∇v ∂βα1 (I − P)f 2 + wl ∂βα (I − P)f 2 ) x t,x  ≤ Cε wl ∂βα11 (I − P)f 2 + Cε wl ∂βα (I − P)f 2 ; |α1 |+|β1 |≤N

|(w (∂ α−α1 ∇x Φ + ∂ α−α1 E) · ∇v ∂βα1 Pf, ∂βα (I − P)f )| 2l

≤ C( ∂ α−α1 ∇x Φ L3 (R3 ) + |x|∂ α−α1 E L∞ )( wl ∇x ∇v ∂βα1 Pf 2 + wl ∂βα (I − P)f 2 ) t,x  ≤ Cε ∂ α1 Pf 2 + Cε wl ∂βα (I − P)f 2 . 1≤|α1 |≤N

No.4

Yang & Yu: OPTIMAL CONVERGENCE RATES OF LANDAU EQUATION

1045

Similarly, when |α − α1 | ≥ N/2, this term has the same bound. Thus, we have |(w2l (∂ α−α1 ∇x Φ + ∂ α−α1 E) · ∇v ∂βα1 f, ∂βα (I − P)f )|   ≤ Cε wl ∂βα11 (I − P)f 2 + Cε ∂ α1 Pf 2 + Cε wl ∂βα (I − P)f 2 . |α1 |+|β1 |≤N

1≤|α1 |≤N

For the fourth term on the right hand side of (3.3), by using the decomposition (1.6) again, the condition (H1 ) and the relation |g|σ ≥ C|(1 + |v|)1/2 g|2 , we get 2l α−α1 α (I − P)f, ∂ (I − P)f ) (w ∂β1 v · ∂ α1 E∂β−β β 1 α−α1 ≤ C ∂ α1 E L∞ ( wl (1 + |v|)1/2 ∂β−β (I − P)f 2 + wl (1 + |v|)1/2 ∂βα (I − P)f 2 ) t,x 1 α−α1 ≤ Cε wl ∂β−β (I − P)f 2σ + Cε wl ∂βα (I − P)f 2σ ; 1 2l α−α1 α Pf, ∂ (I − P)f ) (w ∂β1 v · ∂ α1 E∂β−β β 1

≤ C |x|∂ α1 E L∞ ∇x ∂ α−α1 Pf 2 + Cε wl ∂βα (I − P)f 2σ t,x ≤ Cε ∇x ∂ α−α1 Pf 2 + Cε wl ∂βα (I − P)f 2σ , where we have used |β| ≥ 1 and the Hardy inequality. By using (1.4), (3.5) and the condition (H1 ), we have 2l α  α (w ∂β S, ∂β (I − P)f ) ≤ Cη wl−1/2 ∂βα (eΦ/2 μ−1/2 S) 2 + Cη ∂ α (e−Φ/2 E) 2 + 2η wl ∂βα (I − P)f 2σ ≤ 2Cη ε(1 + t)−1− + 2η wl ∂βα (I − P)f 2σ . Since |β| ≥ 1, we have 2l (w ∂β−β1 v · ∇x ∂βα1 Pf, ∂βα (I − P)f ) ≤ C ∇x ∂ α Pf 2 + η wl ∂βα (I − P)f 2σ . And for any η > 0, the last term on the right hand side of (3.3) satisfies |(w2l [∂t + v · ∇x ]∂βα Pf, ∂βα (I − P)f )| ≤ η wl ∂βα (I − P)f 2σ + Cη [ ∂t ∂ α Pf 2 + ∇x ∂ α Pf 2 ]. Hence, based on the above inequality, the last term on the right hand side of (3.3) satisfies (w2l [∂t + v · ∇x ]∂βα Pf, ∂βα (I − P)f ) ≤ η wl ∂βα (I − P)f 2σ + Cη [ε(1 + t)−1− +





∂ α Pf 2 +

1≤|α|≤N

∂ α (I − P)f 2 ]. (3.6)

1≤|α|≤N

On the other hand, applying ∂ α (|α| ≤ N − 1) on (3.1) gives 1 [∂t + v · ∇x ]∂ α Pf = ∂ α (e−Φ/2 Γ(f, f )) − ∂ α ((∇x Φ + E) · ∇v f ) − ∂ α (v · Ef ) + ∂ α S 2  α α1 −Φ α−α1 ∂ e L[∂ (I − P)f ]. (3.7) −[∂t + v · ∇x ]∂ (I − P)f − α1 ≤α

Since

P∂t ∂ α f, ∂ α Γ(f, f ) − [∂t + L]∂ α1 (I − P)f  = 0,

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by multiplying ∂t ∂ α Pf to (3.7) and then integrating over R3 × R3 , (H1 ) implies that 1  − ∂ α (v · Ef ) − ∂ α ((∇x Φ + E) · ∇v f ), ∂t ∂ α Pf 2 ≤ η ∂t ∂ α Pf 2 + Cη ∂ α1 (∇x Φ + E) · ∂ α−α1 f 2 ≤ η ∂t ∂ α Pf 2 + Cη ε ∇x ∂ α−α1 f 2 , where we have the Hardy inequality and H¨ older inequality. On the other hand, by using (1.4), (3.5) and the condition (H1 ), we have

 α ∂ S, ∂t ∂ α Pf ≤ η ∂t ∂ α Pf 2 + Cη ε(1 + t)−1− . Thus, we can obtain from (3.7) that 

∂t ∂ α Pf 2 ≤ C[ε(1 + t)−1− +



∂ α Pf 2 +

1≤|α|≤N

∂ α (I − P)f 2 ].

(3.8)

1≤|α|≤N

By combining the above inequalities, we have d wl ∂βα (I − P)f 2 + wl ∂βα (I − P)f 2σ dt ≤ Cε(1 + t)−1− + C(η + ε) wl ∂βα (I − P)f 2σ + Cε 

+C

∂ α Pf 2 + C

+Cη



wl ∂βα (I − P)f 2σ

1≤|β  |,|α |+|β  |≤N

∂ α (I − P)f 2σ

|α|≤N

1≤|α|≤N







w

l

α ∇x ∂β−β (I 1

l (t). − P)f 2 + C El (t)D

(3.9)

|β1 |=1

For any 0 < |β| ≤ N and any l ≥ 0, by taking η > 0 and ε > 0 small enough, the summation of (3.9) over |α| + |β| ≤ N by a suitable linear combination gives (3.2). And this completes the proof of the lemma. For the decay rate estimate, we also need the following weighted energy estimate. Lemma 3.2 Under the assumptions (H0 ) and (H1 ), for any l ≥ 0 and small ε, we have d dt

 wl ∂ α f 2 + wl (I − P)f 2 + wl ∂ α (I − P)f 2σ

 1≤|α|≤N

−1−

≤ Cε(1 + t) +C

|α|≤N



+ Cε

 wl ∂βα (I

− P)f 2σ

1≤|β  |,|α |+|β  |≤N



∂ α Pf 2 + C

1≤|α|≤N



l (t). ∂ α (I − P)f 2σ + C El (t)D

(3.10)

1≤|α|≤N

Proof We apply ∂ α on equation (1.3) with 1 ≤ |α| ≤ N to have  [∂t + v · ∇x ]∂ α f + e−Φ L∂ α f + Cαα1 ∂ α−α1 e−Φ L[∂ α1 (I − P)f ] − (∇x Φ + E) · ∇v ∂ α f 

−

α1 <α

Cαα1 (∂ α1 ∇x Φ + ∂ α1 E) · ∇v ∂ α−α1 f +

0<α1 ≤α α

= ∂ (e

−Φ/2

1  α1 Cα v · ∂ α1 E∂ α−α1 f 2 α1 ≤α

α

Γ(f, f )) + ∂ S.

(3.11)

No.4

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1047

In the inner product of (3.11) over R3 × R3 with w2l ∂ α f , the first term on the left hand d side is equal to 12 dt wl ∂ α f 2 . The other terms can be estimated as follows. First, (3.5) and Lemma 2.4 imply that (e−Φ L∂ α f, w2l ∂ α f ) ≥ 12 wl ∂ α f 2σ − C ∂ α f 2σ . For the third term on the left hand side of (3.11), it suffices to consider the case |α−α1 | = 1 because the other cases are easier. For this, by Lemma 2.4, we obtain 

|(w2l ∂ α−α Φe−Φ L[(∂ α1 (I − P)f ], ∂ α f ))|   |∂ α−α1 Φ| · |wl ∂ α1 (I − P)f |σ |wl ∂ α f |σ dx ≤C |α−α1 |=1

R3

≤ Cε wl ∂ α f 2σ + Cε



wl ∂ α1 (I − P)f 2σ .

α1 <α

By using the condition (H1 ), the fourth term on the left hand side of (3.11) is bounded by |((∇x Φ + E) · ∇v ∂ α f, w2l ∂ α f )| ≤ Cε wl ∂ α f 2σ . For the fifth term on the left hand side of (3.11), we can apply the decomposition (1.6) on f and use condition (H1 ) to obtain |((∂ α1 ∇x Φ + ∂ α1 E) · ∇v ∂ α−α1 Pf, w2l ∂ α f )| ≤ C( ∂ α1 ∇x Φ L3 (R3 ) + |x|∂ α1 E L∞ ) wl ∇x ∂ α−α1 Pf · wl ∂ α f t,x ≤ Cε ∇x ∂ α−α1 Pf 2 + Cε wl ∂ α f 2 ; |((∂ α1 ∇x Φ + ∂ α1 E) · ∇v ∂ α−α1 (I − P)f, w2l ∂ α f )| ≤ C( ∂ α1 ∇x Φ L∞ + ∂ α1 E L∞ ) wl ∇v ∂ α−α1 (I − P)f · wl ∂ α f t,x ≤ Cε wl ∇v ∂ α−α1 (I − P)f 2 + Cε wl ∂ α f 2 , when |α1 | ≤ N/2. On the other hand, when |α1 | ≥ N/2, we have |((∂ α1 ∇x Φ + ∂ α1 E) · ∇v ∂ α−α1 (I − P)f, w2l ∂ α f )| ≤ C( ∂ α1 ∇x Φ L3 + |x|∂ α1 E L∞ ) wl ∇v ∇x ∂ α−α1 (I − P)f · wl ∂ α f t,x ≤ Cε wl ∇x ∇v ∂ α−α1 (I − P)f 2 + Cε wl ∂ α f 2 , where we have used the Hardy inequality. We thus have the following estimate on the fifth term on the left hand side of (3.11) as |((∂ α1 ∇x Φ + ∂ α1 E) · ∇v ∂ α−α1 f, w2l ∂ α f )|  ≤ Cε ∇x ∂ α−α1 Pf 2 + Cε wl ∂βα11 (I − P)f 2 + Cε wl ∂ α f 2 . |α1 |+|β1 |≤N

The similar argument yields that the sixth term satisfies (v · ∂ α1 E∂ α−α1 f, w2l ∂ α f )   ≤ Cε ∇x ∂ α−α1 Pf 2 + Cε wl ∇x ∂ α−α1 (I − P)f 2σ + Cε wl ∂ α f 2σ . α1 =0

α1 =0

By using (1.4), (3.5) and the condition (H1 ), we have  w2l ∂ α f )| ≤ Cη wl−1/2 ∂ α (eΦ/2 μ−1/2 S) 2 + Cη ∂ α (e−Φ/2 E) 2 + 2η wl ∂ α f 2 |(∂ α S, σ ≤ 2Cη ε(1 + t)−1− + 2η wl ∂ α f 2σ .

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Moreover, by (2.9), (1.11) and (3.5), the nonlinear collision operator satisfies l (t). |(∂ α (e−Φ/2 Γ(f, f )), w2l ∂ α f )| ≤ η wl ∂ α f 2σ + Cη El (t)D Notice that there exists a constant C > 0 such that wl ∂ α f 2σ ≤ C( ∂ α Pf 2 + wl ∂ α (I − P)f 2σ ). By combining the above estimates and taking η > 0 and ε > 0 small enough, we have from (3.11) that d dt

 1≤|α|≤N

wl ∂ α (I − P)f 2σ

1≤|α|≤N

−1−

≤ Cε(1 + t)



+ Cε



wl ∂βα (I − P)f 2σ

1≤|β  |,|α |+|β  |≤N



+C



wl ∂ α f 2 +

∂ α Pf 2 + C

1≤|α|≤N



l (t). ∂ α (I − P)f 2σ + Cη El (t)D

(3.12)

1≤|α|≤N

If we take the inner product of (3.1) with w2l (I − P)f , by the smallness assumption on the external force and the source term, the standard energy estimation leads to 1 d wl (I − P)f 2 + wl (I − P)f 2σ 2 dt  ≤ Cε(1 + t)−1− + C ∂ α Pf 2 + C 1≤|α|≤N



l (t). (3.13) ∂ α (I − P)f 2σ + Cη El (t)D

1≤|α|≤N

Finally, a suitable combination of (3.12) and (3.13) gives (3.10) and this completes the proof of the lemma. The inequalities (3.2) and (3.10) give the estimate of the microscopic component in Section 4. As the second part in this section, we will recall the compensating function introduced by Kawashima [13] for the Boltzmann equation. For this, consider [∂t + v · ∇x + L]∂ α f = g,

(3.14)

where g is given and |α| ≤ N . Let v · ξ (ξ ∈ R3 ) be the symbol of the streaming operator v · ∇x . Note that v · ξ is a  as the subspace of L2 (R3 ) spanned by the thirteen linear operator in L2 (R3 ) from N onto W  = span{ϕj μ1/2 |j = 1, · · · , 13}. Here, functions(moments) ϕj μ1/2 , j = 1, 2, · · · , 13, namely, W ϕ1 = 1; ϕ8 = v1 v2 ;

ϕj+1 = vj (j = 1, 2, 3); ϕj+4 = vj2 (j = 1, 2, 3); ϕ9 = v2 v3 ; ϕ10 = v3 v1 ; ϕj+10 = |v|2 vj (j = 1, 2, 3).

Notice that any collision invariant is a linear combination of ψk = ϕk , k = 1, · · · , 4, and ψ5 = ϕ5 + ϕ6 + ϕ7 . Thus, we can rewrite the thirteen moments as: ψ1 = 1;

ψj+1 = vj (j = 1, 2, 3); ψ5 = |v|2 ; ψj+4 = vj2 (j = 2, 3);

ψ8 = v1 v2 ;

ψ9 = v2 v3 ; ψ10 = v3 v1 ; ψj+10 = |v|2 vj (j = 1, 2, 3).

No.4

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1049

 by ej , 1 ≤ j ≤ 13 as in [7, 13]. Let Denote an orthogonal basis for this 13 dimensional space W 13 {ψk μ1/2 }13 k=1 A13×13 = [ej ]j=1 ,

where detA = 0 and [ej ]5j=1 is the orthogonal basis of the null space of L. : Moreover, let P0 be the orthogonal projection from L2 (R3 ) onto W P0 f =

13 

f, ek ek .

(3.15)

k=1

Set Wk = ∂ α f, ek . Then (3.14) implies that ∂t W + (j = 1, 2, 3) and L are symmetric matrices L = { L[el ], ek }13 k,l=1 ,

V (ξ) =

3 

 j

V j ∂xj W + LW = g + R, where V j

V j ξj = { (v · ξ)ek , el }13 k,l=1 ,

(3.16)

j=1

g is the vector with components g, ej , and R contains the terms having (I − P0 )f . Write Rz for the real part of z ∈ C and W = [WI , WII ]T , WI = [W1 , · · · , W5 ]T , WII = [W6 , · · · , W13 ]T . With the above preparation and notations, the following definition of compensating function comes from [13]. Definition 3.3 S(ω) with ω ∈ S2 is called a compensating function for (3.14) if (i) S(·) is a C ∞ function on S2 with value taken in the space of bounded linear operators on L2 (R3 ) satisfying S(−ω) = −S(ω) for all ω ∈ S2 . (ii) iS(ω) is self-adjoint on L2 (R3 ) for all ω ∈ S2 . (iii) There exists c0 > 0 such that for all ∂ α f ∈ D(L) and ω ∈ S2 , R S(ω)(v · ω)∂ α f, ∂ α f  + L[∂ α f ], ∂ α f  ≥ c0 (|∂ α Pf |22 + |∂ α (I − P)f |2σ ). The construction of the compensating function for (3.14) needs the following lemma proved in [7, 13]. Lemma 3.4 There exist three 13 × 13 real constant skew-symmetric matrices Rj (j = 3  Rj ωj , satisfies 1, 2, 3) and positive constants C1 and C2 such that R(ω) ≡ j=1

R

R(ω)V (ω)W, W  ≥ C1 |WI |2 − C2 |WII |2 for all W ∈ C13 . Here

·, · is the inner product on C13 . Then a compensating function for the equation (3.14) can be defined as follows. For any given ω ∈ S2 , set R(ω) ≡ {rij (ω)}13 i,j=1 and define S(ω)∂ α f ≡

13 

λrk (ω) ∂ α f, e ek

for some λ > 0, ∂ α f ∈ L2 (R3 ).

(3.17)

k,=1

The following lemma showing that S(ω) is a compensating function was proved in [13] and we include the proof here for the convenience of the readers. Notice that here we also include the dissipation on the microscopic component in · σ -norm as in the Definition 3.3 for the analysis in this paper.

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Lemma 3.5 If λ > 0 is small enough, the operator S(ω) defined (3.17) is a compensating . function for (3.14). Moreover, S(ω): L2 (R3 ) → W 3   Rj ωj , where Proof Firstly, recall that [ek ]13 k=1 is an orthogonal basis of W and R(ω) ≡ j=1

Rj is a constant 13×13 real skew-symmetric matrix. Thus S(·) is C ∞ (S 2 ) and S(−ω) = −S(ω). This verifies (i) of the definition. The conditions (ii) and (iii) can be checked as follows. Let g1 , g2 ∈ L2 (R3v ). Then 13 

S(ω)g1 , g2  =

λrk (ω) g1 , e  g2 , ek .

(3.18)

k,=1 13 13 13 Write w = {wk }13 k=1 = { g1 , ek }k=1 ; u = {uk }k=1 = { g2 , ek }k=1 . Then

S(ω)g1 , g2  = λ

R(ω)w, u. Hence

iS(ω)g1 , g2  = λ

iR(ω)w, u. Since R(ω) is skew symmetric, iS(ω) is self-adjoint so that condition (ii) holds. For the condition (iii), let f ∈ D(L). From (3.17), we have

S(ω)(v · ω)∂ α f, ∂ α f  ≡

13 

λrk (ω) (v · ω)∂ α f, e  ∂ α f, ek .

k,=1

We substitute the decomposition ∂ α f = P0 ∂ α f + (I − P0 )∂ α f into the right hand side of (3.19), where P0 is defined in (3.15). Then

S(ω)(v · ω)∂ α f, ∂ α f  = λ

R(ω)V (ω)W, W  +

13 

λrk (v · ω)(I − P0 )∂ α f, e  ∂ α f, ek ,

k,=1

where V (ω) is the matrix defined in (3.16) and W is the vector in C13 whose k−th component is ∂ α f, ek . By virtue of Lemma 3.4 and the exponential decay of ek in v, we have Rλ

R(ω)V (ω)W, W  ≥ C1 λ|WI |2 − λC2 |WII |2 ≥ C1 λ|P∂ α f |22 − C2 λ|(I − P)∂ α f |2σ . It is obvious that there exists some positive constant C3 such that | (v · ω)(I − P0 )∂ α f, e | ≤ C3 |(I − P0 )∂ α f |σ ≤ C3 |(I − P)∂ α f |σ . Therefore, we obtain R S(ω)(v · ω)∂ α f, ∂ α f  ≥ λ[C1 |∂ α Pf |22 − C2 |∂ α (I − P)f |2σ ] − C3 λ|∂ α f |σ |(I − P)∂ α f |σ ≥ λ(C1 − ε)|∂ α Pf |22 − λCε |(I − P)∂ α f |2σ ,

(3.19)

for any ε > 0 small enough where Cε > 0 is a constant depending on ε. Here, we have used the fact that |∂ α f |2σ ≤ C(|(I − P)∂ α f |2σ + |∂ α Pf |22 ). Recall

L[∂ α f ], ∂ α f  ≥ δ0 |(I − P)∂ α f |2σ

for some δ0 > 0.

(3.20)

No.4

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Then a suitable combination of (3.19) and (3.20) shows that there exists c0 > 0 such that R S(ω)(v · ω)∂ α f, ∂ α f  + L[∂ α f ], ∂ α f  ≥ c0 (|∂ α Pf |22 + |∂ α (I − P)f |2σ ). And this completes the proof of the lemma. In the following, we will use the above compensating function to estimate the macroscopic component of the solution to (1.3). Lemma 3.6 Under the assumptions (H0 ) and (H1 ), for the equation (1.3), we have the following estimate: 

 d  α f ), ∂ α f dξ  ∂ α f 2 − κ |ξ| iS(ω)(∂ dt 3 1≤|α|≤N 1≤|α|≤N −1 R   +δ2 P∂ α f 2 + δ1 (I − P)∂ α f 2σ 2≤|α|≤N

1≤|α|≤N −1−

≤ C ∇x Pf + Cε(1 + t) 2

and

l (t), + C El (t)D

(3.21)



d  ∂ α f 2 − κ (1 + |ξ|2 )N −1 |ξ| iS(ω)fˆ, fˆdξ dt R3 |α|≤N   +δ2 P∂ α f 2 + δ1 (I − P)∂ α f 2σ |α|≤N

1≤|α|≤N −1−

≤ Cε(1 + t)

l (t), + C El (t)D

(3.22)

where κ > 0 is small. Proof Let ω = ξ/|ξ| and take the Fourier transform in x of (3.14) to have α f ) + i|ξ|(v · ω)∂ α f + L[∂ αf ] = g   ∂t (∂ ˆ.

(3.23)

1  α f ], ∂ α f  = R ∂ αf , g   ∂t |∂ α f |22 + L[∂ ˆ. 2

(3.24)

Hence,

Applying −i|ξ|S(ω) to (3.23) gives α f ) + |ξ|2 S(ω)((v · ω)∂ α f ) − i|ξ|S(ω)L[∂ α f ] = −i|ξ|S(ω)ˆ   −i|ξ|S(ω)∂t (∂ g.

(3.25)

α f , we have By taking the inner product of the above equation with ∂ α f ), ∂ α f  + |ξ|2 R S(ω)((v · ω)∂ α f ), ∂ αf     R −i|ξ|S(ω)∂t (∂   α f ], ∂ α f  − iS(ω)ˆ αf  .  = |ξ|R iS(ω)L[∂ g , ∂

(3.26)

α f ), ∂ α f ]. (3.24) × (1 + |ξ|2 ) +  Since iS(ω) is self-adjoint, the first term is − 21 ∂t [|ξ| iS(ω)(∂ (3.26) × κ with a positive constant κ yields

(1 + |ξ|2 )

κ|ξ| α f ), ∂ α f  + (1 + |ξ|2 − κ|ξ|2 ) 

iS(ω)(∂ 2 2 2 α α α f ), ∂ α f  + L[∂ α f ], ∂ α f }      × L[∂ f ], ∂ f  + κ|ξ| {R S(ω)((v · ω)∂   αf , g α f ], ∂ α f  − iS(ω)ˆ αf  .  = (1 + |ξ|2 )R ∂ ˆ + κ|ξ|R iS(ω)L[∂ g, ∂ ∂t

α f |2 − |∂ 2

(3.27)

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For 0 < κ < 1, the second term on the left hand side of (3.27) is bounded from below by α f ], ∂ α f  ≥ (1 − κ)(1 + |ξ|2 ) · δ |(I − P)∂ α f |2 .   (1 + |ξ|2 − κ|ξ|2 ) L[∂ 0 σ

By Lemma 3.5, the third term on the left hand side of (3.27) is bounded by α f ), ∂ α f  + L[∂ α f ], ∂ α f } ≥ κ|ξ|2 · c (|P∂ α f |2 + |(I − P)∂ α f |2 ).      κ|ξ|2 {R S(ω)((v · ω)∂ 0 2 σ

In addition, we have 13 

αf ] = S(ω)L[∂

α f ], e e = λrk (ω) L[∂  k

k,=1

13 

α f ], e e , λrk (ω) L[(I − P)∂  k

k,=1

and α f ], e | ≤ C|(I − P)∂ αf | ,  | L[(I − P)∂  σ

where we have used Lemma 2.4 and the fact that Lg = Γ(μ, g) + Γ(g, μ). Thus, the absolute value of the second term on the right hand side of (3.27) satisfies   α f ], ∂ α f | + | iS(ω)ˆ α f |  g , ∂ cκ|ξ| | iS(ω)L[∂ 13    α f | |∂ αf | + α f , e |  | ˆ g , e | · | ∂ ≤ cκ|ξ| |(I − P)∂ σ 2 k k,=1 α f |2 + κε|ξ|2 |∂ α f |2 + c  ≤ cε κ|(I − P)∂ ε σ 2

13 

| ˆ g , e |2 .

=1

If we choose κ, ε > 0 small enough, the above inequalities imply that there exist δ1 , δ2 > 0 such that

α f |2 − κ|ξ| iS(ω)(∂ α f ), ∂ α f  + δ (1 + |ξ|2 )|(I − P)∂ α f |2 + δ |ξ|2 |P∂ α f |2     ∂t (1 + |ξ|2 )|∂ 1 2 2 σ 2 αf , g ≤ (1 + |ξ|2 )R ∂ ˆ + cε

13 

| ˆ g , e |2 .

(3.28)

=1

Integrating (3.28) over ξ and summing over 1 ≤ |α| ≤ N − 1 give  

 α 2 α f ), ∂ α f dξ  ∂ f − κ |ξ| iS(ω)(∂ ∂t 3 1≤|α|≤N −1 R

1≤|α|≤N

+δ1



(I − P)∂ α f 2σ + δ2

1≤|α|≤N

≤

 1≤|α|≤N −1



R3



P∂ α f 2

2≤|α|≤N αf , g (1 + |ξ|2 )R ∂ ˆdξ + cε



13  

1≤|α|≤N −1 =1

R3

| ˆ g, e |2 dξ.

(3.29)

Back to the Landau equation with external forcing, set 1 g = ∂ α (e−Φ/2 Γ(f, f )) + ∂ α S + ∂ α ((∇x Φ + E) · ∇v f ) − v · ∂ α (Ef ) − ∂ α ((e−Φ − 1)Lf ). 2

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We first consider the second term on the right hand side of (3.29). By (3.5) and the condition (H1 ), we obtain    | ∂ α (e−Φ/2 Γ(f, f )), e |2 dξ 1≤|α|≤N −1



=

R3



1≤|α|≤N −1

| ∂ α (e−Φ/2 Γ(f, f )), e |2 dx

R3







≤C

R3

1≤|α|≤N −1 α +α ≤α

 2  2  2  2 { ∂ α f ∂ α f + ∂ α f ∂ α f ]dx σ

2

σ

2

l (t), ≤ C El (t)D and

 R3



 | ∂ α S, e |2 dξ =

 e |2 dx | ∂ α S,

R3

≤ C w−1/2 ∂ α (eΦ/2 μ−1/2 S) 2 + C ∂ α (e−Φ/2 E) 2 ≤ 2Cε(1 + t)−1− , and



 1≤|α|≤N −1



≤C

R3

| v ·  ∂ α (Ef ) − ∂ α ((∇x Φ + E) · ∇v f ), e |2 dξ

  R3

1≤|α|≤N α ≤α



≤ Cε





(I − P)∂ α f 2σ + Cε

1≤|α|≤N

and 





R3

P∂ α f 2 ,

| (e−Φ − 1)Lf, e |2 dξ

 

1≤|α|≤N −1 α ≤α

≤ Cε

 1≤|α|≤N

1≤|α|≤N −1

=



(|∂ α ∇x Φ|2 + |∂ α E|2 )|∂ α−α f |22 dx



R3





| ∂ α−α (e−Φ − 1)L∂ α f, e |2 dx

(I − P)∂ α f 2σ .

1≤|α|≤N

For the first term on the right hand side of (3.29), we have    αf , g αf , g (1 + |ξ|2 )R ∂ ˆdξ ≤

∂ ˆdξ + R3

R3

R3

αf , g |ξ|2 ∂ ˆdξ .

By using (1.11), (3.5) and Lemma 2.5, we get   α f , ∂ α (e−Φ/2

∂ Γ(f, f ))dξ = (∂ α (e−Φ/2 Γ(f, f )), ∂ α f ) 3 R = (∂ α (e−Φ/2 Γ(f, f )), (I − P)∂ α f ) l (t). ≤ η (I − P)∂ α f 2σ + C El (t)D

(3.30)

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Similarly, we have    α f , ∂ α (e−Φ/2 δi ∂ α f , ∂ δi ∂ α (e −Φ/2 Γ(f, f ))dξ |ξ|2 ∂ Γ(f, f ))dξ =

∂ 3 R3 R δi α = (∂ ∂ Γ(f, f ), ∂ δi ∂ α f ) l (t), ≤ η (I − P)∂ δi ∂ α f 2σ + C El (t)D where |α| ≤ N − 1. The other terms in g can be estimated similarly by using the condition (H1 ). In summary, we have  1≤|α|≤N −1

 R3

13  



αf , g (1 + |ξ|2 )R ∂ ˆdξ + cε

1≤|α|≤N −1 =1

l (t) + Cε(1 + t) ≤ C El (t)D

−1−



+ C(ε + η)

R3

| ˆ g , e |2 dξ

(I − P)∂ α f 2σ + C(ε + η)

1≤|α|≤N



P∂ α f 2 .

1≤|α|≤N

If we take η > 0 and ε > 0 small enough, (3.29) and the above estimate give the desired estimate (3.21). To prove (3.22), let α = 0 in (3.28) and set 1 g = e−Φ/2 Γ(f, f ) + S + (∇x Φ + E) · ∇v f − v · Ef − (e−Φ − 1)Lf. 2 Multiplying the resulting equation by (1 + |ξ|2 )N −1 and integrating over ξ give  

2 N ˆ2 ∂t (1 + |ξ| ) |f|2 dξ − κ (1 + |ξ|2 )N −1 |ξ| iS(ω)fˆ, fˆdξ 3 R3 R  2 N 2 (1 + |ξ| ) |(I − P)fˆ|σ dξ + δ2 (1 + |ξ|2 )N −1 |ξ|2 |Pfˆ|22 dξ +δ1  ≤

R3

R3

(1 + |ξ|2 )N R fˆ, gˆdξ + cε

13   =1

R3

R3

(1 + |ξ|2 )N | ˆ g , e |2 dξ.

(3.31)

By using Lemmas 2.4–2.5 and the assumptions in Theorem 1.1, the same argument as above implies that (3.22) holds. And this completes the proof of the lemma. For the study on the optimal time decay, we need the following estimate on the solution operator of the linearized Landau equation: ∂t f + v · ∇x f + Lf = 0,

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

(3.32)

whose solution can be written as f (t, x, v) = U (t, 0)f0 (x, v),

(3.33)

with U (t, s) being the solution operator. The solution operator has the following decay estimates. Lemma 3.7 Let k ≥ k1 ≥ 0 and f0 ∈ H N ∩ Zq . If f (t, x, v) ∈ C 0 ([0, ∞); H N ) ∩ C 1 ([0, ∞); H N −1 ) is a solution of (3.32), we have ∇kx U (t, 0)f0 ≤ C(1 + t)−σq,m ( ∇kx1 f0 Zq + ∇kx f0 ),

(3.34)

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  for any integer m = k − k1 ≥ 0, where q ∈ [1, 2] and σq,m = 32 1q − 12 + m 2. Proof By letting |α| = 0 in (3.28) and g = 0, we have

ˆ + δ1 (1 + |ξ|2 )|(I − P)fˆ|2 + δ2 |ξ|2 |Pfˆ|2 ≤ 0. ∂t (1 + |ξ|2 )|fˆ|22 − κ|ξ| iS(ω)fˆ, f 2 σ Hence, ∂t E[f ] + δ

(3.35)

|ξ|2 E[f ] ≤ 0, 1 + |ξ|2

(3.36)

where

κ|ξ|

iS(ω)fˆ(t, ξ, ·), fˆ(t, ξ, ·). 1 + |ξ|2 Since κ > 0 is small enough and S(ω) is a bounded operator, it is clear that for small κ, E[f ] = |fˆ(t, ξ, ·)|22 −

1 ˆ |f (t, ξ, ·)|22 ≤ E[f ] ≤ 2|fˆ(t, ξ, ·)|22 . 2 From (3.36) and (3.37), we have ˆ ξ, ·)|2 ≤ ce |f(t, 2

2

|ξ| −δt 1+|ξ| 2

Multiplying this by |ξ|2k and integrating over ξ give   k 2 2k ˆ 2 ∇x f = |ξ| |f (t, ξ, ·)|2 dξ ≤ c R3

R3



Set I0 =

2

|ξ|≤1

|ξ|2k e

|ξ| −δt 1+|ξ| 2

|ξ|≤1

≤



|ξ|2p m e

|ξ|≤1

2 −δp t |ξ| 2

|ξ|2k e

|ξ|≥1

dξ

|ξ|2k e

|ξ α−α |2 e−δt 

|ξ|≤1

|fˆ0 (ξ)|22 dξ.

|ξ| −δt 1+|ξ| 2



|ξ|≤1

1/p 

|ξ|2

−δt 1+|ξ|2

2

For the first term in I0 , we have   |ξ|2 2k −δt 1+|ξ|2 ˆ 2 |ξ| e |f0 (ξ)|2 dξ ≤



|fˆ0 (ξ)|22 .

 |fˆ0 (ξ)|22 dξ +

(3.37)

 |ξ α fˆ0 (ξ)|2q 2 dξ

|ξ|2 2

(3.38)

|fˆ0 (ξ)|22 dξ.

(3.39)

 |ξ α fˆ0 (ξ)|22 dξ

1/q

,

where |α| = k, |α | = k1 , m = |α − α | and p ∈ [1, ∞) with p1 + q1 = 1. Note that  2   |ξ|  |ξ|2p m e−δp t 2 dξ ≤ C(1 + t)−3/2−p m , |ξ|≤1

and





|ξ|≤1

Thus,

 |ξ α fˆ0 (ξ)|2q 2 dξ



2

|ξ|≤1

|ξ|2k e

|ξ| −δt 1+|ξ| 2

1/q



≤ ∂ α f0 2Zq ,

1 1 +  = 1. q 2q 



|fˆ0 (ξ)|22 dξ ≤ c(1 + t)−3/2−p m ∂ α f0 2Zq .

For the second term in I0 , we directly have Hence, 



2

2k |ξ|≥1 |ξ| e 

|ξ| −δt 1+|ξ| 2

I0 ≤ C(1 + t)−3/2p −m ∂ α f0 2Zq + ce

−δt 2

|fˆ0 (ξ)|22 dξ ≤ ce

−δt 2

∂ α f0 2 .

∂ α f0 2

≤ C(1 + t)−2σq,m ( ∇kx1 f0 Zq + ∇kx f0 )2 . Therefore, (3.34) follows from (3.38) and the above estimate on I0 . And this completes the proof of the lemma.

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Optimal Time Decay Rate

In this section, we shall derive a refined energy estimate which is used in constructing global solutions and proving the optimal time decay. For this, we first recall the local existence result proved in [14] as follows. Lemma 4.1 For any l ≥ 0, under the conditions (H0 ) and (H1 ), if E () ≤ ε, there exists T ∗ = T ∗ (ε) > 0 such that there is a unique solution f (t, x, v) to (1.3) in [0, T ∗ ) × R3 × R3 satisfying   t El (t) + wl ∂ α f (s) 2σ ds ≤ Cε. |α|≤N

0

√ √ Moreover, if F0 (x, v) = μ + μf0 (x, v) ≥ 0, then F (t, x, v) = μ + μf (t, x, v) ≥ 0. We are now ready to prove Theorem 1.1 and we split the proof into two parts, namely for existence and optimal decay for clear presentation. Proof of global existence By Lemma 3.6, we have 

d  α 2 ∂ f − κ (1 + |ξ|2 )N −1 |ξ| iS(ω)fˆ, fˆdξ dt R3 |α|≤N   +δ1 (I − P)∂ α f 2σ + δ2 P∂ α f 2 |α|≤N

1≤|α|≤N −1−

≤ Cε(1 + t)

l (t), + C El (t)D

(4.1)

where κ > 0 is a small constant. From Lemma 3.2, we have d dt

 wl ∂ α f 2 + wl (I − P)f 2 + wl ∂ α (I − P)f 2σ

 1≤|α|≤N

−1−

≤ Cε(1 + t)



+C

|α|≤N



+ Cε

 wl ∂βα (I

− P)f 2σ

1≤|β  |,|α |+|β  |≤N

∂ α Pf 2 + C

1≤|α|≤N



l (t), ∂ α (I − P)f 2σ + C El (t)D

(4.2)

1≤|α|≤N

where ε > 0 is also small constant. And by Lemma 3.1, we have d

 1≤|β|,|α|+|β|≤N

dt

wl ∂βα (I − P)f 2 + wl ∂βα (I − P)f 2σ

≤ Cε(1 + t)−1− + C



∂ α Pf 2 + C



l (t). (4.3) wl ∂ α (I − P)f 2σ + C El (t)D

|α|≤N

1≤|α|≤N

A suitable linear combination of (4.1), (4.2) and (4.3) yields  d  α 2 ∂ f − κ (1 + |ξ|2 )N −1 |ξ| iS(ω)fˆ, fˆdξ dt 3 R |α|≤N    + wl ∂ α f 2 + wl (I − P)f 2 + wl ∂βα (I − P)f 2 1≤|α|≤N

 +



1≤|α|≤N

∂ α Pf 2 +

 |α|≤N

1≤|β|,|α|+|β|≤N

wl ∂ α (I − P)f 2σ +



|α|≤N

∂ α (I − P)f 2σ

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+

wl ∂βα (I − P)f 2σ

1057



1≤|β|,|α|+|β|≤N

l (t). ≤ Cε(1 + t)−1− + C El (t)D

(4.4)

On the other hand, the boundedness of the operator S(ω) implies   (1 + |ξ|2 )N −1 |ξ| iS(ω)fˆ, fˆdξ ≤ C ∂ α f 2 + C R3

|α|≤N



∂ α f 2 .

|α|≤N −1

Therefore, we can define the functionals in (1.8) and (1.9) as   α 2 ∂ f − κ (1 + |ξ|2 )N −1 |ξ| iS(ω)fˆ, fˆdξ El (t) = R3

|α|≤N



+



wl ∂ α f 2 + wl (I − P)f 2 +

1≤|α|≤N

wl ∂βα (I − P)f 2 ,

1≤|β|,|α|+|β|≤N

and 

Dl (t) =

∂ α Pf 2 +

 |α|≤N

1≤|α|≤N



∂ α (I − P)f 2σ +

wl ∂βα (I − P)f 2σ .

|α|+|β|≤N

Finally, we obtain d l (t) ≤ Cε(1 + t)−1− + CEl (t)Dl (t). El (t) + Dl (t) ≤ Cε(1 + t)−1− + C El (t)D dt

(4.5)

If we assume El (0) ≤ ε for a sufficiently small constant ε > 0, based on the local existence in Lemma 4.1, the global existence follows from the standard continuity argument. The following proof on the optimal decay estimate is motivated by the work of Duan-UkaiYang-Zhao [6] on the Boltzmann equation by using an energy-spectrum approach. Proof of optimal decay Lemma 3.6, we have d dt +δ2



Take  = 3/2 in the assumption on the source term. By

1≤|α|≤N −1

1≤|α|≤N







∂ α f 2 − κ



P∂ α f 2 + δ1

2≤|α|≤N

(I − P)∂ α f 2σ

1≤|α|≤N −5/2

≤ C ∇x Pf + Cε(1 + t) 2

R3

α f ), ∂ α f dξ  |ξ| iS(ω)(∂

l (t), + C El (t)D

(4.6)

where κ > 0 is a small constant. From Lemma 3.2, we have d dt



 wl ∂ α f 2 + wl (I − P)f 2 + wl ∂ α (I − P)f 2σ

1≤|α|≤N

≤ Cε(1 + t)−5/2 + Cε +C

 1≤|α|≤N

|α|≤N





wl ∂βα (I − P)f 2σ

1≤|β  |,|α |+|β  |≤N

∂ α Pf 2 + C



1≤|α|≤N

l (t), ∂ α (I − P)f 2σ + C El (t)D

(4.7)

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where ε > 0 is a small constant. Lemma 3.1 implies that

d  wl ∂βα (I − P)f 2 + wl ∂βα (I − P)f 2σ dt 1≤|β|,|α|+|β|≤N   l (t). ≤ Cε(1 + t)−5/2 + C ∂ α Pf 2 + wl ∂ α (I − P)f 2σ + C El (t)D

(4.8)

|α|≤N

1≤|α|≤N

In addition, the standard energy estimate gives d (I − P)f 2 + (I − P)f 2σ dt l (t). ≤ Cε(1 + t)−5/2 + C ∇x Pf 2 + C ∇x (I − P)f 2σ + C El (t)D

(4.9)

A suitable linear combination of (4.6), (4.7), (4.8) and (4.9) yields   d  α 2 α f ), ∂ α f dξ  ∂ f − κ |ξ| iS(ω)(∂ dt 3 R 1≤|α|≤N 1≤|α|≤N −1    l α 2 w ∂ f + wl (I − P)f 2 + wl ∂βα (I − P)f 2 + (I − P)f 2 + 1≤|α|≤N

 +



∂ α Pf 2 +



wl ∂βα (I

−



wl ∂ α (I − P)f 2σ +

|α|≤N

1≤|α|≤N

+

1≤|β|,|α|+|β|≤N



∂ α (I − P)f 2σ

|α|≤N

P)f 2σ

+ (I −

P)f 2σ



1≤|β|,|α|+|β|≤N

l (t) + C ∇x Pf 2 . ≤ Cε(1 + t)−5/2 + C El (t)D On the other hand, the boundedness of the operator S(ω) implies that    α f ), ∂ α f dξ ≤ C  |ξ| iS(ω)(∂ ∂ α f 2 + C 1≤|α|≤N −1

R3



∂ α f 2 .

1≤|α|≤N −1

1≤|α|≤N

Therefore, we can define a functional Hl (t) equivalent to E l (t) in Lemma 2.2 as    α f ), ∂ α f dξ  E l (t) ∼ Hl (t) = ∂ α f 2 − κ |ξ| iS(ω)(∂ 1≤|α|≤N −1

1≤|α|≤N



+

R3

wl ∂ α f 2 + wl (I − P)f 2

1≤|α|≤N



+

wl ∂βα (I − P)f 2 + (I − P)f 2 ,

1≤|β|,|α|+|β|≤N

which is bounded by CDl (t). Thus, we obtain d l (t) + C ∇x Pf 2 Hl (t) + Dl (t) ≤ Cε(1 + t)−5/2 + C El (t)D dt ≤ Cε(1 + t)−5/2 + CEl (t)Dl (t) + C ∇x Pf 2 . For El (t) < ε with ε > 0 being small enough, we have d Hl (t) + Dl (t) ≤ Cε(1 + t)−5/2 + C ∇x Pf 2 , dt

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1059

which gives from the fact that E l (t) ≤ CDl (t) that, d Hl (t) + Hl (t) ≤ Cε(1 + t)−5/2 + C ∇x Pf 2 . dt

(4.10)

If we set 1 G = e−Φ/2 Γ(f, f ) + S + (∇x Φ + E) · ∇v f − v · Ef − (e−Φ − 1)Lf, 2 where S = eΦ/2 μ−1/2 S + eΦ/2 μ1/2 v · Φ, then equation (1.5) can be written as ∂t f + v · ∇x f + Lf = G. With (3.33), the solution to equation (1.3) can be written in the mild form 

t

f (t) = U (t, 0)f0 +

U (t, s)G(s)ds. 0

By using Lemma 3.7, we can obtain ∇x Pf (t) ≤ ∇x f (t) ≤ Cλ0 (1 + t)−5/4 + C



t

0

(1 + t − s)−5/4 ( G(s) Z1 + ∇x G(s) )ds,

where λ0 = f0 Z1 + ∇x f0 . If l ≥ 1, by (3.5), Lemma 2.1 and Lemma 2.2, we have e−Φ/2 Γ(f, f ) 2Z1 ≤ C



∂β1 f 2 ·

|β1 |≤2

≤C





w∂β2 f 2

|β2 |≤2

w∂β (I − P)f 4 + C Pf 4 ≤ CEl (t)Hl (t) + C Pf 4 ,

|β|≤2

and ∇x (e−Φ/2 Γ(f, f )) 2 ≤ C ∇x Γ(f, f ) 2 + C ∇x ΦΓ(f, f ) 2 ≤ CEl (t)Hl (t), where we have used the fact that El (t) ∼ El (t) and E l (t) ∼ Hl (t). By (1.11), (3.5) and the condition (H2 ), we have  Z1 + 1 v · Ef Z1 + (∇x Φ + E) · ∇v f Z1 G(s) Z1 ≤ e−Φ/2 Γ(f, f ) Z1 + S 2 + sup |e−Φ − 1| · Lf Z1 x∈R3

√  2 v∇x f ≤ C ε Hl (s) + C Pf 2 + Cε(1 + t)−5/4 + |x|E L∞ t (Lx )  −Φ 2 ) ) ∇x ∇v f + C sup |e +( ∇x Φ L6/5 + |x|E L∞ − 1| w∂β (I − P)f (L t x x∈R3

√  ≤ C ε Hl (s) + C Pf 2 + Cε( ∇v ∇x f + w∇x f  + w∂β (I − P)f ) + Cε(1 + t)−5/4

|β|≤2

|β|≤2

√  ≤ C ε Hl (s) + C Pf 2 + Cε(1 + t)−5/4

  +Cε ∇x Pf + ∇v ∇x (I − P)f + w∇x (I − P)f + w∂β (I − P)f , |β|≤2

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where we have used Lemma 2.1, the Hardy inequality and the fact that El (t) ≤ ε. Notice that by an argument similar to those for Lemma 2.1 and Lemma 2.2, we have Lf ≤ C



w∂β (I − P)f .

|β|≤2

Similarly, for l ≥ 1, it holds that

√  √ ∇x G(s) ≤ C ε Hl (s) + C ε(1 + t)−5/4 + Cε ∇x Pf + ∇v ∇x (I − P)f   + w∇x (I − P)f + w∂βα (I − P)f . |α|≤1,|β|≤2

Define   (1 + s)5/2 Hl (s) ,

M (t) = sup 0≤s≤t

M0 (t) = sup

0≤s≤t



 (1 + s)3/2 f (s) 2 .

(4.11)

Since M (t) and M0 (t) are non-decreasing, when l ≥ 1, we have √  √ G(s) Z1 + ∇x G(s) ≤ C ε Hl (s) + C Pf (s) 2 + C ε(1 + s)−5/4  √ √ ≤ C ε(1 + s)−5/4 M (s) + C(1 + s)−3/2 M0 (t) + C ε(1 + s)−5/4 , for any 0 ≤ s ≤ t. With this, we have ∇x Pf (t) ≤ ∇x f (t)

 t √  ε M (t) + M0 (t)) (1 + t − s)−5/4 (1 + s)−5/4 ds 0 √ √  (4.12) ≤ C(1 + t)−5/4 (λ0 + ε + ε M (t) + M0 (t)).

≤ Cλ0 (1 + t)−5/4 + C(ε +

On the other hand, by the Gronwall inequality, (4.10) gives Hl (t) ≤ e

−ct



t

Hl (0) + C

e−c(t−s) ∇x Pf (s) 2 ds + Cε(1 + t)−5/2 .

0

Then (4.12) yields Hl (t) ≤ e−ct Hl (0) + C

 0

t

e−c(t−s) (1 + s)−5/2 ds(λ20 + ε + εM (t) + M02 (t)) + Cε(1 + t)−5/2

≤ C(1 + t)−5/2 (Hl (0) + λ20 + ε + εM (t) + M02 (t)). Hence, for any t ≥ 0, M (t) ≤ C(Hl (0) + λ20 + ε + εM (t) + M02 (t)). That is, when ε > 0 is small enough, one has M (t) ≤ C(Hl (0) + λ20 + ε + M02 (t)). By (4.11), this gives Hl (t) ≤ C(1 + t)−5/2 (Hl (0) + λ20 + ε + M02 (t)).

(4.13)

By using Lemma 3.7, it holds that f (t) ≤ C(1 + t)−3/4 ( f0 Z1 + f0 ) + C

 0

t

(1 + t − s)−3/4 ( G(s) Z1 + G(s) )ds.

No.4

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Yang & Yu: OPTIMAL CONVERGENCE RATES OF LANDAU EQUATION

Since  + 1 v · Ef + (∇x Φ + E) · ∇v f + (e−Φ − 1)Lf G(s) ≤ e−Φ/2 Γ(f, f ) + S 2 √ √  2 ≤ C ε Hl (t) + Pf + C ε(1 + t)−5/4

  +Cε ∇x Pf + ∇v ∇x (I − P)f + w∇x (I − P)f + w∂β (I − P)f , |β|≤2

we have  √ √ G(s) Z1 + G(s) ≤ C( ε + ε) Hl (s) + Pf 2 + C ε(1 + s)−5/4  √ ≤ C( ε + λ0 + Hl (0) + M0 (s))(1 + s)−5/4 + (1 + s)−3/2 M0 (s). Finally, it follows that f (t) ≤ C(1 + t)−3/4 ( f0 Z1 + f0 )  t  √ +C (1 + t − s)−3/4 (1 + s)−5/4 ds( ε + λ0 + Hl (0) + M0 (t)) 0  √ ≤ C(1 + t)−3/4 ( f0 Z1 + f0 + ε + λ0 + Hl (0) + M0 (t)).

(4.14)

Notice that ε > 0 can be taken small enough and  √ √ f0 Z1 + f0 + ε + λ0 + Hl (0) ≤ C ε.  √ From (4.14), we obtain M0 (t) ≤ C ε + CM0 (t). Thus, when M0 (0) ≤ Cε, we have M0 (t) ≤ Cε, which implies f (t) ≤ Cε(1 + t)−3/4 .

(4.15)

From (4.13) and (4.15), we obtain Hl (t) ≤ Cε(1 + t)−5/2 . Since   ∂ α Pf (t) 2 + wl ∂βα (I − P)f (t) 2 , Hl (t) ∼ 1≤|α|≤N

|α|+|β|≤N

(1.18) is proved. And this completes the proof of Theorem 1.1. Proof of Theorem 1.2 In the proof of Theorem 1.1, we have used the fact that when γ ≥ −1, |g|σ ≥ C|w1/2 g|2 ≥ C|(1 + |v|)1/2 g|2 to control the terms involving the multiplication of the velocity v. If E(t, x) = 0 and S(t, x, v) = 0, we do not have these terms in the equation. Therefore, the same argument for Theorem 1.1 holds when γ ≥ −2 so that the Theorem 1.2 follows. References [1] Degond P, Lemou M. Dispersion relations for the linearized Fokker-Planck equation. Arch Ration Mech Anal, 1997, 138(2): 137–167 [2] Desvillettes L, Villani C. On the spatially homogeneous Landau equation for hard potentials (I-II). Comm P D E, 2000, 25 (1/2): 179–298 [3] Desvillettes L, Villani C. On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent Math, 2005, 159 (2): 245–316

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Vol.29 Ser.B

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