- Email: [email protected]
Regularizing effects for the classical solutions to the Landau equation in the whole space
J. Math. Anal. Appl. 417 (2014) 123–143
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Regularizing effects for the classical solutions to the Landau equation in the whole space Shuangqian Liu a , Xuan Ma b,∗ a b
Department of Mathematics, Jinan University, Guangzhou 510632, China Department of Mathematics, Guangdong University of Education, Guangzhou 510303, China
a r t i c l e
i n f o
Article history: Received 6 April 2013 Available online 13 March 2014 Submitted by D. Wang Keywords: Landau equation Hypo-elliptic estimate Regularizing effect
a b s t r a c t The Landau equation describes the binary collisional effects (through long range coulombian interaction) in a plasma. In this paper, we prove that the known classical solutions to the Landau equation near Maxwellian in the whole space have a regularizing effect in all (time, space and velocity) variables, that is, become immediately smooth with respect to all variables. © 2014 Elsevier Inc. All rights reserved.
1. Introduction and statement of the main result We consider the following generalized Landau equation: ∂t F + v · ∇x F = ∇v ·
ψ(v − u) F (u)∇v F (v) − F (v)∇u F (u) du ,
(1.1)
R3
with the initial data F (0, x, v) = F0 (x, v). Here F (t, x, v) 0 is the distribution function for the particles at time t 0, with spatial variable x ∈ R3 and velocity v ∈ R3 . The non-negative matrix ψ is defined as ψ ij (v) =
vi vj |v|γ+2 . δ ij − |v|2
The index γ is a parameter leading to the standard classification of hard potential (γ > 0), the Maxwellian molecule (γ = 0) or soft potential (γ < 0), cf. [9,23]. The original Landau collision operator for the Coulombic interaction corresponds to the case γ = −3. In this paper, we restrict our discussion to the case −3 γ < −2. * Corresponding author. E-mail address: [email protected] (X. Ma). http://dx.doi.org/10.1016/j.jmaa.2014.03.006 0022-247X/© 2014 Elsevier Inc. All rights reserved.
124
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
It is well known that the classical Landau collision operator can be formally derived from the Boltzmann operator when the collision between particles become grazing. As in the Boltzmann equation, we denote a global Maxwellian by M (v) = (2π)−3/2 e−|v|
2
/2
,
with the standard perturbation F (t, x, v) to M as F = M (v) +
√
M f.
Then the Landau equation (1.1) for f (t, x, v) takes the form ∂t f + v · ∇x f + Lf = Γ (f, f ),
f (0, x, v) = f0 (x, v).
(1.2)
The Landau collision frequency is σ
ij
ψ ij (v − u)M (u) du.
= R3
The linearized collision operator L in (1.2) is defined as [12,21] √ √ 1 Lf = − √ Q(M, M f ) + Q( M f, M ) = −Af − Kf, M
(1.3)
where √ 1 vi vj f + ∂i σ ij vj f, Af = √ Q(M, M f ) = ∂i σ ij ∂j f − σ ij 4 M
√ 1 1 1 vj ij 2 2 √ Kf = Q( M f, M ) = −∂i M (v) ψ ∗ M ∂j f + f 2 M
1 vi 1 vj + M 2 (v) ψ ij ∗ M 2 ∂j f + f , 2 2
(1.4)
(1.5)
and the collision operator Γ (f, g) is given by Γ [f, g] = M −1/2 Q M 1/2 f, M 1/2 g
vi 1/2 ij M f ∂j g = ∂i ψ ∗ M f ∂j g − ψ ∗ 2
vi 1/2 M ∂j f g, − ∂i ψ ij ∗ M 1/2 ∂j f g + ψ ij ∗ 2
ij
1/2
(1.6)
where we have used Einstein’s summation for the repeated indices. For notational simplicity, we use ·, · to denote the L2 inner product in Rv3 , with its L2 norm given by | · |2 , and (·, ·) is L2 inner product in Rx3 × Rv3 with corresponding L2 norm · . For s ∈ R, m ∈ Z+ , p 0, we use the standard notation H s or W m,p to denote the usual Sobolev space. Given any k ∈ R3 and function f (x), let k f = f (x + k) − f (x).
(1.7)
We shall also use (·, ·) to denote the L2 inner product in Rx3 × Rv3 × Rk3 with corresponding L2 norm |·|.
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
125
We introduce a weight function as
−(γ+2) w = w(v) = 1 + |v| ,
−3 γ < −2.
For l 0, we denote the weighted L2 -norm 3 l 2 2l ij ij vi vj 2 w g = g dv, w σ ∂i g∂j g + σ σ 4 i,j=1 R3
3 l 2 w g = σ i,j=1
R3 ×R3
vi vj 2 g dx dv. w2l σ ij ∂i g∂j g + σ ij 4
Let α = [α1 , α2 , α3 ], β = [β1 , β2 , β3 ] denote multi-indices with length |α| and |β| respectively, and let ∂βα = ∂xα11 ∂xα22 ∂xα33 ∂vβ11 ∂vβ22 ∂vβ33 . Furthermore, define β β if no component of β is greater than the component of β, and β < β if β β
and |β | < |β|. We also use Cββ to denote the usual binomial coefficient ββ . From now on, we use C to denote a generic positive constant may be different from line to line. Let −3 γ < −2, N 8 and l ∈ R. For any function f (t, x, v), we define the energy functional EN,l (t) as EN,l (t) ∼
l−|β| α 2 w ∂β f ,
|α|+|β|N
where A ∼ B means A B and B A, and A B means that there is a generic constant C > 0 such that A CB. We also define the energy dissipation rate DN,l (t) as DN,l (t) =
∂ α Pf 2 + 1|α|N
2 l−|β| α w ∂β {I − P}f , σ
|α|+|β|N
where P is the usual macroscopic projection, which maps f to the null space of L. Throughout this paper, we shall use the following weighted (with respect to the velocity variable v ∈ R3 ) Sobolev space. For s, l ∈ R, set
Hls = f (t, x, v): wl (v)f ∈ H s Rx3 × Rv3 . Since the regularity property to be proved here is local in time, for convenient, we define the following local version of weighted Sobolev space. For −∞ < T1 < T2 < +∞, we denote by W m,p ((T1 , T2 ); Hls ) the set of all distribution functions on (T1 , T2 ) with values in Hls . L2 ((T1 , T2 ); L2x,v,k ) and L2 ((T1 , T2 ); L2v,k ) are defined similarly. We now summarize the known results about global existence of solution to the Landau equation (1.1) near global Maxwellian in [14] as follows: Theorem 1.1. Let −2 > γ −3 and F0 (x, v) = M (v) + M (v)f0 (x, v) 0 with x ∈ R3 . If the initial energy EN,l (0) is small enough for any l 0, then there exists a unique global solution f (t, x, v) to
126
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
the Cauchy problem (1.2) such that F (t, x, v) = M (v) + N 8,
M (v)f (t, x, v) 0 and for any l 0 and
T sup T ∈[0,+∞)
EN,l (t) +
DN,l (t) dt CEN,l (0).
(1.8)
0
Remark 1.1. It is worthy to pointing out that although the additional weight wl is not considered in [14], one can get (1.8) by the same argument, and Guo [13] obtained the similar results in the case of the spatially periodic. We also emphasize that here l 0 can be arbitrary large. In this paper our main result states that the classical solutions obtained in Theorem 1.1 satisfy the following smoothness. Theorem 1.2. Let −2 > γ −3. There exists a small constant 0 > 0 such that for any l 0 and large enough, E8,l (0) 0 . Then for any 0 < τ∗ < T < +∞ and > 0, the classical solutions to Eq. (1.2) given by Theorem 1.1 satisfy the following regularizing effect: f ∈ W ∞,∞ [τ∗ , T ];
∞
H∞ R3 × R3 .
>0
Remark 1.2. It is shown in [6] that the global classical solution to the Landau equation (1.1) near global Maxwellian in the periodic box obtained by Guo [12] has the smoothing effect with respect to all variables. In the whole space, the solution F (t, x, v) to (1.1) does not satisfy the condition of the main results in [6], because F (t, x, v) does not belong to L2x (R3 ). In fact our main results are similar as ones in [2], where the regularizing effect on the non-cutoff Boltzmann equation was proved by using some polynomial weights. Remark 1.3. In order to obtain the fractional regularity with respect to the space variables, the standard hypo-elliptic estimate for the kinetic equation is employed. In general, to do this, we have to pay the price that high moment bounds are necessary for the solution itself. Therefore, it seems that to obtain the smoothing effect of the solution to the Landau equation, “any l 0 and large enough” will be needed in Theorem 1.2. It will be a very interesting problem if one can obtain the same results for some fixed l > 0. By neglecting the bilinear term Γ (f, f ), one can obtain the linear Landau equation. For the linear Landau equation we have the following smoothness: Corollary 1.1. Let −2 > γ −3. For all l 0, the initial energy E8,l (f0 ) C, Eq. (1.2) without the bilinear term Γ has a unique global solution satisfying E8,l (f ) C. Such a solution satisfies the following regularizing effect (for any 0 < τ∗ < T < ∞ and > 0): f ∈ W ∞,∞ [τ∗ , T ];
∞
H∞ R3 × R3 .
>0
Remark 1.4. It is shown in [1] that the linear Landau equation with γ = 0 (Eq. (1.2) dropping the bilinear term Γ ) has the same regularizing effect as our results. Here our results include the very soft potentials. There have been great investigations about the Landau equation, see [3,5,6,9,10,12,13,15,16,18,19,23,22, 26,27]. Desvillettes and Villani [10] proved global existence, uniqueness and smoothness of classical solutions for spatially homogeneous Landau equation for hard potentials and a large class of initial data. Degond and
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
127
Lemou [9] studied the spectral properties and dispersion relation of linearized Landau operator. Guo [12] constructed global classical solutions near a global Maxwellian for a general Landau equation in a periodic box. Hsiao and Yu extended Guo’s results to the whole space in [14]. The large time behaviors of the Landau equation (even with external forces) were investigated in [17,20,21,24,25]. Recently, a lot of progress has been made on the study of the regularity for the solutions of Landau equations, cf. [1,8,6,10] and references therein, which shows that in some sense the Landau equation can be seen as a non-linear and non-local analog of the hypo-elliptic Fokker–Planck equation [7]. We shall point out that Chen, Desvillettes and He [6] proved that the classical solution to the Landau equation (1.1) in the periodic box obtained by Guo [12] becomes immediately smooth with respect to all variables by some hints of [4,5] and [10]: Roughly speaking, smoothness in the velocity variable is produced by the elliptic property of diffusive matrix to the Landau collision operator [10] and smoothness in the position variable is produced by the classical averaging lemma [4,5,11]. However, regularizing effect of global solution to the Landau equation near global Maxwellian in whole space has been remained open. In this paper, we are concerned with the regularizing effect of the solution obtained in [14,24,25]. In detail, since the solution F (t, x, v) to (1.1) in the whole space is not in L2x (R3 ), we study the regularizing effect of the solution to the linearized Landau equation (1.2) instead of the original Landau equation (1.1). We obtain the smoothness in the velocity variable by using energy method and the dissipative property of the linearized Landau collision operator in [9,12], which is different from [6,10], where smoothness in the velocity variable was obtained by the elliptic property of diffusive matrix to the Landau collision operator. It is known that the linearized operator L and the bilinear operator Γ can be rewritten as the sum of divergence form and some error terms, smoothness in the position variable can be shown by using the classical averaging lemma [4,5,11]. Lastly, we prove the smoothness in time variable as [6] and deduce the smoothness in all variables by the iterative method as [1,2,6,10]. However, there is now a new difficulty, and its resolution is the main originality of this paper: since the smallness condition on the “energy” is required in the proof of the global existence, see Theorem 1.1, but for the proof of our Theorem 1.2, this will lead to the disaster when one tried to improve the regularity by iterative method (see Proposition 3.1). As a consequence, we improved the estimates for the nonlinear operator Γ (see Lemma 2.5) and employed Gronwall’s inequality to fix up this gap, see Remark 3.2. The rest of the paper is outlined as follows. Section 2 is devoted to some basic estimates for the operators L and Γ , note that a crucial estimate on Γ is proved at the end of this section. In Section 3, we establish one step of the main induction argument for the proof of Theorem 1.2. More precisely, we lift the regularity of the variables x and v by four elementary lemmas and a proposition which combines the previous estimates. As a consequence of Section 3, the proof of Theorem 1.2 is given in Section 4. 2. Preliminaries In this section, we give some preliminary lemmas which will be used in the proof of our main theorems. We show the following lemma about fractional order Sobolev space. Lemma 2.1. Let 0 < δ < 1, then f (x) ∈ H δ (Rn ), if and only if f (x) ∈ L2 (Rn ) and k f |k|− n2 −δ 2 L
< +∞,
x,k
where k is defined as (1.7). Then there exists c(n) > 0 such that n f L2x + k f |k|− 2 −δ L2
x,k
2δ = c(n) 1 + |ξ| fˆL2 . ξ
(2.1)
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
128
n Moreover, if we denote fδ,k = k f |k|− 2 −δ and fˆ is the Fourier transform of f , then we have
|fˆδ,k |2 dk = c(n)|ξ|2δ |fˆ|2 .
(2.2)
Rn
Proof. We only prove (2.2), since (2.1) is well-known to us. To do this we directly compute that
|fˆδ,k |2 dk = Rn
Rn
|e2πiξ·k − 1|2 ˆ 2 dk f (ξ) . |k|n+2δ
Then by applying the rotation invariance, one can get that Rn
|e2πiξ·k − 1|2 dk = c(n)|ξ|2δ , |k|n+2δ
which implies that (2.2) is true. Thus Lemma 2.1 is proved. 2 In what follows, we summarize some basic estimates for the operators L and Γ which were shown in [12,13,20,21]. Lemma 2.2. Let γ −3, l ∈ R, then there exists C > 0 such that 2 γ+2 γ+2 2 C wl 1 + |v| 2 ∇v g 2 + wl 1 + |v| 2 g 2 2 2 γ γ+2 2 wl g σ C wl 1 + |v| 2 ∇v g 2 + wl 1 + |v| 2 g 2 . Lemma 2.3. Let γ ∈ [−3, −2) and |β| > 0. For any l 0 and η > 0, there exists Cη > 0 such that 2 wl ∂β g 2 − C M 1/8 g 2 . w2l ∂β [Lg], ∂β g wl ∂β g σ − η 1 σ 2
(2.3)
|β1 ||β|
Moreover, for any l > 0, we also have
2 2 w2l Lg, g wl g σ − C M 1/8 g 2 .
Lemma 2.4. Let γ ∈ [−3, −2) and |α| + |β| N . For any l 0, we have M 1/8 ∂ α1 f wl ∂ α−α1 g w2l ∂βα Γ (f, g), ∂βα h C β1 β2 2 σ 1/8 α l α−α l α 1 1 + M ∂β1 f σ w ∂β2 g 2 w ∂β hσ ,
where the summation is over α2 + α1 = α, β1 + β2 β. Proof. The proof of (2.4) is similar as that of Theorem 3 in [12], we omit the details for brevity. 2 Lastly, making use of Lemma 2.4, we have Lemma 2.5. Let γ ∈ [−3, −2), l 0 and N 8, then for any η > 0, it holds that
(2.4)
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
129
2l α w ∂β Γ (f, g), ∂βα h
|α|+|β|N
C
l α l α w ∂β f + w ∂β g + η
|α|+|β|7
+C
|α|+|β|7
+ Cη
|α|+|β|N
l α 2 w ∂β h
σ
l α 2 w ∂β g + C σ
|α|+|β|N
l α 2 w ∂β f
|α|+|β|N
l α w ∂β f
|α|+|β|7
l α 2 w ∂β g .
l α w ∂β g
l α 2 w ∂β f
σ
|α|+|β|N
(2.5)
|α|+|β|N
Proof. Invoking (2.4), to prove (2.5), it suffices to estimate
1/8 α l α−α l α 1 M ∂ 1 f w ∂ g σ w ∂β hσ dx β1 β2 2
R3
+
J1
1/8 α l α−α l α 1 M ∂ 1 f w ∂ g 2 w ∂β hσ dx . β1 β2 σ
R3
J2
Our goal therefore is now to compute J1 and J2 , for this, we divide our calculations into following two cases. Case 1. |α1 | + |β1 | 5. For J1 , Sobolev’s imbedding theorem and Hölder’s inequality imply that 1 l α J1 C M 1/8 ∂βα11 f 2 H 2 wl ∂βα−α g σ w ∂β h σ 2 x l α 2 l α 2 w ∂β g + w ∂β h . wl ∂βα f C σ σ |α|+|β|7
(2.6)
|α|+|β|N
Applying Cauchy–Schwartz’s inequality with η and Sobolev’s inequality, one can see that 2 J2 η wl ∂βα hσ + Cη 2 η wl ∂βα hσ + Cη
1/8 α 2 l α−α 2 1 M ∂ 1 f w ∂ g dx β1
R3
β2
σ
2
l α 2 l α 2 w ∂β f w ∂β g .
(2.7)
|α|+|β|N
Case 2. |α1 | + |β1 | > 5. Note first from Lemma 2.2 that 2 l α−α 2 1 1 w ∂ g σ C wl−1/2 ∂βα−α g 2+ β2 2
l−1/2 α−α 2 1 w . ∂β g 2
|β2 |=|β2 |+1
From which and the fact that |α − α1 | + |β2 | < N − 5, and performing the similar calculations as (2.7), we see that 2 2 2 1 J1 η wl ∂βα hσ + Cη M 1/8 ∂βα11 f 2 wl ∂βα−α g σ dx 2 2 η wl ∂βα hσ + Cη
R3
|α|+|β|N
l α 2 w ∂β f
l−1/2 α 2 w ∂β g .
|α |+|β |N −2
(2.8)
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
130
It remains to estimate J2 for |α1 | + |β1 | > 5, if |α1 | + |β1 | = |α| + |β| = N , then |α − α1 | + |β2 | = 0, we get from Sobolev’s inequality and Lemma 2.2 that J2 C wl g 2 H 2 M 1/8 ∂βα11 f σ wl ∂βα hσ x l α 2 l α 2 wl ∂ α g w ∂β f + w ∂β h . C σ σ |α|2
(2.9)
|α|+|β|N
If 5 < |α1 | + |β1 | N − 1, following the same way as (2.8), one can see that
2 J2 η wl ∂βα hσ + Cη 2 η wl ∂βα hσ + Cη
1/8 α 2 l α−α 2 1 M ∂ 1 f w ∂ g dx β1
R3
β2
σ
l α 2 w ∂β f
|α|+|β|N
2
l α 2 w ∂β g .
(2.10)
|α|+|β|N −3
Therefore (2.5) follows from (2.6), (2.7), (2.8), (2.9) and (2.10). This completes the proof of Lemma 2.5. 2 3. Gain of regularity in v and x Since the proof of Theorem 1.2 is based on an induction on the number of derivatives (in x and v) that can be controlled (see Section 4). In this section, we will show how to get one step of this induction. To make our presentation more clear, we shall divide our discussion into following two parts. The first one is devoted to the study of the smoothness of f (t, x, v) with respect to v. 3.1. Gain of regularity in v In this subsection, we intend to prove the regularity of velocity with the aid of energy method. Lemma 3.1. Let −2 > γ −3, N 8 be a given integer, and f be a smooth solution of Eq. (1.2) given by Theorem 1.1. Suppose that for any +∞ > T > τ1 > τ0 > 0, and any l1 N and sufficiently large,
sup t∈[τ0 ,T ]
2 l1 +1−N + γ w 2(γ+2) ∂ α f + β
|α|+|β|N −1
T
l +1−N α 2 w 1 ∂β f dt C0 ,
(3.1)
|α|+|β|N τ0
and
l +1−N α 2 w 1 ∂β f (τ1 ) C0 ,
(3.2)
|α|+|β|N
where C0 = C0 ( 0 , T, γ, N, l1 ) is a constant. Then there exists a constant C˜1 > 0, which depends on N , l1 , γ, T , 0 , such that for any +∞ > T > τ1 > 0, and all > 0 and large enough, sup t∈[τ1 ,T ]
|α|+|β|N
+ γ w 2(γ+2) ∂βα f 2 +
w ∇v ∂βα f 2 dx dv dt C˜1 .
(3.3)
|α|+|β|N [τ ,T ]×R3 ×R3 1
Proof. Taking ∂βα to Eq. (1.2) with |α| + |β| N and taking the inner product with w2l0 ∂βα f (l0 0) over Rx3 × Rv3 , we obtain
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
131
[∂t + v · ∇x ]∂βα f, w2l0 ∂βα f + ∂βα [Lf ], w2l0 ∂βα f β
α = ∂βα Γ (f, f ), w2l0 ∂βα f − f, w2l0 ∂βα f . Cβ 1 ∂β1 v · ∇x ∂β−β 1
(3.4)
|β1 |=1
We will estimate each term of (3.4). From (2.3) in Lemma 2.3, one has 2
2l0 α wl0 ∂βα f 2 − C w−1/2 ∂ α f 2 . w ∂β [Lf ], ∂βα f wl0 ∂βα f σ − η 1 σ
(3.5)
|β1 ||β|
In light of (2.5) and recalling
√ E8,l0 (t) C 0 , we see that
α √ ∂β Γ (f, f ), w2l0 ∂βα f C( 0 + η)
l α 2 w 0 ∂β f + C σ
|α|+|β|N
l α 4 w 0 ∂β f .
(3.6)
|α|+|β|N
And for β > 0, it is obvious that 2 2
β1 α α Cβ ∂β1 v · ∇x ∂β−β f, w2l0 ∂βα f C wl0 ∂βα f σ + C wl0 ∇x ∂β−β f , 1 1
(3.7)
where |β − β1 | = |β| − 1. Letting η > 0 small enough, an appropriate combination of (3.4), (3.5), (3.6) and (3.7) yields d wl0 ∂βα f 2 + wl0 ∂βα f 2 C σ dt
l α 2 w 0 ∂ 1 f + η wl0 ∂βα f 2 β1 1 σ
|α1 |+|β1 |N
+C
l α 4 w 0 ∂β f .
|β1 ||β|
(3.8)
|α|+|β|N
For |α| + |β| N and any l0 0, a suitable summation of (3.8) over 0 |β| N gives |α|+|β|N
d wl0 ∂βα f 2 + wl0 ∂βα 2 C σ dt
l α 2 w 0 ∂β f + C
|α|+|β|N
l α 4 w 0 ∂β f .
(3.9)
|α|+|β|N
On the other hand, utilizing (3.1) with l0 = l1 + 1 − N , we see that T
l α 2 w 0 ∂β f dt C0 .
|α|+|β|N τ0
With this, we get from (3.9), (3.2) and Gronwall’s inequality that e
−
T
|α|+|β|N
τ1
α wl0 ∂β f 2 dt
l α w 0 ∂β f (T )2
|α|+|β|N
T +
e
−
|α|+|β|N
t τ1
α wl0 ∂β f (τ )2 dτ
T e τ1
+
l α w 0 ∂β f (t)2 dt σ
|α|+|β|N
τ1
C
−
|α|+|β|N
|α|+|β|N
t τ1
α wl0 ∂β f (τ )2 dτ
l α w 0 ∂β f (t)2 dt
|α|+|β|N
l α w 0 ∂β f (τ1 )2 .
(3.10)
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
132
Then (3.10) and Lemma 2.2 imply that
sup t∈[τ1 ,T ]
l α 2 w 0 ∂β f +
|α|+|β|N
T
1 + |v| γ/2 wl0 ∇v ∂βα f (t)2 dt CeC0 .
(3.11)
|α|+|β|N τ1
γ With (3.11) in hand, by setting = l0 − 2(γ+2) = l1 + 1 − N − This concludes the proof of Lemma 3.1. 2
γ (γ+2)
0, one can see that (3.3) holds true.
Remark 3.1. The constant C˜1 in Lemma 3.1 depends on T , which yields that the high regularity property obtained in the following results is in the pointwise sense with respect to time t. This explains why our main result in Theorem 1.2 is restricted to local time [τ∗ , T ]. Remark 3.2. We have succeeded in applying fairly straightforward “energy” method estimates to produce the smoothness of the variable v, however unlike the usual energy method developed by Guo and Strain in [12,13,21,20], thanks to Lemma 2.5 and Gronwall’s inequality, here we only need the smallness of E8,l (t) with sufficiently large l but EN,l (t) with N 8, which means that neither C0 or C˜1 in Lemma 3.1 is required to be small. This observation pays the way for the iterative technique utilized later. Remark 3.3. Noticing that the assumption (3.1) is coincide with (1.8) while T is finite, in particular, this shows that the result of Theorem 1.2 is not empty. 3.2. Gain of regularity in x We now turn to prove the regularity for position variable x. We can not hope to get it directly by an energy estimate like we did for variable v since no diffusion term in x is available. Motivated by the work [6] and the case near vacuum about the Boltzmann equation [2], we will resort to the averaging lemma. Lemma 3.2. Let −2 > γ −3, N 8 be a given integer, and f be a smooth solution of Eq. (1.2) given by Theorem 1.1. Suppose that for any +∞ > T > τ1 > 0, and any 0 and sufficiently large,
sup t∈[τ1 ,T ]
α 2 w ∂β f C˜1 ,
|α|+|β|N
for some constant C˜1 = C˜1 ( , T, γ, N, C0 ). Then, for any > 0 and large enough, any 0 < τ1 < T < +∞ and |α| + |β| N , there exists a constant C˜2 > 0, which depends on N , , γ, τ1 , T and C˜1 , such that
α 2 w ∂β f
1/20
Hx
dv dt C˜2 .
(3.12)
[τ1 ,T ]×R3
Proof. Let p(t, x, v) = w ∂βα f (1 + |v|2 )2 . To prove (3.12), we first need to prove
−4 1 1 + |v|2 1 + |ξ| 10 |ˆ p(t, ξ, v)|2 dξ dv dt C˜2 ,
(3.13)
[τ1 ,T ]×R6
where pˆ(t, ξ, v) is the Fourier transform of p(t, x, v) with respect to variable x. Noticing that (3.13) is obviously true for |ξ| 1 according to the assumption, therefore we focus on the case |ξ| 1. For this, let
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
χ := χ(v) ∈ Cc∞ (R3 ) be a test function which satisfies χ(v) 0 and write
R3
133
χ(v) dv = 1. Set χ = −3 χ( v ) and
pˆ(t, ξ, v) = pˆ(t, ξ, v) − pˆ(t, ξ, ·) ∗ χ (v) + pˆ(t, ξ, ·) ∗ χ (v). Here, will be chosen later (and will depend on ξ). We get from Minkowski’s inequality that R3
−4 pˆ(t, ξ, v) − pˆ(t, ξ, ·) ∗ χ (v)2 dv 1 + |v|2 2 pˆ(t, ξ, v) − pˆ(t, ξ, v − u) χ (u) du dv R3 R3
R3
pˆ(t, ξ, v) − pˆ(t, ξ, v − u)2 dv
R3
∇v pˆ(t, ξ, v)2 dv
R3
C
2
12
2 χ (u) du
2 χ (u)|u| du
R3
∇v pˆ(t, ξ, v)2 dv,
R3
so that
−4 1 2
1 + |v|2 1 + |ξ| 10 pˆ(t, ξ, v) − pˆ(t, ξ, ·) ∗ χ (v) dv dξ dt
[τ1 ,T ]×R6
2
1 2 1 + |ξ| 10 ∇v pˆ(t, ξ, v) dv dξ dt.
C [τ1
(3.14)
,T ]×R6
Remembering that p(t, x, v) = w ∂βα f (1 + |v|2 )2 , we see that p satisfies the following equation (using Einstein’s summation for the repeated indices): ∂t p + v · ∇x p = (I) + (II ) + (III ) + (IV ),
(3.15)
where (I) =
0,
β1 |β1 |=1 Cβ w (1
for β = 0,
− + |v| ) ∂β1 v · for |β| > 0, ⎧ ∂i [w (1 + |v|2 )2 ∂βα (σ ij ∂j f )] ⎪ ⎪ ⎪ ⎪ ⎨ − ∂i [w (1 + |v|2 )2 ∂ α (M 1/2 [ψ ij ∗ (M 1/2 (∂j f + vj f ))])] β 2 (II ) = ⎪ + ∂i [w (1 + |v|2 )2 ∂βα ([ψ ij ∗ (M 1/2 f )]∂j f )] ⎪ ⎪ ⎪ ⎩ − ∂i [w (1 + |v|2 )2 ∂βα ([ψ ij ∗ (M 1/2 ∂j f )]f )], for |β| 0,
2
2 2 vi vj f − ∂i w 1 + |v|2 ∂βα σ ij ∂j f , (III ) = w 1 + |v|2 ∂βα ∂i σ ij vj f − w 1 + |v|2 ∂βα ∂i σ ij 4
2 vj (IV ) = ∂i w 1 + |v|2 ∂βα M 1/2 ψ ij ∗ M 1/2 ∂j f + f 2 2 2
α ∇x ∂β−β f, 1
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
134
vi 1/2 ij vj 1/2 M , ψ ∗ M ∂j f + f +w 1+ 2 2 2
(V ) = −∂i w 1 + |v|2 ∂βα ψ ij ∗ M 1/2 f ∂j f
2 + ∂i w 1 + |v|2 ∂βα ψ ij ∗ M 1/2 ∂j f f
2 vi 1/2 M f ∂j f − w 1 + |v|2 ∂βα ψ ij ∗ 2
vi 1/2 2 2 α ij M ∂j f f . + w 1 + |v| ∂β ψ ∗ 2
2 |v|2 ∂βα
The terms in (II )–(V ) come from (1.4), (1.5) and (1.6) after taking ∂βα derivatives on −Lf and Γ (f, f ). We use (I)i , (II )i , (III )i , (IV )i , (V )i (i = 1, 2, . . . , ) to denote the corresponding terms in (I), (II ), (III ), (IV ), (V ) for simplicity of our presentation. In fact, we can write the equation satisfied by p under the form ∂t p + v · ∇ x p = p 1 + ∇ v · p 2 . Here, p1 is the sum of the terms (I), (III ), (IV ) and (V ), while ∇v · p2 is the sum of the terms in (II ). We claim that p1 , p2 ∈ L2 ([τ1 , T ]; L2x,v ). We only present here the estimates for the terms in (II ), while the other terms can be shown similarly. For the term (II )1 , since |σ ij | C(1 + |v|)γ+2 , we have from (3.3) that
w 1 + |v|2 2 ∂βα σ ij ∂j f
L2 ([τ1 ,T ];L2x,v )
α w 1 + |v| γ+6 ∂j ∂β−β f L2 ([τ 1
C
|β1 ||β|
γ+6 α w− γ+2 ∂j ∂β−β f L2 ([τ 1
C
|β1 ||β|
2 1 ,T ];Lx,v )
2 1 ,T ];Lx,v )
C
# C˜1 .
As to the term (II )2 , applying Lemma 3.1 again, we obtain
w 1 + |v|2 2 ∂βα M 1/2 ψ ij ∗ M 1/2 ∂j f + vj f 2 2 L ([τ1 ,T ];L2
x,v )
C
1 M 8 ∂j ∂βα f 2 2 L ([τ
β2 β
2 1 ,T ];Lx,v
1 + M 8 ∂βα2 f L2 ([τ )
2 1 ,T ];Lx,v )
# C( C˜1 + C0 ),
where we have used the basic estimate for the convolution of ψ ij : 12 12 1 ij
ψ ij (v − u)M 18 (u)2 du M 8 ∂j ∂ α2 f (u)2 du ψ ∗ ∂βα M 12 ∂j f β2 1 β2 β1
β2 β1
R3
R3
γ+2 1 C 1 + |v|2 2 M 8 ∂j ∂βα2 f 2 .
For the term (II )3 , we first see that
∂βα ψ ij ∗ M 1/2 f ∂j f =
α1 α,β1 β
α−α1 Cαα1 Cββ1 ψ ij ∗ ∂βα11 M 1/2 f ∂j ∂β−β f. 1
(3.16)
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
Then if |α1 | + |β1 |
N 2,
135
we have from (3.16) and Lemma 3.1 that
w 1 + |v|2 2 ψ ij ∗ ∂ α1 M 1/2 f ∂j ∂ α−α1 f 2 β1 β−β1 L ([τ1 ,T ];L2x,v ) w 1 + |v| γ+6 ∂j ∂ α−α1 f M 18 ∂ α1 f 2 C β−β1 β2 2 2 L ([τ ,T ];L2 ) 1
β2 β1
C
1 M 8 ∂ α1 +α f
L∞ ([τ1 ,T ];L2x,v )
β2
β2 β1 |α |2
Similarly if |α1 | + |β1 |
N 2,
x
− γ+6 w γ+2 ∂j ∂ α−α1 f 2 β−β1 L ([τ
2 1 ,T ];Lx,v )
C C˜1 .
by (3.16) and Lemma 3.1, it follows that
1 + |v|2 2 w ψ ij ∗ ∂ α1 M 1/2 f ∂j ∂ α−α1 f 2 β1 β−β1 L ([τ1 ,T ];L2x,v ) w 1 + |v| γ+6 ∂j ∂ α−α1 f M 18 ∂ α1 f 2 C β−β1 β 2 2 L ([τ ,T ];L2 ) 2 1
β2 β1
C
γ+6 w− γ+2 ∂j ∂ α−α1 +α f ∞ β−β1 L ([τ
β2 β1 |α |2
2 1 ,T ];Lx,v )
x
α w ∂ 1 f β2
L2 ([τ1 ,T1 ];L2x,v )
C
# C˜1 C0 .
The term (II )4 can be treated as the term (II )3 . Finally, we conclude that p1 , p2 ∈ L2 ([τ1 , T ]; L2x,v ). Now, employing the variant average lemma in [4], we can prove that
|ξ|
1 2
T
pˆ(t, ξ, v) ∗ χ (v)2 dt
τ1
2 C χ (v − u) 1 + |u|2 L∞ + ∇v χ (v − u) 1 + |u|2 L∞ u u 2 2
× pˆ(τ1 , ξ, v)L2 + pˆ(t, ξ, v)L2 ([τ ,T ];L2 ) 1 v v 2 2 pˆ2 (t, ξ, v) 2 + pˆ1 (t, ξ, v) 2 , 2 + 2 L ([τ1 ,T ];Lv )
L ([τ1 ,T ];Lv )
Since χ (v − u)(1 + |u|2 )L∞ C −3 (1 + |v|2 ), ∇v χ (v − u)(1 + |u|2 )L∞ C −4 (1 + |v|2 ), we see that u u
1 + |v|2
−4
2 1 |ξ| 10 pˆ(t, ξ, v) ∗ χ (v) dξ dv dt
[τ1 ,T ]×R6
C
2 2 1 1 |ξ| 10 − 2 −6 + −8 pˆ(τ1 , ξ, v)L2 + pˆ(t, ξ, v)L2 ([τ v
R3
2 + pˆ1 (t, ξ, v)L2 ([τ
2 1 ,T ];Lv )
2 + pˆ2 (t, ξ, v)L2 ([τ
2 1 ,T ];Lv )
dξ.
2 1 ,T ];Lv )
(3.17)
If we choose = (|ξ|) = |ξ|− 20 (|ξ| 1), we can bound (3.17) remembering that (1 + |v|2 )−4 ∈ L1v , p(τ1 , x, v) ∈ L2x,v and p, p1 , p2 ∈ L2 ([τ1 , T ]; L2x,v ), which implies that (3.12) holds true. Thus the proof of Lemma 3.2 is completed. 2 1
Roughly speaking, Lemma 3.2 together with Lemma 3.1 shows that when f is a solution of Eq. (1.2) conN +1/20 ). To improve the regularity about position variable x, structed in Theorem 1.1, then f ∈ L2 ([τ1 , T ]; H we will first show a preliminary lemma.
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
136
1 Lemma 3.3. Let −2 > γ −3, N 8 be a given integer, δ = 20 and f be a smooth solution of Eq. (1.2) given by Theorem 1.1. Suppose that for any +∞ > T > τ2 > τ1 > 0, and any 0 large enough,
|α|+|β|N
α w ∂β f (τ2 )2 1/20 C˜2 , Hx
α 2 w ∂β f
1/20
Hx
|α|+|β|N [τ ,T ]×R3 1
dv dt C˜2 .
(3.18)
Then there exists a constant C˜3 > 0, which depends on N , , γ, T and C˜2 , such that for any > 0 large enough and any 0 < τ2 < T < +∞
+ γ w 2(γ+2) k ∂βα f |k|−δ− 32 2 ∞ L
|α|+|β|N
+
([τ2 ,T ];L2x,v,k )
w ∇v k ∂βα f |k|−δ− 32 2 2
L ([τ2 ,T ];L2x,v,k )
|α|+|β|N
C˜3 .
Proof. Taking |k|−δ− 2 k on Eq. (1.2), we have (denoting g˜ = k f |k|−δ− 2 ) 3
3
∂t g˜ + v · ∇x g˜ + L˜ g = Γ g˜, f (x + k) + Γ (f, g˜),
(3.19)
where we have used the fact that for any functions f1 and f2 , k f1 (x)f2 (x) = k f1 (x)f2 (x + k) + f1 (x) k f2 (x). Now we perform energy estimates for Eq. (3.19). Taking ∂βα (|α| + |β| N ) to Eq. (3.19), and taking the inner product with w2 ∂βα g˜ over Rx3 × Rv3 × Rk3 , one has
1 d w ∂βα g˜2 + ∂βα L˜ g , w2 ∂βα g˜ = ∂βα Γ g˜, f (x + k) , w2 ∂βα g˜ + ∂βα Γ (f, g˜), w2 ∂βα g˜ 2 dt β α Cβ 1 ∂β1 v · ∇x ∂β−β g˜, w2 ∂βα g˜ . − 1
(3.20)
|β1 |=1
From (2.3), we can obtain
2 w ∂βα g˜2 − C ∂ α g˜2 . ∂βα L˜ g , w2 ∂βα g˜ w ∂βα g˜σ − η 1 σ
(3.21)
|β1 ||β|
Now we consider the first and the second term of the right-hand side in (3.20). For this, we only compute the first term, since estimates for the second one being similar. It follows from (2.4) that
α ∂β Γ g˜, f (x + k) , w2 ∂βα g˜ M 1/8 ∂ α1 g˜ w ∂ α2 f (x + k) C β1 β2 2 σ R6
+ w ∂βα22 f (x + k)2 M 1/8 ∂βα11 g˜σ w ∂βα g˜σ dx dk,
(3.22)
where the summation is over α2 + α1 = α, β1 + β2 β. We only present here the estimates for the first part of the right-hand side of (3.22), because the second one can be treated similarly. Using the idea explained in the proof of (2.5), we carry out the following computations.
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
137
If |α1 | + |β1 | 5, by Hölder’s inequality and Sobolev’s inequality, we find
1/8 α α M ∂ 1 g˜ w ∂ 2 f (x + k) w ∂βα g˜ dx dk β1 β2 2 σ σ
R6
C
supM 1/8 ∂βα11 g˜2 w ∂βα g˜σ dk w ∂βα22 f σ x
R3
C
M 1/8 ∂ α1 +α g˜w ∂βα g˜ w ∂ α2 f , β1 β2 σ σ
|α |2
where we have used the transition invariance property of the Lebesgue integral. If |α1 | + |β1 | 5, since |α2 | + |β2 | < N − 5, owing to Cauchy–Schwartz’s inequality with η and Sobolev’s inequality, we obtain
1/8 α α M ∂ 1 g˜ w ∂ 2 f (x + k) w ∂βα g˜ dx dk β1 β2 2 σ σ
R6
2 2 η w ∂βα g˜σ + Cη supw ∂βα22 f (x + k)σ
1/8 α 2 M ∂ 1 g˜ dx dk β1
x
2 η w ∂βα g˜σ + Cη
R3
−1/2 α 2 1/8 α 2 w ∂ 2 f M ∂ 1 g˜ . β2
|α |+|β2 |N −2
β1
We thus have a uniform estimate as:
∂βα Γ g˜, f (x + k) , w2 ∂βα g˜ + ∂βα Γ (f, g˜), w2 ∂βα g˜ 2 √ w ∂βα g˜4 . C(η + 0 )w ∂βα g˜σ + Cη
(3.23)
|α|+|β|N |α|+|β|N
And it is straightforward to check that 2 2
α α g˜, w2 ∂βα g˜ C w ∂βα g˜ + C w ∇x ∂β−β g˜ , ∂β1 v · ∇x ∂β−β 1 1
(3.24)
where |β − β1 | = |β| − 1. By choosing η > 0 small enough and combing (3.20), (3.21), (3.23), (3.24) and the assumption (3.18), we obtain d w ∂βα g˜2 + C w ∂βα g˜4 . w ∂βα g˜2 + w ∂βα g˜2 C σ dt |α|+|β|N
|α|+|β|N
|α|+|β|N
Recalling Lemmas 2.2, 3.1 and 3.2 and applying Gronwall’s inequality, we have that
+ γ w 2(γ+2) ∂βα g˜(t)2 +
t∈[τ2 ,T ] |α|+|β|N
T
w ∇v ∂βα g˜2 dt CeC˜2 .
|α|+|β|N τ2
This concludes the proof of Lemma 3.3. 2 In what follows we will lift the regularity of position variable x again, for results in this direction, we have
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
138
Lemma 3.4. Let −2 > γ −3, N 8 be a given integer, and f be a smooth solution of Eq. (1.2) given by Theorem 1.1. Suppose that for any +∞ > T > τ2 > τ1 > 0, and any 0 and large enough,
α w ∂β f (τ2 )2
1/20
Hx,v
|α|+|β|N
C˜3 .
Then there exists a constant C˜4 > 0, which depends on N , , γ, T , C˜3 , such that for any 0 and large enough,
α 2 w ∂β f
1/10
Hx
|α|+|β|N [τ ,T ]×R3 2
dv dt C˜4 .
Proof. In view of (2.2) in Lemma 2.1, we know that when |ξ| 1 and δ = (3.25)
⇔
(3.25)
1 20 ,
2 α ˜ |ξ|2δ+1/10 w ∂$ β f (ξ) dξ dv dt C4
[τ2 ,T ]×R6
⇔ [τ2
−δ− 32 2 α |ξ|1/10 w k ∂$ dv dk dξ dt C˜4 . β f |k|
(3.26)
,T ]×R9
In order to prove estimate (3.26), by analogy with approach taken in the proof of Lemma 3.2, we define 3 pδ,k = w (1 + |v|2 )2 k ∂βα f |k|−δ− 2 , and write
pˆδ,k (t, ξ, v) = pˆδ,k (t, ξ, v) − pˆδ,k (t, ξ, ·) ∗ χ (v) + pˆδ,k (t, ξ, ·) ∗ χ (v), the parameter being chosen later. Following the proof of the estimates for (3.14), we get
[τ2 ,T ]×R6
1 + |v|2
−4 1 2 1 + |ξ| 10 pˆδ,k (t, ξ, v) − pˆδ,k (t, ξ, ·) ∗ χ (v) dv dξ dt
2
1 2 1 + |ξ| 10 ∇v pˆδ,k (t, ξ, v) dv dξ dt.
C
(3.27)
[τ2 ,T ]×R6
Noticing that g˜ = k f |k|−δ− 2 , we see that pδ,k also satisfies the following equation 3
∂t pδ,k + v · ∇x pδ,k = (VI ) + (VII ) + (VIII ) + (IX ) + (X), where the right-hand side is ⎧ ⎨ 0, for β = 0, (VI ) = − C β1 w (1 + |v|2 )2 ∂β v · ∇x ∂ α f |k|−δ− 32 , for |β| > 0, 1 β−β1 β ⎩ |β1 |=1
(VII ) =
⎧ ⎪ ∂i [w (1 + |v|2 )2 ∂βα (σ ij ∂j g˜)] ⎪ ⎪ ⎪ v ⎪ ⎪ − ∂i [w (1 + |v|2 )2 ∂βα (M 1/2 [ψ ij ∗ (M 1/2 (∂j g˜ + 2j g˜))])] ⎪ ⎪ ⎪ ⎪ ⎨ + ∂i [w (1 + |v|2 )2 ∂ α ([ψ ij ∗ (M 1/2 g˜)]∂j f (x + k))] β
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
+ ∂i [w (1 + |v|2 )2 ∂βα ([ψ ij ∗ (M 1/2 f )]∂j g˜)] − ∂i [w (1 + |v|2 )2 ∂βα ([ψ ij ∗ (M 1/2 ∂j g˜)]f (x + k))] g )], − ∂i [w (1 + |v|2 )2 ∂βα ([ψ ij ∗ (M 1/2 ∂j f )]˜
for |β| 0,
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
(VIII ) = w 1 +
2 |v|2 ∂βα ∂i σ ij vj g˜
−w 1+
2 |v|2 ∂βα
∂i σ
ij vi vj
139
g˜
4 α ij 2 2 ∂β σ ∂j g˜ , − ∂i w 1 + |v|
α vj 2 2 1/2 ij 1/2 ψ ∗ M ∂j g˜ + g˜ ∂β M (IX ) = ∂i w 1 + |v| 2
2 vi 1/2 ij vi M ψ ∗ M 1/2 ∂j g˜ + g˜ , + w 1 + |v|2 ∂βα 2 2 2
(X) = −∂i w 1 + |v|2 ∂βα ψ ij ∗ M 1/2 g˜ ∂j f (x + k) 2
+ ∂i w 1 + |v|2 ∂βα ψ ij ∗ M 1/2 ∂j g˜ f (x + k) 2
− ∂i w 1 + |v|2 ∂βα ψ ij ∗ M 1/2 f ∂j g˜
2 + ∂i w 1 + |v|2 ∂βα ψ ij ∗ M 1/2 ∂j f g˜
vi 1/2 2 2 α ij − w 1 + |v| ∂β ψ ∗ M g˜ ∂j f (x + k) 2
2 vi 1/2 M ∂j g˜ f (x + k) + w 1 + |v|2 ∂βα ψ ij ∗ 2
vi 1/2 2 2 α ij M f ∂j g˜ − w 1 + |v| ∂β ψ ∗ 2
2 vi 1/2 M ∂j f g˜ . + w 1 + |v|2 ∂βα ψ ij ∗ 2
The terms in (VI )–(X ) can be obtained by using the operator k and then timing |k|−δ− 2 from (3.15). We use (VI )i , (VII )i , (VIII )i , (IX )i , (X)i (i = 1, 2, . . . , ) to denote the corresponding terms in (VI )–(V ) for simplicity of our presentation. In fact, we can write the equation satisfied by pδ,k under the form 3
(1)
(2)
∂t pδ,k + v · ∇x pδ,k = pδ,k + ∇ · pδ,k , (2)
(1)
where pδ,k consists in the sum of terms (VII ), while pδ,k is the sum of other terms. (1)
(2)
We claim that pδ,k , pδ,k ∈ L2 ([τ2 , T ]; L2x,v,k ). We only present here the estimates for the terms (VII )1 , (VII )3 and (VII )6 while the other terms can be estimated similarly. For the term (VII )1 , we have from Lemma 3.3 that,
w 1 + |v|2 2 ∂βα σ ij ∂j g˜ 2 L ([τ
2 2 ,T ];Lx,v,k )
C
α w 1 + |v| γ+6 ∂j ∂β−β g˜L2 ([τ 1 |β1 ||β|
2 2 ,T ];Lx,v,k )
# C C˜3 . Now we turn to estimate (VII )3 , we first write
∂βα ψ ij ∗ M 1/2 g˜ ∂j f (x + k) =
α−α1 Cαα1 Cββ1 ψ ij ∗ ∂βα11 M 1/2 g˜ ∂j ∂β−β f (x + k). 1
α1 α,β1 β
If |α1 | + |β1 | N/2, we compute
w 1 + |v|2 2 ψ ij ∗ ∂ α1 M 1/2 g˜ ∂j ∂ α−α1 f (x + k) 2 β1 β−β1 L ([τ
2 2 ,T ];Lx,v,k )
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
140
C
M 1/8 ∂ α0 +α1 g˜
L∞ ([τ2 ,T ];L2x,v,k )
β0
β0 β1 |α0 |2
C
w 1 + |v| γ+6 ∂j ∂ α−α1 f β−β1
L2 ([τ2 ,T ];L2x,v )
# C˜1 C˜3 .
Here we have used (3.16), Lemmas 3.1 and 3.3. Likewise, for |α1 | + |β1 | N/2, it follows that
w 1 + |v|2 2 ψ ij ∗ ∂ α1 M 1/2 g˜ ∂j ∂ α−α1 f (x + k) 2 β1 β−β1 L ([τ2 ,T ];L2x,v,k )
M 1/8 ∂ α1 g˜ ∞ w 1 + |v| γ+6 ∂j ∂ α0 +α−α1 f 2 C β0 β−β1 L ([τ ,T ];L2 ) L ([τ 2
β0 β1 |α0 |2
C
x,v,k
2 2 ,T ];Lx,v )
# C˜1 C˜3 .
Therefore, in any case the term (VII )3 can be bounded by C As to the term (VII )6 , noticing that
∂βα ψ ij ∗ M 1/2 ∂j f g˜ =
C˜1 C˜3 .
α−α1 Cαα1 Cββ1 ψ ij ∗ ∂βα11 M 1/2 ∂j f ∂β−β g˜. 1
α1 α,β1 β
If |α1 | + |β1 | N/2, we know that
w 1 + |v|2 2 ψ ij ∗ ∂ α1 M 1/2 ∂j f ∂ α−α1 g˜ 2 β1 β−β1 L ([τ2 ,T ];L2x,v,k ) M 1/8 ∂j ∂ α0 +α1 f ∞ w 1 + |v| γ+6 ∂ α−α1 g˜ 2 C β0 β−β1 L ([τ ,T ];L2 ) L ([τ 2
β0 β1 |α0 |2
x,v
2 2 ,T ];Lx,v,k )
# C C˜1 C˜3 .
Here we have also used (3.16), Lemmas 3.1 and 3.3. If |α1 | + |β1 | N/2, we deduce from (3.16), Lemmas 3.1 and 3.3 that
w 1 + |v|2 2 ψ ij ∗ ∂ α1 M 1/2 ∂j f ∂ α−α1 g˜ 2 β1 β−β1 L ([τ2 ,T ];L2x,v,k ) M 1/8 ∂j ∂ α1 f 2 w 1 + |v| γ+6 ∂ α0 +α−α1 g˜ ∞ C β0 β−β1 L ([τ ,T ];L2 ) L ([τ β0 β1 |α0 |2
2
x,v
2 2 ,T ];Lx,v,k )
# C C˜1 C˜3 .
We thus conclude that the term (VII )6 can be also bounded by C Proceeding like the estimates for (3.17), we discover
C˜1 C˜3 .
2
−4 1 1 + |v|2 |ξ| 10 pˆδ,k (t, ξ, v, k) ∗ χ (v) dξ dk dv dt
[τ2 ,T ]×R9
C
2 2 1 1 |ξ| 10 − 2 −6 + −8 pˆδ,k (τ2 , ξ, v, k)L2 + pˆδ,k (t, ξ, v, k)L2 ([τ v,k
R3
(1) 2 + pδ,k (t, ξ, v, k)L2 ([τ
2 2 ,T ];Lv,k )
(2) 2 + pδ,k (t, ξ, v, k)L2 ([τ
2 2 ,T ];Lv,k )
dξ.
2 2 ,T ];Lv,k )
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
141
Choosing = |ξ|−1/20 and recalling (3.27), we obtain
1 + |v|2
−4
2 1 |ξ| 10 pˆδ,k (t, ξ, v) dξ dk dv dt C˜4 ,
[τ2 ,T ]×R9
which implies (3.25). This completes the proof of Lemma 3.4.
2
We can now iterate Lemmas 3.3 and 3.4 nineteen times to deduce the higher order regularity of f . Proposition 3.1. Let −2 > γ −3, N 8 be a given integer, and f be a smooth solution of Eq. (1.2) given by Theorem 1.1. Suppose that for any +∞ > T > τ0 > 0, and any l N and sufficiently large,
sup t∈[τ0 ,T ]
2 l+1−N + γ w 2(γ+2) ∂ α f + β
|α|+|β|N −1
T
l+1−N α 2 w ∂β f dt C0 ,
|α|+|β|N τ0
where C0 = C0 ( 0 , T, γ, N, l) is some constant. Then for any 0 < τ1 < τ20 < T < +∞ and 0 and large enough, there exists a constant C˜20 > 0, which depends on N , l, γ, T , C0 such that f L∞ ([τ20 ,T ];H N +1 ) C˜20 .
4. The proof of Theorem 1.2 Now we are in a position to complete the Proof of Theorem 1.2. By applying Proposition 3.1 repeatedly, we get by induction on N that for any 0 < τ∗ < T + ∞ and 0 and large enough, ˜ f L∞ ([τ∗ ,T1 ];H∞ ) C,
(4.1)
where C˜ is a positive constant. We now prove by induction on m that ∂tm f ∈ L∞ ([τ, T1 ]; H∞ ) in the sense of distribution. In light of (4.1), this is true for m = 0. Assume that the induction hypothesis holds for any integer which is less than m. Then for all multi-indices α and β and any 0, we have m α m+1 α
m
m
α k α ∂β ∂t f w = −w ∂β v · ∇x ∂t f + w ∂β L ∂t f + Cm w ∂β Γ ∂tk f, ∂tm−k f k=0
= D1 + D2 + D3 . From the induction hypothesis, we obtain ˜ D1 L∞ ([τ∗ ,T1 ];L2x,v ) C. For D2 , one can get from (1.3)–(1.5) and induction hypothesis that for some ˜1 0 D2 L∞ ([τ∗ ,T1 ];L2x,v ) C
˜ α m w 1 ∂ ¯ ∂t f β
¯ |β||β|+2
L∞ ([τ∗ ,T1 ];L2x,v )
˜ C.
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
142
To estimate D3 , by the definition (1.6), we have to compute the following four terms:
vj 1/2 k ψ ij ∗ M ∂t f ∂i ∂tm−k f 2
ij 1/2 k m−k vj 1/2 m−k α α ij k M ∂i ∂t f ∂t f − w ∂β ψ ∗ f . w ∂j ∂β ψ ∗ M ∂i ∂t f ∂t 2
w ∂j ∂βα
ψ ij ∗ M 1/2 ∂tk f ∂i ∂tm−k f − w ∂βα
We see that for ej is the jth standard coordinate vector in R3 , ij 1/2 k m−k ψ ∗ M ∂t f ∂i ∂t f
α−α1 m−k β1 ij = Cαα1 Cβ+e ψ ∗ ∂βα11 M 1/2 ∂tk f ∂i ∂β−β ∂t f. j 1
α ∂β+e j
α1 α,β1 β+ej
Applying the estimate (3.16), Sobolev’s inequality and the induction hypothesis, one can get that for some
˜2 0 ij
w ψ ∗ ∂ α1 M 1/2 ∂tk f ∂i ∂ α−α1 ∂tm−k f 2 ∞ β1 β−β1 L ([τ∗ ,T ];L2x,v ) M 1/8 ∂tk ∂ α0 +α1 f ∞ w 1 + |v| γ+2 ∂i ∂ α−α1 ∂tm−k f ∞ C β2 β−β1 L ([τ ,T ];L2 ) L ([τ ∗
|β2 ||β1 | |α0 |2
C
˜ w2 ∂tk ∂ α0 +α1 f ∞ β2 L ([τ
|β2 ||β1 | |α0 |2
x,v
2 ∗ ,T ];Lx,v )
˜ w 2 ∂i ∂ α−α1 ∂tm−k f ∞ β−β1 L ([τ
2 ∗ ,T ];Lx,v )
2 ∗ ,T ];Lx,v )
˜ C C.
The other terms can be treated in the same way. Therefore we know that m+1 ∂t f
L∞ ([τ∗ ,T ];H∞ )
˜ C.
This ends the proof of Theorem 1.2. 2 Acknowledgments SQL was supported by the NSFC grants 11101188 and 11271160. XM was supported by ARF (GDEI) (Project No. 11ARF05). The authors would like to thank Professor Hongjun Yu for his fruitful discussion on the topic. References [1] R. Alexandre, S. Ukai, Y. Morimoto, C.-J. Xu, T. Yang, Uncertainty principle and kinetic equations, J. Funct. Anal. 255 (8) (2008) 2013–2066. [2] R. Alexandre, S. Ukai, Y. Morimoto, C.-J. Xu, T. Yang, Regularizing effect and local existence for the non-cutoff Boltzmann equation, Arch. Ration. Mech. Anal. 198 (1) (2010) 39–123. [3] A. Arsen’ev, O. Buryak, On the connection between a solution of the Boltzmann equation and a solution of the Landau– Fokker–Planck equation, Math. USSR Sb. 69 (2) (1991) 465–478. [4] F. Bouchut, L. Desvillettes, Averaging lemmas without time Fourier transform and application to discretized kinetic equations, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 19–36. [5] L. Boudin, L. Desvillettes, On the singularities of the global small solutions of the full Boltzmann equation, Monatsh. Math. 131 (2000) 91–108. [6] Y. Chen, L. Desvillettes, L. He, Smoothing effects for classical solutions of the full Landau equation, Arch. Ration. Mech. Anal. 193 (1) (2009) 21–55. [7] H. Chen, W.-X. Li, C.-J. Xu, Gevrey hypoellipticity for linear and non-linear Fokker–Planck equations, J. Differential Equations 246 (1) (2009) 320–339. [8] H. Chen, W.-X. Li, C.-J. Xu, Analytic smoothness effect of solutions for spatially homogeneous Landau equation, J. Differential Equations 248 (1) (2010) 77–94.
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
143
[9] P. Degond, M. Lemou, Dispersion relations for the linearized Fokker–Planck equation, Arch. Ration. Mech. Anal. 138 (2) (1997) 137–167. [10] L. Desvillettes, C. Villani, On the spatially homogeneous Landau equation for hard potentials (I–II), Comm. Partial Differential Equations 25 (1–2) (2000) 179–298. [11] R. Diperna, P.-L. Lions, Global weak solutions of Vlasov–Maxwell systems, Comm. Pure Appl. Math. 42 (6) (1989) 729–757. [12] Y. Guo, The Landau equation in periodic box, Comm. Math. Phys. 231 (2002) 391–434. [13] Y. Guo, Boltzmann diffusive limit beyond the Navier–Stokes approximation, Comm. Pure Appl. Math. 59 (5) (2006) 626–687. [14] L. Hsiao, H.-J. Yu, On the Cauchy problem of the Boltzmann and Landau equations with soft potentials, Quart. Appl. Math. 65 (2) (2007) 281–315. [15] P.-L. Lions, On Boltzmann and Landau equations, Philos. Trans. R. Soc. Lond. Ser. A 346 (1994) 191–3204. [16] S.-Q. Liu, Smoothing effects for the classical solutions to the Landau–Fermi–Dirac equation, Chin. Ann. Math. Ser. B 33 (6) (2012) 857–876. [17] S.-Q. Liu, H.-X. Liu, Optimal convergence rate of the Landau equation with frictional force, Acta Math. Sci. Ser. B 32 (5) (2012) 1781–1804. [18] S.-Q. Liu, X. Ma, Exponential decay of the Landau equation with potential forces, J. Math. Anal. Appl. 394 (1) (2012) 159–176. s stability of the Boltzmann equation and Landau equation with soft potentials, Appl. Math. Comput. [19] X. Ma, S.-Q. Liu, Hx,v 215 (10) (2010) 3721–3730. [20] R.M. Strain, Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations 31 (3) (2006) 417–429. [21] R.M. Strain, Y. Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal. 187 (2) (2008) 287–339. [22] C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Ration. Mech. Anal. 143 (3) (1998) 273–307. [23] C. Villani, A review of mathematical topics in collisional kinetic theory, in: Handbook of Mathematical Fluid Dynamics, vol. I, North-Holland, Amsterdam, 2002, pp. 71–305. [24] T. Yang, H.-J. Yu, Optimal convergence rates of Landau equation with external forcing in the whole space, Acta Math. Sci. Ser. B 4 (2009) 1035–1062. [25] H.-J. Yu, Convergence rate for the Boltzmann and Landau equations with soft potentials, Proc. Roy. Soc. Edinburgh Sect. A 139 (2) (2009) 393–416. [26] H.-J. Yu, Smoothing effects for classical solutions of the relativistic Landau–Maxwell system, J. Differential Equations 246 (10) (2009) 3776–3817. [27] M. Zhan, Local existence of classical solutions to the Landau equations, Transport Theory Statist. Phys. 23 (4) (1994) 479–499.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Regularizing effects for the classical solutions to the Landau equation in the whole space Shuangqian Liu a , Xuan Ma b,∗ a b
Department of Mathematics, Jinan University, Guangzhou 510632, China Department of Mathematics, Guangdong University of Education, Guangzhou 510303, China
a r t i c l e
i n f o
Article history: Received 6 April 2013 Available online 13 March 2014 Submitted by D. Wang Keywords: Landau equation Hypo-elliptic estimate Regularizing effect
a b s t r a c t The Landau equation describes the binary collisional effects (through long range coulombian interaction) in a plasma. In this paper, we prove that the known classical solutions to the Landau equation near Maxwellian in the whole space have a regularizing effect in all (time, space and velocity) variables, that is, become immediately smooth with respect to all variables. © 2014 Elsevier Inc. All rights reserved.
1. Introduction and statement of the main result We consider the following generalized Landau equation: ∂t F + v · ∇x F = ∇v ·
ψ(v − u) F (u)∇v F (v) − F (v)∇u F (u) du ,
(1.1)
R3
with the initial data F (0, x, v) = F0 (x, v). Here F (t, x, v) 0 is the distribution function for the particles at time t 0, with spatial variable x ∈ R3 and velocity v ∈ R3 . The non-negative matrix ψ is defined as ψ ij (v) =
vi vj |v|γ+2 . δ ij − |v|2
The index γ is a parameter leading to the standard classification of hard potential (γ > 0), the Maxwellian molecule (γ = 0) or soft potential (γ < 0), cf. [9,23]. The original Landau collision operator for the Coulombic interaction corresponds to the case γ = −3. In this paper, we restrict our discussion to the case −3 γ < −2. * Corresponding author. E-mail address: [email protected] (X. Ma). http://dx.doi.org/10.1016/j.jmaa.2014.03.006 0022-247X/© 2014 Elsevier Inc. All rights reserved.
124
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
It is well known that the classical Landau collision operator can be formally derived from the Boltzmann operator when the collision between particles become grazing. As in the Boltzmann equation, we denote a global Maxwellian by M (v) = (2π)−3/2 e−|v|
2
/2
,
with the standard perturbation F (t, x, v) to M as F = M (v) +
√
M f.
Then the Landau equation (1.1) for f (t, x, v) takes the form ∂t f + v · ∇x f + Lf = Γ (f, f ),
f (0, x, v) = f0 (x, v).
(1.2)
The Landau collision frequency is σ
ij
ψ ij (v − u)M (u) du.
= R3
The linearized collision operator L in (1.2) is defined as [12,21] √ √ 1 Lf = − √ Q(M, M f ) + Q( M f, M ) = −Af − Kf, M
(1.3)
where √ 1 vi vj f + ∂i σ ij vj f, Af = √ Q(M, M f ) = ∂i σ ij ∂j f − σ ij 4 M
√ 1 1 1 vj ij 2 2 √ Kf = Q( M f, M ) = −∂i M (v) ψ ∗ M ∂j f + f 2 M
1 vi 1 vj + M 2 (v) ψ ij ∗ M 2 ∂j f + f , 2 2
(1.4)
(1.5)
and the collision operator Γ (f, g) is given by Γ [f, g] = M −1/2 Q M 1/2 f, M 1/2 g
vi 1/2 ij M f ∂j g = ∂i ψ ∗ M f ∂j g − ψ ∗ 2
vi 1/2 M ∂j f g, − ∂i ψ ij ∗ M 1/2 ∂j f g + ψ ij ∗ 2
ij
1/2
(1.6)
where we have used Einstein’s summation for the repeated indices. For notational simplicity, we use ·, · to denote the L2 inner product in Rv3 , with its L2 norm given by | · |2 , and (·, ·) is L2 inner product in Rx3 × Rv3 with corresponding L2 norm · . For s ∈ R, m ∈ Z+ , p 0, we use the standard notation H s or W m,p to denote the usual Sobolev space. Given any k ∈ R3 and function f (x), let k f = f (x + k) − f (x).
(1.7)
We shall also use (·, ·) to denote the L2 inner product in Rx3 × Rv3 × Rk3 with corresponding L2 norm |·|.
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
125
We introduce a weight function as
−(γ+2) w = w(v) = 1 + |v| ,
−3 γ < −2.
For l 0, we denote the weighted L2 -norm 3 l 2 2l ij ij vi vj 2 w g = g dv, w σ ∂i g∂j g + σ σ 4 i,j=1 R3
3 l 2 w g = σ i,j=1
R3 ×R3
vi vj 2 g dx dv. w2l σ ij ∂i g∂j g + σ ij 4
Let α = [α1 , α2 , α3 ], β = [β1 , β2 , β3 ] denote multi-indices with length |α| and |β| respectively, and let ∂βα = ∂xα11 ∂xα22 ∂xα33 ∂vβ11 ∂vβ22 ∂vβ33 . Furthermore, define β β if no component of β is greater than the component of β, and β < β if β β
and |β | < |β|. We also use Cββ to denote the usual binomial coefficient ββ . From now on, we use C to denote a generic positive constant may be different from line to line. Let −3 γ < −2, N 8 and l ∈ R. For any function f (t, x, v), we define the energy functional EN,l (t) as EN,l (t) ∼
l−|β| α 2 w ∂β f ,
|α|+|β|N
where A ∼ B means A B and B A, and A B means that there is a generic constant C > 0 such that A CB. We also define the energy dissipation rate DN,l (t) as DN,l (t) =
∂ α Pf 2 + 1|α|N
2 l−|β| α w ∂β {I − P}f , σ
|α|+|β|N
where P is the usual macroscopic projection, which maps f to the null space of L. Throughout this paper, we shall use the following weighted (with respect to the velocity variable v ∈ R3 ) Sobolev space. For s, l ∈ R, set
Hls = f (t, x, v): wl (v)f ∈ H s Rx3 × Rv3 . Since the regularity property to be proved here is local in time, for convenient, we define the following local version of weighted Sobolev space. For −∞ < T1 < T2 < +∞, we denote by W m,p ((T1 , T2 ); Hls ) the set of all distribution functions on (T1 , T2 ) with values in Hls . L2 ((T1 , T2 ); L2x,v,k ) and L2 ((T1 , T2 ); L2v,k ) are defined similarly. We now summarize the known results about global existence of solution to the Landau equation (1.1) near global Maxwellian in [14] as follows: Theorem 1.1. Let −2 > γ −3 and F0 (x, v) = M (v) + M (v)f0 (x, v) 0 with x ∈ R3 . If the initial energy EN,l (0) is small enough for any l 0, then there exists a unique global solution f (t, x, v) to
126
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
the Cauchy problem (1.2) such that F (t, x, v) = M (v) + N 8,
M (v)f (t, x, v) 0 and for any l 0 and
T sup T ∈[0,+∞)
EN,l (t) +
DN,l (t) dt CEN,l (0).
(1.8)
0
Remark 1.1. It is worthy to pointing out that although the additional weight wl is not considered in [14], one can get (1.8) by the same argument, and Guo [13] obtained the similar results in the case of the spatially periodic. We also emphasize that here l 0 can be arbitrary large. In this paper our main result states that the classical solutions obtained in Theorem 1.1 satisfy the following smoothness. Theorem 1.2. Let −2 > γ −3. There exists a small constant 0 > 0 such that for any l 0 and large enough, E8,l (0) 0 . Then for any 0 < τ∗ < T < +∞ and > 0, the classical solutions to Eq. (1.2) given by Theorem 1.1 satisfy the following regularizing effect: f ∈ W ∞,∞ [τ∗ , T ];
∞
H∞ R3 × R3 .
>0
Remark 1.2. It is shown in [6] that the global classical solution to the Landau equation (1.1) near global Maxwellian in the periodic box obtained by Guo [12] has the smoothing effect with respect to all variables. In the whole space, the solution F (t, x, v) to (1.1) does not satisfy the condition of the main results in [6], because F (t, x, v) does not belong to L2x (R3 ). In fact our main results are similar as ones in [2], where the regularizing effect on the non-cutoff Boltzmann equation was proved by using some polynomial weights. Remark 1.3. In order to obtain the fractional regularity with respect to the space variables, the standard hypo-elliptic estimate for the kinetic equation is employed. In general, to do this, we have to pay the price that high moment bounds are necessary for the solution itself. Therefore, it seems that to obtain the smoothing effect of the solution to the Landau equation, “any l 0 and large enough” will be needed in Theorem 1.2. It will be a very interesting problem if one can obtain the same results for some fixed l > 0. By neglecting the bilinear term Γ (f, f ), one can obtain the linear Landau equation. For the linear Landau equation we have the following smoothness: Corollary 1.1. Let −2 > γ −3. For all l 0, the initial energy E8,l (f0 ) C, Eq. (1.2) without the bilinear term Γ has a unique global solution satisfying E8,l (f ) C. Such a solution satisfies the following regularizing effect (for any 0 < τ∗ < T < ∞ and > 0): f ∈ W ∞,∞ [τ∗ , T ];
∞
H∞ R3 × R3 .
>0
Remark 1.4. It is shown in [1] that the linear Landau equation with γ = 0 (Eq. (1.2) dropping the bilinear term Γ ) has the same regularizing effect as our results. Here our results include the very soft potentials. There have been great investigations about the Landau equation, see [3,5,6,9,10,12,13,15,16,18,19,23,22, 26,27]. Desvillettes and Villani [10] proved global existence, uniqueness and smoothness of classical solutions for spatially homogeneous Landau equation for hard potentials and a large class of initial data. Degond and
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
127
Lemou [9] studied the spectral properties and dispersion relation of linearized Landau operator. Guo [12] constructed global classical solutions near a global Maxwellian for a general Landau equation in a periodic box. Hsiao and Yu extended Guo’s results to the whole space in [14]. The large time behaviors of the Landau equation (even with external forces) were investigated in [17,20,21,24,25]. Recently, a lot of progress has been made on the study of the regularity for the solutions of Landau equations, cf. [1,8,6,10] and references therein, which shows that in some sense the Landau equation can be seen as a non-linear and non-local analog of the hypo-elliptic Fokker–Planck equation [7]. We shall point out that Chen, Desvillettes and He [6] proved that the classical solution to the Landau equation (1.1) in the periodic box obtained by Guo [12] becomes immediately smooth with respect to all variables by some hints of [4,5] and [10]: Roughly speaking, smoothness in the velocity variable is produced by the elliptic property of diffusive matrix to the Landau collision operator [10] and smoothness in the position variable is produced by the classical averaging lemma [4,5,11]. However, regularizing effect of global solution to the Landau equation near global Maxwellian in whole space has been remained open. In this paper, we are concerned with the regularizing effect of the solution obtained in [14,24,25]. In detail, since the solution F (t, x, v) to (1.1) in the whole space is not in L2x (R3 ), we study the regularizing effect of the solution to the linearized Landau equation (1.2) instead of the original Landau equation (1.1). We obtain the smoothness in the velocity variable by using energy method and the dissipative property of the linearized Landau collision operator in [9,12], which is different from [6,10], where smoothness in the velocity variable was obtained by the elliptic property of diffusive matrix to the Landau collision operator. It is known that the linearized operator L and the bilinear operator Γ can be rewritten as the sum of divergence form and some error terms, smoothness in the position variable can be shown by using the classical averaging lemma [4,5,11]. Lastly, we prove the smoothness in time variable as [6] and deduce the smoothness in all variables by the iterative method as [1,2,6,10]. However, there is now a new difficulty, and its resolution is the main originality of this paper: since the smallness condition on the “energy” is required in the proof of the global existence, see Theorem 1.1, but for the proof of our Theorem 1.2, this will lead to the disaster when one tried to improve the regularity by iterative method (see Proposition 3.1). As a consequence, we improved the estimates for the nonlinear operator Γ (see Lemma 2.5) and employed Gronwall’s inequality to fix up this gap, see Remark 3.2. The rest of the paper is outlined as follows. Section 2 is devoted to some basic estimates for the operators L and Γ , note that a crucial estimate on Γ is proved at the end of this section. In Section 3, we establish one step of the main induction argument for the proof of Theorem 1.2. More precisely, we lift the regularity of the variables x and v by four elementary lemmas and a proposition which combines the previous estimates. As a consequence of Section 3, the proof of Theorem 1.2 is given in Section 4. 2. Preliminaries In this section, we give some preliminary lemmas which will be used in the proof of our main theorems. We show the following lemma about fractional order Sobolev space. Lemma 2.1. Let 0 < δ < 1, then f (x) ∈ H δ (Rn ), if and only if f (x) ∈ L2 (Rn ) and k f |k|− n2 −δ 2 L
< +∞,
x,k
where k is defined as (1.7). Then there exists c(n) > 0 such that n f L2x + k f |k|− 2 −δ L2
x,k
2δ = c(n) 1 + |ξ| fˆL2 . ξ
(2.1)
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
128
n Moreover, if we denote fδ,k = k f |k|− 2 −δ and fˆ is the Fourier transform of f , then we have
|fˆδ,k |2 dk = c(n)|ξ|2δ |fˆ|2 .
(2.2)
Rn
Proof. We only prove (2.2), since (2.1) is well-known to us. To do this we directly compute that
|fˆδ,k |2 dk = Rn
Rn
|e2πiξ·k − 1|2 ˆ 2 dk f (ξ) . |k|n+2δ
Then by applying the rotation invariance, one can get that Rn
|e2πiξ·k − 1|2 dk = c(n)|ξ|2δ , |k|n+2δ
which implies that (2.2) is true. Thus Lemma 2.1 is proved. 2 In what follows, we summarize some basic estimates for the operators L and Γ which were shown in [12,13,20,21]. Lemma 2.2. Let γ −3, l ∈ R, then there exists C > 0 such that 2 γ+2 γ+2 2 C wl 1 + |v| 2 ∇v g 2 + wl 1 + |v| 2 g 2 2 2 γ γ+2 2 wl g σ C wl 1 + |v| 2 ∇v g 2 + wl 1 + |v| 2 g 2 . Lemma 2.3. Let γ ∈ [−3, −2) and |β| > 0. For any l 0 and η > 0, there exists Cη > 0 such that 2 wl ∂β g 2 − C M 1/8 g 2 . w2l ∂β [Lg], ∂β g wl ∂β g σ − η 1 σ 2
(2.3)
|β1 ||β|
Moreover, for any l > 0, we also have
2 2 w2l Lg, g wl g σ − C M 1/8 g 2 .
Lemma 2.4. Let γ ∈ [−3, −2) and |α| + |β| N . For any l 0, we have M 1/8 ∂ α1 f wl ∂ α−α1 g w2l ∂βα Γ (f, g), ∂βα h C β1 β2 2 σ 1/8 α l α−α l α 1 1 + M ∂β1 f σ w ∂β2 g 2 w ∂β hσ ,
where the summation is over α2 + α1 = α, β1 + β2 β. Proof. The proof of (2.4) is similar as that of Theorem 3 in [12], we omit the details for brevity. 2 Lastly, making use of Lemma 2.4, we have Lemma 2.5. Let γ ∈ [−3, −2), l 0 and N 8, then for any η > 0, it holds that
(2.4)
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
129
2l α w ∂β Γ (f, g), ∂βα h
|α|+|β|N
C
l α l α w ∂β f + w ∂β g + η
|α|+|β|7
+C
|α|+|β|7
+ Cη
|α|+|β|N
l α 2 w ∂β h
σ
l α 2 w ∂β g + C σ
|α|+|β|N
l α 2 w ∂β f
|α|+|β|N
l α w ∂β f
|α|+|β|7
l α 2 w ∂β g .
l α w ∂β g
l α 2 w ∂β f
σ
|α|+|β|N
(2.5)
|α|+|β|N
Proof. Invoking (2.4), to prove (2.5), it suffices to estimate
1/8 α l α−α l α 1 M ∂ 1 f w ∂ g σ w ∂β hσ dx β1 β2 2
R3
+
J1
1/8 α l α−α l α 1 M ∂ 1 f w ∂ g 2 w ∂β hσ dx . β1 β2 σ
R3
J2
Our goal therefore is now to compute J1 and J2 , for this, we divide our calculations into following two cases. Case 1. |α1 | + |β1 | 5. For J1 , Sobolev’s imbedding theorem and Hölder’s inequality imply that 1 l α J1 C M 1/8 ∂βα11 f 2 H 2 wl ∂βα−α g σ w ∂β h σ 2 x l α 2 l α 2 w ∂β g + w ∂β h . wl ∂βα f C σ σ |α|+|β|7
(2.6)
|α|+|β|N
Applying Cauchy–Schwartz’s inequality with η and Sobolev’s inequality, one can see that 2 J2 η wl ∂βα hσ + Cη 2 η wl ∂βα hσ + Cη
1/8 α 2 l α−α 2 1 M ∂ 1 f w ∂ g dx β1
R3
β2
σ
2
l α 2 l α 2 w ∂β f w ∂β g .
(2.7)
|α|+|β|N
Case 2. |α1 | + |β1 | > 5. Note first from Lemma 2.2 that 2 l α−α 2 1 1 w ∂ g σ C wl−1/2 ∂βα−α g 2+ β2 2
l−1/2 α−α 2 1 w . ∂β g 2
|β2 |=|β2 |+1
From which and the fact that |α − α1 | + |β2 | < N − 5, and performing the similar calculations as (2.7), we see that 2 2 2 1 J1 η wl ∂βα hσ + Cη M 1/8 ∂βα11 f 2 wl ∂βα−α g σ dx 2 2 η wl ∂βα hσ + Cη
R3
|α|+|β|N
l α 2 w ∂β f
l−1/2 α 2 w ∂β g .
|α |+|β |N −2
(2.8)
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
130
It remains to estimate J2 for |α1 | + |β1 | > 5, if |α1 | + |β1 | = |α| + |β| = N , then |α − α1 | + |β2 | = 0, we get from Sobolev’s inequality and Lemma 2.2 that J2 C wl g 2 H 2 M 1/8 ∂βα11 f σ wl ∂βα hσ x l α 2 l α 2 wl ∂ α g w ∂β f + w ∂β h . C σ σ |α|2
(2.9)
|α|+|β|N
If 5 < |α1 | + |β1 | N − 1, following the same way as (2.8), one can see that
2 J2 η wl ∂βα hσ + Cη 2 η wl ∂βα hσ + Cη
1/8 α 2 l α−α 2 1 M ∂ 1 f w ∂ g dx β1
R3
β2
σ
l α 2 w ∂β f
|α|+|β|N
2
l α 2 w ∂β g .
(2.10)
|α|+|β|N −3
Therefore (2.5) follows from (2.6), (2.7), (2.8), (2.9) and (2.10). This completes the proof of Lemma 2.5. 2 3. Gain of regularity in v and x Since the proof of Theorem 1.2 is based on an induction on the number of derivatives (in x and v) that can be controlled (see Section 4). In this section, we will show how to get one step of this induction. To make our presentation more clear, we shall divide our discussion into following two parts. The first one is devoted to the study of the smoothness of f (t, x, v) with respect to v. 3.1. Gain of regularity in v In this subsection, we intend to prove the regularity of velocity with the aid of energy method. Lemma 3.1. Let −2 > γ −3, N 8 be a given integer, and f be a smooth solution of Eq. (1.2) given by Theorem 1.1. Suppose that for any +∞ > T > τ1 > τ0 > 0, and any l1 N and sufficiently large,
sup t∈[τ0 ,T ]
2 l1 +1−N + γ w 2(γ+2) ∂ α f + β
|α|+|β|N −1
T
l +1−N α 2 w 1 ∂β f dt C0 ,
(3.1)
|α|+|β|N τ0
and
l +1−N α 2 w 1 ∂β f (τ1 ) C0 ,
(3.2)
|α|+|β|N
where C0 = C0 ( 0 , T, γ, N, l1 ) is a constant. Then there exists a constant C˜1 > 0, which depends on N , l1 , γ, T , 0 , such that for any +∞ > T > τ1 > 0, and all > 0 and large enough, sup t∈[τ1 ,T ]
|α|+|β|N
+ γ w 2(γ+2) ∂βα f 2 +
w ∇v ∂βα f 2 dx dv dt C˜1 .
(3.3)
|α|+|β|N [τ ,T ]×R3 ×R3 1
Proof. Taking ∂βα to Eq. (1.2) with |α| + |β| N and taking the inner product with w2l0 ∂βα f (l0 0) over Rx3 × Rv3 , we obtain
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
131
[∂t + v · ∇x ]∂βα f, w2l0 ∂βα f + ∂βα [Lf ], w2l0 ∂βα f β
α = ∂βα Γ (f, f ), w2l0 ∂βα f − f, w2l0 ∂βα f . Cβ 1 ∂β1 v · ∇x ∂β−β 1
(3.4)
|β1 |=1
We will estimate each term of (3.4). From (2.3) in Lemma 2.3, one has 2
2l0 α wl0 ∂βα f 2 − C w−1/2 ∂ α f 2 . w ∂β [Lf ], ∂βα f wl0 ∂βα f σ − η 1 σ
(3.5)
|β1 ||β|
In light of (2.5) and recalling
√ E8,l0 (t) C 0 , we see that
α √ ∂β Γ (f, f ), w2l0 ∂βα f C( 0 + η)
l α 2 w 0 ∂β f + C σ
|α|+|β|N
l α 4 w 0 ∂β f .
(3.6)
|α|+|β|N
And for β > 0, it is obvious that 2 2
β1 α α Cβ ∂β1 v · ∇x ∂β−β f, w2l0 ∂βα f C wl0 ∂βα f σ + C wl0 ∇x ∂β−β f , 1 1
(3.7)
where |β − β1 | = |β| − 1. Letting η > 0 small enough, an appropriate combination of (3.4), (3.5), (3.6) and (3.7) yields d wl0 ∂βα f 2 + wl0 ∂βα f 2 C σ dt
l α 2 w 0 ∂ 1 f + η wl0 ∂βα f 2 β1 1 σ
|α1 |+|β1 |N
+C
l α 4 w 0 ∂β f .
|β1 ||β|
(3.8)
|α|+|β|N
For |α| + |β| N and any l0 0, a suitable summation of (3.8) over 0 |β| N gives |α|+|β|N
d wl0 ∂βα f 2 + wl0 ∂βα 2 C σ dt
l α 2 w 0 ∂β f + C
|α|+|β|N
l α 4 w 0 ∂β f .
(3.9)
|α|+|β|N
On the other hand, utilizing (3.1) with l0 = l1 + 1 − N , we see that T
l α 2 w 0 ∂β f dt C0 .
|α|+|β|N τ0
With this, we get from (3.9), (3.2) and Gronwall’s inequality that e
−
T
|α|+|β|N
τ1
α wl0 ∂β f 2 dt
l α w 0 ∂β f (T )2
|α|+|β|N
T +
e
−
|α|+|β|N
t τ1
α wl0 ∂β f (τ )2 dτ
T e τ1
+
l α w 0 ∂β f (t)2 dt σ
|α|+|β|N
τ1
C
−
|α|+|β|N
|α|+|β|N
t τ1
α wl0 ∂β f (τ )2 dτ
l α w 0 ∂β f (t)2 dt
|α|+|β|N
l α w 0 ∂β f (τ1 )2 .
(3.10)
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
132
Then (3.10) and Lemma 2.2 imply that
sup t∈[τ1 ,T ]
l α 2 w 0 ∂β f +
|α|+|β|N
T
1 + |v| γ/2 wl0 ∇v ∂βα f (t)2 dt CeC0 .
(3.11)
|α|+|β|N τ1
γ With (3.11) in hand, by setting = l0 − 2(γ+2) = l1 + 1 − N − This concludes the proof of Lemma 3.1. 2
γ (γ+2)
0, one can see that (3.3) holds true.
Remark 3.1. The constant C˜1 in Lemma 3.1 depends on T , which yields that the high regularity property obtained in the following results is in the pointwise sense with respect to time t. This explains why our main result in Theorem 1.2 is restricted to local time [τ∗ , T ]. Remark 3.2. We have succeeded in applying fairly straightforward “energy” method estimates to produce the smoothness of the variable v, however unlike the usual energy method developed by Guo and Strain in [12,13,21,20], thanks to Lemma 2.5 and Gronwall’s inequality, here we only need the smallness of E8,l (t) with sufficiently large l but EN,l (t) with N 8, which means that neither C0 or C˜1 in Lemma 3.1 is required to be small. This observation pays the way for the iterative technique utilized later. Remark 3.3. Noticing that the assumption (3.1) is coincide with (1.8) while T is finite, in particular, this shows that the result of Theorem 1.2 is not empty. 3.2. Gain of regularity in x We now turn to prove the regularity for position variable x. We can not hope to get it directly by an energy estimate like we did for variable v since no diffusion term in x is available. Motivated by the work [6] and the case near vacuum about the Boltzmann equation [2], we will resort to the averaging lemma. Lemma 3.2. Let −2 > γ −3, N 8 be a given integer, and f be a smooth solution of Eq. (1.2) given by Theorem 1.1. Suppose that for any +∞ > T > τ1 > 0, and any 0 and sufficiently large,
sup t∈[τ1 ,T ]
α 2 w ∂β f C˜1 ,
|α|+|β|N
for some constant C˜1 = C˜1 ( , T, γ, N, C0 ). Then, for any > 0 and large enough, any 0 < τ1 < T < +∞ and |α| + |β| N , there exists a constant C˜2 > 0, which depends on N , , γ, τ1 , T and C˜1 , such that
α 2 w ∂β f
1/20
Hx
dv dt C˜2 .
(3.12)
[τ1 ,T ]×R3
Proof. Let p(t, x, v) = w ∂βα f (1 + |v|2 )2 . To prove (3.12), we first need to prove
−4 1 1 + |v|2 1 + |ξ| 10 |ˆ p(t, ξ, v)|2 dξ dv dt C˜2 ,
(3.13)
[τ1 ,T ]×R6
where pˆ(t, ξ, v) is the Fourier transform of p(t, x, v) with respect to variable x. Noticing that (3.13) is obviously true for |ξ| 1 according to the assumption, therefore we focus on the case |ξ| 1. For this, let
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
χ := χ(v) ∈ Cc∞ (R3 ) be a test function which satisfies χ(v) 0 and write
R3
133
χ(v) dv = 1. Set χ = −3 χ( v ) and
pˆ(t, ξ, v) = pˆ(t, ξ, v) − pˆ(t, ξ, ·) ∗ χ (v) + pˆ(t, ξ, ·) ∗ χ (v). Here, will be chosen later (and will depend on ξ). We get from Minkowski’s inequality that R3
−4 pˆ(t, ξ, v) − pˆ(t, ξ, ·) ∗ χ (v)2 dv 1 + |v|2 2 pˆ(t, ξ, v) − pˆ(t, ξ, v − u) χ (u) du dv R3 R3
R3
pˆ(t, ξ, v) − pˆ(t, ξ, v − u)2 dv
R3
∇v pˆ(t, ξ, v)2 dv
R3
C
2
12
2 χ (u) du
2 χ (u)|u| du
R3
∇v pˆ(t, ξ, v)2 dv,
R3
so that
−4 1 2
1 + |v|2 1 + |ξ| 10 pˆ(t, ξ, v) − pˆ(t, ξ, ·) ∗ χ (v) dv dξ dt
[τ1 ,T ]×R6
2
1 2 1 + |ξ| 10 ∇v pˆ(t, ξ, v) dv dξ dt.
C [τ1
(3.14)
,T ]×R6
Remembering that p(t, x, v) = w ∂βα f (1 + |v|2 )2 , we see that p satisfies the following equation (using Einstein’s summation for the repeated indices): ∂t p + v · ∇x p = (I) + (II ) + (III ) + (IV ),
(3.15)
where (I) =
0,
β1 |β1 |=1 Cβ w (1
for β = 0,
− + |v| ) ∂β1 v · for |β| > 0, ⎧ ∂i [w (1 + |v|2 )2 ∂βα (σ ij ∂j f )] ⎪ ⎪ ⎪ ⎪ ⎨ − ∂i [w (1 + |v|2 )2 ∂ α (M 1/2 [ψ ij ∗ (M 1/2 (∂j f + vj f ))])] β 2 (II ) = ⎪ + ∂i [w (1 + |v|2 )2 ∂βα ([ψ ij ∗ (M 1/2 f )]∂j f )] ⎪ ⎪ ⎪ ⎩ − ∂i [w (1 + |v|2 )2 ∂βα ([ψ ij ∗ (M 1/2 ∂j f )]f )], for |β| 0,
2
2 2 vi vj f − ∂i w 1 + |v|2 ∂βα σ ij ∂j f , (III ) = w 1 + |v|2 ∂βα ∂i σ ij vj f − w 1 + |v|2 ∂βα ∂i σ ij 4
2 vj (IV ) = ∂i w 1 + |v|2 ∂βα M 1/2 ψ ij ∗ M 1/2 ∂j f + f 2 2 2
α ∇x ∂β−β f, 1
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
134
vi 1/2 ij vj 1/2 M , ψ ∗ M ∂j f + f +w 1+ 2 2 2
(V ) = −∂i w 1 + |v|2 ∂βα ψ ij ∗ M 1/2 f ∂j f
2 + ∂i w 1 + |v|2 ∂βα ψ ij ∗ M 1/2 ∂j f f
2 vi 1/2 M f ∂j f − w 1 + |v|2 ∂βα ψ ij ∗ 2
vi 1/2 2 2 α ij M ∂j f f . + w 1 + |v| ∂β ψ ∗ 2
2 |v|2 ∂βα
The terms in (II )–(V ) come from (1.4), (1.5) and (1.6) after taking ∂βα derivatives on −Lf and Γ (f, f ). We use (I)i , (II )i , (III )i , (IV )i , (V )i (i = 1, 2, . . . , ) to denote the corresponding terms in (I), (II ), (III ), (IV ), (V ) for simplicity of our presentation. In fact, we can write the equation satisfied by p under the form ∂t p + v · ∇ x p = p 1 + ∇ v · p 2 . Here, p1 is the sum of the terms (I), (III ), (IV ) and (V ), while ∇v · p2 is the sum of the terms in (II ). We claim that p1 , p2 ∈ L2 ([τ1 , T ]; L2x,v ). We only present here the estimates for the terms in (II ), while the other terms can be shown similarly. For the term (II )1 , since |σ ij | C(1 + |v|)γ+2 , we have from (3.3) that
w 1 + |v|2 2 ∂βα σ ij ∂j f
L2 ([τ1 ,T ];L2x,v )
α w 1 + |v| γ+6 ∂j ∂β−β f L2 ([τ 1
C
|β1 ||β|
γ+6 α w− γ+2 ∂j ∂β−β f L2 ([τ 1
C
|β1 ||β|
2 1 ,T ];Lx,v )
2 1 ,T ];Lx,v )
C
# C˜1 .
As to the term (II )2 , applying Lemma 3.1 again, we obtain
w 1 + |v|2 2 ∂βα M 1/2 ψ ij ∗ M 1/2 ∂j f + vj f 2 2 L ([τ1 ,T ];L2
x,v )
C
1 M 8 ∂j ∂βα f 2 2 L ([τ
β2 β
2 1 ,T ];Lx,v
1 + M 8 ∂βα2 f L2 ([τ )
2 1 ,T ];Lx,v )
# C( C˜1 + C0 ),
where we have used the basic estimate for the convolution of ψ ij : 12 12 1 ij
ψ ij (v − u)M 18 (u)2 du M 8 ∂j ∂ α2 f (u)2 du ψ ∗ ∂βα M 12 ∂j f β2 1 β2 β1
β2 β1
R3
R3
γ+2 1 C 1 + |v|2 2 M 8 ∂j ∂βα2 f 2 .
For the term (II )3 , we first see that
∂βα ψ ij ∗ M 1/2 f ∂j f =
α1 α,β1 β
α−α1 Cαα1 Cββ1 ψ ij ∗ ∂βα11 M 1/2 f ∂j ∂β−β f. 1
(3.16)
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
Then if |α1 | + |β1 |
N 2,
135
we have from (3.16) and Lemma 3.1 that
w 1 + |v|2 2 ψ ij ∗ ∂ α1 M 1/2 f ∂j ∂ α−α1 f 2 β1 β−β1 L ([τ1 ,T ];L2x,v ) w 1 + |v| γ+6 ∂j ∂ α−α1 f M 18 ∂ α1 f 2 C β−β1 β2 2 2 L ([τ ,T ];L2 ) 1
β2 β1
C
1 M 8 ∂ α1 +α f
L∞ ([τ1 ,T ];L2x,v )
β2
β2 β1 |α |2
Similarly if |α1 | + |β1 |
N 2,
x
− γ+6 w γ+2 ∂j ∂ α−α1 f 2 β−β1 L ([τ
2 1 ,T ];Lx,v )
C C˜1 .
by (3.16) and Lemma 3.1, it follows that
1 + |v|2 2 w ψ ij ∗ ∂ α1 M 1/2 f ∂j ∂ α−α1 f 2 β1 β−β1 L ([τ1 ,T ];L2x,v ) w 1 + |v| γ+6 ∂j ∂ α−α1 f M 18 ∂ α1 f 2 C β−β1 β 2 2 L ([τ ,T ];L2 ) 2 1
β2 β1
C
γ+6 w− γ+2 ∂j ∂ α−α1 +α f ∞ β−β1 L ([τ
β2 β1 |α |2
2 1 ,T ];Lx,v )
x
α w ∂ 1 f β2
L2 ([τ1 ,T1 ];L2x,v )
C
# C˜1 C0 .
The term (II )4 can be treated as the term (II )3 . Finally, we conclude that p1 , p2 ∈ L2 ([τ1 , T ]; L2x,v ). Now, employing the variant average lemma in [4], we can prove that
|ξ|
1 2
T
pˆ(t, ξ, v) ∗ χ (v)2 dt
τ1
2 C χ (v − u) 1 + |u|2 L∞ + ∇v χ (v − u) 1 + |u|2 L∞ u u 2 2
× pˆ(τ1 , ξ, v)L2 + pˆ(t, ξ, v)L2 ([τ ,T ];L2 ) 1 v v 2 2 pˆ2 (t, ξ, v) 2 + pˆ1 (t, ξ, v) 2 , 2 + 2 L ([τ1 ,T ];Lv )
L ([τ1 ,T ];Lv )
Since χ (v − u)(1 + |u|2 )L∞ C −3 (1 + |v|2 ), ∇v χ (v − u)(1 + |u|2 )L∞ C −4 (1 + |v|2 ), we see that u u
1 + |v|2
−4
2 1 |ξ| 10 pˆ(t, ξ, v) ∗ χ (v) dξ dv dt
[τ1 ,T ]×R6
C
2 2 1 1 |ξ| 10 − 2 −6 + −8 pˆ(τ1 , ξ, v)L2 + pˆ(t, ξ, v)L2 ([τ v
R3
2 + pˆ1 (t, ξ, v)L2 ([τ
2 1 ,T ];Lv )
2 + pˆ2 (t, ξ, v)L2 ([τ
2 1 ,T ];Lv )
dξ.
2 1 ,T ];Lv )
(3.17)
If we choose = (|ξ|) = |ξ|− 20 (|ξ| 1), we can bound (3.17) remembering that (1 + |v|2 )−4 ∈ L1v , p(τ1 , x, v) ∈ L2x,v and p, p1 , p2 ∈ L2 ([τ1 , T ]; L2x,v ), which implies that (3.12) holds true. Thus the proof of Lemma 3.2 is completed. 2 1
Roughly speaking, Lemma 3.2 together with Lemma 3.1 shows that when f is a solution of Eq. (1.2) conN +1/20 ). To improve the regularity about position variable x, structed in Theorem 1.1, then f ∈ L2 ([τ1 , T ]; H we will first show a preliminary lemma.
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
136
1 Lemma 3.3. Let −2 > γ −3, N 8 be a given integer, δ = 20 and f be a smooth solution of Eq. (1.2) given by Theorem 1.1. Suppose that for any +∞ > T > τ2 > τ1 > 0, and any 0 large enough,
|α|+|β|N
α w ∂β f (τ2 )2 1/20 C˜2 , Hx
α 2 w ∂β f
1/20
Hx
|α|+|β|N [τ ,T ]×R3 1
dv dt C˜2 .
(3.18)
Then there exists a constant C˜3 > 0, which depends on N , , γ, T and C˜2 , such that for any > 0 large enough and any 0 < τ2 < T < +∞
+ γ w 2(γ+2) k ∂βα f |k|−δ− 32 2 ∞ L
|α|+|β|N
+
([τ2 ,T ];L2x,v,k )
w ∇v k ∂βα f |k|−δ− 32 2 2
L ([τ2 ,T ];L2x,v,k )
|α|+|β|N
C˜3 .
Proof. Taking |k|−δ− 2 k on Eq. (1.2), we have (denoting g˜ = k f |k|−δ− 2 ) 3
3
∂t g˜ + v · ∇x g˜ + L˜ g = Γ g˜, f (x + k) + Γ (f, g˜),
(3.19)
where we have used the fact that for any functions f1 and f2 , k f1 (x)f2 (x) = k f1 (x)f2 (x + k) + f1 (x) k f2 (x). Now we perform energy estimates for Eq. (3.19). Taking ∂βα (|α| + |β| N ) to Eq. (3.19), and taking the inner product with w2 ∂βα g˜ over Rx3 × Rv3 × Rk3 , one has
1 d w ∂βα g˜2 + ∂βα L˜ g , w2 ∂βα g˜ = ∂βα Γ g˜, f (x + k) , w2 ∂βα g˜ + ∂βα Γ (f, g˜), w2 ∂βα g˜ 2 dt β α Cβ 1 ∂β1 v · ∇x ∂β−β g˜, w2 ∂βα g˜ . − 1
(3.20)
|β1 |=1
From (2.3), we can obtain
2 w ∂βα g˜2 − C ∂ α g˜2 . ∂βα L˜ g , w2 ∂βα g˜ w ∂βα g˜σ − η 1 σ
(3.21)
|β1 ||β|
Now we consider the first and the second term of the right-hand side in (3.20). For this, we only compute the first term, since estimates for the second one being similar. It follows from (2.4) that
α ∂β Γ g˜, f (x + k) , w2 ∂βα g˜ M 1/8 ∂ α1 g˜ w ∂ α2 f (x + k) C β1 β2 2 σ R6
+ w ∂βα22 f (x + k)2 M 1/8 ∂βα11 g˜σ w ∂βα g˜σ dx dk,
(3.22)
where the summation is over α2 + α1 = α, β1 + β2 β. We only present here the estimates for the first part of the right-hand side of (3.22), because the second one can be treated similarly. Using the idea explained in the proof of (2.5), we carry out the following computations.
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
137
If |α1 | + |β1 | 5, by Hölder’s inequality and Sobolev’s inequality, we find
1/8 α α M ∂ 1 g˜ w ∂ 2 f (x + k) w ∂βα g˜ dx dk β1 β2 2 σ σ
R6
C
supM 1/8 ∂βα11 g˜2 w ∂βα g˜σ dk w ∂βα22 f σ x
R3
C
M 1/8 ∂ α1 +α g˜w ∂βα g˜ w ∂ α2 f , β1 β2 σ σ
|α |2
where we have used the transition invariance property of the Lebesgue integral. If |α1 | + |β1 | 5, since |α2 | + |β2 | < N − 5, owing to Cauchy–Schwartz’s inequality with η and Sobolev’s inequality, we obtain
1/8 α α M ∂ 1 g˜ w ∂ 2 f (x + k) w ∂βα g˜ dx dk β1 β2 2 σ σ
R6
2 2 η w ∂βα g˜σ + Cη supw ∂βα22 f (x + k)σ
1/8 α 2 M ∂ 1 g˜ dx dk β1
x
2 η w ∂βα g˜σ + Cη
R3
−1/2 α 2 1/8 α 2 w ∂ 2 f M ∂ 1 g˜ . β2
|α |+|β2 |N −2
β1
We thus have a uniform estimate as:
∂βα Γ g˜, f (x + k) , w2 ∂βα g˜ + ∂βα Γ (f, g˜), w2 ∂βα g˜ 2 √ w ∂βα g˜4 . C(η + 0 )w ∂βα g˜σ + Cη
(3.23)
|α|+|β|N |α|+|β|N
And it is straightforward to check that 2 2
α α g˜, w2 ∂βα g˜ C w ∂βα g˜ + C w ∇x ∂β−β g˜ , ∂β1 v · ∇x ∂β−β 1 1
(3.24)
where |β − β1 | = |β| − 1. By choosing η > 0 small enough and combing (3.20), (3.21), (3.23), (3.24) and the assumption (3.18), we obtain d w ∂βα g˜2 + C w ∂βα g˜4 . w ∂βα g˜2 + w ∂βα g˜2 C σ dt |α|+|β|N
|α|+|β|N
|α|+|β|N
Recalling Lemmas 2.2, 3.1 and 3.2 and applying Gronwall’s inequality, we have that
+ γ w 2(γ+2) ∂βα g˜(t)2 +
t∈[τ2 ,T ] |α|+|β|N
T
w ∇v ∂βα g˜2 dt CeC˜2 .
|α|+|β|N τ2
This concludes the proof of Lemma 3.3. 2 In what follows we will lift the regularity of position variable x again, for results in this direction, we have
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
138
Lemma 3.4. Let −2 > γ −3, N 8 be a given integer, and f be a smooth solution of Eq. (1.2) given by Theorem 1.1. Suppose that for any +∞ > T > τ2 > τ1 > 0, and any 0 and large enough,
α w ∂β f (τ2 )2
1/20
Hx,v
|α|+|β|N
C˜3 .
Then there exists a constant C˜4 > 0, which depends on N , , γ, T , C˜3 , such that for any 0 and large enough,
α 2 w ∂β f
1/10
Hx
|α|+|β|N [τ ,T ]×R3 2
dv dt C˜4 .
Proof. In view of (2.2) in Lemma 2.1, we know that when |ξ| 1 and δ = (3.25)
⇔
(3.25)
1 20 ,
2 α ˜ |ξ|2δ+1/10 w ∂$ β f (ξ) dξ dv dt C4
[τ2 ,T ]×R6
⇔ [τ2
−δ− 32 2 α |ξ|1/10 w k ∂$ dv dk dξ dt C˜4 . β f |k|
(3.26)
,T ]×R9
In order to prove estimate (3.26), by analogy with approach taken in the proof of Lemma 3.2, we define 3 pδ,k = w (1 + |v|2 )2 k ∂βα f |k|−δ− 2 , and write
pˆδ,k (t, ξ, v) = pˆδ,k (t, ξ, v) − pˆδ,k (t, ξ, ·) ∗ χ (v) + pˆδ,k (t, ξ, ·) ∗ χ (v), the parameter being chosen later. Following the proof of the estimates for (3.14), we get
[τ2 ,T ]×R6
1 + |v|2
−4 1 2 1 + |ξ| 10 pˆδ,k (t, ξ, v) − pˆδ,k (t, ξ, ·) ∗ χ (v) dv dξ dt
2
1 2 1 + |ξ| 10 ∇v pˆδ,k (t, ξ, v) dv dξ dt.
C
(3.27)
[τ2 ,T ]×R6
Noticing that g˜ = k f |k|−δ− 2 , we see that pδ,k also satisfies the following equation 3
∂t pδ,k + v · ∇x pδ,k = (VI ) + (VII ) + (VIII ) + (IX ) + (X), where the right-hand side is ⎧ ⎨ 0, for β = 0, (VI ) = − C β1 w (1 + |v|2 )2 ∂β v · ∇x ∂ α f |k|−δ− 32 , for |β| > 0, 1 β−β1 β ⎩ |β1 |=1
(VII ) =
⎧ ⎪ ∂i [w (1 + |v|2 )2 ∂βα (σ ij ∂j g˜)] ⎪ ⎪ ⎪ v ⎪ ⎪ − ∂i [w (1 + |v|2 )2 ∂βα (M 1/2 [ψ ij ∗ (M 1/2 (∂j g˜ + 2j g˜))])] ⎪ ⎪ ⎪ ⎪ ⎨ + ∂i [w (1 + |v|2 )2 ∂ α ([ψ ij ∗ (M 1/2 g˜)]∂j f (x + k))] β
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
+ ∂i [w (1 + |v|2 )2 ∂βα ([ψ ij ∗ (M 1/2 f )]∂j g˜)] − ∂i [w (1 + |v|2 )2 ∂βα ([ψ ij ∗ (M 1/2 ∂j g˜)]f (x + k))] g )], − ∂i [w (1 + |v|2 )2 ∂βα ([ψ ij ∗ (M 1/2 ∂j f )]˜
for |β| 0,
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
(VIII ) = w 1 +
2 |v|2 ∂βα ∂i σ ij vj g˜
−w 1+
2 |v|2 ∂βα
∂i σ
ij vi vj
139
g˜
4 α ij 2 2 ∂β σ ∂j g˜ , − ∂i w 1 + |v|
α vj 2 2 1/2 ij 1/2 ψ ∗ M ∂j g˜ + g˜ ∂β M (IX ) = ∂i w 1 + |v| 2
2 vi 1/2 ij vi M ψ ∗ M 1/2 ∂j g˜ + g˜ , + w 1 + |v|2 ∂βα 2 2 2
(X) = −∂i w 1 + |v|2 ∂βα ψ ij ∗ M 1/2 g˜ ∂j f (x + k) 2
+ ∂i w 1 + |v|2 ∂βα ψ ij ∗ M 1/2 ∂j g˜ f (x + k) 2
− ∂i w 1 + |v|2 ∂βα ψ ij ∗ M 1/2 f ∂j g˜
2 + ∂i w 1 + |v|2 ∂βα ψ ij ∗ M 1/2 ∂j f g˜
vi 1/2 2 2 α ij − w 1 + |v| ∂β ψ ∗ M g˜ ∂j f (x + k) 2
2 vi 1/2 M ∂j g˜ f (x + k) + w 1 + |v|2 ∂βα ψ ij ∗ 2
vi 1/2 2 2 α ij M f ∂j g˜ − w 1 + |v| ∂β ψ ∗ 2
2 vi 1/2 M ∂j f g˜ . + w 1 + |v|2 ∂βα ψ ij ∗ 2
The terms in (VI )–(X ) can be obtained by using the operator k and then timing |k|−δ− 2 from (3.15). We use (VI )i , (VII )i , (VIII )i , (IX )i , (X)i (i = 1, 2, . . . , ) to denote the corresponding terms in (VI )–(V ) for simplicity of our presentation. In fact, we can write the equation satisfied by pδ,k under the form 3
(1)
(2)
∂t pδ,k + v · ∇x pδ,k = pδ,k + ∇ · pδ,k , (2)
(1)
where pδ,k consists in the sum of terms (VII ), while pδ,k is the sum of other terms. (1)
(2)
We claim that pδ,k , pδ,k ∈ L2 ([τ2 , T ]; L2x,v,k ). We only present here the estimates for the terms (VII )1 , (VII )3 and (VII )6 while the other terms can be estimated similarly. For the term (VII )1 , we have from Lemma 3.3 that,
w 1 + |v|2 2 ∂βα σ ij ∂j g˜ 2 L ([τ
2 2 ,T ];Lx,v,k )
C
α w 1 + |v| γ+6 ∂j ∂β−β g˜L2 ([τ 1 |β1 ||β|
2 2 ,T ];Lx,v,k )
# C C˜3 . Now we turn to estimate (VII )3 , we first write
∂βα ψ ij ∗ M 1/2 g˜ ∂j f (x + k) =
α−α1 Cαα1 Cββ1 ψ ij ∗ ∂βα11 M 1/2 g˜ ∂j ∂β−β f (x + k). 1
α1 α,β1 β
If |α1 | + |β1 | N/2, we compute
w 1 + |v|2 2 ψ ij ∗ ∂ α1 M 1/2 g˜ ∂j ∂ α−α1 f (x + k) 2 β1 β−β1 L ([τ
2 2 ,T ];Lx,v,k )
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
140
C
M 1/8 ∂ α0 +α1 g˜
L∞ ([τ2 ,T ];L2x,v,k )
β0
β0 β1 |α0 |2
C
w 1 + |v| γ+6 ∂j ∂ α−α1 f β−β1
L2 ([τ2 ,T ];L2x,v )
# C˜1 C˜3 .
Here we have used (3.16), Lemmas 3.1 and 3.3. Likewise, for |α1 | + |β1 | N/2, it follows that
w 1 + |v|2 2 ψ ij ∗ ∂ α1 M 1/2 g˜ ∂j ∂ α−α1 f (x + k) 2 β1 β−β1 L ([τ2 ,T ];L2x,v,k )
M 1/8 ∂ α1 g˜ ∞ w 1 + |v| γ+6 ∂j ∂ α0 +α−α1 f 2 C β0 β−β1 L ([τ ,T ];L2 ) L ([τ 2
β0 β1 |α0 |2
C
x,v,k
2 2 ,T ];Lx,v )
# C˜1 C˜3 .
Therefore, in any case the term (VII )3 can be bounded by C As to the term (VII )6 , noticing that
∂βα ψ ij ∗ M 1/2 ∂j f g˜ =
C˜1 C˜3 .
α−α1 Cαα1 Cββ1 ψ ij ∗ ∂βα11 M 1/2 ∂j f ∂β−β g˜. 1
α1 α,β1 β
If |α1 | + |β1 | N/2, we know that
w 1 + |v|2 2 ψ ij ∗ ∂ α1 M 1/2 ∂j f ∂ α−α1 g˜ 2 β1 β−β1 L ([τ2 ,T ];L2x,v,k ) M 1/8 ∂j ∂ α0 +α1 f ∞ w 1 + |v| γ+6 ∂ α−α1 g˜ 2 C β0 β−β1 L ([τ ,T ];L2 ) L ([τ 2
β0 β1 |α0 |2
x,v
2 2 ,T ];Lx,v,k )
# C C˜1 C˜3 .
Here we have also used (3.16), Lemmas 3.1 and 3.3. If |α1 | + |β1 | N/2, we deduce from (3.16), Lemmas 3.1 and 3.3 that
w 1 + |v|2 2 ψ ij ∗ ∂ α1 M 1/2 ∂j f ∂ α−α1 g˜ 2 β1 β−β1 L ([τ2 ,T ];L2x,v,k ) M 1/8 ∂j ∂ α1 f 2 w 1 + |v| γ+6 ∂ α0 +α−α1 g˜ ∞ C β0 β−β1 L ([τ ,T ];L2 ) L ([τ β0 β1 |α0 |2
2
x,v
2 2 ,T ];Lx,v,k )
# C C˜1 C˜3 .
We thus conclude that the term (VII )6 can be also bounded by C Proceeding like the estimates for (3.17), we discover
C˜1 C˜3 .
2
−4 1 1 + |v|2 |ξ| 10 pˆδ,k (t, ξ, v, k) ∗ χ (v) dξ dk dv dt
[τ2 ,T ]×R9
C
2 2 1 1 |ξ| 10 − 2 −6 + −8 pˆδ,k (τ2 , ξ, v, k)L2 + pˆδ,k (t, ξ, v, k)L2 ([τ v,k
R3
(1) 2 + pδ,k (t, ξ, v, k)L2 ([τ
2 2 ,T ];Lv,k )
(2) 2 + pδ,k (t, ξ, v, k)L2 ([τ
2 2 ,T ];Lv,k )
dξ.
2 2 ,T ];Lv,k )
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
141
Choosing = |ξ|−1/20 and recalling (3.27), we obtain
1 + |v|2
−4
2 1 |ξ| 10 pˆδ,k (t, ξ, v) dξ dk dv dt C˜4 ,
[τ2 ,T ]×R9
which implies (3.25). This completes the proof of Lemma 3.4.
2
We can now iterate Lemmas 3.3 and 3.4 nineteen times to deduce the higher order regularity of f . Proposition 3.1. Let −2 > γ −3, N 8 be a given integer, and f be a smooth solution of Eq. (1.2) given by Theorem 1.1. Suppose that for any +∞ > T > τ0 > 0, and any l N and sufficiently large,
sup t∈[τ0 ,T ]
2 l+1−N + γ w 2(γ+2) ∂ α f + β
|α|+|β|N −1
T
l+1−N α 2 w ∂β f dt C0 ,
|α|+|β|N τ0
where C0 = C0 ( 0 , T, γ, N, l) is some constant. Then for any 0 < τ1 < τ20 < T < +∞ and 0 and large enough, there exists a constant C˜20 > 0, which depends on N , l, γ, T , C0 such that f L∞ ([τ20 ,T ];H N +1 ) C˜20 .
4. The proof of Theorem 1.2 Now we are in a position to complete the Proof of Theorem 1.2. By applying Proposition 3.1 repeatedly, we get by induction on N that for any 0 < τ∗ < T + ∞ and 0 and large enough, ˜ f L∞ ([τ∗ ,T1 ];H∞ ) C,
(4.1)
where C˜ is a positive constant. We now prove by induction on m that ∂tm f ∈ L∞ ([τ, T1 ]; H∞ ) in the sense of distribution. In light of (4.1), this is true for m = 0. Assume that the induction hypothesis holds for any integer which is less than m. Then for all multi-indices α and β and any 0, we have m α m+1 α
m
m
α k α ∂β ∂t f w = −w ∂β v · ∇x ∂t f + w ∂β L ∂t f + Cm w ∂β Γ ∂tk f, ∂tm−k f k=0
= D1 + D2 + D3 . From the induction hypothesis, we obtain ˜ D1 L∞ ([τ∗ ,T1 ];L2x,v ) C. For D2 , one can get from (1.3)–(1.5) and induction hypothesis that for some ˜1 0 D2 L∞ ([τ∗ ,T1 ];L2x,v ) C
˜ α m w 1 ∂ ¯ ∂t f β
¯ |β||β|+2
L∞ ([τ∗ ,T1 ];L2x,v )
˜ C.
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
142
To estimate D3 , by the definition (1.6), we have to compute the following four terms:
vj 1/2 k ψ ij ∗ M ∂t f ∂i ∂tm−k f 2
ij 1/2 k m−k vj 1/2 m−k α α ij k M ∂i ∂t f ∂t f − w ∂β ψ ∗ f . w ∂j ∂β ψ ∗ M ∂i ∂t f ∂t 2
w ∂j ∂βα
ψ ij ∗ M 1/2 ∂tk f ∂i ∂tm−k f − w ∂βα
We see that for ej is the jth standard coordinate vector in R3 , ij 1/2 k m−k ψ ∗ M ∂t f ∂i ∂t f
α−α1 m−k β1 ij = Cαα1 Cβ+e ψ ∗ ∂βα11 M 1/2 ∂tk f ∂i ∂β−β ∂t f. j 1
α ∂β+e j
α1 α,β1 β+ej
Applying the estimate (3.16), Sobolev’s inequality and the induction hypothesis, one can get that for some
˜2 0 ij
w ψ ∗ ∂ α1 M 1/2 ∂tk f ∂i ∂ α−α1 ∂tm−k f 2 ∞ β1 β−β1 L ([τ∗ ,T ];L2x,v ) M 1/8 ∂tk ∂ α0 +α1 f ∞ w 1 + |v| γ+2 ∂i ∂ α−α1 ∂tm−k f ∞ C β2 β−β1 L ([τ ,T ];L2 ) L ([τ ∗
|β2 ||β1 | |α0 |2
C
˜ w2 ∂tk ∂ α0 +α1 f ∞ β2 L ([τ
|β2 ||β1 | |α0 |2
x,v
2 ∗ ,T ];Lx,v )
˜ w 2 ∂i ∂ α−α1 ∂tm−k f ∞ β−β1 L ([τ
2 ∗ ,T ];Lx,v )
2 ∗ ,T ];Lx,v )
˜ C C.
The other terms can be treated in the same way. Therefore we know that m+1 ∂t f
L∞ ([τ∗ ,T ];H∞ )
˜ C.
This ends the proof of Theorem 1.2. 2 Acknowledgments SQL was supported by the NSFC grants 11101188 and 11271160. XM was supported by ARF (GDEI) (Project No. 11ARF05). The authors would like to thank Professor Hongjun Yu for his fruitful discussion on the topic. References [1] R. Alexandre, S. Ukai, Y. Morimoto, C.-J. Xu, T. Yang, Uncertainty principle and kinetic equations, J. Funct. Anal. 255 (8) (2008) 2013–2066. [2] R. Alexandre, S. Ukai, Y. Morimoto, C.-J. Xu, T. Yang, Regularizing effect and local existence for the non-cutoff Boltzmann equation, Arch. Ration. Mech. Anal. 198 (1) (2010) 39–123. [3] A. Arsen’ev, O. Buryak, On the connection between a solution of the Boltzmann equation and a solution of the Landau– Fokker–Planck equation, Math. USSR Sb. 69 (2) (1991) 465–478. [4] F. Bouchut, L. Desvillettes, Averaging lemmas without time Fourier transform and application to discretized kinetic equations, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 19–36. [5] L. Boudin, L. Desvillettes, On the singularities of the global small solutions of the full Boltzmann equation, Monatsh. Math. 131 (2000) 91–108. [6] Y. Chen, L. Desvillettes, L. He, Smoothing effects for classical solutions of the full Landau equation, Arch. Ration. Mech. Anal. 193 (1) (2009) 21–55. [7] H. Chen, W.-X. Li, C.-J. Xu, Gevrey hypoellipticity for linear and non-linear Fokker–Planck equations, J. Differential Equations 246 (1) (2009) 320–339. [8] H. Chen, W.-X. Li, C.-J. Xu, Analytic smoothness effect of solutions for spatially homogeneous Landau equation, J. Differential Equations 248 (1) (2010) 77–94.
S. Liu, X. Ma / J. Math. Anal. Appl. 417 (2014) 123–143
143
[9] P. Degond, M. Lemou, Dispersion relations for the linearized Fokker–Planck equation, Arch. Ration. Mech. Anal. 138 (2) (1997) 137–167. [10] L. Desvillettes, C. Villani, On the spatially homogeneous Landau equation for hard potentials (I–II), Comm. Partial Differential Equations 25 (1–2) (2000) 179–298. [11] R. Diperna, P.-L. Lions, Global weak solutions of Vlasov–Maxwell systems, Comm. Pure Appl. Math. 42 (6) (1989) 729–757. [12] Y. Guo, The Landau equation in periodic box, Comm. Math. Phys. 231 (2002) 391–434. [13] Y. Guo, Boltzmann diffusive limit beyond the Navier–Stokes approximation, Comm. Pure Appl. Math. 59 (5) (2006) 626–687. [14] L. Hsiao, H.-J. Yu, On the Cauchy problem of the Boltzmann and Landau equations with soft potentials, Quart. Appl. Math. 65 (2) (2007) 281–315. [15] P.-L. Lions, On Boltzmann and Landau equations, Philos. Trans. R. Soc. Lond. Ser. A 346 (1994) 191–3204. [16] S.-Q. Liu, Smoothing effects for the classical solutions to the Landau–Fermi–Dirac equation, Chin. Ann. Math. Ser. B 33 (6) (2012) 857–876. [17] S.-Q. Liu, H.-X. Liu, Optimal convergence rate of the Landau equation with frictional force, Acta Math. Sci. Ser. B 32 (5) (2012) 1781–1804. [18] S.-Q. Liu, X. Ma, Exponential decay of the Landau equation with potential forces, J. Math. Anal. Appl. 394 (1) (2012) 159–176. s stability of the Boltzmann equation and Landau equation with soft potentials, Appl. Math. Comput. [19] X. Ma, S.-Q. Liu, Hx,v 215 (10) (2010) 3721–3730. [20] R.M. Strain, Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations 31 (3) (2006) 417–429. [21] R.M. Strain, Y. Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal. 187 (2) (2008) 287–339. [22] C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Ration. Mech. Anal. 143 (3) (1998) 273–307. [23] C. Villani, A review of mathematical topics in collisional kinetic theory, in: Handbook of Mathematical Fluid Dynamics, vol. I, North-Holland, Amsterdam, 2002, pp. 71–305. [24] T. Yang, H.-J. Yu, Optimal convergence rates of Landau equation with external forcing in the whole space, Acta Math. Sci. Ser. B 4 (2009) 1035–1062. [25] H.-J. Yu, Convergence rate for the Boltzmann and Landau equations with soft potentials, Proc. Roy. Soc. Edinburgh Sect. A 139 (2) (2009) 393–416. [26] H.-J. Yu, Smoothing effects for classical solutions of the relativistic Landau–Maxwell system, J. Differential Equations 246 (10) (2009) 3776–3817. [27] M. Zhan, Local existence of classical solutions to the Landau equations, Transport Theory Statist. Phys. 23 (4) (1994) 479–499.