The non-cutoff boltzmann equation with potential force in the whole space

Acta Mathematica Scientia 2014,34B(5):1519–1539 http://actams.wipm.ac.cn

THE NON-CUTOFF BOLTZMANN EQUATION WITH POTENTIAL FORCE IN THE WHOLE SPACE∗

X#)

Yuanjie LEI (

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China E-mail : [email protected] Abstract This paper is concerned with the non-cutoff Boltzmann equation for full-range interactions with potential force in the whole space. We establish the global existence and optimal temporal convergence rates of classical solutions to the Cauchy problem when initial data is a small perturbation of the stationary solution. The analysis is based on the timeweighted energy method building also upon the recent studies of the non-cutoff Boltzmann equation in [1–3, 15] and the non-cutoff Vlasov-Poisson-Boltzmann system [6]. Key words

non-cutoff Boltzmann; potential force; global existence; convergence rates

2010 MR Subject Classification

1

35A05; 35B65; 35Q35

Introduction

This paper is concerned with the non-cutoff Boltzmann equation with potential force in the whole space R3 ∂t F + v · ∇x F − ∇x φ · ∇v F = Q(F, F ) (1.1) with prescribed initial data F (0, x, v) = F0 (x, v).

(1.2)

Here F = F (t, x, v) ≥ 0 stands for the velocity distribution function for the particles with position x = (x1 , x2 , x3 ) ∈ R3 and velocity v = (v1 , v2 , v3 ) ∈ R3 at time t ≥ 0, φ ≡ φ(x) is the potential function of the external forces which is a given function, and the bilinear collision operator Q(F, G) acting only on the velocity variable is defined by Z n o Q(F, G)(v) = B(v − u, σ) F (u′ )G(v ′ ) − F (u)G(v) dσdu. (1.3) R3 ×S2

Here in terms of velocities u and v before the collision, velocities v ′ and u′ after the collision are defined by

v + u |v − u| v + u |v − u| + σ, u′ = − σ, σ ∈ S2 2 2 2 2 and the Boltzmann collision kernel B(v − u, σ) depends only on the relative velocity |v − u| and on the deviation angle θ given by cos θ = hσ, (v − u)/|v − u|i, where h·, ·i is the usual v′ =

∗ Received March 13, 2013; revised May 20, 2013. This work was supported by the Fundamental Research Funds for the Central Universities.

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dot product in R3 . As in [1–3, 15], without loss of generality, we suppose that B(v − u, σ) is supported on cos θ ≥ 0. Notice also that all the physical parameters, such as the particle mass and all other involving constants, have been chosen to be unit for simplicity of presentation. Throughout the paper, the collision kernel is further supposed to satisfy the following assumptions: (A1) B(v − u, σ) takes the product form in its argument as B(v − u, σ) = Φ(|v − u|)b(cos θ) with Φ and b being non-negative functions; (A2) The angular function σ → b(hσ, (v − u)/|v − u|i) is not integrable on S2 , i.e., Z Z π/2 b(cos θ)dσ = 2π sin θb(cos θ)dθ = ∞. S2

0

Moreover, there are two positive constants cb > 0, 0 < s < 1 such that cb 1 ≤ sin θb(cos θ) ≤ ; 1+2s θ cb θ1+2s (A3) The kinetic function z → Φ(|z|) satisfies Φ(|z|) = CΦ |z|γ

for some positive constant CΦ > 0. Here we should notice that the exponent γ > −3 is determined by the intermolecular interactive mechanism. It is convenient to call hard potentials when γ + 2s ≥ 0 and soft potentials when −3 < γ < −2s. In what follows we will reformulate the problem as in [11]. Denote a normalized 2 Maxwellian µ(v) and µφ (v) by µ(v) = (2π)−3/2 e−|v| /2 and µφ (v) = e−φ µ(v). It is easy to check that µφ (v) is a stationary solution of Equation (1.1). Set F (t, x, v) = µφ (v) + µφ (v)1/2 f (t, x, v), then the Cauchy problem (1.1)–(1.2) can be reformulated as ∂t f + v · ∇x f − ∇x φ · ∇v f + e−φ Lf = e−φ/2 Γ(f, f ) (1.4) with given initial data f (0, x, v) = f0 (x, v). (1.5) 


Here Lf = −µ−1/2 Q µ, µ1/2 f + Q µ1/2 f, µ and Γ(f, g) = µ−1/2 Q µ1/2 f, µ1/2 g denote the linearized and nonlinear collision operators, respectively. As in [12], the null space of L is given by n o N = span µ1/2 , vi µ1/2 (1 ≤ i ≤ 3), |v 2 |µ1/2 .

For given f (t, x, v), one can decompose f (t, x, v) uniquely as f = Pf + {I − P}f

with P being the orthogonal projection from L2v × L2v to ker L defined by   Pf = {a(t, x) + v · b(t, x) + (|v|2 − 3)c(t, x)}µ1/2 ,     Z Z   a= µ1/2 f dv, b = vµ1/2 f dv,  R3 R3  Z     c = 16 (|v|2 − 3)µ1/2 f dv, R3

(1.6)

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Pf and {I − P}f are called the macroscopic component and the microscopic component of f (t, x, v), respectively. The problem on the construction of global solutions near the local Maxwellian µφ (v) for the Cauchy problem (1.4), (1.5) were studied in [21, 22] and [24]. All these results are concentrated on the hard potential case under Grad’s angular cutoff assumption. For the non-cutoff case, although the spectrum of the linearized operator without angular cut-off was analyzed a long time ago by Pao in [19], it is only well-understood recently that, which is due to the study of the non-cutoff Boltzmann equation independently in [1–3] and Gressman-Strain [15, 16], the linearized Boltzmann operator without angular cutoff has the similar anisotropic dissipation phenomenon with the cutoff case. The main purpose of our present manuscript is to see whether or not the approach in [1–3] and [15] for the Boltzmann equation can be applied to the non-cutoff Boltzmann equation with potential force. Before stating our main results, we first introduce some notations used throughout the paper. C denotes some positive constant (generally large) and λ denotes some positive constant (generally small), where both C and λ may take different values in different places. A . B means that there is a generic constant C > 0 such that A ≤ CB. A ∼ B means A . B and m B . A. For an integer m ≥ 0, we use Hx,v , Hxm , Hvm to denote the usual Hilbert spaces H m (R3x × R3ξ ), H m (R3x ), H m (R3v ), respectively, and L2x,v , L2x , L2v are used for the case when m = 0. We denote (·, ·) by the inner product over L2x,v . For q ≥ 1, we also define the mixed velocity-space Lebesgue space Zq = L2v (Lqx ) = L2 (R3v ; Lq (R3x )) with the norm Z Z 2/q 1/2 q kf kZq = |f (x, v)| dx dv R3

R3

for f = f (x, v) ∈ Zq . Moreover, we introduce the norms k · kH˙ m with m ≥ 0 given by kf k2H˙ m ≡ kf kL2v (H˙ xm ) , kf k2H m ≡ kf kL2v (Hxm ) , kf k2L2 ≡ kf k2H 0 . Here H˙ xm = H˙ xm (R3x ) is the standard homogeneous L2x based Sobolev space equipped with norm Z 2 |k|2m |ˆ g (k)|2 dk. kgkH˙ m (R3 ) ≡ x

R3x

For multi-indices α = (α1 , α2 , α3 ) and β = (β1 , β2 , β3 ), we denote ∂βα = ∂xα ∂vβ = ∂xα11 ∂xα22 ∂xα33 ∂vβ11 ∂vβ22 ∂vβ33 . As usual, the length of α is |α| = α1 + α2 + α3 , and β ≤ α means that βj ≤ αj for each j = 1, 2, 3. We now borrow some notations introduced in [15]. The first one is the following sharp weighted geometric fractional Sobolev norm ZZ γ+2s+1 (f ′ − f )2 |f |2Nγs ≡ |f |2L2 + (hvihv ′ i) 2 1d(v,v′ )≤1 dvdv ′ . γ+2s d(v, v ′ )n+2s Here 1Ω is the standard indicator function of the set Ω, hvi = (1 + |v|2 )1/2 , L2l denotes the R weighted space with norm |f |2L2 ≡ R3 |f (v)|2 hvil dv, and the anisotropic metric d(v, v ′ ) meav l q suring the fractional differentiation effects is given by d(v, v ′ ) = |v − v ′ |2 + 14 (|v|2 − |v ′ |2 )2 .

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It is shown in [15, 16] that |f |2L2

γ+2s

Here Hlm (R3v ) =

+ |f |2Hγs . |f |2Nγs . |f |2Hγ+2s . s

  Z f ∈ L2l (R3v ) : |f |2Hlm = hvil |(I − ∆v )m/2 f (v)|2 < ∞ . R3v

In R3x × R3v , k · kHγs = k| · |Hγs kL2x is used, BC ⊂ R3 denotes the ball of radius C centered at the origin, and L2 (BC ) stands for the space L2 over BC and likewise for other spaces. Finally, to state our main result, we define the following velocity weight function w = w(v) by   hvi, γ + 2s ≥ 0, w(v) = (1.7)  hvi−γ , −1 ≤ γ + 2s < 0 and the velocity weight function w(α, β) is given by

w(α, β) = w(v)l−|α|−|β| , l ≥ K.

(1.8)

Corresponding to f = f (t, x, v), we define the instant energy functional El,K (t) and high-order h instant energy functional El,K (t) as functionals satisfying the equivalent relations X

w(α, β)∂βα f (t) 2 , El,K (t) ∼ (1.9) |α|+|β|≤K

h El,K (t) ∼

X

1≤|α|≤K

2

k∂ α Pf (t)k +

X

|α|+|β|≤K



w(α, β)∂ α {I − P}f (t) 2 , β

and the corresponding entropy dissipation rate functional Dl,K (t) is defined by X X

2

w(α, β)∂ α {I − P}f (t) 2 s k∂ α Pf (t)k + Dl,K (t) ∼ β N 1≤|α|≤K

|α|+|β|≤K

(1.10)

(1.11)

γ

for any l ≥ K. With the above notations in hand, our main result for the global solvability of the Cauchy problem (1.4), (1.5) is stated as follows. Theorem 1.1 Assume that one of the following two conditions (i) γ + 2s ≥ 0, K ≥ 2, k∇φkH 5 ≤ δ, (ii) −1 ≤ γ + 2s < 0, 12 ≤ s < 1, K ≥ 4, k∇φkH 7 ≤ δ holds for some sufficiently small positive constant δ. Let l ≥ K and F0 (x, ξ) = µ(v) + µ1/2 (v)f0 (x, v) ≥ 0, there are functionals El,K (t) and Dl,K (t) in the sense of (1.9) and (1.11) such that if the initial data El,K (0) is sufficiently small, then there exists a unique global solution f (t, x, v) to the Cauchy problem (1.4), (1.5) satisfying F (t, x, v) = µ(v) + µ1/2 (v)f (t, x, v) ≥ 0 and the Lyapunov inequality d El,K (t) + λDl,K (t) ≤ 0, t ≥ 0 (1.12) dt holds for some positive constant λ > 0. h Moreover, there is a high-order instant energy functional El,K (t) in the sense of (1.10) such that d h E (t) + λDl,K (t) . k∇x Pf k2 . (1.13) dt l,K

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Our second result in this paper is to show that for the hard potential case, the nonlinear energy estimates in Theorem 1.1 together with time-decay estimates on the linearized system h will lead us to the time-decay rates of the instant energy functionals El,K (t) and El,K (t) provided that the initial perturbation satisfies certain additional integrability assumptions. Theorem 1.2 For the hard potentials, l and K are given in Theorem 1.1. Let f (t, x, v) be the global solution constructed in Theorem 1.1. If we assume further that  max3 k[|x|∇φ, |x|∇2 φ]kL∞ , k|x|∇φkL2 ≤ δ1 (1.14) x∈R

1/2

holds for some sufficiently small positive constant δ1 and that ǫ0 = El,K (0)+kf0 kZ1 is sufficiently small, then we have the following time decay estimates El,K (t) . (1 + t)−3/2 ǫ20 ,

(1.15)

h El,K (t) . (1 + t)−5/2 ǫ20 .

(1.16)

Remark 1.3 Several remarks concerning Theorems 1.1 and 1.2 are listed below: • For particles interacting according to a spherical intermolecular repulsive potential of the form φ(r) = r(1−p) , p ∈ (2, ∞), and the collision kernel B satisfies the conditions above with γ = (p − 5)/(p − 1) and s = 1/(p − 1). In such a case, we have γ = 1 − 4s which satisfy all the conditions listed in Theorems 1.1 and 1.2. • It is worth to pointing out that to guarantee the global solvability of the Cauchy problem (1.4), (1.5), we need to assume that γ + 2s ≥ −1 and the optimal temporal decay estimates on such a global solutions are obtained only for the hard potential case. The global solvability for γ + 2s < −1 and temporal decay for γ + 2s < 0 will be considered in our further study. Now we outline the main ideas to deduce our results. Our analysis is based on the nonlinear energy method developed by Liu-Yang-Yu [18] and Guo [13] for the Boltzmann equation and the weighted energy type estimates on the non cutoff linearized Boltzmann operator and the corresponding nonlinear term obtained independently in [1–3] and Gressman-Strain [15, 16], especially the anisotropic dissipation phenomenon of the linearized Boltzmann operator without angular cutoff similar to that of the cutoff case, play an essential role in our analysis. Compared with the Boltzmann equation, the main difficulty lies in how to control the possible growth induced by the external force and for the soft potential case, another problem is how to compensate the degeneracy of the dissipation phenomenon of the linearized Boltzmann operator. To overcome this difficulty, our trick is to use the weighted energy method with the weight is related to both the spatial variable x and the velocity variable v (for the soft potential case, the weight is also related to the exponent γ), which is motivated by Guo’s work [14] on the construction of global solutions near Maxwellians to the Vlasov-Poisson-Landau systems. But unlike the cases for the Vlasov-Poisson-Landau system and the Vlasov-PoissonBoltzmann system, here the potential function φ(x) is a given function and can not be part of the entropy dissipation rate. A consequence of such an observation is that for the soft potential case, when we deal with the trilinear inner product term (∂ ei φ∂eαi f, w(α, 0)2 ∂ α f ), it can not be bounded by CE 1/2 (t)D(t), what we found is that such a term can be bounded by Ck∇2 φkH 1 kw(α, 0)∂ α f k2L2 if γ + 2s ≥ −1. For the proof of Theorem 1.2, to obtain the time γ+2s decay property of the solution f (t, x, v) for the hard potentials, since the potential force ∇φ

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does not decay with respect to t, we need to assume that  max k[|x|∇φ, |x|∇2 φ]kL∞ , k|x|∇φkL2 ≤ δ1

holds for some sufficiently small δ1 > 0 and we can obtain the time decay of El,K (t) through h time decay of El,K (t) by using the Hardy-Littlewood-Sobolev inequality. The rest of the paper is organized as follows. In Section 2, we will list basic lemmas concerning the properties of L and Γ in the functional framework of [6] and [15], Section 3 and Section 4 are devoted to the proofs of Theorems 1.1 and 1.2, respectively.

2

Preliminaries

In this section, we cite some fundamental results concerning the weighted energy type estimates on the linearized collision operator L and the nonlinear term Γ for non cutoff cases, whose proofs can be found in [6, 15]. We should notice that we suppose −3 < γ < −2s with 1 2 ≤ s < 1 for the soft potentials as in [6] in the following two lemmas. The first lemma concerns the coercivity estimate on the linearized collision operators L. Lemma 2.1 (cf. [6, 15]) For the hard potentials and soft potentials, it holds that (i) (Lg, g) & k{I − P}gk2Nγs ;

(2.1)

(ii) Let l ∈ R. There is C > 0 such that (w2l Lg, g) & kwl gk2Nγs − Ckgk2L2 (BC ) ; (iii) Let β > 0, l ∈ R. For any Cη > 0, C > 0 such that X (w2l ∂β Lg, ∂β g) & kwl ∂β gk2Nγs − η kwl ∂β gk2Nγs − Cη kgk2L2 (BC ) .

(2.2)

(2.3)

β1 ≤β

The second lemma concerns the estimates on the nonlinear collision operator Γ. Lemma 2.2 (cf. [6, 15]) For the hard potentials, we have the basic estimate |hΓ(f, g), hi| . |f |L2v |g|Nγs |h|Nγs . Moreover, suppose that |α| + |β| ≤ K, for any l ≥ K ≥ 2, we have X |hw2l ∂βα Γ(f, g), ∂βα hi| . |wl ∂βα11 f |L2v |wl ∂βα22 g|Nγs |wl ∂βα h|Nγs . α1 +α2 =α, (β1 ,β2 )≤β

(2.4)

(2.5)

For the soft potentials, let gi ∈ C0∞ (R3 , R) (1 ≤ i ≤ 3) and Let |α| + |β| ≤ K (K ≥ 4) with α = α1 + α2 and (β1 , β2 ) ≤ β, then for any l ≥ 0 and m ≥ 0, one has (i) If |α1 | + |α2 | ≤ K/2, X Z |(w2l ∂βα Γ(g1 , g2 ), ∂βα g3 )| . |∂βα11 g1 |2 |wl ∂βα22 g2 |Nγs |wl ∂βα g2 |Nγs dx α1 +α2 =α, (β1 ,β2 )≤β

+

X

R3

α1 +α2 =α, (β1 ,β2 )≤β

Z

R3

|wl ∂βα11 g1 |2 |∂βα22 g2 |Nγs |wl ∂βα g2 |Nγs dx

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+

X

α1 +α2 =α, (β1 ,β2 )≤β

Z

R3

1525

|w−m ∂βα11 g1 |H 2 |wl ∂βα22 g2 |Nγs |wl ∂βα g2 |Nγs dx; (2.6)

(ii) If |α1 | + |α2 | ≥ K/2, |(w2l ∂βα Γ(g1 , g2 ), ∂βα g3 )|

X

.

α1 +α2 =α, (β1 ,β2 )≤β

+

Z

X

Z

|wl ∂βα11 g1 |2 |∂βα22 g2 |Nγs |wl ∂βα g2 |Nγs dx

X

Z

l α s |w ∂ g2 |N s dx. (2.7) |w−m ∂βα11 g1 |2 |wl ∂βα22 g2 |Nγ,2 β γ

α1 +α2 =α, (β1 ,β2 )≤β

+

α1 +α2 =α, (β1 ,β2 )≤β

3

|∂βα11 g1 |2 |wl ∂βα22 g2 |Nγs |wl ∂βα g2 |Nγs dx

R3

R3

R3

Global Solvability

This section is devoted to the proof of Theorem 1.1. To do so, we only need to deduce certain uniform-in-time a priori estimates on the solution to the Cauchy problem (1.4), (1.5). For this purpose, suppose that the Cauchy problem (1.4), (1.5) admits a smooth solution f (t, x, v) over 0 ≤ t ≤ T for some 0 < T ≤ ∞ and the solution f (t, x, v) satisfies the a priori assumption sup El,K (t) ≤ δ0 ,

(3.1)

0≤t≤T

what we want to do is to deduce certain energy type estimates on f (t, x, ξ) under the assumption that δ0 > 0 is a suitably small constant. To make the presentation easy to follow, we divide the analysis into two subsections and the first one is concerned with the macro dissipation. 3.1

Macro Dissipation

This subsection is concerned with the analysis of the macro dissipation. As in [5], by introducing the high-order moment functions A = (Ajm )3×3 and B = (B1 , B2 , B3 ) by E D E 1 D 2 (|v| − 5)vj µ1/2 , f , (3.2) Ajm (f ) = (vj vm − 1)µ1/2 , f , Bj (f ) = 10 one can derive the corresponding local conservation laws. In fact, by multiplying (1.4) by µ1/2 , vi µ1/2 (i = 1, 2, 3) and 16 (|v|2 − 3)µ1/2 and then integrating them over v ∈ R3 , one has   ∂t a + ∇x · b = 0,     ∂t b + ∇x (a + 2c) + ∇x A({I − P}f ) = h∇x φ · ∇v f, vµ1/2 i, (3.3)    1 5 1  ∂t c + ∇x · b + ∇x · B({I − P}f ) = h∇x φ · ∇v f, (|v|2 − 3)µ1/2 i. 3 3 6 Moreover, we need the equations of high-order moments. For this purpose, one can get by 1 multiplying (1.4) by (vj vm − 1)µ1/2 and 10 (|v|2 − 5)vj µ1/2 and then integrating them over 3 v ∈ R that    ∂t Ajj ({I − P}f + 2c) + 2∂j bj = Ajj (r + h), ∂t Ajm {I − P}f + ∂j bm + ∂m bj = Ajm (r + h),

  

∂t Bj {I − P}f + ∂j c = Bj (r + h)

j 6= m,

(3.4)

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with r = −v · ∇x {I − P}f + e−φ L{I − P}f,

(3.5)

h = ∇x φ · ∇v f + Γ(f, f ).

(3.6)

For fixed j ∈ {1, 2, 3}, one can deduce from (3.3)1 and (3.3)2 ,   X X 1 ∂j Amm ({I − P}f ) − ∂m Ajm {I − P}f −∆x bj − ∂j ∂j bj = −∂t 2 m m6=j

X 1 X + ∂j Amm (r + h) − ∂m Ajm (r + h). 2 m

(3.7)

m6=j

Now we focus on (3.3)–(3.4) to estimate the higher order derivatives of the macroscopic coefficients (a, b, c) in L2 norm. Lemma 3.1 For any |α| ≤ K − 1 and 1 ≤ j, m ≤ 3, it holds that 2

k∂ α Ajm {I − P}f, ∂ α Bj {I − P}f k . k∂ α {I − P}f k2Nγs . Moreover, for any |α| ≤ K − 1 and 1 ≤ j, m ≤ 3, we have X ′ k∂ α {I − P}f k2Nγs k(∂ α Ajm (r), ∂ α Bj (r))k2 .

(3.8)

(3.9)

|α′ |≤|α|+1

and k(∂ α Ajm (∇x φ · ∇v f ), ∂ α Bj (∇x φ · ∇v f ))k2 . δ 2 Dl,K (t),

Z

2

∂ α Γ(f, f )dv . El,K (t)Dl,K (t).

(3.10) (3.11)

Proof For brevity, we only prove (3.10) with α = 0, since the other cases in (3.10) and (3.8), (3.9) can be also proved in the similar ways and (3.11) was proved in [6]. Z D E2 2 kAjm (∇x φ · ∇v f )k = (vj vm − 1)µ1/2 , (∇x φ · ∇v f ) dx 3 ZR D E2 = (vj vm − 1)µ1/2 , (∇x φ · ∇v Pf ) dx R3 Z D E2 + (vj vm − 1)µ1/2 , (∇x φ · ∇v {I − P}f ) dx, R3

and the terms on the right hand side of the above equation can be estimated as follows respectively: Z D E2 (vj vm − 1)µ1/2 , (∇x φ · ∇v Pf ) dx 3 ZR D E2 = (vj vm − 1)µ1/2 , (∇x φ · ∇v [(a + v · b + (|v|2 − 3)c)µ1/2 ] dx 3 ZR . |∇x φ|2 |(a, b, c)|2 dx . k(a, b, c)k2L∞ k∇x φk2 . δ 2 Dl,K (t), R3

Z = .

3

D

ZR D ZR

E2 (vj vm − 1)µ1/2 , (∇x φ · ∇v {I − P}f ) dx (vj vm − 1)µ1/2 , (∇x φ · ∇v {I − P}f

3

R3

|∇x φ|2 |{I − P}f |2 dx . δ 2 Dl,K (t).

E2

dx

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Collecting the above two estimates we get (3.10) with α = 0, this completes the proof of Lemma 3.1.  Based on Lemma 3.1, our main result in this subsection is K Theorem 3.2 There is an interactive energy functional Eint (t) such that X K |Eint (t)| . k∂ α f (t)k2

(3.12)

|α|≤K

and d K E (t) + λ dt int

X

k∂ α ∇x (a, b, c)(t)k2 .

|α|≤K−1

X

k∂ α {I − P}f k2Nγs + (δ 2 + El,K (t))Dl,K (t),

|α|≤K

(3.13)

K where Eint (t) is the linear combination of the following terms over |α| ≤ K − 1 and 1 ≤ j ≤ 3, X Z X Z K Eint (t) ∼ ∂ α Bj {I − P}f · ∂j ∂ α cdx + ∂ α b · ∇x ∂ α adx |α|≤K−1

+

X

R3

|α|≤K−1

Z

R3

X

|α|≤K−1

∂j ∂ α Amm {I − P}f −

X

R3

∂m ∂ α Ajm {I − P}f

m

m6=j



· ∂ α bj dx.

K,h Moreover, there is also a time-frequency interaction functionals Eint (t) satisfying X X K,h k∂ α {I − P}f k2 , k∂ α Pf k2 + Eint (t) . 1≤|α|≤K

(3.14)

|α|≤K

and X d K,h k∂ α ∇x (a, b, c)(t)k2 Eint (t) + λ dt 1≤|α|≤K−1 X α k∂ {I − P}f k2Nγs + (δ 2 + El,K (t))Dl,K (t), .

(3.15)

|α|≤K

K,h K where Eint (t) has the similar form as Eint (t).

The proof of Theorem 3.2 is divided into the following four steps:

Proof

Estimate on c. For any ǫ > 0, it holds that X d X X α (∂ Bj {I − P}f, ∂j ∂ α c) + k∇x ∂ α ck2 dt |α|≤K−1 j |α|≤K−1 X X α 2 α ≤ǫ k∂ ∇x bk + Cǫ k∂ {I − P}uk2Nγs + (δ 2 + Cǫ El,K (t))Dl,K (t).

Step 1

|α|≤K−1

(3.16)

|α|≤K

Applying ∂xα with |α| ≤ K − 1 to the macroscopic equation (3.3)3 , multiplying it by ∂j ∂xα c, and then integrating it over R3 , we have k∂ α ∂j ck2 = (∂ α ∂j c, ∂ α ∂j c) = (∂ α ∂j c, −∂ α ∂t Bj {I − P}f + ∂ α Bj (r + h)) = −∂t (∂ α ∂j c, ∂ α Bj {I − P}f ) + (∂ α ∂j ∂t C, ∂ α Bj {I − P}f ) + (∂ α ∂j c, ∂ α Bj (r + h)) ≤ −∂t (∂ α ∂j c, ∂ α Bj {I − P}f ) + (∂ α ∂j ∂t c, ∂ α Bj {I − P}f ) + ǫk∂ α ∂j ck2 X ′ +Cǫ k∂xα {I − P}uk2Nγs + (δ 2 + El,K (t))Dl,K (t), |α′ |≤|α|+1

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and one can also employ (3.3)3 to get (∂ α ∂j ∂t c, ∂ α Bj {I − P}f )      1 5 1 α α 2 1/2 , ∂ Bj {I − P}f = ∂ ∂j − ∇x · b − ∇x · B({I − P}u + ∇x φ · ∇v f, (|v| − 3)µ 3 3 6 X ′ . ǫk∂ α ∂j bk2 + Cǫ k∂xα {I − P}f k2Nγs + (δ 2 + El,K (t))Dl,K (t). |α′ |≤|α|+1

So we get ∂t (∂ α ∂j c, ∂ α Bj {I − P}f ) + k∂ α ∂j ck2 X ′ . ǫk∂ α ∂j bk2 + k∂xα {I − P}uk2Nγs + (δ 2 + El,K (t))Dl,K (t),

(3.17)

|α′ |≤|α|+1

and (3.16) follows by taking summation over 1 ≤ |α| ≤ K. Step 2 Estimate on b. For any ǫ > 0, it holds that   X X d X ∂j ∂ α Amm {I − P}f − ∂m ∂ α Ajm {I − P}f, ∂ α bj + k∇x ∂ α bk2 dt m m6=j |α|≤K−1 X X k∂ α {I − P}uk2Nγs + (δ 2 + Cǫ El,K (t))Dl,K (t). (3.18) k∂ α ∇x bk2 + Cǫ ≤ǫ |α|≤K

|α|≤K−1

In fact, for fixed j ∈ {1, 2, 3}, apply ∂ α to the elliptic-type equation (3.7), multiply it by ∂xα bj , and then integrate it over R3 to find   X d X α α α ∂j ∂ Amm {I − P}f − ∂m ∂ Ajm {I − P}f, ∂ bj + k∇x ∂ α bj k2 + k∂j ∂ α bj k2 dt m m6=j  X  X 1 α α α = ∂j ∂ Amm ({I − P}u) − ∂m ∂ Ajm {I − P}f, ∂x ∂t bj 2 m m6=j   X X 1 ∂j ∂ α Amm (r + h) − ∂m ∂ α Ajm (r + h), ∂ α bj + 2 m m6=j

≡ R1 + R2 .

Using (3.3)2 and the above lemma, R1 ≤ ǫk∂ α ∂t bj k2 + Cǫ

2 X

α′

∂ A({I − P}f )

|α′ |≤K

≤ 4ǫk∂ α ∇x (a, c)k2 + Cǫ

2 X

α′

∂ {I − P}f .

|α′ |≤K

Nγs

For R2 , using the above lemma and integrating by parts imply X 1 X α R2 = − (∂ Amm (r + h), ∂j ∂ α bj ) + (∂ α Ajm (r + h), ∂m ∂ α bj ) 2 m m6=j X α 2 α ≤ ǫk∇x ∂ bj k + Cǫ k∂ Ajm (r + h)k2 m

α

2

≤ ǫk∇x ∂ bj k + Cǫ

X

k∂ α {I − P}uk2Nγs + (δ 2 + Cǫ El,K (t))Dl,K (t),

|α|≤K

and (3.18) follows from the above estimates and by taking summation over 1 ≤ j ≤ 3.

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1529

Step 3 Estimate on a. Let |α| ≤ K − 1. Applying ∂ α to(3.3)2 , multiplying it by ∂ α ∇x a, and then integrating it over R3 , we have ∂t (∂ α b, ∂ α ∇x a) + k∂ α ∇x ak2 = −(2∂ α ∇x c + ∂ α ∇x A({I − P}f ) + h∇x φ · ∇v f, vµ1/2 i, ∂ α ∇x a) + (∂ α b, ∂t ∂ α ∇x a) = −(2∂ α ∇x c + ∂ α ∇x A({I − P}f ) + h∇x φ · ∇v f, vµ1/2 i, ∂ α ∇x a) + (∂ α ∇x b, ∂ α ∇x b) X ≤ ǫk∂xα ∇x ak2 + Ck∂ α ∇x (b, c)k2 + Cǫ k∂ α {I − P}uk2Nγs + (δ 2 + Cǫ El,K (t))Dl,K (t). |α|≤K

(3.19)

Here we used (3.4)1 . Step 4 For sufficiently small positive constants κ1 , κ2 , if we set X Z X Z K Eint (t) = ∂ α B{I − P}f · ∇x ∂ α cdx + κ1 ∂ α b · ∇x ∂ α adx |α|≤K−1

+κ2

R3

X

|α|≤K−1

X Z

|α|≤K−1 j,m=1

R3

X

∂j ∂ α Amm {I − P}f −

R3

X m

m6=j

 ∂m ∂ α Ajm {I − P}f · ∂ α bj dx,

then a suitable linear combination of (3.16), (3.17), (3.18), and (3.19) and by taking summation over |α| ≤ K − 1, we get that X d K Eint (t) + λ k∂ α ∇x (a, b, c)(t)k2 dt |α|≤K−1 X α k∂ {I − P}f k2Nγs + (δ 2 + El,K (t))Dl,K (t). . |α|≤K

This is exactly (3.13) and the proofs of (3.14) and (3.15) can be obtained by a similar way. This completes the proof of Theorem 3.2.  3.2

Weighted Energy Estimates

The goal of this subsection is to deduce the desired weighted energy type estimates based on the a priori assumption (3.1). To this end, we need some estimates on the nonlinear term Γ(f, f ). Lemma 3.3 Let l and K be given in Theorem 1.2. It holds that when |α| ≤ K 1/2

|(∂ α Γ(f, f ), ∂ α {I − P}f )| . El,K (t)Dl,K (t),

(3.20)

2
w (0, 0)Γ(f, f ), {I − P}f . E 1/2 (t)Dl,K (t), l,K

(3.21)

and if α = 0, one has

while if 1 ≤ |α| ≤ K, one has 2
w (α, 0)∂ α Γ(f, f ), ∂ α f . E 1/2 (t)Dl,K (t). l,K

Moreover, when |α| + |β| ≤ K with |β| ≥ 1, we can obtain   2 1/2 w (α, β)∂βα Γ(f, f ), ∂βα {I − P}f . El,K (t)Dl,K (t).

(3.22)

(3.23)

Proof We only prove (3.21) for the hard potentials since for the soft potential case, the desired estimates can be deduced similarly by employing (2.6) and (2.7). (w2 (0, 0)Γ(f, f ), {I − P}f )

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= (w(0, 0)Γ(Pf, Pf ), {I − P}f ) + (w2 (0, 0)Γ(Pf, {I − P}f ), {I − P}f ) +(w2 (0, 0)Γ({I − P}f, Pf ), {I − P}f ) + (w2 (0, 0)Γ({I − P}f, {I − P}f ), {I − P}f ) ≡

4 X

(3.24)

Gi .

i=1

Using (2.5), we have Z G1 . |w(0, 0)Pf |L2v |w(0, 0)Pf |Nγs |w(0, 0){I − P}f |Nγs dx R3

1/2

. k∇x (a, b, c)kH 1 k(a, b, c)kkw(0, 0){I − P}f kNγs . El,K (t)Dl,K (t). Similar to G1 , we can easily obtain that 1/2

G2 + G3 . k∇x (a, b, c)kH 1 kw(0, 0){I − P}f kNγs kw(0, 0){I − P}f kNγs . El,K (t)Dl,K (t). 1/2

Using (2.5), G4 can be bounded by CEl,K (t)Dl,K (t), plugging the above estimates into (3.24), we obtain (3.21). The other two estimates can be proved in a similar way. Here we should notice that the trinomial nonlinear estimates for hard potentials and soft potentials is different, so the K ≥ 2 for hard potentials and K ≥ 4 for soft potentials, the detail computation is trivial, we omit it.  The following two lemmas concern the estimates on ∇x φ · ∇v f . For convenience sake, we use ei to denote the multi-index with the ith element unit and the rest ones zeros. Lemma 3.4 Let l and K be given in Theorem 1.2. If 1 ≤ |α| ≤ K, it holds that X 1 f, w2 (α, 0)∂ α f ) . δDl,K (t). (3.25) (∂ α1 +ei φ∂eα−α i α1 ≤α

Proof

For the hard potentials, when α1 = 0, (∂ ei φ∂eαi f, w2 (α, 0)∂ α f ) = −(2∂ ei φw(α, 0)∂ei w(α, 0), (∂ α f )2 ) . k∇2 φkH 1 kw(α, 0)∂ α f k2L2

γ+2s

,

when α1 6= 0, X

1 f, w2 (α, 0)∂ α f ) (∂ α1 +ei φ∂eα−α i

1≤|α1 |≤|α|

X

.

1 f kL2γ+2s kw(α, 0)∂ α f kL2γ+2s k∂ α1 ∇2 φkH 1 kw(α − α1 , ei )∂eα−α i

1≤|α1 |≤|α|

. δDl,K (t).

(3.26)

For the soft potentials, in this paper we restrict γ + 2s ∈ [−1, 0), so one has |∂ei w(α, 0)| . |w(α, 0)|hvi−1 ≤ |w(α, 0)|hviγ+2s . Hence, (∂ ei φ∂eαi f, w2 (α, 0)∂ α f ) = −(2∂ ei φw(α, 0)∂ei w(α, 0), (∂ α f )2 ) . k∇2 φkH 1 kw(α, 0)∂ α f kL2γ+2s . When α1 = 6 0, X

1 (∂ α1 +ei φ∂eα−α f, w2 (α, 0)∂ α f ) i

1≤|α1 |≤|α|

No.5

Y.J. Lei: NON-CUTOFF BOLTZMANN EQUATION WITH POTENTIAL FORCE

X

=

1531

1 1 {(∂ α1 +ei φ∂eα−α Pf, w2 (α, 0)∂ α f ) + (∂ α1 +ei φ∂eα−α {I − P}f, w2 (α, 0)∂ α Pf ) i i

1≤|α1 |≤|α|

1 +(∂ α1 +ei φ∂eα−α {I − P}f, w2 (α, 0)∂ α {I − P}f )}. i

(3.27)

The first term on the right hand side of (3.27) can be estimated as follows: X 1 (∂ α1 +ei φ∂eα−α Pf, w2 (α, 0)∂ α f ) i 1≤|α1 |≤|α|

X

.

k∂ α1 ∇φkH 1 k∇∂ α−α1 (a, b, c)kkw(α, 0)∂ α f kL2γ+2s

1≤|α1 |≤|α|

. δDl,K (t). The second term on the right hand side of (3.27) can be bounded by CδDl,K (t) as the above estimates. For the third term on the right hand side of (3.27), X 1 (∂ α1 +ei φ∂eα−α {I − P}f, w2 (α, 0)∂ α {I − P}f ) i 1≤|α1 |≤|α|

=

X

{(∂ α1 +ei φ∂ei [w(α − α1 )w−|α1 |/2 ∂ α−α1 {I − P}f ], w(α, 0)w−|α1 |/2 ∂ α f )

1≤|α1 |≤|α|

−(∂ α1 +ei φ∂ei [w(α − α1 , 0)w−|α1 |/2 ]∂ α−α1 {I − P}f, w(α, 0)w−|α1 |/2 ∂ α f )} ≡ J1 + J2 . For J2 , it is straightforward to estimate it X −(∂ α1 +ei φ∂ei [w(α − α1 , 0)w−|α1 |/2 ]∂ α−α1 {I − P}f, w(α, 0)w−|α1 |/2 ∂ α f )} J2 = 1≤|α1 |≤|α|

.

X

k∂ α1 ∇2 φkH 1 kw(α − α1 , 0)∂ α−α1 {I − P}f kL2γ+2s kw(α, 0)∂ α {I − P}f kL2γ+2s

1≤|α1 |≤|α|

. δDl,K (t). For J1 , noticing 1/2 < s < 1, by the Parseval identity, one can obtain Z Z −|α1 |/2 α−α1 ∂ (I − P)f ] J1 ≤ 3 iξi Fv [wl (α − α1 , 0)w 3 R R Fv [wl (α, 0)w−|α1 |/2 ∂ α (I − P)f ]dξ |∂ α1 +ei φ|dx Z 1/2 . hξi Fv [wl (α − α1 , 0)w−|α1 |/2 ∂ α−α1 (I − P)f ] 2 Lξ R3 1/2 hξi Fv [wl (α, 0)w−|α1 |/2 ∂ α (I − P)f ] 2 |∂ α1 +ei φ|dx Lξ Z . wl (α − α1 , 0)w−|α1 |/2 ∂ α−α1 (I − P)f s Hv 3 R wl (α, 0)w−|α1 |/2 ∂ α (I − P)f |∂ α1 +ei φ|dx . k∂

α1

2

∇ φkH 1 kwl (α − α1 , 0)∂

Hvs α−α1

(I − P)f kHγs kwl (α, 0)∂ α (I − P)f kHγs

. δDl,K (t), where Fv is the usual Fourier transform with respect to v-variable, ξ denotes the corresponding √ frequency variable, ¯· denotes the complex conjugate, and i = −1 is the pure imaginary unit. Collecting the estimates above, this completes the proof of Lemma 3.4. 

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Lemma 3.5 Let l and K be given in Theorem 1.2, and take 1 ≤ |α| + |β| ≤ K with |β| ≥ 1.Then, it holds that X α−α1 (∂ α1 +ei φ∂β+e {I − P}f, w2 (α, β)∂βα {I − P}f ) . δDl,K (t). (3.28) i α1 ≤α

This lemma can be proved by using the same argument as Lemma 3.4.

Proof



The next lemma is concerned with the energy estimates based on the a priori assumption (3.1). Lemma 3.6 Let l and K be given in Theorem 1.2, there is El,K (t) satisfying (1.9) such that d El,K (t) + λDl,K (t) ≤ 0 (3.29) dt holds for any 0 ≤ t ≤ T . The proof is divided into three steps. For convenience sake, we write (1.4) as

Proof

∂t f + vi ∂ ei f + e−φ Lf = e−φ/2 Γ(f, f ) + ∂ ei φ∂ei f.

(3.30)

Step 1 Non-weighted energy estimates: Applying ∂ α with |α| ≤ K to (3.30) and taking the inner product with ∂ α f over R3 × R3 , we have (∂t ∂ α f, ∂ α f ) + (e−φ ∂ α Lf , ∂ α f ) X (∂ α1 (e−φ )∂ α−α1 Lf, ∂ α f ) = −(vi ∂ ei +α f, ∂ α f ) − 1≤|α1 |≤|α|

X

+

(∂

α1 −φ/2 α−α1

e

∂

Γ(f, f ), ∂ α f ) +

X

1 f, ∂ α f ). (3.31) (∂ α1 +ei φ∂eα−α i

0≤|α1 |≤|α|

0≤|α1 |≤|α|

1 d α 2 2 dt k∂ f k +λk{I

The two terms on the left hand side is just − P}∂ α f k2Nγs from Lemma 2.1. Now we estimate the terms on the right hand side. It is easy to see that −(vi ∂ ei +α f, ∂ α f ) = 0. 1/2 From Lemma 3.3, the other terms on the right hand side are bounded by C(δ + El,K (t))Dl,K (t). Thus we can obtain 1 d α 2 2 dt k∂ f k

1/2

+ λk{I − P}∂ α f k2Nγs . (δ + El,K (t))Dl,K (t).

(3.32)

By combining (3.13) and the above estimates, we conclude in this step that  X  X X d K k{I − P}∂ α f k2Nγs k∂ α f k2 + κEint k∂∇(a, b, c)k2 + λ (t) + λ dt |α|≤K 2

. (δ + δ + Step 2 have

1/2 El,K (t)

|α|≤K−1

|α|≤K

+ El,K (t))Dl,K (t).

(3.33)

Energy estimates with weight function w(α, β). Applying {I − P} to (3.30), we

∂t {I − P}f + vi ∂ ei {I − P}f + e−φ L{I − P}f = ∂ ei φ∂ei {I − P}f + P(vi ∂ ei f ) − vi ∂ ei Pf − P(∂ ei φ∂ei f ) + ∂ ei φ∂ei Pf + e−φ/2 Γ(f, f ). (3.34) First, multiplying the above equation by w2 (0, 0){I − P}f and taking the integration over R3 × R3 , one has 1 d kw(0, 0){I − P}f k2 + (e−φ L{I − P}f, w2 (0, 0){I − P}f ) 2 dt

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1533

= (∂ ei φ∂ei {I − P}f, w2 (0, 0){I − P}f ) + (P(vi ∂ ei f ), w2 (0, 0){I − P}f ) −(vi ∂ ei Pf, w2 (0, 0){I − P}f ) − (P(∂ ei φ∂ei f ), w2 (0, 0){I − P}f ) +(∂ ei φ∂ei Pf, w2 (0, 0){I − P}f ) + (e−φ/2 Γ(f, f ), w2 (0, 0){I − P}f ) 6 X ≡ Ii .

(3.35)

i=1

The second left hand term can be estimated as follows by Lemma 2.1.

(e−φ L{I − P}f, w2 (0, 0){I − P}f ) & kw(0, 0){I − P}f k2Nγs − k{I − P}f k2L2 (BC ) . Now we estimate the terms on the right hand side of (3.35). We use the Cauchy-Schwartz inequality with ǫ, we obtain I2 + I3 . Cǫ k∇x f k2L2

γ+2s

+ ǫkw(0, 0){I − P}f kL2γ+2s .

For I4 and I5 , I4 + I5 . k∇φkH 2 k∇f kL2γ+2s kw(0, 0){I − P}f kL2γ+2s . δDl,K (t). 1/2

I1 and I6 have the upper bound CδDl,K (t) and CEl,K (t)Dl,K (t) by Lemma 3.4 and Lemma 3.3, respectively. Thus we conclude in this step that d kw(0, 0){I − P}f k2 + λkw(0, 0){I − P}f k2Nγs dt 1/2 . k{I − P}f k2Nγs + k∇x f k2L2 + {δ + El,K (t)}Dl,K (t). (3.36) γ+2s

Second, For the weighted estimate on the spatial derivatives, we start with (3.30). In fact, take 1 ≤ |α| ≤ K, by applying ∂ α to (3.30) and taking the inner product with w2 (α, 0)∂ α f over R3 × R3 , one has d kw(α, 0)∂ α f k2 + λkw(α, 0)∂ α f k2Nγs − Ck∂ α f k2L2 BC dt X α1 −φ α−α1 2 α .− (∂ (e )L(∂ f ), w (α, 0)∂ f ) 0<α1 ≤α

+(∂ α (e−φ/2 Γ(f, f )), w2 (α, 0)∂ α f ) + (∂ α (∂ ei φ∂ei f ), w2 (α, 0)∂ α f )

≡

9 X

(3.37)

Ii .

i=7

1/2

From Lemma 3.3, I7 and I8 have the upper bound C{δ + El,K (t)}Dl,K (t), and I9 can be bounded by CδDl,K (t) from Lemma 3.4. Hence we take summation over 1 ≤ |α| ≤ K and conclude in this step that X d X kw(α, 0)∂ α f k2 + λ kw(α, 0)∂ α f k2Nγs dt 1≤|α|≤K 1≤|α|≤K X 1/2 . k∂ α f k2Nγs + {δ + El,K (t)}Dl,K (t). (3.38) 1≤|α|≤K

Third, For the weighted estimate on the mixed derivatives, we start with (3.34), where |α| + |β| ≤ K with 1 ≤ m ≤ K, |β| = m, applying ∂βα to (3.34), taking the inner product with w2 (α, β)∂βα {I − P}f over R3 × R3 , one obtains d kw(α, β)∂βα {I − P}f k2 + (∂βα (e−φ L{I − P}f ), w2 (α, β)∂βα {I − P}f ) dt

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= −(∂βα (vi ∂ ei {I − P}f ), w2 (α, β)∂βα {I − P}f ) +(∂βα (∂ ei φ∂ei {I − P}f ), w2 (α, β)∂βα {I − P}f ) + (∂βα (P(vi ∂ ei f )), w2 (α, β)∂βα {I − P}f ) −(∂βα (vi ∂ ei Pf ), w2 (α, β)∂βα {I − P}f ) − (∂βα [P(∂ ei φ∂ei f )], w2 (α, β)∂βα {I − P}f ) +(∂βα (∂ ei φ∂ei Pf ), w2 (α, β)∂βα {I − P}f ) + (∂βα (e−φ/2 Γ(f, f )), w2 (α, β)∂βα {I − P}f ) ≡

16 X

(3.39)

Ii .

i=10

The second left hand-side term can be estimated as follows by Lemma 2.1: (∂βα (e−φ L({I − P}f )), w2 (α, β)∂βα {I − P}f ) X & λkw(α, β){I − P}f k2Nγs − η kw(α, β1 )∂βα {I − P}f k2Nγs − Cη k{I − P}f k2L2

BC

β1 ≤β

+

X

(∂ α1 e−φ ∂βα−α1 L{I − P}f, w2 (α, β)∂βα {I − P}f ),

1≤|α1 |≤|α|

and X

(∂ α1 e−φ ∂βα−α1 L{I − P}f, w2 (α, β)∂βα {I − P}f ) . δDl,K (t).

1≤|α1 |≤|α|

For I10 , we have

−(∂βα (vi ∂ ei {I − P}f ), w2 (α, β)∂βα {I − P}f ) = −(vi ∂βα+ei {I − P}f, w2 (α, β)∂βα {I − P}f ) X α+ei − Cβei (∂β−e {I − P}f, w2 (α, β)∂βα {I − P}f ) i X ′ kw(α′ , β ′ )∂βα′ {I − P}f k2Nγs . ≤ |α′ |+|β ′ |≤K, |β ′ |=|β|−1

I11 can be bounded by CδDl,K (t) from Lemma 3.5. For I12 , we use Cauchy-Schwartz inequality with ǫ, we obtain that I12 + I13 ≤ ǫkw(α, β)∂βα {I − P}f k2Nγs + Cǫ k∇|α|+1 f kL2γ+2s . And other terms, I14 + I15 . δDl,K (t). 1/2 CEl,K (t)Dl,K (t)

I16 has the upper bound by Lemma 3.3. Therefore, by plugging all the above estimates into (3.39), taking the summation over {|β| = m, |α| + |β| ≤ K} for 1 ≤ m ≤ K,and then taking the proper linear combination of those K estimates with properly chosen constants Cm > 0 (1 ≤ m ≤ K) small enough, we have K d X Cm dt m=1

.

X

|α|≤K−1

X

kw(α, β)∂βα {I − P}f k2 + λkw(α, β)∂βα {I − P}f k2Nγs

|α|+|β|≤K |β|=m

k∂ α ∇(a, b, c)k2 +

X

1/2

k∂ α {I − P}f k2Nγs + {δ + El,K (t)}Dl,K (t).

(3.40)

|α|≤K

Step 3 We are in a position to prove (3.29) by taking the proper linear combination of those estimates obtained in the previous two steps as follows. d 1/2 El,K (t) + λDl,K (t) . {δ + δ 2 + El,K (t) + El,K (t)}Dl,K (t), (3.41) dt

No.5

Y.J. Lei: NON-CUTOFF BOLTZMANN EQUATION WITH POTENTIAL FORCE

where El,K (t) is given by   X  K El,K (t) = C2 C1 k∂ α f k2 + κEint (t) + kw(0, 0){I − P}f k2 + |α|≤K

+

K X

m=1

Cm

X

X

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kw(α, 0)∂ α f k2

1≤|α|≤K



kw(α, β)∂βα {I − P}f k2 .

|α|+|β|≤K |β|=m

It is easy to see El,K (t) ∼

X

k∂ α Pf k2 +

|α|≤K

X

kw(α, β)∂βα {I − P}f k2 .

|α|+|β|≤K

Recalling the a priori assumption (3.1), we can obtain d El,K (t) + λDl,K (t) ≤ 0. dt Thus we complete the proof of the lemma and close the a priori estimates (3.1). 3.3



The Proof of Theorem 1.1

Fix l, K as stated in Theorem 1.1, the local existence and uniqueness of the solution f (t, x, v) to the Cauchy problem (1.4)–(1.5) can be proved in terms of the energy functional El,K (t) which are given in (1.9), and the details are omitted for simplicity; see [12] with a little modification. Now we have obtained the unform-in-time estimate over 0 ≤ t ≤ T with 0 < T ≤ ∞. By the standard continuity argument, the global existence follows provided the initial energy functional El,K (0) is sufficiently small.

4

Temporal Decay for the Hard Potential Case

In this section we study the time-decay properties of solutions to the Cauchy problem (1.4)–(1.5). 4.1

Linear Time Decay We consider the linearized Boltzmann equation with a nonhomogeneous source g = g(t, x, v):  ∂ f + v · ∇ f + Lf = g, t x (4.1) f |t=0 = f0 .

So the solution of (5.1) formally take the following form Z t f (t) = S(t)f0 + S(t − s)g(s)ds, S(t) ≡ e−tA , A ≡ L + v · ∇x .

(4.2)

0

The following theorem concerns the time decay of the solution to the linearized system (4.1) with g = 0 whose proof can be found in [20]. Theorem 4.1 (cf. [20]) Fix 1 ≤ r ≤ 2 and l ∈ R. Consider the Cauchy problem (5.1) with g = 0, the solution of the linearized homogenous systems satisfies (m > 0) kwl S(t)f0 kH˙ m . (1 + t)−σr,m kwl f0 kH˙ m T Zr

(4.3)

σr,m ≡ 3/2(1/r − 1/2) + m/2.

(4.4)

for the hard potentials and any t ≥ 0 with σr,m being given by

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Nonlinear Time Decay To deduce the nonlinear time decay, we first give the following estimate

h Lemma 4.2 Let l and K be given in Theorem 1.1, there is El,K (t) satisfying (1.10) such that d h E (t) + λDl,K (t) . k∇Pf k2 . (4.5) dt l,K holds for any 0 ≤ t ≤ T .

Proof By letting |α| ≥ 1 in (3.31), repeating those computations in (3.31)-(3.33), one can instead obtain X d X 1/2 k∂ α f k2 + λ k∂ α {I − P}f k2Nγs . (δ + El,K (t))Dl,K (t). (4.6) dt 1≤|α|≤K

1≤|α|≤K

Moreover, taking the inner product of (3.34) with {I − P}f over R3 × R3 gives 1 d k{I − P}f k2 + k(I − P)f k2Nγs 2 dt . |(P(vi ∂ ei f ), {I − P}f )| + |(vi ∂ ei Pf, {I − P}f )| + |(P(∂ ei φ∂ei f ), {I − P}f )| +|(∂ ei φ∂ei Pf, {I − P}f )| + |(e−φ/2 Γ(f, f ), {I − P}f )| 5 X ≡ Hi .

(4.7)

H=1

Now we estimate the right hand side term of (4.7) term by term. We use Cauchy-Schwartz inequality with ǫ, we obtain that H1 + H2 . Cǫ k∇f kL2γ+2s + ǫk{I − P}f k2Nγs . For H3 and H4 , H3 + H4 . δDl,K (t). 1/2

The term H5 can be bounded by CEl,K (t)Dl,K (t) from Lemma 3.3. Collecting all the above estimates, we have d 1/2 k{I − P}f k2 + k{I − P}f k2Nγs . k∇f kL2γ+2s + (δ + El,K (t))Dl,K (t). dt Combine the above estimate with (4.6) and (3.15), we get   X d K,h α 2 2 k∂ f k + k{I − P}f k + κEint dt 1≤|α|≤K X X +λ k∂ α ∇(a, b, c)k2 + λ k∂ α {I − P}f k2Nγs 1≤|α|≤K−1

. k∇Pf k2 + (δ + δ 2 +

(4.8)

|α|≤K

1/2 El,K (t)

+ El,K (t))Dl,K (t).

(4.9)

As in Step 3 in the proof of Lemma 3.6, a suitably linear combination of (4.9), (3.36), (3.38) and (3.40), we have d h E (t) + λDl,K (t) . k∇Pf k2 . dt l,K Here we have used the a priori assumption (3.1). Thus we complete this proof of Lemma 4.2.  To get the time-decay property of the solution f (t, x, v), we also need the the following lemma whose proof can be found in [20].

No.5

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Lemma 4.3 (cf. [20]) For hard potentials, fix real numbers b+ , b− , b′ ≥ 0, and b = b+ −b− , we have the following uniform estimates +

|wb Γ(g, h)|L2 . |wb 4.3

−b−

′

1 . g|L2γ+2s |wb+b h|H2(γ+2s)

(4.10)

Decay Rates of the Instant Energy Functional for Hard Potentials

In this subsection, we will prove the faster time-decay estimate for instant energy functional h El,K (t) and the high-order instant energy functional El,K (t). To this end, we define 3

5

h X(t) = sup (1 + s) 2 El,K (t) + sup (1 + s) 2 El,K (t). 0≤s≤t

(4.11)

0≤s≤t

and assume that sup X(t) ≤ δ0 .

(4.12)

0≤t≤T

Here δ0 is a sufficiently small positive constant. For the hard potentials, we notice that El,K (t) . Dl,K (t) + kPf k2 .

(4.13)

Now we can conclude from (3.29) d El,K (t) + λEl,K (t) . kPf k2 , dt which by solving the ODE inequality above gives Z t El,K (t) . e−λt El,K (0) + e−λ(t−s) kPf k2 ds.

(4.14)

(4.15)

0

kPf k . kw−b Pf k . kw−b f (t)k −3/4

. (1 + t)

kw

−b

f0 k L

Here

T 2

Z1

+

Z

t

0

(1 + t − s)−3/4 kw−b g(s)kL2 T Z1 ds.

g(t, x, v) = ∇x φ · ∇v f + (1 − e−φ )Lf + e−φ/2 Γ(f, f ). and b will be chosen a suitably large positive constant later. We first estimate the term kw−b e−φ/2 Γ(f, f )kL2 T Z1 , by the Lemma 4.3, one has X kw(0, β)∂β f k2 . El,K (t). kw−b e−φ/2 Γ(f, f )kL2 T Z1 . |β|≤1

Using a similar way, one has kw−b (1 − e−φ )Lf kL2 T Z1 = kw−b (1 − e−φ )L{I − P}f kL2 T Z1 X h . k∇φkH 2 kw(0, β)∂β {I − P}f k . δ[El,K (t)]1/2 . |β|≤1

For kw−b ∇x φ · ∇v f kL2 T Z1 , kw

−b

∇x φ∇v f kL2





∇v f

. k∇x φ|x|kL∞

x

|x| L2 x

L2v

h . k∇φ|x|kL∞ k∇x ∇v f k . δ1 [El,K (t)]1/2 x

and kw−b ∇x φ∇v f kZ1





∇v f ∇v f

kL1x kL2v = k∇x φ|x|kL2x . kk∇x φ|x| ·

|x| |x| L2 x

L2v

(4.16)

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h . k∇x φ|x|kL2x k∇x ∇v f k . δ1 [El,K (t)]1/2 .

Plug the above estimates into (4.16) , we have kPf k2 . (1 + t)−3/2 (ǫ20 + (δ 2 + δ12 )X(t) + X 2 (t)).

(4.17)

Combining (4.17) with (4.15), we have El,K (t) . (1 + t)−3/2 (ǫ20 + (δ 2 + δ12 )X(t) + X(t)2 ).

(4.18)

Next to close the a priori estimates (4.12), we need to estimate the time decay of the high-order energy functional, first we notice that h El,K (t) . Dl,K (t).

Now we can conclude from (4.5) that d h h E (t) + λEl,K (t) . k∇x Pf k2 , dt l,K which by solving the above ODE inequality gives Z t h h El,K (t) . e−λt El,K e−λ(t−s) k∇x Pf k2 ds. (0) +

(4.19)

0

It follows from Theorem 4.1 that

k∇x Pf k . kw−b ∇x Pf k . kw−b ∇x f (t)k −5/4

. (1 + t)

kw

−b

f0 kH˙ 1 T Z1

+

Z

t 0

(1 + t − s)−5/4 kw−b g(s)kH˙ 1 T Z1 ds, (4.20)

By a similar way as the time decay of kPf k, we can obtain

k∇x Pf k2 ≤ (1 + t)−5/2 (ǫ20 + (δ 2 + δ12 )X(t) + X 2 (t)). Then we plug the above estimate into (4.19), we can obtain h El,K (t) . (1 + t)−5/2 (ǫ20 + (δ 2 + δ12 )X(t) + X 2 (t)),

(4.21)

for any 0 ≤ t ≤ T . Then combining (4.18) and (4.21), we have X(t) . ǫ20 + X 2 (t).

(4.22)

It is immediate to follow from the a priori estimate (4.12) that X(t) . ǫ20 holds true for any 0 ≤ t ≤ T , as long as ǫ0 is sufficiently small. Thus we close the a priori estimates (4.12). Hence, El,K (t) . (1 + t)−3/2 ǫ20 , and h El,K (t) . (1 + t)−5/2 ǫ20 .

Thus we have completed the proof of Theorem 1.2. References [1] Alexandre R, Morimoto Y, Ukai S, Xu C J, Yang T. Regularizing effect and local existence for non-cutoff Boltzmann equation. Arch Ration Mech Anal, 2010, 198(1): 39–123 [2] Alexandre R, Morimoto Y, Ukai S, Xu C J, Yang T. Global existence and full regularity of the Boltzmann equation without angular cutoff. Comm Math Phys, 2011, 304(2): 513–581

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