Pointwise Behavior of the Linearized Boltzmann Equation on Torus

0.2 MB·English

by  Kung-Chien Wu

POINTWISE BEHAVIOR OF THE LINEARIZED BOLTZMANN
EQUATION ON TORUS
KUNG-CHIEN WU
3
Abstract. We study the pointwise behavior of the linearized Boltzmann equation on
1
torus for non-smooth initial perturbation. The result reveals both the fluid and kinetic
0
2 aspects of this model. The fluid-like waves are constructed as part of the long-wave ex-
pansion in the spectrum of the Fourier mode for the space variable, the time decay rate
n
of the fluid-like waves depends on the size of the domain. We design a Picard-type itera-
a
J tion for constructing the increasingly regular kinetic-like waves, which are carried by the

4 transportequations and have exponential time decay rate. Moreover,the mixture lemma
plays an important role in constructing the kinetic-like waves, we supply a new proof

] of this lemma to avoid constructing explicit solution of the damped transport equations
h
p (compare Liu-Yu’s proof [7, 10]).
-
h
t
a
m
[ 1. Introduction

1 The Boltzmann equation for the hard sphere model reads
v
4 1
3 ∂ F +ξ ·∇ F = Q(F,F),
t x
7 (1)  ε
0 
. F(x,0,ξ) = F (x,ξ),
1 0
0 
3 where
1
1
:
v Q(g,h) = −g(ξ)h(ξ )−g(ξ )h(ξ)+g(ξ′)h(ξ′)+g(ξ′)h(ξ′) |(ξ −ξ )·Ω|dξ dΩ,
i  2 ZU (cid:2) ∗ ∗ ∗ ∗ (cid:3) ∗ ∗
X 


r  U = (ξ ,Ω) ∈ R3 ×S2 : (ξ −ξ )·Ω ≥ 0 ,
a 
∗ ∗
(cid:8) (cid:9)

 ξ = ξ −[(ξ −ξ )·Ω]Ω, ξ = ξ +[(ξ −ξ )·Ω]Ω.
 ′ ′
 ∗ ∗ ∗

Date: December 11, 2013.
2010 Mathematics Subject Classification. 35Q20;82C40.
Key words and phrases. Boltzmann equation; Fluid-like wave; Kinetic-like wave; Maxwellian states;
Pointwise estimate.
This paper is supervised by Cl´ement Mouhot in Cambridge. It is a pleasure to thank I-Kun Chen,
Seung-Yeal Ha, Chanwoo Kim, Hung-Wen Kuo, Chi-Kun Lin, Tai-Ping Liu and Se Eun Noh for stimu-
lating discussion concerning this paper. This work is supported by the Tsz-Tza Foundation in institute
of mathematics, Academia Sinica, Taipei, Taiwan. Part of this work was written during the stay at
Department ofMathematics, StanfordUniversity;the author thanks Tai-PingLiufor his kind hospitality.
1
2 K.-C. WU
Here ε is the Knudsen number, the microscopic velocity ξ ∈ R3 and the space variable
x ∈ T3, the 3-dimensional torus with unit size of each side. In order to remove the
1
parameter ε from the equation, we introduce the new scaled variables:
1 1
x = x, t = t,
ε ε
then after dropping the tilde, the eqeuation (1)ebecomes
∂ F +ξ ·∇ F = Q(F,F), (x,t,ξ) ∈ T3 ×R+ ×R3,
t x 1/ε
(2) 

F(x,0,ξ) = F (x,ξ),
0
where T3 denotes the 3-dimensional torus with size 1/ε of each side. The conservation
1/ε
laws of mass, momentum, as well as energy, can be formulated as
d
(3) 1,ξ,|ξ|2 F(t,x,ξ)dξdx = 0.
dt ZT3 ZR3 n o
1/ε
It is well-know that Maxwellians are steady states to the Boltzmasnn equation. Thus, it
is natural to linearize the Boltzmann equation (2) around a global Maxwellian
1 −|ξ|2
w(ξ) = exp ,
(2π)3/2 (cid:16) 2 (cid:17)
with the standard perturbation f(t,x,ξ) to w as
F = w+w1/2f .
Then after substituting into (2) and dropping the nonlinear term, we have the linearized
Boltzmann equation
∂ f +ξ ·∇ f = 2Q(w,w1/2f) = Lf ,
t x
(4) 

f(x,0,ξ) = I(x,ξ),
here we define f(x,t,ξ) =GtI(x,ξ), i.e. Gt is the solution operator (Green function) of
ε ε
the linearized Boltzmann equation (4). Assuming the initial density distribution function
F (x,ξ) has the same mass, momentum and total energy as the Maxwellian w, we can
0
further rewrite the conservation laws (3) as
(5) w1/2(ξ) 1,ξ,|ξ|2 I(x,ξ)dξdx = 0.
ZT3 ZR3 n o
1/ε
This means that the initial condition I satisfies the zero mean condition.
Before the presentation of the properties of the collision operator L, let us define some
notations in this paper. For microscopic variable ξ, we shall use L2 to denote the classical
ξ
Hilbert space with norm
1/2
kfkL2ξ = (cid:16)ZR3 |f|2dξ(cid:17) ,
BOLTZMANN EQUATION 3
the Sobolev space of functions with all its s-th partial derivatives in L2 will be denoted by
ξ
Hs. The L2 inner product in R3 will be denoted by ·,· . We denote the weighted sup
ξ ξ ξ
norm as (cid:10) (cid:11)
kfk = sup|f(ξ)|(1+|ξ|)β.
L∞
ξ,β ξ R3
∈
For space variable x, we shall use L2 to denote the classical Hilbert space with norm
x
1 1/2
kfkL2x = (cid:16)|T3 | ZT3 |f|2dx(cid:17) ,
1/ε 1/ε
the Sobolev space of functions with all its s-th partial derivatives in L2 will be denoted by
x
Hs. We denote the sup norm as
x
kfk = sup |f(x)|.
L∞
x x T3
∈ 1/ε
Proposition 1. ([2]) The collision operator L consists of a multiplicative operator ν(ξ)
and an integral operator K:
Lf = −ν(ξ)f +Kf ,
where
Kf = W(ξ,ξ )f(ξ )dξ ,
 ZR3 ∗ ∗ ∗



 W(ξ,ξ∗) = √2π|2ξ−ξ∗| exp(cid:8)− (|ξ8||2ξ−−|ξξ∗∗||22)2 − |ξ−8ξ∗|2(cid:9)− |ξ−2ξ∗| exp(cid:8)− |ξ|2+4|ξ∗|2(cid:9),
 ν(ξ) = √12πh2e−|ξ2|2 +(cid:0)|ξ|+|ξ|−1(cid:1)R0|ξ|e−u22dui.

For multiplicative operator ν(ξ), there exists positive lower bound ν such that ν(ξ) ≥ ν
0 0
for all ξ ∈ R3. Moreover, the derivatives of ν(ξ) in ξ is bound, i.e. for all multi index α,
(6) |∂αν(ξ)| ≤ C .
ξ α
The integral operator K has smoothing property in ξ, i.e.,
(7) kKhkH1 ≤ CkhkL2, kKhkL∞ ≤ CkhkL∞ ,
ξ ξ ξ,β ξ,β
for any β ≥ 0.
The integral operator K can be decomposed into the singular part and the regular part,
i.e., K = K +K , K ≡ K and K ≡ K :
s r s s,D r r,D
4 K.-C. WU
|ξ −ξ |
 Ksf = ZR3χ(cid:16) Dν0∗ (cid:17)W(ξ,ξ∗)f(ξ∗)dξ∗,




 Krf = Kf −Ksf ,



χ(r) = 1 for r ∈ [−1,1],




 supp(χ) ⊂ [−2,2], χ ∈ C (R), χ ≥ 0.
 c∞


Proposition 2. ([10]) The singular part K shares the same smoothing properties as K
r
and has strength of the order of the cut-off parameter D:
(8) kKshkH1 ≤ DkhkL2, kKshkL∞ ≤ DkhkL∞ .
ξ ξ ξ,β ξ,β
The regular part K has better smoothing property in ξ: for all s > 0,
r
(9) kKrhkHs ≤ CkhkL2.
ξ ξ
In order to estimate the Green function of the linearized Boltzmann equation in next
section, we need to recall the spectrum Spec(εk), k ∈ Z3, of the operator −iπεξ ·k +L
Proposition 3. ([3]) There exists δ > 0 and τ = τ(δ) > 0 such that
(i) For any |εk| > δ,
(10) Spec(εk) ⊂ {z ∈ C : Re(z) < −τ}.
(ii) For any |εk| < δ, the spectrum within the region {z ∈ C : Re(z) > −τ} consisting of
exactly five eigenvalues {σ (εk)}4 ,
j j=0
(11) Spec(εk)∩{z ∈ C : Re(z) > −τ} = {σ (εk)}4 ,
j j=0
and the corresponding eigenvectors {e (εk)}4 , where
j j=0
3
σ (εk) = a (i|εk|)n +O(εk)4, a > 0,
j j,n j,2
X
n=1
3
e (εk) = e (i|εk|)n +O(εk)4, e ,e = δ .
j j,n j k ξ jk
Xn=1 (cid:10) (cid:11)
(iii)
4
e(−iπεξ·k+L)tf = Πδf +χ{|εk|<δ}Xj=0 eσj(−εk)t(cid:10)ej(εk),f(cid:11)ξej(εk)
where kΠδkL2 = O(1)e−a(τ)t, a(τ) > 0, χ is the indicator function.
ξ {·}
BOLTZMANN EQUATION 5
The fluid behavior is studied by constructing the Green function represented as the
Fourier series in space variable x:
1
Gt = eiπεkx+( iπεξk+L)t,
ε |T3 | · − ·
Xk Z 1/ε
∈
where the Green function is served as a function of ξ. The analysis of the Green function
is equivalent to the analysis of the spectrum of the operator −iπεξ·k+L. Notice that the
spectrum includes five curves which bifurcate from the origin. The origin is the multiple
zero eigenvalues of L, the operator at k = 0. The kernel of L are the fluid variables and
the fluid-like waves are constructed from these curves near the origin.
The kinetic aspect of the solution is described by the damped transport equation:
∂ g +ξ ·∇ g +ν(ξ)g = 0.
t x
The operator K is a smooth operator in ξ variable, we will use this smooth property to
design a Picard-type iteration for constructing the increasingly regular kinetic-like waves.
Once the kinetic-like waves and fluid-like waves were constructed, the rest of the solution
is sufficiently smooth and it has exponential time decay rate.
Theorem 4. Given β > 3, for any I ∈ L with compact support in x and satisfying the
2 ∞ξ,β
zero mean conditions (5), the solution of (4)
f = GtI = Gt I +Gt I +Gt I ,
ε ε,F ε,K ε,R
consists of the fluid part Gt I: smooth in space variable x and the time decay rate depends
ε,F
on the size of the domain, i.e. there exist δ,δ ,C > 0 such that for all s > 0
0 0
(i) If ε > δ,
kGtε,FIkHxsL2ξ = 0,
(ii) If δ < ε < δ,
0
kGtε,FIkHsL2 = O(1)e−O(1)ε2t,
x ξ
(iii) If 0 < ε < δ ,
0
C
kGtε,FIkHxsL2ξ = O(1)(1+t0)3/2e−O(1)ε2t;
the kinetic part Gt I: nonsmooth in space variable x and time decay exponentially
ε,K
kGt Ik = O(1)e O(1)t;
ε,K L∞x L∞ξ,β −
and the smooth remainder part Gt I:
ε,R
kGtε,RIkH2L2 = O(1)e−O(1)t.
x ξ
6 K.-C. WU
The spectrum analysis of the Boltzmann equation was introduced by Ellis-Pinsky [3].
Recently, Mouhot [14] gives the explicit coercivity estimates for the linearized Boltzmann
operator. The spectrum analysis of thelinearized Boltzmann equation hasbeen carried out
by many authors. In particular, the exponential time decay rates for the Boltzmann equa-
tion with hard potentials on torus was firstly provided by Ukai [15]. The time-asymptotic
nonlinear stability was obtained in[9, 16]. Using Nishida’s approach, [8] obtained the time-
asymptotic equivalent of Boltzmann solutions and Navier-Stokes solutions. These works
yield the L2 theory, since the Fourier transform is isometric in L2.
The mixture lemma plays an important role in constructing the kinetic-like waves. It
states that the mixture ofthe two operatorsS andK in Mt (see section 4 below) transports
j
the regularity in the microscopic velocity ξ to the regularity of the space and time (x,t).
This idea was firstly introduced by Liu-Yu [10, 11, 12, 13] to construct the Green function
of the Boltzmann equation. In Liu-Yu’s paper [7, 10, 13], the proof of the mixture lemma
relies on the explicit solution of the damped transport equations. However, in this paper,
we introduce a differential operator to avoid constructing explicit solution, this operator
commutes with free transport operator and can transports the microscopic velocity reg-
ularity to space regularity, this idea will help us to consider more complicated problems,
such as the Fokker-Planck equation or the Landau equation. Recently, we can apply co-
ercivity estimates [5, 14] to prove mixture lemma for Landau equation on soft potential
[17]. Actually, the mixture lemma is similar in spirit to the well-known Averaging Lemma,
see [1, 4, 6]. These two lemmas have been introduced independently and used for different
purposes.
The pointwise description of the one-dimensional linearized Boltzmann equation with
hard sphere was firstly provided by Liu-Yu [10], the fluid-like waves can be constructed
by both complex and spectrum analysis, it reveals the dissipative behavior of the type
of the Navier-Stokes equation as usually seems by the Chapman-Enskog expansion. The
kinetic-like waves can be constructed by Picard-type iteration and mixture lemma. In this
paper, we apply similar ideas on torus, we can also construct the kinetic-like waves and
fluid-like waves, which are both time decay exponentially. Moreover, the decay rate of the
fluid-like waves depend on the size of the domain.
The rest of the paper is organized as follows. In section 2, we construct the Green
function of the linearized Boltzmann equation on torus. We use the long wave short
wave decomposition and the spectrum analysis to obtain time decay rate. In section 3, we
improve the estimate of the fluid-like waves. In section 4, we design a Picard-type iteration
for constructing the increasingly regular kinetic-like waves. Finally, we supply a new proof
of the mixture lemma in the appendix.
2. Long Wave Short Wave Decomposition
Consider the linearized Boltzmann equation
∂ f +ξ ·∇ f = Lf , (x,t,ξ) ∈ (T3 ,R+,R3),
t x 1/ε
(12) 

f(x,0,ξ) = I(x,ξ),

BOLTZMANN EQUATION 7
where I satisfies the zero mean conditions (5). Hereafter, we will use just one index to
denote the 3-dimensional sums with respect to the vector k = (k ,k ,k ) ∈ Z3, hence we
1 2 3
set
= .
X X
k Z (k1,k2,k3) Z3
∈ ∈
Consider the Fourier series of initial condition I in x
I(x,ξ) = (Iˆ) (ξ)eiπεkx,
k ·
 X
k Z
 ∈
(13) 

 1
(Iˆ) (ξ) = I(·,ξ)e iπεkxdx,
 k |T31/ε| ZT31/ε − ·


rewrite the solution f(x,t,ξ) of (12) as Fourier series
f(x,t,ξ) = (fˆ) (t,ξ)eiπεkx,
k ·
 X
k Z
 ∈
(14) 

 1
(fˆ) (t,ξ) = f(·,t,ξ)e iπεkxdx,
 k |T31/ε| ZT31/ε − ·


the Fourier modes of (13)–(14) satisfy the following equations
ˆ ˆ ˆ
∂ f +iπεξ ·kf −Lf = 0,
t k k k

 fˆ(0,ξ) = (Iˆ) .
k k

Hence
fˆ(t,ξ) = e( iπεξk+L)t(Iˆ) (ξ),
k − · k
the solution of (12) is given by
f(x,t,ξ) = eiπεkx+( iπεξk+L)t(Iˆ) (ξ),
· − · k
X
k Z
∈
1
= eiπεk(x y)+( iπεξk+L)tI(y,ξ)dy,
Xk Z |T31/ε| ZT31/ε · − − ·
∈
where the Green function Gt(x,ξ) can be expressed as
ε
1
(15) Gt(x,ξ) = eiπεkx+( iπεξk+L)t.
ε |T3 | · − ·
Xk Z 1/ε
∈
Note that if ε → 0, the Green function Gt(x,ξ) becomes an integral
0
Gt(x,ξ) = eiηx+( iπξη+L)tdη.
0 Z · − ·
R3
8 K.-C. WU
We can decompose the Green function (15) into the long wave part Gt and the short
ε,L
wave part Gt respectively
ε,S
1
Gt (x,ξ) = eiπεkx+( iπεξk+L)t,
ε,L |T3 | · − ·
εXk<δ 1/ε
| |
(16)
1
Gt (x,ξ) = eiπεkx+( iπεξk+L)t.
ε,S |T3 | · − ·
εXk>δ 1/ε
| |
The following long wave short wave analysis relies on spectrum analysis (Proposition 3).
Lemma 5. (Short wave Gt ) For any s > 0, I ∈ HsL2, we have
ε,S x ξ
kGtε,SIkL2xL2ξ ≤ e−O(1)tkIkL2xL2ξ ,
(17)
kGtε,SIkHxsL2ξ ≤ e−O(1)tkIkHxsL2ξ .
Proof. Note that
Gt I = e( iπεξk+L)teiπεkxIˆ (ξ).
ε,S − · · k
X
εk>δ
| |
For L2L2 estimate, we have
x ξ
kGt Ik2 = |e( iπεξk+L)tIˆ (ξ)|2,
ε,S L2 − · k
x X
εk>δ
| |
and hence by spectrum property (10), we obtain
kGt Ik2 = ke( iπεξk+L)tIˆ (ξ)k2 ≤ e O(1)t kIˆ(ξ)k2
ε,S L2L2 − · k L2 − k L2
x ξ X ξ X ξ
εk>δ εk>δ
| | | |
= e O(1)t |Iˆ(ξ)|2dξ ≤ e O(1)tkIk2 .
− ZR3 X k − L2xL2ξ
εk>δ
| |
For the high order estimate, we have
kGt Ik2 ≤ (1+|πεk|2)s|e( iπεξk+L)tIˆ (ξ)|2,
ε,S Hs − · k
x X
εk>δ
| |
and hence
kGt Ik2 ≤ e O(1)t (1+|πεk|2)skIˆ k2 ≤ e O(1)tkIk2 .
ε,S HsL2 − k L2 − HsL2
x ξ X ξ x ξ
εk>δ
| |
(cid:3)
BOLTZMANN EQUATION 9
In order to study the long wave part Gt , we need to decompose the long wave part as
ε,L
the fluid part and non-fluid part, i.e. Gt = Gt +Gt , where
ε,L ε,F ε,L;
⊥
4
Gε,FI = eσj(−εk)teiπεk·x ej(εk),Iˆk ξej(εk),
εXk<δXj=0 (cid:10) (cid:11)
(18) | |
Gt I = eiπεkxΠ Iˆ .
ε,L; · δ k
⊥ X
εk<δ
| |
Lemma 6. (Long wave Gt ) For any s > 0, I ∈ L2L2 and satisfying the zero mean
ε,L x ξ
condition (5), we have
kGtε,L;⊥IkHxsL2ξ ≤ e−O(1)tkIkL2xL2ξ ,
(19)
kGε,FIkHsL2 ≤ e−O(1)ε2tkIkL2L2 .
x ξ x ξ
Proof. For the non-fluid part, using the spectrum property (10), we have
kGt Ik2 ≤ 1+|πεk|2 s|Π Iˆ (ξ)|2 ≤ 1+|πδ|2 s |Π Iˆ (ξ)|2,
ε,L; Hs δ k δ k
⊥ x X (cid:0) (cid:1) (cid:0) (cid:1) X
εk<δ εk<δ
| | | |
and hence
kGt Ik2 ≤ e O(1)tkIk2 .
ε,L; HsL2 − L2L2
⊥ x ξ x ξ
For the fluid part, by (11) and the zero mean conditions (5), we have
4
kGε,FIk2HxsL2ξ ≤ (cid:0)1+|πεk|2(cid:1)s εXk<δXj=0 |eσj(−εk)t||(cid:10)ej(εk),Iˆk(cid:11)ξ|2
| |
4
≤ C |eσj(−εk)t|kIˆkk2L2
X X ξ
εk<δ j=0
| |
4
≤ C e−aj,2|kε|2[1+O(δ2)]tkIˆkk2L2
X X ξ
εk<δ j=0
| |
≤ Ce O(1)ε2tkIk2 .
− L2L2
x ξ
(cid:3)
Remark (i) If ε > δ, we do not have long wave part, i.e. Gt = 0.
ε,L
(ii) The high order estimate of short wave part requires regularity in x. This is because
|πεk| may not be bounded. One needs the regularity of I to ensure the decay of Gt in
ε,S
time.
(iii) In order to remove the regularity assumption in x, we need the Picard-type iteration
for constructing the increasingly regular kinetic-like waves in section 4.
10 K.-C. WU
Theorem 7. For any I ∈ H2L2 and satisfies the zero mean conditions (5), we have the
x ξ
following exponential time decay estimate about the linearized Boltzmann equation (12)
kGtεIkL∞L2 ≤ e−λStkIkH2L2 +e−λLtkIkL2L2 .
x ξ x ξ x ξ
(i) If ε > δ, then λ = O(1) and λ = ∞,
S L
(ii) If ε ≤ δ, then λ = O(1) and λ = O(1)ε2.
S L
3. Fluid Part
In this section, we improve the estimate of the fluid part. Recall the fluid part of the
Boltzmann equation (18)
4
Gtε,FI = eσj(−εk)teiπεk·x ej(εk),Iˆk ξej(εk).
εXk<δXj=0 (cid:10) (cid:11)
| |
We have
4
2
kGtε,FIk2L2ξ = Xj=0 (cid:12)(cid:12) εXk<δeσj(−εk)teiπεk·x(cid:10)ej(εk),Iˆk(cid:11)ξ(cid:12)(cid:12)
(cid:12)| | (cid:12)
4
1 2
= Xj=0 (cid:12)(cid:12)|T31/ε| εXk<δZT31/εZR3eσj(−εk)teiπεk·(x−y)ej(εk)I(y,ξ)dξdy(cid:12)(cid:12) .
(cid:12) | | (cid:12)
Note that
1 1 1 1
e (εk)2t = e (εk)2 → e y2dy
|T3 | − | | t3/2|T3 | − | | t3/2 Z −
1/ε εXk<δ 1/ε εkX<δt1/2 Bδt1/2(0)
| | | |
as ε → 0, this means for any α > 0 there exists δ > 0 such that if ε < δ , then
0 0 0
1
e (εk)2 < α + e y2dy ≡ C .
|T3 | − | | 0 Z − 0
1/ε εkX<δt1/2 R3
| |
Hence
1
kGtε,FIkL2ξ ≤ (cid:16)|T31/ε| εXk<δe−(ε|k|)2t(cid:17)ZT31/ε kIkL2ξdx
| |
1
≤ O(1) e (εk)2t e ε2t
(cid:16)|T31/ε| εXk<δ − | | (cid:17) −
| |
C
≤ O(1) 0 e ε2t.
−
(1+t)3/2
Theorem 8. Assume that ε < δ, I ∈ L2 with compact support in x and satisfies the zero
ξ
mean conditions (5), then there exist C ,δ > 0 such that if 0 < ε < δ , then
0 0 0
C
kGtε,FIkL2ξ ≤ O(1)(1+t0)3/2e−ε2t.