Table Of ContentAcoustic Limit for the Boltzmann equation
in Optimal Scaling
Yan Guo
Brown University
9
0 [email protected]
0
2 Juhi Jang
n Courant Institute
a [email protected]
J
5 Ning Jiang
1
Courant Institute
] [email protected]
P
A
January 15, 2009
.
h
t
a
m Abstract
[
BasedonarecentL2-L∞ framework,weestablishtheacousticlimitoftheBoltzmann
1
equation for general collision kernels. The scaling of the fluctuations with respect to
v
Knudsennumberisoptimal. Ourapproachisbasedonanewanalysisofthecompressible
0
9 Euler limit of the Boltzmann equation, as well as refined estimates of Euler and acoustic
2 solutions.
2
.
1
1 Introduction and Main Results
0
9
0 We study the Boltzmann equation
:
v
Xi ∂ Fε+v Fε = 1 (Fε,Fε) (1.1)
t x
·∇ εQ
r
a
where Fε(t,x,v) 0 is the density of particles of velocity v R3, and position x Ω = R3
≥ ∈ ∈
or T3, a periodic box. The positive parameter ε is the Knudsen number. Throughout this
paper, the collision operator takes the form
(F ,F )(v) = v uγF (u)F (v )B(θ)dωdu
1 2 1 ′ 2 ′
Q R3 S2| − |
Z Z (1.2)
v uγF (u)F (v)B(θ)dωdu,
1 2
− R3 S2| − |
Z Z
where 3 < γ 1, u = u+[(v u) ω]ω, v = v [(v u) ω]ω, cosθ = (u v) ω/v u,
′ ′
− ≤ − · − − · − · | − |
and 0 < B(θ) C cos(θ). Such collision operators cover both the hard-sphere interaction
≤ | |
and inverse power law with an angular cutoff. The hard potential means 0 γ 1, while
≤ ≤
soft potential means 3< γ < 0.
−
1
1.1 Hilbert Expansion
We define a family of special distribution functions µ(t,x,v) the local Maxwellians by
ρ(t,x) [v u(t,x)]2
µ(t,x,v) exp − (1.3)
≡ [2πT(t,x)]3/2 − 2T(t,x)
(cid:26) (cid:27)
which are equilibrium of the collision process:
(µ,µ) = 0.
Q
(ρ,u,T) represent the macroscopic density, bulk velocity, and temperature respectively. If
(ρ,u,T) are constant in t and x, µ is called a global Maxwellian. It was shown in [6, 15]
that for hard-sphere interaction, namely γ = 1, as ε 0, Fε solutions to the Boltzmann
→ { }
equation (1.1) converge to a local Maxwellian µ induced by a solution to the compressible
Euler system:
∂ ρ+ (ρu) = 0
t x
∇ ·
∂ (ρu)+ (ρu u)+ p = 0 (1.4)
t x x
∇ · ⊗ ∇
∂ ρ(e+ 1 u2) + ρu(e+ 1 u2) + (pu) = 0
t 2| | ∇x· 2| | ∇x·
with the equation of st(cid:2)ate (cid:3) (cid:2) (cid:3)
p = ρRT = 2ρe (1.5)
3
as long as the solution stays smooth. Let (ρ(t,x),u(t,x),T(t,x)) be a smooth solution of the
Euler equations (1.4) for t [0,τ], x Ω. Consider the local Maxwellian µ from (ρ,u,T) as
∈ ∈
in (1.3). As in [6], we take the Hilbert expansion of solutions around F µ with the form
0
≡
5
Fε = εnF +ε3Fε ,
n R
n=0
X
where F ,...,F are the first 6 terms of the Hilbert expansion, independent of ε, which solve
0 5
the equations
0 = (F ,F ),
0 0
Q
∂ +v F = (F ,F )+ (F ,F ),
t x 0 0 1 1 0
{ ·∇ } Q Q
∂ +v F = (F ,F )+ (F ,F )+ (F ,F ),
t x 1 0 2 2 0 1 1
{ ·∇ } Q Q Q
...
∂ +v F = (F ,F )+ (F ,F )+ (F ,F ).
t x 5 0 6 6 0 i j
{ ·∇ } Q Q Q
i+j=6
X
1 i 5,1 j 5
≤≤ ≤ ≤
We can construct smooth F (t,x,v),F (t,x,v),...,F (t,x,v) for 0 t τ. For more detailed
1 2 6
discussion, see [6]. Now we put Fε = 5 εnF +ε3Fε into the≤Bol≤tzmann equation (1.1)
n=0 n R
P
2
to derive the remainder equation for Fε
R
1
∂ Fε +v Fε (µ,Fε)+ (Fε,µ)
t R ·∇x R− ε{Q R Q R }
5 (1.6)
=ε2 (Fε,Fε)+ εi 1 (F ,Fε)+ (Fε,F ) +ε2A
Q R R − {Q i R Q R i }
i=1
X
where
A= ∂ +v F + εi+j 6 (F ,F ). (1.7)
t x 5 − i j
−{ ·∇ } Q
i+j 6,1 i,j 5
≥X≤ ≤
The acoustic system is the linearization about the homogeneous state of the compressible
Euler system. After a suitable choice of units, the fluid fluctuations (σ,u,θ) satisfy
∂ σ+ u = 0, σ(x,0) = σ0(x),
t x
∇ ·
∂ u+ (σ+θ)= 0, u(x,0) = u0(x), (1.8)
t x
∇
3∂ θ+ u = 0, θ(x,0) = θ0(x).
2 t ∇x·
Such acoustic system (1.8) can be formally derived from the Boltzmann equation (1.1) by
letting
Fε = µ0+δGε (1.9)
where µ0 is the global Maxwellian which corresponds to ρ= T = 1 and u = 0:
µ0 1 exp( |v|2)
≡ (2π)3/2 − 2
and the fluctuation amplitude δ is a function of ε satisfying
δ 0 as ε 0. (1.10)
→ →
For instance, one can take
δ = εm for any m > 0.
With the above scalings, Gε formally converges to
G = σ+v u+ |v|2−3 θ µ0 (1.11)
· 2
n (cid:16) (cid:17) o
as ε 0, where σ,u,θ satisfy the acoustic system (1.8). For detailed formal derivation, see
→
[1, 10].
1.2 Main Theorems
The endeavor to understand how fluid dynamical equations for both compressible and in-
compressible flows can be derived from kinetic theory goes back to the founding work of
Maxwell [26] and Boltzmann [5]. Most of these derivations are well understood at several
formal levels by now, and yet their full mathematical justifications are still incomplete. In
fact, the purpose of the Hilbert’s sixth problem [19] is to seek a unified theory of the gas
dynamics including various levels of descriptions from a mathematical standpoint. So far,
there are basically three different approaches mathematically. The first is based on spectral
analysis of the semi-group generated by the linearized Boltzmann equation, see [4, 21, 27].
3
The second is based on Hilbert or Chapman-Enskog expansions [6, 7], see more recent work
in [13, 16], [12, 14]. The third approach was the program initiated from [1, 2], working in the
framework of global renormalized solutions after the celebrated work of DiPerna-Lions [8], to
justify global weak solutions of incompressible flows (Navier-Stokes, Stokes, and Euler), and
(compressible) acoustic system, see [1, 2, 3, 10, 11, 18, 22, 23, 24, 28].
The authors in [10] proved the convergence of the acoustic limit from DiPerna-Lions
solutions of the Boltzmann equation (1.1) with the restriction on the size of fluctuations:
m > 1. Recently, in [18], this restriction has been relaxed to the borderline case m =
2
1 by employing some new nonlinear estimates developed in [22] and a new L1 averaging
2
lemma in [11]. However, due to some technical difficulties mainly caused by the lack of
local conservation laws and regularity of renormalized solutions, the case for m < 1 remains
2
an open question. On the other hand, in the framework of classical solutions, in [17], the
authors have established the global-in-time uniform energy estimates and proven the strong
convergence for m = 1, by adapting the nonlinear energy method of [13, 16]. Although this
method displays in a clear way how the dissipation disappears in the acoustic limit in terms
of instant energies and dissipation rates, it does not cover other interesting cases 0 < m < 1
due to weak dissipations.
The purpose of this article is to establish the acoustic limit for 0 < m < 1 via a recent
L2-L framework. We will use δ instead of εm to denote the fluctuation amplitude. Since
∞
our interest is the case of 0 < m < 1 towards the optimal scaling, throughout the paper, we
assume that in addition to (1.10),
ε
0 as ε 0. (1.12)
δ → →
Theorem 1.1. Let τ > 0 be any given finite time and let
σ(0,x) = σ0(x), u(0,x) = u0(x), θ(0,x) = θ0(x) Hs, s 4 (1.13)
∈ ≥
be any given initial data to the acoustic system (1.8). Then there exist an ε >0 and a δ > 0
0 0
such that for each 0 < ε ε and 0 < δ δ , there exists a constant C > 0 so that
0 0
≤ ≤
ε
sup Gε(t) G(t) + sup Gε(t) G(t) C +δ (1.14)
2
0 t τk − k∞ 0 t τk − k ≤ {δ }
≤ ≤ ≤ ≤
where ε 0 as ε 0, and Gε and G are defined in (1.9) and (1.11), and C depends only
δ → →
on τ and the initial data σ0,u0,θ0.
Our proof is different from the previous approach. Instead of estimating Gε G directly,
−
we make a detour to control of Gε G in two steps. The first step (Section 2) is to show as
−
ε 0, Fε is close to the local Maxwellian µδ, constructed from the smooth solution of the
→
compressible Euler equation. In fact, we are able to establish (Theorem 1.2)
Fε µδ = O(ε),
−
before the time of possible shock formation, which is of the order of 1 in the acoustic scaling
δ
(longer than any fixed time τ!). The second step (Section 3) is to show that (Lemma 3.3),
within the time scale of 1,
δ
µδ µ0 = δG+O(δ2).
−
Such an estimate confirms that the solution of the acoustic equation G is the first order
4
(linear) approximation of that to the Euler equations. Combining these two estimates and
comparing with (1.9), we deduce our theorem by dividing δ. Our proof relies on the existence
of global in-time smooth solutions to the linear acoustic system (1.8).
Ourmain technical contribution is a new analysis of the classical compressible Euler limit
to complete step one above. To precisely state our result, we usethestandard notation Hs to
denote the Sobolev space Ws,2(Ω) with correspondingnorm . We also usethe standard
Hs
k·k
notation and to denote L2 norm and L norm in both (x,v) Ω R3 variables.
2 ∞
We use k,·kto denko·tke∞the standard L2 inner product. We also define a w∈eigh×ted L2 norm
h· ·i
g 2 = g2(x,v)ν(v)dxdv,
k kν ZΩ×R3
where the collision frequency ν(v) ν(µ)(v) is defined as
≡
ν(µ)= B(θ)v v γµ(v )dv dω.
′ ′ ′
R3 | − |
Z
Note that for given 3 < γ 1,
− ≤
ν(µ) ρ(1+ v )γ.
∼ | |
Define the linearized collision operator by
L
1
g = (µ,√µg)+ (√µg,µ) .
L −√µ{Q Q }
Let Pg be the L2 projection with respect to [õ,võ, v 2õ]. Then it is well-known that
v | |
there exists a positive number c > 0 such that
0
g,g c I P g 2. (1.15)
hL i ≥ 0k{ − } kν
The solutions to the Boltzmann equation (1.1) are constructed near the local Maxwellian of
the compressible Euler system. So it is natural to rewrite the remainder
Fε = √µfε. (1.16)
R
SinceµisalocalMaxwellian, theequationoftheremainderincludesthenewterm√µ−1(∂t+
v x)√µfε. At large velocities, the distribution functions may be growing rapidly due to
·∇
streaming. To remedy this difficulty, following Caflisch [6], we introduce a global Maxwellian
1 v 2
µ = exp | | .
M
(2πT )3/2 −2T
M (cid:26) M(cid:27)
where T satisfies the following condition
M
T < max T(t,x) < 2T . (1.17)
M M
t [0,τ],x Ω
∈ ∈
Note that under the assumption (1.17), there exist constants c ,c such that for some 1/2 <
1 2
α < 1 and for each (t,x,v) [0,τ] Ω R3, the following holds
∈ × ×
c µ µ c µα . (1.18)
1 M ≤ ≤ 2 M
5
We further define
1
FRε = {1+|v|2}−β√µMhε ≡ w(v)√µMhε (1.19)
for any fixed
9 2γ
β − .
≥ 2
We now state the result on the compressible Euler limit:
Theorem 1.2. Assume that the solution to the Euler equations [ρ(t,x),u(t,x),T(t,x)] is
smooth and ρ(t,x) has a positive lower bound for 0 t τ. Furthermore, assume that the
≤ ≤
temperature T(t,x) satisfies the condition (1.17). Let
5
Fε(0,x,v) = µ(0,x,v)+ εnF (0,x,v)+ε3Fε(0,x,v) 0. (1.20)
n R ≥
n=1
X
Then there is an ε0 > 0 such that for 0 < ε ≤ ε0, and for any β ≥ 9−22γ, there exists a
constant C (µ,F ,F ,..F ) such that
τ 0 1 6
0≤sut≤pτε23 (cid:13)√µ−1(1+|v|2)βFRε(t)(cid:13)∞+0≤sut≤pτ(cid:13)√µ−1FRε(t)(cid:13)2 (1.21)
≤ Cτ ε(cid:13)(cid:13)23 √µ−1(1+|v|2)βFRε(0(cid:13)(cid:13)) + √µ(cid:13)(cid:13)−1FRε(0) 2+(cid:13)(cid:13)1 ,
n (cid:13) (cid:13)∞ (cid:13) (cid:13) o
where Fε is the solution to(cid:13)(cid:13)the remainder equatio(cid:13)(cid:13)n (1.6(cid:13)(cid:13)). (cid:13)(cid:13)
R
Remark: Applying the bound (1.18), Lemma A.1 and Lemma A.2 of [13], we can carefully
choose Fε(0,x,v) in (1.20) so that the initial data (1.20) are non-negative. Because the
R
argument is quite similar, we omit the details here.
Based on the a priori estimates given in Theorem 1.2, following the arguments in the
pioneering work of Caflisch [6], we can immediately derive the compressible Euler limit as
well as the existence of the solutions to the Boltzmann equation. As in [6], the Hilbert
expansion provides a natural way to establish an uniform in ε control for the Euler limit.
However, itwaswell-known[6]thatan v 3fε termduetostreamingintheL2 estimatecreates
| |
an unpleasant analytical difficulty. We employ both L2 and L estimate with polynomial
∞
velocity weight [14, 15] to control such a term with a high power of velocity v. On the
one hand, our analysis requires an additional assumption of moderate temperature variation
(1.17). On the other hand, we do not need the truncation of the Hilbert expansion as in [6],
so that the positivity of the solution is guaranteed. In particular, our theorem is designed to
applytotheacoustic limitbecausethetemperaturevariation isonlyof theorderδ. Moreover,
a cutoff trick used in [30] enables us to treat all soft potentials 3 < γ 1 with an angular
− ≤
cutoff.
2 Compressible Euler Limit
Inthis section, weproveTheorem1.2. Note thatitsufficestoestimate fε(t) and hε(t)
2
to conclude the theorem. The proof relies on an interplay between L2kand Lk estimkateskfo∞r
∞
the Boltzmann equation [14, 15]: L2 norm of fε is controlled by the L norm of the high
∞
6
velocity part and vice versa. These uniform L2-L estimates are stated in the following two
∞
lemmas:
Lemma 2.1. (L2-Estimate): Let (ρ,u,T) be a smooth solution to the Euler equations such
that ρ has a positive lower bound and T satisfies the condition (1.17). Let fε,hε be defined
in (1.16) and (1.19), and c > 0 be as in the coercivity estimate (1.15). Then there exists
0
ε > 0 and a positive constant C = C(µ,F ,F , ,F )> 0, such that for all ε < ε
0 0 1 6 0
···
d c
fε 2+ 0 I P fε 2 C √ε ε3/2hε +1 ( fε 2+ fε ). (2.1)
dtk k2 2εk{ − } kν ≤ { k k∞ } k k2 k k2
Lemma 2.2. (L -Estimate): Let (ρ,u,T) be a smooth solution to the Euler equations such
∞
that ρ has a positive lower bound and T satisfies the condition (1.17). Let fε,hε and c >
0
0 be the same as in Lemma 2.1. Then there exist ε > 0 and a positive constant C =
0
C(µ,c ,F , ,F ) > 0, such that for all ε< ε
0 1 6 0
···
sup ε3/2 hε(s) C ε3/2h + sup fε(s) +ε7/2 . (2.2)
0 2
0 s τ{ k k∞} ≤ {k k∞ 0 s τk k }
≤ ≤ ≤ ≤
The proof of Theorem 1.2 is a direct consequence of Lemmas 2.1 and 2.2.
Proof. of Theorem 1.2:
d c
fε 2+ 0 I P fε 2
dtk k2 2εk{ − } kν
C √ε ε3/2h + sup fε(s) +ε7/2 +1 fε 2+ fε .
≤ (cid:26) (cid:20)k 0k∞ 0≤s≤τk k2 (cid:21) (cid:27) k k2 k k2
(cid:0) (cid:1)
A simple Gronwall inequality yields
fε(t) 2 +1 ( fε(0) 2 +1)eCt{2+√εkε3/2h0k∞+√εsup0≤s≤τkfε(s)k2}.
k k ≤ k k
For ε small, using the Taylor expansion of the exponential function in the above inequality,
we have
fε C ( fε(0) +1) 1+√ε ε3/2h +√ε sup fε(s) . (2.3)
2 1 2 0 2
k k ≤ k k (cid:26) k k∞ 0≤s≤τk k (cid:27)
For t τ, letting ε small, we conclude the proof of our main theorem as:
≤
sup fε(t) C 1+ fε(0) + ε3/2h .
2 τ 2 0
0 t τk k ≤ { k k k k∞}
≤ ≤
2.1 L2 Estimate For fε
Proof. of Lemma 2.1: In terms of fε, we obtain
1
∂ fε+v fε+ fε
t x
·∇ εL
5
= {∂t+v·∇x}√µfε+ε2Γ(fε,fε)+ εi 1 Γ( Fi ,fε)+Γ(fε, Fi ) +ε2A¯
−
õ { õ õ }
i=1
X
7
where A¯= −{∂t+v√·∇µx}F5 + i+j 6,i 5,j 5εi+j−6Γ(√Fµi ,√Fjµ).
≥ ≤ ≤
Taking L2 inner producPt with fε on both sides, since {∂t+v·∇x}√µ is a cubic polynomial
õ
in v, we have for any κ > 0 and a = 1/(3 γ),
−
{∂t+v·∇x}√µfε,fε
õ
(cid:28) (cid:29)
= +
Z|v|≥εκa Z|v|≤εκa
≤ {k∇xρk2+k∇xuk2+k∇xTk2}×k{1+|v|2}3/2fε1|v|≥εκak∞×kfεk2
+{k∇xρk∞+k∇xuk∞ +k∇xTk∞}×k{1+|v|2}3/4fε1|v|≤εκak22
C ε2 hε fε
κ 2
≤ k k∞k k
+Ck{1+|v|2}3/4Pfε1|v|≤εκak22+Ck{1+|v|2}3/4{I−P}fε1|v|≤εκak22
Cκ3 γ
C ε2 hε fε +C fε 2 + − I P fε 2.
≤ κ k k∞k k2 k k2 ε k{ − } kν
Here we have used the fact 1+ v 2 3/2fε 1+ v 2 γ 3hε, for β 3/2+(3 γ) in (1.19),
−
{ | | } ≤{ | | } ≥ −
and the fact µ < Cµ in (1.18) under the assumption (1.17).
M
By the same proof as in Lemma 2.3 of [12] and (1.19),
ε2 Γ(fε,fε),fε Cε2 ν(µ)fε fε 2 C√ε ε3/2hε fε 2.
h i≤ {k k∞}k k2 ≤ k k∞k k2
Similarly, by the same proof as in Lemma 2.3 of [12] and (1.19),
5
F F
εi 1 Γ( i ,fε),fε + Γ(fε, i ),fε
−
{h õ i h õ i}
i=1
X
5
F
C εi 1 fε 2 i dv
≤ − k kνk R3 √µ k∞
i=1 Z
X
C Pfε 2 + I P fε 2
≤ {k kν k{ − } kν}
C fε 2+ I P fε 2 .
≤ {k k2 k{ − } kν}
Clearly, ε2A¯,fε C fε . We therefore conclude our lemma by choosing κ small.
2
h i ≤ k k
2.2 L Estimate For hε
∞
As in [6], we define
1
g = (µ,√µ g)+ (√µ g,µ) = ν(µ)+K g,
M M M
L −√µM{Q Q } { }
8
where Kg = K g K g with
1 2
−
µ(v)
K g = B(θ)u v γ µ (u) g(u)dudω
1 M
ZB3×S2 | − | p µM(v)
K g = B(θ)u v γµ(u) µpM(v′)g(v )dudω
2 ′ ′
ZB3×S2 | − | pµM(v)
+ B(θ)u v γµ(v ) pµM(u′)g(u)dudω.
′ ′
ZB3×S2 | − | pµM(v)
Consider a smooth cutoff function 0 χ 1 suchpthat for any m > 0,
m
≤ ≤
χ (s) 1, for s m; χ (s) 0, for s 2m.
m m
≡ ≤ ≡ ≥
Then define
µ(v)
Kmg = B(θ)u v γχ (u v ) µ (u) g(u)dudω
m M
ZB3×S2 | − | | − | p µM(v)
B(θ)u v γχ (u v )µ(u) µMp(v′)g(v )dudω
m ′ ′
−ZB3×S2 | − | | − | pµM(v)
B(θ)u v γχ (u v )µ(v ) pµM(u′)g(u)dudω,
m ′ ′
−ZB3×S2 | − | | − | pµM(v)
and also define p
Kcg = K Km.
−
Lemma 2.3.
Kmg(v) Cm3+γν(µ) g . (2.4)
| | ≤ || ||∞
And Kcg(v) = l(v,v )g(v )dv where the kernel l satisfies for some c > 0,
R3 ′ ′ ′
R exp cv v 2
′
l(v,v′) ≤Cm v v ({1−+|v−+ v| })1 γ. (2.5)
′ ′ −
| − | | | | |
Proof. Since µ Cµα for α > 1 and u2+ v 2 = u 2+ v 2, we first have
≤ M 2 | | | | | ′| | ′|
µ(v) α 1
µ (u) C µ (u)µ −2(v),
M µ (v) ≤ M M
M
p p
µ(u′) µµM((vv′)) +µ(v′) pµµM((uv′)) ≤ C{µα−21(u′)µM12 (u)+µMα−21(v′)µM12 (u)}.
p M p M
p p 1
Since v u 2m, µ (u) µ (v) and thus µ2 (u) Cν(µ). And since γ > 3, (2.4)
| − | ≤ M ∼ M M ≤ −
follows.
To show (2.5), clearly the kernel for Kc satisfies (2.5), since α > 1. For Kc, we can use
1 2 2
the Carleman change of variable and apply the proof of Lemma 1 in [30] (one can extend the
result to cover all 3< γ 1).
− ≤
We are now ready to prove Lemma 2.2.
9
Proof. of Lemma 2.2: Letting K g wK(g), from (1.6) and (1.19), we obtain
w ≡ w
ν(µ) 1
∂ hε +v hε+ hε + K hε
t x w
·∇ ε ε
= ε2w (hε√µM,hε√µM)+ 5 εi 1 w (F ,hε√µM)+ (hε√µM,F )
− i i
õMQ w w õM{Q w Q w }
i=1
X
+ε2A˜,
wheBreyAD˜u=h−amwe{l∂’ts+√pvµ·rM∇inx}cFip5le+, wei+hja≥v6e,i≤hε5,(jt≤,5xε,vi+)j=−6√wµMQ(Fi,Fj).
P
νt t ν(t s) 1
exp hε(0,x vt,v) exp − Kmhε (s,x v(t s),v)ds
{− ε } − − {− ε } ε w − −
Z0 (cid:18) (cid:19)
t ν(t s) 1
exp − Kchε (s,x v(t s),v)ds
− {− ε } ε w − −
Z0 (cid:18) (cid:19)
t ν(t s) ε2w hε√µM hε√µM
+ exp − ( , ) (s,x v(t s),v)ds
Z0 {− ε }(cid:18)√µMQ w w (cid:19) − −
+ texp ν(t−s) 5 εi−1 w (Fi, hε√µM) (s,x v(t s),v)ds (2.6)
Z0 {− ε } i=1 √µMQ w ! − −
X
+ texp ν(t−s) 5 εi−1 w (hε√µM,Fi) (s,x v(t s),v)ds
Z0 {− ε } i=1 √µMQ w ! − −
X
t ν(t s)
+ exp − ε2A˜(s,x v(t s),v)ds.
{− ε } − −
Z0
First note that
ν(µ) v uγµdu (1+ v )γρ(t,x) ν (v),
M
∼ | − | ∼ | | ∼
Z
t ν(µ)(t s) t cν (t s)
M
exp − ν(µ)ds c exp − ν ds = O(ε).
M
{− ε } ≤ {− ε }
Z0 Z0
Then from (2.4), the second term in (2.6) is bounded by
t ν(t s)
Cm3+γ exp − νds sup hε(t) Cm3+γε sup hε(t) .
Z0 {− ε } 0≤t≤τ|| ||∞ ≤ 0≤t≤τ|| ||∞
By µ Cµ, and since w Q(hε√µM,hε√µM) Cν(µ) hε 2 from the same proof as in
M ≤ |√µM w w | ≤ k k∞
Lemma 10 of [14], the third line in (2.6) is bounded by
t ν(µ)(t s)
Cε2 exp − ν(µ) hε(s) 2 ds
Z0 {− ε } k k∞ (2.7)
Cε3 sup hε(s) 2 .
≤ 0 s tk k∞
≤ ≤
10
