Pseudospectral and spectral bounds for the Oseen vortices operator

0.31 MB·

by  Te Li

PSEUDOSPECTRAL AND SPECTRAL BOUNDS FOR THE OSEEN
VORTICES OPERATOR
TE LI, DONGYIWEI, ANDZHIFEI ZHANG
7
1
0
2
n Abstract. In this paper, we solve Gallay’s conjecture on the spectral lower bound and
a pseudospecrtalboundforthelinearizedoperatoroftheNavier-StokesequationinR2 around
J rapidly rotating Oseen vortices.

3
2
1. introduction

] In this paper, we consider the Navier-Stokes equations in R2
P
A
∂ v ν∆v+v v+ p = 0,
t
− ·∇ ∇
. (1.1) divv = 0,
h

t v(0,x) = v (x),
a  0
m
where v(t,x) denotes the velocity, p(t,x) denotes the pressure and ν > 0 is the viscosity

[ coefficient. Let ω(t,x) = ∂ v1 ∂ v2 be the vorticity. The vorticity formulation of (1.1)
1 takes 2 − 1
v
(1.2) ∂ ω ν∆ω+v ω = 0, ω(0,x) = ω (x).
9 t 0
− ·∇
6
Given the vorticity ω, the velocity can be recovered by the Biot-Savart law
2
6 1 (x y)⊥
.0 (1.3) v(t,x) = 2π R2 x− y 2 ω(t,y)dy = KBS ∗ω.
1 Z | − |
0 Itis well knownthattheNavier-Stokes equations (1.2)hasafamily of self-similar solutions
7 called Lamb-Oseen vortices of the form
1
α x α x
: (1.4) ω(t,x) = , v(t,x) = vG ,
v νtG √νt √νt √νt
i (cid:16) (cid:17) (cid:16) (cid:17)
X
where the vorticity profile and the velocity profile are given by
r
a 1 1 ξ⊥
(ξ) = e−|ξ|2/4, vG(ξ) = 1 e−|ξ|2/4 .
G 4π 2π ξ 2 −
| | (cid:16) (cid:17)
It is easy to see that ω(t,x)dx = α for any t > 0. The parameter α R is called the
R2 ∈
circulation Reynolds number.
R
Toinvestigate thelong-timebehaviourof (1.2),itisconvenienttointroducetheself-similar
variables
x
ξ = , τ = logt,
√νt
and the rescaled vorticity w and the rescaled velocity u
1 x ν x
ω(t,x) = w logt, , v(t,x) = u logt, .
t √νt t √νt
r
(cid:16) (cid:17) (cid:16) (cid:17)
Date: January 24, 2017.
1
2 TELI,DONGYIWEI,ANDZHIFEIZHANG
Then (w,u) satisfies
(1.5) ∂ w+u w =Lw,
τ
·∇
where the linear operator L is given by
ξ
(1.6) L =∆+ +1.
2 ·∇
For any α R,theLamb-Oseen vortex α (ξ) isasteady solution of (1.5). Gallay andWayne
∈ G
[11, 12] proved that for the integrable initial vorticity, the long-time behaviour of the 2-D
Navier-Stokes equations can be described by the Lamb-Oseen vortex. More precisely, for any
initial data w L1(R2), the solution of (1.5) satisfies
0
∈
lim w(τ) α = 0, α = w (ξ)dξ.
τ→+∞ − G L1(R2) R2 0
Z
(cid:13) (cid:13)
This result suggests that α(cid:13) is a stab(cid:13)le equilibrium of (1.5) for any α R. This situation is
G ∈
very similar to the Couette flow (y,0) in a finite channel, which is stable for any Reynolds
number [8]. Recently, there are many important works [1, 2, 3, 16, 25] devoted to the study
of long-time behaviour of the Navier-Stokes(Euler) equations around the Couette flow.
Tostudythestability ofαG, itis naturaltoconsiderthelinearized equation aroundα (ξ),
G
which takes as follows
(1.7) ∂ w = (L αΛ)w,
τ
−
where Λ is a nonlocal linear operator defined by
(1.8) Λw = vG w+u = Λ w+Λ w, u= K w.
1 2 BS
·∇ ·∇G ∗
The operator L αΛ in the weighted space Y = L2(R2, −1dx) defined in section 2
− G
has a compact resolvent. Thus, the spectrum of L αΛ in Y is a sequence of eigenvalues
−
λn(α) n∈N satisfying Reλn(α) 0 for any n,α. A very important problem is to study
{ } ≤
how the spectrum changes as α + , which corresponds to the high Reynolds number
| | → ∞
limit(the most relevant regime for turbulent flows).
Theeigenvalues which correspondto the eigenfunctions in the kernel of Λdonot change as
α varies. We denote by L and Λ therestriction of theoperators L and Λ to the orthogonal
⊥ ⊥
complement of kerΛ in Y. Then we define the spectral lower bound
(1.9) Σ(α) = inf Rez : z σ L +αΛ
⊥ ⊥
∈ −
n o
and pseudospectral bound (cid:0) (cid:1)
−1
−1
(1.10) Ψ(α) = sup L αΛ iλ .
⊥ ⊥ Y→Y
λ∈Rk − − k
(cid:16) (cid:17)
(cid:0) (cid:1)
For selfadjoint operators, spectral and pesudospectral bounds are the same. Here L αΛ
is a non-selfadjoint operator. It is easy to see that Σ(α) Ψ(α) for any α R. In−fact,
≥ ∈
Σ(α) and Ψ(α) are different. Moreover, the pseudo-spectrum plays an important role in the
hydrodynamic stability [22], and the spectrum theory of non-selfadjoint operator is also a
very active topic [4, 5, 20, 21].
Maekawa [17] proved that Σ(α) and Ψ(α) tend to infinity as α + . However, the
| | → ∞
proof does not provide explicit bounds on Σ(α) and Ψ(α). Numerical calculations performed
1
by Prochazka and Pullin [18, 19] indicate that Σ(α) = O(α 2) as α + . Based on the
| | | | → ∞
analysis for a model problem, Gallay [9] proposed the following conjecture.
THE OSEEN VORTICES OPERATOR 3
Conjecture: there exists C > 0 independent of α so that as α + ,
| | → ∞
Σ(α) C−1 α 12, C−1 α 13 Ψ(α) C α 13.
≥ | | | | ≤ ≤ | |
If this conjecture is true, then it shows that the linearized operator L αΛ becomes highly
−
non-selfadjoint in the fast rotating limit, and the fast rotation has a strong stabilizing effect
on vortices.
To solve this conjecture, Gallagher and Gallay suggested the following model problem (see
Villani [24] P. 53 and [23]).
Model problem: identify sufficient condition on f : R R, so that the real parts of the
→
eigenvalues of
H = ∂2+x2+iαf(x)
α − x
in L2(R) go to infinity as α + , and estimate this rate.
| | → ∞
Let Σ(α) be the infimum of the real part of σ(H ) and Ψ(α)−1 be the supremum of the
α
normoftheresolvent ofH along theimaginary axis. Undertheappropriateconditions onf,
α
Gallagher, Gallay and Nier [13] proved that Σ(α) and Ψ(α) go to infinity as α + , and
| | → ∞
presented the precise estimate of the growth rate of Ψ(α). Their proof used the hypocoercive
method, localization techniques, and semiclassical subelliptic estimates.
1
For the simplified linearized operator L αΛ1, Deng [6] proved that Ψ(α) = O(α 3). The
− | |
same result was proved by Deng [7] for the full linearized operator restricted to a smaller
subspace than ker(Λ)⊥. Deng used the multiplier method based on Weyl calculus [15].
The goal of this paper is to give a positive answer on Gallay’s conjecture. The main
difficulty comes from the nonlocal operator Λ so that the hypocoercive method introduced
2
by Villani [24] does not work. In fact, the linearized operator with the nonlocal skew-adjoint
operator often appears in the linear stability theory of the incompressible fluids.
In this paper, we develop a method to handle the nonlocal operator. The most key idea is
to reduce the nonlocal operator to a model operator by constructing the wave operator. This
is motivated by the following simple fact in the scattering theory. Let A,B be two selfadjoint
operators in theHilbertspace H. Let U(t) = eitA and V(t) = eitB bethe strongly continuous
groups of unitary operators. The wave operator is defined by
W = lim W(t), W(t)= U( t)V(t).
±
t→±∞ −
Then it holds that
(1.11) AW = W B.
± ±
In fact, we have
eisAe−itAeitB = e−i(t−s)Aei(t−s)BeisB,
which gives by taking t that
→ ±∞
eisAW = W eisB.
± ±
Then the identity (1.11) follows by taking the derivative in s at s = 0.
In a joint work of the last two authors and Zhao [26], we use similar ideas to prove the
optimal enhanced dissipation rate for the linearized Navier-Stokes equations in T2 around
the Kolmogorov flow.
4 TELI,DONGYIWEI,ANDZHIFEIZHANG
2. Spectral analysis of the linearized operator
In this section, we recall some facts about the spectrum of the linearized operator L αΛ
−
from [11, 12, 9, 10]. Although these facts will not be used in our proof, they will be helpful
to understand this spectral problem.
Let ρ(ξ) be a nonnegative function. We introduce the weighted L2 space
L2(R2,ρdξ) = w L2(R2) : w 2 = w(ξ)2ρ(ξ)dξ < + ,
∈ k kL2(ρ) R2| | ∞
n Z o
which is a (real) Hilbert space equipped with the scalar product
w ,w = w (ξ)w (ξ)ρ(ξ)dξ.
1 2 L2(ρ) 1 2
h i R2
Z
We denote Y = L2(R2, −1dξ).
G
Lemma 2.1. It holds that
1. the operator L is selfadjoint in Y with compact resolvent and purely discrete spectrum
n
(2.1) σ(L) = : n = 0,1,2, .
− 2 ···
2. the operator Λ is skew-symm(cid:8)etric in Y. (cid:9)
The first fact follows from the following observation:
ξ 2 1
(2.2) = −12L 21 = ∆+ | |
L −G G − 16 − 2
is a two-dimensional harmonic oscillator, which is self-adjoint in L2(R2) with compact resol-
vent and discrete spectrum given by σ(L). Furthermore, we know that
−
1. λ = 0 is a simple eigenvalue of L with the eigenfunction ;
0
G
2. λ = 1 is an eigenvalue of L of multiplicity two with the eigenfunctions ∂ and
1 −2 1G
∂ ;
2
G
3. λ = 1 is an eigenvalue of L of multiplicity three with the eigenfunctions ∆ ,(∂2
1 − G 1 −
∂2) and ∂ ∂ .
2 G 1 2G
Now we consider the spectrum of L αΛ in Y for any fixed α R. Since Λ is a relatively
− ∈
compact perturbation of L in Y, L αΛ has a compact resolvent in Y by the classical
−
perturbationtheory[14]. So,thespectrumof L αΛisasequenceofeigenvalues λn(α) n∈N.
− { }
Using the fact that
Λw = 0 for w = ,∂ ,∂ ,∆ ,
1 2
G G G G
we deduce that 0, 1, 1 are also eigenvalues of L αΛ for any α R. Let us introduce the
−2 − − ∈
following subspaces of X:
⊥
Y = w Y : w(ξ)dξ = 0 = ,
0
∈ R2 G
n Z o
(cid:8) (cid:9)
⊥
Y = w Y : ξw(ξ)dξ = 0 = ,∂ ,∂ ,
1 0 1 2
∈ R2 G G G
n Z o
(cid:8) (cid:9)
Y = w Y : ξ 2w(ξ)dξ = 0 = ,∂ ,∂ ,∆ ⊥.
2 1 1 2
∈ R2| | G G G G
n Z o
These spaces are invariant under the linear evolution gene(cid:8)rated by L αΛ(cid:9).
−
The following proposition shows that the Oseen vortex α is spectrally stable in Y for any
α R. G
∈
THE OSEEN VORTICES OPERATOR 5
Proposition 2.2. For any α R, the spectrum of L αΛ satisfies
∈ −
σ(L αΛ) z C : Rez 0 in Y,
− ⊂ ∈ ≤
1
σ(L αΛ) (cid:8)z C : Rez (cid:9) in Y ,
0
− ⊂ ∈ ≤ −2
σ(L αΛ) (cid:8)z C : Rez 1(cid:9) in Y ,
1
− ⊂ ∈ ≤ −
σ(L αΛ) z C : Rez < 1 in Y .
(cid:8) (cid:9) 2
− ⊂ ∈ −
The operator L αΛ is invariant un(cid:8)der rotations with(cid:9)respect to the origin. Thus, it is
natural to introduc−e the polar coordinates (r,θ) in R2. Let us decompose
(2.3) Y = n∈NXn,
⊕
where X denote the subspace of all w Y so that
n
∈
w(rcosθ,rsinθ)= a(r)cos(nθ)+b(r)sin(nθ)
for some radial functions a,b : R+ R.
→
Lemma 2.3. kerΛ= X α∂ +β∂ . In particular, kerΛ⊥ X .
0 1 2 n>0 n
⊕ G G ⊂ ⊕
3. Redu(cid:8)ction to on(cid:9)e-dimensional operators
Following Deng’s work[7], wereducethelinearized operatortoafamily of one-dimensional
operators.
1
We conjugate the linearized operator L αΛ with 2, and then obtain a linear operator
in L2(R2,dξ): − G
α
H
(3.1) α = −12L 21 +α −12Λ 21 = +α ,
H −G G G G L M
where is defined by (2.2) and is defined by
L M
1
w = vG w 12ξ KBS ( 12w) .
M ·∇ − 2G · ∗ G
Let us introduce some notations: (cid:0) (cid:1)
1 +∞ r s
(3.2) [h] = min( , )|k|sh(s)ds,
k
K 2k s r
| | Z0
1 e−r2/4
(3.3) σ(r)= − , g(r) = e−r2/8.
r2/4
Then for w = w (r)eikθ, we have
k∈Z∗ k
P ( iλ)w (rcosθ,rsinθ)= ( w )(r)eikθ,
α α,k,λ k
H − H
k∈Z∗
(cid:0) (cid:1) X
where the operator acts on L2(R ,rdr) and is given by
α,k,λ +
H
1 k2 r2 1
(3.4) = ∂2 ∂ + + +iβ (σ(r) ν ) iβ g [g ],
Hα,k,λ − r − r r r2 16 − 2 k − k − k Kk ·
where
αk
(3.5) β = , λ = β ν R.
k k k
8π ∈
Without loss of generality, we assume β 1 for any k 1.
k
| |≥ | | ≤
We introduce the operator
(3.6) α,k,λ = r12 α,k,λr−21 := k.
H H H
e e
6 TELI,DONGYIWEI,ANDZHIFEIZHANG
Then acts on L2(R ,dr) and is given by
k +
H
k2 1 r2 1
(3.7) e = ∂2+ − 4 + +iβ (σ(r) ν ) iβ g [g ],
Hk − r r2 16 − 2 k − k − k Kk ·
1 +∞ r s
(3.8) ek[h] = min( , )|k|(rs)21h(s)ds, e
K 2k s r
| | Z0
and C∞(R ) is aecore of the operator with domain
0 + Hk
(3.9) D( )= ω H2 (R ,dr) L2(R ,dr) : ω L2(R ,dr) .
Hk ∈ loc + e ∩ + Hk ∈ +
That is, (cid:8) (cid:9)
e e
w
D = D( )= w L2(R ,dr) :∂2w, ,r2w L2(R ,dr) k 2,
Hk ∈ + r r2 ∈ + | | ≥
D1 = D(eHk)=n w ∈L2(R+,dr) :r12∂r2(w/r12),r12∂r(w/r32),or2w ∈ L2(R+,dr) |k| = 1.
n o
Then the resolvent estimate is reduced to the following estimate
e
1
Hku L2(R+,dr) & |βk|3 u L2(R+,dr).
We also write (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13)e (cid:13) (cid:13) (cid:13)
(3.10) = A +iβ B iλ,
k k k k
H −
where
e e e
k2 1 r2 1
A = ∂2+ − 4 + ,
k − r r2 16 − 2
B = σ(r) g [g ].
ek k
− K ·
It is easy to see that
e e
3
(3.11) Ker(B1) = span r2g(r) , Ker(Bk) = 0 for k 2.
| | ≥
Thus, L−αΛ|(kerΛ)⊥ iseunitary eq(cid:8)uivalen(cid:9)t to Hek|(ker(cid:8)Be1)(cid:9)⊥ ⊕ Hk.
|k|=1 |k|≥2
In the sequel, we denote by , the L2(R L,dr) inner productL, and by the norm of
+ e e
L2(R ,dr), the norm ofh·Lp·i(R ,dr). The notation a & b or a . b mke·akns that there
+ Lp +
k·k
exists a constant C > 0 independent of α,k,λ so that
a C−1b or a Cb.
≥ ≤
4. Resolvent estimate of
1
H
As w = w, it is enough to prove the following resolvent estimate for .
H−1 H1 e H1
Theoreem 4.1.eFor any λ R and w r23g(r) ⊥ D1, we have e
∈ ∈ { } ∩
1
1w & β1 3 w .
kH k | | k k
Moveover, there exist λ R and v er32g(r) ⊥ D1 so that
∈ ∈{ } ∩
1
1v . β1 3 v .
kH k | | k k
e
THE OSEEN VORTICES OPERATOR 7
4.1. Reduction to the model operator . Let us introduce the operator T defined by
1
L
I [w](r)g(r)
1
(4.1) Tw(r) = w(r)+ ,
σ′(r)r32
where
r
3
(4.2) I1[w](r) = s2g(s)w(s)ds.
Z0
It is easy to check that T is a bounded linear operator in L2(R ,dr). The adjoint operator
+
T∗ is also a bounded linear operator in L2(R ,dr) given by
+
+∞ ω(s)g(s)
(4.3) T∗ω(r)= ω(r)+r23g(r) ds.
Zr s23σ′(s)
Lemma 4.2. It holds that
3
1. Tw 2 = w 2 hw,r2gi2;
k k k k − kr23gk2
2. T∗T = P, where P is the projection to r32g(r) ⊥ L2(R+,dr);
{ } ∩
3. TT∗ = IL2(R+,dr).
Proof. The first one is equivalent to the second one. Thanks to
r
(r3σ′(r))′ = r3g(r)2 and I1[s32g(s)](r) = s3g(s)2ds = r3σ′(r),
− −
Z0
we find that
3
3 3 I1[s2g(s)](r)g(r)
T(r2g)(r) = r2g(r)+ = 0.
σ′(r)r32
Thus, it suffices to check that for any w ∈ {r32g(r)}⊥ ∩L2(R+,dr), u ∈ C0∞(R+),
T∗Tw,u = w,u ,
h i h i
which is equivalent to verifying that
I [w](r)g(r) I [u](r)g(r) I [w](r)g(r) I [u](r)g(r)
1 1 1 1
,u + w, + , = 0.
σ′(r)r32 σ′(r)r32 σ′(r)r32 σ′(r)r32
D E D E D E
3 ⊥
Using the facts that for w r2g(r) ,
∈
I [ω](0) = lim I [ω](r) = 0,
(cid:8) 1 (cid:9) 1
r→+∞
and (r3σ′(r))′ = r3g(r)2, we get by integration by parts that
−
I [w](r)g(r) +∞ I [w]
1 1
,u = dI [u]
D σ′(r)r32 E Z0 σ′(r)r3 1
+∞ I [w]
= I [u]( 1 )′dr
− 1 σ′(r)r3
Z0
+∞ r32g(r)w(r)σ′(r)r3 I1[ω](σ′(r)r3)′
= I [u] − dr
− 1 (σ′(r)r3)2
Z0
+∞ I [u]g(r) +∞ g2(r)
1
= w(r) dr I [w]I [u] dr.
−Z0 σ′(r)r32 −Z0 1 1 (σ′(r)r32)2
This shows that T∗T = P.
8 TELI,DONGYIWEI,ANDZHIFEIZHANG
On the other hand, we have
+∞ w(s)g(s)
T∗w 2 w 2 = ∂ (r3σ′(r)f 2), f (r)= ds,
| | −| | − r | 1| 1 Zr s23σ′(s)
which gives T∗w 2 = w 2, thus TT∗ = I. (cid:3)
k k k k
We have the following important relationship between T and B .
1
Lemma 4.3. It holds that
e
TB = σ(r)T.
1
Proof. Direct calculation gives
e
I [B w]g(r)
1 1
TB w = B w+
1 1 σ′(r)r32
e
e e I [σw]g(r) I [g [gw]]g(r)
1 1 1
= σ(r)w g [gw]+ K .
− K1 σ′(r)r32 − σ′(r)r32
e
Thus, it suffices to show that e
(4.4) I1[σw] = σ′(r)r32 1[gw]+I1[g 1[gw]]+σ(r)I1[w].
K K
Direct calculation shows that
e e
σ′(r)r32 1[gw] = r12(e−r42 σ(r)) +∞min(r, s)(rs)12g(s)w(s)ds
K − s r
Z0
e = r12(e−r42 σ(r))[ rr−21s23g(s)w(s)ds+ +∞r32s−21g(s)w(s)ds]
−
Z0 Zr
= (e−r42 σ(r))I1[w]+r2(e−r42 σ(r)) +∞s−21g(s)w(s)ds,
− −
Zr
and
r
I1[g 1[gw]](r) = s23g2(s) 1[gw](s)ds
K K
Z0
r 1 +∞ t s
= e s32g2(s)ds min(e , )(ts)12g(t)w(t)dt
2 s t
Z0 Z0
= 1 rse−s42ds st32g(t)w(t)dt+ 1 rs3e−s42ds +∞t−21g(t)w(t)dt
2 2
Z0 Z0 Z0 Zs
= 1 rt32g(t)w(t)dt rse−s42ds+ 1 rs3e−s42ds rt−21g(t)w(t)dt
2 2
Z0 Zt Z0 Zs
+ 1 rs3e−s42ds +∞t−21g(t)w(t)dt
2
Z0 Zr
= rt23g(t)w(t)(e−t42 e−r42)dt+ 1 rt−21g(t)w(t)dt ts3e−s42ds
− 2
Z0 Z0 Z0
+[4(1 e−r42) r2e−r42] +∞t−21g(t)w(t)dt
− −
Zr
= e−r42I1[w]+I1[σw]+r2(σ(r) e−r42) +∞t−12g(t)w(t)dt,
− −
Zr
which give (4.4). (cid:3)
THE OSEEN VORTICES OPERATOR 9
Lemma 4.4. It holds that
[T,A ]w = TA w A Tw = f(r)Tw,
1 1 1
−
where
e e e
g(r)4 g(r)2 6
f(r)= 2 + r 0.
(σ′(r)2) σ′(r) r − ≥
(cid:0) (cid:1)
Proof. First of all, we have
[T,A ]w = I1[(−∂r2 + 43r12 + 1r62)w]g(r) ∂2+ 3 1 + r2 I1[w]g(r) .
1 σ′(r)r32 − − r 4r2 16 σ′(r)r32
(cid:16) (cid:17)(cid:16) (cid:17)
Using the facets that
r
I1[−∂r2w] = −r32g(r)w′(r)+(r32g(r))′w(r)− (s32g(s))′′w(s)ds,
Z0
3 1 r2
(−∂r2+ 4r2 + 16)r32g(r) = r32g(r),
we deduce that
3 1 r2
I1 (−∂r2+ 4r2 + 16)w = −r23g(r)ω′(r)+(r32g(r))′ω(r)+I1[ω],
Direct calculatio(cid:2)n gives (cid:3)
∂r2 Iσ1[′ω(r])gr(r32) = ω′(r)gσ2′((rr)) +ω(r) gσ2′((rr)) ′+r32g(r)ω(r) σg(′(rr))rr323 ′+I1[ω] σg(′(rr))rr323 ′′.
(cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Let F = r32g(r) and G(r) = σ′(r)r3. We have
3 1 r 3 1 r2
F′ = F, F′′ = 1+ F, G′ = F2.
2 · r − 4 4 · r2 − 16 −
Summing up, we ob(cid:0)tain (cid:1) (cid:0) (cid:1)
F′F F2 F
[T,A ]w = +( )′+F( )′ w
1
G G G
(cid:2) 3 1 r2 I [w(cid:3)]F F
e + 1 1 +I [w]( )′′.
− 4 · r2 − 16 G 1 G
On the other hand, we have (cid:0) (cid:1)
F 3 1 r2 F F F′F (G′)2
′′
= 1+ + 4 +2 ,
G 4 · r2 − 16 G G G G2
(cid:0)F′F(cid:1) (cid:0)F2 F (cid:1) F′F (cid:0) (G′)2 (cid:1)
+( )′+F ′ = 4 +2 ,
G G G G G2
Then we infer that (cid:0) (cid:1)
F′F (G′)2 I [w]F
1
[T,A ]w = 4 +2 (w+ )
1 G G2 G
(cid:0) g4 g2 (cid:1)6
e = 2 + ( r) Tw = f(r)Tw.
(σ′)2 σ′(r) r −
(cid:0) (cid:1)
It remains to prove that f(r) 0. We have
≥
g4 g2 6 r6+r2(6 r2)(r2+4 4er42)
2 + ( r)= − − ,
(σ′)2 σ′(r) r − 32(r2 +1 er42)2
4 −
10 TELI,DONGYIWEI,ANDZHIFEIZHANG
while by Taylor expansion, we have
r6+r2(6 r2)(r2+4 4er42) =2r4+24r2 24r2er42 +4r4er42
− − −
+∞ 1 r2 +∞ 1 r2
=2r2 r2+2r2 ( )n 12 ( )n
n! 4 − n! 4
h nX=0 nX=1 i
+∞ 1 r2 +∞ 1 r2
=2r2 8 ( )n 12 ( )n 0.
(n 1)! 4 − n! 4 ≥
h nX=2 − nX=2 i
This completes the proof. (cid:3)
It follows from Lemma 4.3 and Lemma 4.2 that for w r32g(r) ⊥ D1,
∈ { } ∩
T w = TA w+iβ TB w iλTw
1 1 1 1
H −
= TA T∗Tw+iβ σ(r)Tw iλTw.
1 1
e e e −
Lemma 4.2 ensures that T : r23g(re) ⊥ → L2(R+,dr) is invertible and T−1 = T∗. Let
w = T−1u. We infer from Lemma 4.4 that
(cid:8) (cid:9)
T T−1u=TA T−1u+iβ σ(r)u iλu
1 1 1
H −
(4.5) =A u+f(r)u+iβ σ(r)u iλu= u,
1 1 1
e e − L
where
e
g(r)4 g(r)2 6
(4.6) f(r)= 2 + r .
(σ(r)′)2 σ(r)′ r −
So, the operator T plays a role of wave operator. Le(cid:0)t (cid:1)
D( )= ω H2 (R ,dr) L2(R ,dr) : ω L2(R ,dr) .
L1 ∈ loc + ∩ + L1 ∈ +
Then u D( 1) T∗u(cid:8) D( 1) r32g(r) ⊥, and D( 1) = D( 3) = D. (cid:9)
∈ L ⇔ ∈ H ∩ L H
Moreover, we have
(cid:8) (cid:9)
w,w = T−1u,T∗u = u,u . e
1 1 1
hH i hH i hL i
On the other hand, w = Tw = u for any w r32g(r) ⊥ D1. Thus, we reduce the
k k ek k kek ∈ { } ∩
resolvent estimate of to one of the model operator .
1 1
H L
4.2. Coercive estimates.
e
Lemma 4.5. The operator A can be represented as
1
1
(4.7) A1−e2 w = −r−23g−1∂r r3g2∂r(r−32g−1w) .
In particular, we have (cid:0) (cid:1) (cid:2) (cid:3)
e
1
(4.8) A for k 1.
k
≥ 2 ≥
3
Proof. Let F(r)= r2g(r). Then weehave
r−23g−1∂r r3g2∂r(r−23g−1w) = F−1∂r F2∂r(F−1w)
− −
F′′ 3 1 r2 1
(cid:2) (cid:3) = ∂2+ (cid:2) w = ( ∂2(cid:3)+ 1+ )w = A w,
− r F − r 4 · r2 − 16 1− 2
here we used F′′ = (34 · r12 −1+ 1r(cid:0)26)F. (cid:1) (cid:0)e (cid:1)