We consider the asymptotic behavior of the linearized Boltzmann equation for soft potentials with cut-off

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ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

ASYMPTOTIC BEHAVIOR OF LINEARIZED BOLTZMANN EQUATIONS FOR SOFT POTENTIALS WITH CUT-OFF

YAKUI WU, JIAWEI SUN

Abstract. We consider the asymptotic behavior of the linearized Boltzmann equation for soft potentials with cut-off. By introducing a new decomposition of the linearized Boltzmann operator, we analyze the spectrum of the linearized Boltzmann operator and obtain the asymptotic behaviors of the linearized Boltzmann equation forγ∈(−3,0), extending the result in [12] forγ∈(−1,0).

1. Introduction We consider the Boltzmann equation

∂F

∂t +v· ∇xF=Q(F, F), (1.1) where F = F(t, x, v) is the density distribution function of the particles with (t, x, v)∈R⁺×R³×R³,Q(F, G) is a bilinear collision operator given by

Q(F, G) = Z

R³

Z

S²

q(|u−v|, ω) (F(u⁰)G(v⁰)−F(u)G(v))du dω

withu⁰=u−[(u−v)·ω]ω,v⁰ =v+ [(u−v)·ω]ω,ω∈S². For the case with inverse power interactions between particles in [4], the collision kernelq(|u−v|, ω) is taken as

q(|u−v|, ω) =|u−v|^γ|cosθ|^−γ⁰q₀(θ) (1.2) forγ = 1−⁴_s, γ⁰ = 1 +²_s, s >1, where the functionq₀(θ) is bounded, q₀(θ)6= 0 nearθ=π/2, and

cosθ=(u−v)·ω

|u−v| .

We study the Boltzmann equation (1.1) for soft potentials with cut-off. Namely, the collision kernelq(|u−v|, ω) is chosen as

q(|u−v|, ω) =q(θ)|u−v|^γ, γ∈(−3,0), (1.3) whereq(θ) satisfies 0< q(θ)≤C|cosθ|.

Considering the perturbationf ofF around the global MaxwellianM as follows F =M+M^1/2f,

2010Mathematics Subject Classification. 76P05, 35B20, 35B40, 35P20.

Key words and phrases. Linearized Boltzmann operator; soft potentials; spectrum; semigroup;

time decay; estimates.

c

2021 Texas State University.

Submitted December 9, 2020. Published May 27, 2021.

1

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where

M =M(v) = 1

(2π)^3/2e^−|v|²^/2, v∈R³, (1.4) then the Boltzmann equation (1.1) forF is reformulated in terms off into

∂f

∂t =Bf+ Γ(f, f), whereB the linearized Boltzmann operator

B=−v· ∇x+L (1.5)

with the linearized collision operator Lf =M^−1/2

Q(M, M^1/2f) +Q(M^1/2f, M)

, (1.6)

and the nonlinear term Γ(f, f) is

Γ(f, f) =M^−1/2Q(M^1/2f, M^1/2f).

There is a much important progress on the time decay estimates based on the spectral analysis for the linearized Boltzmann equation for hard potentials in [11, 13, 14, 15]. There have been a few researches on the time decay estimates with the help of the spectral analysis of the linearized Boltzmann equation for soft potentials with cut-off. The spectrum theory and time decay estimates for the linearized Boltzmann equation for γ∈(−1,0) with cut-off in spatially-periodic case were established in [1, 2]. The asymptotic behaviors of the semigroup based on the spectral analysis of the linearized Boltzmann equation forγ∈(−1,0) with cut-off in whole space were studied in [12].

In this article, we are concerned with the asymptotic behavior of the semigroup based on the spectral analysis of the linearized Boltzmann equation forγ∈(−3,0) with cut-off. The linearized Boltzmann collision operatorLdefined by (1.6) can be written as

L=−ν+K,

where the operatorsν andKwith the kernelk(u, v) are defined by (2.1) and (2.2) respectively. Ukai and Asano applied the upper bound of the kernel k(u, v) for γ∈(−1,0) to obtain the following important estimate, cf. [12],

Z

R³

|k(u, v)|²(1 +|u|)^−βdu≤Cβ(1 +|v|)^−(β+1) for anyβ ≥0, which implies that the integral operatorK satisfies

K∈C(L²_θ(R³v), L²_ς(R³v)), ifς > θ+2

γ. (1.7)

The compactness of the integral operator K plays an important role in the spectral analysis of the linearized Boltzmann operator. Inspired by the work [5], we introduce a new decomposition of the linearized Boltzmann collision operator

L=−ν+Ks

| {z }

as a whole

+Kc, (1.8)

whereK_c is compact and the norm ofK_s is small. In particular, it holds that Kc∈C(L²_θ₁(R³v), L²_θ₂(R³v))

for anyθ₁, θ₂∈R. Under the help of the decomposition, we can establish the time decay estimates of the semigroupe^tB.

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We take the Fourier transform in (1.5) with respect tox, the linearized Boltz- mann operatorB is turned into

B(ξ) =b −iv·ξ+L. (1.9)

By the Plancherel theorem, we only need to consider the time decay estimates of the semigroupe^t^B(ξ)^b . To this end, we need to establish the estimates of the resolvent (λI−B(ξ))b ⁻¹. We define the operator

Bb0(ξ) =−iv·ξ+L−P,

where the projection operator P is defined by (2.5). According to the result on the spectral analysis ofBb₀(ξ) in Proposition 3.7 and the properties of the resolvent (λI−Bb0(ξ))⁻¹ in Lemma 3.10, we can study the spectrum of the operatorB(ξ) inb L²_θ(R³v) for anyθ∈Randξ6= 0, and prove that (refer to Proposition 4.2)

σ(B(ξ))b ∈C−, σp(B(ξ))b ∈C−,

which is different from the spectrum of the linearized Boltzmann operator for hard potentials with cut-off in the case withθ= 0 as [3, 14]. Combining the decomposition

(λI−B(ξ))b ⁻¹= (I−(λI−Bb0(ξ))⁻¹P)⁻¹(λI−Bb0(ξ))⁻¹,

and the properties of the resolvent (λI−Bb0(ξ))⁻¹ given by Lemma 3.10, we can obtain the properties of the resolvent (λI−B(ξ))b ⁻¹(refer to Lemma 4.7 for details).

By the inverse Laplace transform and the properties of the resolvent (λI−B(ξ))b ⁻¹, we can obtain the time decay estimates of the semigroupe^t^B(ξ)^b for any|ξ| ≥rand r > 0 in a weighted velocity space, which is described by Theorem 4.8. Using resolvent identity, we have

(λI−B(ξ))b ⁻¹

= (λI−Bb₀(ξ))⁻¹+ (λI−Bb₀(ξ))⁻¹P(I−P(λI−Bb₀(ξ))⁻¹P)⁻¹P(λI−Bb₀(ξ))⁻¹. Then we analyze the singularities of (λI−B(ξ))b ⁻¹ nearξ= 0, and point out that the singularities of (λI−B(ξ))b ⁻¹nearξ= 0 arise from (I−P(λI−Bb0(ξ))⁻¹P)⁻¹. We compute the eigenvalues of P(λI−Bb0(ξ))⁻¹P near ξ = 0, and find that the singular points of

(I−P(λI−Bb0(ξ))⁻¹P)⁻¹

near ξ = 0 are µ_j(κ) = σ_j(κ) +iτ_j(κ), j = ±1,0,2,3, where σ_j(κ), τ_j(κ) ∈ C^∞[−r₀, r₀] for some sufficiently small constant r₀ > 0 and κ = |ξ|, which sat- isfy the following asymptotic expansions for anyκ∈[−r₀, r₀],

σj(κ) =σ_j⁽²⁾κ²+O(κ³), j=±1,0,2,3, τ_j(κ) =τ_j⁽¹⁾κ+O(κ³), j=±1,0,2,3,

where σ⁽²⁾_j <0 and τ_j⁽¹⁾ are constants. For more details, we refer to Proposition 4.9. We obtain the time decay estimates of the semigroupe^t^B(ξ)^b nearξ= 0 under the help of the asymptotic analysis ofe^t^B(ξ)^b nearξ= 0 given in Theorem 4.10.

For any ξ∈R³ and θ∈R,B(ξ) generates a semigroupb e^t^B(ξ)^b onL²_θ(R³v) (refer to Lemma 4.4). Since the absence of the spectral gap for the linearized Boltzmann operator for soft potentials, we obtain the time decay estimates of the semigroup e^tB in a weighted Sobolev space. We state our main result below.

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Theorem 1.1. Let γ ∈(−3,0). For any p∈[1,⁶₅), n∈[1,³₂(¹_p −¹₂) + ¹₂),θ∈R andl∈N, it holds that

ke^tBf0kl,θ,2

≤C

(1 +t)⁻ⁿ(kf0k_l,θ−2n,2+kf0k_Lp,2) + (1 +t)⁻³²⁽¹^p⁻¹²⁾kP f0k_Lp,2

(1.10) for any t≥0 andf0∈Hl,θ−2n,2∩L^p,2, whereP is defined by (2.5).

Remark 1.2. IfP f0= 0, then the time decay rate in (1.10) could reach (1 +t)⁻ⁿ, which is faster than (1 +t)⁻³²⁽¹^p⁻¹²⁾ for anyn∈[1,³₂(¹_p−¹₂) +¹₂) andp∈[1,⁶₅).

Notation. We will use C as a general positive constant. Denote h·,·i as the inner product onL²(R³v). We writeT^∗ for the adjoint operator of the operatorT. B(X, Y) stands for the class of linear bounded operators defined on the space X with the range inY, the norm ofT ∈B(X, Y) is expressed as kTk_B(X,Y₎, we will useB(X) forB(X, X). C(X, Y) represents the class of compact operators defined on the space X with the range in Y, we will write C(X) forC(X, X). Let Σ be a metric space and L be a normed space, we defineL^∞(Σ,L) and C⁰(Σ,L) as follows

L^∞(Σ,L) ={f : Σ→L : sup

x∈Σ

kfk_L <∞}, C⁰(Σ,L) ={f : Σ→L :f is continuous from Σ toL}.

We denote byσ(T), σp(T) and σe(T) the spectrum, point spectrum and essential spectrum for the operatorT. We denote by%(T) the resolvent set, and by (λI−T)⁻¹ the resolvent with λ∈%(T). We defineC+ ={λ∈C: Reλ >0} andC− ={λ∈ C: Reλ <0}. We define the Fourier transform ˆf(ξ) off(x) as

fb(ξ) = 1 (2π)^3/2

Z

R³

e^−ix·ξf(x)dx.

Forθ∈R, we define a weightedL²-Lebesgue spaceL²_θ(R³v) ={f(v) :ν^θ/2(v)f(v)∈ L²(R³v)} with the norm

kfk_L2

θ(R³v)=Z

R³

ν(v)^θ|f(v)|²dv1/2

,

where ν(v) is given by (2.1). For θ∈R, we introduce the weighted Sobolev space of the functionf(x, v) byHl,θ,2=L²_θ(R³v;H^l(R³x)) with the norm

kfkl,θ,2=Z

R³

Z

R³

ν(v)^θ(1 +|ξ|)^2l|fˆ(ξ, v)|²dξdv1/2

. Forp≥1, we also need the spaceL^p,2=L²(R³v;L^p(R³x)) with the norm

kfk_Lp,2=Z

R³

Z

R³

|f(x, v)|^pdx2/p

dv1/2

.

The rest of the paper is organized as follows. In Section 2, we introduce some properties of the linear collision operator. In Section 3, we present the results on spectral analysis of the operator Bb0(ξ) for anyξ∈R³ and some properties of the resolvent (λ−Bb0(ξ))⁻¹. In Section 4, we give the spectral analysis of the operator B(ξ) for anyb ξ∈R³ and the time decay estimates of the semigroupe^tB.

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2. Preliminaries

In this section, we introduce a new decomposition of the linearized Boltzmann collision operator, then give some properties of the collision operator and some lemmas, which will be used later.

The linearized collision operatorLdefined by (1.6) satisfies (Lf)(v) =−ν(v)f(v) + (Kf)(v), where

ν(v) = Z

R³

Z

S²

q(|u−v|, ω)M(u)du dω, (2.1) (Kf)(v) =

Z

R³

k(u, v)f(u)du

= Z

R³

Z

S²

q(|u−v|, ω)M^1/2(u)

×

M^1/2(u⁰)f(v⁰) +M^1/2(v⁰)f(u⁰)−M^1/2(v)f(u) du dω.

(2.2)

We will describe some properties of the operator L. For more details, we refer to [5]. The null spaceN₀ of the operatorLis a subspace spanned by the orthonormal basis{Mj, j= 0,1,2,3,4}with

M0=M^1/2, Mj =vjM^1/2(j= 1,2,3), M4= (|v|²−3)

√6 M^1/2, (2.3) where M is defined by (1.4). The operator −Lis nonnegative and self-adjoint on L²(R³v), and satisfies

h−Lf, fi ≥δk(I−P)fk²_L2

1 (2.4)

for some constantδ >0, where the projection operatorP is defined inL²(R³v) as P f =

4

X

j=0

hf, MjiMj. (2.5)

ν(v) is called the collision frequency, and satisfies

C1(1 +|v|)^γ≤ν(v)≤C2(1 +|v|)^γ (2.6) forγ∈(−3,0) and some constantsC1, C2>0.

We use a crucial decomposition of the operator K introduced by [5]. For con- venience to the readers, we write it here. For any >0, define a smooth cut-off functionχ(r) satisfying

χ(r) = 1 forr≥2, χ(r) = 0 forr≤. (2.7) The operatorK is decomposed as follows

K=K_c+K_s, K_c=K_2c−K_1c,

K_s=K_2s−K_1s= (K_2s^1−χ+K_2s^χ)−(K_1s^1−χ+K_1s^χ), (2.8) where

K_1cf = Z

|u|+|v|≤m

Z

S²

|u−v|^γχ(|u−v|)q(θ)M^1/2(u)M^1/2(v)f(u)du dω, K_1s^1−χf =

Z

R³

Z

S²

|u−v|^γ{1−χ(|u−v|)}q(θ)M^1/2(u)M^1/2(v)f(u)du dω,

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K_1s^χf = Z

|u|+|v|≥m

Z

S²

|u−v|^γχ(|u−v|)q(θ)M^1/2(u)M^1/2(v)f(u)du dω, K_2s^1−χf =

Z

R³

Z

S²

|u−v|^γ{1−χ(|u−v|)}q(θ)M^1/2(u)

×

M^1/2(u⁰)f(v⁰) +M^1/2(v⁰)f(u⁰) du dω, K_2cf = 4

Z

|v|+|v+uk|≤m

1

|u_k|e⁻¹⁴^|u^k^|²^−|ζ^k^|²f(v+u_k)k(u_k, ζ_⊥)du_k, K_2s^χf = 4

Z

|v|+|v+u_k|≥m

1

|u_k|e⁻¹⁴^|u^k^|²^−|ζ^k^|²f(v+u_k)k(u_k, ζ_⊥)du_k with

k(u_k, ζ⊥) = Z

R²

e^−|u^⊥^+ζ^⊥^|²[|u_k|²+|u⊥|²]^γ−1² χq

|u_k|²+|u⊥|² q(θ)

|cosθ|du⊥, and the integration variables

u_k= (u·ω)ω, u⊥=u−(u·ω)ω, (2.9) ζ_k+ζ_⊥= 1

2(2v+u_k), ζ_kku_k, ζ_⊥ku⊥. (2.10) We list some properties of the operatorsK andP, which will be used later. For the simplicity of expression, for anyθ∈R, we writeL²_θ forL²_θ(R³v).

Lemma 2.1 ([5]). Forθ∈R, it holds that

|hν^θKf, gi| ≤Ckν^θ/2fk_L2

1kν^θ/2gk_L2

1, (2.11)

whereν(v)is given by (2.1).

Lemma 2.2 ([5]). It holds that

|hν^θKsf, gi| ≤ηkν^θ/2fk_L2

1kν^θ/2gk_L2

1 (2.12)

for any θ∈Randη >0.

Lemma 2.3. ForP defined by (2.5), we have

P∈C(L²_θ₁, L²_θ₂) (2.13) for any θ₁, θ₂∈R.

A proof of the above lemma can be found in [12, Lemma 4.3], we omit it here.

Lemma 2.4. Let γ∈(−3,0). We have (i) For any θ∈R, it holds that

K∈B(L²_θ, L²_θ−2). (2.14)

(ii) For any θ∈Randη >0, it holds that kK_sk_B(L2

θ,L²_θ−2)≤η. (2.15)

(iii) For any θ1, θ2∈R, it holds that K_c∈C(L²_θ

1, L²_θ

2). (2.16)

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Proof. (i) From (2.11), for anyθ∈R, we have

|hν^−1/2ν^θ/2(v)Kf, ν^1/2ν^θ/2(v)gi| ≤Ckν^1/2ν^θ/2(v)fkL²kν^1/2ν^θ/2(v)gkL², which implies that

kKfk_L2

θ−1≤Ckfk_L2 θ+1. Thus, we can get (2.14).

(ii) By (2.12), for anyθ∈Randη >0, we have

|hν^−1/2ν^θ/2(v)Ksf, ν^1/2ν^θ/2(v)gi| ≤ηkν^1/2ν^θ/2(v)fkL²kν^1/2ν^θ/2(v)gkL², which implies that

kKsfk_L2

θ−1 ≤ηkfk_L2 θ+1. Thus, we can obtain (2.15).

(iii) Since _|u¹

k| ∈L²_loc(R³), the kernelk(u_k, ζ_⊥) is bounded for the chosen > 0 and any given m > 0. The Hilbert-Schmidt theorem clearly shows that K_c is a compact operator fromL²_θ

1 toL²_θ

2 for anyθ1,θ2∈R. Thus, we have proved (2.16).

The proof of Lemma 2.4 is complete.

3. Spectrum and resolvent

To analyze the spectrum of the operatorBb(ξ) onL²_θ for anyξ6= 0 and θ∈R, we introduce some auxiliary operators as follows

Ab0(ξ) =−iv·ξ, (3.1)

A(ξ) =b −iv·ξ−ν(v), (3.2) Abs(ξ) =−iv·ξ−ν(v) +Ks, (3.3) Bb0(ξ) =B(ξ)b −P =Abs(ξ) +K0 (3.4) with

K0=Kc−P, (3.5)

whereP,Ks, andKc are defined by (2.5) and (2.8). Let

D(T(ξ)) ={f ∈L²_θ:v·ξf(v)∈L²_θ}, (3.6) where T(ξ) =Ab₀(ξ),A(ξ),b Ab_s(ξ), Bb₀(ξ) orB(ξ) for anyb θ ∈Rand ξ ∈R³. It is obvious that

D(B(ξ)) =b D(Bb0(ξ)) =D(Abs(ξ)) =D(A(ξ)) =b D(Ab0(ξ)).

Lemma 3.1. The operatorA(ξ)b generates a strongly continuous contraction semigroup onL²_θ for anyθ∈Randξ∈R³.

Proof. It holds forf ∈D(A(ξ)) thatb

Rehν^θA(ξ)f, fb i= Rehν^θ(−iv·ξ−ν)f, fi=hν^θ(−ν)f, fi ≤0, Rehν^θAb^∗(ξ)f, fi= Rehν^θ(iv·ξ−ν)f, fi=hν^θ(−ν)f, fi ≤0,

which implies that the operators A(ξ) andb Ab^∗(ξ) are dissipative on L²_θ. Since D(Ab^∗(ξ)) and D(A(ξ)) are dense inb L²_θ, then A(ξ) is a densely defined closed op-b erator onL²_θ by [9, Theorem VIII.1]. Thus, with the help of Corollary 4.4 on p.15 of [8], we obtain that the operatorA(ξ) generates a strongly continuous contractionb

semigroup onL²_θ. The proof is complete.

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Let

Σ = Σ(λ,ξ)=C+×R³. (3.7)

Based on Lemma 3.1, we can obtain the following properties for the resolvent (λI−A(ξ))b ⁻¹.

Lemma 3.2. Let γ∈(−3,0). For anyθ∈R, the following statements hold.

(i) (λI−A(ξ))b ⁻¹∈L^∞(Σ, B(L²_θ−2, L²_θ)).

(ii) (λI−A(ξ))b ⁻¹∈C⁰(Σ, B(L²_θ−2−ζ, L²_θ))for anyζ >0.

(iii) For any fixedr >0 andf ∈L²_θ−2, it holds that sup

λ∈C+,|λ|≥a,|ξ|≤r

k(λI−A(ξ))b ⁻¹fk_L2

θ →0, asa→ ∞.

(iv) Writeλ=σ+iτ withσ, τ ∈Rand letf ∈L²_θ−1. Then sup

σ≥0,ξ∈R³

Z +∞

−∞

k((σ+iτ)I−A(ξ))b ⁻¹fk²_L2

θdτ ≤Ckfk²_L2 θ−1. Here Σis defined by (3.7).

A proof of the above lemma can be found in [12, Lemma 5.1], we omit it here.

Remark 3.3. SinceL²_θ

1 is dense inL²_θ

2 forθ₁, θ₂∈Randθ₁< θ₂, by the aid of (i) and (ii) in Lemma 3.2, it holds that (λI−A(ξ))b ⁻¹f ∈C⁰(Σ, L²_θ) for anyf ∈L²_θ−2. Thanks to (2.15), we can analyze the resolvent set of the operatorAbs(ξ) onL²_θ for anyξ∈R³ andθ∈R.

Lemma 3.4. Let γ∈(−3,0). We have

%(Abs(ξ))⊃C+, σ(Abs(ξ))⊂C−. (3.8) Proof. Forλ∈%(A(ξ)), we decomposeb λI−Abs(ξ) as follows

(λI−Abs(ξ)) = (λI−A(ξ))(Ib −(λI−A(ξ))b ⁻¹Ks). (3.9) Combining (i) in Lemma 3.2 and (2.15), and choosing sufficiently small η, we are able to show that

k(λI−A(ξ))b ⁻¹K_sk_B(L2 θ)

≤ sup

(λ,ξ)∈C+×R³

k(λI−A(ξ))b ⁻¹k_B(L2

θ−2,L²_θ)· kKsk_B(L2

θ,L²_θ−2)≤1 2, which implies that

k(I−(λI−A(ξ))b ⁻¹Ks)⁻¹k_B(L2

θ)≤2. (3.10)

According to Lemma 3.1, (3.9) and the Hille-Yosida theorem, we have C+⊂%(A(ξ))b ⊂%(Abs(ξ)).

The proof is complete.

For anyλ∈%(Ab_s(ξ))∩%(A(ξ)), we haveb

(λI−Ab_s(ξ))⁻¹= (I−(λI−A(ξ))b ⁻¹K_s)⁻¹(λI−A(ξ))b ⁻¹. (3.11) From Lemma 3.4, we can obtain the following properties for the resolvent of (λI− Ab_s(ξ))⁻¹.

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(i) (λI−Ab_s(ξ))⁻¹∈L^∞(Σ, B(L²_θ−2, L²_θ)).

(ii) (λI−Abs(ξ))⁻¹∈C⁰(Σ, B(L²_θ−2−ζ, L²_θ))for anyζ >0.

λ∈C+,|λ|≥a,|ξ|≤r

k(λI−Ab_s(ξ))⁻¹fk_L2

θ →0, asa→ ∞.

σ≥0,ξ∈R³

Z +∞

−∞

k((σ+iτ)I−Abs(ξ))⁻¹fk²_L2 θ

dτ ≤Ckfk²_L2 θ−1. Here Σis defined by (3.7).

Proof. By (3.10), it holds that

(I−(λI−A(ξ))b ⁻¹K_s)⁻¹∈L^∞(Σ, B(L²_θ)). (3.12) Combining (3.11) and (3.12), we can respectively obtain (i), (iii) and (iv) from (i), (iii) and (iv) in Lemma 3.2. We next prove (ii).

LetS1(λ, ξ) = (λI−Abs(ξ))⁻¹. For any (λ0, ξ0),(λ1, ξ1)∈Σ, we have kS1(λ1, ξ1)f−S1(λ0, ξ0)fk_L2

θ

≤ kS1(λ1, ξ1)χm(|v|)f−S1(λ0, ξ0)χm(|v|)fk_L2 θ

+kS1(λ1, ξ1){1−χm(|v|)}f−S1(λ0, ξ0){1−χm(|v|)}fk_L2 θ

=:I₁+I₂,

(3.13)

whereχm(|v|) is defined by (2.7). For any >0 andζ >0, it holds that I₁≤2 sup

(λ,ξ)∈C+×R³

kS₁(λ, ξ)χ_m(|v|)fk_L2 θ

≤C(1 +m)^ζγ²kfk_L2

θ−2−ζ < ,

(3.14)

where m >0 is chosen large enough. For any >0, assuming |λ1−λ0|< and

|ξ1−ξ0|< , we have

I2=k(λ1I+iv·ξ1+ν(v)−Ks)⁻¹(λ0−λ1+iv·ξ0−iv·ξ1)

×(λ0I+iv·ξ0+ν(v)−Ks)⁻¹{1−χm(|v|)}fk_L2 θ

≤Ck(λ1I+iv·ξ₁+ν(v)−K_s)⁻¹{1−χ_m(|v|)}k_B(L²

θ)(|λ0−λ₁| +m|ξ₀−ξ₁|)k(λ₀I+iv·ξ₀+ν(v)−K_s)⁻¹{1−χ_m(|v|)}fk_L2 θ

≤Ckfk_L²

θ,

(3.15)

which together with (3.13) and (3.14) yields (ii). The proof is complete.

ForBb0(ξ), we have the following similar result to [12, Lemma 5.2].

Lemma 3.6. Let γ∈(−3,0). For any ξ∈R³,Bb₀(ξ) generates a strongly continuous contraction semigroup onL².

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Proof. SinceD(Bb₀^∗(ξ)) =D(Bb0(ξ)) is dense inL², it holds thatBb0(ξ) is a densely defined closed operator on L² by [9, Theorem VIII.1]. Thanks to (2.4), for any f ∈D(Bb₀(ξ)), it holds that

RehBb0(ξ)f, fi= RehBb₀^∗(ξ)f, fi=h(L−P)f, fi

=−(h−Lf, fi+hP f, fi)

≤ −(δk(I−P)fk²_L2

1+kP fk²_L2)<0,

(3.16)

which implies thatBb₀(ξ) andBb^∗₀(ξ) are dissipative operators onL². Thus, with the help of [8, Corollary 4.4 on p.15], the operatorBb₀(ξ) generates a strongly continuous contraction semigroup onL². The proof is complete.

Based on Lemma 3.6, we can analyze the spectrum of the operatorBb0(ξ) inL²_θ for anyξ∈R³ andθ∈R.

Proposition 3.7. Let γ∈(−3,0). We have the following results.

(i) σ(Bb0(ξ))⊂C−,%(Bb0(ξ))⊃C+. (ii) σe(Bb0(ξ)) =σe(Abs(ξ)).

(iii) σp(Bb0(ξ))⊂C−.

Proof. According to Lemma 2.3 and (2.16), we know that the operatorK0:L²_θ→ L²_θ is compact. By [6, Theorem 5.35 on p.244], we have σ_e(Bb₀(ξ)) = σ_e(Ab_s(ξ)).

Thus, we have proved (ii). By Lemma 3.4, we have σe(Abs(ξ))⊂σ(Abs(ξ))⊂C−. Combining this and (ii), (iii), we can gain (i). We next prove (iii). Let λ ∈ σ_p(Bb₀(ξ)), there exists f ∈D(Bb₀(ξ)) andf 6= 0, we have

λf=Bb₀(ξ)f. (3.17)

Forθ≤0, thenf ∈L², we can apply (3.16) to derive Reλ <0. Forθ >0, assume Reλ ≥0. According to Lemma 2.3, (2.16) and (3.5), K0 is bounded from L²_θ to L²₋₂, which together with Lemma 3.5 leads to

λf =Bb0(ξ)f ⇒f = (λI−Abs(ξ))⁻¹K0f ∈L².

Then, by (3.16), we haveλ∈C−, which is a contraction to the assumption. Thus,

we have proved (iii). The proof is complete.

For anyλ∈%(Bb0(ξ))∩%(Abs(ξ)), we have

(λI−Bb₀(ξ))⁻¹= (I−(λI−Ab_s(ξ))⁻¹K₀)⁻¹(λI−Ab_s(ξ))⁻¹. (3.18) Let

M(λ, ξ) = (λI−Abs(ξ))⁻¹K0, (3.19) where Abs(ξ) and K0 is defined by (3.3) and (3.5) respectively. We state some properties ofM(λ, ξ) below.

(i) M(λ, ξ)∈L^∞(Σ, C(L²_θ)).

(ii) M(λ, ξ)∈C⁰(Σ, C(L²_θ)).

(iii) For any r >0, it holds that sup

λ∈C+,|λ|≥a,|ξ|≤r

kM(λ, ξ)k_B(L2

θ)→0, asa→ ∞.

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(iv) It holds that sup

λ∈C+,|ξ|≥r

kM(λ, ξ)k_B(L²

θ)→0, asr→ ∞.

Here Σis defined by (3.7).

Proof. (i) According to Lemma 2.3, (2.16) and (3.5), we have

K0∈C(L²_θ, L²_θ−2). (3.20) Combining (i) in Lemma 3.5 and (3.20), we can obtain (i).

(ii) By Lemma 2.3, (2.16) and (3.5), for anyζ >0, it holds that

K₀∈C(L²_θ, L²_θ−2−ζ), (3.21) which together with (ii) in Lemma 3.5 and (3.21) leads to (ii).

(iii) For anyf ∈L²_θ, it holds that kM(λ, ξ)k_B(L²

θ)= sup

kfk_L2 θ

=1

kM(λ, ξ)fk_L²

θ. (3.22)

Combining (iii) in Lemma 3.5, (3.20) and (3.22), we can get (iii).

(iv) For anyf ∈L²_θ, we have kM(λ, ξ)fk_L²

θ ≤ k(λI−Abs(ξ))⁻¹χm(|v|)K0fk_L²

θ

+k(λI−Abs(ξ))⁻¹{1−χm(|v|)}K0fk_L2 θ

=:J₁+J₂,

whereχm(|v|) is defined by (2.7). By (3.21), it holds for any >0 andζ >0 that J₁≤C(1 +m)^ζδ²k(λI−Ab_s)⁻¹k_B(L2

θ−2,L²_θ)kK₀k_B(L2

θ,L²_θ−2−ζ)kfk_L2

θ ≤, (3.23) where m >0 is chosen large enough. We next estimate J2. WriteS1={v ∈R³ :

|v| ≤m, |Imλ+v·ξ| ≤ ^√^|ξ|_r}, S2 ={v ∈R³:|v| ≤m}\S1. We use _|ξ|^ξ , ξ₁, ξ₂ as an orthonormal basis, then

v=hv, ξ

|ξ|i ξ

|ξ|+

v− hv, ξ

|ξ|i ξ

|ξ|

=L ξ

|ξ| +L1ξ1+L2ξ2. It holds that

measS1= Z

S1

1dv≤ Z m

−m

dL1

Z m

−m

dL2

Z ^√¹_r−^Im_|ξ|^λ

−^√¹_r−^Im_|ξ|^λ

dL≤8m²

√r .

For anyλ∈C+ and|ξ| ≥r, we have k(λI−A(ξ))b ⁻¹{1−χm(|v|)}K0fk²_L2

θ

= Z

S₁

ν^θ(v) 1

|Reλ+ν(v)|²+|Imλ+v·ξ|²{1−χ_m(|v|)}|K0f|²dv +

Z

S2

ν^θ(v) 1

|Reλ+ν(v)|²+|Imλ+v·ξ|²{1−χm(|v|)}|K0f|²dv

≤Ckfk²_L2

θ(S₁)+1 rkfk²_L2

θ

→0, asr→ ∞.

(3.24)

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Combining (3.24) and (3.10) yields that

J2=k(I−(λI−A(ξ))b ⁻¹Ks)⁻¹(λI−A(ξ))b ⁻¹{1−χm(|v|)}K0fk_L²

θ

≤ k(I−(λI−A(ξ))b ⁻¹Ks)⁻¹k_B(L2

θ)· k(λI−A(ξ))b ⁻¹{1−χm(|v|)}K0fk_L2 θ

→0, asr→ ∞,

which together with (3.23) verifies (iv). The proof is complete.

Lemma 3.9. Let γ∈(−3,0). For anyθ∈R, we have the following results.

(i) 1∈%((λI−Ab_s(ξ))⁻¹K₀) for any(λ, ξ)∈Σ.

(ii) (I−(λI−Ab_s(ξ))⁻¹K₀)⁻¹∈C⁰(Σ, B(L²_θ)).

(iii) (I−(λI−Abs(ξ))⁻¹K0)⁻¹∈L^∞(Σ, B(L²_θ)).

Proof. From (i) in Lemma 3.8, (λI−Ab_s(ξ))⁻¹K₀:L²_θ→L²_θis compact. By the aid of the spectral theory of the compact operator, if 1∈σ((λI−Abs(ξ))⁻¹K0), then 1∈σ_p((λI−Ab_s(ξ))⁻¹K₀). There existsf ∈L²_θandf 6= 0, it holds that

(λI−Abs(ξ))⁻¹K0f =f ⇒Bb0(ξ)f =λf,

which implies thatλ∈σp(Bb0(ξ)). It is a contradiction to (iii) in Proposition 3.7.

Thus, we have proved (i). Combining (i), (ii) in Lemma 3.8 and (i) in Lemma 3.9, it holds that (I−(λI−Abs(ξ))⁻¹K0)⁻¹∈C⁰(Σ, B(L²_θ)). Thus, we obtain (ii). Making use of (iii), (iv) in Lemma 3.8, there exists a constantr0 which is large enough, it holds for (λ, ξ)∈Σ and|λ|+|ξ| ≥r0that

k(λI−Abs(ξ))⁻¹K0k_B(L2 θ)≤ 1

2. Then

k(I−(λI−Abs(ξ))⁻¹K0)⁻¹k_B(L2

θ)≤2. (3.25)

In view of (ii) in Lemma 3.9, we know that (I−(λI−Abs(ξ))⁻¹K0)⁻¹is uniformly bounded for (λ, ξ) ∈ Σ and |λ|+|ξ| ≤ r₀. Combining this and (3.25), we have

proved (iii). The proof is complete.

With the help of (3.18), Lemma 3.5, Lemma 3.8, and Lemma 3.9, we can obtain the following properties of the resolvent (λI−Bb0(ξ))⁻¹. The proof is omitted here.

(i) (λI−Bb₀(ξ))⁻¹∈L^∞(Σ, B(L²_θ−2, L²_θ)).

(ii) (λI−Bb0(ξ))⁻¹∈C⁰(Σ, B(L²_θ−2−ζ, L²_θ))for any ζ >0.

λ∈C+,|λ|≥a,|ξ|≤r

k(λI−Bb0(ξ))⁻¹k_L²

θ →0, asa→ ∞.

σ≥0,ξ∈R³

Z +∞

−∞

k((σ+iτ)I−Bb0(ξ))⁻¹fk²_L2

θdτ ≤Ckfk²_L2 θ−1. Here Σis defined by (3.7).

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4. Decay estimates of semigroup

In this section, we give the spectrum structure of the operatorB(ξ) onb L²_θ for anyξ∈R³andθ∈R, and some properties of the resolvent (λI−B(ξ))b ⁻¹. Finally, we obtain the time decay estimates of the semigroupe^tB on the spaceHl,θ,2. 4.1. Estimates at high frequency. We recall the definition of B(ξ) given byb (1.9) and (3.4)

Bb(ξ) =−iv·ξ+L=Bb0(ξ) +P. (4.1) We first state the result on the spectrum of the operatorB(ξ) onb L²for anyξ∈R³. Lemma 4.1. Let γ∈(−3,0). For anyξ∈R³, the following statements hold.

(i) B(ξ)b generates a strongly continuous contraction semigroup onL². Conse- quently,

%(B(ξ))b ⊃C+, σ(B(ξ))b ⊂C−. (4.2) (ii)

σp(B(ξ))b ∩ {Reλ= 0}=

(∅, if ξ6= 0,

{0}, if ξ= 0. (4.3)

Proof. Since D(Bb^∗(ξ)) = D(B(ξ)) is dense inb L², it holds that B(ξ) is a denselyb defined closed operator on L² by [9, Theorem VIII.1]. Thanks to (2.4), for any f ∈D(B(ξ)), we haveb

RehBb(ξ)f, fi= RehBb^∗(ξ)f, fi=hLf, fi ≤ −(δk(I−P)fk²_L2

1)≤0, (4.4) which implies that B(ξ) andb Bb^∗(ξ) are dissipative operators on L². Thus, with the help of Corollary 4.4 on p.15 of [8], the operator B(ξ) generates a stronglyb continuous contraction semigroup on L². Thus, we have proved (i). Let λ ∈ σ_p(B(ξ)), there existsb f ∈L² andf 6= 0, it holds that

B(ξ)fb =λf. (4.5)

By (4.4), we have

Reλhf, fi= RehB(ξ)f, fb i=hLf, fi ≤0, (4.6) which implies Reλ ≤0. If Reλ= 0, it holds from (4.6) thathLf, fi= 0, which implies thatf ∈kerL. Then (4.5) is turned into

(Imλ+v·ξ)P f = 0,

which is impossible for f 6= 0 unless Imλ = 0 andξ = 0. Thus, we have proved

(ii). The proof is complete.

We have the following results about the spectrum of the operator B(ξ) onb L²_θ for anyξ∈R³ andθ∈R.

Proposition 4.2. Let γ∈(−3,0). The following statements hold.

(i) σ(B(ξ))b ⊂C−,%(B(ξ))b ⊃C+. (ii) σe(B(ξ)) =b σe(Bb0(ξ)).

(iii) σ_p(B(ξ))b ⊂C− forξ6= 0.

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Proof. We only give the proof of (iii). The proof of (i) and (ii) can be given using arguments similar to those in (i) and (ii) of Proposition 3.7. For any ξ 6= 0, let λ∈σ_p(B(ξ)), there existsb f ∈D(B(ξ)) andb f 6= 0. It holds that

λf=B(ξ)f.b

For θ ≤0, then f ∈ L², by (ii) in Lemma 4.1, we have Reλ < 0 for ξ 6= 0. For θ >0, assume Reλ≥0. By Lemma 2.3 and applying the boundness of the operator P fromL²_θ toL²₋₂and (i) in Lemma 3.10, we have

λf=B(ξ)fb ⇒f = (λI−Bb0(ξ))⁻¹P f ∈L².

By (4.3),λ∈C− forξ6= 0, which is a contradiction to the assumption. Thus, we

have proved (iii). The proof is complete.

Remark 4.3. By Lemma 4.1 and applying similar arguments to those in the proof of Proposition 4.2, we have the following result about the spectrum of the operator B(ξ) onb L²_θ for anyθ∈Randξ∈R³,

σ_p(B(ξ))b ⊂C−∪ {0}.

Lemma 4.4. B(ξ)b generates a strongly continuous semigroup onL²_θfor anyθ∈R with

ke^t^B(ξ)^b k_B(L2

θ)≤e^tkKk^B(L²^θ⁾. (4.7) Proof. Based on Lemma 3.1,A(ξ) generates a strongly continuous contraction semi-b group onL²_θ for any θ∈Rand ξ∈R³, which implies thatke^t^A(ξ)^b k_B(L²

θ)≤1. By (2.14), we have K ∈B(L²_θ). By the theory of the bounded perturbation of semigroup in [8], we obtain that B(ξ) =b A(ξ) +b K generates a strongly continuous semigroup onL²_θ ande^t^B(ξ)^b satisfies (4.7). The proof is complete.

For anyλ∈%(Bb0(ξ))∩%(B(ξ)), we haveb

(λI−B(ξ))b ⁻¹= (I−(λI−Bb₀(ξ))⁻¹P)⁻¹(λI−Bb₀(ξ))⁻¹. (4.8) We define the set

Σr={(λ, ξ)∈C+×R³:|λ|+|ξ| ≥r} (4.9) for anyr >0. Let

M1(λ, ξ) = (λI−Bb0(ξ))⁻¹P. (4.10) Then we obtain a similar results as in Lemma 3.8.

Lemma 4.5. Let γ∈(−3,0). For anyθ∈R, we have the following results.

(i) M1(λ, ξ)∈L^∞(Σ, C(L²_θ)).

(ii) M1(λ, ξ)∈C⁰(Σ, C(L²_θ)).

(iii) For any r >0, it holds that sup

λ∈C+,|λ|≥a,|ξ|≤r

kM1(λ, ξ)k_B(L2

θ)→0, asa→ ∞.

(iv) It holds that sup

λ∈C+,|ξ|≥r

kM1(λ, ξ)k_B(L2

θ)→0, asr→ ∞.

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Proof. It holds that

M1(λ, ξ) = (I−(λI−Abs(ξ))⁻¹K0)⁻¹(λI−Abs(ξ))⁻¹P

for anyλ∈%(Ab_s(ξ)). Thus, under the help of Lemma 2.3, Lemma 3.8 and Lemma 3.9, we can prove Lemma 4.5. We omit the details here.

Similar to Lemma 3.9, we have the following results.

(i) 1∈%((λI−Bb0(ξ))⁻¹P)for any (λ, ξ)∈Σr. (ii) (I−(λI−Bb₀(ξ))⁻¹P)⁻¹∈C⁰(Σ_r, B(L²_θ)).

(iii) (I−(λI−Bb0(ξ))⁻¹P)⁻¹∈L^∞(Σr, B(L²_θ)).

Here Σr is defined by (4.9).

Proof. The proof is similar to the one of Lemma 3.9. We just sketch it. In terms of the compactness of the operator (λI−Bb0(ξ))⁻¹P and the spectrum of the operator B(ξ) stated in Proposition 4.2 and Remark refrem41, we can prove (i). By the aidb of (i), (ii) in Lemma 4.5 and (i) in Lemma 4.6, we can obtain the continuity of (I−(λI−Bb0(ξ))⁻¹P)⁻¹ on Σr. Finally, combining this and (iii), (vi) in Lemma 4.5, we obtain the uniformly boundness of (I−(λI−Bb₀(ξ))⁻¹P)⁻¹ on Σ_r. According to Lemma 3.10, Lemma 4.5, Lemma 4.6, Proposition 4.2, and (4.8), we can obtain the following properties of the resolvent of (λI−B(ξ))b ⁻¹. The details are omitted here.

Lemma 4.7. For any γ∈(−3,0) andθ∈R, the following statements hold.

(i) (λI−B(ξ))b ⁻¹∈L^∞(Σ_r, B(L²_θ−2, L²_θ)).

(ii) (λI−B(ξ))b ⁻¹∈C⁰(Σr, B(L²_θ−2−ζ, L²_θ))for any ζ >0.

λ∈C+,|λ|≥a,|ξ|≤r

k(λI−B(ξ))b ⁻¹fk_L2

θ →0, asa→ ∞.

(iv) Write λ=σ+iτ with σ, τ ∈R and let f ∈L²_θ−1. It holds for any r > 0 that

sup

σ≥0,|ξ|≥r

Z +∞

−∞

k((σ+iτ)I−B(ξ))b ⁻¹fk²_L2 θ

dτ ≤Ckfk²_L2 θ−1

with a constantC >0depending onr.

Here Σ_r is defined by (4.9).

With the help of Lemma 4.7, we can evaluate the time decay estimates of the semigroupe^t^B(ξ)^b for any |ξ| ≥randr >0.

Theorem 4.8. Let γ∈(−3,0). For any |ξ| ≥r,r >0,θ∈Randn≥1, it holds that

ke^t^B(ξ)^b k_B(L2

θ−2n,L²_θ)≤C(1 +t)⁻ⁿ (4.11) for any t≥0.

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Proof. Denote the semigroupe^t^B(ξ)^b by the inverse Laplace transform of the resolvent (λI−B(ξ))b ⁻¹ as follows

e^t^B(ξ)^b f₀= lim

a→∞

1 2πi

Z σ+ia σ−ia

e^λt(λI−B(ξ))b ⁻¹f₀dλ (4.12) for anyf0∈D(B(ξ)), whereb σ >0 can be chosen arbitrarily.

LetS₂(λ, ξ) = (λI−B(ξ))b ⁻¹ andλ=s+iτ. According to Proposition 4.2 and Lemma 4.7, we can use the Cauchy’s theorem in (4.12) to shift the path of the integration froms=σto s= 0 and obtain

e^t^B(ξ)^b f₀

= lim

a→∞

1 2πi

Z +ia

−ia

e^λt(λI−B(ξ))b ⁻¹f0dλ + lim

a→∞

1 2πi

Z 0 σ

e^(s−ia)tS2(s−ia, ξ)f0ds+ Z σ

0

e^(s+ia)tS2(s+ia, ξ)f0ds . (4.13) From (iii) in Lemma 4.7, for anyf₀∈L²_θ−2, we have

kS2(s∓ia, ξ)f0k_L2

θ →0, asa→ ∞. (4.14)

Thus, the last two terms on the right-hand side of (4.13) vanish, and (4.13) is reduced to

e^t^B(ξ)^b f0= lim

a→∞

1 2π

Z a

−a

e^{iτ t}(iτ I−B(ξ))b ⁻¹f0dτ, (4.15) where we make the variable substitutionλ=iτ. Applying the integration by parts on the right-hand side of (4.15) yields

e^t^B(ξ)^b f0= lim

a→∞

1 2π

Z a

−a

e^{iτ t}(iτ I−B(ξ))b ⁻¹f0dτ

= lim

a→∞

1 2π

n

X

k=1

e^{iτ t}(k−1)!

it^k (iτ I−B(ξ))b ^−kf0

τ=a τ=−a

+ lim

a→∞

1 2π

n!

tⁿ Z a

−a

e^{iτ t}(iτ I−B(ξ))b ⁻⁽ⁿ⁺¹⁾f0dτ

(4.16)

for anyf₀∈L²_θ−2(n+1), where we have used d^l

ds^lS₂(s+iτ, ξ)f₀= 1 i^l

d^l

dτ^lS₂(s+iτ, ξ)f₀= (−1)^ll!S₂(λ, ξ)^l+1f₀,

which is valid ats= 0 for anyf0∈L²_θ−2(l+1)from (i) in Lemma 4.7. Owing to (iii) in Lemma 4.7, the first term on the right-hand side of (4.16) tends to 0 asa→ ∞, (4.16) is reduced to

e^t^B(ξ)^b f₀= lim

a→∞

1 2π

n!

tⁿ Z a

−a

e^{iτ t}(iτ I−B(ξ))b ⁻⁽ⁿ⁺¹⁾f₀dτ. (4.17) For anyf₀∈D(B(ξ))b ∩L²_θ−2(n+1)andg∈L²_θ, by (4.17) and (iv) in Lemma 4.7, it holds that

|hν^θe^t^B(ξ)^b f0, gi|= lim

a→∞

Z

R³

ν^θ n!

2πtⁿg Z a

−a

e^{iτ t}(iτ I−B(ξ))b ⁻⁽ⁿ⁺¹⁾f0dτ dv

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≤ C tⁿ

Z ∞

−∞

|hν^θ(iτ I−B(ξ))b ⁻⁽ⁿ⁺¹⁾f0, gi|dτ

≤ C tⁿ

Z ∞

−∞

|hν^θ(iτ I−B(ξ))b ⁻ⁿf0,(−iτ I−B(−ξ))b ⁻¹gi|dτ

≤ C tⁿ

Z ∞

−∞

k(iτ I−B(ξ))b ⁻ⁿf0k_L2

θ−1k(−iτ I−B(−ξ))b ⁻¹g)k_L2 θ+1dτ

≤ C tⁿ

Z ∞

−∞

k(iτ I−B(ξ))b ⁻¹f0k²_L2

θ−2n+1dτ1/2

×Z ∞

−∞

k(−iτ I−B(−ξ))b ⁻¹g)k²_L2 θ+1

dτ1/2

≤ C tⁿkf0k_L²

θ−2nkgk_L²

θ, which implies that

ke^t^B(ξ)^b k_B(L2

θ−2n,L²_θ)≤Ct⁻ⁿ. (4.18) By Lemma 4.4 and (4.18), we can obtain for anyn∈N^∗ andt≥0 that

ke^t^B(ξ)^b k_B(L²

θ−2n,L²_θ)≤C(1 +t)⁻ⁿ. (4.19) By applying the interpolation theorem, we can obtain (4.19) for any n ≥1. The

proof is complete.

4.2. Estimates at low frequency. In this subsection, we analyze the singularities of (λI−B(ξ))b ⁻¹near ξ= 0. We decompose (λI−Bb(ξ))⁻¹as follows

(λI−B(ξ))b ⁻¹= (λI−Bb0(ξ))⁻¹

+ (λI−Bb0(ξ))⁻¹(I−P(λI−Bb0(ξ))⁻¹)⁻¹P(λI−Bb0(ξ))⁻¹. (4.20) We will check that

(I−P(λI−Bb₀(ξ))⁻¹)⁻¹P f =P(I−P(λI−Bb₀(ξ))⁻¹P)⁻¹P f. (4.21) Write

g= (I−P(λI−Bb0(ξ))⁻¹)⁻¹P f. (4.22) By (4.22), it holds that

g=P(λI−Bb0(ξ))⁻¹g+P f ∈kerL, which, from (4.22), implies

P g=P(λI−Bb₀(ξ))⁻¹P g+P f.

Thus, we obtain

g=P g=P(I−P(λI−Bb₀(ξ))⁻¹P)⁻¹P f. (4.23) Substituting (4.21) into (4.20), we have

(λI−B(ξ))b ⁻¹= (λI−Bb₀(ξ))⁻¹

+ (λI−Bb0(ξ))⁻¹P(I−P(λI−Bb0(ξ))⁻¹P)⁻¹P(λI−Bb0(ξ))⁻¹. (4.24) Combining (i) in Lemma 3.10 and (i) in Lemma 4.5, we obtain that the singularities of the resolvent (λI −B(ξ))b ⁻¹ nearξ = 0 arise from (I−P(λI−Bb₀(ξ))⁻¹P)⁻¹.