Local Gevrey Regularity for Linearized Homogeneous Boltzmann Equation

Abstract

The local Gevrey regularity of the solutions of the linearized spatially homogeneous Boltzmann equation has been shown in the non-Maxwellian case with mild singularity.

1. Introduction

This paper focuses on the Gevrey class smoothing property of solutions of the following linear Cauchy problems of the spatially homogeneous Boltzmann equation:𝜕𝑓𝜕𝑡=𝐿𝑓=𝑄𝜇,𝑓+𝑄𝑓,𝜇,𝑣∈ℝ³,𝑡>0,𝜇(𝑣)=(2𝜋)^−3/2𝑒^−|𝑣|2/2,𝑓|_𝑡=0=𝑓₀, (1.1) where the initial datum 𝑓₀≢0 satisfies the natural boundedness on mass, energy, and entropy: 𝑓₀≥0,_ℝ3𝑓₀(𝑣)1+|𝑣|²+log1+𝑓₀(𝑣)𝑑𝑣<+∞. (1.2)𝑄(𝑔,𝑓) is the Boltzmann quadratic operator which has the following form: 𝑄𝑔,𝑓=_ℝ3𝕊2𝐵𝑣−𝑣_∗,𝜎𝑔𝑣^′_∗𝑓𝑣^′−𝑔𝑣_∗𝑓(v)𝑑𝜎𝑑𝑣_∗, (1.3) where 𝜎∈𝕊² (unit sphere of ℝ³); the post- and precollisional velocities are given as follows: 𝑣^′=𝑣+𝑣_∗2+𝑣+𝑣_∗2𝜎,𝑣^′_∗=𝑣+𝑣_∗2−𝑣+𝑣_∗2𝜎. (1.4) The Boltzmann collision cross-section 𝐵(|𝑧|,𝜎) is a nonnegative function which depends only on |𝑧| and the scalar product ⟨𝑧/|𝑧|,𝜎⟩. To capture its main properties, we usually assume 𝐵𝑣−𝑣_∗,𝜎=Φ𝑣−𝑣_∗𝑏(cos𝜃),cos𝜃=𝑣−𝑣_∗𝑣−𝑣_∗𝜋,𝜎,𝜃∈0,2. (1.5)𝜇 is called the normalized Maxwellian distribution in (1.1). Notice that 𝑄(𝜇,𝜇)≡0.

Recall that the inverse power law potential 1/𝜌^𝑠, where 𝑠>1, and 𝜌 denotes the distance between two particles, has the form (1.5) with the corresponding kinetic factors: Φ𝑣−𝑣_∗≈𝑣−𝑣_∗^1−4/𝑠,𝑏()≈𝐾cos𝜃𝜃^2+𝜈,𝜃→0, (1.6) for a constant 𝐾>0 and 0<𝜈=2/𝑠<2. The cases 1<𝑠<4,𝑠=4, and 𝑠>4 correspond to so-called soft, Maxwellian, and hard potentials, respectively.

We will concentrate on the modified hard potentials as follows: Φ𝑣−𝑣_∗=1+𝑣−𝑣_∗^2𝛾/2𝑏()≈𝐾,0<𝛾<1,cos𝜃𝜃^2+𝜈,𝜃→0,0<𝜈<2, (1.7) where the singularity is called the mild singularity when 0<𝜈<1 and the strong singularity when 1≤𝜈<2. In this paper, we consider only the case of the mild singularity. Before making the discussion, we start by introducing the norms of the weighted function spaces: 𝑓_𝐿^𝑝_𝑟=⟨|𝑣|⟩^𝑟𝑓(𝑣)_𝐿𝑝,𝑓_𝐻𝑠𝑟=⟨|𝐷|⟩^𝑠⟨|𝑣|⟩^𝑟𝑓(𝑣)_𝐿2, (1.8) where ⟨|𝑣|⟩=(1+|𝑣|²)^1/2 and ⟨|𝐷|⟩ is the corresponding pseudodifferential operator. And then, we list the definition of the weak solution in the Cauchy problem (1.1); compare [1].

Definition 1.1. For an initial datum 𝑓₀(𝑣)∈𝐿¹₂(ℝ³), 𝑓(𝑡,𝑣) is called a weak solution of the Cauchy problem (1.1) if it satisfies 𝑓(𝑡,𝑣)∈𝐶ℝ⁺;𝒟^′ℝ³∩𝐿²[0,𝑇];𝐿¹₂ℝ³∩𝐿^∞[0,𝑇];𝐿¹ℝ³,𝑓(0,𝑣)=𝑓₀,_ℝ3𝑓(𝑡,𝑣)𝜑(𝑡,𝑣)𝑑𝑣−_ℝ3𝑓(0,𝑣)𝜑(0,𝑣)𝑑𝑣−^𝑡₀𝑑𝜏_ℝ3𝑓(𝜏,𝑣)𝜕_𝜏𝜑()=𝜏,𝑣𝑑𝑣_𝑡0𝑑𝜏_ℝ3𝐿𝑓(𝜏,𝑣)𝜑(𝜏,𝑣)𝑑𝑣, (1.9) for any test function 𝜑∈𝐿^∞([0,𝑇];𝑊^2,∞(ℝ³)).

For the definition of the Gevrey class functions, compare [1–5].

Definition 1.2. Suppose that 𝑊 is a bounded open set on ℝ³, for 𝑠≥1, 𝑢∈𝐺^𝑠(𝑊) which is the Gevrey class function space with index 𝑠, if 𝑢∈𝐶^∞(𝑊) and for any compact subset 𝑈⊂𝑊, there exists a constant 𝐶=𝐶(𝑈)>0 such that for any 𝑘∈ℕ, 𝐷^𝑘𝑢_𝐿2(𝑈)≤𝐶^𝑘+1(𝑘!)^𝑠, (1.10) or equivalently, ⟨|𝐷|⟩^𝑘𝑢_𝐿2(𝑈)≤𝐶^𝑘+1(𝑘!)^𝑠, (1.11) where 𝐷^𝑘𝑢²_𝐿2(𝑈)=_|𝛽|=𝑘𝐷^𝛽𝑢²_𝐿2(𝑈),⟨|𝐷|⟩=1+𝐷_𝑣^21/2. (1.12) Particularly, 𝑢∈𝐺^𝑠(ℝ³), that is, ||𝐷^𝑘𝑢||_𝐿²_(ℝ³₎≤𝐶^𝑘+1(𝑘!)^𝑠, is equivalent to the fact that there exists 𝜖₀>0 such that 𝑒^{𝜖0⟨|𝐷|⟩1/𝑠}𝑢∈𝐿²(ℝ³).

Notice that 𝐺¹(ℝ³) is the usual analytic function space. When 0<𝑠<1, we call 𝐺^𝑠(ℝ³) the ultra-analytic function space, cpmpare [4, 5].

There have been some results about the Gevrey regularity of the solutions for the Boltzmann equation; compare [1, 4, 6–8]. Among them, unique local solutions having the same Gevrey regularity as the initial data are first constructed in [8]. This implies the propagation of the Gevrey regularity. In 2009, Desvillettes et al. improved this result for the nonlinear spatially homogeneous Boltzmann equation, they showed in [6] that, for the Maxwellian molecules model, the Gevrey regularity can propagate globally in time. Other results for the nonlinear case can be found in [4], where the Gevrey regularity of the radially symmetric weak solutions has been proved. Meanwhile, this issue is also considered in [7] for the Maxwellian decay solutions. For the linear case, the best result so far is obtained by the work of Morimoto et al. in [1]; they proved the propagation of Gevrey regularity of the solutions, without any extra assumption for the initial data. We mention that the crucial tools in [1, 6] are the following pseudodifferential operator: 𝐺_𝛿𝑡,𝐷_𝑣=1𝛿+𝑒^{−𝑡⟨|𝐷𝑣|⟩𝜈/2},0<𝜈<2. (1.13)

In the Maxwellian case, this pseudodifferential operator can be used successfully, but it seems unsuitable for the non-Maxwellian model. The difficulty comes from the commutator of the kinetic factor Φ and the pseudodifferential operator (1.13) which lacks of the effective estimations. In this paper, we apply a new method which is based on the mathematical induction to overcome it. Compared with [7], we consider only the local space; however, we discuss this issue by using the much weaker preconditions (actually, we do not need any smooth assumption for the initial data). Concerning the same issue for the other related equations, such as the Landau equation and the Kac equation, compare [2–5].

Now we can state our main result.

Theorem 1.3. Suppose Φ, 𝑏 have the forms in (1.7), 0<𝜈<1. Let 𝑊 be a bounded open set of ℝ³, and 𝑓(𝑡,𝑣) be the weak solution of the Cauchy problem (1.1) satisfying sup_{𝑡∈(0,𝑇]}𝑓(𝑡,⋅)_𝐿2(𝑊)<+∞. (1.14) Then for any 𝑡∈(0,𝑇], there exists a number 𝑠=𝑠(𝑡)>3 satisfying 𝑓(𝑡,⋅)∈𝐺^𝑠(𝑊). More precisely, for any fixed 0<𝑡₀≤𝑇 and compact subset 𝑈⊂𝑊, there exists a constant 𝐶=𝐶(𝑈)>0 and a number s>3 such that for any 𝑘∈ℕ, sup_{𝑡∈[𝑡0,𝑇]}𝐷^𝑘𝑓(𝑡,⋅)_𝐿2(𝑈)≤𝐶^𝑘+1(𝑘!)^𝑠. (1.15)

From Theorem 1.3, we have the following remark.

Remark 1.4. Suppose that Φ, 𝑏 have the forms in (1.7), 0<𝜈<1. If the weak solution 𝑓(𝑡,𝑣) satisfies that sup_{𝑡∈(0,𝑇]}𝑓(𝑡,⋅)_𝐿2(ℝ3)<+∞, (1.16) then for any 𝑡∈(0,𝑇], any bounded open set 𝑈⊂ℝ³, there exists a constant 𝑠=𝑠(𝑡) satisfying 𝑓(𝑡,⋅)∈𝐺^𝑠(𝑈).

2. Useful Lammas for the Main Result

In order to gain the main result, we need to prove the following lemmas in this section.

Lemma 2.1. Suppose Φ(𝑣)=⟨|𝑣|⟩^𝛾=(1+|𝑣|²)^𝛾/2 where 𝛾∈(0,1),𝑣∈ℝ^𝑛, and 𝑛∈ℕ. Then the 𝑘th order derivative of Φ satisfies Φ^(𝑘)(𝑣)≤4^𝑘𝑘!Φ(𝑣)⟨|𝑣|⟩^−𝑘. (2.1)

Proof. Without loss of generality, we only consider the case of 𝑛=1; the other cases are similar. By direct calculation, we have Φ^(2𝑚)(𝑣)=^𝑚_𝑖=0𝐶_𝑖,2𝑚𝛾𝛾−2⋯𝛾−2𝑖−2𝑚+21+𝑣^{2𝛾/2−𝑖−𝑚}𝑣^2𝑖,Φ^(2𝑚+1)(𝑣)=^𝑚_𝑖=0𝐴_𝑖,2𝑚+1𝛾𝛾−2⋯𝛾−2𝑖−2𝑚1+𝑣^{2𝛾/2−𝑖−𝑚−1}𝑣^2𝑖+1. (2.2) In addition, 𝐶_𝑖,2𝑚+2(𝑖+1)𝐶_𝑖+1,2𝑚=𝐴_𝑖,2𝑚+1,(2𝑖+1)𝐴_𝑖,2𝑚+1+𝐴_{𝑖−1,2𝑚+1}=𝐶_𝑖,2𝑚+2. (2.3) Thus we obtain 𝐶_𝑖,2𝑚+2=𝐶_{𝑖−1,2𝑚}+(4𝑖+1)𝐶_𝑖,2𝑚+(2𝑖+1)(2𝑖+2)𝐶_𝑖+1,2𝑚 (2.4) and then we will prove the following inequality: 𝐶_𝑖,2𝑚≤2^2𝑚𝛾−2𝑖−2𝑚⋯𝛾−4𝑚+2. (2.5) The inequality is obviously true for 𝑚=1. Suppose it is valid for 1≤𝑚≤𝑀, then 𝐶_𝑖,2𝑀+2=𝐶_{𝑖−1,2𝑀}+(4𝑖+1)𝐶_𝑖,2𝑀+(2𝑖+1)(2𝑖+2)𝐶_𝑖+1,2𝑀≤2^2𝑀++(⋯(4𝑖+1)𝛾−2𝑖−2𝑀𝛾−2𝑖+2−2𝑀𝛾−2𝑖−2𝑀2𝑖+1)(2𝑖+2)𝛾−2𝑖−2−2𝑀𝛾−4𝑀+2≤2^2(𝑀+1)𝛾−2𝑖−2−2𝑀⋯𝛾−4𝑀𝛾−4𝑀−2 (2.6) which proves (2.5) by induction. Therefore, we have Φ^(2𝑚)(𝑣)≤2^2𝑚𝛾𝛾−2⋯𝛾−4𝑚+2^𝑚_𝑖=01+𝑣^{2𝛾/2−𝑖−𝑚}𝑣^2𝑖≤(𝑚+1)2^4𝑚−1(2𝑚−1)!Φ(𝑣)⟨|𝑣|⟩^−2𝑚≤4^2𝑚(2𝑚)!Φ(𝑣)⟨|𝑣|⟩^−2𝑚. (2.7) The case of (2𝑚+1)th order derivative is similar. This completes the proof of Lemma 2.1.

Setting 𝑀_𝑁(𝜉)=(1+|𝜉|²)^𝑁𝑡/2 for any 𝜉∈ℝ³ and 𝑁∈ℕ, by using the similar technique of Lemma 2.1, we conclude the following.

Remark 2.2. For 𝑡∈(0,1], 𝜕^𝑘_𝜉𝑀_𝑁(𝜉)≤4^𝑘𝜉^{𝑁𝑡−𝑘}|𝑁(𝑁−1)⋯(𝑁−𝑘+1)|≤4^𝑘𝜉^{(𝑁−𝑘)𝑡}|𝑁(𝑁−1)⋯(𝑁−𝑘+1)|, (2.8) where 𝑘∈ℕ,1≤𝑘≤𝑁.

Lemma 2.3. There exists a constant 𝐶 such that for any 𝑘∈ℕ, 𝜕^𝑘_𝑣𝜇(𝑣)≤𝐶^𝑘⋅𝑘!⋅max1,|𝑣|^𝑘⋅𝜇(𝑣), (2.9) where 𝜇 is the absolute Maxwellian distribution in (1.1).

Proof. Without loss of generality, we also only consider the case in the real space ℝ¹. Putting 𝜕^𝑘_𝑣𝜇(𝑣)=𝜕^𝑘_𝑣𝑒^−𝑣2/2=^𝑘_𝑗=0𝑎^′_𝑗,𝑘𝑣^𝑗𝑒^−𝑣2/2,𝜕^𝑘_𝑣𝜇⁻¹(𝑣)=𝜕^𝑘_𝑣𝑒^𝑣2/2=^𝑘_𝑗=0𝑎_𝑗,𝑘𝑣^𝑗𝑒^𝑣2/2. (2.10) Evidently, 0≤𝑎^′_𝑗,𝑘≤𝑎_𝑗,𝑘,𝑎^′_𝑘,𝑘=𝑎_𝑘,𝑘≡1,_1𝑗=0𝑎_𝑗,1=1≤8¹𝑎⋅1!,_𝑗,𝑘+1=𝑎_{𝑗−1,𝑘}+𝑗+1𝑎_𝑗+1,𝑘. (2.11) Therefore, fixed a number  𝑚≥0, together with the following assumption (𝐹_𝑚):_𝑗𝑎_𝑗,𝑚≤8^𝑚⋅𝑚!, (2.12) we can obtain (𝐹_𝑚+1)_𝑗𝑎_𝑗,𝑚+1=_𝑗𝑎_{𝑗−1,𝑚}+𝑗+1𝑎_𝑗+1,𝑚≤(𝑚+2)⋅_𝑗𝑎_{𝑗−1,𝑚}+_𝑗𝑎_𝑗+1,𝑚≤2(𝑚+2)⋅8^𝑚⋅𝑚!≤8^𝑚+1⋅(𝑚+1)!. (2.13) This completes the proof of Lemma 2.3 by induction.

Setting 𝐻^∗(𝑣)=1+𝑣_∗²⁴𝜇^∗(𝑣)=1+𝑣_∗²⁴⋅𝜇𝑣+𝑣_∗, (2.14) where 𝑣 is belong to a bounded set 𝑈. Then we state Lemma 2.4 as below.

Lemma 2.4. There exists a constant 𝐶=𝐶(𝑈)>0, which satisfies that for any 𝑘∈ℕ, sup_𝑣∗𝜕^𝑘_𝑣𝐻^∗(𝑣)≤𝐶^𝑘⋅(𝑘!)². (2.15)

Proof. Since 𝑒^{−(𝑣+𝑣∗)2/4}≤𝑒^𝑣2/4⋅𝑒^{−𝑣2∗/8}, and the fact that when |𝑣|≥1, 𝑣^𝑘𝑒^−𝑣2/4=|𝑣|^𝑘∑^+∞_𝑛=0|𝑣|^2𝑛/2^2𝑛≤|𝑣|⋅𝑛!|𝑣|^𝑘/2^𝑘⋅^𝑘(𝑘/2)!≤2^𝑘⋅𝑘!, (2.16) by using Lemma 2.3, we have 𝜕^𝑘_𝑣𝐻^∗(𝑣)=1+𝑣_∗²⁴𝜕^𝑘_𝑣𝜇^∗(𝑣≤𝑣)1+_∗²⁴⋅𝐶^𝑘⋅𝑘!⋅max1,𝑣+𝑣_∗^𝑘⋅𝜇𝑣+𝑣_∗≤1+𝑣_∗²⁴⋅𝐶^𝑘⋅𝑘!⋅max1,𝑣+𝑣_∗^𝑘⋅𝑒^{−(𝑣+𝑣∗)2/4}⋅𝑒^{−(𝑣+𝑣∗)2/4}≤1+𝑣_∗²⁴𝑒^{−𝑣2∗/8}⋅𝐶^𝑘⋅𝑘!⋅max1,𝑣+𝑣_∗^𝑘⋅𝑒^{−(𝑣+𝑣∗)2/4}⋅𝑒^𝑣2/4≤𝐶^𝑘⋅𝑘!²≤[𝐶(𝑈)]^𝑘⋅(𝑘!)². (2.17) This completes the proof of Lemma 2.4

By applying the Cauchy integral theorem, we will prove the helpful estimates as follows.

Lemma 2.5. Suppose the Fourier transform for 𝑣_∗, ℱΦ𝑣−𝑣_∗𝜇𝑣_∗(𝜉)=ℎ(𝑣,𝜉)𝜇(𝜉), (2.18) where 𝜇 is the absolute Maxwellian distribution in (1.1). Then we have ℎ(𝑣,𝜉)=(2𝜋)^−3/2_ℝ3𝑒^{−|𝑣∗|2/2}1+𝑣−𝑣_∗²−𝜉²+2𝑖𝑣−𝑣_∗⋅𝜉^𝛾/2𝑑𝑣_∗. (2.19)

Proof. First we consider the case of 𝑛=1, ℱΦ𝑣−𝑣_∗𝜇𝑣_∗(𝜉)=_ℝ11+𝑣−𝑣_∗^2𝛾/2(2𝜋)^−3/2𝑒^{−|𝑣∗|2/2−𝑖𝑣∗⋅𝜉}𝑑𝑣_∗=(2𝜋)^−3/2𝑒^−𝜉2/2_ℝ11+𝑣−𝑣_∗^2𝛾/2𝑒^{−(𝑣∗+𝑖𝜉)2/2}𝑑𝑣_∗=(2𝜋)^−3/2𝑒^−𝜉2/2_𝐶𝑒^−𝑧2/21+(𝑣−𝑧+𝑖𝜉)^2𝛾/2𝑑𝑧, (2.20) where 𝑧=𝑣_∗+𝑖𝜉, and 𝐶 denotes the curve: 𝑣_∗+𝑖𝜉,−∞<𝑣_∗<∞. By Cauchy integral theorem [9], it follows that _𝐶𝑒^−𝑧2/21+(𝑣−𝑧+𝑖𝜉)^2𝛾/2𝑑𝑧=_ℝ1𝑒^{−|𝑣∗|2/2}1+𝑣−𝑣_∗+𝑖𝜉^2𝛾/2𝑑𝑣_∗. (2.21) Now we turn to consider the case of 𝑛=3. Letting 𝑣=(𝑣₁,𝑣₂,𝑣₃), and 𝑣_∗=(𝑣_∗1,𝑣_∗2,𝑣_∗3) and using the previous result, we have ℱΦ𝑣−𝑣_∗𝜇𝑣_∗(𝜉)=_ℝ31+𝑣−𝑣_∗^2𝛾/2(2𝜋)^−3/2𝑒^{−|𝑣∗|2/2−𝑖𝑣∗⋅𝜉}𝑑𝑣_∗=(2𝜋)^−3/2_ℝ2ℝ11+𝑣₁−𝑣_∗1²+𝑣₂−𝑣_∗2²+𝑣₃−𝑣_∗3^2𝛾/2𝑒^{−𝑣2∗1/2−𝑖𝑣∗1𝜉1}𝑑𝑣_∗1×𝑒^{−(𝑣2∗2+𝑣2∗3)/2−𝑖(𝑣∗2𝜉2+𝑣∗3𝜉3)}𝑑𝑣_∗2𝑑𝑣_∗3=(2𝜋)^−3/2_ℝ2ℝ11+𝑣₁−𝑣_∗1+𝑖𝜉₁²+𝑣₂−𝑣_∗2²+𝑣₃−𝑣_∗3^2𝛾/2𝑒^{−𝑣2∗1/2−𝜉21/2}𝑑𝑣_∗1×𝑒^{−(𝑣2∗2+𝑣2∗3)/2−𝑖(𝑣∗2𝜉2+𝑣∗3𝜉3)}𝑑𝑣_∗2𝑑𝑣_∗3=(2𝜋)^−3/2𝑒^−|𝜉|2/2_ℝ3𝑒^{−|𝑣∗|2/2}1+³_𝑗=1𝑣_𝑗−𝑣_∗𝑗+𝑖𝜉_𝑗^2𝛾/2𝑑𝑣_∗=(2𝜋)^−3/2𝜇(𝜉)_ℝ3𝑒^{−|𝑣∗|2/2}1+𝑣−𝑣_∗²−𝜉²+2𝑖𝑣−𝑣_∗⋅𝜉^𝛾/2𝑑𝑣_∗. (2.22) Thus we conclude the result of Lemma 2.5.

Lemma 2.6. For the expression of ℎ(𝑣,𝜉) in Lemma 2.5, we have ℎ(𝑣,𝜉)≤𝐶⋅⟨|𝑣|⟩^𝛾𝜉^𝛾,∇²_𝜉ℎ(𝑣,𝜉)≤𝐶⋅⟨|𝑣|⟩^𝛾𝜉^𝛾,ℎ𝑣,𝜉⁺−ℎ(𝑣,𝜉)≤𝐶⋅⟨|𝑣|⟩^𝛾𝜉^1+𝛾𝜃sin2𝜉,𝜃=arccos𝜉,𝜎, (2.23) where 𝜉⁺=(𝜉+|𝜉|𝜎)/2, and 𝐶 is a constant independent of 𝑣 and 𝜉.

Proof. The first inequality is obvious. To prove the third one, set 𝜉=(𝜉₁,𝜉₂,𝜉₃). Since ℎ(𝑣,𝜉)𝜇(𝜉)=ℱΦ𝑣−𝑣_∗𝜇𝑣_∗(𝜉)=_ℝ31+𝑣−𝑣_∗^2𝛾/2(2𝜋)^−3/2𝑒^{−|𝑣∗|2/2−𝑖𝑣∗⋅𝜉}𝑑𝑣_∗, (2.24) proceeding as in the proof of Lemma 2.5, we can get 𝜕_𝜉𝑖ℎ(𝑣,𝜉)𝜇(𝜉)=𝜇(𝜉)𝜕_𝜉𝑖ℎ(𝑣,𝜉)−𝜉_𝑖ℎ(𝑣,𝜉)=_ℝ31+𝑣−𝑣_∗^2𝛾/2(2𝜋)^−3/2𝑒^{−|𝑣∗|2/2−𝑖𝑣∗⋅𝜉}−𝑖𝑣_∗𝑖𝑑𝑣_∗=(2𝜋)^−3/2𝜇(𝜉)_ℝ31+𝑣−𝑣_∗²−𝜉²+2𝑖𝑣−𝑣_∗⋅𝜉^𝛾/2⋅𝑒^{−|𝑣∗|2/2}⋅−𝜉_𝑖−𝑖𝑣_∗𝑖𝑑𝑣_∗. (2.25) Therefore, 𝜕_𝜉𝑖ℎ(𝑣,𝜉)=(2𝜋)^−3/2_ℝ3𝑒^{−|𝑣∗|2/2}1+𝑣−𝑣_∗²−𝜉²+2𝑖𝑣−𝑣_∗⋅𝜉^𝛾/2⋅−𝜉_𝑖−𝑖𝑣_∗𝑖𝑑𝑣_∗+𝜉_𝑖ℎ()=()𝑣,𝜉2𝜋^−3/2_ℝ3𝑒^{−|𝑣∗|2/2}1+𝑣−𝑣_∗²−𝜉²+2𝑖𝑣−𝑣_∗⋅𝜉^𝛾/2⋅−𝑖𝑣_∗𝑖𝑑𝑣_∗ (2.26) which implies that ∇_𝜉ℎ(𝑣,𝜉)≤𝐶⋅⟨|𝑣|⟩^𝛾𝜉^𝛾. (2.27) By the mean value theorem of differentials, we have ℎ𝑣,𝜉⁺−ℎ(𝑣,𝜉)≤𝐶⋅∇_𝜉ℎ𝑣,𝜂⋅𝜉⁺−𝜉≤𝐶^′⋅⟨|𝑣|⟩^𝛾𝜂^𝛾𝜉⁺−𝜉≤𝐶^′′⋅⟨|𝑣|⟩^𝛾𝜉^1+𝛾𝜃sin2, (2.28) where 𝜃=arccos⟨𝜉/|𝜉|,𝜎⟩. Thus the third inequality has been obtained.
Finally, the above way can also be used in estimating the second one. Similarly, 𝜕²_{𝜉𝑖𝜉𝑗}ℎ(𝑣,𝜉)𝜇(𝜉)=𝜕_𝜉𝑗𝜕_𝜉𝑖ℎ(𝑣,𝜉)𝜇(𝜉)=−_ℝ31+𝑣−𝑣_∗^2𝛾/2(2𝜋)^−3/2𝑒^{−|𝑣∗|2/2−𝑖𝑣∗⋅𝜉}𝑣_∗𝑖𝑣_∗𝑗𝑑𝑣_∗=−(2𝜋)^−3/2𝜇(𝜉)_ℝ31+𝑣−𝑣_∗²−𝜉²+2𝑖𝑣−𝑣_∗⋅𝜉^𝛾/2⋅𝑒^{−|𝑣∗|2/2}𝑣_∗𝑖−𝑖𝜉_𝑖𝑣_∗𝑗−𝑖𝜉_𝑗𝑑𝑣_∗. (2.29) On the other hand, 𝜕²_{𝜉𝑖𝜉𝑗}ℎ(𝑣,𝜉)𝜇(𝜉)=𝜕_𝜉𝑗𝜇(𝜉)𝜕_𝜉𝑖ℎ(𝑣,𝜉)−𝜉_𝑖ℎ(=𝜇(𝜉𝜉𝑣,𝜉))_𝑗𝜉_𝑖ℎ(𝑣,𝜉)+𝜕²_{𝜉𝑗𝜉𝑖}ℎ(𝑣,𝜉)−𝜉_𝑗𝜕_𝜉𝑖ℎ(𝑣,𝜉)−𝜉_𝑖𝜕_𝜉𝑗ℎ(𝑣,𝜉). (2.30) Combining with the above expressions of ℎ(𝑣,𝜉) and 𝜕_𝜉𝑖ℎ(𝑣,𝜉), we get 𝜕²_{𝜉𝑗𝜉𝑖}ℎ(𝑣,𝜉)=𝜉_𝑖𝜕_𝜉𝑗ℎ(𝑣,𝜉)+𝜉_𝑗𝜕_𝜉𝑖ℎ(𝑣,𝜉)−𝜉_𝑗𝜉_𝑖ℎ()−()𝑣,𝜉2𝜋^−3/2_ℝ31+𝑣−𝑣_∗²−𝜉²+2𝑖𝑣−𝑣_∗⋅𝜉^𝛾/2⋅𝑒^{−|𝑣∗|2/2}𝑣_∗𝑖−𝑖𝜉_𝑖𝑣_∗𝑗−𝑖𝜉_𝑗𝑑𝑣_∗=−(2𝜋)^−3/2_ℝ31+𝑣−𝑣_∗²−𝜉²+2𝑖𝑣−𝑣_∗⋅𝜉^𝛾/2⋅𝑒^{−|𝑣∗|2/2}𝑣_∗𝑖𝑣_∗𝑗𝑑𝑣_∗. (2.31) Therefore, ∇²_𝜉ℎ(𝑣,𝜉)≤𝐶⋅⟨|𝑣|⟩^𝛾|𝜉|^𝛾. (2.32) This completes the proof of the second inequality.

Lemma 2.7. Suppose that 0<𝜈<1 in (1.7). Then for any 𝑟>0,𝑓∈𝐿¹_2+𝛾(ℝ³)∩𝐻^+∞(ℝ³), there exists a constant 𝐶 independent of 𝑟 satisfying 𝐼₀(𝜏)=𝑄𝑓,𝜇,⟨|𝐷|⟩^𝑟𝑓_𝐿2≤𝐶𝑓_𝐿12+𝛾𝑓_𝐿1(𝑟+3)!. (2.33)

Proof. Let 𝜉^±=(𝜉±|𝜉|𝜎)/2, from Bobylev’s formula (see [10]), we have 𝐼₀(𝜏)=_ℝ3𝜉^𝑟𝑓(𝜉)_ℝ6𝕊2Φ𝑣−𝑣_∗𝑏𝜉𝜉⋅𝜎𝜇𝑣_∗𝑓(𝑣)×𝑒^{−𝑖(𝑣∗⋅𝜉++𝑣⋅𝜉−)}−𝑒^{−𝑖𝑣∗⋅𝜉}𝑑𝜎𝑑𝑣𝑑𝑣_∗=𝑑𝜉_ℝ3𝜉^𝑟𝑓(𝜉)_ℝ3𝕊2ℱΦ𝑣−𝑣_∗𝜇𝑣_∗𝜉⁺𝑏𝑓(𝑣)𝑒^{−𝑖𝑣⋅𝜉−}−ℱΦ𝑣−𝑣_∗𝜇𝑣_∗𝜉)(𝑣=(𝑏𝑓)𝑑𝜎𝑑𝑣𝑑𝜉_ℝ3𝜉^𝑟𝑓(𝜉)_ℝ3𝕊2ℎ𝑣,𝜉⁺−ℎ(𝑣,𝜉)𝜇𝜉⁺𝑏𝑓(𝑣)𝑒^{−𝑖𝑣⋅𝜉−}+𝑑𝜎𝑑𝑣𝑑𝜉_ℝ3𝜉^𝑟𝑓(𝜉)_ℝ3𝕊2ℎ(𝑣,𝜉)𝜇𝜉⁺−𝜇(𝜉)𝑏𝑓(𝑣)𝑒^{−𝑖𝑣⋅𝜉−}+𝑑𝜎𝑑𝑣𝑑𝜉_ℝ3𝜉^𝑟𝑓(𝜉)_ℝ3𝕊2ℎ(𝑣,𝜉)𝜇(𝜉)𝑏𝑓(𝑣)𝑒^{−𝑖𝑣⋅𝜉−}−𝑒⁰𝑑𝜎𝑑𝑣𝑑𝜉=𝐼₀₁+𝐼₀₂+𝐼₀₃. (2.34) In [1], it is shown that 𝜇𝜉⁺−𝜇(𝜉)≤𝜇𝜉⁺𝜉²sin²𝜃2,𝑒^−|𝜉|2/2=𝜇(𝜉)≤𝜇𝜉⁺≤𝑒^−|𝜉|2/4,|𝜉|^𝑟+𝛾+2≤(𝑟+3)!𝑒^{⟨|𝜉|⟩(𝑟+𝛾+2)/(𝑟+3)}≤(𝑟+3)!𝑒^⟨|𝜉|⟩. (2.35) Together with Lemma 2.6, we have 𝐼₀₂≤𝐶⋅𝑓_{𝐿11+𝛾ℝ3}|𝜉|^𝑟+𝛾+2𝑒^−|𝜉|2/4𝑓(𝜉)𝑓𝑑𝜉≤𝐶⋅_𝐿11+𝛾𝑓_𝐿1ℝ3|𝜉|^𝑟+𝛾+2𝑒^−|𝜉|2/4𝑓𝑑𝜉≤𝐶⋅_𝐿11+𝛾𝑓_𝐿1(𝑟+3)!_ℝ3𝑒^{⟨|𝜉|⟩−|𝜉|2/4}𝑑𝜉≤𝐶^′𝑓_𝐿11+𝛾𝑓_𝐿1(𝑟+3)!. (2.36) Now we turn to estimate the terms in 𝐼₀₁ and 𝐼₀₃. For the case 0<𝜈<1 in (1.7), it is easy to see that 𝑒^{−𝑖𝑣⋅𝜉−}−𝑒⁰=−2sin𝑣⋅𝜉⁻2sin𝑣⋅𝜉⁻2+𝑖cos𝑣⋅𝜉⁻2≤𝐶|𝑣|𝜉⁻|𝑣𝜉𝜃≤𝐶|sin2. (2.37) Therefore, applying the above estimates and Lemma 2.6, we also conclude that 𝐼_0𝑖≤𝐶^′𝑓_𝐿11+𝛾𝑓_𝐿1(𝑟+3)! (2.38) for any 𝑖∈{1,3}. This completes the proof of Lemma 2.7.

3. Related Analysis

Let 𝑓 be the weak solution of the Cauchy problem (1.1). For any 𝑘∈ℕ, the compact support supp𝑀_𝑘𝑓⊆supp𝑓, (3.1) which implies that for any compact subset 𝑈⊂𝑊, 𝑓^∗=𝑓,if𝑣∈𝑈,0,if𝑣∉𝑈, (3.2) is also a weak solution of the following equation 𝜕𝑓𝜕𝑡,𝑀²_𝑘𝑓_𝐿2(ℝ3)=𝑄𝜇,𝑓+𝑄𝑓,𝜇,𝑀²_𝑘𝑓_𝐿2(ℝ3). (3.3) Since Theorem 1.3 is mainly concerned with the Gevrey smoothness property of the solution 𝑓 on 𝑊, we need only to study the solution of the above equation on any fixed compact subset of 𝑊. That is, we can suppose that 𝑓 has compact support in 𝑈 for any 𝑡∈[0,𝑇], supp𝑓⊆𝑈,𝑓𝑈^𝑐≡0. (3.4) Thus, for any 𝑝≥0, 𝑓_{𝐿1𝑝(ℝ3)}≤𝑂(1)𝑓_𝐿1(𝑈)𝑓<+∞,_{𝐻𝑝(ℝ3)}=𝑓_{𝐻𝑝(𝑈)}. (3.5) Together with Lemma 2.6, we can get the fact that 𝑓∈𝐻^+∞(ℝ³). This proof is similar as the proof of [11, Theorem 1.1] and hence omitted. Clearly, ||𝑓||_𝐻^𝑟_(ℝ³₎=||𝑓||_𝐻^𝑟_(𝑈). Moreover, without loss of generality, we restrict 𝑇≤1, then for any 𝑘∈ℕ, it is assumed that 𝐸_𝑘:forany𝑖∈[0,𝑘−1],sup_{𝑡∈(0,𝑇]}𝑓(𝑡,⋅)_𝐻𝑖𝑡≤𝐶₀^𝑖+1(𝑖!)^𝑠, (3.6) where 𝐶₀ is a sufficiently large constant satisfying 𝐶₀≥16⁶maxsup_{(]𝑡∈0,𝑇}𝑓_𝐿𝑖,𝑖=1,2. (3.7)

In the following discussion, we will use 𝐶 and 𝐶_𝑖,𝑖∈ℕ to denote the positive constants independent of 𝑘 and 𝑡. Let 𝑀_𝑘(𝐷_𝑣)=⟨|𝐷_𝑣|⟩^𝑘𝑡 and Φ^∗(𝑣)=⟨|𝑣−𝑣_∗|⟩^𝛾. In order to prove Theorem 1.3, we need the propositions as below.

Proposition 3.1. One has sup_{𝑡∈(0,𝑇]}𝑀_𝑘𝐷_𝑣,Φ^∗𝑓(𝑡,𝑣)_𝐿2≤𝐶⋅𝐶₀^𝑘+1(𝑘!)^𝑠. (3.8)

Proposition 3.2. One has sup_{𝑡∈(0,𝑇]}∇_𝑣𝑀_𝑘𝐷_𝑣,Φ^∗𝑓(𝑡,𝑣)_𝐿2≤𝐶(𝑘+1)𝑀_𝑘𝑓(𝑡,𝑣)_𝐿2+𝐶₀^𝑘+1(𝑘!)^𝑠,sup_{𝑡∈(0,𝑇]}𝑀_𝑘𝐷_𝑣,𝐻^∗𝑓(𝑡,𝑣)_𝐿2≤𝐶⋅𝐶₀^𝑘+1(𝑘!)^𝑠, (3.9)(3.10)
where 𝐻^∗ is the function which has the form (2.14).

The proof of the above propositions will be given in Section 5.

4. Proof of Theorem 1.3

Now we will prove the main result in this section. For any 𝑡∈(0,𝑇], we state the following identity from [11]: 𝑄𝜇,𝑓,𝑀²_𝑘𝑓_𝐿2−𝑄𝜇,𝑀_𝑘𝑓,𝑀_𝑘𝑓_𝐿2=𝐼₁+𝐼₂+𝐼₃, (4.1) where 𝐼₁=_ℝ6𝕊2𝑏(cos𝜃)𝜇𝑣_∗𝑀_𝑘(𝜉)−𝑀_𝑘𝜉⁺Φ^∗𝑓𝜉⁺𝑒^{−𝑖𝑣∗𝜉−}𝑀_𝑘(𝜉)𝑓(𝜉)𝑑𝜎𝑑𝑣_∗𝐼𝑑𝜉,₂=_ℝ6𝕊2𝑏(cos𝜃)𝜇𝑣_∗𝑀_𝑘,Φ^∗𝑓𝑣^′⋅𝑀_𝑘𝑓𝑣^′−𝑀_𝑘,Φ^∗𝑓(𝑣)⋅𝑀_𝑘𝑓(𝑣)𝑑𝜎𝑑𝑣_∗𝐼𝑑𝑣,₃=_ℝ6𝕊2𝑏(cos𝜃)𝜇𝑣_∗𝑀_𝑘,Φ^∗𝑓(𝑣)−𝑀_𝑘,Φ^∗𝑓𝑣^′𝑀_𝑘𝑓𝑣^′𝑑𝜎𝑑𝑣_∗𝑑𝑣. (4.2)

Our purpose is to obtain the estimations of 𝐼₁, 𝐼₂ and 𝐼₃. Setting 𝜂=|𝜉|² and 𝜂⁺=|𝜉⁺|², since |𝜉⁺|=|𝜉|cos(𝜃/2) and |𝜉⁺|²−|𝜉|²=|𝜉|²sin²(𝜃/2), applying the mean value theorem and the fact that 0<𝑡≤𝑇≤1, we have 𝑀_𝑘(𝜉)−𝑀_𝑘𝜉⁺=1+𝜂^𝑘𝑡/2−1+𝜂^+𝑘𝑡/2≤𝐶⋅𝑘𝑡⋅𝜂−𝜂⁺1+𝜂₀^{𝑘𝑡/2−1}≤𝐶⋅𝑘𝑡⋅sin²𝜃21+𝜂^𝑘𝑡/2≤𝐶^′⋅𝑘⋅sin²𝜃𝑀_𝑘(𝜉), (4.3) where 𝜂₀ is a number between 𝜂 an 𝜂⁺. Therefore, 𝐼₁≤𝐶_ℝ3𝕊2𝑏sin²𝜃𝑀_𝑘(𝜉)_ℝ6Φ𝑣−𝑣_∗𝜇𝑣_∗𝑓(𝑣)𝑒^{−𝑖𝑣∗⋅𝜉−−𝑖𝑣⋅𝜉+}𝑑𝑣_∗×𝑑𝑣𝑘𝑀_𝑘(𝜉)𝑓(𝜉)𝑑𝜎𝑑𝜉=𝐶_ℝ3𝕊2𝑏sin²𝜃𝑀_𝑘(𝜉)_ℝ6Φ𝑣_∗𝜇𝑣_∗+𝑣𝑓(𝑣)𝑒^{−𝑖𝑣∗⋅𝜉−−𝑖𝑣⋅𝜉}𝑑𝑣_∗×𝑑𝑣𝑘𝑀_𝑘(𝜉)𝑓(𝜉)𝑑𝜎𝑑𝜉≤𝐶_ℝ6𝕊2𝑏sin²𝜃𝑣_∗^𝛾−8⋅𝑀_𝑘(𝜉)𝐻^∗𝑓(𝜉)⋅𝑘𝑀_𝑘(𝜉)𝑓(𝜉)𝑑𝜎𝑑𝑣_∗𝑑𝜉≤𝐶_ℝ6𝕊2𝑏sin²𝜃𝑣_∗^𝛾−8⋅𝑀_𝑘(𝜉)𝐻^∗𝑓(𝜉)²𝑑𝜎𝑑𝑣_∗𝑑𝜉+𝐶_ℝ6𝕊2𝑏sin²𝜃𝑣_∗^𝛾−8⋅𝑘𝑀_𝑘(𝜉)𝑓(𝜉)²𝑑𝜎𝑑𝑣_∗𝑑𝜉=𝐼₁₁+𝐼₁₂. (4.4) Here 𝐻^∗ is the function which has the form (2.14). It is clear that 𝐼₁₂≤𝐶^′𝑘²𝑀_𝑘𝑓²_𝐿2,𝐼₁₁≤𝐶_ℝ3𝑣_∗^𝛾−8⋅𝑀_𝑘𝐷_𝑣𝐻^∗𝑓²_𝐿2𝑑𝑣_∗≤𝐶_ℝ3𝑣_∗^𝛾−8𝑀_𝑘,𝐻^∗𝑓²_𝐿2𝑑𝑣_∗+𝐶_ℝ3𝑣_∗^𝛾−8𝐻^∗𝑀_𝑘𝑓²_𝐿2𝑑𝑣_∗=𝐼₁₁₁+𝐼₁₁₂. (4.5) By the hypothesis (3.4), 𝑓 has compact support in 𝑈, we obtain 𝐼₁₁₂≤𝐶_ℝ3𝑣_∗^𝛾+8⋅𝜇𝑣+𝑣_∗𝑀_𝑘𝑓²_𝐿2𝑑𝑣_∗≤𝐶_ℝ3𝑣_∗^𝛾+8𝑒^{−|𝑣∗|2/4}⋅𝑒^|𝑣|2/2𝑀_𝑘𝑓²_𝐿2𝑑𝑣_∗≤𝐶^′𝑀_𝑘𝑓²_𝐿2. (4.6) Here we use the fact that 𝑒^{−|𝑣+𝑣∗|2/2}≤𝑒^|𝑣|2/2⋅𝑒^{−|𝑣∗|2/4}. By (3.10) of Proposition 3.2, we get 𝐼₁₁₁≤𝐶^′⋅𝐶₀^𝑘+1(𝑘!)^𝑠2. (4.7) This, together with (4.5)-(4.6), implies 𝐼₁≤𝐶₁⋅𝑘²𝑀_𝑘𝑓²_𝐿2+𝐶₀^𝑘+1(𝑘!)^𝑠2. (4.8) The cancellation lemma gives (cf. [10, 11]) 𝐼₂=𝑆_ℝ6𝜇𝑣_∗𝑀_𝑘,Φ^∗𝑓(𝑣)⋅𝑀_𝑘𝑓(𝑣)𝑑𝑣𝑑𝑣_∗, (4.9) where 𝑆 is a constant function. Therefore, 𝐼₂≤𝐶𝜇_𝐿1⋅𝑀_𝑘,Φ^∗𝑓_𝐿2⋅𝑀_𝑘𝑓_𝐿2≤𝐶⋅𝐶₀^𝑘+1(𝑘!)^𝑠𝜇_𝐿1⋅𝑀_𝑘𝑓_𝐿2≤𝐶₂𝐶₀^𝑘+1(𝑘!)^𝑠2+𝑀_𝑘𝑓²_𝐿2. (4.10) Since |𝑣^′−𝑣|≤𝐶⟨|𝑣^′|⟩⟨|𝑣_∗|⟩sin(𝜃/2), by using (3.4), Proposition 3.2, and the change of variables 𝑣→𝑧=𝑣^′+𝜏𝑣−𝑣^′ (4.11) whose Jacobian is bounded uniformly for 𝑣_∗,𝜎,𝜏 (see [11]), we have 𝐼₃≤𝐶¹_0ℝ6𝕊2𝑏(cos𝜃)𝑣^′−𝑣⋅𝑀_𝑘𝑓𝑣^′⋅∇_𝑣𝑀_𝑘,Φ^∗𝑓𝑣^′+𝜏𝑣−𝑣^′×𝜇𝑣_∗𝑑𝜎𝑑𝑣𝑑𝑣_∗𝑑𝜏≤𝐶^′1_0ℝ6𝕊2𝑏()𝜃cos𝜃sin2𝑀_𝑘𝑓𝑣^′⋅∇_𝑣𝑀_𝑘,Φ^∗𝑓𝑣^′+𝜏𝑣−𝑣^′×𝑣_∗𝜇𝑣_∗𝑑𝜎𝑑𝑣𝑑𝑣_∗𝑑𝜏≤𝐶^′1_0ℝ6𝕊2𝑏()𝜃cos𝜃sin2𝑣_∗𝜇𝑣_∗×𝑀_𝑘𝑓𝑣^′2+∇_𝑣𝑀_𝑘,Φ^∗𝑓𝑣^′+𝜏𝑣−𝑣^′2𝑑𝜎𝑑𝑣𝑑𝑣_∗𝑑𝜏≤𝐶^′′𝑀_𝑘𝑓²_𝐿2+∇_𝑣𝑀_𝑘,Φ^∗𝑓²_𝐿2≤𝐶₃𝐶₀^𝑘+1(𝑘!)^𝑠2+𝑘²𝑀_𝑘𝑓²_𝐿2. (4.12) Combining (4.8), (4.10), and (4.12), we obtain 𝑄𝜇,𝑓,𝑀²_𝑘𝑓_𝐿2−𝑄𝜇,𝑀_𝑘𝑓,𝑀_𝑘𝑓_𝐿2≤𝐶₄𝐶₀^𝑘+1(𝑘!)^𝑠2+𝑘²𝑀_𝑘𝑓²_𝐿2. (4.13) Moreover, by [11, Lemma 2.2] and [11, page 467], we have ⟨⋅⟩^𝛾/2𝑀_𝑘𝑓²_𝐻𝜈/2=𝑂(1)⟨|𝐷|⟩^𝜈/2𝑀_𝑘𝑓²_𝐿2𝛾/2,𝑄𝜇,𝑀_𝑘𝑓,𝑀_𝑘𝑓_𝐿2≤−𝐶_𝜇,1⟨⋅⟩^𝛾/2𝑀_𝑘𝑓²_𝐻𝜈/2+𝐶_𝜇,2𝑀_𝑘𝑓²_𝐿2𝛾/2, (4.14) where 𝐶_𝜇,1 and 𝐶_𝜇,2 are the constants depending only on 𝜇. Therefore, by (3.4) and (4.14), we get 𝑄𝜇,𝑀_𝑘𝑓,𝑀_𝑘𝑓_𝐿2≤−𝐶₅𝑀_𝑘𝑓²_𝐻𝜈/2+𝐶₆𝑀_𝑘𝑓²_𝐿2. (4.15) Together with (4.13), we thus have 𝑄𝜇,𝑓,𝑀²_𝑘𝑓_𝐿2≤𝐶₇⋅𝐶₀^𝑘+1(𝑘!)^𝑠2+𝐶₈⋅𝑘²𝑀_𝑘𝑓²_𝐿2−𝐶₅𝑀_𝑘𝑓²_𝐻𝜈/2. (4.16) Let 𝑀²_𝑘𝑓 be the test function in the Cauchy problem (1.1), for any 𝑡∈(0,𝑇], we have 𝑀_𝑘𝑓(𝑡,𝑣)²_𝐿2=𝑓₀(𝑣)²_𝐿2+2^𝑡_0ℝ3𝐿𝑓(𝜏,𝑣)𝑀²_𝑘𝑓()+𝜏,𝑣𝑑𝑣𝑑𝜏_𝑡0ℝ3𝑓(𝜏,𝑣)𝜕_𝜏𝑀²_𝑘(𝜏)𝑓(𝜏,𝑣)𝑑𝑣𝑑𝜏=2^𝑡₀𝑄𝜇,𝑓,𝑀²_𝑘𝑓_𝐿2+𝑄𝑓,𝜇,𝑀²_𝑘𝑓_𝐿2𝑑𝜏+^𝑡_0ℝ3𝑓(𝜏,𝑣)𝜕_𝜏𝑀²_𝑘(𝜏)𝑓(𝜏,𝑣)𝑑𝑣𝑑𝜏+𝑓₀(𝑣)²_𝐿2. (4.17) Since 𝑂(1)(𝑘!)^2𝑠≥2^2(𝑘+2)[(𝑘+2)!]²≥()𝜕2𝑘+3!,_𝑡𝑀²_𝑘(𝑡,𝜉)=2𝑘𝑀²_𝑘(𝑡,𝜉)log𝜉, (4.18) by Lemma 2.7 and (4.16), it holds that 𝑀_𝑘𝑓(𝑡,𝑣)²_𝐿2+𝐶₅^𝑡₀𝑀_𝑘𝑓²_𝐻𝜈/2𝑑𝜏≤2k^𝑡₀log𝐷_𝑣^1/2𝑀_𝑘𝑓(𝜏)²_𝐿2𝑑𝜏+𝐶₈^𝑡₀𝑘²𝑀_𝑘𝑓²_𝐿2𝑑𝜏+𝐶₉𝐶₀^𝑘+1(𝑘!)^𝑠2+𝑓₀(𝑣)²_𝐿2. (4.19) The Young’s inequality gives 𝐶₈𝑘²≤4𝐶₈𝐶₅⋅(𝜈+2)^2/𝜈⋅𝐶₈𝜈𝜈+2⋅𝑘^2+4/𝜈𝜉⁻²+𝐶₅2𝜉^𝜈,2𝑘log𝜉=4𝑘𝜈log𝜉^𝜈/2≤4𝑘𝜈⋅𝜉^𝜈/2≤4⋅(4+𝜈)𝐶₈⋅𝜈(𝜈+2)^{(4+𝜈)/𝜈}2𝜈+2⋅𝑘^2+4/𝜈𝜉⁻²+𝐶₅2𝜉^𝜈, (4.20) which implies 2𝑘^𝑡₀log𝐷_𝑣^1/2𝑀_𝑘𝑓(𝜏)²_𝐿2𝑑𝜏+𝐶₈^𝑡₀𝑘𝑀_𝑘𝑓²_𝐿2𝑑𝜏≤𝐶₅^𝑡₀𝑀_𝑘𝑓²_𝐻𝜈/2𝑑𝜏+𝐶₁₀⋅𝑘^2+4/𝜈⋅^𝑡₀𝑀_𝑘−1𝑓²_𝐿2𝑑𝜏. (4.21) Taking (4.21) into (4.19), and applying the assumption (𝐸_𝑘), we have 𝑀_𝑘𝑓(𝑡,𝑣)²_𝐿2≤𝑓₀(𝑣)²_𝐿2+𝐶₉𝐶₀^𝑘+1(𝑘!)^𝑠2+𝐶₁₀⋅𝑘^2+4/𝜈𝑡₀𝑀_𝑘−1𝑓²_𝐿2≤𝑓𝑑𝜏₀(𝑣)²_𝐿2+𝐶₉𝐶₀^𝑘+1(𝑘!)^𝑠2+𝐶₁₀⋅𝑘^2+4/𝜈𝐶^𝑘₀⋅[(𝑘−1)!]^𝑠2≤𝐶₁₁𝐶₀^𝑘+1(𝑘!)^𝑠2, (4.22) which implies that sup_{𝑡∈(0,𝑇]}𝑀_𝑘𝑓(𝑡,𝑣)_𝐿2=sup_{𝑡∈(0,𝑇]}𝑓(𝑡,⋅)_𝐻𝑘𝑡≤𝐶₁₁⋅𝐶₀^𝑘+1(𝑘!)^𝑠. (4.23) In other words, it follows from (𝐸_𝑘) that 𝐸_𝑘+1:forany𝑖∈[0,𝑘],sup_{𝑡∈(0,𝑇]}𝑓(𝑡,⋅)_𝐻𝑖𝑡≤𝐶₁₁⋅𝐶₀^𝑖+1(𝑖!)^𝑠. (4.24) Taking the same procedures as above, we can also gain (𝐸_𝑘+2) from (𝐸_𝑘+1), which is described as below: 𝐸_𝑘+2:forany𝑖∈[0,𝑘+1],sup_{𝑡∈(0,𝑇]}𝑓(𝑡,⋅)_𝐻𝑖𝑡≤𝐶²₁₁⋅𝐶₀^𝑖+1(𝑖!)^𝑠, (4.25) that is,sup_{𝑡∈(0,𝑇]}𝑓(𝑡,⋅)_𝐻0𝑡≤𝐶¹₀(0!)^𝑠⇒sup_{𝑡∈(0,𝑇]}𝑓(𝑡,⋅)_𝐻1𝑡≤𝐶¹₁₁⋅𝐶²₀(1!)^𝑠⇒sup_{𝑡∈(0,𝑇]}𝑓(𝑡,⋅)_𝐻2𝑡≤𝐶²₁₁⋅𝐶³₀(2!)^𝑠⋮⇒sup_{𝑡∈(0,𝑇]}𝑓(𝑡,⋅)_𝐻𝑘𝑡≤𝐶^𝑘₁₁⋅𝐶₀^𝑘+1(𝑘!)^𝑠. (4.26) Let 𝐶₁₂=𝐶₀⋅𝐶₁₁, we thus conclude that for any 𝑘∈ℕ, sup_{𝑡∈(0,𝑇]}𝑓(𝑡,⋅)_{𝐻𝑘𝑡(𝑈)}≤𝐶^𝑘+1₁₂(𝑘!)^𝑠. (4.27) For any fixed number 0<𝑡≤𝑇≤1, suppose that 𝑠₀=1𝑡𝑠+1𝑠,₁=^⌈1/𝑡⌉_𝑖=0(1−𝑖𝑡), (4.28) where ⌈1/𝑡⌉ denotes the smallest integer bigger than 1/𝑡. With a convention that 𝑘!=1 if 0≥𝑘∈ℤ, we have 2^𝑘𝑠1=2^𝑘⋅2^{𝑘(1−𝑡)}⋅2^{𝑘(1−2𝑡)}⋯2^{𝑘(1−{1/𝑡}𝑡)}≥𝑘!()!)𝑘]!⋅)𝑘]!𝑘𝑡[(1−𝑡[(1−𝑡()!)𝑘]!⋯⌈⌉𝑡)𝑘]!𝑘𝑡[(1−2𝑡[(1−1/𝑡()!⌉)𝑡)𝑘]!≥𝑘𝑡[(1−(⌈1/𝑡+1𝑘![(𝑘𝑡)!]^{(⌈1/𝑡⌉+1)}≥𝑘![(𝑘𝑡)!]^𝑠0/𝑠. (4.29) This, together with (4.27), implies that sup_{𝑡∈(0,𝑇]}𝑓(𝑡,⋅)_{𝐻𝑘𝑡(𝑈)}≤𝐶^𝑘+1₁₂(𝑘!)^𝑠≤2^{𝑘𝑠𝑠1}𝐶^𝑘+1₁₂[(𝑘𝑡!)]^𝑠0≤𝐶^𝑘+1₁₃[(𝑘𝑡!)]^𝑠0, (4.30) where 𝑘∈ℕ, and 𝐶₁₆ is a constant only depending on 𝑡. Furthermore, for any fixed number 𝑡₀>0, put 𝑠^′₀=1𝑡₀𝑠+1𝑠,_′1=^{⌈1/𝑡0⌉}_𝑖=01−𝑖𝑡₀. (4.31) Then for any 𝑘∈ℕ, we can choose 𝐶^′₁₃=2^𝑠𝑠′1𝐶₁₂ and have the fact that sup_{𝑡∈[𝑡0,𝑇]}𝑓(𝑡,⋅)_{𝐻𝑘𝑡(𝑈)}≤𝐶^′₁₃^𝑘+1[(𝑘!)]^𝑠′0. (4.32) This completes the proof of Theorem 1.3.

5. Proof of Propositions 3.1 and 3.2

Proof of Proposition 3.1. We first notice that 𝑀_𝑘𝐷_𝑣,Φ^∗𝑓(𝑣)=𝑀_𝑘𝐷_𝑣Φ^∗𝑓(𝑣)−Φ^∗(𝑣)𝑀_𝑘𝐷_𝑣𝑓(𝑣)=ℱ⁻¹𝑀_𝑘(𝜉)∗Φ^∗𝑓(𝑣)−Φ^∗(𝑣)ℱ⁻¹𝑀_𝑘(𝜉)(𝑣)=∗𝑓_ℝ6𝑒^{𝑖(𝑣−𝑦)𝜉}𝑀_𝑘(𝜉)𝑑𝜉𝑓𝑦Φ^∗𝑦−Φ^∗(𝑣)𝑑𝑦. (5.1) Using the Taylor formula of order 𝑘+6, we get Φ^∗𝑦−Φ^∗(𝑣)=^𝑘+5_𝑗=1𝑦−𝑣^𝑗𝜕𝑗!^𝑗𝑣Φ^∗(𝑣)+𝑦−𝑣^𝑘+6()!𝜕𝑘+6^𝑣𝑘+6Φ^∗(𝑐) (5.2) for some 𝑐∈(𝑦,𝑣). Hence, 𝑀_𝑘𝐷_𝑣,Φ^∗𝑓(𝑣)=^𝑘+5_𝑗=1Γ_𝑗𝑓(𝑣)+Γ_𝑘+6𝑓, (5.3) where Γ_𝑗𝑓(𝑣)=_ℝ6𝑒^{𝑖(𝑣−𝑦)𝜉}𝑀_𝑘(𝜉)𝑑𝜉𝑓𝑦𝑦−𝑣^𝑗𝜕𝑗!^𝑗𝑣Φ^∗(𝑣)=()𝑑𝑦−𝑖^𝑗𝑗!_ℝ⁶𝑒^{𝑖(𝑣−𝑦)𝜉}𝜕^𝑗_𝜉𝑀_𝑘(𝜉)𝑑𝜉𝑓𝑦𝜕^𝑗_𝑣Φ^∗(𝑣)=()𝑑𝑦−𝑖^𝑗ℱ𝑗!⁻¹𝜕^𝑗_𝜉𝑀_𝑘∗𝑓(𝑣)⋅𝜕^𝑗_𝑣Φ^∗(𝑣),Γ_𝑘+6𝑓=_ℝ6𝑒^{𝑖(𝑣−𝑦)𝜉}𝑀_𝑘(𝜉)𝑑𝜉𝑓𝑦𝑦−𝑣^𝑘+6()!𝜕𝑘+6^𝑣𝑘+6Φ^∗(𝑐)=()𝑑𝑦−𝑖^𝑘+6(𝑘+6)!_ℝ⁶𝑒^{𝑖(𝑣−𝑦)𝜉}𝜕_𝜉^𝑘+6𝑀_𝑘(𝜉)𝑑𝜉𝑓𝑦𝜕_𝑣^𝑘+6Φ^∗(𝑐)𝑑𝑦. (5.4) From Lemma 2.1, Remark 2.2, (3.6), and (3.7), it follows that ^𝑘_𝑗=1Γ_𝑗𝑓(𝑣)_𝐿2=^𝑘_𝑗=1(−𝑖)^𝑗ℱ𝑗!⁻¹𝜕^𝑗_𝜉𝑀_𝑘∗𝑓(𝑣)⋅𝜕^𝑗_𝑣Φ^∗(𝑣)_𝐿2≤^𝑘_𝑗=11𝜕𝑗!^𝑗𝜉𝑀_𝑘(𝜉)ℱ𝑓(𝜉)_𝐿2⋅𝜕^𝑗_𝑣Φ^∗(𝑣)_𝐿∞≤𝐶⋅^𝑘_𝑗=116^𝑗𝑘⋯𝑘−𝑗+1𝑘−𝑗!^𝑠⋅𝐶₀^{𝑘−𝑗+1}≤𝐶⋅𝐶₀^𝑘+1{(𝑘−1)!}^𝑠𝑘²≤𝐶₁⋅𝐶₀^𝑘+1(𝑘!)^𝑠,_{𝑘+5𝑗=𝑘+1}Γ_𝑗𝑓(𝑣)_𝐿2≤^𝑘+5_𝑗=𝑘+11𝜕𝑗!^𝑗𝜉𝑀_𝑘(𝜉)ℱ𝑓(𝜉)_𝐿2⋅𝜕^𝑗_𝑣Φ^∗(𝑣)_𝐿∞≤𝐶⋅(𝑘+5)!sup_{𝑡∈(0,𝑇]}𝑓(𝑡,𝑣)_𝐿2≤𝐶₂⋅𝐶₀^𝑘+1(𝑘!)^𝑠,Γ_𝑘+6𝑓_𝐿2=(−𝑖)^𝑘+6(𝑘+6)!_ℝ⁶𝑒^{𝑖(𝑣−𝑦)𝜉}𝜕_𝜉^𝑘+6𝑀_𝑘(𝜉)𝑑𝜉𝑓𝑦𝜕_𝑣^𝑘+6Φ^∗(𝑐)𝑑𝑦_𝐿2≤1(𝑘+6)!_ℝ⁶𝜕_𝜉^𝑘+6𝑀_𝑘(𝜉)𝑑𝜉⋅𝑓𝑦⋅𝜕_𝑣^𝑘+6Φ^∗(𝑐)𝑑𝑦_𝐿2≤𝐶⋅𝐶₀^𝑘+1(𝑘!)^𝑠_ℝ31+𝜉²⁻³𝑑𝜉≤𝐶₃⋅𝐶₀^𝑘+1(𝑘!)^𝑠. (5.5) Combining (5.5), we complete the proof of Proposition 3.1.

Proof of Proposition 3.2. One has ∇_𝑣𝑀_𝑘𝐷_𝑣,Φ^∗𝑓(𝑡,𝑣)_𝐿2≤𝐶⋅𝜉ℱ𝑀_𝑘,Φ^∗𝑓(𝑡,𝜉)_𝐿2≤𝐶⋅𝐷_𝑣𝑀_𝑘,Φ^∗𝑓(𝑡,𝑣)−𝐷_𝑣,Φ^∗𝑀_𝑘𝑓(𝑡,𝑣)_𝐿2. (5.6)
Similar to the proof of Proposition 3.1, we obtain 𝐷_𝑣𝑀_𝑘,Φ^∗𝑓(𝑡,𝑣)_𝐿2≤𝐶(𝑘+1)𝑀_𝑘𝑓(𝑡,𝑣)_𝐿2+𝐶₀^𝑘+1(𝑘!)^𝑠,𝐷_𝑣,Φ^∗𝑀_𝑘𝑓(𝑡,𝑣)_𝐿2≤𝐶𝑀_𝑘𝑓(𝑡,𝑣)_𝐿2+𝐶₀^𝑘+1(𝑘!)^𝑠. (5.7) Then (3.9) is obtained. The proof of (3.10) is similar so is omitted. This completes the proof of Proposition 3.2.

Acknowledgments

This work was supported by the Tian Yuan Specialized Research Fund for Mathematics (no. 11226167), the Natural Science Foundation of Hainan Province (no. 111005), the Scientific Research Foundation of Hainan Province Education Bureau (no. Hjkj2011-19), and the Ph.D. Scientific Research Starting Foundation of Hainan Normal University (no. HSBS1016).