速度平均引理及其在波茲曼方程的應用

國立臺灣大學理學院數學系 碩士論文

Department of Mathematics College of Science

National Taiwan University Master Thesis

速度平均引理及其在波茲曼方程的應用 Velocity Averaging Lemmas and Their Application to Boltzmann Equation

莊秉翰 Ping-Han Chuang

指導教授﹕陳逸昆 博士 Advisor: I-Kun Chen, Ph. D.

中華民國 108 年 7 月 July, 2019

誌謝

時光飛逝，畢業在即，轉眼間兩年研究所生涯將邁入尾聲，回首這兩年日常 一點一滴，雖免不了惆悵失落，但也為我生命寫下一首清雅的詩。台大校園蓊蓊 鬱鬱的景色，湖畔邊鴨鵝鳥獸無憂無慮覓食，師生行政人員們的幽默風趣，在我 埋頭苦思研究數學時添了不少愜意。感謝陳逸昆指導老師，協助指導我完成這篇 論文，並提供論文研究方向，感謝口試委員夏俊雄老師和沈俊嚴老師，口試期間 出現重感冒症狀，表現不如預期，仍包容我的疏失，感謝助教王雅真和幹事趙怡 茹，提醒我重要修業程序，感謝 408 研究室同學，無拘無束分享各自研究數學心 得，最後感謝我父母，給予我支持和鼓勵。

莊秉翰 誌於 國立台灣大學數學系 民國 108 年 7 月

doi:10.6342/NTU201901751

中文摘要

1988 年時，Golse、Lions、Perthame 和 Sentis 證明了速度平均會增加函數正 則性，這個現象後來被稱作「速度平均引理」，速度平均引理在DiPerna 和 Lions 的波茲曼方程之柯西問題整體解存在性理論有重要的應用，而其自身也有許多富 有意義的延伸，在此論文中，我們首先回顧經典的速度平均引理，再透過速度平 均效應導出我們關於穩態線性化波茲曼方程正則性的新結果。

關鍵詞:速度平均，正則化效應，索博列夫空間、傳遞方程、波茲曼方程

Abstract

In 1988, Golse, Lions, Perthame and Sentis jointly proved that velocity averaging has regularizing effects. This phenomenon was later called “Velocity Averaging

Lemmas.” The Velocity Averaging Lemmas have significant applications in the global existence theory of the Cauchy problem for Boltzmann equations by DiPerna and Lions and also have many meaningful extensions themselves. In this thesis, we first review classical Velocity Averaging Lemmas, and then we present our new regularity results for the stationary linearized Boltzmann equation by velocity averaging effects.

Keywords: velocity average, regularizing effect, Sobolev space, transport equation, Boltzmann equation.

doi:10.6342/NTU201901751

目 錄

誌謝 ... i

中文摘要 ... ii

Abstract ... iii

目錄 ... iv

1 Introduction ... 1

2 Preliminaries ... 3

3 Velocity Averaging Lemmas ... 5

4 Regularity of the Stationary Linearized Boltzmann Equation ... 22

References ... 30

1 Introduction

In this thesis, we first review famous Velocity Averaging Lemmas by Golse, Lions, Perthame and Sentis [16] and then we introduce our new regularity results on the stationary linearized Boltzmann equation. Velocity Averaging Lemmas play impor- tant roles in the study of regularity of solutions to nonlinear transport equations and in the celebrated theory of global existence on Boltzmann equation [12] by Diperna and Lions. Many of their extensions are also studied and can be found in the works of [9, 10, 13, 18, 19].

The mechanism of Velocity Averaging Lemmas is described as follows. Let u(x, v) be a real-valued function defined on R^N_x × R^N_v and ψ(v) a bounded function with compact support on R^N_v . The average of u with respect to the velocity is

¯ u(x) =

Z

R^N

u(x, v)ψ(v)dv.

Suppose u and its weak directional derivative (along v) v · ∂xu(x, v) both belong to some L^p(dxdv) (1 < p < ∞) space, then, generally speaking, the velocity average of u is more regular than u(·, v) for almost every fixed v. Note that v · ∂_xu(x, v) only gives information about the smoothness of u(·, v) in the direction v.

Now we turn to the stationary linearized boltzmann equation with a source term v · ∇_xf (x, v) = L(f )(x, v) + φ(x, v) (1) for (x, v) ∈ R³× R³. Here L denotes the linearization of the collision operator. The collision operator Q is defined by

Q(F, G) = Z

R³

Z 2π 0

Z ^π₂

0

(F (v⁰)G(v_∗⁰) − F (v)G(v∗)) B(|v∗− v|, θ)dθdεdv∗, where v, v∗ denote the velocities before the collision of two particles, v⁰, v⁰_∗ their ve- locities after the collision and B is the cross section, depending on the intermolecular force or potential. If we linearize Q around the standard Maxwellian

M (v) = π⁻³²e^−|v|2 in the form F = M + M¹²f , then L reads

L(f ) = M⁻¹² h

Q(M¹²f, M ) + Q(M, M¹²f ) i

. Under a suitable cutoff assumption by Grad [17] by assuming

0 ≤ B ≤ C|v − v∗|^γcos θ sin θ, (2) where γ ∈ [0, 1] is a fixed number (hard sphere model, cutoff hard potential, and cutoff Maxwellian molecular gases). L can be decomposed into

L(f ) = −ν(v)f + K(f ), where ν is a function of v and K is an integral operator

K(f )(x, v) = Z

3

f (x, v∗)k(v∗, v)dv∗.

doi:10.6342/NTU201901751 We may consider Picard-type iterations for (1) and then consider the following

system of differential equations consequently ν(v)f₀+ v · ∇_xf₀ = φ,

ν(v)f_m+ v · ∇_xf_m = K(f_m−1), for m ≥ 1.

Therefore f = Σ^∞_m=0fm is a solution to (1) formally. Observe the left hand side is a transport operator consisting of ν(v)f_m and v · ∇_xf_m and the right hand side K(f_m−1) is a velocity average. It is reasonable to anticipate f_m has regularity in x by an iterative application of velocity averaging theory. The precise statement of our result is Theorem 4.3. Liu and Yu have proved a theorem regarding this aspect, also known as Mixture Lemma, for the evolutionary (with time t) Boltzmann equation in [21]. However Mixture Lemma requires regularity in v of initial data.

By comparison, our result doesn’t need any regularity.

For the bounded domain case, although we do not have the same result, the velocity averaging effect still works in this case. Consider the stationary linearized Boltzmann equation in C¹ bounded convex domain X with incoming boundary data g

ν(v)f (x, v) + v · ∇_xf (x, v) = K(f )(x, v), for (x, v) ∈ X × R³, f |_Γ−(q, v) = g(q, v), for (q, v) ∈ Γ₋,

where we have used the fact L(f ) = −ν(v)f + K(f ). It can be shown that, if f and g belong to L², the above boundary value problem can be formulated as

f (x, v) = e^{−ν(v)τx,v}g(x − τ_x,v, v) + Z τx,v

0

e^−ν(v)sK(f )(x − sv, v)ds, where τ_x,v := inf{t > 0 : x − tv /∈ X}. If we denote

S_X(f )(x, v) :=

Z τx,v

0

e^−ν(v)sf (x − sv, v)ds,

our other new result (Theorem 4.10) shows that the mixed operator KSXK has reg- ularizing effect in space variable x. This result has more uses than the previous one, since we more often consider stationary linearized Boltzmann equation in bounded domain. We also illustrate how it applies to the study of regularity of Boltzmann equation. In the same spirit for pointwise estimates, Chen proved that solutions to the stationary linearized Boltzmann equation in a bounded convex domain are H¨older continuous in [7] by observing velocity averaging effects via change of vari- ables method. Along the same lines, Chen, Hsia and Kawagoe improved on the result and showed that solutions are indeed differentiable in [8].

The velocity averaging phenomenon was first discussed in [2] and [15, 16]. The classical results are reviewed in this thesis. We provide two proofs for the case p = 2. The first proof was originally given in [16] and it relies on Fourier analysis.

The second proof is similar to the first one, but its technique can be used to prove the main theorems in Section 4. As for the case p 6= 2, [16] proved that the velocity average ¯u belongs to each Sobolev space W^s,p, where 0 < s < min(1/p, 1/p⁰) and p⁰ is the H¨older conjugate exponent of p, by interpolation method. In 2000, Devore and Petrova [10] improved on this result. They finally settled the borderline case

¯

u ∈ W^s,p for s = min(1/p, 1/p⁰) and they also proved the sharpness of their result in terms of Besov spaces. The discussion for evolutionary (with time t) transport oper- ators is parallel to the stationary case, so [16] only mentioned the evolutionary case briefly without explicit statements and proofs. However, we state the corresponding theorems for completeness and provide a new proof for the case p = 2.

The above Velocity Averaging Lemmas fail when p = 1 or p = ∞. A counterex- ample is also provided. In order to yield a regularity result for p = 1, we need to make some further assumptions. In this case, we consider a set of functions such that their directional derivatives and themselves are bounded and uniformly inte- grable, then the set of their velocity averages is precompact. This fact is skillfully applied in [12].

After considering the case for whole space, the case for bounded domain X ⊂ R^N_x is discussed. The key of its proof is Extension Lemma, which says if a function u, its trace and directional derivative belong to L^p, then u can be extended to the whole space R^N without loss of regularity. Therefore Velocity Averaging Lemmas can be established in bounded domains by Extension Lemma.

An outline of this thesis is as follows. In Section 2, we recall some basic notions, definitions and theorems of real analysis. In Section 3, we review Velocity Averag- ing Lemmas. Section 4 is devoted to our new regularity results on the stationary linearized Boltzmann equation. To our knowledge, no research exists addressing Theorem 4.3 and Theorem 4.10.

2 Preliminaries

In this section, we recall some basic notions and theorems in real analysis. First of all, we define f ractional Sobolev spaces. Fractional Sobolev spaces are often used to measure the regularity of a function. Assume readers are familiar with the classical Sobolev spaces of integer order. For those not familiar, see [14].

Definition 2.1. Let Ω be an open subset of R^N. For 1 ≤ p ≤ ∞, θ ∈ (0, 1) and f ∈ L^p(Ω), the Slobodeckij seminorm is defined by

[f ]_θ,p :=

Z

Ω

Z

Ω

|f (x) − f (y)|^p

|x − y|^θp+N dxdy

1/p

, for 1 ≤ p < ∞, [f ]_θ,∞:= ess sup

Ω×Ω

|f (x) − f (y)|

|x − y|^θ , for p = ∞.

Let s > 0 be not an integer and θ = s − bsc ∈ (0, 1). The fractional Sobolev space is defined as

W^s,p(Ω) :=f ∈ W^bsc,p : [D^αf ]_θ,p < ∞ for each |α| = bsc with the norm

kf k_W^s,p = kf k_W^bsc,p+ Σ|α|=bsc[D^αf ]_θ,p.

For p = 2, let us denote W^s,p by H^s. Sometimes, we only interested in the regularity in one variable, so the concept of M ixed spaces arises.

doi:10.6342/NTU201901751 Definition 2.2. Let Ω ⊂ R^N_x be an open subset.

L²_vH_x^s(Ω × R^N_v ) := {f ∈ L²(Ω × R^N_v ) : kf k_L²

vH_x^s(Ω×R^N_v) < ∞}, with the norm

kf k_L²_vHx^s_(Ω×R^N_v) :=

Z

R^N

kf (·, v)k²_Hs(Ω)dv

1/2

.

In this thesis, we also denote the Fourier transform of f in x variable byF f and the corresponding Fourier variable by ξ. Bessel potential spaces are another class of Sobolev-type spaces of noninteger order. They are defined as follows.

Definition 2.3. For 1 < p < ∞,

He^s,p :=f ∈ L^p(R^N) :F⁻¹(1 + |ξ|²)^s/2F f ∈ L^p(R^N) equipped with the norm

kf k_Hes,p :=

F⁻¹(1 + |ξ|²)^s/2F f

L^p(R^N).

It turns out that, for p = 2, the Sobolev space and the Bessel potential space are the same for any integer and noninteger order s > 0. The following theorem is well-known. We refer the interested readers to [11] for its proof and details.

Theorem 2.4. For any s > 0, eH^s(R^N) = H^s(R^N) and the norm of eH^s(R^N) and H^s(R^N) are equivalent.

Therefore, to verify whether a function f ∈ L² belongs to H^s or not, it is equivalent to verifying

Z

R^N

(1 + |ξ|^2s)|F f(ξ)|²dξ < ∞.

Now we can define the following mixed spaces without using Slobodeckij seminorm.

Definition 2.5. For s ≥ 0, we define

L²_vH_x^s := {f ∈ L²(R^N_x × R^N_v ) : kf k_L²_vHx^s < ∞}, with the norm

kf k_L²_vH^s_x :=

Z

R^N

Z

R^N

(1 + |ξ|^2s)|F f(ξ, v)|²dξdv

1/2

.

The concept of uniform integrability is important in real analysis. We will use the following definition in L¹ Velocity Averaging Lemma.

Definition 2.6. A set K ⊂ L¹(R^N_y ) is called uniformly integrable if, for each ε > 0, there exists δ > 0 such that

sup

f ∈K

Z

A

|f (y)|dy < ε whenever |A| < δ.

The following criterion is useful in the proof of L¹ Velocity Averaging Lemma.

Theorem 2.7. Let 1 ≤ p < ∞. A bounded subset K ⊂ L^p(Ω) (Ω is an open subset of R^N) is precompact in L^p(Ω) if and only if for each ε > 0 there exists a number δ > 0 and a subset G b Ω such that for every u ∈ K and h ∈ R^N with |h| < δ both of the following inequalities hold:

Z

Ω

|u(x + h) − u(x)|^pdx < ε (let u = 0 outside Ω). (3) Z

Ω− ¯G

|u(x)|^p < ε. (4)

A proof for the above theorem can be found in [1]. Sobolev spaces have compact embedding property. This property allows us to utilize compactness in Sobolev spaces.

Theorem 2.8. Let s ∈ (0, 1), 1 ≤ p < ∞, 1 ≤ q ≤ p and Ω ⊂ R^N be a C¹ bounded open set. Then,

W^s,p(Ω) ⊂⊂ L^q(Ω).

In other words, if K is a bounded subset in L^p(Ω) and sup

f ∈K

Z

Ω

Z

Ω

|f (x) − f (y)|^p

|x − y|^{N +sp} dxdy < ∞, then K is precompact in L^q(Ω).

A proof for the above theorem can be found in [11].

3 Velocity Averaging Lemmas

We revisit the classical Velocity Averaging Lemmas in this section. Throughout the section, ψ ∈ L^∞_c (R^N) is any fixed bounded function with compact support. We denote by C various positive constants which may depend on ψ and we use the notation

¯ u =

Z

R^N

u(v)ψ(v)dv.

Before we introduce Velocity Averaging Lemmas, we first prove a basic fact regarding transport equation., which is fundamental in velocity averaging theory.

Proposition 3.1. Let 1 ≤ p ≤ ∞. For any f ∈ L^p(R^N_x × R^N_v ), the transport equation

u + v · ∂_xu = f in R^N_x × R^N_v (5) has a unique solution uf in L^p(R^N_x × R^N_v ). The linear operator T : L^p(R^N_x × R^N_v ) → L^p(R^N_x) defined by

T (f ) = ¯u_f = Z

R^N

u(·, v)ψ(v)dv is continuous.

doi:10.6342/NTU201901751 Proof. Given f ∈ L^p(R^N_x × R^N_v ). We claim

u_f(x, v) = Z ∞

0

e^−sf (x − sv, v)ds

is the unique solution in L^p(R^N_x × R^N_v ). Given φ ∈ C_c^∞(R^N × R^N). We have Z

R^N

Z

R^N

uf(φ − v · ∂xφ)dxdv

= Z

R^N

Z

R^N

Z ∞ 0

e^−sf (x − sv, v) (φ(x, v) − v · ∂xφ(x, v)) dsdxdv.

Change of variables (s, x, v) → (s, x + sv, v) gives Z

R^N

Z

R^N

Z ∞ 0

e^−sf (x − sv, v) (φ(x, v) − v · ∂xφ(x, v)) dsdxdv

= Z

R^N

Z

R^N

Z ∞ 0

e^−sf (x, v) (φ(x + sv, v) − v · ∂xφ(x + sv, v)) dsdxdv

= − Z

R^N

Z

R^N

f (x, v) Z ∞

0

∂

∂s e^−sφ(x + sv, v)
dsdxdv

= Z

R^N

Z

R^N

f (x, v)φ(x, v)dxdv.

Hence u_f is a weak solution to (5). By H¨older’s inequality, ku_fk^p_Lp(R^N_x×R^N_v)

= Z Z 

Z ∞ 0

e^−sf (x − sv, v)ds    

p

dxdv

≤

Z Z Z ∞ 0

e^−p0s2 ds

_p0^p Z ∞ 0

e^−ps2 |f (x − sv, v)|^pds

 dxdv

≤C Z

R^N

Z

R^N

Z ∞ 0

e^−ps2 |f (x − sv, v)|^pdsdxdv

=C Z

R^N

Z

R^N

Z ∞ 0

e^−ps2 |f (x, v)|^pdsdxdv

≤Ckf k^p_Lp(R^N_x×R^N_v).

We also have, by H¨older’s inequality again, k ¯u_fk^p_Lp(R^N_x) =

Z     Z

u_f(x, v)ψ(v)dv    

p

dx

≤ Z Z

|ψ(v)|^p0dv

_p0^p Z

|u_f(x, v)|^pdv

 dx

≤ Cku_fk^p_Lp(R^N_x×R^N_v)

≤ Ckf k^p_Lp(R^N_x×R^N_v).

Now it only remains to prove the uniqueness. Suppose u₁ ∈ L^p(R^N_x × R^N_v ) and u₂ ∈ L^p(R^N_x × R^N_v ) are two solutions to (5). Let w = u₁− u₂. Thus

w + v · ∂_xw = 0.

Pick a φ ∈ C_c^∞(R^N × R^N). Then e^−sφ(x + sv, v) ∈ C_c^∞(R^N × R^N) for each s > 0.

Therefore, Z

R^N

Z

R^N

w(x, v) e^−sφ(x + sv, v) − v · ∂_xe^−sφ(x + sv, v)
dxdv = 0.

We have

∂

∂s

 e^−s

Z

R^N

Z

R^N

w(x, v)φ(x + sv, v)dxdv



= 0, which implies

e^−s Z

R^N

Z

R^N

w(x, v)φ(x + sv, v)dxdv = C. (6) By H¨older’s inequality,

Z

R^N

Z

R^N

w(x, v)φ(x + sv, v)dxdv ≤ kwk_L^pkφk_Lp0. Hence letting s → ∞ in (6) implies C = 0. In particular,

Z

R^N

Z

R^N

w(x, v)φ(x, v)dxdv = 0.

Since φ is arbitrary, we conclude w(x, v) = 0 almost everywhere. In other words, u₁ = u₂.

Remark. The transport equation

u + v · ∂_xu = 0 has nontrivial solutions in the space L¹_loc(R^N_x × R^N_v ).

Next, we consider L² Stationary Velocity Averaging Lemma, the main theorem in [16]. We provide two proofs here. Both proofs rely on Fourier analysis.

Theorem 3.2. Let u = u(x, v) be such that u and v · ∂_xu (= _∂s^∂u(x + sv, v) s=0

in the weak sense) both belong to L²(R^N_x × R^N_v ). Then ¯u(x) = R

R^Nu(x, v)ψ(v)dv belongs to H¹²(R^N) and

k¯uk

H¹²(R^N)≤ C(kuk_L²_(R^N_x×R^N_v)+ kv · ∂_xuk_L²_(R^N_x×R^N_v)). (7) Proof 1 of Theorem 3.2. The assumption on u may be formulated as F u and (v · ξ)F u belong to L²(R^N_ξ × R^N_v ). Notice that

Z

R^N

|ξ|

    Z

R^N

F u(ξ, v)ψ(v)dv    

2

dξ

= Z

|ξ|

    Z

|v·ξ|<1

F u(ξ, v)ψ(v)dv +Z

|v·ξ|≥1

F u(ξ, v)ψ(v)dv    

2

dξ

≤2 Z

|ξ|

    Z

|v·ξ|<1

F u(ξ, v)ψ(v)dv    

2

dξ

+ 2 Z

|ξ|

    Z

|v·ξ|≥1

F u(ξ, v)ψ(v)dv    

2

dξ

=:I + J. (8)

doi:10.6342/NTU201901751 By Cauchy-Schwarz inequality, we obtain

    Z

|v·ξ|<1

F u(ξ, v)ψ(v)dv    

2

≤

Z

|v·ξ|<1

|ψ(v)|²dv

 Z

R^N

|F u(ξ, v)|²dv

 . Note that

Z

|v·ξ|<1

|ψ(v)|²dv = Z

|v·_|ξ|^ξ|<_|ξ|¹

|ψ(v)|²dvkξdv⊥ξ, where vkξ = (v ·_|ξ|^ξ)_|ξ|^ξ and dv⊥ξ = v − vkξ. Therefore,

Z

|v·_|ξ|^ξ|<_|ξ|¹

|ψ(v)|²dvkξdv⊥ξ = Z

R^{N −1}

Z

|v_kξ|<_|ξ|¹

|ψ(v)|²dvkξdv⊥ξ

≤C 1

|ξ|. Hence,

I ≤ C Z

R^N

Z

R^N

|F u(ξ, v)|²dvdξ. (9)

By Cauchy-Schwarz inequality again, we have 

   Z

|v·ξ|≥1

F u(ξ, v)ψ(v)dv    

2

≤

Z

|v·ξ|≥1

1

|v · ξ|²|ψ(v)|²dv

 Z

R^N

|v · ξ|²|F u(ξ, v)|²dv

 . Note that

Z

|v·ξ|≥1

1

|v · ξ|²|ψ(v)|²dv = 1

|ξ|² Z

|v·_|ξ|^ξ|≥_|ξ|¹

1   v · _|ξ|^ξ

2|ψ(v)|²dvkξdv⊥ξ

= 1

|ξ|² Z

R^{N −1}

Z

|v_kξ|≥_|ξ|¹

1

|vkξ|²|ψ(v)|²dvkξdv⊥ξ

≤C 1

|ξ|² · 1

|ξ|

−1

=C 1

|ξ|. Therefore,

J ≤ C Z

R^N

Z

R^N

|v · ξ|²|F u(ξ, v)|²dvdξ. (10) By Plancherel’s identity,

Z Z

|u(x, v)|²dxdv = Z Z

|F u(ξ, v)|²dξdv, Z Z

|v · ∂_xu(x, v)|²dxdv = Z Z

|v · ξ|²|F u(ξ, v)|²dξdv.

From this, (8), (9) and (10), we conclude k¯uk

H¹²(R^N) =

Z

R^N

(1 + |ξ|)|F ¯u(ξ)|²dξ

1/2

≤C(k¯uk_L²_(R^N₎+ I¹² + J¹²)

≤C(kuk_L²_(R^N_x×R^N_v)+ kv · ∂_xuk_L²_(R^N_x×R^N_v)).

Proof 2 of Theorem 3.2. Since u ∈ L²(R^N_x × R^N_v ) and v · ∂_xu ∈ L²(R^N_x × R^N_v ), we have

u + v · ∂_xu = f ∈ L²(R^N_x × R^N_v ).

By Proposition 3.1, we have

u(x, v) = Z ∞

0

e^−sf (x − sv, v)ds and

¯ u(x) =

Z

R^N

Z ∞ 0

e^−sf (x − sv, v)ψ(v)dsdv ∈ L²(R^N_x).

Considering the Fourier transform of ¯u, we have F ¯u(ξ) =Z

R^N

Z ∞ 0

e^−sF f(· − sv, v)(ξ)ψ(v)dsdv

= Z

R^N

Z ∞ 0

e^−se^−i(v·ξ)sF f(ξ, v)ψ(v)dsdv

= Z

R^N

F f(ξ, v)

1 + i(v · ξ)ψ(v)dv.

Thus, Z

R^N

|ξ||F ¯u(ξ)|²dξ = Z

|ξ|

    Z

R^N

F f(ξ, v)

1 + i(v · ξ)ψ(v)dv    

2

dξ

≤ Z

|ξ|

Z

R^N

|F f(ξ, v)|²dv

 Z

R^N

|ψ(v)|² 1 + (v · ξ)²dv



dξ (11) by Cauchy-Schwarz inequality. Notice that

Z

R^N

|ψ(v)|²

1 + (v · ξ)²dv = Z ∞

−∞

Z

R^{N −1}

|ψ(v)|²

1 + |ξ|²(v · _|ξ|^ξ )²dv⊥ξdv_kξ

≤Ckψk²_L∞

Z ∞

−∞

1

1 + |ξ|²(vkξ)²dv_kξ. Change of the variable s = |ξ|v_kξ gives

Z ∞

−∞

1

1 + |ξ|²(vkξ)²dv_kξ = 1

|ξ|

Z ∞

−∞

1 1 + s²ds

≤C 1

|ξ|. Substituting this into (11) gives

Z

R^N

|ξ||F ¯u(ξ)|²dξ ≤C Z Z

|F f(ξ, v)|²dvdξ

=C Z Z

|f (x, v)|²dxdv

doi:10.6342/NTU201901751 by Plancherel’s identity. Consequently,

k¯uk

H¹²(R^N)=

Z

R^N

(1 + |ξ|)|F ¯u(ξ)|²dξ

1/2

≤C k¯uk_L²_(R^N₎+

Z

R^N

|ξ||F ¯u(ξ)|²dξ

1/2!

≤C(kuk_L² + kf k_L²)

=C(kuk_L² + ku + v · ∂_xuk_L²)

≤C(kuk_L² + kv · ∂_xuk_L²).

Notice that L² Velocity Averaging Lemma can be formulated as follows.

Corollary 3.3. Under the hypotheses of Proposition 3.1, T : L²(R^N_x × R^N_v ) → H¹²(R^N) is continuous.

Proof. If f ∈ L²(R^N_x × R^N_v ), then u_f ∈ L²(R^N_x × R^N_v ) and v · ∂_xu_f = f − u_f ∈ L²(R^N_x × R^N_v ).

Furthermore,

kv · ∂_xu_fk_L² =kf − u_fk_L²

≤kf k_L² + ku_fk_L² ≤ Ckf k_L². According to Theorem 3.2, ¯u_f ∈ H¹²(R^N) and

k ¯u_fk

H¹² ≤ C(ku_fk_L² + kv · ∂_xu_fk_L²) ≤ Ckf k_L².

With the above corollary, we are able to generalize Theorem 3.2 to any L^p space with 1 < p < ∞ via the real method of interpolation.

Theorem 3.4. Let u = u(x, v) be such that u and v · ∂_xu both belong to L^p(R^N_x × R^N_v ) with 1 < p < ∞. Then ¯u belongs to W^s,p(R^N) with s = min(¹_p,_p¹0) and

k¯uk_W^s,p_(R^N₎≤ C(kuk_L^p+ kv · ∂_xuk_L^p). (12) Proof. The proof is based on the following real interpolation theorem in [10].

Theorem 3.5. We have the following relations between interpolation spaces and Sobolev spaces.



L¹(R^N), H¹²(R^N)

2

p0,p = W^p01,p(R^N), for 1 < p < 2, (13)

and 

L^∞(R^N), H¹²(R^N)

2 p,p

⊂ W^1p,p(R^N), for 2 < p < ∞. (14) Moreover, the norm for the interpolation space on the left hand side of (13) is equivalent to the norm of the Sobolev space on the right side. Likewise, the norm of the Sobolev space on the right side of (14) is less than a fixed multiple of the norm of the interpolation space on the left side.

Consider the operator T in Corollary 3.3. We have T : L¹(R^N_x × R^N_v ) → L¹(R^N), T : L²(R^N_x × R^N_v ) → H¹²(R^N).

By Theorem 3.5 and the fact (L¹, L²)²

p0,p= L^p for 1 < p < 2, T : L^p(R^N_x × R^N_v ) → W^p01,p(R^N)

is bounded for 1 < p < 2. Hence if u + v · ∂_xu =: f ∈ L^p(R^N_x × R^N_v ), then

¯

u = T (f ) ∈ W^p01,p(R^N) and k¯uk

W

1 p0,p

(R^N)≤ Ckf k_L^p ≤ C(kuk_L^p+ kv · ∂_xuk_L^p).

The case 2 < p < ∞ follows similarly.

Unfortunately, the above L^p Velocity Averaging Lemma fails for the endpoint cases p = 1 and p = ∞. [16] provided an example to illustrate this phenomenon.

Theorem 3.6. Under the hypothesis of Proposition 3.1. Suppose also ψ = 1{|v|≤1}. There exists a bounded subset K ⊂ L¹(R^N_x × R^N_v ) such that T (K) is not weakly precompact in L¹(B₁).

Proof. Let v₀ be an arbitrary point in {0 < |v| < 1}. Consider a sequence of functions

f_n(x, v) = n^2Nψ₁(n(0 − x)) ψ₂(n(v₀− v)) , where ψ₁ and ψ₂ are two mollifiers in C_c^∞(B₁). Therefore,

T (f_n)(x) = Z

{|v|≤1}

Z ∞ 0

f_n(x − tv, v)e^−tdtdv.

If T (f_n) converges weakly in L¹(B₁) to a function w. Given θ ∈ C_c^∞(B₁). On the one hand, as n → ∞,

Z

|x|≤1

T (fn)(x)θ(x)dx = Z

|x|≤1

Z

|v|≤1

Z ∞ 0

fn(x − tv, v)e^−tθ(x)dtdvdx

→ Z ∞

0

e^−tθ(tv0)dt.

On the other hand, as n → ∞, Z

|x|≤1

T (f_n)(x)θ(x)dx → Z

|x|≤1

w(x)θ(x)dx.

Therefore,

Z ∞ 0

e^−tθ(tv₀)dt = Z

|x|≤1

w(x)θ(x)dx (15)

for any θ ∈ C_c^∞(B1). Let θ be a mollifier in C_c^∞(B1) and y a Lebesgue point of w not on the line {tv₀ : t > 0}. Consider θ_m(x) = m^Nθ (N (y − x)). Hence, as m → ∞,

Z

|x|≤1

w(x)θ_m(x)dx → w(y).

doi:10.6342/NTU201901751 We also have, as m → ∞,

Z ∞ 0

e^−tθ_m(tv₀)dt → 0.

Therefore, (15) implies w(y) = 0 for any Lebesgue point y /∈ {tv₀ : t > 0}, which means w = 0 almost everywhere, a contradiction to (15). Hence we let K = {f_n}_n∈N.

Remark. Theorem 3.4 implies

T : L^p(R^N_x × R^N_v ) → W^s,p(R^N_x) (16) is bounded with 1 < p < ∞ and s = min(¹_p,_p¹0). However, Theorem 3.6 shows that (16) fails when p = 1 no matter how small s > 0 is. For if for some s > 0 such that (16) holds, then consider K in Theorem 3.6. Therefore, T (K) ⊂ W^s,1(R^N_x) is bounded. Let M = { T (f )|_B

1 : f ∈ K}. Hence, M ⊂ W^s,1(B1) is bounded. We have W^s,1(B₁) ⊂⊂ L¹(B₁) by compact embedding property, so M is precompact in L¹(B₁), which contradicts Theorem 3.6.

For the case p = 1, nonetheless, we can still make additional assumptions to yield a regularity result via compact embedding property of Sobolev space.

Theorem 3.7. Under the hypotheses of Proposition 3.1, if K ⊂ L¹(R^N_x × R^N_v ) is bounded and uniformly integrable, then T (K) is precompact in L¹(S) for any C¹ bounded open set S ⊂ R^N.

Proof. For any f ∈ K and α > 0, we define

χ_α,f = 1{(x,v):|f (x,v)|<α}, ωα,f = 1 − χα,f.

Note that f · ω_α,f and f · χ_α,f are the large part and the small part of f respectively.

We can write

¯

u_f = T (f ) = T (f · ω_α,f) + T (f · χ_α,f)

=: φf + ψf.

The idea is that we deal with large parts by uniform integrability and small parts by L² Velocity Averaging Lemma. Clearly, we have

Z

|φ_f(x + h) − φ_f(x)|dx ≤ 2 Z

|φ_f(x)|dx

= 2 Z

|T (f · ω_α,f)|dx

≤ 2 Z Z

|f · ω_α,f|dvdx.

We claim

sup

f ∈K

Z Z

|f · ω_α,f|dvdx → 0

as α → ∞. For given ε > 0, choose δ as in Definition 2.6. Since K is bounded in L¹(R^N_x × R^N_v ),

sup

f ∈K

|{|f (x, v)| ≥ α}| ≤ sup

f ∈K

1 α

Z Z

{|f |≥α}

|f |dxdv

≤1 α sup

f ∈K

Z Z

|f |dxdv

<δ,

if α is large enough. The uniform integrability implies

sup

f ∈K

Z Z

|f · ω_α,f|dxdv = sup

f ∈K

Z Z

{|f (x,v)|≥α}

|f |dxdv

<ε.

Thus we can choose α > 0 such that Z

|φ_f(x + h) − φ_f(x)|dx ≤ 2 Z

|φ_f(x)|dx < 2ε (17)

for every f ∈ K and h ∈ R^N. With the α chosen above, sup

f ∈K

Z Z

|f · χ_α,f|²dxdv ≤ sup

f ∈K

α Z Z

|f |dxdv

< ∞.

In other words, {f · χ_α,f : f ∈ K} is bounded in L²(R^N_x × R^N_v ). According to Corollary 3.3, the set {ψ_f : f ∈ K} is bounded in H¹². Since H¹²(S) ⊂⊂ L¹(S) for any C¹ bounded open set S ⊂ R^N. By Theorem 2.7,

sup

f ∈K

Z Z

S

|ψ_f(x + h) − ψ_f(x)|dx → 0 (18)

as h → 0. Hence T (K) satisfies (3) by (17) and (18). For (4), we note that K is uniformly integrable and

Z

S− ¯G

|¯u(x)|dx ≤ C Z

S− ¯G

Z

D

|f (x, v)|dvdx < ε,

where D is the support of ψ, if |S − ¯G| is small enough. Finally, we conclude that T (K) is precompact in L¹(S) for any C¹ bounded open set S.

The Velocity Averaging Lemmas for evolutionary case are parallel to those for stationary case. However, since the transport operators ∂_t+ v · ∂_x and v · ∂_x have different (but similar) structures, the proofs for stationary case can not be applied to evolutionary case. Therefore, we choose to modify the second proof for Theorem 3.2. As in the stationary case, we first consider the velocity averaging operator with time t and then introduce L² Evolutionary Velocity Averaging Lemma.

doi:10.6342/NTU201901751 Proposition 3.8. Let 1 ≤ p ≤ ∞. For any f ∈ L^p(R_t× R^N_x × R^N_v ), the transport

equation

u + ∂_tu + v · ∂_xu = f in R_t× R^N_x × R^N_v (19) has a unique solution u_f in L^p(R_t× R^N_x × R^N_v ). The linear operator T : L^p(R_t× R^N_x × R^N_v ) → L^p(R_t× R^N_x) defined by T (f ) = ¯u_f is continuous.

Proof. First, claim

uf(t, x, v) = Z ∞

0

e^−sf (t − s, x − sv, v)ds

is a solution to (19). The rest is similar to the proof in Proposition 3.1.

Theorem 3.9. Let u = u(t, x, v) be such that u and ∂tu + v · ∂xu (=

∂

∂su(t + s, x + sv, v)

_s=0 in the weak sense) both belong to L²(R_t × R^N_x × R^N_v ).

Then ¯u(t, x) = R

R^Nu(t, x, v)ψ(v)dv belongs to H¹²(R_t× R^N_x) and k¯uk

H¹²(Rt×R^N_x) ≤ C(kuk_L² + k∂_tu + v · ∂_xuk_L²). (20) Proof. In this proof, we denote byF f the Fourier transform of f in t and x variable only, and by τ and ξ the corresponding Fourier variables. Since u ∈ L²(Rt× R^N_x × R^N_v ) and ∂_tu + v · ∂_xu ∈ L²(R_t× R^N_x × R^N_v ), it follows that

u + ∂_tu + v · ∂_xu = f ∈ L²(R_t× R^N_x × R^N_v ).

By Proposition 3.8, we have u(t, x, v) =

Z ∞ 0

e^−sf (t − s, x − sv, v)ds and

¯

u(t, x) = Z

R^N

Z ∞ 0

e^−sf (t − s, x − sv, v)ψ(v)dsdv.

Therefore,

F ¯u(τ, ξ) =Z

R^N

Z ∞ 0

e^−s· e^{−iτ s}· e^−i(v·ξ)sF f(τ, ξ, v)ψ(v)dsdv

= Z

R^N

F f(τ, ξ, v)ψ(v) 1 + i (τ + (v · ξ))dv.

By Cauchy-Schwarz inequality, Z

R^N

Z

R^N

(|τ |²+ |ξ|²)^1/2     Z

R^N

F f(τ, ξ, v)ψ(v) 1 + i (τ + (v · ξ))dv

2

dξdτ

≤ Z

R^N

Z

R^N

(|τ | + |ξ|)

Z

R^N

|F f(τ, ξ, v)|²dv

  |ψ(v)|²dv 1 + (τ + (v · ξ))²

 dξdτ

= Z Z

|ξ|

Z

R^N

|F f(τ, ξ, v)|²dv

  |ψ(v)|²dv 1 + (τ + (v · ξ))²

 dξdτ +

Z Z

|τ |

Z

R^N

|F f(τ, ξ, v)|²dv

  |ψ(v)|²dv 1 + (τ + (v · ξ))²

 dξdτ

=: I + J.

Pick R > 0 sufficiently large so that Z |ψ(v)|²

1 + (τ + (v · ξ))²dv ≤ Z

[−R,R]^{N −1}

Z R

−R

|ψ(v)|²

1 + (τ + (v · ξ))²dv_⊥ξdv_kξ

≤C Z R

−R

1 1 + τ + |ξ|vkξ

2dv_kξ

≤C 1

|ξ|

Z ∞

−∞

1 1 + λ²dλ

≤C 1

|ξ|, where λ = τ + |ξ|v_kξ. Hence,

I ≤ C Z Z Z

|F f(τ, ξ, v)|²dvdξdτ. (21) For J ,

|τ |

Z |ψ(v)|²

1 + (τ + (v · ξ))²dv ≤ C Z R

−R

|τ |

1 + τ + |ξ|v_kξ
2dvkξ

follows similarly. If |ξ| = 0, Z R

−R

|τ | 1 + τ + |ξ|vkξ

2dv_kξ ≤2R · |τ | 1 + τ²

≤C.

If |ξ| 6= 0, then we have the following lemma.

Lemma 3.10. There exists a constant C > 0 depending only on R > 0 such that Z R

−R

1

1 + (τ + |ξ|s)²ds ≤ C · 1

|τ |. Proof of Lemma 3.10. Clearly,

Z R

−R

|τ |

1 + (τ + |ξ|s)²ds =|τ |

|ξ|

Z τ +|ξ|R τ −|ξ|R

1 1 + λ²dλ

=R · |τ |

|ξ|R(arctan(τ + |ξ|R) − arctan(τ − |ξ|R))

=R sgn(τ )D

 τ

|ξ|R, |ξ|R

 , where λ = τ + |ξ|s and

D(a, b) = a (arctan b(a + 1) − arctan b(a − 1)) .

We show that D is a bounded function in R². If |a| ≤ 2, then |D(a, b)| ≤ 2π. Now we fix an a with |a| > 2 and consider D(a, b) as a function of b. Since D(a, b) → 0 as |b| → ∞, the maximum and the minimum of D(a, ·) must occur at critical points b = ±1/√

a²− 1. However, D

a, ±^{√ 1}

a²−1



is a bounded function for |a| ≥ 2. Thus Lemma 3.10 follows.

doi:10.6342/NTU201901751 Go back to the proof of Theorem 3.9. Hence, according to Lemma 3.10, we have

|τ |

Z |ψ(v)|²

1 + (τ + (v · ξ))²dv ≤ C and

J ≤ C Z Z Z

|F f(τ, ξ, v)|²dvdξdτ. (22) By (21) and (22), we conclude that

k¯uk

H¹²(Rt×R^N_x)≤C(k¯uk_L²_(Rt×R^N

x)+ kf k_L²_(Rt×R^N

x×R^N_v))

≤C(kuk_L² + k∂_tu + v · ∂_xuk_L²).

Other proofs for Theorem 3.9 can be found in, for example, [4, 9].

Also notice that Theorem 3.9 can be formulated as follows.

Corollary 3.11. Under the hypotheses of Proposition 3.8, T : L² Rt× R^N_x × R^N_v
→ H¹²(Rt× R^N_x) is continuous.

By Corollary 3.11, we can prove L^p (1 < p < ∞) and L¹ Evolutionary Veloc- ity Averaging Lemmas as in the stationary case. We omit proofs since they are essentially the same.

Theorem 3.12. Let u = u(t, x, v) be such that u and ∂_tu + v · ∂_xu both belong to L^p(R_t× R^N_x × R^N_v ). Then ¯u(t, x) belongs to W^s,p(R_t× R^N_x) with s = min(¹_p,_p¹0) and k¯uk_W^s,p_(Rt×R^N_x)≤ C(kuk_L^p + k∂_tu + v · ∂_xuk_L^p). (23) Theorem 3.13. Under the hypotheses of Proposition 3.8, if K ⊂ L¹(R_t×R^N_x ×R^N_v ) is bounded and uniformly integrable, then T (K) is precompact in L¹((0, T ) × S) for any T > 0 and C¹ bounded open set S ⊂ R^N.

So far, we only dealt with functions defined on whole space R^N_x . Now we consider a localized version of the above results. Let X be a C¹ bounded convex domain in R^N_x. Denote by dΣ the surface measure on ∂X, and by n(q) the unit outward normal vector to X at q ∈ ∂X. We define

Γ := ∂X × R^N_v ,

Γ₊ := {(q, v) ∈ Γ |n(q) · v > 0 } , Γ₋ := {(q, v) ∈ Γ |n(q) · v < 0 } , Γ₀ := {(q, v) ∈ Γ |n(q) · v = 0 } ,

and also define the backward exit time τ_x,v > 0 and the f orward exit time γ_x,v > 0 by relations

x − τ_x,vv ∈∂X, x + γ_x,vv ∈∂X,

where (x, v) ∈ ¯X × R^N. τ_x,v and γ_x,v are well-defined since X is convex. Denote by dσ the measure

dσ = |v · n(q)|dΣ(q)dv.

We have the following change of variables formulas.

Lemma 3.14. For any h ∈ L¹(X × R^N_v ), the following hold.

Z

R^N

Z

X

h(x, v)dxdv = Z

Γ−

Z γq,v

0

h(q + tv, v)dtdσ, (24) and, for almost every v ∈ R^N,

Z

X

h(x, v)dx = Z

Γ−,v

Z γq,v

0

h(q + tv, v)|v · n(q)|dtdΣ(q), (25) where Γ−,v := {x ∈ X : (x, v) ∈ Γ−}.

Proof. For (x, v) ∈ X × R^N, write x = q + tv, where (q, v) ∈ Γ− and 0 < t < γ_q,v. Consider a chart θ : D ⊂ R^{N −1}→ ∂X, where D is an open subset of R^{N −1}. Denote q = θ(α). Now we consider the change of variable (x, v) → (α, t, v). Clearly, we have

x = θ(α) + tv,

xαi = θαi for 1 ≤ i ≤ N − 1, x_t= v.

Therefore, the Jacobian for the change of vatiables is

J (α, t, v) = det









 θ_α1

... θ_{αN −1}

v









 .

Hence, it follows that

dxdv =|J (α, t, v)|dαdtdv

=|v · n(q)|dtdΣ(q)dv

=dtdσ.

Consequently, we have (24) and (25).

For 1 < p < ∞, we define

W^p(X) :=u(x, v) : u and v · ∂_xu both belong to L^p(X × R^N) , kuk_W^p_(X) := kuk_L^p_(X×R^N₎+ kv · ∂_xuk_L^p_(X×R^N₎.

Let us discuss the boundary value u|_Γ− of u ∈ W^p(X) on Γ− with measure dσ. The idea follows from [5, 6] and [20]. From the identity (24), we have

Z

R^N

Z

X

|u(x, v)|^pdxdv = Z

Γ−

Z γq,v

0

|u(q + tv, v)|^pdtdσ, Z

R^N

Z

X

|v · ∂_xu(x, v)|^pdxdv = Z

Γ−

Z γq,v

0

|∂

∂tu(q + tv, v)|^pdtdσ.

Thus, for almost every (q, v) ∈ Γ− (with respect to dσ), u(q + tv, v), as a function of t, belongs to W^1,p(0, γ_q,v), which means u(q + tv, v) is an absolutely continuous