KANIEL-SHINBROT ITERATION AND GLOBAL SOLUTIONS OF THE CAUCHY PROBLEM FOR THE BOLTZMANN EQUATION

OF THE CAUCHY PROBLEM FOR THE BOLTZMANN EQUATION

CLAUDE BARDOS, IRENE GAMBA, AND C. DAVID LEVERMORE

Abstract. Kaniel-Shinbrot iteration can be used to establish the existence of solutions to the Boltzmann equation over the spatial domain R^D for some initial data that is pointwise bounded above and below by a class of global Maxwellians larger than previously considered.

When these solutions are global in time, we obain results on their large-time asymptotics.

1. Introduction

We consider a kinetic density F(v, x, t) over the velocity-position domain R^D×R^D that is governed by the Cauchy problem for the Boltzmann equation:

(1.1) ∂tF +v·∇^xF =B(F, F), F!

!t=0 =Fⁱⁿ.

We assume that the initial data Fⁱⁿ(v, x) is nonnegative and satisfies the bounds

(1.2) 0<

" "

R^D×R^D

#1 +|v|²+|x|²$

Fⁱⁿdvdx <∞. The collision operatorB(F, F) has the form

(1.3) B(F, F) =

" "

S^D−1×R^D

(F_∗^"F^"−F_∗F) bdωdv_∗,

where ω ∈ S^D−1, b(ω, v −v∗) is the collision kernel, while F∗, F^", and F_∗^" denote F(·, x, t) evaluated at v∗, v^" =v−ω ω·(v−v∗), andv_∗^" =v∗+ω ω·(v−v∗) respectively.

1.1. Collision Kernels. We will assume that the collision kernel has the separable form (1.4) b(ω, v−v∗) = ˆb(ω·n)|v−v∗|^β, where n= v−v∗

|v−v∗|,

for someβ ∈(−D,2) while ˆb(ω·n) is positive almost everywhere and satisfies the weak small- deflection cutoff condition

(1.5) %

%ˆb%

%L¹(dω) =

"

S^D−1

ˆb(ω·n) dω <∞.

The conditions β > −D and (1.5) are required for b(ω, v −v∗) given by (1.4) to be locally integrable with respect to dωdv∗. They allow us to decompose the collision operator (1.3) as (1.6) B(F, F) =G(F, F)−A(F)F ,

where the attenuation operator A(F) and the gain operatorG(F, F) are defined by (1.7) A(F) =

" "

S^D−1×R^D

F∗bdωdv∗, G(F, F) =

" "

S^D−1×R^D

F_∗^"F^"bdωdv∗.

Notes on our join work started at ICERM, draft December 4, 2011.

1

The form (1.4) arises from the classical scattering cross section calculation for identical hard spheres of massm and diameter do, which yields

(1.8) b(ω, v−v∗) = d_o^D−1

2m |ω·(v−v∗)|. This corresponds to the case β = 1 and ˆb(ω·n) = ^do_2m^D−1 |ω·n| in (1.4).

The form (1.4) also arises from the classical scattering cross-section calculation for an inverse- power interparticle potential after a small-deflection cutoff is imposed. For potentials propor- tional to r^−k, where r is the distance between the center of masses, one has

(1.9) β= 1−2D−1

k for k > 2D−1 D + 1.

The cases β ∈ (−D,0), β = 0, β ∈ (0,1], and β ∈ (1,2) are called respectively the “soft”,

“Maxwell”, “hard”, and “super-hard” cases. The super-hard cases do not arise from an inverse- power interparticle potential.

1.2. Conservation Laws. Any solution F of the Boltzmann equation (1.1) formally satisfies the local conservation law

∂t

"

R^D

ξF dv+∇^x·

"

R^D

vξF dv = 0, when ξ(v, x, t) is any quantity that satisfies

(1.10) ξ(·, x, t)∈Span{1, v1, v2,· · · , vD,|v|²}, ∂tξ+v·∇^xξ = 0.

It has been known essentially since Boltzmann [4], who worked out the case D = 3, that the only such quantities ξ are linear combinations of the 4 + 2D + ^D(D−1)₂ quantities

(1.11) 1, v , x−vt , ¹₂|v|², v∧x v·(x−vt), ¹₂|x−vt|²,

where v ∧x = v x^T −x v^T is the skew tensor product. By integrating the coresponding local conservation laws over space and time, we formally obtain the global conservation laws

(1.12)

" "

R^D×R^D

















 1 v x−vt

1 2|v|² v∧x v·(x−vt)

1

2|x−vt|²



















F(v, x, t) dvdx=

" "

R^D×R^D

















 1 v x

1 2|v|² v∧x v·x

1 2|x|²



















Fⁱⁿ(v, x) dvdx ,

where the right-hand sides above exist by the bounds (1.2). These conserved quantities are associated respectively with the conservation laws of mass, momentum, initial center of mass, energy, angular momentum, scalar momentum moment, and scalar inertial moment. The last two are not general physical laws, but are shared by the solutions of other kinetic equations.

1.3. Global Maxwellians. One consequence of Boltzmann’s celebratedH-theorem [4] is that if f(v) is any nonnegative integrable function that has appropriate behavior as |v| → ∞then B(f, f) = 0 if and only if

(1.13) f = ρ

(2πθ)^D2 exp ,

− |v−u|² 2θ

-

for some (ρ, u,θ)∈R₊×R^D×R₊.

Functions of the form (1.13) where ρ, u, and θ functions of (x, t) are called local Maxwellians.

Any local Maxwellian that also satisfies the kinetic equation (1.1) is called aglobal Maxwellian.

The family of all global Maxwellians over the spatial domain R^D with positive mass, zero net momentum, and center of mass at the origin has the form

(1.14a) M= m

(2π)^D

.det(Q) exp#

−q(v, x, t)$

, Q= (ac−b²)I+B²,

(1.14b) q(v, x, t) = 1 2

, v

x−vt -T ,

cI bI +B bI−B aI

- , v

x−vt -

,

with m >0 and (a, b, c, B)∈Ω whereΩ is the open cone in R×R×R×R^D∧D defined by

(1.15) Ω=/

(a, b, c, B)∈R₊×R×R₊×R^D∧D : (ac−b²)I +B² >00 .

HereR₊ denotes the positive real numbers and R^D∧D denotes the skew-symmetric D×D real matrices. The form (1.14) can be derived from the fact that log(M) must satisfy

log(M)∈Span{1, v1, v2,· · · , vD,|v|²}, #

∂t+v·∇^x$

log(M) = 0,

whereby log(M) must be a linear combination of the quantities (1.11). The form (1.14) then comes from the requirements thatMhave finite mass, zero net momentum, and center of mass at the origin. The larger family of global Maxwellians with positive mass is obtained from (1.14) by introducing translations inv and x; whereby it has a total of 4 + 2D + ^D(D−1)₂ parameters.

Remark. The positive definiteness condition (1.15) can be thought of as a bound on b and B in terms of ac by expressing it as

(1.16) b²+(B(² < ac ,

where (B( is the(²-norm, which equals the spectral radius for the skew-symmetric matrix B.

We can bring Minto the local Maxwellian form (1.13). By expanding (1.14b) we find that q(v, x, t) = ¹₂#

at²−2bt+c$

|v|²−#

axt−bx−Bx$

·v+¹₂a|x|².

Because a, c >0 and ac > b² by the remark above, we know that at²−2bt+c > 0 for everyt.

We can therefore complete the square in v to obtain q(v, x, t) = 1

2θ(t)|v−u(x, t)|²+ θ(t)

2 x^TQx , where θ(t) and u(x, t) are given by

(1.17) θ(t) = 1

at²−2bt+c, u(x, t) =θ(t)#

axt−bx−Bx$ . We thereby see that Mhas the local Maxwellian form

(1.18) M= ρ(x, t)

#2πθ(t)$^D₂ exp ,

−|v−u(x, t)|² 2θ(t)

- ,

where the temperatureθ(t) and bulk velocityu(x, t) are given by (1.17), while the mass density ρ(x, t) is given by

(1.19) ρ(x, t) =m

,θ(t) 2π

-^D₂

.det(Q) exp ,

−θ(t) 2 x^TQx

- . Because Q >0 by (1.15), we see that ρ(x, t) is integrable over R^D.

2. Properties of Global Maxwellians

2.1. Moments of Global Maxwellians. We can evaluate the zeroth, first, second, and any other moment of the global Maxwellian familyMbecause they are Gaussian densities over the velocity-position space R^D×R^D. We will use some basic facts about matrices in the form

(2.1) A=

, cI bI+B

bI−B aI -

, where (a, b, c, B)∈R×R×R×R^D∧D.

Specifically, if c)= 0 then the matrices A and Q= (ac−b²)I+B² are related by the facts

(2.2)

(i) det(A) = det(Q),

(ii) A is invertible ⇐⇒ Qis invertible, in which case A⁻¹ =

,Q⁻¹ 0 0 Q⁻¹

- , aI −bI −B

−bI +B cI -

, (iii) A ≥0 ⇐⇒ Q≥0 and c >0,

(iv) A >0 ⇐⇒ Q >0 and c >0 ⇐⇒ (a, b, c, B)∈Ω. These facts can all be seen from the block LDU factorization

A=

, I 0

1

c(bI −B) I

- ,cI 0 0 ¹_cQ

- ,I ¹_c(bI +B)

0 I

- .

This factorization holds for anyAof the form (2.1) withc)= 0 andB ∈R^D×D. Fact (ii) follows from fact (i). The formula for A⁻¹ can be checked by direct computation. Facts (iii) and (iv) require that B ∈ R^D∧D, which insures that A and Qare symmetric. Fact (iv) shows that Ω is a cone. Because it is defined by strict inequalites, Ω⊂R₊×R×R₊×R^D∧D is open.

By evaluating the Gaussian integral and using facts (i) and (iv) of (2.2) we see that

" "

R^D×R^DMdvdx=

"

R^D

ρ(x, t) dx=m . Hence, m is the total mass ofM. Because Mis an even function of #

v x−vt$T

, we see that

" "

R^D×R^D

, v

x−vt -

Mdvdx= ,0

0 -

.

This reflects the facts thatMhas no net momentum and that its center of mass is at the origin.

In fact, every odd-order moment will vanish because M has even symmetry.

The second moments of M may be computed by evaluating the Gaussian integrals and by using the formula forA⁻¹ given in fact (ii) of (2.2) as

(2.3)

" "

R^D×R^D

, v v^T v(x−vt)^T

(x−vt)v^T (x−vt) (x−vt)^T -

Mdvdx

=

" "

R^D×R^D

, v

x−vt

- , v

x−vt -T

Mdvdx

=mA⁻¹ =m

, aQ⁻¹ −bQ⁻¹−Q⁻¹B

−bQ⁻¹+Q⁻¹B cQ⁻¹

- .

In particular, the values of the quadratic conserved quantities from (1.12) are given by

(2.4)

" "

R^D×R^D|v|²Mdvdx=mtr# Q⁻¹$

a ,

" "

R^D×R^D

v·(x−vt)Mdvdx=−mtr# Q⁻¹$

b ,

" "

R^D×R^D|x−vt|²Mdvdx=mtr# Q⁻¹$

c ,

" "

R^D×R^D

v∧xMdvdx=−2mQ⁻¹B .

Every even-order moment of Mmay also be computed, but we will not do so here.

2.2. Conserved Quantities and Global Maxwellians. Every Cauchy problem (1.1) with initial data Fⁱⁿ(v, x) that satisfies the bounds (1.2) can be associated with a unique global Maxwellian determined by the values of the conserved quantities computed from Fⁱⁿ. By choosing an appropriate rescaling and Galilean frame, we may assume without loss of generality that

(2.5)

" "

R^D×R^D

Fⁱⁿdvdx= 1,

" "

R^D×R^D

v Fⁱⁿdvdx=

" "

R^D×R^D

x Fⁱⁿdvdx= 0. The following theorem was proved in [8].

Theorem 2.1. Let Fⁱⁿ(v, x) be a nonnegative function that satisfies the bounds (1.2) and the normalizations (2.5). Let a∗, b∗, c∗, and B∗ be given by

(2.6)

" "

R^D×R^D|v|²Fⁱⁿdvdx=a∗,

" "

R^D×R^D

v·x Fⁱⁿdvdx=b∗,

" "

R^D×R^D|x|²Fⁱⁿdvdx=c∗,

" "

R^D×R^D

v∧x Fⁱⁿdvdx=B∗. Then(a∗, b∗, c∗, B∗)∈Ω∗, where Ω∗ is the open cone in R×R×R×R^D∧D defined by (2.7) Ω∗ =1

(a∗, b∗, c∗, B∗)∈R₊×R×R₊×R^D∧D : ¹₂tr(|B∗|)<.

a∗c∗−b_∗²2 , with |B∗|=.

B_∗^TB∗ =.

−B_∗².

Conversely, if (a_∗, b_∗, c_∗, B_∗) ∈ Ω_∗ then there exists a unique (up to a time shift) global MaxwellianMgiven by (1.14) withm= 1 and(a, b, c, B)∈Ωsuch that the quadratic converved quantities associated with M through (2.4) have values (a_∗, b_∗, c_∗, B_∗) — i.e. such that

(2.8) a∗ = tr# Q⁻¹$

a , b∗ =−tr# Q⁻¹$

b , c∗ = tr# Q⁻¹$

c , B∗ =−2Q⁻¹B , where Q= (ac−b²)I+B² and the set Ω was defined by (1.15).

Remark. The first part of this theorem states that Ω_∗ contains the set of values that can be realized by the conserved quantities given by (2.6). The second part asserts that every point inΩ_∗ can be so realized. Therefore Ω_∗ characterizes all such values.

Remark. Because B∗ is skew-symmetric, its nonzero eigenvalues will come in conjugate pairs.

When B∗ has at most two nonzero eigenvalues then (B∗( = ¹₂tr(|B∗|), where (B∗( is the (² matrix norm, which equals the spectral radius of B∗. If D = 2 or D = 3 then B∗ can have at most two nonzero eigenvalues and we see by (1.16) that Ω∗ = Ω. If D > 3 then Ω∗ will be a proper subset ofΩbecause (B∗(< ¹₂tr(|B∗|) when B∗ has more than two nonzero eigenvalues.

2.3. Ordering Global Maxwellians. The following lemma characterizes when one global Maxwellian with finite mass bounds another that has the same net momentum and center of mass. By choosing an appropriate Galilean frame, we can assume without loss of generality that the net momentum is zero and the center of mass is the origin. In that case the global Maxwellians will have the form (1.14).

Lemma 2.1. Let M¹ and M² be global Maxwellians of the form (1.14) with parameters given by (m1, a1, b1, c1, B1)∈R₊×Ω and (m2, a2, b2, c2, B2)∈R₊×Ω respectively. Then M¹ ≤M² for every (v, x, t) if and only if

, c2I b2I+B2

b2I−B2 a2I -

≤

, c1I b1I +B1

b1I −B1 a1I -

. (2.9)

m1

3 det##

a1c1−b₁²$

I+B₁²$

≤m2

3 det##

a2c2−b₂²$

I+B₂²$ , (2.10)

Similarly, M¹ <M² for every (v, x, t) if and only if (2.9) holds and

(2.11) m1

3 det##

a1c1−b₁²$

I+B₁²$

< m2

3 det##

a2c2−b₂²$

I+B₂²$ , Proof. It follows from (1.14) that

log ,M²

M¹ -

= log



 m₂3

det##

a₂c₂ −b₂²$

I +B₂²$ m₁3

det##

a₁c₁ −b₁²$

I +B₁²$





+ 1 2

, v

x−vt -T ,

(c1 −c2)I (b1 −b2)I+ (B1−B2) (b1−b2)I−(B1−B2) (a1 −a2)I

- , v

x−vt -

.

The assertions of the lemma can be read off from this. !

Remark. The matrix inequality (2.9) is equivalent to a2 ≤a1,c2 ≤c1, and

(2.12) #

(a1−a2)(c1 −c2)−(b1−b2)²$

I+ (B1−B2)² ≥0.

This positive definiteness condition can be thought of as a bound on b1−b2 and B1 −B2 in terms of (a₁−a₂)(c₁−c₂) by expressing it as

(b1 −b2)²+(B1−B2(² <(a1−a2)(c1−c2),

where (B1−B2( is the matrix (²-norm, which equals the spectral radius for B1−B2. Remark. The matrix inequality (2.9) implies that

3 det##

a₂c₂−b₂²$

I+B₂²$

≤ 3

det##

a₁c₁−b₁²$

I+B₁²$ , with equality if and only if

, c2I b2I +B2

b2I −B2 a2I -

=

, c1I b1I +B1

b1I−B1 a1I -

. Therefore ifM1 ≤M2 then m₁ < m₂ unless M1 =M2.

Remark. In this section we assumed that M¹ and M² have the same net momentum and the same center of mass. It is clear that if M¹ ≤ M² then M¹ and M² must have the same net momentum, but they need not have the same same center of mass. Allowing for different centers of mass does not change condition (2.9), but does modify condition (2.10).

2.4. Bounds by Global Maxwellians. Let M¹ and M² be global Maxwellians of the form (1.14) with parameters given by (m1, a1, b1, c1, B1) ∈R₊×Ω and (m2, a2, b2, c2, B2)∈R₊×Ω respectively such thatM¹ ≤M² for every (v, x, t). Let M¹(t) and M²(t) denote their values at timet. Let Fⁱⁿ(v, x) satisfy the pointwise bounds

(2.13) M¹(s)< Fⁱⁿ<M²(s) over R^D×R^D for some s∈R. and the normalizations

(2.14)

" "

R^D×R^D

Fⁱⁿdvdx= 1,

" "

R^D×R^D

v Fⁱⁿdvdx=

" "

R^D×R^D

x Fⁱⁿdvdx= 0. We would like to show that under certain conditions there exists a unique solutionF(v, x, t) of the kinetic equation posed over the spatial domain R^D with initial data Fⁱⁿ(v, x).

In addition, we hope to show that there exists a global Maxwellian M of the form (1.14) with m= 1 and (a, b, c, B)∈Ω such that

F(v, x, t)∼M(v, x, t) as t→ ∞ in some topology. If we let (a∗, b∗, c∗, B∗)∈Ω∗ be given by

(2.15)

" "

R^D×R^D|v|²Fⁱⁿdvdx=a∗,

" "

R^D×R^D

v·x Fⁱⁿdvdx =b∗,

" "

R^D×R^D

|x|²Fⁱⁿdvdx=c_∗,

" "

R^D×R^D

v∧x Fⁱⁿdvdx =B_∗, then by Theorem 2.1 there exists (a, b, c, B)∈Ωsuch that

(2.16) a∗ = tr# Q⁻¹$

a , b∗ =−tr# Q⁻¹$

b , c∗ = tr# Q⁻¹$

c , B∗ =−2Q⁻¹B , where Q= (ac−b²)I+B².

The global Maxwellian bounds (2.13) implies that (a∗, b∗, c∗, B∗) can be bounded by the corresponding quantities associated withmⁱⁿ₁ Mⁱⁿ1 and mⁱⁿ₂Mⁱⁿ2, which are

(2.17)

a1∗ =m1tr# Q⁻¹₁ $

a1, a2∗ =m2tr# Q⁻¹₂ $

a2, b1∗ =−m1tr#

Q⁻¹₁ $

b1, b2∗ =−m2tr# Q⁻¹₂ $

b2, c1∗ =m1tr#

Q⁻¹₁ $

c1, c2∗ =m2tr# Q⁻¹₂ $

c2, B1∗ =−2m1Q⁻¹₁ B1, B2∗ =−2m2Q⁻¹₂ B2,

where Q₁ = (a₁c₁−b₁²)I+B₁² and Q₂ = (a₂c₂−b₂²)I +B₂². Theorem 2.1 implies that

(2.18)

#a∗ −a1∗, b∗ −b1∗, c∗−c1∗, B∗−B1∗

$∈Ω∗,

#a2∗−a∗, b2∗−b∗, c2∗−c∗, B2∗ −B∗

$∈Ω∗,

#a_2∗ −a_1∗, b_2∗−b_1∗, c_2∗ −c_1∗, B_2∗−B_1∗$

∈Ω_∗, which yields the bounds

(2.19)

a1∗ < a∗ < a2∗, c1∗ < c∗ < c2∗,

1

2tr(|B∗−B1∗|)<.

(a∗ −a1∗)(c∗−c1∗)−(b∗−b1∗)²,

1

2tr(|B_2∗−B_∗|)<.

(a_2∗−a_∗)(c_2∗−c_∗)−(b_2∗−b_∗)²,

1

2tr(|B2∗−B1∗|)<.

(a2∗−a1∗)(c2∗−c1∗)−(b2∗ −b1∗)².

3. Kaniel-Shinbrot Iteration

3.1. Kaniel-Shinbrot Theorem. Kaniel-Shinbrot iteration can be used to prove the existence of solutions F to the Boltzmann initial-value problem posed over the spatial domain R^D with initial data Fⁱⁿ(v, x) that satisfies certain pointwise bounds. More specifically, it is used to construct two sequences of approximate solutions, /

F_j^L0

j∈N and / F_j^U0

j∈N, the first of which converges monotonically toF from below, while the second converges monotonically toF from above. In other words, for some T ∈(0,∞] these opposing monotone sequences should satisfy the order relationships

(3.1) F_j^L≤F_j+1^L ≤F_j+1^U ≤F_j^U over R^D×R^D×[0, T) for every j ∈N.

The expectation is that these sequences will converge toF(v, x, t) over R^D×R^D×[0, T). When this construction can be carried out forT =∞then the solution F will be global in time.

Kaniel-Shinbrot iteration constructs sequences as follows. GivenF_j−1^L andF_j−1^U for anyj ∈Z₊ we define the Kaniel-Shinbrot iteratesF_j^L and F_j^U to be the solution of the linear system

∂tF_j^U+v·∇^xF_j^U+A# F_j−1^L $

F_j^U =G#

F_j−1^U , F_j−1^U $ ,

∂tF_j^L+v·∇^xF_j^L+A# F_j−1^U $

F_j^L=G#

F_j−1^L , F_j−1^L $ (3.2a) ,

F_j^U!

!t=0 =F_j^L!

!t=0 =Fⁱⁿ. (3.2b)

The existence of F_j^L and F_j^U can be inferred from the mild formulation of system (3.2).

The form of the Kaniel-Shinbrot iteration (3.2) is motivated by the fact that A and G have the monotonicity properties

(3.3) F ≤G =⇒ A(F)≤A(G) and G(F, F)≤G(G, G).

The following lemma shows that this form insures that Kaniel-Shinbrot iterates preserve the order relationships that appear in (3.1).

Lemma 3.1. (Order Preservation Lemma) If for some T ∈ (0,∞] the Kaniel-Shinbrot iterates F_j−1^L , F_j−1^U , F_j^L, andF_j^U satisfy

(3.4) F_j−1^L ≤F_j^L≤F_j^U ≤F_j−1^U over R^D×R^D×[0, T), then the Kaniel-Shinbrot iterates F_j+1^L and F_j+1^U satisfy

(3.5) F_j^L≤F_j+1^L ≤F_j+1^U ≤F_j^U over R^D×R^D×[0, T). Proof. By (3.2a) of the Kaniel-Shinbrot construction we see that

#∂t+v·∇^x +A#

F_j^U$$ #

F_j+1^L −F_j^L$

=G#

F_j^L, F_j^L$

−G#

F_j+1^L , F_j+1^L $ +A#

F_j−1^U −F_j^U$ F_j^L,

#∂t+v·∇^x +A#

F_j^L$$ #

F_j^U −F_j+1^U $

=G#

F_j−1^U , F_j−1^U $

−G#

F_j^U, F_j^U$ +A#

F_j^L−F_j−1^L $ F_j^U. By the monotonicity (3.3) of A and G and hypothesis (3.4) the right-hand sides above are nonnegative over R^D×R^D× [0, T), while (3.2b) of the Kaniel-Shinbrot construction implies that

#F_j+1^L −F_j^L$ !

!t=0 =#

F_j^U−F_j+1^U $ !

!t=0 = 0, over R^D×R^D. The maximum principle then implies that

(3.6) #

F_j+1^L −F_j^L$

≥0, #

F_j^U−F_j+1^U $

≥0, over R^D×R^D×[0, T). This yields two of the three inequalities asserted by (3.5).

Similarly, by (3.2a) of the Kaniel-Shinbrot construction we have

#∂t+v·∇^x+A#

F_j^U$$ #

F_j+1^U −F_j+1^L $

=G#

F_j^U, F_j^U$

−G#

F_j^L, F_j^L$ +A#

F_j^U −F_j^L$ F_j+1^U . By the monotonicity (3.3) of A and G and hypothesis (3.4) the right-hand side above is non- negative over R^D×R^D × [0, T), while by (3.2b) of the Kaniel-Shinbrot construction we see that

#F_j+1^U −F_j+1^L $ !

!t=0 = 0, over R^D×R^D. The maximum principle then implies that

#F_j+1^U −F_j+1^L $

≥0, over R^D×R^D×[0, T).

When this inequality is combimed with the two in (3.6), we see that assertion (3.5) holds. ! Induction, Lemma 3.1, the Lebesgue Monotone Convergence Theorem, and a stability bound can be used [7, 6] to prove the following.

Theorem 3.1. (Kaniel-Shinbrot) If for some T ∈ (0,∞] the Kaniel-Shinbrot iterates F₀^L, F₀^U, F₁^L, and F₁^U satisfy the so-called beginning condition

(3.7) F₀^L≤F₁^L≤F₁^U ≤F₀^U over R^D×R^D×[0, T), then the Kaniel-Shinbrot iteration yields opposing monotone sequences/

F_j^L0

j∈N and / F_j^U0

j∈N

overR^D×R^D×[0, T)— i.e. sequences that satisfy the order relationship (3.1). These sequences converge to a unique mild solution of the initial-value problem (1.1) for the Boltzmann equation.

Remark. The hard part of applying the Kaniel-Shinbrot Theorem is being sure that the first two iterates satisfy the beginning condition (3.7).

3.2. Beginning with Local Maxwellians. When the initial Kaniel-Shinbrot iterates are local Maxwellians then there is a simple criterion that insures the beginning condition (3.7) is satisfied for some T ∈(0,∞].

Proposition 3.1. (Local Maxwellian Beginning Lemma) Suppose M^L and M^U are local Maxwellians that satisfy

(3.8a) M^L!

!t=0 ≤M^U!

!t=0 over R^D×R^D, and for some T ∈(0,∞] satisfy

(3.8b) ∂tM^U +v·∇^xM^U ≥A(M^U −M^L)M^U over R^D×R^D×[0, T),

−∂tM^L−v·∇^xM^L≥A(M^U −M^L)M^L over R^D×R^D×[0, T), Then for every initial data Fⁱⁿ such that

M^L!

!t=0 ≤Fⁱⁿ≤M^U!

!t=0 over R^D×R^D,

the Kaniel-Shinbrot iterates obtained by setting F₀^L=M^L and F₀^U =M^U satisfy the beginning condition (3.7) over[0, T). Moreover, the Kaniel-Shinbrot Theorem (3.1) yields the unique mild solution F(v, x, t) of the initial-value problem (1.1) for the Boltzmann equation that satisfies the bounds

(3.9) M^L(v, x, t)≤F(v, x, t)≤M^U(v, x, t) over R^D×R^D×[0, T).

Proof. Because F₀^L=M^L and F₀^U =M^U, the Kaniel-Shinbrot construction (3.2) yields

∂tF₁^U +v·∇^xF₁^U+A(M^L)F₁^U =G(M^U, M^U),

∂tF₁^L+v·∇^xF₁^L+A(M^U)F₁^L=G(M^L, M^L), F₁^U!

!t=0 =F₁^L!

!t=0 =Fⁱⁿ.

The above right-hand sides satisfy G(M^L, M^L) = A(M^L)M^L and G(M^U, M^U) = A(M^U)M^U because M^L and M^U are local Maxwellians. Therefore

#∂t+v·∇^x+A(M^L)$ #

M^U−F₁^U$

=∂tM^U +v·∇^xM^U−A(M^U −M^L)M^U,

#∂t+v·∇^x+A(M^U)$ #

F₁^L−M^L$

=−∂tM^L−v·∇^xM^L−A(M^U −M^L)M^L.

By hypothesis (3.8b) the right-hand sides above are nonnegative over R^D×R^D×[0, T), while by hypothesis (3.8a)

#M^U −F₁^U$ !

!t=0 ≥0, #

F₁^L−M^L$ !

!t=0 ≥0, over R^D×R^D. The maximum principle then implies that

(3.10) #

M^U −F₁^U$

≥0, #

F₁^L−M^L$

≥0, over R^D×R^D×[0, T).

This yields two of the three inequalities that are required by the beginning condition (3.7).

Similarly, we have

#∂t+v·∇^x+A(M^U)$ #

F₁^U−F₁^L$

=G(M^U, M^U)−G(M^L, M^L) +A(M^U −M^L)F₁^U. The right-hand side above is nonnegative by the monotonicity (3.3) ofA and G, while we see by (3.2b) of the Kaniel-Shinbrot construction that

#F₁^U−F₁^L$ !

!t=0 = 0 overR^D×R^D. The maximum principle then implies that

#F₁^U −F₁^L$

≥0 over R^D×R^D×[0, T).

When this inequality is combined with the two in (3.10), we see that the beginning condition (3.7) is satisfied. The Kaniel-Shinbrot Theorem (3.1) thereby yields a unique mild solution of the initial-value problem (1.1) for the Boltzmann equation that satisfies (3.9). ! Proposition 3.1 requires us to find local Maxwellians M^L and M^U that meet criterion (3.8) for some T ∈(0,∞]. When M^L >0 this criterion may be recast as

#∂t+v·∇^x$ log#

M^U$

≥A#

M^U−M^L$

over R^D×R^D×[0, T),

−#

∂t+v·∇^x$ log#

M^L$

≥A#

M^U−M^L$

over R^D×R^D×[0, T), (3.11a)

M^L!

!t=0 ≤M^U!

!t=0 over R^D×R^D. (3.11b)

When M^L= 0 criterion (3.8) reduces to simply

(3.12) #

∂t+v·∇^x$

log(M^U)≥ A# M^U$

over R^D×R^D×[0, T).

We will construct such local Maxwellians from the family of global Maxwellians given by (1.14).

When we can do this with T =∞, the Kaniel-Shinbrot theorem will yield global solutions.

3.3. Constructing Local Maxwellians from Global Ones. Let M¹ and M² be global Maxwellians in the form (1.14) that are given by parameters (m1, a1, b1, c1, B1)∈ R₊×Ω and (m2, a2, b2, c2, B2) ∈ R₊ ×Ω respectively, and that satisfy M¹ ≤ M² for every (v, x, t). Let M¹(t) andM²(t) denote their values at time t. Set

(3.13)

Q1 = (a1c1−b₁²)I +B₁², Q2 = (a2c2−b₂²)I+B₂², q1(v, x, t) = 1

2

, v

x−vt -T,

c1I b1I+B1

b1I−B1 a1I

- , v

x−vt -

.

q2(v, x, t) = 1 2

, v

x−vt -T,

c₂I b₂I+B₂ b2I−B2 a2I

- , v

x−vt -

, so that

M¹ = m1

(2π)^D

.det(Q1) exp#

−q1(v, x, t)$

, M² = m2

(2π)^D

.det(Q2) exp#

−q2(v, x, t)$ . Because M¹ ≤M², we know thatq1(v, x, t)≥q2(v, x, t).

We will construct local Maxwellians M^U and M^L in the form M^U(v, x, t) =γ(t) m2

(2π)^D

.det(Q2) exp ,

−q2(v, x, s+t) η(t)

- , (3.14a)

M^L(v, x, t) = 1 γ(t)

m1

(2π)^D

.det(Q1) exp ,

− ,

2− 1 η(t)

-

q1(v, x, s+t) -

, (3.14b)

where the functionsγ(t) and η(t) satisfy

γ(0) =η(0) = 1, γ^"(t)>0, η^"(t)≥0. Clearly, inequality (3.11b) is satisfied because

M^L!

!t=0 =M1(s)≤M2(s) = M^U!

!t=0. By direct calculation we find that

#∂t+v·∇^x$ log#

M^U$

= γ^"(t)

γ(t) +η^"(t) η(t)

q2(v, x, s+t) η(t) ,

−#

∂t+v·∇^x$ log#

M^L$

= γ^"(t)

γ(t) +η^"(t) η(t)

q1(v, x, s+t) η(t) . Because q1(v, x, s+t)≥q2(v, x, s+t), we see that

−#

∂t+v·∇^x$ log#

M^L$

≥#

∂t+v·∇^x$ log#

M^U$ , therefore criterion (3.11) will be satisfied if

(3.15) γ^"(t)

γ(t) + η^"(t) η(t)

q2(v, x, s+t) η(t) ≥A#

M^U −M^L$ .

Similarly, if M^L= 0 while M^U is given by (3.14a) then criterion (3.12) will be satisfied if

(3.16) γ^"(t)

γ(t) + η^"(t) η(t)

q2(v, x, s+t) η(t) ≥A#

M^U$ .

This extends the basic inequality derived in [7] to a more general class of local Maxwellians.

We can treat criterion (3.16) as criterion (3.15) with M^L = 0,

The right-hand sides of (3.15) and (3.16) contain expressions of the form A(M) where M is some local Maxwellian. If ρ(x, t), u(x, t), and θ(x, t) are the mass density, bulk velocity, and temperature associated withM. Then

(3.17)

A(M) =

" "

S^D−1×R^D

M∗b(ω, v−v∗) dωdv∗

=%

%ˆb%

%L¹(dω)

"

R^D|v−v∗|^β ρ

(2πθ)^D2 exp ,

−|v_∗−u|² 2θ

- dv∗

=ρ θ^β2 a

,v−u

√θ -

,

where the attenuation coefficient a(w) is defined by

(3.18) a(w) =%

%ˆb%

%L¹(dω)

1 (2π)^D2

"

R^D|w−w∗|^βexp#

−¹2|w∗|²$ dw∗.

By rotation invariancea(w) is a function of only|w|. It is a bounded function whenβ ∈(−D,0], and is an unbounded function when β ∈ (0,2]. This fundamental difference in the behavior of a(w) requires a different analysis for each of the two cases. These will be presented in the subsequent two subsections.

Remark. For any local Maxwellian M with mass density ρ(x, t), bulk velocity u(x, t), and temperature θ one has

(3.19)

#∂t+v·∇^x$

log(M) = ∂tρ+u·∇^xρ+ρ∇^x·u

ρ + (v−u)·ρ(∂tu+u·∇^xu) +θ∇^xρ ρθ

+

,|v−u|² 2θ − ^D2

-∂tθ+u·∇^xθ+ _D²θ∇^x·u θ

+

,(v−u)×(v−u)

θ − D¹

|v−u|²

θ I

- :∇^xu +

,|v−u|² 2θ − ^D2

-(v−u)·∇^xθ

θ .

For this to be positive everywhere, we must impose the constraints

∇^xθ = 0, ∇^xu+ (∇^xu)^T −D¹∇^x·uI = 0. For this to be a function of|v−u|, we must impose the constraint

ρ(∂tu+u·∇^xu) +θ∇^xρ= 0. The form of (3.19) thereby simplifies to

#∂t+v·∇^x$

log(M) = ∂tρ+u·∇^xρ+ρ∇^x·u

ρ +

,|v−u|² 2θ − ^D₂

-∂tθ+_D²θ∇^x·u

θ .

This quadratic function of |v −u| is similar to those obtained in (3.15) and (3.16). Moreover, the constraints we imposed above imply that the spatial dependence of ρ, u, and θ has the same form as for a global Maxwellian. This means the forms that we have assumed in (3.14) are close to being general.

3.4. Soft and Maxwell Cases. We now complete the construction of Section 3.3 for cases when the collision kernel has the separable form (1.4) with β ∈ (−D,0]. In other words, for cases when the kernel arises from either soft or Maxwell potentials. Because (3.18) shows the right-hand sides of (3.15) and (3.16) are bounded functions of v when β ∈ (−D,0], we take η(t) = 1 in the forms of M^U and M^L given by (3.14). Criterion (3.15) then reduces to

(3.20) γ^"(t)

γ(t) ≥A#

M^U −M^L$

, γ(0) = 1. Below we show how this criterion can be met in certain cases.

3.4.1. Near Vacuum Case. The easiest case to treat is when the lower bound is the vacuum.

Proposition 3.2. Let M(t) be the global Maxwellian in the form (1.14) given by m >0 and (a, b, c, B)∈Ω. Let Fⁱⁿ(v, x) be any initial data such that

(3.21) 0≤Fⁱⁿ≤M(s) for some s∈R.

Let β ∈(−D,0]. Let

(3.22) Ns(t) =(a(^L∞(dw)

3 det# ₁

2πQ$

" t 0

θ(s+t^")^D+β2 dt^", where Q= (ac−b²)I+B² and θ(t) is given by (1.17).

Then there exists a mild solutionF(v, x, t)to the initial-value problem (1.1) for the Boltzmann equation over [0, T) that satisfies the bounds

(3.23) 0≤F(t)≤γ(t)M(s+t) over R^D×R^D×[0, T), where

(3.24) γ(t) = 1

1−mNs(t) over[0, T), T = sup{t >0 : mNs(t)<1}. In particular, this solution is global when β ∈(1−D,0] and

(3.25) m N_s^∞≤1, where N_s^∞ = lim

t→∞Ns(t).

and is globally bounded by a global Maxwellian when strict inequality holds.

Remark. The local result above is in the spirit of Kaniel and Shinbrot [7], while the global result is in the spirit of Illner and Shinbrot [6]. The bounds obtained here are different because of our use of the global Maxwelian family (1.14).

Proof. Set M^L = 0 andM^U =γ(t)M(s+t). Then criterion (3.20) reduces to γ^"

γ ≥γA(M(s+t)), γ(0) = 1. Because by (3.18) and (1.19)

A(M(s+t)) =mθ(s+t)^D+β2 3

det# ₁

2πQ$ exp

,

−θ 2x^TQx

- a

,v −u

√θ -

≤mθ(s+t)^D+β2 3

det# ₁

2πQ$

(a(^L∞(dw) =mN_s^"(t), this criterion will be satisfied if

γ^" =γ²mN_s^"(t), γ(0) = 1.

But (3.24) gives the solutionγ(t) of this initial-value problem along with its interval of existence [0, T), whereby criterion (3.20) is satisfied over this interval. The bounds (3.23) thereby follow from (3.9) of Proposition 3.1.

Because θ(t) given by (1.17) decays like t⁻² as t → ∞, the integrand in definition (3.22) of Ns(t) decays like t^−(D+β) as t → ∞. It is therefore clear that N_s^∞ = limt→∞Ns(t) <∞ if and only ifβ ∈(1−D,0]. The bound (3.25) on m then follows from (3.24). ! 3.4.2. Near Global Maxwellian Case. The next easiest case to treat is when the lower and upper bounds are proportional to the same global Maxwellian.

Proposition 3.3. Let M(t) be the global Maxwellian in the form (1.14) given by m = 1 and some (a, b, c, B)∈Ω. Let m2 > m1 >0. Let Fⁱⁿ(v, x) be any initial data such that

(3.26) m1M(s)≤Fⁱⁿ≤m2M(s) for somes ∈R. Let γmin =.

m1/m2. Let β ∈(−D,0]. Let Ns(t) be given by (3.22).

Then there exists a mild solutionF(v, x, t)to the initial-value problem (1.1) for the Boltzmann equation over [0, T) that satisfies the bounds

(3.27) m1

γ(t)M(s+t)≤F(t)≤γ(t)m2M(s+t) over R^D×R^D×[0, T), where

(3.28) γ(t) =γmin

1 +γmin+ (1−γmin)e^{2γminm2Ns(t)}

1 +γmin−(1−γmin)e^{2γminm2Ns(t)} over [0, T), T = sup{t >0 : (1−γmin)e^{2γminm2Ns(t)}<1 +γmin}. In particular, this solution is global when β ∈(1−D,0] and

(3.29) m2N_s^∞ ≤ 1 2γmin

log

,1 +γmin

1−γmin

-

, where N_s^∞= lim

t→∞Ns(t), and is globally bounded by a global Maxwellian when strict inequality holds.

Remark. Condition (3.29) yields global solutions with larger mass by pickingγ_min closer to 1.

Remark. This result is in the spirit of Toscani [11]. The bounds obtained here are different because of our use of the global Maxwelian family (1.14). In particular, we can treat initial data with significant rotation that are excluded by the hypotheses in [11].

Proof. Set M^L = _γ(t)^m1M(s+t) and M^U =γ(t)m₂M(s+t). Criterion (3.20) then reduces to γ^"

γ ≥ ,

γ− γ_min² γ

-

m2A(M(s+t)), γ(0) = 1. Because A(M(s+t))≤N_s^"(t), this criterion will be satisfied if

γ^" =#

γ²−γ_min² $

m2N_s^"(t), γ(0) = 1.

But (3.28) gives the solutionγ(t) of this initial-value problem along with its interval of existence [0, T), whereby criterion (3.20) is satisfied over this interval. The bounds (3.27) thereby follow from (3.9) of Proposition 3.1.

By arguing as in the proof of Proposition 3.2, we see that N_s^∞ = limt→∞Ns(t) <∞ if and only ifβ ∈(1−D,0]. The bound (3.29) on m2 then follows from (3.28). !

3.4.3. Between Global Maxwellians Case. We now treat the case is when the lower and upper bounds are proportional to possibly different global Maxwellians. The question addressed below is whether or not the only bounds on global solutions that one can obtain are the ones already recovered by the arguments in the foregoing subsections.

Let β ∈ (−D,0]. Let M1 and M2 be the global Maxwellians in the form (1.14) that are given by some (m1, a1, b1, c1, B1) and (m2, a2, b2, c2, B2) inR₊×Ωrespectively. We assume that M¹ ≤M². This implies thatm1

.det(Q1)≤m2

.det(Q2), where Q1 = (a1c1−b₁²)I+B₁² >0 and Q₂ = (a₂c₂−b₂²)I+B₂² >0. It also implies that q₁(v, x, t)≥q₂(v, x, t), which means that a₁ ≥a₂, c₁ ≥c₂, and

(b1−b2)²+(B1−B2(² ≤(a1−a2)(c1−c2). Because by (1.17) we have

θ1(t) = 1

a₁t²−2b₁t+c₁ >0, θ1(t) = 1

a₂t² −2b₂t+c₂ >0, while by (2.12) we have a₁ ≥a₂ and (a₁−a₂)(c₁−c₂)≥(b₁−b₂)², we see that

1

θ1(t)− 1

θ2(t) = (a1−a2)t²−2(b1−b2)t+ (c1−c2)≥0. Hence, θ₁(t)≤θ₂(t). It is also clear that ρ₁(x, t)≤ρ₂(x, t).

Now set M^L= _γ(t)¹ M1(s+t) and M^U =γ(t)M2(s+t). Then because

#∂t+v·∇^x$

log(M^U) =−#

∂t+v·∇^x$

log(M^L) = γ^"

γ , A#

M^U−M^L$

=γA(M2(s+t))− 1

γ A(M1(s+t)), criterion (3.20) becomes

γ^"

γ ≥γA#

M²(s+t)$

− 1 γ A#

M¹(s+t)$

, γ(0) = 1. The above differential inequality has the form

γ^" ≥ 4

γ²− A#

M¹(s+t)$ A#

M²(s+t)$ 5

A#

M2(s+t)$ .

By using (3.18) followed by (1.17) and (1.19), we find that the ratio above is A(M¹)

A(M2) = ρ1θ

β 2

1 a

,v−u1

√θ1

-

ρ₂θ

β 2

2 a

,v−u2

√θ2

- =

m1

.det(Q1)θ

D+β 2

1 exp

,

−θ1

2x^TQ1x -

a

,v −u1

√θ1

-

m₂.

det(Q₂)θ

D+β 2

2 exp

,

−θ2

2x^TQ₂x -

a

,v −u2

√θ2

-.

We would like to know when this ratio is bounded below by a positive number.

The ratio is bounded below by a positive number when (a1, b1, c1, B1) = (a2, b2, c2, B2), in which case

A#

M¹(s+t)$ A#

M²(s+t)$ = m1

m2

. This is just the “Toscani case” treated in the last subsection.