COMPACTNESS OF THE GAIN PARTS OF THE LINEARIZED
BOLTZMANN OPERATOR WITH WEAKLY CUTOFF KERNELS
C. DAVID LEVERMORE AND WEIRAN SUN
A. We prove an L p compactness result for the gain parts of the linearized Boltzmann collision
operator associated with weakly cutoff collision kernels that derive from a power-law intermolecular
potential. We replace the Grad cutoff assumption previously made by Caflisch [1], Golse and Poupaud
[7], and Guo [11] with a weaker local integrability assumption. This class includes all classical kernels
to which the DiPerna-Lions theory applies that derive from a repulsive inverse-power intermolecular
potential. In particular, our approach allows the treatment of both hard and soft potential cases.
1. I
The linearized Boltzmann collision operator arises in the study of fluid dynamical approximations
to solutions of the Boltzmann equation. That equation governs the kinetic density F(v, x, t) of a gas
composed of identical particles with velocities v ∈ RD and positions x ∈ RD at time t ≥ 0 as [3, 4]
(1.1)
∂t F + v · ∇x F = B(F, F) ,
where the collision operator B(F, F) is given by
ZZ
(1.2)
B(F, F) =
(F 1′ F ′ − F 1 F) b(ω, v1 − v) dω dv1 .
SD−1 ×RD
Here ω ∈ SD−1 is a unit vector, dω is the usual rotationally invariant Lebesgue measure on SD−1 , while
F 1′ , F ′ , F 1 , and F are the density F(·, x, t) evaluated at the velocities v′1 , v′ , v1 , and v respectively. Here
(v′1 , v′ ) are the velocities after an elastic binary collision between two molecules that had the velocities
(v1 , v) before the collision, or vice versa. Because both momentum and energy are conserved in an
elastic collision, one can express v′1 and v′ in terms of v1 and v as
(1.3)
v′1 = v1 − ω ω · (v1 − v) ,
v′ = v + ω ω · (v1 − v) ,
where the unit vector ω ∈ SD−1 is parallel to the deflections v′1 − v1 and v′ − v, and is thereby normal
to the plane of reflection. The part of B(F, F) involving F 1′ and F ′ is called the gain term, while the
part involving F 1 and F is called the loss term.
By Galilean invariance and rotational invariance, every classical collision kernel b(ω, v1 − v) has
the general form [3]
v1 − v
,
n=
b(ω, v1 − v) = |v1 − v| Σ(|ω · n|, |v1 − v|) ,
|v1 − v|
where Σ ≥ 0 is the differential scattering cross-section. When the molecules are hard spheres of mass
m and radius ro then b has the form
(1.4)
b(ω, v1 − v) = |ω · (v1 − v)|
Date: July 23, 2009.
1
(2ro )D−1
.
2m
2
C. DAVID LEVERMORE AND WEIRAN SUN
When the molecules are point particles with a repulsive intermolecular potential proportional to r−k
for some k > 0, where r is the intermolecular distance, then the kernel b has the factored form
(1.5)
b(ω, v1 − v) = b̂(ω · n) |v1 − v|β ,
where b̂ is an even function of ω · n and β = 1 − 2(D−1)
< 1. Notice that the hard sphere kernel (1.4)
k
can be put into this form with β = 1. The cases β < 0, β = 0, and β > 0 are respectively referred to as
the “soft”, “Maxwell”, and “hard” potential cases.
In this paper we assume that the collision kernel b has the factored form (1.5) where b̂ and β satisfy
(1.6)
b̂(ω · n) ∈ L1 (dω) ,
−D < β .
These assumptions are necessary for b to be locally integrable in all of its variables. This is needed
to seperately make sense of the gain and loss parts of B(F, F) for all continuous, rapidly decaying F.
They include the so-called “super hard” kernels coresponding to β > 1, which do not arise classically.
The first assumption in (1.6) is a so-called small deflection cutoff condition because the b̂ that is
derived from microscopic physics has a nonintegrable singularity at ω · n = 0 due to the large number
of grazing collisions. Grad [10] observed that these grazing collisions should not appreciably affect
the macroscopic dynamics, and therefore proposed that these singularities can be cutoff in order to
make analysis of the Boltzmann equation more tractable. The small deflection cutoff condition in (1.6)
is much weaker than the one introduced by Grad, which requires that b̂ vanish like |ω · n| as |ω · n| → 0.
It is therefore sometimes called the weak cutoff condition. In order to apply the DiPerna-Lions theroy
of global renormalized solutions [5] to the Boltzmann equation with kernels in the factored form (1.5),
it is necessary to impose our assumptions (1.6) and also to require that β < 2.
Now let M denote the Maxwellian given by
(1.7)
M(v) =
1
1 2
e− 2 |v| .
D/2
(2π)
We consider the linearized Boltzmann operator L defined by
ZZ
2
(1.8)
L f = − B(M, M f ) =
f + f1 − f ′ − f1′ b(ω, v1 − v) dω M1 dv1 .
M
SD−1 ×RD
This case is general because any Maxwellian can be put into the form (1.7) by applying a Galilean
transformation to make its bulk velocity zero and a rescaling of units to make its mass density and
temperature equal to unity, and because the collision operator B(F, F) commutes with Galilean boosts
and is homogeneous under rescalings of density and temperature units. This last fact is a consequence
of the factored form (1.5) of b. Finally, also without loss of generality, we normalize b̂ so that
Z
(1.9)
b̂(ω · n) dω = 1 .
SD−1
We decompose the linearized Boltzmann operator L as
(1.10)
L f = a(v) I + K1 − K2 − K3 f ,
where the attenuation coefficient a(v) is given by
ZZ
(1.11)
a(v) =
b(ω, v1 − v) dω M1 dv1 ,
SD−1 ×RD
COMPACTNESS OF THE GAIN PARTS OF THE LINEARIZED BOLTZMANN OPERATOR
3
while the loss operator K1 and the gain operators K2 and K3 are given by
ZZ
1
f1 b(ω, v1 − v) dω M1 dv1 ,
K1 f =
a(v)
SD−1 ×RD
ZZ
1
K2 f =
f ′ b(ω, v1 − v) dω M1 dv1 ,
(1.12)
a(v)
D−1
D
Z ZS ×R
1
K3 f =
f1′ b(ω, v1 − v) dω M1 dv1 .
a(v)
D−1
D
S ×R
The main result of this paper is the following.
Main Theorem. Let the collision kernel b have the factored form (1.5) and satisfy conditions (1.6).
Then for every p ∈ (1, ∞) and every j = 1, 2, 3 the operator
(1.13)
K j : L p (aMdv) → L p (aMdv)
is compact .
The first result of this kind was given by Hilbert in the same paper in which he introduced what we
now call the Hilbert expansion [13]. For the hard sphere case in D = 3 he applied his new theory of
integral operators to essentially show that for the operator K = K1 − K2 − K3 one has
(1.14)
aK : L2 (Mdv) → L2 (Mdv)
is compact .
More precisely, what he showed was isometrically equivalent to this. Also for the hard sphere case
in D = 3, Hecke [12] gave a variant of Hilbert’s result by showing that the kernel of K 5 is HilbertSchmidt in L2 (aMdv), and Carleman [2] improved upon this by showing that the kernel of K 2 is
Hilbert-Schmidt in L2 (aMdv). Grad [10] extended Hilbert’s result to the Maxwell and hard potential
cases, β ∈ [0, 1], by introducing his small deflection cutoff condition. This allowed him to adapt
Hecke’s argument and show that the kernel of (aK)3 is Hilbert-Schmidt in L2 (Mdv). By using similar
methods, Caflisch [1] extended Grad’s result (1.14) to the soft potential case by treating Grad cutoff
kernels with β ∈ (−1, 1] in D = 3, and Golse and Poupaud [7] established that K is compact over
L2 (aMdv) for Grad cutoff kernels with β ∈ (−2, 1] in D = 3. By using an approach that is closer to
our own, Guo [11] extended Caflisch’s result for Grad cutoff kernels to the full range β ∈ (−3, 1] in
D = 3. Our approach allows us to replace the Grad cutoff condition with the weaker cutoff condition
given in (1.6). We do so for general dimension D.
The most important application of such compactness results has been to establish Fredholm alternative results for the linearized collision operator L. Indeed, Grad showed that for Grad cutoff kernels
with β ∈ [0, 1] the operator L satisfied a Fredholm alternative in L2 (Mdv). For soft potential case this
does not hold, but it was pointed out by Golse and Poupaud in [7] that a1 L still satisfies a Fredholm
alternative in L2 (aMdv). Fredholm alternatives and their related coercivity bounds play an important role in establishing hydrodynamic limits of the Boltzmann equation [4]. We mention that the
Fredholm alternative has been established in L2 (aMdv) for an even broader class of collision kernels
without any small deflection cutoff assumption by Mouhot and Strain [16] using a different approach,
thereby improving upon an earlier result of Pao [17]. Our result yields a Fredholm alternative in
L p (aMdv) for every p ∈ (1, ∞), but we require the weak cutoff condition. This L p Fredholm alternative is used in [15] to prove an incompressible Navier-Stokes limit for the Boltzmann equation for the
class of collision kernels considered here, thereby extending the results of Golse and Saint-Raymond
[8, 9] for Grad cutoff kernels with β ∈ [0, 1].
2. C P
2.1. Preliminaries. We begin with a basic fact that illustrates why the operators K j defined by (1.12)
and the spaces L p (aMdv) are natural for this study.
4
C. DAVID LEVERMORE AND WEIRAN SUN
Lemma 2.1. Let the collision kernel b have the factored form (1.5) and satisfy conditions (1.6). Then
for every p ∈ [1, ∞] and every j = 1, 2, 3 the operator
K j : L p (aMdv) → L p (aMdv)
(2.1)
is bounded with kK j kB(L p ) ≤ 1 .
Proof. First let p ∈ (1, ∞) and set p∗ ∈ (1, ∞) such that 1p + p1∗ = 1. For every f, g ∈ Cc (RD ) we have
Z
ZZZ
(2.2)
g K1 f aMdv =
g f1 b dω M1dv1 Mdv .
RD
SD−1 ×RD ×RD
An application of the Hölder inequality then yields
! p1∗ Z Z Z
Z
ZZZ
p∗
|g K1 f | aMdv ≤
|g| b dω M1dv1 Mdv
RD
= kgkL
SD−1 ×RD ×RD
p∗
(aMdv)
k f kL p (aMdv) .
! 1p
| f1 | b dω M1dv1 Mdv
p
SD−1 ×RD ×RD
∗
This inequality extends to every f ∈ L p (aMdv) and g ∈ L p (aMdv) by a density argument. By the
Reisz Representation Theorem we see that assertion (2.1) for K1 holds with kK1 kB(L p ) ≤ 1 for every
p ∈ (1, ∞). The extension to every p ∈ [1, ∞] follows because (2.2) holds for every f ∈ L1 (aMdv)
and g ∈ L∞ (aMdv) or for every f ∈ L∞ (aMdv) and g ∈ L1 (aMdv). The proofs of assertion (2.1) for
K2 and K3 go similarly.
Remark. This proof does not require the kernel b(ω, v1 − v) to have the factored form (1.5). Rather,
it only requires that the attenuation coefficient a(v) given by (1.11) exists.
Given Lemma 2.1, our Main Theorem follows by a straightforward interpolation argument once we
show that the compactness assertion (1.13) holds for p = 2. We will use the following compactness
criterion [14], which is a generalization of the classical Hilbert-Schmidt criterion.
Lemma 2.2. Let K be an integral operator given by
Z
K f (v) =
K(v, v′ ) f (v′ ) dµ(v′ ) ,
RD
where dµ(v) is a σ-finite, positive measure over RD . Let the kernel K(v, v′ ) be symmetric in v and v′
and for some r ∈ [1, 2] satisfy the bound
Z Z
 r1∗
! rr∗


|K(v, v′ )|r dµ(v′ ) dµ(v) < ∞ ,
(2.3)
kKkLr∗(Lr ) = 
D
D
R
1
r
R
where r∗ ∈ [2, ∞] satisfies + = 1. Let p, p∗ ∈ [r, r∗ ] satisfy 1p + p1∗ = 1. Then for every f ∈ L p (dµ)
∗
and g ∈ L p (dµ) one has
Z
ZZ
K(v, v′ ) f (v′) g(v) dµ(v′ ) dµ(v) ≤ kKkLr∗ (Lr ) k f kL p kgkL p∗ ,
(2.4)
|g(v) K f (v)| dµ(v) ≤
RD
p
p
1
r∗
RD ×RD
whereby K : L (dµ) → L (dµ) is bounded with kKkB(L p ) ≤ kKkLr∗ (Lr ) . Moreover, if r ∈ (1, 2] then
K : L p (dµ) → L p (dµ) is compact.
The bounded (2.4) for either p = r or p = r∗ is simply obtained by two applications of the Hölder
inequality. Its extension to every p ∈ [r, r∗ ] then follows by interpolation. The assertion that K maps
L p (dµ) into itself and the bound on kKkB(L p) are consequences of the Riesz Representation Theorem.
The compactness assertion holds because when r ∈ (1, 2] the finite-rank kernels are dense in the space
∗
Lr (dµ; Lr (dµ)) whose norm is defined by (2.3). The classical Hilbert-Schmidt compactness criterion
is the special case r = 2. The above criterion often becomes easier to meet as r gets closer to 1.
COMPACTNESS OF THE GAIN PARTS OF THE LINEARIZED BOLTZMANN OPERATOR
5
2.2. Compactness of the Loss Operator. In this section we establish that the loss operator K1 is
compact from L2 (aMdv) to L2 (aMdv). We begin with the following lemma, which plays a central
role in our compactness proofs for the operators K1 , K2 , and K3 .
Lemma 2.3. Let H ∈ L∞ (dv) such that H ≥ 0, H , 0, and (1 + |v|)r H ∈ L∞ (dv) for every r > 0. Let
γ > −D and define
Z
(2.5)
h(v) =
|v1 − v|γ H(v1 ) dv1 .
RD
Then h ∈ C(RD ) and there exist positive constants C and C such that
C (1 + |v|)γ ≤ h(v) ≤ C (1 + |v|)γ ,
(2.6)
for every v ∈ RD .
Proof. First consider the case when γ ≥ 0. From the elementary bounds
(2.7)
|v1 − v|2 ≤ (1 + |v1 |2 ) (1 + |v|2 ) ≤ (1 + |v1 |)2 (1 + |v|)2 ,
for every v ∈ RD ,
and the fact H > 0, we directly obtain the upper bound
Z
Z
γ
h(v) =
|v1 − v| H(v1 ) dv1 ≤
(1 + |v1 |)γ H(v1 ) dv1 (1 + |v|)γ ,
RD
RD
which yields the upper bound of (2.6).
Next, the bounds (2.7) and the fact H > 0 imply that (1 + |v|)−γ |v1 − v|γ H(v1 ) ≤ (1 + |v1 |)γ H(v1 ).
The Lebesgue Dominated Convergence Theorem therefore implies that the positive function
Z
1
h(v)
=
|v1 − v|γ H(v1 ) dv1 ,
(2.8)
v 7→
(1 + |v|)γ (1 + |v|)γ RD
is continuous over RD and satisfies
Z
Z
1
h(v)
γ
= lim
lim
|v1 − v| H(v1 ) dv1 =
H(v1 ) dv1 > 0 .
|v|→∞ (1 + |v|)γ
|v|→∞ (1 + |v|)γ
RD
RD
The function (2.8) is thereby bounded away from zero, whereby the lower bound of (2.6) follows.
Now consider the case when γ ∈ (−D, 0). From the elementary bounds (2.7) and the fact H > 0,
we directly obtain the lower bound
Z
Z
γ
γ
(1 + |v1 |) H(v1 ) dv1 (1 + |v|) ≤
|v1 − v|γ H(v1 ) dv1 = h(v) ,
RD
RD
which yields the lower bound of (2.6).
∗
∗
D
D
) and q ∈ ( |γ|
, ∞), classical
Next, because |v|γ ∈ (L p + Lq ) while H ∈ (L p ∩ Lq ) for every p ∈ (1, |γ|
D
results on convolutions [6] show that h ∈ C0 (R ). The function (2.8) is therefore continuous over RD .
Let s > D and C s < ∞ such that k(1 + |v|)s H(v)kL∞ ≤ C s . Then for every nonzero v ∈ RD we have
Z
Z
h(v)
1
|v|γ
Cs
C s |v|D
γ
γ
≤
dv
=
dw1 ,
|v
−
v|
|w
−
v̂|
1
1
1
(1 + |v|)γ (1 + |v|)γ RD
(1 + |v1 |)s
(1 + |v|)γ RD
(1 + |v||w1 |)s
where v̂ = v/|v| and v1 = |v|w1 . By rotational invariance, the integral on the right-hand side above is
independent of v̂. Moreover, by classical “δ-function” approximation estimates we can show
Z
Z
C s |v|D
Cs
h(v)
γ
|w1 − v̂|
≤ lim
dw1 =
dv1 < ∞ .
lim sup
γ
s
s
|v|→∞ RD
(1 + |v||w1 |)
|v|→∞ (1 + |v|)
RD (1 + |v1 |)
The function (2.8) is thereby uniformly bounded, whereby the upper bound of (2.6) follows.
Compactness of K1 is a direct result of Lemmas 2.2 and 2.3.
6
C. DAVID LEVERMORE AND WEIRAN SUN
Theorem 2.1. Let the collision kernel b have the factored form (1.5) and satisfy conditions (1.6).
Then
(2.9)
K1 : L2 (aMdv) → L2 (aMdv)
is compact .
Proof. When b has the factored form (1.5) and satisfies (1.6) then with the normalization (1.9) the
attenuation coefficient defined by (1.11) takes the form
Z
(2.10)
a(v) =
|v1 − v|β M1 dv1 ,
RD
while the loss operator K1 defined by (1.12) takes the form
Z
(2.11)
K1 f (v) =
K1 (v, v1 ) f (v1 ) a1 M1 dv1 ,
RD
where the kernel K1 (v, v1 ) is given by
1
K1 (v, v1 ) =
a(v) a(v1 )
Z
SD−1
b(ω, v1 − v) dω =
|v1 − v|β
.
a(v) a(v1 )
By Lemma 2.3 with H = M and γ = β there exist positive constants C a and C a such that
(2.12)
C a (1 + |v|)β ≤ a(v) ≤ C a (1 + |v|)β ,
for every v ∈ RD .
Because β ∈ (−D, ∞), there exists r ∈ (1, 2] such that βr ∈ (−D, ∞). Direct calculation shows that
! 1r
! 1r
Z
Z
1
r
βr 1−r
(2.13)
|K1 (v, v1 )| a1 M1 dv1 =
|v1 − v| a1 M1 dv1 .
a(v) RD
RD
By Lemma 2.3 with H = a1−r M and γ = βr there exist positive constants C r and C r such that
Z
βr
C r (1 + |v|) ≤
|v1 − v|βr a11−r M1 dv1 ≤ C r (1 + |v|)βr , for every v ∈ RD .
RD
When this estimate is combined with estimate (2.12), we see from (2.13) that
1
1
! 1r
Z
r
C rr
C
0<
≤
|K1 (v, v1 )|r a1 M1 dv1 ≤ r < ∞ , for every v ∈ RD .
Ca
Ca
RD
∗
In particular, we see that K1 ∈ L∞ (aMdv; Lr (aMdv)) ⊂ Lr (aMdv; Lr (aMdv)), where 1r + r1∗ = 1.
Because the kernel K1 (v, v1 ) thereby satisfies condition (2.3) for some r ∈ (1, 2], upon applying
Lemma 2.2 with p = 2 we see that assertion (2.9) holds.
2.3. Compactness of the Gain Operators. The remainder of this paper establishes that the gain
operators K2 and K3 are compact from L2 (aMdv) to L2 (aMdv). In this section we reduce the proof to
technical lemmas that will be proved in later sections.
Theorem 2.2. Let the collision kernel b have the factored form (1.5) and satisfy conditions (1.6).
Then for j = 2, 3
(2.14)
K j : L2 (aMdv) → L2 (aMdv)
is compact .
Proof. We see from definition (1.12) of K2 and K3 and the factored form (1.5) of b that
ZZ
1
K2 f (v) =
f (v′ ) b̂(ω · n) |v1 − v|β dω M1 dv1 ,
a(v)
D−1
D
Z ZS ×R
(2.15)
1
K3 f (v) =
f (v′1 ) b̂(ω · n) |v1 − v|β dω M1 dv1 .
a(v)
D−1
D
S ×R
COMPACTNESS OF THE GAIN PARTS OF THE LINEARIZED BOLTZMANN OPERATOR
7
We employ expressions for the kernels of these operators similar to those introduced by Grad [10].
Because v′ , v′1 and b̂ are even functions of ω, the integrals over ω in (2.15) are just twice the integrals
over the region ω · n > 0. It then follows from (1.3) that in this region we have the identities
|v1 − v|2 = |v′1 − v|2 + |v′ − v|2 ,
(v′1 − v) · (v′ − v) = 0 ,
|v′ − v|
(v′ − v) · (v1 − v)
,
=
ω·n=
p ′
|v′ − v| |v1 − v|
|v1 − v|2 + |v′ − v|2
(2.16)
|v1 |2 = |v|2 + 2v · (v′1 − v) + 2v · (v′ − v) + |v′1 − v|2 + |v′ − v|2 .
From these identities it can be shown that the gain operators K2 and K3 have the forms
Z
Z
′
′
′ ′ ′
(2.17)
K2 f (v) =
K2 (v, v ) f (v ) a M dv ,
K3 f (v) =
K2 (v, v′1 ) f (v′1 ) a′1 M1′ dv′1 ,
RD
RD
where, off their diagonals, the symmetric kernels K2 (v, v′ ) and K3 (v, v′1 ) are given by


β
Z
2
′
2 2
′


|y|
+
|v
−
v|
2
|v
−
v|
1 2
′

− 2 |y| −y·w 
 dy ,
 p
K2 (v, v′ ) =
b̂
e
a(v) a(v′ ) y⊥(v′ −v)
|v′ − v|D−1
|y|2 + |v′ − v|2
(2.18)


β
Z

|v′1 − v|2 + |z|2 2 − 1 |z|2 −z·w′ 
2
|z|
′
 dz ,
2
1 b̂
K3 (v, v1 ) =
e

p

′
a(v) a(v1 ) z⊥(v′1 −v)
|z|D−1
|v′1 − v|2 + |z|2
with the vectors w′ (v, v′ ) and w′1 (v, v′1 ) defined by
(2.19)
w′ =
|v|2 − v′ · v ′ |v′ |2 − v′ · v
v +
v,
|v′ − v|2
|v′ − v|2
w′1 =
|v|2 − v′1 · v ′ |v′1 |2 − v′1 · v
v +
v.
|v′1 − v|2 1
|v′1 − v|2
These vectors have minimum norm on the lines {v + s(v′ − v) : s ∈ R} and {v + s(v′1 − v) : s ∈ R}
respectively. In (2.18) the variables of integration y and z are identified as y = v′1 − v and z = v′ − v.
The compactness of K2 and K3 is established by showing that they are each the limit of a sequence
of compact operators. We construct the approximating sequences by truncations. Specifically, for
every R > 0 and ǫ > 0 we construct the operators K2ǫ,R and K3ǫ,R by replacing the kernels in (2.17) by
the doubly truncated symmetric kernels that are respectively given by
β
Z
|y|2 + |v′ − v|2 2 − 1 |y|2 −y·w′
2
ǫ,R
′
e 2
K2 (v, v ) = 1{|v|<R} 1{|v′ |<R}
a(v) a(v′ ) y⊥(v′ −v)
|v′ − v|D−1


′


|v
−
v|
 1{|v′ −v|>ǫ|v1 −v|} dy ,
× b̂ p
|y|2 + |v′ − v|2
(2.20)
β
Z
′
2
2 2
|v
−
v|
+
|z|
1 2
2
′
1
e− 2 |z| −z·w1
K3ǫ,R (v, v′1 ) = 1{|v|<R} 1{|v′1 |<R}
′
D−1
a(v) a(v1 ) z⊥(v′1 −v)
|z|




|z|
 1{|v′ −v|>ǫ|v1 −v|} 1{|z|>ǫ|v1 −v|} dz ,
× b̂ p
1
|v′1 − v|2 + |z|2
where 1S denotes the indicator function of a set S . The fact that the operators K2ǫ,R and K3ǫ,R are
compact over L2 (aMdv) for every R > 0 and ǫ > 0 will follow from Lemma 2.4, which is stated and
proved in Section 2.4. The fact that the operators K2ǫ,R and K3ǫ,R approximate K2 and K3 as ǫ → 0 and
R → ∞ will follow from Lemma 2.5, which is stated and proved in Section 2.5. Because the compact
operators are a closed subspace of B(L2 ), we thereby obtain the compactness of K2 and K3 .
8
C. DAVID LEVERMORE AND WEIRAN SUN
Remark. When this result is combined with the compactness of the loss operator K1 over L2 (aMdv)
provided by Theorem 2.1, and the boundedness of K1 , K2 , and K3 over L p (aMdv) for every p ∈ [1, ∞)
provided by Lemma 2.1 then our Main Theorem follows by a standard interpolation argument.
2.4. Compactness of the Approximating Operators. In this section we establish the compactness
of the approximating operators K2ǫ,R and K3ǫ,R . We have the following.
Lemma 2.4. Let K2ǫ,R and K3ǫ,R be the operators with kernels K2ǫ,R (v, v′ ) and K3ǫ,R (v, v′1 ) respectively
given by (2.20). Then
(a) K2ǫ,R : L2 (aMdv) → L2 (aMdv) is compact;
(b) K3ǫ,R : L2 (aMdv) → L2 (aMdv) is compact.
Proof. The normalization (1.9) and a change of variables yields
Z
D−2 Z π2
(2.21)
1=
b̂(cos(θ)) sin(θ)D−2 dθ .
b̂(ω · n) dω = 2S SD−1
0
K2ǫ,R (v, v′ )
We will use this fact to bound the kernels
and K3ǫ,R (v, v′1 ) in such a way that the compactness
of the operators K2ǫ,R and K3ǫ,R will follow from Lemma 2.2.
1 ′2
1 2
′
We first establish assertion (a). By employing the pointwise bound e− 2 |y| −y·w ≤ e 2 |w | on the
integrand in definition of K2ǫ,R (v, v′ ) given by (2.20), we reduce the resulting bounding integral to a
single radial integral over |y|. By introducing the change of variables
dθ
|v′ − v|
d|y|
,
(2.22)
cos(θ) = p
=
,
cos(θ) sin(θ)
|y|
|y|2 + |v′ − v|2
we may use (2.21) to express the resulting pointwise bound as
2|v′ − v|β 21 |w′ |2 D−2 S K2ǫ,R (v, v′ ) ≤ 1{|v|<R} 1{|v′ |<R}
e
a(v) a(v′ )
Z π2
dθ
sin(θ) D−1
cos(θ)−β b̂(cos(θ))
1{cos(θ)>ǫ}
×
cos(θ)
cos(θ) sin(θ)
0
′
β
1
1 ′2
|v − v|
2 |w |
e
.
≤ 1{|v|<R} 1{|v′ |<R}
a(v) a(v′ )
ǫ D+β
Because |w′ | ≤ |v|, we therefore have the pointwise bounds
1
2
|v′ − v|β e 2 R
0≤
≤
.
a(v) a(v′ ) ǫ D+β
Up to a constant factor, this upper bound has the same form as K1 (v, v1 ). Following the proof of
Theorem 2.1, there exists r ∈ (1, 2] such that βr ∈ (−D, ∞) and
1
! 1r
Z
1 2
r
2R
C
r e
ǫ,R
′ r ′ ′
′
< ∞ , for every v ∈ RD ,
|K2 (v, v )| a M dv ≤
D+β
C
ǫ
D
R
a
K2ǫ,R (v, v′ )
where C r and C a are the same constants appearing in the proof of Theorem 2.1. Because the symmetric kernel K2ǫ,R (v, v′ ) thereby satisfies condition (2.3) for some r ∈ (1, 2], upon applying Lemma 2.2
with p = 2 we see that assertion (a) holds.
We now establish (b), which asserts the compactness of K3ǫ,R : L2 (aMdv) → L2 (aMdv) in a similar
1 ′ 2
1 2
′
way. By employing the pointwise bound e− 2 |y| −y·w1 ≤ e 2 |w1 | on the integrand in definition of K3ǫ,R (v, v′1 )
COMPACTNESS OF THE GAIN PARTS OF THE LINEARIZED BOLTZMANN OPERATOR
9
given by (2.20), we reduce the resulting bounding integral to a single radial integral over |z|. By
introducing the change of variables
(2.23)
sin(θ) = p
|v′1 − v|
|v′1 − v|2 + |z|2
−
,
d|z|
dθ
=
,
cos(θ) sin(θ)
|z|
we may use (2.21) to express the resulting pointwise bound as
K3ǫ,R (v, v′1 ) ≤ 1{|v|<R} 1{|v′1 |<R}
×
Z
π
2
0
2|v′1 − v|β 1 |w′ |2 D−2 e 2 1 S a(v) a(v′1 )
sin(θ)−β b̂(cos(θ)) 1{cos(θ)>ǫ} 1{sin(θ)>ǫ}
≤ 1{|v|<R} 1{|v′1 |<R}
dθ
cos(θ) sin(θ)
|v′1 − v|β 1 |w′ |2
1
e 2 1 max{D+β,1} .
′
a(v) a(v1 )
ǫ
Because |w′1 | ≤ |v|, we therefore have the pointwise bounds
1
0 ≤ K3ǫ,R (v, v′1 ) ≤
2
|v′1 − v|β
e2R
.
a(v) a(v′1 ) ǫ max{D+β,1}
This upper bound has the same form as the upper bound we obtained for K2 (v, v′ )ǫ,R . By arguing as
we did to establish (a), we see that assertion (b) also follows from Lemma 2.2.
2.5. Convergence of the Approximating Operators. In this section we show that the remainder
operators K2 − K2ǫ,R and K3 − K3ǫ,R can be made arbitrarily small. Their kernels have the form
ǫ
eǫ,R (v, v′ ) ,
K2 (v, v′ ) − K2ǫ,R (v, v′ ) = K 2 (v, v′ ) + K
2
ǫ
where
eǫ,R (v, v′1 ) ,
K3 (v, v′1 ) − K3ǫ,R (v, v′1 ) = K 3 (v, v′1 ) + K
3


Z
′


|v
−
v|
2|v′ − v|−(D−1)
β − 12 |y|2 −y·w′ 
 p
 1{|v′ −v|≤ǫ|v1 −v|} dy ,
b̂
|v
−
v|
e
=
1
a(v) a(v′ )
y⊥(v′ −v)
|y|2 + |v′ − v|2
−(D−1)
′
eǫ,R (v, v′ ) = 1 − 1{|v|<R} 1{|v′ |<R} 2|v − v|
K
2
a(v) a(v′ )


Z
′


|v
−
v|
β − 12 |y|2 −y·w′ 
 1{|v′ −v|>ǫ|v1 −v|} dy ,
b̂  p
×
|v1 − v| e
y⊥(v′ −v)
|y|2 + |v′ − v|2


Z
β


|z|
2
ǫ
1 2
|v
−
v|

′
1
− 2 |z| −z·w1 

 p
K 3 (v, v′1 ) =
b̂
e
a(v) a(v′1 ) z⊥(v′1 −v) |z|D−1
|z|2 + |v′1 − v|2
× 1 − 1{|v′1 −v|>ǫ|v1 −v|} 1{|z|>ǫ|v1 −v|} dz
2
eǫ,R (v, v′1 ) = 1 − 1{|v|<R} 1{|v′ |<R}
K
3
1
a(v) a(v′1 )


Z

|z|
|v1 − v|β − 12 |z|2 −z·w′ 
 1{|v′ −v|>ǫ|v1 −v|} 1{|z|>ǫ|v1 −v|} dz .
1 b̂  p
×
e

1
D−1
′
2
2
′
|z|
|v1 − v| + |z|
z⊥(v1 −v)
ǫ
K 2 (v, v′ )
(2.24)
10
C. DAVID LEVERMORE AND WEIRAN SUN
ǫ
e ǫ,R , K ǫ , and K
e ǫ,R denote the operators with the kernels K ǫ (v, v′ ), K
eǫ,R (v, v′ ), K ǫ3 (v, v′ ), and
Let K 2 , K
3
2
1
2
3
2
eǫ,R (v, v′ ) respectively. In the following lemma we show that
K
1
3
ǫ ǫ,R ǫ,R ǫ R↑∞
R↑∞
ǫ↓0
ǫ↓0
e 2 −→
e 2 −→
K
K
K 2 B(L2 ) −→ 0 ,
−→
K
0
,
0.
0
,
2
3 B(L )
2
3
B(L )
B(L )
Its statement and proof employs the L2 (aMdv) inner product, which we denote
Z
△
hg, hi =
g h aMdv .
RD
Lemma 2.5. For every η > 0 there exists ǫ0 > 0 such that for every g, f ∈ L2 (aMdv) and ǫ ∈ (0, ǫ0)
we haveD
ǫ E
(i) g, K 2 f ≤ η kgkL2 (aMdv) k f kL2 (aMdv) ,
D
ǫ E
(ii) g, K f ≤ η kgkL2 (aMdv) k f kL2 (aMdv) .
3
2
Moreover,
D for every
ǫ > 0 there exists Rǫ > 0 such that for every g, f ∈ L (aMdv) and R > Rǫ we have
E
e ǫ,R f ≤ η kgkL2 (aMdv) k f kL2 (aMdv) ,
(iii) g, K
2
D
E
ǫ,R
e
(iv) g, K3 f ≤ η kgkL2 (aMdv) k f kL2 (aMdv) .
Proof. (i) Define the nonnegative measure dµ = b(ω, v1 − v) dω M1 dv1 M dv. Then
D
E Z Z Z
g, K ǫ f ≤
|g(v)| | f (v′)| 1{|v′ −v|≤ǫ|v1 −v|} dµ
2
SD−1 ×RD ×RD
(2.25)
≤
ZZZ
! 21
| f (v )| 1{|v′ −v|≤ǫ|v1 −v|} dµ kgkL2 (aMdv) .
′ 2
SD−1 ×RD ×RD
In order to estimate the first factor on the right-hand side of the above inequality, we use the change
of variables (v, v1 ) 7→ (v′ , v′1 ) and the symmetries of the measure dµ to obtain
ZZZ
ZZZ
′ 2
| f (v )| 1{|v′ −v|≤ǫ|v1 −v|} dµ =
| f (v)|2 1{|v′ −v|≤ǫ|v1 −v|} dµ
D−1
D
D
D−1
D
D
S ×R ×R
S ×R ×R
Z
(2.26)
=
| f (v)|2 aǫ (v) M dv ,
RD
where because dµ = |v1 − v|β b̂(ω · n) dω M1 dv1 M dv, we can express aǫ (v) as
ZZ
ǫ
(2.27)
a (v) =
|v1 − v|β b̂(ω · n) 1{|v′ −v|≤ǫ|v1 −v|} dω M1 dv1 .
SD−1 ×RD
By assumption (1.6) that b̂(ω · n) ∈ L1 (dω), we have for every η > 0, there exists ǫ0 > 0 such that
Z
Z
b̂(ω · n) 1{|v′ −v|≤ǫ|v1 −v|} dω =
b̂(ω · n) 1{|ω·n|≤ǫ} dω ≤ η2
for every ǫ ∈ (0, ǫ0 ) .
SD−1
SD−1
We then see from (2.27) that for every ǫ ∈ (0, ǫ0 ) we have
Z
ǫ
2
a (v) ≤ η
|v1 − v|β M1 dv1 = η2 a(v) ,
RD
which by (2.26) implies that
ZZZ
SD−1 ×RD ×RD
| f (v′ )|2 1{|v′ −v|≤ǫ|v1 −v|} dµ ≤ η2 k f kL22(aMdv) .
Upon placing this bound into (2.25), we establish (i).
COMPACTNESS OF THE GAIN PARTS OF THE LINEARIZED BOLTZMANN OPERATOR
11
(ii) The proof of (ii) is similar to that of (i). From (2.24), we see that
ZZZ
D
ǫ E
′
ǫ
ǫ
′
′
(2.28)
f
K
≤
|g(v)|
|
f
(v
)|
1
−
1
1
g,
{|v −v|>ǫ|v1 −v|} {|v1 −v|>ǫ|v1 −v|} dµ ≤ I1 + I2 ,
3
1
SD−1 ×RD ×RD
where
I1ǫ
=
(2.29)
I2ǫ
=
ZZZ
ZZZ
SD−1 ×RD ×RD
SD−1 ×RD ×RD
|g(v)| | f (v′1 )| 1{|v′ −v|≤ǫ|v1 −v|} dµ ,
|g(v)| | f (v′1 )| 1{|v′1 −v|≤ǫ|v1 −v|} dµ .
We use the Schwarz inequality and the symmetries of dµ to estimate I1ǫ as
I1ǫ
ZZZ
≤
Z
=
RD
! 12 Z Z Z
|g(v)| 1|v′ −v|≤ǫ|v1 −v| dµ
2
SD−1 ×RD ×RD
! 21
|g(v)|2 aǫ (v) Mdv k f kL2 (aMdv) ,
SD−1 ×RD ×RD
| f (v′1)|2
! 21
dµ
where aǫ (v) is defined by (2.27). By arguing as in the proof of (i), there exists ǫ1 > 0 such that for
every ǫ ∈ (0, ǫ1), we have aǫ (v) ≤ 41 η2 a(v). Therefore, for every ǫ ∈ (0, ǫ1 ), we have
I1ǫ ≤ 21 η kgkL2 (aMdv) k f kL2 (aMdv) .
(2.30)
The estimate for I2ǫ is similar to that for I1ǫ . The smallness of I2ǫ comes from the fact that there exists
ǫ2 > 0 such that for every ǫ ∈ (0, ǫ2 ) we have
Z
Z
b̂(ω · n) 1{|v′1 −v|≤ǫ|v1 −v|} dω =
b̂(ω · n) 1{1−|ω·n|2 ≤ǫ 2 } dω ≤ 14 η2 ,
SD−1
SD−1
which implies that
I2ǫ ≤ 21 η kgkL2 (aMdv) k f kL2 (aMdv) .
(2.31)
Upon setting ǫ0 = min{ǫ1 , ǫ2} and placing bounds (2.30) and (2.31) into (2.28), we establish (ii).
e ǫ,R f i with ǫ fixed.
(iii) Next, we estimate hg, K
(2.32)
where
D
E Z Z Z
g, K
e ǫ,R f ≤
2
I3ǫ,R
=
(2.33)
I4ǫ,R
=
2
SD−1 ×RD ×RD
ZZZ
ZZZ
|g(v)| | f (v′)| 1 − 1{|v|<R} 1{|v′ |<R} 1{|v′ −v|>ǫ|v1 −v|} dµ ≤ I3ǫ,R + I4ǫ,R ,
SD−1 ×RD ×RD
SD−1 ×RD ×RD
|g(v)| | f (v′ )| 1{|v|≥R} 1{|v′ −v|>ǫ|v1 −v|} dµ ,
|g(v)| | f (v′ )| 1{|v′ |≥R} 1{|v′ −v|>ǫ|v1 −v|} dµ .
By the change of variables (v, v′ ) 7→ (v′ , v), it is clear that bounding I4ǫ,R is equivalent to bounding I3ǫ,R .
Therefore, we will only show how to bound I3ǫ,R .
For every m > 0, we can bound I3ǫ,R as
(2.34)
I3ǫ,R ≤ J1m + J2ǫ,R,m ,
12
C. DAVID LEVERMORE AND WEIRAN SUN
where
J1m
=
(2.35)
J2ǫ,R,m
=
ZZZ
ZZZ
SD−1 ×RD ×RD
SD−1 ×RD ×RD
|g(v)| | f (v′ )| 1{|v1 |≥m} dµ ,
|g(v)| | f (v′ )| 1{|v′ −v|>ǫ|v1 −v|} 1{|v|≥R} 1{|v1 |<m} dµ .
We bound J1m and J2ǫ,R.m separately.
1 2 √
We use the Schwarz inequality, the normalization (1.9), and the fact that M1 1{|v1 |≥m} ≤ e− 4 m M1 to
estimate J1m as
! 21
ZZZ
m
2
J1 ≤
|g(v)| 1{|v1 |≥m} dµ k f kL2 (aMdv)
(2.36)
=
Z
RD
− 41 m2
≤e
SD−1 ×RD ×RD
|g(v)|
Z
2
RD
Z
RD
β
|v1 − v| 1{|v1 |≥m} M1 dv1
|g(v)|2
Z
|v1 − v|β
RD
p
!
M1 dv1
! 12
M dv k f kL2 (aMdv)
!
! 21
M dv k f kL2 (aMdv) .
By Lemma 2.3 there exists a positive constant C1 such that
Z
p
|v1 − v|β M1 dv1 ≤ C12 a(v) .
RD
Therefore, estimate (2.36) yields
1
2
J1m ≤ C1 e− 4 m kgkL2 (aMdv) k f kL2 (aMdv) .
1
2
If we choose m large enough so that C1 e− 4 m < 14 η, then
(2.37)
J1m ≤ 41 η kgkL2 (aMdv) k f kL2 (aMdv) .
Next, we show that for every fixed ǫ and m we can make J2ǫ,R,m arbitrarily small by taking R large
enough. First observe that in the set {|v′ − v| > ǫ|v1 − v|} we have
|v′1 − v1 | = |v′ − v| > ǫ|v1 − v| ,
which implies that |v′1 | > ǫ|v1 − v| − |v1 |. We can thereby choose R large enough so that in the set
{|v′ − v| > ǫ|v1 − v|} ∩ {|v| ≥ R} ∩ {|v1 | < m} we have
|v′1 | > ǫ|v1 − v| − |v1 | ≥ ǫ|v| − (1 + ǫ)|v1 | ≥ ǫR − (1 + ǫ)m > 12 ǫR .
p
(ǫR)2
1 ′ 2
For every such R we use the fact that M1′ M1 1{|v′1 |> ǫR2 } ≤ e− 32 e− 8 |v1 | to estimate J2ǫ,R,m as
ZZZ
√ √ dµ
ǫ,R,m
J2
=
|g(v)| M | f (v′ )| M ′ 1{|v′ −v|>ǫ|v1 −v|} 1{|v|≥R} 1{|v1 |<m} √
D−1
D
D
M′ M
Z Z ZS ×R ×R √
√ dµ
≤
|g(v)| M | f (v′ )| M ′ 1{|v′1 |> ǫR2 } 1{|v1 |<m} √
D−1
D
D
M′ M
(2.38)
Z Z ZS ×R ×R q
√
√ =
|g(v)| M | f (v′ )| M ′ 1{|v′1 |> ǫR2 } 1{|v1 |<m} M1′ M1 b dω dv1 dv
D−1
D
D
S ×R ×R
ZZZ
√ √ (ǫR)2
1 ′ 2
− 32
≤e
|g(v)| M | f (v′ )| M ′ 1{|v1 |<m} e− 8 |v1 | b dω dv1 dv .
SD−1 ×RD ×RD
COMPACTNESS OF THE GAIN PARTS OF THE LINEARIZED BOLTZMANN OPERATOR
13
The Schwarz inequality, the change of variable (v′1 , v′ ) 7→ (v1 , v), and the normalization (1.9) give
! 21
ZZZ
(ǫR)2
J2ǫ,R,m ≤ e− 32
|g(v)|2 M 1{|v1 |<m} b dω dv1 dv
SD−1 ×RD ×RD
×
(2.39)
− (ǫR)
32
ZZZ
Z
2
=e
×
SD−1 ×RD ×RD
RD
Z
RD
′ − 41 |v′1 |2
′ 2
|g(v)|
Z
2
β
RD
| f (v)|
2
Z
| f (v )| M e
! 12
b dω dv1 dv
|v1 − v| 1{|v1 |<m} dv1
β − 14 |v1 |2
RD
|v1 − v| e
dv1
!
! 21
M dv
!
! 12
M dv .
By Lemma 2.3 there exists a positive constant C2 such that
Z
Z
1
2
β
|v1 − v| 1{|v1 |<m} dv1 ≤ C2 a(v) ,
|v1 − v|β e− 4 |v1 | dv1 ≤ C2 a(v) .
RD
RD
Therefore, estimate (2.39) yields
J2ǫ,R,m ≤ C2 e−
2
− (ǫR)
32
If we choose R large enough so that C2 e
(ǫR)2
32
kgkL2 (aMdv) k f kL2 (aMdv) .
< 14 η, then
J2ǫ,R,m ≤ 41 η kgkL2 (aMdv) k f kL2(aMdv) .
(2.40)
By combining estimates (2.37) and (2.40) into (2.34), we bound I3ǫ,R as
I3ǫ,R ≤ 21 η kgkL2 (aMdv) k f kL2 (aMdv) .
(2.41)
The bound on I4ǫ,R is identical to that for I3ǫ,R . It therefore follows from (2.32) that for every ǫ > 0
there exists Rǫ > 0 such that (iii) holds for every R > Rǫ .
(iv) The proof of (iv) closely follows that of (iii). First, we have
D
E Z Z Z
′
g, K
e ǫ,R f ≤
′
|g(v)|
|
f
(v
)|
1
−
1
1
{|v|<R} {|v1 |<R} 1{|v′1 −v|>ǫ|v1 −v|} 1{|z|>ǫ|v1 −v|} dµ
1
3
(2.42)
SD−1 ×RD ×RD
≤ I5ǫ,R + I6ǫ,R ,
where
I5ǫ,R
=
SD−1 ×RD ×RD
(2.43)
I6ǫ,R
ZZZ
=
ZZZ
SD−1 ×RD ×RD
|g(v)| | f (v′1 )| 1{|v|≥R} 1{|v′1 −v|>ǫ|v1 −v|} dµ ,
|g(v)| | f (v′1 )| 1{|v′1 |≥R} 1{|v′1 −v|>ǫ|v1 −v|} dµ .
The change of variables (v, v′1 ) 7→ (v′1 , v) shows that bounding I6ǫ,R is equivalent to bounding I5ǫ,R .
Therefore we only need to bound I5ǫ,R . As we did in (2.34–2.35), this estimate begins with the bound
ZZZ
ǫ,R
I5 ≤
|g(v)| | f (v′1 )| 1{|v1 |≥m} dµ
D−1
D
D
Z ZS Z ×R ×R
+
|g(v)| | f (v′1 )| 1{|v′1 −v|>ǫ|v1 −v|} 1{|v|≥R} 1{|v1 |<m} dµ .
SD−1 ×RD ×RD
The above integrals are bounded as in (2.36–2.37) and (2.38–2.40) respectively. Hence, for any ǫ > 0,
there exists Rǫ > 0 such that (iv) holds for every R > Rǫ . To avoid repetition, we omit these details.
14
C. DAVID LEVERMORE AND WEIRAN SUN
We have now shown that all the remainders associated with the approximations of K2 and K3 can
be made arbitrarily small. This completes the proof of Lemma 2.5, and thereby finishes the proof of
Theorem 2.2. By the remark at the end of Section 2.3, this also establishes our Main Theorem.
Remark. The Main Theorem remains true if one replaces the assumption that the collision kernel b
has the factored form (1.5-1.6) with the assumption that b satisfies the bounds
b(ω, v1 − v) ≤ b̂(ω · n) |v1 − v|α + |v1 − v|β ,
a(v) ≥ â (1 + |v|)β ,
for some β > α > −D, b̂ ∈ L1 (dω) and â > 0, where the attenuation coefficient a is given by (1.11).
Because we do not know of any additional physical collision kernels (ones derived from a classical
intermolecular potential) that would be included by this generalization, we do not present its proof.
The proof is similar to the one given above, only longer because there are more terms to estimate.
R
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(C.D. Levermore) D  M & I  S  T, U  M,
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