Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular

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Spectral gap and coercivity estimates for linearized
Boltzmann collision operators without angular cutoff
Clément Mouhot, Robert Strain
To cite this version:
Clément Mouhot, Robert Strain. Spectral gap and coercivity estimates for linearized Boltzmann
collision operators without angular cutoff. Journal de Mathématiques Pures et Appliquées, 2007, 87
(5), pp.515-535. ￿10.1016/j.matpur.2007.03.003￿. ￿hal-00086958v2￿
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SPECTRAL GAP AND COERCIVITY ESTIMATES FOR
LINEARIZED BOLTZMANN COLLISION OPERATORS WITHOUT
ANGULAR CUTOFF
CLE´MENT MOUHOT & ROBERT M. STRAIN
Abstract. In this paper we prove new constructive coercivity estimates for the
Boltzmanncollisionoperatorwithoutcutoff,thatisforlong-rangeinteractions. In
particular we give a generalizedsufficient condition for the existence of a spectral
gapwhichinvolvesboththegrowthbehaviorofthecollisionkernelatlargerelative
velocitiesandits singularbehavioratgrazingandfrontalcollisions. Itprovidesin
particular existence of a spectralgap andestimates on it for interactionsderiving
from the hard potentials φ(r) = r−(s−1), s 5 or the so-called moderately soft
potentials φ(r) = r−(s−1), 3<s < 5, (withou≥t angular cutoff). In particular this
paper recovers (by constructive means), improves and extends previous results of
6
Pao[46]. WealsoobtainconstructivecoercivityestimatesfortheLandaucollision
0
0 operator for the optimal coercivity norm pointed out in [34] and we formulate a
2 conjecture about a unified necessary and sufficient condition for the existence of
a spectral gap for Boltzmann and Landau linearized collision operators.
l
u
J

1
Mathematics Subject Classification (2000): 76P05Rarefiedgasflows, Boltz-
2
mann equation [See also 82B40, 82C40, 82D05].
-
Keywords: coercivity estimates, linearized Boltzmann operator, linearized Lan-
2
dau operator, quantitative, long-range interaction, non-cutoff, soft potentials, spec-
n
o tral gap.
i
s
r
e
v

, Contents
8
5
1. Introduction 2
9
6 2. A technical estimate on L 11
B
8
3. Proof of Theorem 1.1 for ε > 0 16
0
0 4. Proof of Theorem 1.1 for ε = 0 20
0 5. Proof of Theorem 1.2 23
-
d References 26
s
c
c
Date: July 21, 2006.
1
2 CLE´MENT MOUHOT& ROBERTM. STRAIN
1. Introduction
This paper deals with the question of spectral gap and coercivity estimates for the
Boltzmann and Landau integro-differential collision operators. This is motivated
by the question of obtaining explicit constant in the recent new energy methods
in [33, 34, 35, 36, 37, 47, 48, 45, 49, 50]. The starting points are the recent new
constructive toolsofworks [11,44]. Aswe shall explainthis work recovers, improves,
makes explicit and clarifies the works of Pao [46] three decades ago. Before entering
into the details, let us introduce the mathematical objects.
1.1. The Boltzmann equation. The Boltzmann equation (Cf. [17, 19]) describes
the behavior of a dilute gas when the only interactions taken into account are binary
collisions. It reads in some space domain Ω RN:
⊂
∂f
(1.1) +v f = Q (f,f), x Ω, v RN, t 0,
x B
∂t ·∇ ∈ ∈ ≥
where N 2 is the dimension. In equation (1.1), Q is the quadratic Boltzmann
≥
collision operator, defined by
Q (f,f) = f(v′)f(v′) f(v)f(v ) B( v v ,cosθ)dv dσ
B ∗ − ∗ | − ∗| ∗
ZRN×SN−1
(cid:2) (cid:3)
in the so-called “σ-representation” (see [55, Chapter 1, Section 4.6]. In this repre-
sentation the parametrization of the collision is
v +v v v v +v v v
v′ = ∗ + | − ∗|σ, v′ = ∗ | − ∗|σ, σ SN−1
2 2 ∗ 2 − 2 ∈
and the deviation angle is defined by cosθ = (v′ v′) (v v)/ v v 2. Up to a
∗ − · ∗ − | ∗ − |
jacobian factor 2N−2 sinN−2(θ/2) (see again [55, Chapter 1, Section 4.6]), one can
also define the alternative so-called “ω-representation”
Q (f,f) = f(v′)f(v′) f(v)f(v ) 2N−2 sinN−2(θ/2)B( v v ,cosθ)dv dω
B ∗ − ∗ | − ∗| ∗
ZRN×SN−1
(cid:2) (cid:3)
with the formula
v′ = v +((v v) ω)ω, v′ = v ((v v) ω)ω, ω SN−1.
∗ − · ∗ ∗ − ∗ − · ∈
Remark that this operator is local in x, and therefore all its functional study in the
sequel shall be done with no space variable x.
Boltzmann’scollisionoperatorhasthefundamentalpropertiesofconserving mass,
momentum and energy
Q (f,f)φ(v)dv = 0, φ(v) = 1,v, v 2
B
| |
RN
Z
COERCIVITY ESTIMATES FOR NON-CUTOFF BOLTZMANN OPERATORS 3
and satisfies well-known Boltzmann’s H theorem
d
f logf dv = Q(f,f)log(f)dv 0.
−dt − ≥
RN RN
Z Z
The functional f logf is the entropy of the solution. Boltzmann’s H theorem
−
implies that at some point x Ω, any equilibrium distribution function, i.e., any
R ∈
function which is a maximum of the entropy, has the form of a locally Maxwellian
distribution
ρ u v 2
(1.2) M(ρ,u,T)(v) = exp | − | ,
(2πT)N/2 − 2T
(cid:18) (cid:19)
where ρ, u, T are the density, mean velocity and temperature of the gas at the point
x, defined by
1 1
(1.3) ρ = f(v)dv, u = vf(v)dv, T = u v 2f(v)dv.
ρ Nρ | − |
RN RN RN
Z Z Z
For further details on the physical background and derivation of the Boltzmann
equation we refer to [17, 19] and [55].
We consider collision kernels of the form
(1.4) B( v v ,cosθ) = v v γb(cosθ), γ ( N,+ )
∗ ∗
| − | | − | ∈ − ∞
with
(1.5) b(cosθ) b∗(θ) (sinθ/2)−(N−1)−α, α [0,2),
θ∼0
∼ ∈
where b∗(θ) is non-negative, bounded and non-zero near θ 0. When α 0 the
∼ ≥
angular singularity is not integrable, the operator is said to be non-cutoff.
An important remark on the quadratic collision operator is: by using the change
of variable σ σ (which changes θ into π θ) one can replace b by
→ − −
1
˜
b(cosθ) = 1 [b(cosθ)+b(cos(π θ))]
θ∈[0,π/2]
2 −
(where 1 denotes the usual characteristic function of a set E. Therefore we shall
E
consider without restriction in the sequel that b satisfies the singularity condi-
tion (1.5) above, and is 0 on [π/2,π].
For particles interacting according to some spherical repulsive potential
φ(r) = r−(s−1), s [2,+ ),
∈ ∞
the collision kernel is not explicit but it can be shown that for the dimension N = 3,
B satisfies (1.4) with γ = (s 5)/(s 1) and (1.5) with α = 2/(s 1) (see for
− − −
instance [17, 19, 5]). Therefore as a convention one shall speak in (1.4) of hard
potentials when s 5, maxwellian potential when s = 5, soft potentials when 2 <
≥
4 CLE´MENT MOUHOT& ROBERTM. STRAIN
s < 5. Moreover among soft potentials we shall denote by moderately soft potentials
the case when 3 s < 0.
≤
Let us mention also for the sake of completeness that in the case of contact inter-
actions (the so-called hard spheres model), the collision kernel is locally integrable
and explicit: it takes the form (in dimension N = 3) B( v v ,cosθ) = v v (up
∗ ∗
| − | | − |
to a normalization constant).
1.2. Linearized Boltzmann collision operator. We denote by
µ = µ(v) := (2π)−N/2e−|v|2/2
the normalized unique equilibrium with mass 1, momentum 0 and temperature 1.
We consider fluctuations around this equilibrium of the form
f = µ+µ1/2g
which results the following linearized collision operator (note the sign convention):
L (g) = µ−1/2 Q (µ,µ1/2g)+Q (µ1/2g,µ) .
B B B
−
For the sake of simplification we shall always consider in the sequel the linearized
(cid:2) (cid:3)
collision operator around this normalized equilibrium. This is no restriction: a
detailed discussion of the dependence of the spectral gap and coercivity estimates
on this operator in terms of the mass, mean velocity and temperature can be found
in [44].
It is well-known (see [17] for instance) that L (acting in the velocity space) is an
B
unbounded symmetric operator on L2, such that its Dirichlet form satisfies
D (g) := L g,g 0,
B B
h i ≥
and that D (g) = 0 if and only if g = Pg where
B
Pg = a+b v +c v 2 µ1/2
· | |
(with a,c R and b RN) is the(cid:0) L2 orthogonal(cid:1)projection onto the space of the
∈ ∈
so-called “collisional invariants”
Span µ1/2,v µ1/2,...,v µ1/2, v 2µ1/2 .
1 N
| |
1.3. The Landau equatio(cid:8)n. The Landau equation was wr(cid:9)itten by Landau in 1936
([42]). It is similar to the Boltzmann equation (1.1) but with a different collision
operatorQ . Indeed in thecase of long-distance interactions, collisions occur mostly
L
for very small θ. When all collisions become concentrated on θ = 0, one obtains by
the so-called grazing collision limit asymptotic (see for instance [10, 21, 22, 52, 7]
for a detailed discussion) the Landau collision operator
(1.6) Q (f,f) = a(v v )[f ( f) f( f) ]dv ,
L ∗ ∗ ∗ ∗
∇· − ∇ − ∇
(cid:18)ZRN (cid:19)
COERCIVITY ESTIMATES FOR NON-CUTOFF BOLTZMANN OPERATORS 5
where = (∂ ,...,∂ ), f = f(v), f = f(v ) etc. The non-negative symmetric
∇ v1 vN ∗ ∗
N N matrix a = a(z) is
×
z z
(1.7) a (z) = δ i j z γ+2.
ij ij
− z 2 | |
(cid:26) | | (cid:27)
This operator is used for instance in plasma physics in the case of a Coulomb po-
tential where Φ( z ) = z −3 in dimension N = 3 (for more details see [55, Chapter 1,
| | | |
Section 1.7] and the references therein). Indeed let us mention that for Coulomb
interactions the Boltzmann collision operator does not make sense anymore (see [54,
Annex I, Appendix]). By analogy with the Boltzmann’s case, and even if the phys-
ical meaning of such collision kernel is not clear for the Landau collision operator,
one shall speak of hard potentials when γ > 0, maxwellian potentials when γ = 0,
and soft potentials when γ < 0. Moreover among soft potentials we shall denote by
moderately soft potentials the case when 2 γ < 0.
− ≤
1.4. Linearized Landau collision operator. Consider again fluctuation around
the equilibrium of the form
f = µ+µ1/2g.
Then the linearized Landau collision operator is defined by
L g = µ−1/2 Q (µ,µ1/2g)+Q (µ1/2g,µ) .
L L L
−
It was proved in [34, 39, 20] that that L (acting on the velocity space) is an
(cid:2) L (cid:3)
unbounded symmetric operator on L2, such that its Dirichlet form satisfies
D (g) := L g,g 0
L L
h i ≥
and D (g) = 0 if and only if g = Pg.
L
1.5. Previous results and motivations. The study of spectral gap estimates for
the linearized Boltzmann and Landau collision operators has a long history, see
for instance [38, 57, 15, 31, 32, 58, 14, 12] and we refer to [11, 44] for a more
detailed discussion of it. Let us just emphasize some references directly related
to this paper. For linearized Boltzmann collision operators with locally integrable
collision kernel (which is satisfied for instance under the so-called Grad’s angular
cutoff), the existence of a spectral gap is equivalent to γ 0 (see [32, 14, 19] for
≥
non-constructive proofs). The question of obtaining polynomial or exponential rate
of relaxation of the form O(e−λtβ) for 0 < β < 1 and for soft potentials with cutoff
was studied in [14, 51, 48, 49], and explicit spectral gap and generalized coercivity
estimates were obtained in [11, 44].
For the sake of completeness let us also mention that Cercignani introduced an
alternative “potential cutoff” to Grad’s angular cutoff in [16] for which results on
the spectrum of the linearized collision operator were obtained in [18].
6 CLE´MENT MOUHOT& ROBERTM. STRAIN
Whilethecutofftheoryofthelinearizedcollisionoperator(withoutspacevariable)
is quite developed, for non locally integrable collision kernels, there are quite few
works. First for the Boltzmann collision operator at the linearized level there are
essentially the two papers by Pao [46]: he proved that the resolvent is compact for
inverse power-laws interaction potentials with s > 3 (in our notation). His proof
was using tools from pseudo-differential operators theory: the idea was to compute
the symbol of the linearized operator L , and then to search for adequate condition
B
for L + C (for some constant C > 0) to be invertible with compact inverse, by
B
reducing to the Maxwell case and decomposing along spherical harmonics. This
fundamental work was critically reviewed in particular for the use of techniques
from pseudo-differential operator theory in [40].
The critics of [40] on [46], together with the fact that the proof in [46] was highly
technical, were probably the reasons for which the results in [46] were not consid-
ered as completely reliable, and this paper was somehow forgotten in the following
decades.
One of our goal in this work is to clarify the discussion about the validity of the
results [46], by recovering and improving strongly these results. Moreover we replace
the question treated in [46] into the unified framework of quantitative coercivity
estimates for the linearized collision operators, which is related to the works [11, 44]
(for the Landau collision operator at the linearized level, let us mention the key
works [20, 34] and the approach developed in [44] which shall be used here). A
general motivation for this framework is to provide explicit rate of convergence to
equilibrium in the energy methods which have emerged recently in the collisional
kinetic theory: see [33, 34, 35, 36, 37, 47, 48, 45, 49, 50].
At the non-linear level, let us mention some breakthroughs related to non-cutoff
interactions: in the spatially homogeneous Cauchy theory: [8, 9, 28, 52, 30], in
the grazing collision limit from Boltzmann to Landau equation: [10, 22, 21], in the
parabolic-like regularizing property in the velocity variable: [23, 24, 25, 41, 53, 5],
in the construction of renormalized solutions (in the spirit of [27]): [6, 7].
InthespiritofthemethodofPao,asystematic approachbyFouriertransformand
pseudo-differential operators for the study of linear and non-linear Boltzmann non-
cutoffcollisionoperatorshasbeendeveloped byAlexandre [1,3,4]. Theapplications
of this technical tools are not clear at now, but they seem to have contributed to
the understanding of the Boltzmann equation in two ways: in the development of
a renormalization process adapted to the non-cutoff case ([2, 6]), and by providing
some sharp estimates from above on the collision operator.
1.6. Main theorems. First we state our coercivity results for the linearized Boltz-
mann collision operator:
Theorem 1.1. Let B be a collision kernel satisfying (1.4,1.5). Then
COERCIVITY ESTIMATES FOR NON-CUTOFF BOLTZMANN OPERATORS 7
For any ε > 0 there is a constant C , constructive from our proof and
B,ε
•
depending on B and ε, such that the Dirichlet form D of the linearized
B
Boltzmann collision operator associated to B satisfies
D (g) C [g Pg](1+ v 2)(γ+α−ε)/4 2 .
B ≥ B,ε − | | L2(RN)
There is a constant C (cid:13)(obtained by non-construc(cid:13)tive means in our proof)
• B,(cid:13)0 (cid:13)
such that
D (g) C [g Pg](1+ v 2)(γ+α)/4 2 .
B ≥ B,0 − | | L2(RN)
Second we state the construc(cid:13)tive version of the coerci(cid:13)vity result in [34] for the
(cid:13) (cid:13)
linearized Landau collision operator:
Theorem 1.2. Let γ [ N,+ ). Then there is some constant C , constructive
γ
∈ − ∞
from our proof and depending on γ, such that the Dirichlet form D of the associated
L
linearized Landau collision operator satisfies:
2
D (g) C [g Pg] .
L ≥ γ − σ
where is the following anisotropic n(cid:13)orm: (cid:13)
σ
k·k (cid:13) (cid:13)
2
g 2 := (1+ v 2)γ/4Π g
k kσ | | v∇v L2(RN)
(cid:13) (cid:13)
(cid:13)(cid:13) + (1+ v 2)(γ+2(cid:13)(cid:13))/4[I Πv] vg 2 + (1+ v 2)(γ+2)/4g 2
| | − ∇ L2(RN) | | L2(RN)
(cid:13) (cid:13) (cid:13) (cid:13)
with (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
v v
Π g = g .
v v v
∇ v ·∇ v
(cid:18)| | (cid:19) | |
1.7. Consequences on the spectrum. In a previous paper the first author has
proved:
Theorem 1.3 ([44]). Let B be a collision kernel satisfying (1.4,1.5) with α > 0.
Then there is a constant C (constructive from the proof and depending on B) such
B
that the Dirichlet form D of the linearized Boltzmann collision operator associated
B
to B satisfies
2
D (g) C [g Pg] .
B ≥ B − Hlαo/c2(RN)
(cid:13) (cid:13)
Therefore as soon as γ + α > 0 an(cid:13)d α > 0(cid:13)it is straightforward by gathering
Theorems 1.1 and 1.3 that the resolvent of L is compact and so that the spectrum
B
is purely discrete and the eigenvectors basis is complete in L2. In dimension N =
3 for interactions deriving from inverse-power laws potentials φ(r) = r−(s−1), the
corresponding condition on s is: (s 5)/(s 1) + 2/(s 1) > 0, that is s > 3.
− − −
8 CLE´MENT MOUHOT& ROBERTM. STRAIN
Hence we can recover completely the results of Pao [46], by constructive means and
without tools of pseudo-differential operators.
Moreover Theorem 1.1 yields the first estimates on this spectral gap in the case
γ+α > 0 and answers to the question of the existence of a spectral gap in the limit
case γ +α = 0.
For the linearized Landau collision operator, it had been already shown in [34]
that
2 2
C [g Pg] D (g) C [g Pg]
1 − σ ≤ L ≤ 2 − σ
by non-constructive me(cid:13)ans (in fa(cid:13)ct the constant C(cid:13) could (cid:13)be made explicit from
2
(cid:13) (cid:13) (cid:13) (cid:13)
the proof whereas the constant C was obtained by a compactness argument). This
1
implies straightforwardly that the resolvent is compact as soon as γ > 2 (which
−
implies in this case that the spectrum is purely discrete and the eigenvectors basis
is complete in L2). Moreover spectral gap exists if and only if γ 2.
≥ −
Inthis context, Theorem 1.2 provides thefirst estimates onit for thiswhole region
(as well as an explicit version of the coercivity estimate).
1.8. Conjecture and perspectives. For the linearized Boltzmann collision op-
erator with angular cutoff, the existence of a spectral gap is equivalent to γ 0
≥
(see [32, 14, 19]). This situation can be loosely thought of as part of the limit case
“α = 0”. For the linearized Landau collision operator, the existence of a spectral
gap is equivalent to γ 2 as discussed above. This situation can be thought of as
≥ −
the limit case “α = 2”.
From the necessary and sufficient condition in these two limit cases, and the
sufficient condition of Theorem 1.1 in the intermediate cases, it is natural to state
the following conjecture:
Conjecture. Let B be a collision kernel satisfying (1.4,1.5) with γ ( N,+ )
∈ − ∞
and α [0,2). Then the linearized Boltzmann collision operator associated to B
∈
admits a spectral gap if and only if γ+α 0. Moreover this statement is still valid
≥
if one includes formally the case of angular cutoff in “α = 0”, and add the linearized
Landau collision operator as the limit case “α = 2”.
Letusremarkthatinthelimitcases ofthelinearizedBoltzmanncollisionoperator
with angular cutoff and the linearized Landau collision operator the conjecture is
proved as discussed before. And Theorem 1.1 proves that the condition γ +α 0
≥
is sufficient for the existence of a spectral gap.
Here are now some open questions linked with this conjecture:
(1) In order to show that the condition is necessary, is it possible to find some
particular sequence of functions g contradicting the spectral gap estimate
n
when γ +α < 0?
COERCIVITY ESTIMATES FOR NON-CUTOFF BOLTZMANN OPERATORS 9
(2) A related and more difficult question is to identify and understand the coer-
civity norm for the non-cutoff linearized Boltzmann collision operator. This
norm is likely to be intricate and anisotropic for the weight on the fractional
derivative part (as for the linearized Landau collision operator). This point
is related to the open question of constructing smooth perturbative solutions
near equilibrium for the Boltzmann equation without angular cutoff (see the
discussion in [45]).
(3) The spectrum is purely discrete in the case γ + α > 0, α > 0 (and we
conjecture that this is also true on the non-cutoff borderline case γ+α > 0,
α = 0). In the angular cutoff case the geometry of the spectrum can be
obtained by perturbation arguments (Cf. [32, 14, 19]). Hence there remains
the region γ+α 0 for which there is no information on the geometry of the
≤
spectrum at now (including the linearized Landau collision operator when
α = 2 and γ < 2).
−
Finally we shall put in perspective this conjecture with the non-linear case. A
non-linear analogous of a spectral gap is provided by Cercignani’s conjecture:
(f) C H(f M )
B f f
D ≥ |
where
f
H(f M ) = H(f) H(M ) := f log dv
f f
| − M
RN f
Z
is the relative entropy between f and its associated Maxwellian equilibrium distri-
bution, and
(f) := Q (f,f) logf dv ( 0)
B B
D − ≥
RN
Z
denotes the entropy production functional (for more details we refer to [56]). It was
shown in [26] that an equivalent version of this conjecture is true for the Landau
collision operator when γ 0, and it was shown in [56] that this conjecture for the
≥
Boltzmann collision operator for B(v v ,cosθ) K (1 + v v )γ with γ 2.
∗ B ∗
− ≥ | − | ≥
From this two limit cases Villani “interpolated” the following conjecture:
Conjecture (Villani [56]). Cercignani’s conjecture is satisfied if and only if
γ +α 2.
≥
(The other part of the conjecture on the question of the existence of a spectral
gap for the linearized collision operator was not correct –as shown by Theorem 1.1–,
which is likely to be explained by the confusion in the field about Pao’s results).
Obviously this conjecture leaves more room than the one we made on the spectral
gap, since it is not stated in which functional space the distribution f lives, but from
the results in [56] and [13], it is reasonable to try to prove that Villani’s conjecture
holds in spaces of nonnegative functions in L1(1+ v q) Hk for any q,k 0.
| | ∩ ≥