Hypoelliptic Estimates for a Linear Model of the Boltzmann Equation without Angular Cutoff

40 Pages·2011·0.46 MB·English

HYPOELLIPTIC ESTIMATES FOR A LINEAR MODEL OF THE
BOLTZMANN EQUATION WITHOUT ANGULAR CUTOFF
NICOLASLERNER,YOSHINORIMORIMOTO,KARELPRAVDA-STAROV
1 Abstract. In this paper, we establish optimal hypoelliptic estimates for a class of
1
kinetic equations, which are simplified linear models for the spatially inhomogeneous
0
2 Boltzmann equation without angular cutoff.

y
a 1. Introduction
M
In this paper, we study the following kinetic operator

4 (1) P =∂ +v ∂ +a(t,x,v)( ∆˜ )σ, t R, x,v Rn,
t x v
1 · − ∈ ∈
where 0 < σ < 1 is a constant parameter, x y stands for the standard dot-product

] on Rn and a denotes a C∞(R2n+1) function s·atisfying
P b
A (2) a >0, (t,x,v) R2n+1, a(t,x,v) a >0.
0 0
∃ ∀ ∈ ≥
.
h Here the notation C∞(R2n+1) stands for the space of C∞(R2n+1) functions whose
t b
a derivatives of any order are bounded on R2n+1 and ( ∆˜ )σ is the Fourier multiplier
v
m −
with symbol
[
(3) F(η)= η 2σw(η)+ η 2 1 w(η) , η Rn,
| | | | − ∈
2
v with being the Euclidean norm, w a C(cid:0)∞(Rn) fu(cid:1)nction satisfying 0 w 1,
|·| ≤ ≤
5 w(η) = 1 if η 2, w(η) = 0 if η 1; and Dt = (2πi)−1∂t, Dx = (2πi)−1∂x,
| | ≥ | | ≤
1 D =(2πi)−1∂ .
v v
9 When σ = 1, this operator reduces to the so-called Vlasov-Fokker-Planck opera-
4
tor, whereas when 0 < σ < 1, it stands for a simplified linear model of the spatially
.
2 inhomogeneousBoltzmannequationwithoutangularcutoff(see the endofthis intro-
1
duction together with section 5.1 in appendix). This is our motivation for studying
0
theregularizingpropertiesofthislinearmodelandestablishinghypoellipticestimates
1
: with optimal loss of derivatives with respect to the exponent 0 < σ < 1 of the frac-
v tional Laplacian ( ∆˜ )σ. This linear model has the familiar structure
i − v
X
Transport part in the (t,x) variables + Elliptic part in the v variable
r
a
and it is easy to get the regularity in the v variable. The non-commutation of the
transport part (the skew-adjoint part) with the self-adjoint elliptic part ( ∆˜ )σ will
v
−
produce the regularizing effect in the x variable.
Regardingthis linear model, the existence and the C∞ regularityfor the solutions
oftheCauchyproblemtolinearandsemi-linearequationsassociatedwiththeoperator
(1) were proved in [31]. H. Chen, W.-X. Li and C.-J. Xu have also recently studied
itsGevreyhypoellipticity. Morespecifically,theyestablishedin[16](Proposition2.1)
the following hypoelliptic estimate. Let P be the operator defined in (1) and K a
2000 Mathematics Subject Classification. 35H10;35B65;82C40.
Key words and phrases. Kinetic equations, Boltzmann equation without angular cutoff, Regu-
larity,hypoellipticestimates,hypoellipticity,Wickquantization.
1
2 NICOLASLERNER,YOSHINORIMORIMOTO,KARELPRAVDA-STAROV
compact subset of R2n+1. For any s 0, there exists a positive constant C > 0
K,s
≥
such that for any u C∞(K),
∈ 0
(4) u C Pu + u ,
s+δ K,s s s
k k ≤ k k k k
with s standing for the Hs(R2n+1) So(cid:0)bolev norm an(cid:1)d
k·k
σ σ 1
(5) δ =max , >0.
4 2 − 6
The notation C∞(K) stands for the se(cid:16)t of C∞(R(cid:17)2n+1) functions with support in K.
0 0
This hypoelliptic estimate with loss of
max(2σ,1) δ >0,
−
derivatives is then a key instrumental ingredientfor their investigationofthe Gevrey
hypoellipticity of the operator P. However,this hypoelliptic estimate (4) is not opti-
mal. Inthepresentwork,weareinterestedinestablishinghypoellipticestimateswith
optimal loss of derivatives with respect to the exponent 0 < σ < 1 of the fractional
Laplacian ( ∆˜ )σ. More specifically, we shall show by using different microlocal
v
−
techniques that the operator P is hypoelliptic with a loss of
max(4σ2,1)
>0,
(2σ+1)
derivatives,thatis,thatthehypoellipticestimates(4)holdwiththenewpositivegain
2σ
(6) δ = >0,
2σ+1
which improves for any 0<σ <1 the gain provided by (5),
2σ σ σ 1
>max , .
2σ+1 4 2 − 6
(cid:16) (cid:17)
Our main result reads as follows.
Theorem1. LetP betheoperatordefinedin (1),K beacompactsubsetofR2n+1 and
s R. Then, there exists a positive constant C >0 such that for all u C∞(K),
∈ K,s ∈ 0
(7) (1+|Dt|2σ2+σ1 +|Dx|2σ2+σ1 +|Dv|2σ)u s ≤CK,s kPuks+kuks ,
with s(cid:13)being the Hs(R2n+1) Sobolev norm. (cid:13) (cid:0) (cid:1)
k·k (cid:13) (cid:13)
The hypoelliptic estimates (7) are optimal in term of the exponents of derivative
terms appearing in their left-hand-side, namely, 2σ/(2σ + 1) for the regularity in
the time and space variables and 2σ for the regularity in the velocity variable. The
exponent2σ for the regularityinthe velocity variablehas indeedthe same growthas
thediffusivepartofthekineticoperator(1). Regardingtheoptimalityoftheexponent
2σ/(2σ+1) for the regularity in the time and space variables, we first notice that
Theorem 1 is a natural extension for the values of the parameter 0 < σ < 1 of the
well-known optimal hypoelliptic estimates with loss of 4/3 derivatives known for the
Vlasov-Fokker-Planckoperator, case σ =1, (see [12, 15, 35]),
(1+ D 2/3+ D 2/3+ D 2)u C Pu + u .
| t| | x| | v| s ≤ K,s k ks k ks
Seealso[21]f(cid:13)orgeneralmicrolocalmethodsfor(cid:13)provingo(cid:0)ptimalhypoelli(cid:1)pticestimates
(cid:13) (cid:13)
with loss of 4/3 derivatives for certain classes of kinetic equations.
Wededucetheoptimalityoftheexponent2σ/(2σ+1)fortheregularityinthetime
and space variables by using a simple scaling argument. Indeed, let us consider the
specific case when the function a appearing in the definition of the kinetic operator
HYPOELLIPTIC ESTIMATES FOR A LINEAR MODEL OF THE BOLTZMANN EQUATION 3
P is constant and assume that the hypoelliptic estimates (4) hold for a positive gain
δ >0. It follows that the estimate
(D δ+ D δ+ D δ)u C iD u+iv D u+ D 2σu + u ,
t x v L2 t x v L2 L2
k | | | | | | k ≤ k · | | k k k
holds for any u ∈ C0∞(R2n+1) with su(cid:0)pport in the closed unit Euclidean ball(cid:1)B1 =
B(0,1)inR2n+1. ThesymplecticinvarianceoftheWeylquantization(Theorem18.5.9
in [22]) shows that for any λ 1,
≥
Tλ−1 iDt+iv·Dx+|Dv|2σ Tλ =λ2σ2+σ1 iDt+iv·Dx+|Dv|2σ
and (cid:0) (cid:1) (cid:0) (cid:1)
Tλ−1 |Dt|δ+|Dx|δ+|Dv|δ Tλ =λ22σσ+δ1|Dt|δ+λδ|Dx|δ+λ2σδ+1|Dv|δ,
where Tλ sta(cid:0)nds for the following u(cid:1)nitary transformation on L2(R2n+1),
Tλu(t,x,v)=λ2(22σσ++n1)+n2u(λ2σ2+σ1t,λx,λ2σ1+1v).
Recalling that λ 1, we notice that the C∞(R) function T u is supported in B if
the C∞(R) functi≥on u is supportedin B . B0y applyingthe pλreviousa prioriestim1ate
0 1
to functions T u, we obtain that
λ
(D δ+ D δ+ D δ)T u C (iD +iv D + D 2σ)T u + T u .
t x v λ L2 t x v λ L2 λ L2
k | | | | | | k ≤ k · | | k k k
By using that T−1 is a unitary transfo(cid:0)rmationon L2(R2n+1), we deduce that for a(cid:1)ny
λ
λ 1,
≥
(λ22σσ+δ1 Dt δ+λδ Dx δ+λ2σδ+1 Dv δ)u L2
k | | | | | | k
C λ2σ2+σ1 (iDt+iv Dx+ Dv 2σ)u L2 + u L2 .
≤ k · | | k k k
This estimate may hold for any λ (cid:0)1 only if (cid:1)
≥
2σ
δ .
≤ 2σ+1
Thisscalingargumentshowsthatthepositivegain2σ/(2σ+1)>0inthehypoelliptic
estimates (4) is the optimal possible one.
As a consequence of these optimal hypoelliptic estimates, we obtain the following
result where we write
f Hs (R2n+1),
∈ loc,(t0,x0) t,x,v
if there exists an open neighborhood U of the point (t ,x ) in Rn+1 such that
0 0
φ(t,x)f Hs(R2n+1) for any φ C∞(U).
∈ t,x,v ∈ 0
Corollary 1. Let P be the operator defined in (1) and N N. If u H (R2n+1)
∈ ∈ −N t,x,v
and Pu Hs (R2n+1) with s 0, then there exists an integer k 1 such that
∈ loc,(t0,x0) t,x,v ≥ ≥
u Hs+2σ2σ+1 (R2n+1),
v k ∈ loc,(t0,x0) t,x,v
h i
where v = (1+ v 2)1/2. In particular, if u H (R2n+1) and Pu H∞(R2n+1)
−N
then uh iC∞(R2n+|1|). ∈ ∈
∈
Corollary 1 allows to recover the C∞ hypoellipticity proved in [31] (Theorem 1.2)
with now optimal loss of derivatives. Notice that the equation (1) is not a classical
pseudodifferential equation. Indeed, the coefficient v in (1) is unbounded and the
fractional Laplacian( ∆˜ )σ is a classicalpseudo-differential operator in the velocity
v
−
variable v but not in all the variables t,x,v. This accounts for parts of the difficul-
ties encountered when studying this kinetic operator in particular when using cutoff
4 NICOLASLERNER,YOSHINORIMORIMOTO,KARELPRAVDA-STAROV
functions in the velocity variable. This also accounts for the weight v −k appearing
h i
in the statement of Corollary 1. Notice that the proof of this result is constructive
and that one may derive anexplicit (possibly not sharp)bound onthe integer k 1.
≥
The proof of Theorem 1 is relying on some microlocal techniques developed by
N. Lerner for proving energy estimates while using the Wick quantization [26]. Let
us mention that some of these techniques were already used in the work [36]. The
strategyforprovingTheorem1isthefollowing. Wewanttoconsiderthisequationas
an evolution equation along the characteristic curves of ∂ +v ∂ ; for that purpose,
t x
·
we straighten this vector field and get the normal form
iD +a(t,D ,D +tD )F(x tx ).
t x2 x1 x2 2− 1
This normal form suggests to derive some a priori estimates for the one-dimensional
first-order differential operator
iD +a(t,ξ ,ξ +tξ )F(x tx ),
t 2 1 2 2 1
−
depending on the parameters x ,x ,ξ ,ξ Rn. We then deduce from those a priori
1 2 1 2
∈
estimates some a priori estimates for the operator
Wick
iD + a(t,ξ ,ξ +tξ )F(x tx ) ,
t 2 1 2 2 1
−
defined by using the Wick q(cid:2)uantization. As a last step(cid:3), we need to control some
remainder terms in order to come back to the standard quantization and derive a
priori estimates for the originaloperator
iD +a(t,D ,D +tD )F(x tx ).
t x2 x1 x2 2− 1
For the sake of completeness and to keep the paper essentially self-contained, the
definitionandallthe mainfeaturesoftheWick quantizationarerecalledinappendix
(Section 5.2). Next section is devoted to the proof of a key hypoelliptic estimate
(Proposition 1) which is the core of the present work. This key estimate is then the
main ingredient in Section 3 for proving Theorem 1. Corollary 1 is established in
Section 4.
Before ending this introduction, we give some references and comments about the
hypoelliptic properties of the non-cutoff Boltzmann equation. Following the rigorous
derivation of the lower bound for the non-cutoff Boltzmann collision operator in [3],
therehavebeenmanyworksontheregularityforthesolutionsoftheBoltzmannequa-
tion in both spatially homogeneous (see [4, 5, 17, 23, 29, 30] and references therein)
and inhomogeneous cases (see [7, 8, 10]). In all those works, it was highlighted that
theBoltzmanncollisionoperatorbehavesessentiallyasafractionalLaplacian( ∆˜ )σ
v
−
undertheangularsingularityassumptiononthecollisioncross-section(see[3,7]). We
refer the reader to Section 5.1 in appendix for comprehensiveexplanations about the
relevanceofthismodel. Inthespatiallyhomogeneouscase,thisdiffusivestructureim-
pliesaC∞ smoothingeffectfortheweaksolutionstotheCauchyproblemconstructed
in [38] (See [11, 23]). Furthermore, as in the case of the heat equation, the spatially
homogeneousBoltzmannequationwithoutangularcutoffenjoysasmoothingeffectin
theGevreyclassoforderσ (see[24,29,30]). Relatedtothis Gevreysmoothingeffect
forthespatiallyhomogeneousBoltzmannequation,anultra-analyticsmoothingeffect
was proved in [32] for both non-linear homogeneous Landau equations and inhomo-
geneous linear Landauequations. Regarding the study of the Boltzmann equation in
Gevreyspaces,wealsoreferthe readerto the seminalwork[37]whichestablishesthe
existence anduniqueness in Gevrey classes of a localsolutionto the Cauchyproblem
fortheBoltzmannequationinbothspatiallyhomogeneousandinhomogeneouscases.
Considering now the spatially inhomogeneous Boltzmann equation without angular
cutoff, the C∞ hypoellipticity was established in [7, 8, 10] by using the coercivity
HYPOELLIPTIC ESTIMATES FOR A LINEAR MODEL OF THE BOLTZMANN EQUATION 5
estimate for proving the regularity with respect to the velocity variable v (see [3, 7])
and a version of the uncertainty principle for proving the regularity in the time and
space variables t,x via bootstrap arguments (see [6]). However, it should be noted
that those works do not provide any optimal hypoelliptic estimates for the spatially
inhomogeneousBoltzmannequationwithoutangularcutoff. Asanattempttounder-
stand further the smoothing effect induced by the Boltzmann collision operator and
to relate exactly the structure of the angular singularity in the collision cross-section
to this regularizing effect, the present work studies the hypoelliptic properties of the
simplified linear model (1) and aims at giving insights on the hypoelliptic properties
what may be expected for the general spatially inhomogeneous Boltzmann equation
without angular cutoff.
2. A key hypoelliptic estimate
Theorem 1 will be derived from the following key hypoelliptic estimate:
Proposition 1. Let P be the operator defined in (1) and T > 0 be a positive con-
stant. Then, there exists a positive constant C >0 such that for all u S(R2n+1)
T ∈ t,x,v
satisfying
(8) supp u(,x,v) [ T,T], (x,v) R2n,
· ⊂ − ∈
we have
(9) (1+|Dx|2σ2+σ1 +|Dv|2σ)u L2(R2n+1) ≤CT kPukL2(R2n+1)+kukL2(R2n+1) .
(cid:13) (cid:13) (cid:0) (cid:1)
(cid:13) (cid:13)
Inthefollowing,weusestandardnotationsforsymbolclasses,see[22](Chapter18)
or [27]. The symbol class S(m,Γ) associated to the order function m and metric
dx2 dξ2
Γ= + ,
ϕ(x,ξ)2 Φ(x,ξ)2
with ϕ, Φ given positive functions, stands for the set of functions a C∞(R2n,C)
satisfying for all α Nn and β Nn, ∈ x,ξ
∈ ∈
(10) C >0, (x,ξ) R2n, ∂α∂βa(x,ξ) C m(x,ξ)ϕ(x,ξ)−|α|Φ(x,ξ)−|β|.
∃ α,β ∀ ∈ | x ξ |≤ α,β
2.1. Some symbol reductions. We begin by few symbol reductions in order to
reducethesymboloftheoperatortoaconvenientnormalform. Forconvenienceonly,
we shall use the Weyl quantization rather than the standard one. Notice that it will
be sufficient to prove the hypoelliptic estimate (9) for the operatorpw defined by the
Weyl quantization of the symbol
(11) p(t,x,v;τ,ξ,η)=2πiτ +2πiv ξ+a(t,x,v)F(2πη),
·
with τ, ξ, η respectively standing for the dual variables of the variables t, x, v and
F being the function defined in (3), while using the following normalization for the
Weyl quantization
x+y
(awu)(x)= e2iπ(x−y)·ξa ,ξ u(y)dydξ.
ZR2n 2
Indeed, symbolic calculus shows that the opera(cid:0)tor (cid:1)
1
R=P pw+ a(t,x,v) ( F)(2πD ),
v v
− 2i∇ · ∇
is bounded on L2. This is a direct consequence of the L2 continuity theorem in the
class S0 (case m = ϕ = Φ = 1 in (10)) after noticing that the Weyl symbol of the
00
operatorR together with allits derivatives ofany order arebounded on R4n+2, since
6 NICOLASLERNER,YOSHINORIMORIMOTO,KARELPRAVDA-STAROV
allthederivativesofordergreaterorequalto2ofthesymbolF areboundedgiventhe
choice of the positive parameter 0 < σ < 1. We refer the reader to formula (2.1.26)
in[27]forexplicitconstantsinthe compositionformulawiththenormalizationofthe
Weyl quantization chosen here. It then remains to notice that for any ε>0, there is
a positive constant C >0 such that
ε
a(t,x,v) ( F)(2πD )u . ( F)(2πD )u
∇v · ∇ v L2(R2n+1) ∇ v L2(R2n+1)
(cid:13)(cid:13) .k(1+|Dv|2σ−1)ukL2(R(cid:13)(cid:13)2n+1) ≤εk(1+(cid:13)(cid:13)|Dv|2σ)ukL2(R(cid:13)(cid:13)2n+1)+CεkukL2(R2n+1),
since the function a is bounded on R2n+1. When studying the operator pw, it is
v
∇
convenient to work on the Fourier side in the variables x,v. This is equivalent to
study the operator defined by the Weyl quantization of the symbol
2πiτ 2πix η+a(t, ξ, η)F(2πv).
− · − −
After relabeling the variables and simplifying the notations, we are reduced to study
the operator P =pw defined by the Weyl quantization of the symbol
(12) p(t,x,v;τ,ξ,η)=iτ iy ξ+a(t,ξ,η)F(x),
− ·
where a stands for a C∞(R2n+1) function satisfying
b
(13) a >0, (t,ξ,η) R2n+1, a(t,ξ,η) a >0,
0 0
∃ ∀ ∈ ≥
and F is the function
(14) F(x)= x2σw(x)+ x2 1 w(x) ,
| | | | −
with w a new function having experienced a sm(cid:0)all homot(cid:1)hetic transformation com-
pared to the function appearing in (3). For the sake of simplicity, we shall keep the
definition given in (3) and assume that w C∞(Rn), 0 w 1, w(η)=1 if η 2,
∈ ≤ ≤ | |≥
and w(η) = 0 if η 1. Using these notations, Proposition 1 is equivalent to the
| | ≤
proofofthefollowingaprioriestimate. ForanyT >0,thereexistsapositiveconstant
C >0 such that for all u S(R2n+1) satisfying
T ∈ t,x,y
(15) supp u(,x,y) [ T,T], (x,y) R2n,
· ⊂ − ∈
we have
(16) (1+|x|2σ +|y|2σ2+σ1)u L2(R2n+1) ≤CT kPukL2(R2n+1)+kukL2(R2n+1) .
In order t(cid:13)o do so, we consider th(cid:13)e new variables(cid:0) (cid:1)
(cid:13) (cid:13)
(x ,x )=(y,x+ty),
1 2
with t R fixed. We define A(x,y)=(y,x+ty). Associated to this linear change of
∈
variables are the real linear symplectic transformation
(x ,x ;ξ ,ξ )=χ(x,y;ξ,η)=(A(x,y);(A−1)T(ξ,η))=(y,x+ty;η tξ,ξ),
1 2 1 2
−
and the two unitary operators on L2(R2n)
(M u)(x ,x )=u(x tx ,x ); (M−1u)(x,y)=u(y,x+ty).
t 1 2 2− 1 1 t
Keeping on considering the t-variable as a parameter and defining the symbol
b (x,y;ξ,η)= iy ξ+a(t,ξ,η)F(x),
t
− ·
we have
(b χ−1)(x ,x ;ξ ,ξ )= ix ξ +a(t,ξ ,ξ +tξ )F(x tx ),
t 1 2 1 2 1 2 2 1 2 2 1
◦ − · −
and the symplectic invariance of the Weyl quantization (Theorem 18.5.9 in [22]) or a
simple direct calculation
M−1awM =(a χ)w,
t t ◦
HYPOELLIPTIC ESTIMATES FOR A LINEAR MODEL OF THE BOLTZMANN EQUATION 7
show that
M bw(x,y,D ,D )M−1 = ix D + a(t,ξ ,ξ +tξ )F(x tx ) w.
t t x y t − 1· x2 2 1 2 2− 1
We then consider the two unitary operators ac(cid:2)ting on L2(R2n+1), (cid:3)
t,x,y
(Mu)(t,x ,x )=u(t,x tx ,x ), (M−1u)(t,x,y)=u(t,y,x+ty),
1 2 2 1 1
−
and notice that
(MiD M−1u)(t,x ,x )=iD u(t,x ,x )+ix D u(t,x ,x ).
t 1 2 t 1 2 1· x2 1 2
It follows that
iD + a(t,ξ ,ξ +tξ )F(x tx ) w =MPM−1.
t 2 1 2 2 1
−
Notice also that (cid:2) (cid:3)
1+ x2 tx1 2σ+ x1 2σ2+σ1 =M(1+ x2σ + y 2σ2+σ1)M−1.
| − | | | | | | |
We can therefore reduce the proof of Proposition 1 to the proof of the following a
priori estimate. For any T >0, there exists a positive constant C >0 such that for
T
all u S(R2n+1 ) satisfying
∈ t,x1,x2
(17) supp u(,x ,x ) [ T,T], (x ,x ) R2n,
1 2 1 2
· ⊂ − ∈
we have
(18) (1+|x2−tx1|2σ+|x1|2σ2+σ1)u L2(R2n+1) ≤CT kPukL2(R2n+1)+kukL2(R2n+1) ,
for th(cid:13)e operator (cid:13) (cid:0) (cid:1)
(cid:13) (cid:13)
w
(19) P =iD + a(t,ξ ,ξ +tξ )F(x tx ) .
t 2 1 2 2 1
−
(cid:2) (cid:3)
2.2. Energy estimates via the Wick quantization. In order to establish the
a priori estimate (18), we shall use some techniques developed in [26] for proving
energy estimates via the Wick quantization. We recall in appendix (see Section 5.2)
the definition and all the main features of the Wick quantization which will be used
here.
Afirststepinadaptingthisapproachistostudythefirst-orderdifferentialoperator
on L2(R ),
t
P˜ =iD +a(t,ξ ,ξ +tξ )F(x tx ),
t 2 1 2 2 1
−
wherethevariablesx ,x ,ξ ,ξ areconsideredasparameters,andprovesomeapriori
1 2 1 2
estimates with respect to those parameters. We begin by noticing from (13) that for
any u S(R ),
t
∈
Re(P˜u,u)L2(Rt) = Ra(t,ξ2,ξ1+tξ2)F(x2−tx1)|u(t)|2dt≥a0 F(x2−tx1)1/2u 2L2(Rt),
Z
sincetheoperatoriD isskew-adjoint. ItfollowsfromtheCau(cid:13)chy-Schwarzine(cid:13)quality
t (cid:13) (cid:13)
that
(20) a0 F(x2−tx1)1/2u 2L2(Rt) ≤kP˜ukL2(Rt)kukL2(Rt).
By denoting H =1lR(cid:13) the Heaviside(cid:13)function, we may write for any T R,
(cid:13)+ (cid:13) ∈
(21) 2Re(P˜u,−H(t−T)u)L2(Rt) =2Re(Dtu,iH(t−T)u)L2(Rt)
2 H(t T)a(t,ξ ,ξ +tξ )F(x tx )u(t)2dt,
2 1 2 2 1
− R − − | |
Z
and obtain by a simple integration by parts that
1
2Re(Dtu,iH(t−T)u)L2(Rt) =([Dt,iH(t−T)]u,u)L2(Rt) = 2π|u(T)|2.
8 NICOLASLERNER,YOSHINORIMORIMOTO,KARELPRAVDA-STAROV
Recalling that a L∞(R2n+1), one may find a positive constant C >0 such that
0
∈
2 H(t T)a(t,ξ ,ξ +tξ )F(x tx )u(t)2dt C F(x tx )1/2u 2 .
R − 2 1 2 2− 1 | | ≤ 0 2− 1 L2(Rt)
It(cid:12)(cid:12)(cid:12)foZllows from (20), (21) and the Cauchy-Schwar(cid:12)(cid:12)(cid:12)z ineq(cid:13)(cid:13)uality that there(cid:13)(cid:13)exists a
constant C >0 such that for all u S(R ) and (x ,x ,ξ ,ξ ) R4n,
1 t 1 2 1 2
∈ ∈
(22) kuk2L∞(Rt) ≤C1kP˜ukL2(Rt)kukL2(Rt).
Following [26], we split-up the L2(R )-norm of u,
t
(23) |x1|2σ2+σ1kuk2L2(Rt) =Z{t∈R: |x1|2σ2+σ1>|x2−tx1|2σ}|x1|2σ2σ+1|u(t)|2dt
+Z{t∈R: |x1|2σ2σ+1≤|x2−tx1|2σ}|x1|2σ2+σ1|u(t)|2dt,
and estimate from above the first integral as
Z{t∈R: |x1|2σ2σ+1>|x2−tx1|2σ}|x1|2σ2+σ1|u(t)|2dt
≤|x1|2σ2+σ1m {t∈R: |x1|2σ2+σ1 >|x2−tx1|2σ} kuk2L∞(Rt) ≤2kuk2L∞(Rt),
with m standing for th(cid:0)e Lebesgue measure on R. It follo(cid:1)ws from (22) that this first
integral can be estimated from above as
(24) Z{t∈R: |x1|2σ2+σ1>|x2−tx1|2σ}|x1|2σ2+σ1|u(t)|2dt≤2C1kP˜ukL2(Rt)kukL2(Rt).
While estimating from above the second integral in the right-hand-side of (23), we
may first write that
Z{t∈R: |x1|2σ2+σ1≤|x2−tx1|2σ}|x1|2σ2+σ1|u(t)|2dt
x tx 2σ u(t)2dt x tx 2σ u(t)2dt,
≤Z{t∈R: |x1|2σ2+σ1≤|x2−tx1|2σ}| 2− 1| | | ≤ZR| 2− 1| | |
and then notice from (14) that there exists a positive constant C > 0 such that for
2
all (t,x ,x ,ξ ,ξ ) R4n+1,
1 2 1 2
∈
x tx 2σ F(x tx )+C .
2 1 2 1 2
| − | ≤ −
It follows from (20) that
(25) Z{t∈R: |x1|2σ2+σ1≤|x2−tx1|2σ}|x1|2σ2σ+1|u(t)|2dt
≤kF(x2−tx1)1/2uk2L2(Rt)+C2kuk2L2(Rt) ≤a−01kP˜ukL2(Rt)kukL2(Rt)+C2kuk2L2(Rt).
We deduce from (23), (24) and (25) that there exist some positive constants C > 0
3
and C >0 such that for all u S(R ) and (x ,x ,ξ ,ξ ) R4n,
4 t 1 2 1 2
∈ ∈
|x1|2σ2+σ1kuk2L2(Rt) ≤C3kP˜ukL2(Rt)kukL2(Rt)+C3kuk2L2(Rt),
that is
|x1|2σ2+σ1kukL2(Rt) ≤C3kP˜ukL2(Rt)+C3kukL2(Rt),
which implies that
(26) khx1i2σ2+σ1uk2L2(Rt) ≤C4kP˜uk2L2(Rt)+C4kuk2L2(Rt),
HYPOELLIPTIC ESTIMATES FOR A LINEAR MODEL OF THE BOLTZMANN EQUATION 9
with x = (1+ x2)1/2. We shall now prove that there exists a positive constant
C >h0 isuch that|fo|r all u S(R ) and (x ,x ,ξ ,ξ ) R4n,
5 t 1 2 1 2
∈ ∈
(27) khx1i2σ2+σ1uk2L2(Rt)+k|x2−tx1|2σuk2L2(Rt) ≤C5kP˜uk2L2(Rt)+C5kuk2L2(Rt).
In order to do so, let ε 1 and write the components of the variables x ,x as
0 1 2
∈{± }
x =(x ,x ,...,x ) and x =(x ,x ,...,x ).
1 1,1 1,2 1,n 2 2,1 2,2 2,n
Let j 1,...,n . We shall first study the case when the real parameter
∈{ }
(28) x =0.
1,j
6
WhileexpandingthefollowingL2(R )dot-productwhereH stilldenotestheHeaviside
t
function
2Re P˜u,|x2,j −tx1,j|2σH(ε0(x2,j −tx1,j)−2hx1i2σ1+1)u L2(Rt)
= 2Re(cid:0)Dtu,−i|x2,j−tx1,j|2σH(ε0(x2,j −tx1,j)−2hx1i2σ1+(cid:1)1)u L2(Rt)
+ 2 (cid:0)a(t,ξ2,ξ1+tξ2)F(x2 tx1)x2,j tx1,j 2σH(ε0(x2,j t(cid:1)x1,j) 2 x1 2σ1+1)u(t)2dt,
R − | − | − − h i | |
Z
we notice that
2Re Dtu,−i|x2,j −tx1,j|2σH(ε0(x2,j −tx1,j)−2hx1i2σ1+1)u L2(Rt)
(cid:0) = [Dt,−i|x2,j−tx1,j|2σH(ε0(x2,j −tx1,j)−2h(cid:1)x1i2σ1+1)]u,u L2(Rt).
(cid:0) (cid:1)
A direct computation gives that
[Dt,−i|x2,j −tx1,j|2σH(ε0(x2,j −tx1,j)−2hx1i2σ1+1)]u,u L2(Rt)
= (cid:0)ε0σ x1,j x2,j tx1,j 2σ−1H(ε0(x2,j tx1,j) 2 x1 2σ1+1(cid:1))u(t)2dt
π R | − | − − h i | |
Z
22σ−1 x
+ε0 1,j x1 2σ2+σ1 u(T)2,
π x h i | |
1,j
| |
with
T =x2,jx−1,1j −2ε0x−1,1jhx1i2σ1+1
and we deduce from (22) that
[Dt,−i|x2,j −tx1,j|2σH(ε0(x2,j −tx1,j)−2hx1i2σ1+1)]u,u L2(Rt)
≤ (cid:12)(cid:12)2(cid:0)2σπ−1C1kP˜ukL2(Rt)khx1i2σ2+σ1ukL2(Rt) (cid:1) (cid:12)(cid:12)
σ
+ x1,j x2,j tx1,j 2σ−1H(ε0(x2,j tx1,j) 2 x1 2σ1+1)u(t)2dt.
π R| || − | − − h i | |
Z
Since from (13),
2a0kF(x2−tx1)1/2|x2,j −tx1,j|σH(ε0(x2,j −tx1,j)−2hx1i2σ1+1)uk2L2(Rt)
2 a(t,ξ2,ξ1+tξ2)F(x2 tx1)x2,j tx1,j 2σH(ε0(x2,j tx1,j) 2 x1 2σ1+1)u(t)2dt,
≤ R − | − | − − h i | |
Z
10 NICOLASLERNER,YOSHINORIMORIMOTO,KARELPRAVDA-STAROV
we deduce from the Cauchy-Schwarz inequality; and all the previous identities and
estimates obtained after (28) that
2a0kF(x2−tx1)1/2|x2,j −tx1,j|σH(ε0(x2,j −tx1,j)−2hx1i2σ1+1)uk2L2(Rt)
22σ−1
≤ π C1kP˜ukL2(Rt)khx1i2σ2+σ1ukL2(Rt)
σ
+ x1,j x2,j tx1,j 2σ−1H(ε0(x2,j tx1,j) 2 x1 2σ1+1)u(t)2dt
π R| || − | − − h i | |
Z
+ 2kP˜ukL2(Rt)k|x2,j −tx1,j|2σH(ε0(x2,j −tx1,j)−2hx1i2σ1+1)ukL2(Rt).
Since from (14), we have
F(x tx )= x tx 2σ x tx 2σ,
2 1 2 1 2,j 1,j
− | − | ≥| − |
1
on the support of the function H(ε0(x2,j tx1,j) 2 x1 2σ+1), we then deduce from
− − h i
the previous estimate that there exists a positive constant C > 0 such that for all
6
u S(R ) and (x ,x ,ξ ,ξ ) R4n, x =0,
t 1 2 1 2 1,j
∈ ∈ 6
(29) C6−1k|x2,j −tx1,j|2σH(ε0(x2,j −tx1,j)−2hx1i2σ1+1)uk2L2(Rt) ≤kP˜uk2L2(Rt)
+khx1i2σ2+σ1uk2L2(Rt)+ R|x1,j||x2,j−tx1,j|2σ−1H(ε0(x2,j−tx1,j)−2hx1i2σ1+1)|u(t)|2dt.
Z
By using that
1
x2,j tx1,j −1 x1 −2σ1+1,
| − | ≤ 2h i
1
onthe supportofthe function H(ε0(x2,j tx1,j) 2 x1 2σ+1), one canestimate from
− − h i
above the following integral as
x1,j x2,j tx1,j 2σ−1H(ε0(x2,j tx1,j) 2 x1 2σ1+1)u(t)2dt
R| || − | − − h i | |
Z
2−1 x1 2σ2+σ1 x2,j tx1,j 2σH(ε0(x2,j tx1,j) 2 x1 2σ1+1)u(t)2dt
≤ Rh i | − | − − h i | |
Z
≤ 2−1khx1i2σ2+σ1ukL2(Rt)k|x2,j −tx1,j|2σH(ε0(x2,j −tx1,j)−2hx1i2σ1+1)ukL2(Rt).
According to (29), this implies that there exists a positive constant C >0 such that
7
for all u S(R ) and (x ,x ,ξ ,ξ ) R4n, x =0,
t 1 2 1 2 1,j
∈ ∈ 6
(30) C7−1k|x2,j −tx1,j|2σH(ε0(x2,j −tx1,j)−2hx1i2σ1+1)uk2L2(Rt)
≤kP˜uk2L2(Rt)+khx1i2σ2+σ1uk2L2(Rt).
Finally, since
x2,j tx1,j 2σ x2,j tx1,j 2σH(x2,j tx1,j 2 x1 2σ1+1)
| − | ≤| − | − − h i
+ x2,j tx1,j 2σH( x2,j +tx1,j 2 x1 2σ1+1)+22σ x1 2σ2+σ1,
| − | − − h i h i
we deduce from (30) that there exists a positive constant C > 0 such that for all
8
u S(R ) and (x ,x ,ξ ,ξ ) R4n, x =0,
t 1 2 1 2 1,j
∈ ∈ 6
(31) k|x2,j −tx1,j|2σuk2L2(Rt) ≤C8kP˜uk2L2(Rt)+C8khx1i2σ2+σ1uk2L2(Rt),
which together with (26) proves the a priori estimate
(32) khx1i2σ2+σ1uk2L2(Rt)+k|x2,j −tx1,j|2σuk2L2(Rt) ≤C9kP˜uk2L2(Rt)+C9kuk2L2(Rt),