A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equation

0.18 MB·English

by  Yoshinori Morimoto

A REMARK ON THE ULTRA-ANALYTIC SMOOTHING PROPERTIES
OF THE SPATIALLY HOMOGENEOUS LANDAU EQUATION
Y.MORIMOTO, K. PRAVDA-STAROV& C.-J. XU
Abstract. We consider the non-linear spatially homogeneous Landau equation with
Maxwellian molecules in a close-to-equilibrium framework and show that the Cauchy
problem for the fluctuation around the Maxwellian equilibrium distribution enjoys a
3 Gelfand-ShilovregularizingeffectintheclassS1/2(Rd),implyingtheultra-analyticityof
1/2
1
both thefluctuation and its Fourier transform, for any positive time.
0
2

n
a
J 1. Introduction

3
2 In the work [13], we consider the spatially homogeneous non-cutoff Boltzmann equation
with Maxwellian molecules in a close-to-equilibrium framework and study the smoothing

]
P properties of the Cauchy problem for the fluctuation around the Maxwellian equilibrium
A distribution. TheBoltzmann equation describesthebehavior ofadilutegas whentheonly
. interactions taken into account are binary collisions [5]. In the spatially homogeneous case
h
with Maxwellian molecules, it reads as the equation
t
a
m
∂ f = Q(f,f),
t
(1.1)
[
(f t=0 = f0,
|
1
v for the density distribution of the particles f = f(t,v) 0, t 0, v Rd, with d 2,
6 where the non-linear term ≥ ≥ ∈ ≥
6
5 (1.2) Q(f,f)= b v−v∗ σ (f′f′ f f)dσdv ,
5 Rd Sd−1 v v∗ · ∗ − ∗ ∗
1. Z Z (cid:16)| − | (cid:17)
stands for the Boltzmann collision operator whose cross section is a non-negative function
0
3 satisfying to the assumption
1
v: (1.3) (sinθ)d−2b(cosθ) ≈ θ−1−2s,
i θ→0+
X
for some 0< s < 1. The notation a b means that a/b is bounded from above and below
r ≈
a by fixed positive constants. The term (1.3) is not integrable in zero
π
2
(sinθ)d−2b(cosθ)dθ = + .
∞
Z0
This non-integrability plays a major role regarding the qualitative behaviour of the solu-
tions to the Boltzmann equation and this feature is essential for the smoothing effect to
be present, see the discussion in [13] and all the references herein.
In [13], we consider the spatially homogeneous non-cutoff Boltzmann equation with
Maxwellian molecules (1.1)intheradiallysymmetriccasewithinitialdensitydistributions
(1.4) f0 = µd+√µdg0, g0 L2(Rd) radial, g0 L2 1,
∈ k k ≪
Date: January 24, 2013.
2000 Mathematics Subject Classification. 35B65.
Keywordsandphrases. Landauequation,Gelfand-Shilovregularity,Ultra-analyticity,Smoothingeffect.
1
2 Y.MORIMOTO,K.PRAVDA-STAROV&C.-J.XU
close to the Maxwellian equilibrium distribution
2
(1.5) µd(v) = (2π)−d2e−|v2| , v Rd, Q(µd,µd) = 0,
∈
where is the Euclidean norm on Rd, in the physical 3-dimensional case d = 3. The
|·|
main result in [13] shows that the Cauchy problem for the fluctuation
f = µ +√µ g,
3 3
around the Maxwellian equilibrium distribution
(1.6) ∂tg+Lg = µ−31/2Q(√µ3g,√µ3g),
(g t=0 = g0 L2(R3),
| ∈
where
Lg = µ−1/2Q(µ ,µ1/2g) µ−1/2Q(µ1/2g,µ ),
− 3 3 3 − 3 3 3
enjoys the same Gelfand-Shilov regularizing effect as the Cauchy problem defined by the
evolution equation associated to the fractional harmonic oscillator
∂ g+ sg = 0, v 2
t
(1.7) H = ∆ + | | ,
(g t=0 = g0 L2(R3), H − v 4
| ∈
where 0 < s < 1 is the positive parameter appearing in the assumption (1.3). More
specifically, we prove that under the assumption (1.4), the Cauchy problem (1.6) admits
a unique global radial solution g L∞(R+,L2(R3)), which belongs to the Gelfand-Shilov
∈ t v
class S1/2s(R3) for any positive time
1/2s
(1.8) t > 0, g(t) S1/2s(R3).
∀ ∈ 1/2s
The definition of the Gelfand-Shilov regularity is recalled in appendix (Section 4.2).
Inthepresentwork,westudythespatiallyhomogeneousLandauequationwithMaxwellian
molecules
∂ f = Q (f,f),
t L
(1.9)
(f t=0 = f0.
|
The Landau collision operator Q (f,f) is understood as the limiting Boltzmann operator
L
in thegrazing collision limit asymptotic [1,2,6,7,18], whens tends to 1 in thesingularity
assumption (1.3). In the physical 3-dimensional case, the linearized non-cutoff Boltzmann
operator with Maxwellian molecules was actually showed to be equal to the fractional
linearized Landau operator with Maxwellian molecules [12] (Theorem 2.3),
L = a(H,∆S2)LLs,
up to a positive bounded isomorphism on L2(R3),
1
∃c> 0,∀f ∈ L2(R3), ckfk2L2 ≤ (a(H,∆S2)f,f)L2 ≤ ckfk2L2,
|v|2
commutingwiththeharmonicoscillator = ∆ + andtheLaplace-Beltramioperator
H − v 4
1
∆S2 = (vj∂k vk∂j)2,
2 −
1≤j,k≤3
X
j6=k
on the unit sphere S2. In view of this link between the linearized Boltzmann and Landau
operators, and in analogy with the Gelfand-Shilov smoothing result proven in [13] for
the spatially homogeneous non-cutoff Boltzmann equation, we may therefore expect that
the spatially homogeneous Landau equation also enjoys specific Gelfand-Shilov smoothing
properties. Thepurposeofthisnoteistoconfirmthisinsightandtocheck thattheCauchy
ON THE ULTRA-ANALYTIC SMOOTHING PROPERTIES OF THE LANDAU EQUATION 3
problem for the fluctuation around the Maxwellian equilibrium distribution associated to
the spatially homogeneous Landau equation with Maxwellian molecules actually enjoys a
Gelfand-Shilov regularizing effect in the class S1/2(Rd), implying the ultra-analyticity of
1/2
both the fluctuation and its Fourier transform, for any positive time.
2. The Landau equation
The Landau equation written by Landau in 1936 [11] is the equation
∂ f +v f = Q (f,f),
t x L
(2.1) ·∇
(f t=0 = f0,
|
for the density distribution of the particles f = f(t,x,v) 0 at time t, having position
x Rd and velocity v Rd, with d 2. The term Q≥(f,f) is the Landau collision
L
∈ ∈ ≥
operator associated to the Landau bilinear operator
Q (g,f) = a(v v ) g(t,x,v )( f)(t,x,v) ( g)(t,x,v )f(t,x,v) dv ,
L v ∗ ∗ v v ∗ ∗
∇ · Rd − ∇ − ∇
(cid:16)Z (cid:17)
(cid:0) (cid:1)
where a = (a ) stands for the non-negative symmetric matrix
i,j 1≤i,j≤d
(2.2) a(v) = v γ(v 2Id v v) M (R), d< γ <+ .
d
| | | | − ⊗ ∈ − ∞
In this work, we study the spatially homogeneous case when the density distribution of
the particles does not depend on the position variable
∂ f = Q (f,f),
t L
(2.3)
(f t=0 = f0,
|
for Maxwellian molecules, that is, when the parameter γ = 0 in the assumption (2.2). At
least formally, it is easily checked that the mass, the momentum and the kinetic energy
are conserved quantities by this evolution equation
1
(2.4) f(t,v)dv = M, f(t,v)vdv = MV, f(t,v)v 2dv = E, t 0,
Rd Rd 2 Rd | | ≥
Z Z Z
with M > 0, V Rd, E > 0. The Cauchy problem (2.3) associated to the spatially
∈
homogeneous Landau equation with Maxwellian molecules and some quantitative features
of the solutions were thoroughly studied by Villani [17]. The propositions 4 and 6 of
the work [17] show that, for each non-negative measurable initial density distribution f
0
having finite mass and finite energy
1
(2.5) f 0, 0 < f (v)dv = M < + , 0 < f (v)v 2dv = E < + ,
0 0 0
≥ Rd ∞ 2 Rd | | ∞
Z Z
the Cauchy problem (2.3) admits a unique global classical solution f(t,v) defined for all
t 0. Furthermore, this solution is showed to bea non-negative boundedsmooth function
≥
f(t) 0, f(t) L∞(Rd) C∞(Rd),
≥ ∈ v ∩ v
for any positive time t > 0.
In this work, we study a close-to-equilibrium framework. To that end, we consider the
linearization of the spatially homogeneous Landau equation
f = µ +√µ g,
d d
around the Maxwellian equilibrium distribution
2
(2.6) µd(v) = (2π)−d2e−|v2| , v Rd.
∈
By using that Q (µ ,µ )= 0, and setting
L d d
(2.7) L g = µ−1/2Q(µ ,µ1/2g) µ−1/2Q(µ1/2g,µ ),
L − d d d − d d d
4 Y.MORIMOTO,K.PRAVDA-STAROV&C.-J.XU
the original spatially homogeneous Landau equation (2.3) is reduced to the Cauchy prob-
lem for the fluctuation
(2.8) ∂tg+LLg =µ−d1/2QL(√µdg,√µdg),
(g t=0 = g0.
|
An explicit computation [12] (Proposition 2.1) shows that the linearized Landau operator
with Maxwellian molecules acting on the Schwartz space is equal to
d d
L = (d 1) ∆ + ∆ (d 1) P
L Sd−1 Sd−1 1
− H− 2 − − − H− 2
(cid:16) (cid:17) h (cid:16) (cid:17)i d
+ ∆ (d 1) P ,
Sd−1 2
− − − H− 2
h (cid:16) (cid:17)i
|v|2
where = ∆ + is the harmonic oscillator,
H − v 4
1
∆ = (v ∂ v ∂ )2,
Sd−1 j k k j
2 −
1≤j,k≤d
X
j6=k
standsfortheLaplace-BeltramioperatorontheunitsphereSd−1 andP aretheorthogonal
k
projections onto the Hermite basis defined in Section 4.1. The linearized Landau operator
is a non-negative operator
(L g,g) 0,
L L2(Rdv) ≥
satisfying
(2.9) (L g,g) = 0 g = Pg,
L L2(Rd) ⇔
where Pg = (a +b v +cv 2)µ1/2, with a,c R, b Rd, stands for the L2-orthogonal
· | | d ∈ ∈
projection onto the space of collisional invariants
(2.10) = Span µ1/2,v µ1/2,...,v µ1/2, v 2µ1/2 .
N d 1 d d d | | d
By elaborating on the solution(cid:8)s constructed by Villani [17], th(cid:9)e purpose of this note is
to study the Gelfand-Shilov regularizing properties of the Cauchy problem (2.8) for the
fluctuation around the Maxwellian equilibrium distribution. For the sake of simplicity,
we may assume without loss of generality that the density distribution satisfies (2.4) with
V = 0. Furthermore, by changing the unknown function f to f˜as
M 2E
(2.11) f = f˜ · , α = ,
αd α Md
r
(cid:16) (cid:17)
we may reduce our study to the case when
(2.12) f(t,v)dv = 1, f(t,v)vdv = 0, f(t,v)v 2dv = d, t 0.
Rd Rd Rd | | ≥
Z Z Z
Let f0 = µd +√µdg0 ≥ 0, with g0 ∈ L1(Rdv)∩L2(Rdv), be a non-negative initial density
distribution having finite mass and finite energy such that
(2.13) f (v)dv = 1, f (v)vdv = 0, f (v)v 2dv = d.
0 0 0
Rd Rd Rd | |
Z Z Z
Such an initial density distribution is rapidly decreasing with a finite temperature tail
1 1 = sup β 0 : f0(v)eβ|v2|2dv < + ,
2 ≤ T(f0) ≥ Rd ∞
n Z o
since
|v|2 1
(2.14) ZRdf0(v)e 4 dv = (2π)d4 ZRd µd(v)+g0(v) dv < +∞,
(cid:0)p (cid:1)
ON THE ULTRA-ANALYTIC SMOOTHING PROPERTIES OF THE LANDAU EQUATION 5
when g L1(Rd). The analysis of the evolution of the temperature tail led in [17]
0 ∈ v
(Section 6, p. 972-974) shows that
2 2
|v| |v|
f0(v)e 4 dv < + t> 0, f(t,v)e 4 dv < + .
Rd ∞ ⇒ ∀ Rd ∞
Z Z
This implies that the fluctuation f = µ +√µ g 0, around the Maxwellian equilibrium
d d
≥
distribution defined by
(2.15) g(t) = µ−1/2(f(t) µ ) L1(Rd) C∞(Rd) S′(Rd), t >0,
d − d ∈ v ∩ v ⊂ v
belongs to L1(Rd) and therefore remains a tempered distribution for all t > 0. The
v
following statement is the main result contained in this note:
Theorem 2.1. Let f0 = µd +√µdg0 ≥ 0, with g0 ∈ L1(Rdv)∩L2(Rdv), be a non-negative
measurable function having finite mass and finite energy such that
(2.16) f (v)dv = 1, f (v)vdv = 0, f (v)v 2dv = d.
0 0 0
Rd Rd Rd | |
Z Z Z
Let f(t) = µd +√µdg(t), with g(t) ∈ L1(Rdv)∩C∞(Rdv) when t > 0, be the unique global
classical solution of the Cauchy problem associated to the spatially homogeneous Landau
equation with Maxwellian molecules
∂ f = Q (f,f),
t L
(f t=0 = f0,
|
constructed by Villani [17]. Then, there exists a positive constant δ > 0 such that
1/2
C > 0, t 0, etδHg(t) = eδ(2k+d)t P g(t) 2 Ced(d−1)t( g +1),
∃ ∀ ≥ k kL2 k k kL2 ≤ k 0kL2
(cid:16)Xk≥0 (cid:17)
with = ∆ + |v|2, where stands for the L2(Rd)-norm and P are the orthogonal
H − v 4 k·kL2 v k
projections onto the Hermite basis defined in Section 4.1. In particular, this implies that
the fluctuation belongs to the Gelfand-Shilov space S1/2(Rd) for any positive time
1/2
t > 0, g(t) S1/2(Rd).
∀ ∈ 1/2
Remark. The orthogonal projection P : S′(Rd) S(Rd) is well-defined on tempered
k v → v
distributions since the Hermite functions are Schwartz functions.
This result shows that the Cauchy problem (2.8) enjoys an ultra-analytic regularizing
effect in the Gevrey class G1/2(Rd) both for the fluctuation and its Fourier transform in
the velocity variable for any positive time
g(t), g(t) G1/2(Rd), t > 0.
∈
Let us recall that the existence, uniqueness, the Sobolev regularity and the polynomial
decay of the weak solutions to the Cbauchy problem (2.3) have been studied by Desvillettes
and Villani for hard potentials [8] (Theorem 6), that is, when the parameter satisfies
0 < γ 1 in the assumption (2.2). Under rather weak assumptions on the initial datum,
≤
e.g. f L1 , with δ > 0, they prove that there exists a weak solution to the Cauchy
0 ∈ 2+δ
problem such that f C∞([t ,+ [,S(Rd)), for all t > 0, and for all t > 0, s > 0,
m N, ∈ 0 ∞ v 0 0
∈
sup f(t, ) < + .
Hm
k · k s ∞
t≥t0
6 Y.MORIMOTO,K.PRAVDA-STAROV&C.-J.XU
TheGevrey regularity f(t, ) Gσ, for any σ > 1, forall positive timet > 0 of thesolution
· ∈
to the Cauchy problem (2.3) with an initial datum f with finite mass, energy and entropy
0
satisfying
t > 0, m 0, sup f(t, ) < + ,
0 Hm
∀ ≥ k · k γ ∞
t≥t0
was later established by Chen, Li and Xu for the hard potential case and the Maxwellian
molecules case [3]. Under the same assumptions on the solution, this result was later
extended to analytic regularity [4]:
t0 > 0, c0,C > 0, t t0, ec0(−∆v)1/2f(t, ) L2 C(t+1),
∀ ∃ ∀ ≥ k · k ≤
in the hard potential case and the Maxwellian molecules case. Regarding specifically the
Maxwellianmoleculescaseγ = 0,MorimotoandXuestablishedintheultra-analyticity[14]
(Theorem 1.1),
0 <t < T, f(t, ) G1/2(Rd),
∀ · ∈
0 < T0 < T, c0 > 0, 0 < t T0, e−c0t∆vf(t, ) L2 ed2t f0 L2,
∀ ∃ ∀ ≤ k · k ≤ k k
of any positive weak solution f(t,x) > 0 to the Cauchy problem (2.3) satisfying f
L∞(]0,T[,L2(Rd) L1(Rd)), with 0 < T + , with an initial datum satisfying f ∈
∩ 2 ≤ ∞ 0 ∈
L2(Rd) L1(Rd). Theresultof Theorem2.1 allows to specifyfurtherthepropertyof ultra-
∩ 2
analyticsmoothingprovenbyMorimotoandXuintheclose-to-equilibriumframework[14].
Thisresultpoints outthe specificdecay of thefluctuation both in thevelocity and its dual
Fourier variable. As for the Boltzmann equation, the Gelfand-Shilov regularity seems
relevant to describe the regularizing properties of the Landau equation in the close-to-
equilibrium framework.
3. Proof of Theorem 2.1
The proof of Theorem 2.1 is elementary and relies only on spectral arguments following
theresultsestablishedbyVillani[17]. Letf0 = µd+√µdg0 ≥ 0,withg0 ∈ L1(Rdv)∩L2(Rdv),
be a non-negative measurable function having finite mass and finite energy such that
(3.1) f (v)dv = 1, f (v)vdv = 0, f (v)v 2dv = d.
0 0 0
Rd Rd Rd | |
Z Z Z
Following [17] (p. 966), we may choose an orthonormal basis of Rd diagonalizing the non-
negative symmetric quadratic form
d
q(x)= f (v)(x v)2dv = x x f (v)v v dv 0,
0 j k 0 j k
Rd · Rd ≥
Z j,k=1 Z
X
where x v = d x v , x = (x ,...,x ), v = (v ,...,v ), stands for the standard dot
· j=1 j j 1 d 1 d
productinRd. Inthis orthonormalbasisofRd, theuniquesolution totheCauchy problem
P v
associated tothespatially homogeneous Landauequation withMaxwellian molecules (2.3)
is showed to satisfy [17] (Section 5),
d
(3.2) ∂ f = (d T (t))∂2f +(d 1) (vf)+∆ f,
t − j j − ∇· Sd−1
j=1
X
with
T (t)= f(t,v)v2dv = 1+(T (0) 1)e−4dt,
j Rd j j −
Z
ON THE ULTRA-ANALYTIC SMOOTHING PROPERTIES OF THE LANDAU EQUATION 7
and
d
(3.3) f(t,v)dv = 1, f(t,v)v dv = 0, f(t,v)v 2dv = T (t) = d,
j j
Rd Rd Rd | |
Z Z Z j=1
X
j = k f(t,v)v v = 0,
j k
6 ⇒ Rd
Z
when t 0. These conditions imply that the fluctuation satisfies g(t) L1(Rd) ⊥,
≥ ∈ v ∩N
that is,
µ (v)g(t,v)dv = 0, v µ (v)g(t,v)dv = 0, v 2 µ (v)g(t,v)dv = 0,
d j d d
Rd Rd Rd| |
Z Z Z
p p p
together with
j = k v v µ (v)g(t,v) = 0,
j k d
6 ⇒ Rd
Z
p
when t 0. The equation (3.2) may be rewritten for the fluctuation as
≥
d
∂tg = µd−1/2 (d−1−αje−4dt)∂j2(µd+√µdg)+(d−1)µ−d1/2∇·(vµd+v√µdg)+∆Sd−1g,
j=1
X
with
d
(3.4) α = v2 µ (v)g (v)dv, α = 0.
j j d 0 j
Rd
Z j=1
p X
It follows that
v 2 d d v2 1
∂tg = − (d−1) −∆v + |4| − 2 −∆Sd−1 g−e−4dt αj ∂j2+ 4j −vj∂j − 2 g
h (cid:16) (cid:17) i Xj=1 h i
d
e−4dt α (v2 1)µ1/2.
− j j − d
j=1
X
By using that d α = 0, we notice that
j=1 j
P
v 2 d d
∂tg = − (d − 1) − ∆v + |4| − 2 − ∆Sd−1 g − e−4dt αj[(A+,j)2g + vj2µ1d/2],
h (cid:16) (cid:17) i Xj=1
where A is the creation operator defined in Section 4.1. We consider
+,j
n
S = P ,
n k
k=0
X
the orthogonal projection onto the n+1 lowest energy levels of the harmonic oscillator,
whereP standsfortheorthogonalprojectionontotheHermitebasisdefinedinSection4.1.
k
Asmentionedabove,theorthogonalprojectionS iswell-definedontempereddistributions
n
since the Hermite functions are Schwartz functions. This gives a sense for the orthogonal
projection of the fluctuation S g(t) S(Rd) as a Schwartz function. Then, a direct
n ∈ v
8 Y.MORIMOTO,K.PRAVDA-STAROV&C.-J.XU
computation shows that for all t 0, δ > 0, n 2,
≥ ≥
1
∂ ( etδHS g 2 ) δ( (etδHS g),etδHS g) = Re(∂ S g,e2δtHS g)
2 t k n kL2 − H n n L2 t n n L2
= (d 1)( (etδHS g),etδHS g) (( ∆ )(etδHS g),etδHS g)
n n L2 Sd−1 n n L2
− − H − −
d
1
+ d(d 1) etδHS g 2 e−4dt α (etδH(v2µ1/2),etδHS g)
2 − k n kL2 − j j d n L2
j=1
X
d
e−4dt α (etδHS (A )2g,etδHS g) ,
j n +,j n L2
−
j=1
X
since the harmonic oscillator and the Laplace-Beltrami operator on Sd−1 are commuting
selfadjoint operators. We deduce from (4.5), (4.6) and (4.7) that
(etδHS (A )2g,etδHS g) = e2δt((A )2etδHS g,etδHS g)
n +,j n L2 +,j n−2 n L2
= e2δt(A etδHS g,A etδHS g)
+,j n−2 −,j n L2
and
etδH(v2µ1/2) = etδH((A +A )2Ψ )= etδH(A2 +A2 +A A +A A )Ψ
j d +,j −,j 0 +,j −,j +,j −,j −,j +,j 0
= (e2δtA2+,j +e−2δtA2−,j +A+,jA−,j +A−,jA+,j)ed2δtΨ0 = √2e(2+d2)δtΨ2ej +ed2δtΨ0.
It follows that
1
∂ ( etδHS g 2 )+(d 1 δ)( (etδHS g),etδHS g)
2 t k n kL2 − − H n n L2
d
1
d(d 1) etδHS g 2 +e−(4d−2δ)t α A etδHS g A etδHS g
≤ 2 − k n kL2 | j|k +,j n−2 kL2k −,j n kL2
j=1
X
d
+e−(4−δ2)dt 2e4δt +1 αj etδHSng L2.
| | k k
p (cid:16)Xj=1 (cid:17)
By using that
d
1
= (A A +A A ),
+,j −,j −,j +,j
H 2
j=1
X
we notice that
d d
1
A u A u ( A u 2 + A u 2 ) =( u,u) .
k +,j kL2k −,j kL2 ≤ 2 k +,j kL2 k −,j kL2 H L2
j=1 j=1
X X
By using that
A etδHS g A etδHS g ,
+,j n−2 L2 +,j n L2
k k ≤ k k
we obtain that
1
∂ ( etδHS g 2 )+(d 1 δ)( (etδHS g),etδHS g)
2 t k n kL2 − − H n n L2
1
d(d 1) etδHS g 2 +e−(4d−2δ)t sup α ( (etδHS g),etδHS g)
≤ 2 − k n kL2 | j| H n n L2
1≤j≤d
(cid:16) (cid:17)
d
+e−(4−δ2)dt 2e4δt +1 αj etδHSng L2.
| | k k
p (cid:16)Xj=1 (cid:17)
ON THE ULTRA-ANALYTIC SMOOTHING PROPERTIES OF THE LANDAU EQUATION 9
We notice that
0 <1+α = v2(µ +√µ g )dv,
j j d d 0
Rd
Z
since the initial density distribution f0 = µd +√µdg0 ≥ 0 satisfies Rdf0dv = 1. On the
other hand, we deduce from (3.4) that
R
d
(1+α ) = d.
j
j=1
X
This implies that 1 < α < d 1, because d 2. We may choose the positive constant
j
− − ≥
0< δ 1 such that
≤
sup α d 1 δ.
j
| |≤ − −
1≤j≤d
It follows that
d
1 1
2∂t(ketδHSngk2L2)≤ 2d(d−1)ketδHSngk2L2+e−(4−δ2)dt 2e4δt +1 |αj| ketδHSngkL2
p (cid:16)Xj=1 (cid:17)
1 9
d(d 1) etδHS g 2 +√3d(d 1) etδHS g d(d 1) etδHS g 2 + d(d 1).
≤ 2 − k n kL2 − k n kL2 ≤ − k n kL2 2 −
We obtain that for all t 0, n 2,
≥ ≥
9
etδHS g(t) 2 e2d(d−1)t g 2 + (e2d(d−1)t 1),
k n kL2 ≤ k 0kL2 2 −
which implies that for all t 0,
≥
9
etδHg(t) 2 e2d(d−1)t g 2 + (e2d(d−1)t 1).
k kL2 ≤ k 0kL2 2 −
It follows that there exists a positive constant C > 0 such that
etδHg(t) Ced(d−1)t( g +1), t 0,
k kL2(Rdv) ≤ k 0kL2 ≥
and we deduce from (4.8) that for any positive time
g(t) S1/2(Rd), t > 0.
∈ 1/2
This ends the proof of Theorem 2.1.
4. Appendix
4.1. The harmonic oscillator. The standard Hermite functions (φ ) are defined for
n n≥0
x R,
∈
(4.1) φn(x) = (2−nn1)!√n πex22 ddxnn(e−x2) = 2n1n!√π x − ddx n(e−x22) = a√n+nφ!0,
(cid:16) (cid:17)
where a is the creation operator
+ p p
1 d
a = x .
+
√2 − dx
(cid:16) (cid:17)
The family (φ ) is an orthonormal basis of L2(R). We set for n 0, α = (α )
n n≥0 j 1≤j≤d
≥ ∈
Nd, x R, v Rd,
∈ ∈
1 x d n
(4.2) ψ (x) = 2−1/4φ (2−1/2x), ψ = ψ ,
n n n 0
√n! 2 − dx
(cid:16) (cid:17)
d
(4.3) Ψ (v) = ψ (v ), = Span Ψ ,
α αj j Ek { α}α∈Nd,|α|=k
j=1
Y
10 Y.MORIMOTO,K.PRAVDA-STAROV&C.-J.XU
with α = α + +α . Thefamily (Ψ ) is anorthonormalbasis of L2(Rd)composed
| | 1 ··· d α α∈Nd
by the eigenfunctions of the d-dimensional harmonic oscillator
v 2 d
(4.4) = ∆ + | | = +k P , Id = P ,
v k k
H − 4 2
Xk≥0(cid:16) (cid:17) Xk≥0
whereP is the orthogonal projection onto whose dimension is k+d−1 . Theeigenvalue
k Ek d−1
d/2 is simple in all dimensions and is generated by the function
0
E (cid:0) (cid:1)
(4.5) Ψ0(v) = 1 d e−|v4|2 = µ1d/2(v),
(2π)4
where µ is the Maxwellian distribution defined in (1.5). Setting
d
v ∂
j
A = , 1 j d,
±,j
2 ∓ ∂v ≤ ≤
j
we have
1
Ψ = Aα1 ...Aαd Ψ , α = (α ,...,α ) Nd,
α √α !...α ! +,1 +,d 0 1 d ∈
1 d
(4.6) A+,jΨα = αj +1Ψα+ej, A−,jΨα = √αjΨα−ej (= 0 if αj = 0),
where (e ,...,e ) standps for the canonical basis of Rd. In particular, we readily notice that
1 d
for all t 0, δ > 0,
≥
(4.7) etδHA = eδtA etδH, etδHA = e−δtA etδH.
+,j +,j −,j −,j
4.2. Gelfand-Shilov regularity. We refer thereader to the works [9, 10, 15, 16] and the
references herein for extensive expositions of the Gelfand-Shilov regularity. The Gelfand-
Shilov spaces Sµ(Rd), with µ,ν > 0, µ + ν 1, are defined as the spaces of smooth
ν
functions f C∞(Rd) satisfying to the estimat≥es
∈
∃A,C > 0, |∂vαf(v)| ≤ CA|α|(α!)µe−A1|v|1/ν, v ∈ Rd, α ∈ Nd,
or, equivalently
A,C > 0, sup vβ∂αf(v) CA|α|+|β|(α!)µ(β!)ν, α,β Nd.
∃ | v | ≤ ∈
v∈Rd
These Gelfand-Shilov spaces Sµ(Rd) may also be characterized as the spaces of Schwartz
ν
functions f S(Rd) satisfying to the estimates
∈
C > 0,ε > 0, f(v) Ce−ε|v|1/ν, v Rd, f(ξ) Ce−ε|ξ|1/µ, ξ Rd.
∃ | | ≤ ∈ | | ≤ ∈
In particular, we notice that Hermite functions belong to the symmetric Gelfand-Shilov
b
space S1/2(Rd). More generally, the symmetric Gelfand-Shilov spaces Sµ(Rd), with µ
1/2 µ ≥
1/2,canbenicelycharacterizedthroughthedecompositionintotheHermitebasis(Ψ ) ,
α α∈Nd
see e.g. [16] (Proposition 1.2),
1
(4.8) f ∈ Sµµ(Rd) ⇔ f ∈ L2(Rd), ∃t0 > 0, (f,Ψα)L2exp(t0|α|2µ) α∈Nd l2(Nd) < +∞
(cid:13)f(cid:0) L2(Rd), t > 0, e(cid:1)t0H1/2(cid:13)µf < + ,
(cid:13) 0 (cid:13) L2
⇔ ∈ ∃ k k ∞
where (Ψ ) stands for the Hermite basis defined in Section 4.1, and where
α α∈Nd
v 2
= ∆ + | | ,
v
H − 4