The Wigner-Fokker-Planck equation: Stationary states and large-time behavior

0.33 MB·English

by  Anton Arnold

THE WIGNER-FOKKER-PLANCK EQUATION: STATIONARY
STATES AND LARGE TIME BEHAVIOR
ANTONARNOLD,IRENEM.GAMBA,MARIAPIAGUALDANI,STE´PHANEMISCHLER,
CLE´MENTMOUHOT,ANDCHRISTOFSPARBER
Abstract. WeconsiderthelinearWigner-Fokker-Planckequationsubjectto
2
confiningpotentialswhicharesmoothperturbationsoftheharmonicoscillator
1
potential. Foracertainclassofperturbations weprovethattheequation ad-
0
mitsauniquestationarysolutioninaweightedSobolevspace. Akeyingredient
2
oftheproofisanewresultontheexistenceofspectralgapsforFokker-Planck
n type operators in certain weighted L2–spaces. In addition we show that the
a steadystatecorrespondstoapositivedensitymatrixoperatorwithunittrace
J and that the solutions of the time-dependent problem converge towards the

0 steadystatewithanexponential rate.
2

]
P
1. Introduction
A
. ThisworkisdevotedtothestudyoftheWigner-Fokker-Planck equation (WFP),
h
considered in the following dimensionless form (where all physical constants are
t
a
normalized to one for simplicity):
m
∂ w+ξ w+Θ[V]w =∆ w+2div (ξw)+∆ w,
[ (1.1) t ·∇x ξ ξ x
3 ( w t=0 = w0(x,ξ),
v where x,ξ Rd, for d 1, and t R(cid:12) . Here, w(t,x,ξ) is the (real valued) Wigner
1 ∈ ≥ ∈ (cid:12)+
transform [37] of a quantum mechanical density matrix ρ(t,x,y), as defined by
9
7 1 η η
2 (1.2) w(t,x,ξ):= ρ t,x+ , x e−iξ·ηdη.
0. (2π)d ZRd (cid:16) 2 − 2(cid:17)
Recallthat,foranytimet R ,aquantummechanical(mixed)stateisgivenbya
101 tphoesisteivteo,fseblof-uanddjoeidntoptrearcaetoc∈rlassos+nopLe2r(aRtodr)ρa(ntd)∈byT1+. HerewedenotebyB(L2(Rd))
:
v T := ρ B(L2(Rd)):tr ρ < ,
1
i { ∈ | | ∞}
X the corresponding set of trace-class operators. We consequently write ρ T+
r T , if in addition ρ 0 (in the sense of non-negative operators). Since T∈ 1T⊂,
a 1 ≥ 1 ⊂ 2
the space of Hilbert-Schmidt operators, i.e.
T2 := ρ B(L2(Rd)):tr(ρ∗ρ)< ,
{ ∈ ∞}
2000 Mathematics Subject Classification. 82C10,35S10,74H40, 81Q15.
Key words and phrases. Wigner transform, Fokker Planck operator, spectral gap, stationary
solution,largetimebehavior.
A. Arnoldacknowledges partial support from the FWF (project “Quantum Transport Equa-
tions: Kinetic, Relativistic, and Diffusive Phenomena” and Wissenschaftskolleg “Differentialgle-
ichungen”),theO¨AD(Amadeusproject),andtheNewtonInstituteofCambridgeUniversity. I.M.
Gamba is supported by nsf-dms0807712. M. P. Gualdani is supported by NSF-DMS-1109682.
C.Mouhotwouldliketothank CambridgeUniversitywhoprovidedrepeated hospitalityin2009
thankstotheAwardNo. KUK-I1-007-43,fundedbytheKingAbdullahUniversityofScienceand
Technology (KAUST). C. Sparber has been supported by the Royal Society through his Royal
SocietyUniversityResearchFellowship. SupportfromtheInstituteofComputationalEngineering
andSciences attheUniversityofTexasatAustinisalsogratefullyacknowledged.
1
2 A.ARNOLD,I.GAMBA,M.P.GUALDANI,S.MISCHLER,C.MOUHOT,ANDC.SPARBER
we can identify the operator ρ(t) with its corresponding integral kernel ρ(t, , )
L2(R2d),theso-calleddensitymatrix. Consequently,ρ(t)actsonanygivenfun·ct·io∈n
ϕ L2(Rd) via
∈
(ρ(t)ϕ)(x)= ρ(t,x,y)ϕ(y)dy.
ZRd
Using the Wigner transformation (1.2), which by definition yields a real-valued
function w(t, , ) L2(R2d), one obtains a phase-space description of quantum
mechanics, re·m·ini∈scent of classical statistical mechanics, with x Rd being the
position and ξ Rd the momentum. However, in contrast to classi∈cal phase space
∈
distributions, w(t,x,ξ) in general also takes negative values.
Equation (1.1) governs the time evolution of w(t,x,ξ) in the framework of so-
called open quantum systems, which model both the Hamiltonian evolution of a
quantum system and its interaction with an environment (see [13], e.g.). Here,
wespecificallydescribetheseinteractionsbythe Fokker-Planck(FP)type diffusion
operatoronther.h.s. of(1.1). Fornotationalsimplicityweusehereonlynormalized
constants in the quantum FP operator. However, all of the subsequent analysis
also applies to the general WFP model presented in [34] (cf. Remark 2.4 below).
Potentialforcesactingonw(t, , )aretakeninto accountby the pseudo-differential
· ·
operator
i
(1.3) (Θ[V]f)(x,ξ):= δV(x,η)f(x,ξ′) eiη·(ξ−ξ′)dξ′dη,
−(2π)d ZZR2d
where the symbol δV is given by
η η
(1.4) δV(x,η)=V x+ V x ,
2 − − 2
(cid:16) (cid:17) (cid:16) (cid:17)
and V is a given real valued function. The WFP equation is a kinetic model
for quantum mechanical charge-transport, including diffusive effects, as needed,
e.g., in the description of quantum Brownian motion [15], quantum optics [18],
and semiconductor device simulations [16]. It can be considered as a quantum
mechanicalgeneralizationoftheusualkineticFokker-Planckequation(orKramer’s
equation), to which it is known to converge in the classical limit ~ 0, after an
→
appropriaterescalingoftheappearingphysicalparameters[10]. TheWFPequation
has been partly derived in [11] as a rigorous scaling limit for a system of particles
interactingwithaheatbathofphonons. Additional“derivations”(basedonformal
arguments from physics) can also can be found in [14, 15, 35, 36].
Inrecentyears,mathematicalstudiesofWFP type equationsmainlyfocusedon
the Cauchy problem (with or without self-consistent Poisson-coupling), see [2, 3,
4, 7, 9, 12]. In these works, the task of establishing a rigorous definition for the
particle density n(t,x) has led to various functional analytical settings. To this
end, it is importantto note that the dynamics induced by (1.1) maps T+(L2(Rd))
1
into itself, since the so-called Lindblad condition is fulfilled (see again Remark 2.4
below). For more details on this we refer to [7, 9] and the references given therein.
In the present work we shall be mainly interested in the asymptotic behavior as
t + of solutions to (1.1). To this end, we first need to study the stationary
→ ∞
problem corresponding to (1.1). Let us remark, that stationary equations for open
quantum systems, based on the Wigner formalism, seem to be rather difficult to
treat as only very few results exist (in spite of significant efforts, cf. [6] where the
stationary, inflow-problem for the linear Wigner equation in d = 1 was analyzed).
In fact the only result for the WFP equation is given in [34], where the existence
of a unique steady state for a quadratic potential V(x) x2 has been proved.
∝ | |
However,the cited work is based on severalexplicit calculations,which cannot be
applied in the case of a more general potential V(x).
STATIONARY STATES AND LARGE TIME BEHAVIOR OF WFP 3
Thegoalofthe presentpaperistwofold: First,weaimtoestablishtheexistence
of a normalized steady state w (x,ξ) for (1.1) in the case of confining potentials
∞
V(x), which are given by a suitable class of perturbations of quadratic potentials
(thus, V(x) can be considered as a perturbed harmonic oscillator potential). The
secondgoalis to study the long-timebehaviorof (1.1). We shallproveexponential
convergence of the time-dependent solution w(t,x,ξ) towards w as t + .
∞ → ∞
In a subsequent step, we shall also prove that the stationary Wigner function w
correspondstoadensitymatrixoperatorρ T+. Remarkably,thisproofexploi∞ts
∞ ∈ 1
the positivity preservation of the time-dependent problem (using results from [9]),
via a stability property of the steady states.
To establish the existence of a (unique) steady state w , the basic idea is to
∞
prove the existence of a spectral gap for the unperturbed Wigner-Fokker-Planck
operator with quadratic potential. This implies invertibility of the (unperturbed)
WFP-operator on the orthogonal of its kernel. Assuming that the perturbation
potential is sufficiently small with respect to this spectral gap, we can set up a
fixed point iteration to obtain the existence of w . The key difficulty in doing
∞
so, is the choice of a suitable functional setting: On the one hand a Gaussian
weightedL2–spaceseemstobe anaturalcandidate,sinceitensuresdissipativityof
theunperturbedWFP-operator(seeSection3). Indeed,thisspaceisclassicalinthe
studyofthelong-timebehavioroftheclassical(kinetic)Fokker-Planckequation,see
[26]. However, it does not allow for feasible perturbations through Θ[V ]. In fact,
0
evenforsmoothandcompactlysupportedperturbationpotentialsV ,the operator
0
Θ[V ]wouldbeunbounded insuchanL2–space(duetothenon-localityofΘ[V ],see
0 0
Remark 5.2). We therefore have to enlarge the functional space and to show that
the unperturbedWFP-operatorthenstill hasa(now smaller)spectralgap. This is
a key step in our approach. It is a result from spectral and semigroup theory (cf.
Proposition4.8)whichisrelatedtoamoregeneralmathematicaltheoryofspectral
gapestimates for kinetic equations, developedinparallelin[24] (see also [29]). We
also remark that for V(x) = x2 the WFP equation corresponds to a differential
| |
operator with quadratic symbol [34] and thus our approach is closely related to
recent results for hypo-elliptic and sub-elliptic operators given in [17, 26, 31].
Comparing our methods to closely related results in the quantum mechanical
literature, we first cite [20], where several criteria for the existence of stationary
density matrices for quantum dynamical semigroups (in Lindblad form) were ob-
tained by means of compactness methods. In [5] the applicability of this general
approach to the WFP equation was established. In [22, 21] sufficient conditions
(basedoncommutatorrelationsforthe Lindbladoperators)forthe large-timecon-
vergenceofopenquantumsystems werederived. However,these techniquesdo not
provide a rate of convergence towards the steady states. In comparison to that,
the novelty of the present work consists in establishing steady states in a kinetic
framework and in proving exponential convergence rates. However, the optimality
of such rates for the WFP equation remains an open problem. In this context one
should also mention the recent work [25], in which explicit estimates on the norm
of a semigroup in terms of bounds on the resolvent of its generator are obtained,
very much along the same lines as in present paper and in [24].
Thepaperisorganizedasfollows: InSection2wepresentthebasicmathematical
setting (in particularthe classof potentials coveredin ourapproach)andstate our
two main theorems. In Section 3 we collect some known results for the case of a
purely quadratic potential and we introduce the Gaussian weighted L2–space for
thisunperturbedWFPoperator. ThisbasicsettingisthengeneralizedinSection4,
which contains the core of our (enlarged) functional framework: We shall prove
4 A.ARNOLD,I.GAMBA,M.P.GUALDANI,S.MISCHLER,C.MOUHOT,ANDC.SPARBER
newspectralgapestimatesforthe WFPoperatorwithaharmonicpotentialinL2–
spaceswith only polynomial weights. InSection 5 we provethe boundedness ofthe
operator Θ[V ] in these spaces. Finally, Section 6 concludes the proof of our main
0
result by combining the previously established elements. Appendix A includes the
rather technical proof of a preliminary step which guarantees the applicability of
the spectral method developed in [24].
2. Setting of the problem and main results
2.1. Basic definitions. In this work we shalluse the following conventionfor the
Fourier transform of a function ϕ(x):
ϕ(k):= ϕ(x)e ikxdx.
− ·
ZRd
From now on we shall assume that the (real valued, time-independent) potential
b
V, appearing in (1.1), is of the form
1
(2.1) V(x)= x2+λV (x),
0
2| |
with V C (Rd;R) and λ R some given perturbation parameter. In other
0 ∞
∈ ∈
words we consider a smooth perturbation V of the harmonic oscillator potential.
0
The precise assumption on V is listed in (2.9). An easy calculation shows that for
0
such a V the stationary equation, corresponding to (1.1), can be written as
(2.2) Lw=λΘ[V ]w,
0
where L is the linear operator
(2.3) Lw := ξ w+x w+∆ w+2div (ξw)+∆ w.
x ξ ξ ξ x
− ·∇ ·∇
Remark 2.1. When considering the slightly more general class of potentials
1
V(x)= x2+α x+λV (x), λ R, α Rd,
0
2| | · ∈ ∈
we would find, instead of (2.3), the following operator: L w := Lw +α w.
α ξ
·∇
Thus, by the change of variables x x+α we are back to (2.3).
7→
The basic idea for establishing the existence of (stationary) solutions to (2.2) is
the use of a fixed point iteration. However, L has a non-trivial kernel. Indeed it
has been proved in [34] that, in the case λ = 0, there exists a unique stationary
solution µ (R2d), satisfying
∈S
(2.4) Lµ=0
and the normalization condition
(2.5) µ(x,ξ)dxdξ =1.
ZZR2d
Explicitly, µ can be written as
(2.6) µ=ce A(x,ξ),
−
where the function A is given by
1
(2.7) A(x,ξ):= x2+2x ξ+3ξ 2 ,
4 | | · | |
and the constant c > 0 is chosen suc(cid:0)h that (2.5) holds. (cid:1)Note that for any ρ T
1
such that w L1(R2d) the following formal identity ∈
∈
trρ= ρ(x,x)dx= w(x,ξ)dxdξ,
ZRd ZZR2d
STATIONARY STATES AND LARGE TIME BEHAVIOR OF WFP 5
can be rigorously justified by a limiting procedure in T , see [1]. Since trρ is
1
proportional to the total mass of the quantum system, we can interpret condition
(2.5) as a mass normalization.
In the following, we shall denote by σ >0 the biggest constant such that
(2.8) HessA σI 0, for all(x,ξ) R2d,
− ≥ ∈
in the sense of positive definite matrices, where I denotes the identity matrix on
R2d. In the analysis of the classical FP equation, condition (2.8) is referred to as
the Bakry-Emery criterion [8]. In our case one easily computes
σ =1 1/√2.
−
In a Gaussian weighted L2–space, σ will be the spectral gap of the unperturbed
WFP-operator and hence the decay rate towards the corresponding steady µ (cf.
(3.9), (3.10) below).
The functional setting of our problem will be based on the following weighted
Hilbertspaces. WhilethestationaryandtransientWignerfunctionarerealvalued,
we need to consider function spaces over C, for the upcoming spectral analysis.
Definition 2.2. For any m N, we define := L2(R2d,ν 1dxdξ), where the
∈ Hm m−
weight is
ν 1 :=1+Am(x,ξ).
m−
We equip with the inner product
m
H
fg¯
f,g = dxdξ.
h iHm ZZR2d νm
Clearly, we have that , for all m N.
m+1 m
H ⊂H ∈
2.2. Main results. With these definitions at hand, we can now state the main
theorems of our work. Note that for the sake of transparency we did not try to
optimize the appearing constants.
Theorem 1. Let m Kd be some fixed integer, where K = K(A) (1,144] is
≥ ∈
a constant depending only on A(x,ξ) (defined in Lemma 4.2). Assume that the
perturbation potential V satisfies
0
(2.9) Γm :=Cm mj amx k∂xjV0kL∞(Rd) <+∞,
| |≤
where C > 0 depends only on m and d, as seen in the proof of Proposition 5.1.
m
Next we fix some γ (0,γ ), where γ > 0 is given in (4.7). Furthermore, let
m m m
∈
the perturbation parameter λ satisfy
e γ
m
(2.10) λ < ,
| | Γ δ
m m
where δ =δ (γ )>1 is defined in (4.15e). Then it holds:
m m m
(i) The stationary Wigner-Fokker-Planck equation (2.2) admits a unique weak
solutionew H1(R2d), satisfying w dxdξ = 1. Moreover,
w is real∞va∈lueHdman∩d satisfies w H2 (R2Rd2)d. ∞
∞ ∞ ∈ loc RR
(ii) Equation (1.1) admits a unique mild solution w C([0, ), ). In addi-
m
∈ ∞ H
tion, for any such mild solution w(t) with initial data w satsifying
0 m
∈ H
w dxdξ =1, we have
R2d 0
RR w(t) w δme−κmt w0 w , t 0,
k − ∞kHm ≤ k − ∞kHm ∀ ≥
with an exponential decay rate
κ :=γ λδ Γ >0.
m m m m
−| |
e
6 A.ARNOLD,I.GAMBA,M.P.GUALDANI,S.MISCHLER,C.MOUHOT,ANDC.SPARBER
(iii) Concerning the continuity of w =w (λ) w.r.t. λ, we have
∞ ∞
λδ Γ
m m
w µ | | µ .
k ∞− kHm ≤ γm λδmΓmk kHm
−| |
Remark 2.3. In this result, the constant σ := γ /δ > 0, roughly speaking,
m m m
playsthesameroleforLon asσ >0edoesinthecaseof (where isdefinedin
m
H H H
Definition 3.1), where itis nothing but the sizeof the spectralgap,seeProposition
e
3.5. For L on , σ also gives the exponential decay rate in the unperturbed case
H
λ = 0. For L on the situation is more complicated. Here, assertion (ii) yields
m
H
an exponential decay of the unperturbed semi-group with rate κ = γ (0,γ )
m m m
∈
and γ = σ (but possibly equal to σ, as can be seen from (4.15)). In addition,
m m
6
one should note that δm >1 may blow-up as γm γm, cf. estimate (e4.11).
ր
Theorem1is formulatedinthe Wignerpicture ofquantummechanics. We shall
e
now turn our attention to the corresponding density matrix operators ρ(t). This
is important since it is a priori not clear that w is physically meaningful – in
∞
the sense of being the Wigner transform of a positive trace class operator. To this
end we denote by ρ the Hilbert-Schmidt operator corresponding to the kernel
∞
ρ (x,y), which is obtained from w (x,ξ) by the inverse Wigner transform, i.e.
∞ ∞
x+y
ρ (x,y)= w ,ξ e iξ(x y)dξ.
− · −
∞ ZRd ∞(cid:18) 2 (cid:19)
Analogously we denote by ρ the Hilbert-Schmidt operator corresponding to the
0
initial Wigner function w .
0 m
∈H
We remarkthat the existence ofa unique mildsolutionof equation(1.1)on
m
H
will be a byproduct of our analysis.
Theorem 2. Let m Kd be some fixed integer. Let V , λ, and w satisfy the
0 0
≥
same assumptions as in Theorem 1. Then we have:
(i) The steady state ρ is a positive trace-class operator on L2(Rd), satisfying
∞
trρ =1.
(ii) Let∞ρ C([0, ),T ) be the unique density matrix trajectory corresponding
2
∈ ∞
to the mild solution of (1.1). Then, the steady state ρ is exponentially
∞
stable, in the sense that
kρ(t)−ρ∞kT2 ≤(2π)d2δme−κmtkw0−w∞kHm , ∀t≥0.
(iii) If the initial state w corresponds to a density matrix ρ T+ (and
0 ∈Hm 0 ∈ 1
hence w is real valued, trρ w dxdξ =1), then we also have
0 0 0
≡
tlim kρ(t)−RRρ∞kT1 =0.
→∞
Notethat,inthepresentedframework,wedonotobtainexponentialconvergence
towards the steady state in the T -norm but only in the sense of Hilbert-Schmidt
1
operators. This is due to the weak compactness methods involved in the proof of
Gru¨mm’s theorem (cf. the proof of Th. 2 in 6).
§
Remark 2.4. Consider now the following, more general quantum Fokker-Planck
type operator replacing the r.h.s. of (1.1):
Qw :=D ∆ w+2D div ( w)+2D div (ξw)+D ∆ w.
pp ξ pq x ξ f ξ qq x
∇
It is straightforward to extend our results to this case as long as the Lindblad
condition holds, i.e.
D2
(2.11) D 0, D D D2 + f 0.
pp ≥ pp qq− pq 4 ≥
(cid:18) (cid:19)
STATIONARY STATES AND LARGE TIME BEHAVIOR OF WFP 7
The modified quadratic function A(x,ξ) is given in [34]. The Lindblad condition
(2.11) implies that discarding in (1.1) the diffusion in x, and hence reducing the
r.h.s.to the classicalFokker-PlanckoperatorQ w :=∆ w+2div (ξw), wouldnot
cl ξ ξ
describe a “correct” open quantum system. Nevertheless, this is a frequently used
model in applications [38], yielding reasonable results in numerical simulations.
3. Basic properties of the unperturbed operator L
3.1. Functional framework. It has been shown in [34] that the operator L, de-
fined in (2.3), can be rewritten in the following form
(3.1) Lw=div( w+w( A+F)),
∇ ∇
with
1
(3.2) div(Fe A)= div(Fµ)=0.
−
c
Here and in the sequel, all differential operators act with respect to both x and ξ
(if not indicated otherwise). In (3.1), the function A is defined by (2.7) and
ξ 1 x 3ξ
(3.3) F := − A= − − .
x+2ξ −∇ 2 x+ξ
(cid:18) (cid:19) (cid:18) (cid:19)
Thereasontodosoisthat(3.1)belongstoaclassofnon-symmetricFokker-Planck
operators considered in [8]. From this point of view, a natural functional space to
study the unperturbed operator L is given by the following definition.
Definition 3.1. Let :=L2(R2d,µ 1 dxdξ), equipped with the inner product
−
H
fg¯
f,g = dxdξ.
h iH ZZR2d µ
We can now decompose L into its symmetric and anti-symmetric part in , i.e.
H
(3.4) L=Ls+Las,
where
(3.5) Lsw :=div( w+w A), Lasw:=div(Fw).
∇ ∇
It has been shown in [34], that the following property holds:
(3.6) Lsµ=0, Lasµ=0.
where µ is the stationary state defined in (2.6). Next we shall properly define
the operator L. To this end we first consider L , which is closable (w.r.t. the
C0∞
-norm) since it is dissipative:
(cid:12)
H (cid:12)
Lemma 3.2. L is dissipative, i.e. it satisfies Re Lw,w 0, for all w
C0∞ h iH ≤ ∈
C0∞(R2d). (cid:12)
(cid:12)
Proof. Using A= µ 1 µ we have, on the one hand
−
∇ − ∇
w¯ w¯ w
Lsw,w = div( w+w A)dxdξ = div µ dxdξ
h iH ZZR2d µ ∇ ∇ ZZR2d µ (cid:18) ∇(cid:18)µ(cid:19)(cid:19)
2
w
= µ dxdξ 0.
−ZZR2d (cid:12)∇(cid:18)µ(cid:19)(cid:12) ≤
(cid:12) (cid:12)
On the other hand, it foll(cid:12)ows from(cid:12)(3.2) that
(cid:12) (cid:12)
w
wdivF = F µ,
−µ ·∇
and thus
w w w
div(Fw)= µF µ ∇ =µF .
− · µ2∇ − µ ·∇ µ
(cid:18) (cid:19) (cid:18) (cid:19)
8 A.ARNOLD,I.GAMBA,M.P.GUALDANI,S.MISCHLER,C.MOUHOT,ANDC.SPARBER
An easy calculation then shows
w¯ w¯ w
Re Lasw,w =Re div(Fw)dxdξ =Re F µdxdξ
h iH ZZR2d µ ZZR2d µ ·∇(cid:18)µ(cid:19)
2
1 w
= div(Fµ)dxdξ =0,
− 2ZZR2d(cid:12)µ(cid:12)
by (3.2). To sum up we have s(cid:12)(cid:12)how(cid:12)(cid:12)n that Re Lw,w 0 holds. (cid:3)
(cid:12) (cid:12) h iH ≤
The operator L L is now closed, densely defined on and dissipative.
≡ C0∞ H
Moreover one easily sees that L = Ls Las. The main goal of this section is to
(cid:12) ∗ −
provethat L admits a sp(cid:12)ectralgapandis invertibleonthe orthogonalcomplement
of its kernel. For the first property, we start showing that L is the generator of a
C -semigroupofcontractionson . For this we recallthe followingresultfrom[9].
0
H
Lemma 3.3. Let the operator P = p (x,ξ, , ), where p is a second order
2 x ξ 2
polynomial, be defined on the domain D(P) =∇C∇(R2d). Then P is closable and
0∞
P is the maximum extension of P in L2(R2d).
C0∞
(cid:12)Severalvariants of such a result (on different functional spaces) can be found in
(cid:12)
[2, 3]. We can use this result now in order to prove that L is the generator of a
C -semigroup.
0
Lemma 3.4. L generates a C -semigroup of contractions on .
0
H
Proof. Defining v :=w/õ transforms the evolution problem
∂ w=Lw, w =w
t t=0 0 ∈H
into its analog on L2(R2d). The new unkn(cid:12)own v(t,x,ξ) then satisfies the following
(cid:12)
equation
∂ v =Hv, v =w /√µ,
t t=0 0
where H is (formally) given by
(cid:12)
(cid:12)
Hv =∆v+F v+Uv,
·∇
and the new “potential” U =U(x,ξ) reads
1 1
U = ∆A A2.
2 − 4|∇ |
Defining H on D(H)=C (R2d), we have
0∞
w
Lw =õH ,
õ
(cid:18) (cid:19)
andthusthedissipativityofLon directlycarriesovertoH H onL2(R2d).
H ≡ C0∞
Next, we consider H∗ C0∞, defined via hHf,giL2 =hf,H∗giL2, for(cid:12)(cid:12)f,g ∈C0∞(R2d).
Due to the definitions(cid:12)(cid:12) (2.7) and (3.3), the operator H∗ C0∞ is exactly of the form
neededin orderto apply Lemma 3.3. Thus, H∗ ≡H∗ C0∞(cid:12)(cid:12) is alsodissipative(onall
of its domain). Hence, the Lumer-Phillips Theorem (see [30], Section 1.4) implies
(cid:12)
that H is the generator of a C -semigroup on L2(R2d(cid:12)), denoted by eHt.
0
Reversing the transformation w v then implies that L is the generator of the
→
C -semigroup U on , given by
0 t
H
w
U w =õeHt 0 .
t 0
õ
(cid:18) (cid:19)
This finishes the proof. (cid:3)
STATIONARY STATES AND LARGE TIME BEHAVIOR OF WFP 9
3.2. Semigrouppropertieson . Theabovelemmashowsthattheunperturbed
H
WFP equation
∂ w =Lw, w =w ,
t t=0 0 ∈H
has, for all w0 , a unique mild soluti(cid:12)on w C([0, ), ), where w(t,x,ξ) =
∈ H (cid:12) ∈ ∞ H
U w (x,ξ), with U defined above. Obviously we also have U µ = µ, by (2.4).
t 0 t t
Moreover,in[34]theGreen’sfunctionofU wascomputedexplicitly. Itshowsthat
t
U conserves mass, i.e.
t
w(t,x,ξ)dxdξ = w (x,ξ)dxdξ, t 0.
0
ZZR2d ZZR2d ∀ ≥
Next, we define
(3.7) := w :w µ ,
⊥
H { ∈H ⊥ }⊂H
which is a closed subset of . Note that w µ simply means that
H ⊥
w,µ w(x,ξ)dxdξ =0.
h iH ≡ZZR2d
Hence, we have for w C (R2d), using (3.5):
∈ 0∞
Lasw,µ Lasw(x,ξ)dxdξ =0.
h iH ≡ZZR2d
Thus, Las : D(Las) . Moreover, Ls : D(Ls) , since
⊥ ⊥ ⊥ ⊥ ⊥
H ∩ → H H ∩ → H H
is spanned by the eigenfunctions of Ls (except of µ). To sum up, the operators Ls
and Las are simultaneously reducible on the two subspaces =span[µ] .
⊥
H ⊕H
We also have that U maps into itself, since for w the conservation
t ⊥ 0 ⊥
H ∈ H
of mass implies
(3.8) U w ,µ w(t,x,ξ)dxdξ = w (x,ξ)dxdξ =0, t 0.
t 0 0
h iH ≡ZZR2d ZZR2d ∀ ≥
Lemma 3.4 allows us to prove that L has a spectral gap in , in the sense that
H
(3.9) σ(L) 0 z C:Rez σ .
\{ }⊂{ ∈ ≤− }
Proposition 3.5. It holds
1
L 1 ,
k − kB(H⊥) ≤ σ
where σ >0 is defined in (2.8).
Proof. Condition (2.8) implies that Ls has a spectral gap of size σ >0 (cf. 3.2 in
§
[8],e.g.). Moreover,[8,Theorem2.19]alsoyieldsexponentialdecay(withthesame
rate) for the non-symmetric WFP equation:
(3.10) Ut(w0 µ) e−σt w0 µ .
k − k ≤ k − k
H H
Here, w has to satisfy w dxdξ = µ dxdξ = 1. By the discussion
0 ∈ H R2d 0 R2d
above, we know that L is the generator of U . Hence, (3.10) implies
RR RRt
H⊥ H⊥
(cid:12) 1 (cid:12)
(3.11) (L z)(cid:12)1 , (cid:12)z C, Rez > σ,
k − − kB(H⊥) ≤ Rez+σ ∀ ∈ −
which proves the assertion for z =0. (cid:3)
Asafinalpreparatorystepinthissection,weshallprovemoredetailedcoercivity
properties of L within . We shall denote 1 := w : w , and 1
⊥ −
H H { ∈ H ∇ ∈ H} H
will denote its dual.
10A.ARNOLD,I.GAMBA,M.P.GUALDANI,S.MISCHLER,C.MOUHOT,ANDC.SPARBER
Lemma 3.6. In the operator L satisfies
⊥
H
(3.12) Re Lw,w σ w 2 .
− h iH ≥ k kH
Similarly, there exists a constant 0<α<σ, such that
(3.13) Re Lw,w α w 21, w ⊥ 1.
− h iH ≥ k kH ∀ ∈H ∩H
Proof. We shall use the weighted Poincar´e inequality (see [8]): For any function
f L2(R2d,µdxdξ), such that fµdxdξ =0, it holds:
∈ R2d
(3.14) f 2µRdRxdξ 1 µ f 2 dxdξ.
ZZR2d| | ≤ σ ZZR2d |∇ |
Estimate (3.12) then readily follows by setting f =w/µ:
w 2 w2
Re Lw,w = µ dxdξ σ | | dxdξ.
h iH −ZZR2d (cid:12)∇(cid:18)µ(cid:19)(cid:12) ≤− ZZR2d µ
(cid:12) (cid:12)
In order to prove assertion (3.1(cid:12)3), we no(cid:12)te that
(cid:12) (cid:12)
w 2 w2 w2 µ
µ = |∇ | 2d| | div w2 ∇ ,
∇ µ µ − µ − | | µ2
(cid:12) (cid:18) (cid:19)(cid:12) (cid:18) (cid:19)
(cid:12) (cid:12)
taking into accoun(cid:12)tthat ∆(cid:12)(logµ)= 2d. Next, let 0<α<1 (to be chosenlater),
(cid:12) (cid:12) −
and write
2 2
w w
Re Lw,w = α µ dxdξ (1 α) µ dxdξ
h iH − ZZR2d (cid:12)∇(cid:18)µ(cid:19)(cid:12) − − ZZR2d (cid:12)∇(cid:18)µ(cid:19)(cid:12)
(cid:12)(cid:12)w2 (cid:12)(cid:12) w2 (cid:12)(cid:12) (cid:12)(cid:12)
= α |∇(cid:12) | dx(cid:12)dξ+2dα | | dxdξ(cid:12) (cid:12)
− ZZR2d µ ZZR2d µ
w 2
(1 α) µ dxdξ.
− − ZZR2d (cid:12)∇(cid:18)µ(cid:19)(cid:12)
(cid:12) (cid:12)
Inequality (3.14) for f =w/µ then(cid:12)implies:(cid:12)
(cid:12) (cid:12)
w 2 w2
(1 α) µ dxdξ σ(1 α) | | dxdξ.
− − ZZR2d (cid:12)∇(cid:18)µ(cid:19)(cid:12) ≤− − ZZR2d µ
(cid:12) (cid:12)
Therefore (cid:12) (cid:12)
(cid:12) (cid:12)
w2 w2
Re Lw,w α |∇ | dxdξ+(2dα σ(1 α)) | | dxdξ.
h iH ≤− ZZR2d µ − − ZZR2d µ
The choice α=σ/(σ+2d+1) yields assertion (3.13). (cid:3)
In the next sectionwe shall study the operator L in the larger functional spaces
(see Definition 2.2). This is necessary since the perturbation operator Θ[V ] is
m 0
Hunbounded in , even for V C (Rd), cf. Remark 5.2.
H 0 ∈ 0∞
4. Study of the unperturbed problem in
m
H
Inthissection,weadaptthegeneralprocedureoutlinedin[24,29]tothespecific
modelathand. Oneofthe maindifferencestothe modelsstudiedin[24]isthe fact
that the WFP operator includes a diffusion in x. Nevertheless, we shall follow the
main ideas of [24]. In a first step, this requires us to gain sufficient control on the
action of U on . After that, we establish a new decomposition of L (not to be
t m
H
confusedwiththedecompositionL=Ls+Las usedabove)inordertoliftresolvent
estimates onto the enlarged space . Together with the Gearhart-Pru¨ss
m
H ⊃ H
Theorem (cf. Theorem V.1.11 in [19]), these estimates will finally allow us to infer
exponential decay of U on .
t m
H