Long-time behaviour of a fully discrete Lagrangian scheme for a family of fourth order

1.1 MB·English

by  Horst Osberger

LONG-TIME BEHAVIOUR OF A FULLY DISCRETE LAGRANGIAN
SCHEME FOR A FAMILY OF FOURTH ORDER
HORSTOSBERGER
Abstract. AfullydiscreteLagrangianschemeforsolvingafamilyoffourthorderequations
numerically is presented. The discretization is based on the equation’s underlying gradient
flowstructurew.r.t. theL2-Wassersteindistance,andadaptsnumerousofitsmostimportant
structuralpropertiesbyconstruction,asconservationofmassandentropy-dissipation.
5 In this paper, the long-time behaviour of our discretization is analyzed: We show that
1 discretesolutionsdecayexponentiallytoequilibriumatthesamerateassmoothsolutionsof
0 the origin problem. Moreover, we give a proof of convergence of discrete entropy minimizers
2 towardsBarenblatt-profilesorGaussians,respectively,usingΓ-convergence.

n
a
J
1. Introduction
2
2
Inthispaper,weproposeandstudyafullydiscretenumericalschemeforafamilyofnonlinear

fourth order equations of the type
] ` ˘
A
B u“´ upuα´1uα q `λpxuq for xPΩ“R, tą0 (1)
N t xx x x x
andup0,.q“u0 onΩatinitialtimet“0. Theinitialdensityu0 ě0isassumedtobecompactly
.
h supported and integrable with total mass M ą 0, and we further require strict positivity of
at u0 on supppu0q “ ra,bs. For the sake of simplicity, let us further assume that M “ 1. We
m are especially interested in the long-time behaviour of discrete solutions and their rate of decay
[ towards equilibrium. For the exponent in (1), we consider values α P r1,1s, and assume λ ě 0.
2
The most famous examples for parabolic equations described by (1) are the so-called DLSS
2
equation for α “ 1, (first analysed by Derrida, Lebowitz, Speer and Spohn in [24, 25] with
v 2
0 application in semi-cunductor physics) and the thin-film equation for α“1 — indeed, for other
0 valuesofα,referencesareveryrareintheliterature,except[45]ofMatthes,McCannandSavar´e.
8 Due to the physically motivated origin of equation (1) (especially for α“ 1 and α“1), it is
4 2
notsurprisingthatsolutionsto(1)carrymanystructuralpropertiesasforinstancenonnegativity,
0
theconservationofmassandthedissipationof(several)entropyfunctionals. Insection2,weare
.
1 goingtolistmore properties ofsolutionsto(1). For thenumericalapproximationof solutions to
0
(1), it is hence natural to ask for structure-preserving discretizations that inherit at least some
5
of those properties. A minimum criteria for such a scheme should be the preservation of non-
1
: negativity,whichcanalreadybeadifficulttask,ifstandarddiscretizationsareused. Sofar,many
v
(semi-)discretizations have been proposed in the literature, and most of them keep some basic
i
X structuralpropertiesoftheequation’sunderlyingnature. Takeforexample[10,16,40,42],where
r positivity appears as a conclusion of Lyaponov functionals — a logarithmic/power entropy [10,
a
16,40]orsomevariantofa(perturbed)informationfunctional. Butthereisonlyalittlenumber
of examples, where structural properties of equation (1) are adopted from the discretization by
construction. AveryfirsttryinthisdirectionwasafullyLagrangiandiscretizationfortheDLSS
equation by Du¨ring, Matthes and Pina [26], which is based on its L2-Wasserstein gradient flow
ThisresearchwassupportedbytheDFGCollaborativeResearchCenterTRR109,“DiscretizationinGeometry
andDynamics”.
1
2 HORSTOSBERGER
representation and thus preserves non-negativity and dissipation of the Fisher-information. A
similar approach was then applied [47], again for the special case α“ 1, where we even showed
2
convergence of our numerical scheme, which was – as far as we know – the first convergence
proof of a fully discrete numerical scheme for the DLSS equation, which additionally dissipates
two Lyapunov functionals.
1.1. Description of the numerical scheme. We are now going to present a scheme, which is
practical, stable and easy to implement. In fact our dicretization seems to be so mundane that
onewouldnotassumeanyspecialpropertiestherein,atthefirstglance. Butwearegoingtoshow
later in section 2, that our numerical approximation can be derived as a natural restriction of
a L2-Wasserstein gradient flow in the potential landscape of the so-called perturbed information
functional
ż ż
` ˘
1 λ
F puq“ B uα 2dx` |x|2upxqdx, (2)
α,λ 2α x 2
Ω Ω
into a discrete Lagrangian setting, thus preserves a deep structure. The starting point for our
discretization is the Lagrangian representation of (1). Since each upt,¨q is of mass M, there is
a Lagrangian map Xpt,¨q : r0,Ms Ñ Ω — the so-called pseudo-inverse distribution function of
upt,¨q — such that
ż
Xpt,ξq
ξ “ upt,xqdx, for each ξ Pr0,Ms. (3)
´8
Written in terms of X, the Wasserstein gradient flow for F turns into an L2-gradient flow for
α,λ
« ff
ż 2 ż
1 M 1 1 λ M
F pu˝Xq“ dξ` X2dξ,
α,λ 2α Xα X 2
0 ξ ξ 0
ξ
that is,
` ˘ ` ˘
2α 1
BtX“ p2α`1q2Bξ Zα`32BξξZα`12 `λX, where Zpt,ξq:“ B Xpt,ξq “u t,Xpt,ξq . (4)
ξ
To build a bridge from (4) to the origin equation (1), remember that (1) can be written as a
transport equation,
ˆ ˙
δF puq
B u`puv q “0, with velocity field v “´ α,λ , (5)
t α x α δu
x
where δF puq{δu denotes the Eulerian first variation. So take the time derivative in equation
α,λ
(3) and use (5), then a formal calculation yields
ż
Xpt,ξq
0“B Xpt,ξqupt,Xpt,ξqq` B upt,Xpt,ξqqdx
t t
ż´8
Xpt,ξq
“B Xpt,ξqupt,Xpt,ξqq´ puv q pt,xqdx“B Xpt,ξqupt,Xpt,ξqq´puv q˝Xpt,ξq.
t α x t α
´8
This is equivalent to
B Xpt,ξq“v ˝Xpt,ξq for pt,ξqPp0,`8qˆr0,Ms,
t α
which is further equivalent to (4).
Beforewecometotheproperdefinitionofthenumericalscheme,wefixaspatio-temporaldis-
cretizationparameter∆“pτ;δq: Givenτ ą0, introducevaryingtimestepsizesτ “pτ1ř,τ2,...q
with τn Pp0,τs, then a time decomposition of r0,`8q is defined by ttnunPN with tn :“ nj“1τn.
LONG-TIME BEHAVIOUR FOR A FAMILY OF FOURTH ORDER 3
As spatial discretization, fix K PN and δ “M{K, and declare an equedistant decomposition of
the mass space r0,Ms through the set tξ uK with ξ :“kδ, k “0,...,K.
k k“0 k
Our numerical scheme is now defined as a standard discretization of equation (4):
Numerical scheme. Fix a discretization parameter ∆“pτ;δq. Then for any pα,λqPr1,1sˆ
2
r0,`8q and any initial density function u0 PL1pΩq satisfying the above requirements, a numer-
ical scheme for (1) is recursively given as follows:
(1) For n “ 0, define an initial sequence of monotone values (cid:126)x0 :“ px0,...,x0 q P RK`1
∆ 0 K
uniquely by x0 “a, x “b and
0 K
ż
x0
k
ξ “ u0pxqdx, for any k “1,...,K´1.
k
x0
k´1
Thevector(cid:126)x0 describesanon-equidistantdecompositionofthesupportra,bsoftheinitial
∆
density function u0. In any interval rx0 ,x0s, k “1,...,K, the density u0 has mass δ.
k´1 k
(2) Forně1,definerecursivelyamonotonevector(cid:126)xn :“pxn,...,xnqPRK`1 asasolution
∆ 0 K
of the system, consisting of pK`1q-many equations
” ı
xn´xn´1 2α
k τnk “ p2α`1q2δ pzkn`12qα`32rD2δ(cid:126)zα`21sk`12 ´pzkn´21qα`32rD2δ(cid:126)zα`12sk´21 `λxk, (6)
with k “0,...,K, where the values zn ě0 are defined by
(cid:96)´1
# 2
δ , for (cid:96)“1,...,K
z(cid:96)n´12 “ 0xn(cid:96)´xn(cid:96)´1 , else , (7)
and
` ˘
rD2δ(cid:126)zα`21sk´12 :“δ´2 zkα``1212 ´2zkα´`1221 `zkα´`3212 .
We later show in Proposition 10, that the solvability of the system (6) is guaranteed.
The above procedure p1q´p2q yields a sequence of monotone vectors(cid:126)x :“p(cid:126)x0 ,(cid:126)x1 ,...,(cid:126)xn,...q,
∆ ∆ ∆ ∆
and any entry (cid:126)xn defines a spatial decomposition of the compact interval rxn,xns Ă Ω, n P N.
∆ 0 K
Fixing k “ 1,...,K, the sequence n ÞÑ xn defines a discrete temporal evolution of spatial grid
k
points in Ω, and if one assigns each interval rxn ,xns a constant mass package δ, the map
k´1 k
n ÞÑ rxn ,xns characterizes the temporal movement of mass. Hence (cid:126)x is uniquely related to
k´1 k ∆
a sequence of local constant density functions u :“ pu0 ,u1 ,...,un,...q, where each function
∆ ∆ ∆ ∆
un :ΩÑR holds
∆ `
ÿK δ
unpxq“u r(cid:126)xns:“ I pxq. (8)
∆ δ ∆ xn´xn pxk´1,xks
k“1 k k´1
We will see later in section 2.1, that the information functional F can be derived using the
α,λ
dissipation of the entropy
#
ż ż
Λ Θ sα`1{2, αPp1,1s
H puq“ ϕ puqdx` α,λ |x|2upxqdx, with ϕ psq:“ αα´1{2 2 ,
α,λ Ω α 2 Ω α Θ1{2slnpsq, α“ 12
? a
withconstantsΘ :“ 2α{p2α`1q, andΛ :“ λ{p2α`1q. Asreplacementsfortheentropy
α α,λ
H and the perturbed information functional F , we introduce
α,λ α,λ
#
ÿK Λ ÿK Θ sα´1{2, αPp1,1s
H p(cid:126)xq:“δ f pz q` α,λδ |x |2, with f psq:“ αα´1{2 2 , (9)
α,λ α k´21 2 k α Θ lnpsq, α“ 1
k“1 k“0 1{2 2
4 HORSTOSBERGER
and
¨ ˛
F p(cid:126)xq:“Θ2δ ÿ ˝zkα``1212 ´zkα´`1221‚2` λδ ÿK |x |2. (10)
α,λ α δ 2 k
kPI0 k“0
K
1.2. Familiarschemes. TheconstructionofnumericalschemesasasolutionofdiscreteWasser-
steingradientflowswithLagrangianrepresentationisnotnewintheliterature. Manyapproaches
in this spirit have been realised for second-order diffusion equation [9, 11, 44, 49], but also for
chemotaxis systems [6], for non-local aggregation equations [17, 20], and for variants of the
Boltzmann equation [33]. We further refer [43] to the reader interested in a very general nu-
merical treatement of Wasserstein gradient flows. In case of fourth order equations, there are
some results for the thin-film equation and its more general version, the Hele-Shaw flow, see
[21, 33], but converegence results are missing. Rigorous stability and convergence results for
fully discrete schemes are rare and can just be found in [32, 46] for second order equations, and
in [47] for the DLSS equation. However, there are results available for semi-discrete Lagrangian
approximations, see e.g. [2, 27].
1.3. Main results. In this section, fix a discretization ∆ “ pτ;δq with τ,δ ą 0. For any
solution (cid:126)x of (11), we will further denote by u “ pu0 ,u1 ,...q the corresponing sequence of
∆ ∆ ∆ ∆
local constant density functions, as defined in (8).
All analytical results that will follow, arise from the very fundamental observation, that solu-
tionstotheschemedefinedinsection1.1canbesuccessivelyderivedasminimizersofthediscrete
minimizing movemend scheme
ÿ`
δ
(cid:126)xÞÑ x ´xn´1q2`F p(cid:126)xq. (11)
2τ k k α,λ
n
k
An immediate consequence of the minimization procedure is, that solutions (cid:126)xn dissipate the
∆
functional F .
α,λ
Concerningthelong-timebehaviourofsolutions(cid:126)x ,remarkablesimilaritiestothecontinuous
∆
case appear. Assuming first the case λ ą 0, it turns out that the unique minimizer (cid:126)xmin of
δ
H is even a minimizer of the discrete information functional F , and the corresponding set
α,λ α,λ
of density functions umin “ u r(cid:126)xmins converges for δ Ñ 0 towards a Barenblatt-profile b or
δ δ δ α,λ
Gaussian b , respectively, that is defined by
1{2,λ
` ˘
b “ a´b|x|2 1{pα´1{2q, b“ α?´1{2Λ if αą1{2 and (12)
α,λ ` 2α α,λ
b1{2,λ “ae´Λ1{2,λ|x|2 if α“1{2, (13)
whereaPRischosentoconserveunitmass. Beyondthis,solutions(cid:126)xn satisfying(6)convergeas
∆
nÑ8towardsaminimizer(cid:126)xminofF withanexponentialdecayrate,whichis”asymptotically
δ α,λ
equal”totheoneobtainedinthecontinuouscase. Theaboveresultsaremergedinthefollowing
theorems:
Theorem 1. Assume λą0. Then the sequence of minimizers umin holds
δ
umin ÝδÝÑÝÑ0 b , strongly in LppΩq for any pě1, (14)
δ α,λ
uˆmin ÝδÝÑÝÑ0 b , uniformly on Ω, (15)
δ α,λ
where uˆmin is a locally affine interpolation of umin defined in Lemma 16.
δ δ
LONG-TIME BEHAVIOUR FOR A FAMILY OF FOURTH ORDER 5
Theorem 2. For λ ą 0, any sequence of monotone vectors (cid:126)x satisfying (11) dissipates the
∆
entropies H and F at least exponential, i.e.
α,λ α,λ
` ˘
Hα,λp(cid:126)xn∆q´Hmα,iλn ď Hα,λp(cid:126)x0∆q´Hmα,iλn e´1`2λλτtn, and (16)
` ˘
Fα,λp(cid:126)xn∆q´Fmα,iλn ď Fα,λp(cid:126)x0∆q´Fmα,iλn e´1`2λλτtn, (17)
withHmin “H p(cid:126)xminqandFmin “F p(cid:126)xminq. Theassociatedsequenceofdensitiesu further
α,λ α,λ δ α,λ α,λ δ ∆
holds
` ˘
}un∆´umδin}2L1pΩq ďcα,λ Hα,λp(cid:126)x0∆q´Hmα,iλn e´1`2λλτtn, (18)
for any time step n“1,2,..., where c ą0 depends only on α,λ.
α,λ
Let us now consider the zero-confinement case λ “ 0. In the continuous setting, the long-
time behaviour of solutions to (1) with λ “ 0 can be studied by a rescaling of solutions to (1)
with λ ą 0. We are able to translate this methode into the discrete case and derive a discrete
counterpartof[45,Corollary5.5],whichdescribestheintermediateasymptoticsofsolutionsthat
approach self-similar Barenblatt profiles as tÑ8.
Theorem 3. Assume λ “ 0 and take a sequence of monotone (cid:126)xn satisfying (11). Then there
∆
exists a constand c ą0 depending only on α, such that
α
b
` ˘
}un∆´bn∆,α,0}L1pΩq ďcα Hα,0p(cid:126)x0∆q´Hmα,i0npR∆nq´1, with R∆n :“ 1`aτp2α`3qtn bτp21α`3q,
where bn is a rescaled discrete Barenblatt profile and a ,b ą 0, such that a ,b Ñ 1 for
∆,α,0 τ τ τ τ
τ Ñ0, see section 3.2 for more details.
Before we come to the analytical part of this paper, we want to point out the following:
The ideas for the proofs of Theorem 2 and 3 are mainly guided by the techniques developd in
[45]. The remarkable observation of this work is the fascinating structure preservation of our
discretization, which allows us to adapt nearly any calculation from the continuous theory for
the discrete setting.
1.4. Structure of paper. In the following section 2, we point out some of the main structural
features of equation (1) and the functionals H and F , and show that our scheme rises
α,λ α,λ
from a discrete L2-Wasserstein gradient flow, so that many properties of the continuous flow
are inherited. Section 3 treats the analysis of discrete equilibria in case of positive confinement
λ ą 0: we prove convergence of discrete stationary states to Barenblatt-profiles or Gaussians,
respectivelly,andanalysetheasymptoticsofdiscretesolutionsforλ“0. Finally,somenumerical
experiments are presented in section 4.
2. Structural properties — continuous vs. discrete case
2.1. Structuralpropertiesofequation(1). Thefamilyoffourthorderequations(1)carriesa
bunchofremarkablestructuralproperies. Themostfundamentaloneistheconservationofmass,
i.e. tÞÑ}upt,¨q} isaconstantfunctionfortPr0,`8qandattainsthevalueM :“}u0} .
L1pΩq L1pΩq
This is a naturally given property, if one interprets solutions to (1) as gradient flows in the
potential landscape of the perturbed information functional
ż ż
` ˘
1 λ
F puq“ B uα 2dx` |x|2upxqdx, (19)
α,λ 2α x 2
Ω Ω
equippedwiththeL2-WassersteinmetricW . Asanimmediateconsequence,F isaLyapunov
2 α,λ
functional, and one can find infinitely many other (formal) Lyapunov functionals at least for
6 HORSTOSBERGER
special choices of α — see [7, 12, 37] for α“ 1 or [3, 18, 29] for α“1. Apart from F , one of
2 α,λ
the most important such Lyapunov functionals is given by the Λ -convex entropy
α,λ
#
ż ż
Λ Θ sα`1{2, αPp1,1s
H puq“ ϕ puqdx` α,λ |x|2upxqdx, ϕ psq:“ αα´1{2 2 . (20)
α,λ Ω α 2 Ω α Θ1{2slnpsq, α“ 12
It turns out that the functionals F and H are not just Lyapunov functionals, but share
α,λ α,λ
numerous remarkable similiarities. One can indeed see (1) as an higher order extension of the
second order porous media/heat equation [36]
B v “´grad H pvq“´Θ B pvαq`Λ pxuq , (21)
s W2 α,λ α xx α,λ x
which is nothing less than the L2-Wasserstein gradient flow of H . Furthermore, the unper-
α,λ
turbed functional F , i.e. λ“Λ “0, equals the dissipation of H along its own gradient
α,0 α,λ α,0
flow,
d
F pvpsqq“´ H pvpsqq. (22)
α,0 ds α,0
In view of of the gradient flow structure, this relation makes equation (1) the “big brother”
of the porous media/heat equation (21), see [23, 45] for structural consequences. Another as-
tonishing common feature is the correlation of F and H by the so-called fundamental
α,λ α,λ
entropy-information relation: For any uPPpΩq with H puqă8, it holds
α,λ
F puq“|grad H |2`p2α´1qΛ H puq, for any λě0, (23)
α,λ W2 α,λ α,λ α,λ
see [45, Corollary 2.3]. This equation is a crucial tool for the analysis of equilibria of both
functionals and the corresponding long-time behaviour of solutions to (1) and (21).
In addition to the above listing, a typical property of diffusion processes like (1) or (21) with
positive confinement λ,Λ ą0 is the convergence towards unique stationary solutions u8 and
α,λ
v8,respectively,independentofthechoiceofinitialdata. Itismaybeoneofthemostsurprising
facts, that both equations (1) and (21) share the same steady state, i.e. the stationary solutions
u8 and v8 are identical. Those stationary states are solutions of the elliptic equations
` ˘
´ P puq `Λ pxuq “0, (24)
α xx α,λ x
with P psq:“Θ sα`1{2, and have the form of Barenblatt profils or Gaussians, respectively, see
α α
definition(12)&(13). ThiswasfirstobservedbyDenzlerandMcCannin[23],andfurtherstudied
in [45] using the Wasserstein gradient flow structure of both equations and their remarkable
relation via (22).
In case of αPt1,1u, the mathematical literature is full of numerous results, which is because
2
of the physical importance of (1) in those limiting cases.
2.1.1. DLSS euqation. As already mentioned at the very beginning, the DLSS equation — (1)
with α “ 1 — rises from the Toom model [24, 25] in one spatial dimension on the half-line
2
r0,`8q, and was used to describe interface fluctuations, therein. Moreover, the DLSS equation
alsofindsapplicationinsemi-conductorphysics,namelyasasimplifiedmodel(low-temperature,
field-free) for a quantum drift diffusion system for electron densities, see [39].
From the analytical point of view, a big variety of results in different settings has been
developed in the last view decades. For results on existence and uniqueness, we refer f.i.
[7,28,35,30,38,39], and[12,19,14,30,38,41,45]forqualitativeandquantitativedescriptions
of the long-time behaviour. The main reason, which makes the research on this topic so non-
trivial, is a lack of comparison/maximum principles as in the theory of second order equations
(21). And, unfortunatelly, the abscents of such analytical tools is not neglectable, as the work
[7] of Bleher et.al shows: as soon as a solution u of (1) with α “ 1 is strictly positive, one can
2
LONG-TIME BEHAVIOUR FOR A FAMILY OF FOURTH ORDER 7
show that it is even C8-smooth, but there are no regularity results available from the moment
when u touches zero. The problem of strictly positivity of such solutions seems to be a difficult
task, since it is still open. This is why alternative theories for non-negative weak solutions have
more and more become matters of great interest, as f.i. an approach based on entropy methodes
developed in [30, 38].
2.1.2. Thin-film equation. The thin-film equation — (1) with α “ 1 — is of similar physically
importance as the DLSS equation, since it gives a dimension-reduced description of the free-
surface problem with the Navier-Stokes equation in the case of laminar flow, [48]. In case of
linearmobility—whichisexactlythecaseinoursituation—thethin-filmequationcanalsobe
usedtodescribethepinchingofthinnecksinaHele-Shaw cellinonespatialdimension,andthus
plays an extraordinary role in physical applications. To this topic, the literature provides some
interesting results in the framework of entropy methods, see [13, 18, 29]. In the (more generel)
case of non-negative mobility functions m, i.e.
B u“´divpmpuqD∆uq, (25)
t
one of the first achievements to this topic available in the mathematical literature was done
by Bernis and Friedman [4]. The same equation is observed in [5], treating a vast number of
results to numerous mobility functions of physical meaning. There are several other references
in this direction, f.i. Gru¨n et. al [3, 22, 34], concerning long-time behaviour of solutions and the
non-trivial question of spreading behaviour of the support.
2.2. Structure-preservation of the numerical scheme. In this section, we try to get a
better intuition of the scheme in section 1.1. Foremost we will derive (6) as a discrete system
of Euler-Lagrange equations of a variational problem that rises from a L2-Wasserstein gradient
flow restricted on a discrete submanifold P pΩq of the space of probability measures PpΩq on Ω.
δ
This is why the numerical scheme holds several discrete analogues of the results discussed in the
previoussection. Asthefollowingsectionshows,someoftheinheritedpropertiesareobtainedby
construction (f.i. preservation of mass and dissipation of the entropy), where others are caused
bytheunderlyingdicsretegradientflowstructureandasmartchoiceofadiscreteL2-Wasserstein
distance. Moreover it is possible to prove that the entropy and the information functional share
the same minimizer (cid:126)xmin even in the discrete case, and solutions of the discrete gradient flow
δ
convergeswithanexponentialratetothisstationarystate. Theproveofthisobservationismore
sophisticated, thatiswhywededicateanownsection(section3)tothetreatmentofthisspecial
property.
2.2.1. Ansatz space and discrete entropy/information functionals. The entropies H and F
α,λ α,λ
as defined in (20)&(2) are non-negative functionals on PpΩq. If we first consider the zero-
confinementcaseλ“0,onecanderiveinanalogyto[46]thediscretizationin(9)ofH justby
α,0
restriction to a finite-dimensional submanifold P pΩq of PpΩq: For fixed K P N, the set P pΩq
δ δ
consists of all local constant density functions u “ u r(cid:126)xs (remember definition (8)), such that
δ
(cid:126)xPRK`1 is a monotone vector, i.e.
(cid:32) ˇ (
(cid:126)xPx :“ px ,...,x qˇx ăx ă...ăx ăx ĎRK`1.
δ 0 K 0 1 K´1 K
Such density functions u “ u r(cid:126)xs P P pΩq bear a one-to-one relation to their Lagrangians or
δ δ
Lagrangian maps, which are defefined on the mass grid r0,Ms with uniform decomposition p0“
ξ ,...,ξ ,...,ξ “ Mq. More precicely, we define for(cid:126)x P x the local affine and monotonically
0 k K δ
increasingfunctionX“X r(cid:126)xs:r0,MsÑΩ, suchthatXpξ q“x foranyk “0,...,K. Itthen
δ k k
holds u˝X “ 1 for u P P pΩq and its corresponding Lagrangian map. For later analysis, we
Xξ δ
8 HORSTOSBERGER
introduceinadditiontothedecompositiontξ uK theintermediatevaluespξ ,ξ ,...,ξ q
k k“0 k´1 3 K´1
2 2 2
by ξ “ 1pξ `ξ q for k “1,...,K.
k´1 2 k k´1
2
In view of the entropy’s discretization, this implies using (9)&(20), a change of variables
x“X r(cid:126)xs, and the definition (7) of the(cid:126)x-dependent vectors(cid:126)z
δ
ż
` ˘ ÿK
H p(cid:126)xq“H pu r(cid:126)xsq“ ϕ u r(cid:126)xs dx“δ f pz q,
α,0 α,0 δ α δ α k´1
Ω k“1 2
which is perfectly compatible with (9). Obviously, one cannot derive the discrete information
functional F in the same way, since F is not defined on P pΩq. So instead of restriction,
α,0 α,0 δ
we mimic property (22) that is for any(cid:126)xPx
δ
F p(cid:126)xq“δ´1B H p(cid:126)xqTB H p(cid:126)xq“x∇ H p(cid:126)xq,∇ H p(cid:126)xqy .
α,0 (cid:126)x α,0 (cid:126)x α,0 δ α,0 δ α,0 δ
Here,thekthcomponentofB fp(cid:126)xqholdsrB fp(cid:126)xqs “B fp(cid:126)xqforanyk “0,...,K andarbitrary
(cid:126)x (cid:126)x k xk
function f : x Ñ R. Moreover, we set ∇ fp(cid:126)xq “ δ´1B fp(cid:126)xq and introduce for (cid:126)v,w(cid:126) P RK`1 the
δ δ (cid:126)x
scalar product x¨,¨y by
δ
ÿK b
x(cid:126)v,w(cid:126)y “δ v w , with induced norm }(cid:126)v} “ x(cid:126)v,(cid:126)vy .
δ k k δ δ
k“0
Example 4. Each component z of(cid:126)z“z r(cid:126)xs is a function on x , and
κ δ δ
e ´e
B z “´z2 κ`12 κ´12, (26)
(cid:126)x κ κ δ
where we denote for k “0,...,K by e PRK`1 the pk`1qth canonical unit vector.
k
Remark 5. One of the most fundamental properties of the L2-Wasserstein metric W on PpΩq
2
in one space dimension is its excplicit representation in terms of Lagrangian coordinates. We
refer [1, 50] for a comprehensive introduction to the topic. This enables to prove the existence of
K-independent constants c ,c ą0, such that
1 2
c }(cid:126)x´(cid:126)y} ďW pu r(cid:126)xs,u r(cid:126)ysqďc }(cid:126)x´(cid:126)y} , for all(cid:126)x,(cid:126)yPx . (27)
1 δ 2 δ δ 2 δ δ
A proof of this statement for Ω“ra,bsĂp´8,`8q is given in [46,Lemma7], and can be easily
recomposed for Ω“R.
Let us further introduce the sets of (semi)-indizes
! )
1 3 1
I0 “t0,1,...,Ku, and I1{2 “ , ,...,K´ .
K K 2 2 2
The calculation (26) in the above example yields the expizit representation of the gradient
B H p(cid:126)xq,
(cid:126)x α,λ
ÿ e ´e
α`1 κ´1 κ`1
B(cid:126)xHα,0p(cid:126)xq“Θαδ zκ 2 2 δ 2, (28)
κPI1{2
K
and further of the discretized information functional
¨ ˛
ÿ zα`21 ´zα`21 2
F p(cid:126)xq“}∇ H p(cid:126)xq}2 “Θ2δ ˝ k`12 k´12 ‚ .
α,0 δ α,0 δ α δ
kPI0
K ş
In the case of positive confinemend λą0, we note that the drift potential uÞÑ |x|2upxqdx
Ω
holds an equivalent representation in terms of Lagrangian coordinates, that is namely X ÞÑ
LONG-TIME BEHAVIOUR FOR A FAMILY OF FOURTH ORDER 9
ş
M|Xpξq|2dξ. In our setting, the simplest discretization of this functional is hence by summing-
0
up over all values x weighted with δ. This yields
k
ÿ ÿ
Λ λ
H p(cid:126)xq“H p(cid:126)xq` α,λδ |x |2, and F p(cid:126)xq“F p(cid:126)xq` δ |x |2
α,λ α,0 2 k α,λ α,0 2 k
kPI0 kPI0
K K
as an extension to the case of positive λ, which is nothing else than (9)&(10). Note in addition,
ř
that δ |x |2 “}(cid:126)x}2.
kPI0 k δ
K
A first structural property of the above simple discretization is convecity retention from the
continuous to the discrete setting:
Lemma 6. The functional(cid:126)xÞÑH is Λ -convex, i.e.
α,λ α,λ
` ˘
Λ
H p1´sq(cid:126)x`s(cid:126)y ďp1´sqH p(cid:126)xq`sH p(cid:126)yq´ α,λp1´sqs}(cid:126)x´(cid:126)y}2 (29)
α,λ α,λ α,λ 2 δ
for any (cid:126)x,(cid:126)y P x and s P p0,1q. It therefore admits a unique minimizer (cid:126)xmin P x . If we further
δ δ δ
assume Λ ą0, then it holds for any(cid:126)xPx
α,λ δ
› ›
Λα,λ ›(cid:126)x´(cid:126)xmin›2 ďH p(cid:126)xq´H p(cid:126)xminqď 1 }∇ H p(cid:126)xq}2. (30)
2 δ δ α,λ α,λ δ 2Λ δ α,λ δ
α,λ
Proof. If we prove (29), then the existence of the unique minimizer is a consequence [46, Propo-
sition 10]. By definition and a change of variables, we get for αPp1,1s
2
#
ż
M Θ s1{2´α, αPp1,1s
H p(cid:126)xq“H pu r(cid:126)xsq“ ψ pX r(cid:126)xs q dξ, with ψ psq“ αα´1{2 2 ,
α,0 α,0 δ 0 α δ ξ α ´Θ1{2lnpsq, α“ 12
hence(cid:126)xÞÑH p(cid:126)xq is convex. Since the functional(cid:126)xÞÑ}(cid:126)x}2 holds trivially
α,0 δ
}p1´sq(cid:126)x`s(cid:126)y}2 ďp1´sq}(cid:126)x}2`s}(cid:126)y}2´p1´sqs}(cid:126)x´(cid:126)y}2
δ δ δ δ
for any(cid:126)x,(cid:126)yPx and sPp0,1q, the functionals H p(cid:126)xq“H p(cid:126)xq` Λα,λ }(cid:126)x}2 hold (29).
δ α,λ α,0 2 δ
Deviding (29) by są0 and passing to the limit as sÓ0 yields
Λ
H p(cid:126)xq´H p(cid:126)yqďB H p(cid:126)xqp(cid:126)x´(cid:126)yq´ α,λ }(cid:126)x´(cid:126)y}2.
α,λ α,λ (cid:126)x α,λ 2 δ
The second inequality of (30) easily follows from Young’s inequality |ab| ď ε|a|2 `p2εq´11|b|2
2
with ε“p2δΛ q´1, and even holds for arbitrary(cid:126)yPx .
α,λ δ
To get the first inequaltiy of (30), we set(cid:126)x“(cid:126)xmin and again devide (29) by są0, then
δ
` ˘
Hα,λ p1´sq(cid:126)xmδin`s(cid:126)y ´Hα,λp(cid:126)xmδinq ďH p(cid:126)yq´H p(cid:126)xminq´ Λα,λp1´sq››(cid:126)xmin´(cid:126)y››2,
s α,λ α,λ δ 2 δ δ
where the left hand side is obviously non-negative for any są0. Since są0 was arbitrary, the
statement is proven. (cid:3)
As a further conclusion of our natural discretization, we get a discrete fundamental entropy-
information relation analogously to the continuous case (23).
Corollary 7. For any λě0, every(cid:126)xPx with H p(cid:126)xqă8 we have
δ α,0
F p(cid:126)xq“}∇ H p(cid:126)xq}2`p2α´1qΛ H p(cid:126)xq, for αPp1,1s and (31)
α,λ › δ α,λ δ› α,λ α,λ 2
F p(cid:126)xq“›∇ H p(cid:126)xq›2`Λ , for α“ 1 (32)
1{2,λ δ 1{2,λ δ 1{2,λ 2
10 HORSTOSBERGER
Remark 8. Note that the above seemingly appearing discontinuity at α “ 1 is not real. For
2
αą 1, the second term in right hand side of (31) is explicitly given by
2 ¨ ˛
p2α´1qΛ H p(cid:126)xq“p2α´1qΛ ˝Θ δ ÿ zκα´1{2 ` Λα,λ }(cid:126)x}2‚
α,λ α,λ α,λ α α´1{2 2 δ
κPI1{2
K
ÿ
Λ
“2Λ Θ δ zα´1{2`p2α´1q α,λ }(cid:126)x}2,
α,λ α κ 2 δ
κPI1{2
K
ř
For α Ó 1, one gets Λ Ñ Λ , Θ Ñ 1 and especially δ zα´1{2 Ñ M “ 1, The
2 α,λ 1{2,λ α 2 κPI1{2 κ
K
drift-term vanishes since p2α´1qÑ0.
Proof of Corollary 7 . Let us first assume α P p1,1s. A straight-forward calculation using the
2
definition of }.} , ∇ and B H in (28) yields
δ δ (cid:126)x α,λ
}∇ H p(cid:126)xq}2 “δ´1xB H p(cid:126)xq,B H p(cid:126)xqy
δ α,λ δ (cid:126)x α,λ (cid:126)x α,λ ÿ ÿ
“}∇δHα,0p(cid:126)xq}2δ ´2ΘαΛα,λδ zκα´12 `Λ2α,λδ |xk|2. (33)
κPI1K{2 kPI0K
Here we used the explicit representation of B H p(cid:126)xq, remember (28),
(cid:126)x α,λ
ÿ e ´e ÿ
α`1 κ´1 κ`1
B(cid:126)xHα,λp(cid:126)xq“Θαδ zκ 2 2 δ 2 `Λα,λδ xkek,
κPI1K{2 kPI0K
and especially the definition of (7), which yields
C G
ÿ e ´e ÿ ÿ x ´x
δ´1 Θαδ zκα`12 κ´12 δ κ`21,Λα,λδ xkek “ΘαΛα,λδ zκα`12 κ´12 δ κ`12
κPI1K{2 kPI0K κPI1Kÿ{2
α´1
“´ΘαΛα,λδ zκ 2
κPI1{2
K
aSince α ‰ 12, we can write 2Θα “ p2α ´ 1qα´Θ1α{2. Further note that the relation Λα,λ “
λ{p2α`1q yields
ˆ ˙ ˆ ˙ ˆ ˙
λ λ 1 λ α´1{2 λ 2α´1
Λ2 “ “ “ 1´ “ 1´ .
α,λ 2α`1 2 α`1{2 2 α`1{2 2 2α`1
Using this information and the definition of H , we proceed in the above calculations by
α,0 ˆ ˙
ÿ
λ 2α´1
}∇ H p(cid:126)xq}2 “F p(cid:126)xq´p2α´1qΛ H p(cid:126)xq` 1´ δ |x |2
δ α,λ δ α,0 α,λ α,0 2 2α`1 k
kPI0
K
λ ÿ Λ2 ÿ
“F p(cid:126)xq´p2α´1qΛ H p(cid:126)xq` δ |x |2´p2α´1q α,λδ |x |2
α,0 α,λ α,0 2 k 2 k
kPI0 kPI0
K K
“F p(cid:126)xq´p2α´1qΛ H p(cid:126)xq.
α,λ α,λ α,λ
a
In case of α“ 1, we see that Θ “ 1, and Λ “ λ{2. We hence conclude in (33)
2 1{2 2 1{2,λ
› › › › ÿ ÿ
›∇ H p(cid:126)xq›2 “›∇ H p(cid:126)xq›2´Λ δ z0` λδ |x |2 “F p(cid:126)xq´Λ
δ 1{2,λ δ δ 1{2,0 δ 1{2,λ κ 2 k α,λ 1{2,λ
κPI1K{2 kPI0K
l