Nonlinear diffusions: extremal properties of Barenblatt profiles, best matching and delays

1.5 MB·English

by  Jean Dolbeault

Nonlinear diffusions: extremal properties of Barenblatt
profiles, best matching and delays
Jean Dolbeault · Giuseppe Toscani
5
January9,2015
1
0
2
Abstract In this paper, we consider functionals based on moments and non-

n
linear entropies which have a linear growth in time in case of source-type so-
a
lutions to the fast diffusion or porous medium equations, that are also known
J
asBarenblattsolutions.Asfunctionsoftime,thesefunctionalshaveconvexity
5
1 properties for generic solutions, so that their asymptotic slopes are extremal
for Barenblatt profiles. The method relies on scaling properties of the evo-

] lution equations and provides a simple and direct proof of sharp Gagliardo-
P
Nirenberg-Sobolev inequalities in scale invariant form. The method also gives
A
refined estimates of the growth of the second moment and, as a consequence,
.
h establishes the monotonicity of the delay corresponding to the best matching
t BarenblattsolutioncomparedtotheBarenblattsolutionwithsameinitialsec-
a
m ondmoment.Herethenotionofbestmatchingisdefinedintermsofarelative
entropy.
[

1
Keywords Nonlineardiffusionequations Source-typesolutions Gagliardo-
v
· ·
6 Nirenberg-Sobolev inequalities Improved inequalities Scalings Second
· · ·
4 moment Temperature R´enyi entropy Best matching Barenblatt profiles
6 Delay · · · ·
3
0
. Mathematics Subject Classification (2010) primary: 35K55 35K65
1 · ·
35B40; secondary: 46E35 39B62 49J40
0
· ·
5
1
: J.Dolbeault
v
Ceremade(UMRCNRSno.7534),Universit´eParis-Dauphine,PlacedeLattredeTassigny,
Xi F-75775ParisC´edex16,France
E-mail:[email protected]
r
a G.Toscani
UniversityofPaviaDepartmentofMathematics,ViaFerrata1,27100Pavia,Italy
E-mail:[email protected]
2 JeanDolbeault,GiuseppeToscani
1 Introduction
Consider the nonlinear diffusion equation in Rd, d 1,
≥
∂u
=∆up, (1)
∂t
with initial datum u(x,t=0)=u (x) 0 such that
0
≥
(cid:90) 1(cid:90)
u dx=1, Θ(0)= x2u dx<+ . (2)
Rd 0 d Rd| | 0 ∞
It is known since the work of A. Friedman and S. Kamin, [14], that the large
timebehaviorofthesolutionsto(1)isgovernedbythesource-typeBarenblatt
solutions
1 (cid:16) x (cid:17)
(t,x):= (3)
U(cid:63) (cid:0)κt1/µ(cid:1)d B(cid:63) κt1/µ
where
(cid:12)2µp(cid:12)1/µ
µ:=2+d(p 1), κ:=(cid:12) (cid:12)
− (cid:12)p 1(cid:12)
−
and is the Barenblatt profile
(cid:63)
B

(x):=(cid:0)C(cid:63)−|x|2(cid:1)1+/(p−1) ifp>1,
(cid:63)
B (cid:0)C + x2(cid:1)1/(p−1) ifp<1.
(cid:63)
| |
Here (s) = max s,0 and, for p > 1 2, the constant C is chosen so that
(cid:82) + { } − d (cid:63)
(x)dx=1.Solution(3)wasfoundaround1950byYa.B.Zel’dovichand
RdB(cid:63)
A.S. Kompaneets. Later G.I. Barenblatt analyzed the solution representing
heatreleasefromapointsourceforp>1.See[24,1],and[22]formoredetails.
The result of [14], subsequently improved in a number of papers like,
e.g.,[13],guaranteestovariousextentsthattheBarenblattsolutioncanfruit-
fullybeusedtoobtainthelarge-timebehaviorofallothersolutions,departing
from an initial value like in (2). Precise estimates of the difference in time of
basic quantities of the real solution with respect to the source-type one are
howeverdifficulttoobtainandadditionalconditionsareneededforlowvalues
of p>0.
Asmainexample,letusconsiderthegrowthofthesecondmomentsofthe
solution to (1), for an initial value satisfying (2). For a Barenblatt solution,
we easily obtain the exact growth
1(cid:90) κ2 (cid:90)
x2 (t,x)dx=t2/µΘ where Θ := x2 (x)dx.
d Rd| | U(cid:63) (cid:63) (cid:63) d Rd| | B(cid:63)
For p > 1 it has been shown in [19] that this time-behaviour is captured
by any other solution to (1) satisfying conditions (2) for large times. Further
studiespublishedin[12]thenrevealedthat,atleastinthefastdiffusionrange,
there is a nondecreasing delay between the propagation in time of the second
ExtremalpropertiesofBarenblattprofilesanddelays 3
moments of the source-type solution and the generic solution to (1). We shall
giveasimpleexplanationforthisfactinTheorem3,andalsoshowthatinthe
porous medium case p>1 the delay is nonincreasing.
A way to look at this phenomenon is the following. The second moment
oftheBarenblattsolution,raisedtopowerµ/2,growslinearlyintime.Hence,
its first variation in time of Θ (t)µ/2 is constant, while the second variation
(cid:63)
is equal to zero. To see how the second moment of any other solution to
equation (1) behaves in time, it comes natural to estimate the time variations
of G:=Θµ/2, where the second moment functional is defined by
1(cid:90)
Θ(t):= x2u(t,x)dx.
d Rd| |
Thisanalysiswillleadtotheinterestingobservationthatthesecondvariation
of G has a fixed sign, which implies concavity or convexity in t depending
on the value of p. Also, since the first variation is invariant with respect to
dilations and the solution to (1) is asymptotically self-similar, it follows at
once that the first variation satisfies a sharp inequality which connects the
second moment to its first variation, and optimality is achieved by Barenblatt
profiles.
Thisideacanalsobeappliedtothestudyofthebehaviorofthederivative
of the second moment, as predicted by the nonlinear diffusion. It holds
Θ(cid:48) =2E
where, up to a sign, the generalized entropy functional is defined by
(cid:90)
E(t):= u(t,x)pdx.
Rd| |
FortheBarenblattsolution(3)onecaneasilycheckthatE (t)σ growslinearly
(cid:63)
intimeifd(1 p)σ =µ.ThisleadstoestimatethetimevariationofF:=Eσ.
−
Asinthepreviouscase,wewillconcludebyshowingthatthesecondvariation
in time of F has a negative sign if p<1, which implies concavity. In this case
also, the first variation is invariant with respect to dilations, which implies a
sharpinequalityofGagliardo-Nirenberg-Sobolevtype,andoptimalityisagain
achieved by Barenblatt profiles.
As the results of this paper clearly indicate, looking at functionals of the
solution to the nonlinear diffusion(1) which grow linearly in time when evalu-
ated along the source-type solution represent a valid and powerful alternative
to the well-known entropy-entropy production method developed in the last
two decades to investigate the large-time behavior of the solution by scaling
the problem in order to obtain a Fokker-Planck equation with a fixed steady
state.
This new way of looking at the problem opens completely new questions,
which at the moment seem to be very difficult to deal with. For instance,
higher order derivatives in time of the second moment of the solution are
easilyevaluatedincorrespondencetoBarenblattsolutions,bygivingaprecise
4 JeanDolbeault,GiuseppeToscani
growth, say t−σn, where n is the order of the derivative. Hence, one can think
to proceed as for the first two cases, by considering the power of order σ of
n
the n-th derivative and by checking if there is a sign.
With Theorem 1, we shall start by a simplified proof of the isoperimet-
ric inequality for R´enyi entropy powers in [18], based on based on the linear
growth of the functional F, that is also stated in [4, Theorem 4.4] and empha-
size the fact that the method is limited to exponents p 1 1. Using this
≥ − d
inequality, J.A. Carrillo and G. Toscani have been able to establish improved
ratesofconvergenceinthenon-asymptoticregimeofthesolutionsto(1)in[4,
Theorem 5.1]. Since R´enyi entropy powers are equivalent to relative entropies
relative to best matching Barenblatt solutions underappropriateconditionson
the second moments as shown in [9,10], improved rates have to be related
with results obtained in rescaled variables in [10], where the scales were de-
finedin termsof second moments.Equivalently,the improved entropy-entropy
production inequality of [10, Theorem 1] is similar in nature to the isoperi-
metric inequality for R´enyi entropy powers. As we shall observe in this paper,
the isoperimetric inequality for R´enyi entropy powers is in fact a Gagliardo-
Nirenberg inequality in scale invariant form, which degenerates into Sobolev’s
inequality in the limit case p=1 1 and explains why it cannot hold true for
p<1 1.Theentropy-entropy pr−odduction inequality of[8]isalsoaGagliardo-
−d
Nirenberg inequality, but not in scale invariant form and it has recently been
established in [11] that the difference between the scale invariant form and
the non scale invariant form is enough to account for improved rates of con-
vergence in Fokker-Planck type equations, i.e., in nonlinear diffusions after a
convenient rescaling.
Theorem 2 is devoted to a rather simple observation on second moments,
which is however at the core of our paper. It is based on the linear growth of
the functional G. By introducing the relative entropy with respect to the best
matching Barenblatt function, we establish the result of Theorem 3 on delays.
This result has been proved in [12] by a much more complicated method,
when p < 1. Here we simply rely on the convexity or concavity of t G(t),
(cid:55)→
depending whether p < 1 or p > 1, and both cases are covered. This analysis
also suggests a method to obtain estimates of the delays, that is investigated
in Section 5.
Let us conclude this introduction with a brief review of the literature.
For considerations on second moment methods in fast diffusion and porous
medium equations, we refer to [19,5,12] and references therein. Sharp Ga-
gliardo-Nirenberg-Sobolev inequalities have been studied in [8,10,18,11] from
the point of view of the rates of convergence of the solutions to (1) in the
intermediateasymptoticsregime,andalsoforobtainingimprovedconvergence
rates in the initial regime. A counterpart of such improved rates is the notion
of delay which was established in [12] in the fast diffusion regime and will be
recovered as a very simple consequence of moment estimates in Theorem 3
and extended to the porous medium case.
ExtremalpropertiesofBarenblattprofilesanddelays 5
Theorems1and2followthesamelineofthought:computethetimeevolu-
tion of a functional which grows linearly when evaluated in the case of Baren-
blatt functions and has some concavity or convexity property otherwise. As
a main consequence, we provide a proof of some Gagliardo-Nirenberg-Sobolev
inequalities which goes along the lines of [3,8,2] on the one hand, of [18,4,21]
on the other hand, and makes a synthetic link between the two approaches.
The first approach is inspired by the entropy functional introduced by J. Ral-
ston and W.I. Newman in [16,17], also known in the literature as the Tsallis
entropy, while the second one is more related with R´enyi entropies and con-
nected with information theory inspired by [7,23]. The reader interested in
further details is invited to refer to [21] and references therein for a more
detailed account.
2 Notations, main results and consequences
The entropy is defined by
(cid:90)
E:= updx
Rd
and the Fisher information by
(cid:90) p
I:= u v 2dx with v = up−1.
Rd |∇ | p 1
−
If u solves (1), then
E(cid:48) =(1 p)I.
−
To compute I(cid:48), we will use the fact that
∂v
=(p 1)v∆v+ v 2 (4)
∂t − |∇ |
and get that
F:=Eσ
has a linear growth asymptotically as t + if
→ ∞
(cid:18) (cid:19)
µ 2 1 2 1
σ = =1+ +p 1 = 1.
d(1 p) 1 p d − d 1 p −
− − −
ThisdefinitionisthesameastheoneofSection1.Thegrowthisexactlylinear
in case of Barenblatt profiles, so that Eσ−1I is independent of t if we denote
(cid:63) (cid:63)
by E , I , F etc. the entropy, the Fisher information, etc. of these Barenblatt
(cid:63) (cid:63) (cid:63)
profiles.
Theorem 1 Assume that p 1 1 if d > 1 and p > 0 if d = 1. With the
≥ − d
above notations, t F(t) is increasing, (1 p)F(cid:48)(cid:48)(t) 0 and
(cid:55)→ − ≤
1
lim F(t)=(1 p)σ lim Eσ−1I=(1 p)σEσ−1I .
t→+∞ t − t→+∞ − (cid:63) (cid:63)
6 JeanDolbeault,GiuseppeToscani
Thisresulthasbeenestablishedin[18].Inthispaperwegiveaslightlysimpler
proof. The result of Theorem 1 amounts to state that t F(t) is concave if
(cid:55)→
p<1andconvexifp>1,withanasymptoticslopegivenby(1 p)σEσ−1I .
− (cid:63) (cid:63)
Moreover, we observe that the inequality
Eσ−1I Eσ−1I (5)
≥ (cid:63) (cid:63)
is equivalent to one of the two following Gagliardo-Nirenberg inequalities:
(i) If 1 1 p<1, then (5) is equivalent to
− d ≤
w θ w 1−θ C w (6)
(cid:107)∇ (cid:107)L2(Rd)(cid:107) (cid:107)Lq+1(Rd) ≥ GN(cid:107) (cid:107)L2q(Rd)
where
d q 1 d
θ = − , 1<q
q d+2 q(d 2) ≤ d 2
− − −
and equality with optimal constant C is achieved by p−1/2.
GN (cid:63)
B
(ii) If p>1, then (5) is equivalent to
(cid:18)(cid:90) (cid:19)1−θ
2q
w θ w2qdx C w (7)
(cid:107)∇ (cid:107)L2(Rd) Rd ≥ GN(cid:107) (cid:107)Lq+1(Rd)
where
d q 1
θ = −
q d+2 q(d 2)
− −
and q takes any value in (0,1).
In both cases, we relate Gagliardo-Nirenberg inequalities and (5) by
w 1
up−1/2 = with q = .
w 2p 1
L2q(Rd)
(cid:107) (cid:107) −
These considerations show that p=1 1 correspond to Sobolev’s inequality:
− d
2q = 2d/(d 2) and θ = 1 if d 3. This case is therefore the threshold case
− ≥
for the validity of the method. Details will be given in Section 3.
Using the moment
1(cid:90)
Θ := x2udx,
d Rd| |
we can introduce the functional
G:=Θ1−η/2 with η =d(1 p).
−
As a function of t, G also has a linear growth in case of Barenblatt profiles.
Again this definition is compatible with the one of Section 1 since
µ η
=1 .
2 − 2
In the general case, we get that
G(cid:48) =µH,
ExtremalpropertiesofBarenblattprofilesanddelays 7
where we define the R´enyi entropy power functional H by
H:=Θ−η/2E=Θd2(p−1)E
and observe that the corresponding functional H for Barenblatt profiles is
(cid:63)
independent of t.
Theorem 2 Assume that p 1 2. With the above notations, t G(t) is
≥ − d (cid:55)→
increasing, (1 p)H(cid:48)(t) 0 and
− ≥
1 (cid:16) η(cid:17) (cid:16) η(cid:17)
lim G(t)= 1 lim H= 1 H .
t→+∞ t − 2 t→+∞ − 2 (cid:63)
The function t G(t) is convex if p < 1 and concave if p > 1, with an
(cid:55)→
asymptotic slope given by (1 η/2)H . Details will be given in Section 4.
(cid:63)
−
We can also consider the relative entropy with respect to the best matching
Barenblatt function defined as
[u]:= inf [u s]
F s>0F |U(cid:63)
where s(x) = (s,x) = s−d/µ 1(s−1/µx) is defined by (3). Here the vari-
U(cid:63) U(cid:63) U(cid:63)
able s plays the role of a scaling parameter, and the relative entropy with
respect to a given function is defined by
U
[u ]:= 1 (cid:90) (cid:104)up p p p−1(cid:0)u (cid:1)(cid:105)dx.
F |U p 1 Rd −U − U −U
−
A key observation in [9] is the fact that [u] = [u s] is achieved by a
F F |U(cid:63)
unique Barenblatt solution which satisfies
(cid:90) (cid:90)
x2udx= x2 sdx.
Rd| | Rd| | U(cid:63)
The proof of this fact is a straightforward computation: as a functional of ,
U
[u ] is concave and has at most one maximum point. Hence if we write
F |U
that s(x)p/(p−1) =a(s)+b(s) x2, then
|U(cid:63) | | |
d [u s]= p (cid:90) (cid:0)a(cid:48)(s)+b(cid:48)(s) x2(cid:1)(u s)dx.
dsF |U(cid:63) p 1 Rd | | −U(cid:63)
−
Since(cid:82) udx=(cid:82) sdx=1,thetermproportionaltoa(cid:48)doesnotcontribute
Rd RdU(cid:63)
and the other one vanishes if and only if the moments are equal.
After undoing the change of variables, we get that
(cid:18) (cid:19)µ
Θ 2
s= . (8)
Θ
(cid:63)
With this choice of s, s is the best matching Barenblatt function in the sense
U(cid:63)
that this Barenblatt function minimizes the relative entropy [u s] among
F |U(cid:63)
all Barenblatt functions ( s) .
U(cid:63) s>0
8 JeanDolbeault,GiuseppeToscani
If u is a solution to (1), for any t 0, we can define s as a function of t
≥
and consider the delay which is defined as
(cid:18) (cid:19)µ
Θ(t) 2
τ(t):= t.
Θ −
(cid:63)
The main result of this paper is that t τ(t) is monotone.
(cid:55)→
Theorem 3 Assume that p 1 1 and p = 1. With the above notations,
≥ − d (cid:54)
the best matching Barenblatt function of a solution u to (1) satisfying (2) is
(t,x) (t+τ(t),x) and the function t τ(t) is nondecreasing if p > 1
(cid:63)
and no(cid:55)→ninUcreasing if 1 1 p<1. (cid:55)→
− d ≤
Withsgivenby(8),wenoticethattherelative entropy with respect to the best
matching Barenblatt function is given by
1 (cid:90) (cid:104) (cid:105)
[u s]= up ( s)p dx
F |U(cid:63) p 1 Rd − U(cid:63)
−
and it is related with the the R´enyi entropy power functional by
H−H(cid:63) =Θd2(p−1)E−Θ(cid:63)d2(p−1)E(cid:63) =(p−1)Θd2(p−1)F[u|U(cid:63)s].
See Figs. 1-2. An estimate of the delay t τ(t) will be given in Section 5.
(cid:55)→
Fast diffusion case
µ/2
s (s + τ(0))Θ
s Θµ/2(s)
µ/2
s (s+ τ(t))Θ
µ/2
s (s+ τ )Θ s
∞
0 t
τ(0)
τ(t)
τ = lim τ(s)
s +
∞ → ∞
Fig.1 Inthefastdiffusioncase,the(bold)curves(cid:55)→G(s)=Θµ/2(s)isconvex,increasing,
anditsasymptoteiss(cid:55)→(s+τ∞)Θ(cid:63)µ/2.Atanytimet≥0,weobservethatitcrossesthe
line s(cid:55)→(s+τ(t))Θµ/2 =(s−t+1)Θµ/2(t) transversally, so that t(cid:55)→τ(t) is monotone
(cid:63)
decreasing,unlessthesolutionisitselfaBarenblattsolutionuptoatimeshift.
ExtremalpropertiesofBarenblattprofilesanddelays 9
µ/2
s (s+ τ )Θ
∞
Porous medium case s Θµ/2(s)
µ/2
s (s+ τ(t))Θ
µ/2
s (s+ τ(0))Θ
s
0 t
τ(0)
τ(t)
τ = lim τ(s)
s +
∞ → ∞
Fig. 2 Intheporousmediumcase,thecurves(cid:55)→Θµ/2(s)isstillincreasingbutconcave,
so that t(cid:55)→τ(t) is monotone increasing, unless the solution is itself a Barenblatt solution
uptoatimeshift.
3 A direct proof of Gagliardo-Nirenberg inequalities
This section is devoted to the proof of Theorem 1 and also to a proof of
Gagliardo-Nirenberg inequalities (6) and (7). We provide a slightly simplified
proof of Lemma 4 below, compared to the existing literature: see [18] and
references therein. Some of the key computations will be reused in Section 5.
3.1 A preliminary computation
According to [18, Appendix B], we have the following result.
Lemma 4 If u solves (1) with initial datum u(x,t=0)=u (x) 0 suchthat
0
(cid:82) u dx=1 and 1(cid:82) x2u dx<+ , then v = p up−1 solv≥es (4) and
Rd 0 d Rd| | 0 ∞ p−1
d (cid:90) (cid:90) (cid:16) (cid:17)
I(cid:48) = u v 2dx= 2 up D2v 2+(p 1)(∆v)2 dx. (9)
dt Rd |∇ | − Rd (cid:107) (cid:107) −
10 JeanDolbeault,GiuseppeToscani
Proof Let us give a simplified proof of this result. Using (1) and (4), we can
compute
d (cid:90)
u v 2dx
dt Rd |∇ |
(cid:90) ∂u (cid:90) ∂v
= v 2dx+ 2 u v dx
Rd ∂t |∇ | Rd ∇ ·∇∂t
(cid:90) (cid:90) (cid:16) (cid:17)
= ∆(up) v 2dx+ 2 u v (p 1)v∆v+ v 2 dx
Rd |∇ | Rd ∇ ·∇ − |∇ |
(cid:90)
= up∆ v 2dx
Rd |∇ |
(cid:90) (cid:90)
+2(p 1) uv v ∆vdx+ 2(p 1) u v v∆vdx
− Rd ∇ ·∇ − Rd ∇ ·∇
(cid:90)
+2 u v v 2dx
Rd ∇ ·∇|∇ |
(cid:90)
= up∆ v 2dx
− Rd |∇ |
(cid:90) (cid:90)
+2(p 1) uv v ∆vdx+ 2(p 1) u v v∆vdx
− Rd ∇ ·∇ − Rd ∇ ·∇
where the last line is given by an integration by parts:
(cid:90) (cid:90) (cid:90)
u v v 2dx= (up) v 2dx= up∆ v 2dx.
Rd ∇ ·∇|∇ | Rd∇ ·∇|∇ | − Rd |∇ |
1) Using the elementary identity
1
∆ v 2 = D2v 2+ v ∆v,
2 |∇ | (cid:107) (cid:107) ∇ ·∇
we get that
(cid:90) (cid:90) (cid:90)
up∆ v 2dx=2 up D2v 2dx+2 up v ∆vdx.
Rd |∇ | Rd (cid:107) (cid:107) Rd ∇ ·∇
2) Since u v = (up), an integration by parts gives
∇ ∇
(cid:90) (cid:90)
u v v∆vdx= (up) v∆vdx
Rd ∇ ·∇ Rd∇ ·∇
(cid:90) (cid:90)
= up(∆v)2dx up v ∆vdx
− Rd − Rd ∇ ·∇
and with uv = p up we find that
p−1
(cid:90) (cid:90)
2(p 1) uv v ∆vdx+ 2(p 1) u v v∆vdx
− Rd ∇ ·∇ − Rd ∇ ·∇
(cid:90) (cid:90)
= 2(p 1) up(∆v)2dx+2 up v ∆vdx.
− − Rd Rd ∇ ·∇
Collecting terms establishes (9).
(cid:116)(cid:117)