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Ann. I. H. Poincaré – AN 30 (2013) 917–934

www.elsevier.com/locate/anihpc

Improved interpolation inequalities, relative entropy and fast diffusion equations

Jean Dolbeault

a,

, Giuseppe Toscani

b

aCeremade (UMR CNRS no. 7534), Université Paris-Dauphine, Place de Lattre de Tassigny, F-75775 Paris Cédex 16, France bUniversity of Pavia Department of Mathematics, Via Ferrata 1, 27100 Pavia, Italy

Received 4 December 2012; received in revised form 21 December 2012; accepted 22 December 2012 Available online 11 January 2013

Abstract

We consider a family of Gagliardo–Nirenberg–Sobolev interpolation inequalities which interpolate between Sobolev’s inequality and the logarithmic Sobolev inequality, with optimal constants. The difference of the two terms in the interpolation inequalities (written with optimal constant) measures a distance to the manifold of the optimal functions. We give an explicit estimate of the remainder term and establish an improved inequality, with explicit norms and fully detailed constants. Our approach is based on nonlinear evolution equations and improved entropy–entropy production estimates along the associated flow. Optimizing a relative entropy functional with respect to a scaling parameter, or handling properly second moment estimates, turns out to be the central technical issue. This is a new method in the theory of nonlinear evolution equations, which can be interpreted as the best fit of the solution in the asymptotic regime among all asymptotic profiles.

©2013 Elsevier Masson SAS. All rights reserved.

MSC:26D10; 46E35; 35K55

Keywords:Gagliardo–Nirenberg–Sobolev inequalities; Improved inequalities; Manifold of optimal functions; Entropy–entropy production method; Fast diffusion equation; Barenblatt solutions; Second moment; Intermediate asymptotics; Sharp rates; Optimal constants

1. Introduction and main results

Consider the following sub-family of the Gagliardo–Nirenberg–Sobolev inequalities

f2pCp,dGNfθ2f1p+θ1 (1)

withθ=θ (p):=pp1d+2dp(d2), 1< pdd2 ifd3 and 1< p <∞ifd=2. Such an inequality holds for any smooth functionf with sufficient decay at infinity and, by density, for any functionf ∈Lp+1(Rd)such that∇f is square integrable. We shall assume thatCp,dGN is the best possible constant in (1). In[16], it has been established that equality holds in (1) iff=Fpwith

Fp(x)=

1+ |x|2p−11

x∈Rd (2)

* Corresponding author.

E-mail addresses:dolbeaul@ceremade.dauphine.fr(J. Dolbeault),giuseppe.toscani@unipv.it(G. Toscani).

0294-1449/$ – see front matter ©2013 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2012.12.004

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and that all extremal functions are equal toFp up to a multiplication by a constant, a translation and a scaling. See Appendix Afor an expression ofCp,dGN. Ifd3, the limit casep=d/(d−2)corresponds to Sobolev’s inequality and one recovers the optimal functions found by T. Aubin and G. Talenti in[3,23]. Whenp→1, the inequality becomes an equality, so that we may differentiate both sides with respect top and recover the euclidean logarithmic Sobolev inequality in optimal scale invariant form (see[20,25,16]for details).

It is rather straightforward to observe that inequality (1) can be rewritten, in a non-scale invariant form, as anon- homogeneous Gagliardo–Nirenberg–Sobolev inequality: for anyf∈Lp+1D1,2(Rd),

Rd

|∇f|2dx+

Rd

|f|p+1dxKp,d Rd

|f|2pdx γ

(3) with

γ=γ (p, d):=d+2−p(d−2)

dp(d−4) . (4)

The optimal constantKp,d can easily be related withCp,dGN. Indeed, by optimizing the left hand side of (3) written for fλ(x):=λd/(2p)f (λx)for anyx∈Rd, with respect toλ >0, one recovers that (3) and (1) are equivalent. The detailed relation betweenKp,dandCp,dGNcan be found in Section7.

Define now CM:=

M M

d2(p−1)p(d4)

, M:=

Rd

1+ |x|2 2p

p1dx=πd2 Γdp(d4)

2(p1)

Γ 2p

p1

.

Consider next a generic, non-negative optimal function, fM,y,σ(p) (x):=σ4pd

CM+1

σ|xy|2 1

p1

x∈Rd

and let us define the manifold of the optimal functions as M(p)d :=

fM,y,σ(p) : (M, y, σ )Md

.

We shall measure the distance toM(p)d with the functional R(p)[f] := inf

gM(p)d

Rd

g1p

|f|2pg2p

− 2p p+1

|f|p+1gp+1 dx.

To simplify our statement, we will introduce a normalization constraint and assume thatf ∈L2p(R2, (1+ |x|2) dx) is such that

Rd|x|2|f|2pdx (

Rd|f|2pdx)γ =d(p−1)σMγ1

d+2−p(d−2), σ(p):=

4d+2−p(d−2) (p−1)2(p+1)

4p

dp(d4)

. (5)

Such a condition is not restrictive, as it is always possible to cover the general case by rescaling the inequality, but significantly simplifies the expressions. As we shall see in the proof, the only goal is to fixσ=1.

Our main result goes as follows.

Theorem 1. Letd 2,p >1 and assume that p < d/(d−2)if d 3. For anyf ∈Lp+1D1,2(Rd)such that condition(5)holds, we have

Rd

|∇f|2dx+

Rd

|f|p+1dxKp,d Rd

|f|2pdx γ

Cp,d

(R(p)[f])2 (

Rd|f|2pdx)γ whereγ is given by(4).

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Although we do not know its optimal value, we are able to give an expression ofCp;d(see Appendix A), which is such that

plim1+Cp,d=0 and lim

pd/(d2)Cp,d=0.

The space Lp+1D1,2(Rd) is the natural space for Gagliardo–Nirenberg inequalities as it can be characterized as the completion of the space of smooth functions with compact support with respect to the norm · such that f2= ∇f22+ f2p+1. In this paper, we shall also use the notations fp,q :=(

Rd|x|p|f|qdx)1/q, so that fq= f0,q.

Under condition (5), we shall deduce from Theorem4that R(p)[f]CCKf2p(γ2p 2) inf

gM(p)d

|f|2pg2p2

1 (6)

withδ=d+2−p(d+6)for some constantCCK whose expression is given in Section3, Eq. (13). Putting this estimate together with the result of Theorem1, with

Cp,d:=Cd,pCCK2, we obtain the following estimate.

Corollary 2.Under the same assumptions as in Theorem1, we have

Rd

|∇f|2dx+

Rd

|f|p+1dxKp,d Rd

|f|2pdx γ

Cp,df2p(γ2p 4) inf

gMd(p)

|f|2pg2p4

1.

The critical casep=d/(d−2)corresponding to Sobolev’s inequality raises a number of difficulties which are not under control at this stage. However, results which have been obtained in such a critical case, by different methods, are the main motivation for the present paper.

In[9, Question (c), p. 75], H. Brezis and E. Lieb asked the question of what kind of distance toM(p)d is controlled by the difference of the two terms in the critical Sobolev inequality written with an optimal constant. Some partial answers have been provided over the years, of which we can list the following ones. First G. Bianchi and H. Egnell gave in[5]a result based on the concentration-compactness method, which determines a non-constructive estimate for a distance to the set of optimal functions. In[15], A. Cianchi, N. Fusco, F. Maggi and A. Pratelli established an improved inequality using symmetrization methods. Also see[14]for an overview of various results based on such methods. Recently another type of improvement, which relates Sobolev’s inequality to the Hardy–Littlewood–

Sobolev inequalities, has been established in[17], based on the flow of a nonlinear diffusion equation, in the regime of extinction in finite time. Theorem1does not provide an answer in the critical case, but gives an improvement with fully explicit constants in the subcritical regime. Our method of proof enlightens a new aspect of the problem. Indeed, Theorem1shows that the difference of the two terms in the critical Sobolev inequality provides a better control under the additional information thatf2,2pis finite. Such a condition disappears in the setting of Corollary2.

In this paper, our goal is to establish an improvement of Gagliardo–Nirenberg inequalities based on the flow of the fast diffusion equationin the regime of convergence towards Barenblatt self-similar profiles, with an explicit measure of the distance to the set of optimal functions. Our approach is based on arelative entropyfunctional. The method relies on a recent paper[19], which is itself based on a long series of studies on intermediate asymptotics of the fast diffusion equation, and on the entropy–entropy production method introduced in [4,2]in the linear case and later extended to nonlinear diffusions: see[21,22,16,12,11]. In this setting, having a finite second moment is crucial. Let us give some explanations.

Consider the fast diffusion equation with exponentmgiven in terms of the exponentpof Theorem1by p= 1

2m−1 ⇐⇒ m=p+1

2p . (7)

More specifically, form(0,1), we shall consider the solutions of

∂u

∂t + ∇ · u

ηum1−2x

=0 t >0, x∈Rd (8)

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with initial datumu(t=0,·)=u0. Hereηis a positive parameter which does not depend ont. Letube the unique stationary solution such thatM=

Rdu dx=

Rdudx. It is given by u(x)=

K+1

η|x|2 1

m1

x∈Rd

for some positive constantKwhich is uniquely determined byM. The following exponents are associated with the fast diffusion equation (8) and will be used all over this paper:

mc:=d−2

d , m1:=d−1

d and m˜1:= d d+2.

To the critical exponent 2p=2d/(d−2)for Sobolev’s inequality corresponds the critical exponentm1for the fast diffusion equation. Ford3, the conditionp(1, d/(d−2))in Theorem1is equivalent tom(m1,1)while for d=2,p(1,)meansm(1/2,1).

It has been established in[21,22]that therelative entropy(orfree energy) F[u|u] := 1

m−1

Rd

umummum1(uu) dx

decays according to d

dtF

u(·, t )|u

= −I

u(·, t )|u ifuis a solution of (8), where

I

u(·, t )|u

:=η m 1−m

Rd

uum1− ∇um12dx

is theentropy production termorrelative Fisher information.Ifm∈ [m1,1), according to[16], these two functionals are related by a Gagliardo–Nirenberg interpolation inequality, namely

F[u|u]1

4I[u|u]. (9)

We shall give a concise proof of this inequality in the next section (see Remark1) based on the entropy–entropy production method, which amounts to relate dtdI[u(·, t )|u]andI[u(·, t )|u]. We shall later replace the diffusion parameter ηin (8) by a time-dependent coefficientσ (t ), which is itself computed using the second moment of u,

Rd|x|2u(x, t ) dx. By doing so, we will be able to capture thebest matchingBarenblatt solution and get improved decay rates in the entropy–entropy production inequality. Elementary estimates allow to rephrase these improved rates into improved functional inequalities forf such that|f|2p=u, for anyp(1, d/(d−2)), as in Theorem1.

This paper is organized as follows. In Section 2, we apply the entropy–entropy production method to the fast diffusion equation as in[11]. The key computation, without justifications for the integrations by parts, is reproduced here since we need it later in Section6, in the case of a time-dependent diffusion coefficient. Next, in Section3, we establish a new estimate of Csiszár–Kullback type. By requiring a condition on the second moment, we are able to produce a new estimate which was not known before, namely to directly control the difference of the solution with a Barenblatt solution in L1(Rd).

Second moment estimates are the key of a recent paper and we shall primarily refer to[19]in which the asymptotic behavior of the solutions of the fast diffusion equation was studied. In Section4we recall the main results that were proved in[19], and that are also needed in the present paper.

With these preliminaries in hand, an improved entropy–entropy production inequality is established in Section5, which is at the core of our paper. It is known since[16]that entropy–entropy production inequalities amount to optimal Gagliardo–Nirenberg–Sobolev inequalities. Such a rephrasing of our result in a more standard form of functional inequalities is done in Section6, which contains the proof of Theorem1. Further observations have been collected in Section7. One of the striking results of our approach is that all constants can be explicitly computed. This is somewhat technical although not really difficult. To make the reading easier, explicit computations have been collected inAppendix A.

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2. The entropy–entropy production method

Consider a solutionu=u(x, t )of Eq. (8) and define z(x, t ):=ηum1−2x

so that Eq. (8) can be rewritten as

∂u

∂t + ∇ ·(uz)=0.

To keep notations compact, we shall use the following conventions. IfA=(Aij)di,j=1andB=(Bij)di,j=1 are two matrices, letA:B=d

i,j=1AijBij and|A|2=A:A. Ifaandbtake values inRd, we adopt the definitions:

a·b= d i=1

aibi, ∇ ·a= d i=1

∂ai

∂xi, ab=(aibj)di,j=1, ∇ ⊗a= ∂aj

∂xi d

i,j=1

.

Later we will need a version of the entropy–entropy production method in case of a time-dependent diffusion coefficient. Before doing so, let us recall the key computation of the standard method. With the above notations, it is straightforward to check that

∂z

∂t =η(1m)

um2∇ ·(uz)

and ∇ ⊗z=η∇ ⊗ ∇um1−2 Id. With these definitions, the time-derivative of 1mmηI[u|u] =

Rdu|z|2dxcan be computed as d

dt

Rd

u|z|2dx=

Rd

∂u

∂t|z|2dx+2

Rd

uz·∂z

∂t dx.

The first term can be evaluated by

Rd

∂u

∂t|z|2dx= −

Rd

∇ ·(uz)|z|2dx

=2

Rd

uzz: ∇ ⊗z dx

=2η

Rd

uzz: ∇ ⊗ ∇um1dx−4

Rd

u|z|2dx

=2η(1−m)

Rd

um2u⊗ ∇ :(uzz) dx−4

Rd

u|z|2dx

=2η(1−m)

Rd

um2(u·z)2dx+2η(1−m)

Rd

um1(u·z)(∇ ·z) dx

+2η(1−m)

Rd

um1(z⊗ ∇u):(∇ ⊗z) dx−4

Rd

u|z|2dx.

The second term can be evaluated by 2

Rd

uz·∂z

∂t dx=2η(1−m)

Rd

(uz· ∇)

um2∇ ·(uz) dx

= −2η(1−m)

Rd

um2

∇ ·(uz)2

dx

(6)

= −2η(1−m)

Rd

um(∇ ·z)2+2um1(u·z)(∇ ·z)+um2(u·z)2 dx.

Summarizing, we have found that

Rd

∂u

∂t|z|2dx+4

Rd

u|z|2dx= −2η(1−m)

Rd

um2

u2(∇ ·z)2+u(u·z)(∇ ·z)u(z⊗ ∇u):(∇ ⊗z) dx.

Using the fact that

2zj

∂xi∂xj =2zi

∂xj2, we obtain that

Rd

um1(u·z)(∇ ·z) dx= −1 m

Rd

um(∇ ·z)2dx− 1 m

Rd

um d i,j=1

zi 2zj

∂xi∂xj

dx

and

Rd

um1(z⊗ ∇u):(∇ ⊗z) dx= 1 m

Rd

um|∇z|2dx+ 1 m

Rd

um d i,j=1

zi2zi

∂xj2 dx can be combined to give

Rd

um2

u(u·z)(∇ ·z)uuz: ∇ ⊗z

dx= −1 m

Rd

um(∇ ·z)2dx+ 1 m

Rd

um|∇z|2dx.

This shows that d dt

Rd

u|z|2dx+4

Rd

u|z|2dx= −2η1−m m

Rd

um

|∇z|2(1m)(∇ ·z)2

dx. (10)

By the arithmetic geometric inequality, we know that

|∇z|2(1m)(∇ ·z)20

if 1−m1/d, that is, ifmm1. Altogether, we have formally established the following result.

Proposition 3.Letd1,m(m1,1)and assume thatuis a non-negative solution of (8)with initial datumu0in L1(Rd)such thatum0 andx→ |x|2u0are both integrable onRd. With the above defined notations, we get that

d dtI

u(·, t )|u

−4I

u(·, t )|u

t >0.

The proof of such a result requires to justify that all integrations by parts make sense. We refer to [12,13]for a proof in the porous medium case (m >1) and to[11]form1m <1. The casem=1 was covered long ago in[4].

Remark 1.Proposition3provides a proof of (9). Indeed, with a Gronwall estimate, we first get that I

u(·, t )|u

I[u0|u]e4tt0

ifI[u0|u]is finite. SinceI[u(·, t )|u]is non-negative, we know that

tlim→∞I

u(·, t )|u

=0,

which proves the convergence ofu(·, t )touast→ ∞. As a consequence, we also have limt→∞F[u(·, t )|u] =0 and since

d dt

I

u(·, t )|u

−4F

u(·, t )|u

= d dtI

u(·, t )|u +4I

u(·, t )|u

0,

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an integration with respect toton(0,)shows that I[u0|u] −4F[u0|u]0,

which is precisely (9) written foru=u0. 3. A Csiszár–Kullback inequality

Letm(m˜1,1)withm˜1=d+d2 and consider the relative entropy Fσ[u] := 1

m−1

Rd

umBσmmBσm1(uBσ) dx

for some Barenblatt function Bσ(x):=σd2

CM+ 1

σ|x|2m11

x∈Rd (11)

whereσis a positive constant andCM is chosen such thatBσ1=M >0. Withpandmrelated by (7), the definition ofCMcoincides with the one of Section1. See details inAppendix A.

Theorem 4. Let d 1, m(m˜1,1) and assume that u is a non-negative function in L1(Rd) such that um and x→ |x|2uare both integrable onRd. Ifu1=Mand

Rd|x|2u dx=

Rd|x|2Bσdx, then Fσ[u]

σd2(1m) m 8

RdB1mdx

CMuBσ1+ 1 σ

Rd

|x|2|uBσ|dx 2

.

Notice that the condition

Rd|x|2u dx=

Rd|x|2Bσdxis explicit and determinesσuniquely:

σ= 1 KM

Rd

|x|2u dx withKM :=

Rd

|x|2B1dx.

For further details, see Lemma5and (20) below, andAppendix Afor detailed expressions ofKM and

RdB1mdx.

With this choice ofσ, sinceBσm1=σd2(1m)CM+σd2(mcm)|x|2, we remark that

RdBσm1(uBσ) dx=0 so that the relative entropy reduces to

Fσ[u] := 1 m−1

Rd

umBσm dx.

Proof of Theorem4. Letv:=u/Bσ andσ :=Bσmdx. With these notations, we observe that

Rd

(v−1) dμσ=

Rd

Bσm1(uBσ) dx

=σd2(1m)CM

Rd

(uBσ) dx+σd2(mcm)

Rd

|x|2(uBσ) dx=0.

Thus

Rd

(v−1) dμσ=

v>1

(v−1) dμσ

v<1

(1v) dμσ =0, which, coupled with

v>1

(v−1) dμσ +

v<1

(1v) dμσ =

Rd

|v−1|σ,

(8)

implies

Rd

|uBσ|Bσm1dx=

Rd

|v−1|σ =2

v<1

|v−1|σ.

On the other hand, a Taylor expansion shows that Fσ[u] = 1

m−1

Rd

vm−1−m(v−1)

σ=m 2

Rd

ξm2|v−1|2σ

for some functionξ taking values in the interval(min{1, v},max{1, v}), thus giving the lower bound Fσ[u]m

2

v<1

ξm2|v−1|2σ m 2

v<1

|v−1|2σ.

Using the Cauchy–Schwarz inequality, we get

v<1

|v−1|σ 2

=

v<1

|v−1|B

m

σ2B

m

σ2 dx 2

v<1

|v−1|2σ

Rd

Bσmdx

and finally obtain that Fσ[u]m

2 (

v<1|v−1|σ)2

RdBσmdx =m 8

(

Rd|uBσ|Bσm1dx)2

RdBσmdx , which concludes the proof. 2

Notice that the inequality of Theorem4can be rewritten in terms of|f|2p=uandg2p=Bσwithp=1/(2m−1).

SeeAppendix Afor the computation of

RdBσmdx,σ,CMandKM in terms of

Rd|x|2u dxandM. In the framework of Corollary2, we observe that condition (5) can be rephrased as

σ= 1

KMf2p2,2p= 1 K1

f2p2,2p

f2pγ2p =σ. (12)

Altogether we find in such a case that R(p)[f] =p−1

p+1Fσ[u]CCK|f|2p− |g|2p2

1

with

CCK=p−1 p+1

d+2−p(d−2)

32p σd

p1

4p M1γ. (13)

Remark 2.Various other estimates can be derived, based on second order Taylor expansions. For instance, as in[16], we can write that

Fσ[u] =

Rd

ψ vm

ψ (1)ψ(1)

vm−1 σ

withv:=u/Bσ andψ (s):=1mms1/m, and get Fσ[u] 1

m22m vm−12L1/m(Rd,dμσ)

max{vmL1/m(Rd,dμσ),1L1/m(Rd,dμσ)}2m1 . UsingvmL1/m(Rd,dμσ)= 1L1/m(Rd,dμσ)= Bσmm1 and

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Rd

umBσmdx=

Rd

umBσmBσm(m1)Bσm(1m)dx

vm−1

L1/m(Rd,dμσ)Bσm1m

1

by the Cauchy–Schwarz inequality, we find Fσ[u]umBσm21

m22mBσm1

.

Withf =um12, this also gives another estimate of Csiszár–Kullback type, namely R(p)[f] κp,d

f2,2pd2(p1)f2p12(d+2p(d2)) inf

gM(p)d

|f|p+1gp+12

1,

for some positive constant κp,d, which is valid for any p(1,) ifd =2 and any p(1,dd2]if d 3. Also see[24,12,10,18]for further results on Csiszár–Kullback type inequalities corresponding to entropies associated with porous media and fast diffusion equations.

4. Recent results on the optimal matching by Barenblatt solutions

Consider onRdthe fast diffusion equation with harmonic confining potential given by

∂u

∂t + ∇ · u

σd2(mmc)um1−2x

=0, t >0, x∈Rd, (14)

with initial datumu0. Hereσ is a function oft. Let us summarize some results obtained in[19]and the strategy of their proofs.

Result 1.At any timet >0, we can choose thebest matching Barenblattas follows. Consider a given functionuand optimizeλFλ[u].

Lemma 5.For any givenu∈L1+(Rd)such thatumand|x|2uare both integrable, ifm(m˜1,1), there is a unique λ=λ>0which minimizesλFλ[u], and it is explicitly given by

λ= 1 KM

Rd

|x|2u dx

whereKM=

Rd|x|2B1dx. Forλ=λ, the Barenblatt profileBλsatisfies

Rd

|x|2Bλdx=

Rd

|x|2u dx.

As a consequence, we know that d

Fλ[u]

λ=λ=0.

Of course, ifuis a solution of (14), the value ofλin Lemma5may depend ont. Now we chooseσ (t )=λ(t ), i.e., σ (t )= 1

KM

Rd

|x|2u(x, t ) dxt0. (15)

This makes (14) a non-local equation.

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Result 2. With the above choice, if we consider a solution of (14) and compute the time derivative of the relative entropy, we find that

d dtFσ (t )

u(·, t )

=σ(t) d

dσFσ[u]

|σ=σ (t )

+ m m−1

Rd

um1Bσ (t )m1∂u

∂t dx.

However, as a consequence of the choice (15) and of Lemma5, we know that d

dσFσ[u]

|σ=σ (t )

=0, and finally obtain

d dtFσ (t )

u(·, t )

= −mσ (t)d2(mmc) 1−m

Rd

u

um1Bσ (t )m12dx. (16)

The computation then goes as in[7,8](also see[21,22,16]for details). With our choice ofσ, we gain an additional orthogonality condition which is useful for improving the rates of convergence (see[19, Theorem 1]) in the asymptotic regimet→ ∞, compared to the results of[8](also see below).

Result 3.Now let us state one more result of[19]which is of interest for the present paper.

Lemma 6.With the above notations, ifuandσ are defined respectively by(14)and(15), then the functiontσ (t ) is positive, decreasing, withσ:=limt→∞σ (t ) >0and

σ(t)= −2d(1m)2

mKM σd2(mmc)Fσ (t )

u(·, t )

0. (17)

The main difficulty is to establish thatσis positive. This can be done with an appropriate change of variables which reduces (14) to the case whereσ does not depend ont. In[19], a proof has been given, based on asymptotic re- sults for the fast diffusion equation that were established in[16,7,6,8]. An alternative proof will be given in Remark3, below.

5. The scaled entropy–entropy production inequality Consider the relative Fisher information

Iσ[u] :=σd2(mmc) m 1−m

Rd

uum1− ∇Bσm12dx.

By applying (9) withu=B1andη=1 toxσd/2u(

σ x)and using the fact thatB1(x)=σd/2Bσ(σ x), we get the inequality

Fσ[u]1 4Iσ[u].

Now, ifσis time-dependent as in Section4, we have the following relations.

Lemma 7.Ifuis a solution of (14)withσ (t )=K1M

Rd|x|2u(x, t ) dx, thenσsatisfies(17). Moreover, for anyt0, we have

d dtFσ (t )

u(·, t )

= −Iσ (t )

u(·, t )

(18) and

d dtIσ (t )

u(·, t )

− 4+1

2(mmc)(mm1) d2|σ(t)| σ (t )

Iσ (t )

u(·, t )

. (19)

(11)

Proof. Eqs. (17) and (18) have already been stated respectively in Lemma6and in (16). They are recalled here only for the convenience of the reader. It remains to prove (19).

For any givenσ=σ (t ), Proposition3gives d

dtIσ (t )

u(·, t )

= d

dtIλ

u(·, t )

|λ=σ (t )

+σ(t) d

dλIλ[u]

|λ=σ (t )

−4Iσ (t )

u(·, t ) +σ(t)

d dλIλ[u]

|λ=σ (t )

. Owing to the definition ofIλ, we obtain

d

dλIλ[u] =d

2(mmc)1

λIλ[u] − m

1−d2(mmc)

Rd

2u

um1− ∇Bλm1

· d

Bλm1 dx.

By definition (11),∇Bλm1(x)=2xλd2(mmc), which implies λd2(mmc) d

Bλm1

= −d

λ(mmc)x.

Substituting this expression into the above computation and integrating by parts, we conclude with the equality d

dλIλ[u] =d

2(mmc)1

λIλ[u] −2d

λ (mmc) 2mλd2(mmc) 1−m

Rd

|x|2u dxd

Rd

umdx

.

A simple computation shows that d

Rd

B1mdx= −

Rd

x· ∇B1mdx= 2m 1−m

Rd

|x|2B1dx= 2m

1−mKM (20)

and, as a consequence, ifλ=σ=K1M

Rd|x|2u dx, then 2mλd2(mmc)

1−m

Rd

|x|2u dx=d

Rd

Bλmdx,

and finally d

dλIλ[u] = d

(mmc)

Iλ[u] −4d(1−m)Fλ[u]

. (21)

Altogether, we have found that d

dt Iσ (t )

u(·, t )

+4Iσ (t )

u(·, t )

d

2(mmc(t) σ (t )

Iσ (t )[u] −4d(1−m)Fσ (t )[u] . The last term of the right hand side is non-positive because by (17) we know thatσ(t)0 and

Iσ (t )[u] −4d(1−m)Fσ (t )[u] =d(1m)

Iσ (t )[u] −4Fσ (t )[u]

+d(mm1)Iσ (t )[u] d(mm1)Iσ (t )[u]0.

This implies (19). 2

To avoid carrying heavy notations, let us write f (t ):=Fσ (t )

u(·, t )

and j (t):=Jσ (t )

u(·, t )

and denotef (0),j (0)andσ (0)respectively byf0,j0andσ0. Estimates (17), (18) and (19) can be rewritten as

⎧⎪

⎪⎪

⎪⎪

⎪⎩

f= −j0,

σ= −κ1σd2(mmc)f 0, j+4jκ2

σ

(22)

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