Stability in Gagliardo-Nirenberg inequalities

Jean Dolbeault

http://www.ceremade.dauphine.fr/_∼dolbeaul

Ceremade, Université Paris-Dauphine October 28, 2020

Online analysis and PDE seminar (Spain)

The stability result of G. Bianchi and H. Egnell

A question:[Brezis, Lieb (1985)]Is there a natural way to bound S_dk∇u_k²_L2(

R^d)− ku_k²

L^2∗(R^d)

from below in terms of a “distance” to the set of optimal [Aubin-Talenti]

functionswhend_≥3?

B[Bianchi, Egnell (1991)]There is a positive constantαsuch that Sdk∇u_k²

L²(R^d)− ku_k²

L^2∗(_R^d)≥α inf

ϕ∈Mk∇u_{− ∇}ϕk²_L2(R^d)

BVarious improvements,e.g.,[Cianchi, Fusco, Maggi, Pratelli (2009)]

there are constantsαandκandu_7→λ(u)such that S_dk∇u_k²_L2(

R^d)≥(1₊κ λ(u)^α)_ku_k²

L^2∗(R^d)

The question ofconstructiveestimates is still widely open

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

From the carré du champ method to stability results

I[u]_≥ΛF[u]

Carré du champ method(D. Bakry and M. Emery) From

∂u

∂t ^=Lu^m(typically), d_I

dt ^{≤ −ΛI} deduce thatI−ΛFis monotone non-increasing with limit0 BImproved constantmeansstability

Under some restrictions on the functions, there is someΛ?≥Λsuch that I−ΛF≥(Λ?−Λ)_F

BImproved entropy – entropy production inequality I≥Λψ(_F)

for someψsuch thatψ(0)₌0,ψ⁰(0)₌1andψ⁰⁰>0 I−ΛF≥Λ(ψ(_F)_−F)_≥0

Outline

Part I:Two examples of stability results by entropy methods BSobolev and Hardy-Littlewood-Sobolev inequalities

joint work with G. Jankowiak

BSubcritical interpolation inequalities on the sphere joint work with M.J. Esteban and M. Loss

Part II:A constructive result based on entropy and parabolic regularity joint work with M. Bonforte, B. Nazaret and N. Simonov

Thefast diffusion flowandentropy methods

BRényi entropy powers:a word on thecarré du champmethod B^theentropy-entropy production inequality

Bspectral gap: theasymptotic time layer

B^theinitial time layer, a backward nonlinear estimate Theuniform convergence in relative error

B^thethreshold time

Ba quantitative global Harnack principle and Hölder regularity Bthe stability result in the entropy framework

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

Part I

Two examples of stability

results by entropy methods

Example 1

Sobolev and Hardy-Littlewood-Sobolev inequalities

BStability in a weaker norm but with explicit constants BFrom duality to improved estimates based on Yamabe’s flow

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

Sobolev and HLS

As it has been noticed byE. Lieb (1983)Sobolev’s inequality inR^d,d_≥3, ku_k²

L^2∗(R^d)≤Sdk∇u_k²

L²(R^d) ∀u_∈D^1,2(_R^d) and the Hardy-Littlewood-Sobolev inequality

S_dkv_k²

Ld+2^2d(_R^d)≥ Z

R^dv(_−∆)⁻¹vdx _∀v_∈L^d+2d2(_R^d)

aredualof each other. HereS_dis the Aubin-Talenti constant and2^∗_=d^2d₋₂

The first step in the entropy method

Proposition

Assume thatd_≥3andm₌^d_d⁻₊²₂. Ifv is a solution the Yamabe flow

∂v

∂t ^=∆v^m t_>0, x_∈R^d, m₌d₋2 d₊2 with nonnegative initial datum inL^2d/(d+2)(_R^d), then

1 2

d dt

"

Z

R^dv(₋∆)⁻¹vdx₋Sdkv_k²

L^d2d+2(R^d)

#

= µZ

R^dv^m+1dx

¶²_d h

Sdk∇u_k²

L²(R^d)− ku_k²

L^2∗(R^d) i≥0

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

An improvement

Jd[v] :₌ Z

R^dv^d2d+2dx and Hd[v] :₌ Z

R^dv(₋∆)⁻¹vdx₋Sdkv_k²

Ld+2^2d (R^d)

Theorem (J.D., G. Jankowiak) Assume thatd_≥3. Then we have

0_≤Hd[v]₊SdJd[v]^1+2dψ³

Jd[v]^2d−1hSdk∇u_k²

L²(R^d)− ku_k²

L^2∗(_R^d) i´

∀u_∈D,v₌u^d+d−22 whereψ(x) :₌^p_C²₊2_Cx_−C for anyx_≥0,with_C=1

... and a consequence: _{C =} 1 is not optimal

Theorem

[JD, G. Jankowiak]In the inequality Sdkw^q_k²

Ld+^2d2(R^d)− Z

R^dw^q(₋∆)⁻¹w^qdx

≤C_dS_dkw_k

8 d−2 L^2∗(_R^d)

hk∇w_k²

L²(R^d)−S_dkw_k²

L^2∗(_R^d) i

we have

d

d₊4^≤C_d<1

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

Example 2

Improved interpolation inequalities on the sphere

The interpolation inequalities on _S

On thed-dimensional sphere, let us consider the interpolation inequality k∇u_k²

L²(S^d)+ d p₋2^ku_k²

L²(S^d)≥ d p₋2^ku_k²

L^p(S^d) ∀u_∈H¹(_S^d,dµ)

where the measuredµis the uniform probability measure onS^d⊂R^d+1 corresponding to the measure induced by the Lebesgue measure onR^d+1, and the exposantp_≥1,p₆₌2, is such that

p_≤2^∗:₌ 2d d₋2

ifd_≥3. We adopt the convention that2^∗_{= ∞}ifd₌1ord₌2. The case p₌2corresponds to the logarithmic Sobolev inequality

k∇u_k²_L2(

S^d)≥d 2

Z

S^d|u_|²log





|u_|² ku_k²

L²(_S^d)



dµ ∀u_∈H¹(_S^d,dµ)\ {0}

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

←

me #M¥0*197esN

two two

#etog oq n

Ça ton:# T

The Bakry-Emery method

Entropy functional

Fp[ρ] :_=p¹₋₂^hR_Sdρ^2p dµ−(^R_Sdρdµ)^2pi if p₆₌2

F2[ρ] :₌^R_Sdρlog

µ ρ

kρk_L1(Sd)

¶ dµ Fisher informationfunctional

Ip[ρ] :₌^R_Sd|∇ρ^1p|²dµ

Bakry-Emery (carré du champ) method: use the heat flow

∂ρ

∂t ^=∆ρ

and compute_dt^d Fp[ρ]_{= −Ip}[ρ]and_dt^dIp[ρ]_{≤ −}d_Ip[ρ]to get d

dt(_Ip[ρ]₋d_Fp[ρ])_≤0 _{=⇒ Ip}[ρ]_≥d_Fp[ρ]

withρ= |u_|^p, ifp_≤2^#:₌^2d2+1

(d−1)²

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

A refined interpolation inequality on the sphere

Theorem Assume that

p₆₌2, and 1_≤p_≤2^# if d_≥2, p_≥1 if d₌1 γ=

µd₋1 d₊2

¶2

(p₋1) (2^#₋p) if d_≥2, γ=p₋1

3 if d₌1 Then for anyu_∈H¹(_S^d),

k∇u_k²

L²(S^d)≥ d 2₋p₋γ

µ ku_k²

L²(S^d)− ku_k²⁻

2γ 2−p L^p(S^d)ku_k

2γ 2−p L²(S^d)

¶

ifγ6=2₋p

k∇u_k²

L²(S^d)≥ 2d p₋2^ku_k²

L²(S^d)log



 ku_k²

L²(_S^d) ku_k²

L^p(_S^d)



 ∀u_∈H¹(_S^d)

Improved interpolation inequalities on the sphere

λ^? :₌ inf v_∈H¹₊(_S^d,dµ)

R

S^dv dµ=1 R

S^dx_|v_|^pdµ=0 R

S^d(∆v)²dµ R

S^d|∇v_|²νdµ>d

For anyf _∈H¹(_S^d,dµ)s.t.R

S^dx_|f_|^pdµ=0, consider the inequality Z

S^d|∇f_|²νdµ+ λ

p₋2^kf_k²_2≥ ^λ p₋2^kf_k²_p

Proposition

Ifp_∈(2,2^#), the inequality holds with

λ≥d₊ (d₋1)²

d(d₊2)(2^#₋p) (λ^?−d)

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

p ₌ 2: the logarithmic Sobolev case

λ^?=d₊ 2(d₊2)

2(d₊3)₊^p2(d₊3) (2d₊3)

Proposition

Letd_≥2. For anyu_∈H¹(_S^d,dµ)\{0}such that^R_Sdx_|u_|²dµ=0, we have Z

S^d|∇u_|²dµ≥ δ 2 Z

S^d|u_|²log Ã|u_|²

ku_k²₂

! dµ

withδ:₌d_+d^{2 4d−1}

2(d₊3)₊p

2(d₊3) (2d₊3)

The evolution under the fast diffusion flow

To overcome the limitationp_≤2^#, one can consider a nonlinear diffusion of fast diffusion / porous medium type

∂ρ

∂t ^=∆ρ

m. (1)

[Demange],[JD, Esteban, Kowalczyk, Loss]: for anyp_∈[1,2^∗] Kp[ρ] :₌ d

dt

³Ip[ρ]₋d_Fp[ρ]^´_≤0

1.0 1.5 2.5 3.0

0.0 0.5 1.5 2.0

(p,m)admissible region,d₌5

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

Stability under antipodal symmetry

With the additional restriction ofantipodal symmetry, that is u(₋x)₌u(x) _∀x_∈S^d

Theorem

Ifp_∈(1,2)_∪(2,2^∗), we have Z

S^d|∇u_|²dµ≥ d p₋2

"

1₊(d²₋4) (2^∗₋p) d(d₊2)₊p₋1

#

³ku_k²_Lp(

S^d)− ku_k²_L2(

S^d)

´

for anyu_∈H¹(_S^d,dµ)with antipodal symmetry. The limit casep₌2 corresponds to the improved logarithmic Sobolev inequality

Z

S^d|∇u_|²dµ≥d 2

(d₊3)² (d₊1)² Z

S^d|u_|²log



 |u_|² ku_k²

L²(_S^d)



dµ

The optimal constant in the antipodal framework

æ

æ æ

æ æ

1.5 2.0 2.5 3.0

6 7 8 9 10 11 12

Numerical computation of the optimal constant whend₌5and 1_≤p_≤10/3_≈3.33. The limiting value of the constant is numerically found

to be equal toλ?=2^1−2/pd_≈6.59754withd₌5andp₌10/3

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

Part II

A constructive result of stability based on entropy

and parabolic regularity

BAn introduction

BThe fast diffusion equation BRegularity and stability

Main results (part II) have been obtained in collaboration with

Matteo Bonforte

BUniversidad Autónoma de Madrid and ICMAT

Bruno Nazaret

BUniversité Paris 1 Panthéon-Sorbonne and Mokaplan team

Nikita Simonov

BCeremade, Université Paris-Dauphine (PSL)

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

Introduction

A special family of Gagliardo-Nirenberg inequalities Optimal functions

A stability result

Gagliardo-Nirenberg inequalities

For any smoothf onR^d with compact support

k∇f_k^θ_2kf_k¹_p⁻₊^θ_1≥CGNkf_k2p (2) [Gagliardo, 1958] [Nirenberg, 1959] θ=_p(d+^d₂^(p_−p⁻_(d¹⁾₋₂₎₎

ifd_≥3, the exponentpis in the range1_<p_≤d^d₋₂ and

2p_=d²₋^d₂₌2^∗₌:2p^∗is the critical Sobolev exponent, corresponding to Sobolev’s inequalitywith (θ=1)[Rodemich, 1968] [Aubin & Talenti, 1976]

k∇f_k²_2≥S_dkf_k²_2∗

B^ifd₌1or2, the exponentpis in the range1_<p_{< +∞ =}:p^∗

the limit case asp_→1₊isEuclidean logarithmic Sobolev inequality in scale invariant form[Blachman, 1965] [Stam, 1959] [Weissler, 1978]

d 2 log

µ 2 πd e

Z

R^d|∇f_|²dx

¶

≥ Z

R^d|f_|²log_|f_|²dx for any functionf _∈H¹(_R^d, dx)such thatkf_k2=1

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

Optimal functions and scalings

k∇f_k^θ_2kf_k¹_p⁻₊^θ_1≥CGNkf_k2p (1) [del Pino, JD, 2002]Equality is achieved by theAubin-Talentitype function

g(x)₌^³1_{+ |}x_|^2´−

p−11

∀x_∈R^d

By homogeneity, translation, scalings, equality is also achieved by g_λ,µ,y(x) :₌µ λ^−2pd g^³x₋y

λ

´

(λ,µ,y)_∈(0,+∞)_×R×R^d

BA non-scale invariant form of the inequality a_k∇f_k²₂₊b_kf_k^p_p⁺₊¹_1≥KGNkf_k²₂^p_p^γ

a₌¹₂(p₋1)²,b₌2^d−p_p^(d₊₁⁻²⁾,KGN= kg_k²₂^p(1_p ^−γ)andγ=^d+_d−²⁻_p^p_(d^(d₋⁻₄₎²⁾ Ifp₌1: standardEuclidean logarithmic Sobolev inequality[Gross, 1975]

Z

|∇f_|²dx_≥1 2 Z

|v_|²log Ã|f_|²

2

! dx₊d

4 log(2πe²)_kf_k²₂

The stability issue

What kind of distance to the manifoldMof the Aubin-Talenti type functions is measured by thedeficit functionalδ?

δ[f] :₌a_k∇f_k²₂₊b_kf_k^p_p⁺₊¹_1−KGNkf_k²₂^p_p^γ Some (not completely satisfactory) answers:

BIn the critical casep₌d/(d₋2),d_≥3,[Bianchi, Egnell, 1991]: there is a positive constantC such that

k∇f_k²₂₋S_dkf_k²_2∗≥C inf

Mk∇f_{− ∇}g_k²₂

B[JD, Jankowiak]Assume thatd_≥3and letq₌^d_d⁺₋²₂. There exists a constantC with1_{<C ≤}1₊⁴_d such that

k∇f_k²₂₋Sdkf_k²_2∗≥ ^C Sdkf_k²⁽²_2∗ ^∗−2)

µ Sd

°

°f^q°° 2

2d d+2−

Z

R^d|f_|^q(₋∆)⁻¹_|f_|^qdx

¶

B[Blanchet, Bonforte, JD, Grillo, Vázquez] [JD, Toscani]... various improvements based on entropy methods and fast diffusion flows

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

A stability result

Therelative entropy F[f] :₌₁²₋^p_p^R_Rd

³|f_|^p+1₋g^p+1₋¹_2p^+pg^1−p¡_|f_|^2p₋g^2p¢´dx Thedeficit functional

δ[f] :₌a_k∇f_k²₂₊b_kf_k^p+1

p₊1−KGNkf_k²₂^pγ

p ≥0 Theorem

Letd_≥1,p_∈(1,p^∗),A_>0andG_>0. There is a _C>0such that δ[f]_{≥C F}[f]

for anyf_∈W :₌^©f_∈L¹(_R^d,(1_{+ |}x_|)²dx) :_∇f_∈L²(_R^d, dx)^ªsuch that Z

R^d|f_|^2pdx₌ Z

R^d|g_|^2pdx, Z

R^dx_|f_|^2pdx₌0 sup

r_>0

r^{d−p(d−4)p−1} Z

|x_|>r|f_|^2pdx_≤A and F[f]_≤G

Some comments

BThe constantC is explicit

BA Csiszár-Kullback inequality. There exists a constantC_p>0such that

°

°°|f_|^2p₋g^2p°°_°

L¹(R^d)≤Cp

qF[f] if kf_kL2p(R^d)= kg_kL2p(R^d)

BLiterature on stability of Sobolev type inequalities is huge:

– Weak L^2∗/2-remainder term in bounded domains[Brezis, Lieb, 1985]

– Fractional versions and(₋∆)^s[Lu, Wei, 2000] [Gazzola, Grunau, 2001]

[Bartsch, Weth, Willem, 2003] [Chen, Frank, Weth, 2013]

– Inverse stereographic projection (eigenvalues):[Ding, 1986] [Beckner, 1993] [Morpurgo, 2002] [Bartsch, Schneider, Weth, 2004]

– Symmetrization[Cianchi, Fusco, Maggi, Pratelli, 2009]and[Figalli, Maggi, Pratelli, 2010]

... to be continued

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

BOn stability and flows (continued)

– Many other papers by Figalli and his collaborators, among which (most recent ones):[Figalli, Neumayer, 2018] [Neumayer, 2020] [Figalli, Zhang, 2020] [Figalli, Glaudo, 2020]

– Stability for Gagliardo-Nirenberg inequalities[Carlen, Figalli, 2013]

[Seuffert, 2017] [Nguyen, 2019]

– Gradient flow issues[Otto, 2001]and many subsequent papers

– Carré du champ applied to the fast diffusion equation[Carrillo, Toscani, 2000] [Carrillo and Vázquez, 2003] [CJMTU, 2001] [Jüngel, 2016]

– Spectral gap properties[Scheffer, 2001] [Denzler, McCann, 2003 & 2005]

BOn entropy methods

– Carré du champ: the semi-group and Markov precesses point of view [Bakry, Gentil, Ledoux, 2014]

– The PDE point of view (+ some applications to numerical analysis) [Jüngel, 2016]

BGlobal Harnack principle:[Vázquez, 2003] [Bonforte, Vázquez, 2006]

[Vázquez, 2006] [Bonforte, Simonov, 2020]

=⇒Our tool: the fast diffusion equation

The fast diffusion equation

∂u

∂t ^=∆u^m (3)

The Rényi entropy powers and the Gagliardo-Nirenberg inequalities Self-similar solutions and the entropy-entropy production method Large time asymptotics, spectral analysis (Hardy-Poincaré inequality)

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

The fast diffusion equation in original variables

Consider thefast diffusionequation inR^d,d_≥1,m_∈(0,1)

∂u

∂t ^=∆u^m (2)

with initial datumu(t₌0,x)₌u₀(x)_≥0such that Z

R^du₀dx_=M>0 and Z

R^d|x_|²u₀dx_{< +∞}

The large time behavior is governed bythe self-similar Barenblatt solutions

U(t,x) :₌ 1

¡κt^{1/µ¢d B} µ x

κt^1/µ

¶

whereµ:₌2₊d(m₋1),κ:₌

¯

¯

¯ 2µm m₋1

¯

¯

¯ 1/µ

andBis the Barenblatt profile B(x) :₌^¡C_{+ |}x_|^2¢−

1 1−m

The Rényi entropy power F

Theentropyis defined by

E:₌ Z

R^du^mdx and theFisher informationby

I:₌ Z

R^du_|∇P_|²dx with P₌ m

m₋1u^m−1is thepressure variable Ifusolves the fast diffusion equation, then

E⁰₌(1₋m)I TheRényi entropy power

F:₌E^σ₌ µZ

R^du^m_dx

¶σ

with σ=2 d

1 1₋m⁻1 applied to self-similar Barenblatt solutions has a linear growth int

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

The concavity property

Theorem

[Toscani, Savaré, 2014]Assume that m_1≤m_<1ifd_>1andm_>1/2if d₌1. ThenF(t)isconcave, increasing, and

t_→+∞lim F⁰(t)₌(1₋m)σ lim

t_→+∞E^σ−1I [Dolbeault, Toscani, 2016]The inequality

E^σ−1I_≥E[_B]^σ−1I[_B] is equivalent to the Gagliardo-Nirenberg inequality

k∇f_k^θ_2kf_k¹_p⁻₊^θ_1≥CGNkf_k2p (1) u^m−1/2_=kf^f_k

2p andp_=2m¹_−1∈(1,p^∗)_⇐⇒max^©1₂,m₁^ª_<m_<1

Self-similar variables: entropy-entropy production method

The fast diffusion equation

∂u

∂t ^=∆u^m has a self-similar solution

U(t,x) :₌ 1 κ^d(µt)^d/µB

à x

κ(µt)^1/µ

!

where B(x) :₌^¡1_{+ |}x_|^2¢−

1 1−m

A time-dependent rescaling based onself-similar variables u(t,x)₌ 1

κ^dR^dv^³τ, x κR

´ where dR

dt ⁼R^1−µ, τ(t) :₌¹₂log µR(t)

R₀

¶

Then the functionvsolvesa Fokker-Planck type equation

∂v

∂τ+ ∇ ·h

v^³_∇u^m−1₋2x^{´ i}₌0 withsame initial datumv₀₌u₀ifR₀₌R(0)₌1

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

Free energy and Fisher information

The functionv_andB(same mass) solve the Fokker-Planck type equation

∂v

∂t ^{+ ∇ ·} h

v^³_∇v^m−1₋2x^{´ i}₌0 (4) A Lyapunov functional[Ralston, Newman, 1984]

Generalized entropyorFree energy F[v] :_{= −}1

m Z

R^d

³

v^m_−B^m₋m_B^m−1(v_−B)^´dx

Entropy production is measured by theGeneralized Fisher information

d

dt^F[v]_{= −I}[v], I[v] :₌ Z

R^dv^¯¯_¯∇v^m−1₊2x^¯¯_¯²dx

The entropy - entropy production inequality

B(x) :₌^¡1_{+ |}x_|^2¢−

1−m1

Theorem

[del Pino, JD, 2002]d_≥3,m_∈[m₁,1),m_>¹₂, ^R_Rdv₀dx₌^R_RdBdx Z

R^dv

¯

¯¯∇v^m−1₊2x

¯

¯

¯

2dx_=I[v]_≥4_F[v]₌4 Z

R^d

µB^m m ⁻

v^m

m ^{+ |}x_|²(v_−B)

¶ dx

p_=2m¹₋₁,v₌f^2p

k∇f_k^θ_2kf_k¹_p⁻₊^θ_1≥CGNkf_k2p⇐⇒δ[f]_=I[v]₋4F[v]_≥0

Corollary

[del Pino, JD, 2002]A solutionv of (4)with initial datav_0∈L¹₊(_R^d)such that_|x_|²v_0∈L¹(_R^d),v₀^m_∈L¹(_R^d)satisfies

F[v(t_,·)]_≤F[v₀]e^−4t

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

A computation on a large ball, with boundary terms

Carré du champ method[Carrillo, Toscani] [Carrillo, Vázquez]

[Carrillo, Jüngel, Toscani, Markowich, Unterreiter]

∂u

∂t ^{+ ∇ ·} h

v^³_∇v^m−1₋2x^{´ i}₌0 t_>0, x_∈B_R

³∇v^m−1₋2x^´_· x

|x_|⁼0 t_>0, x_∈∂B_R d

dt Z

BR

v_|∇v^m−1₋2x_|²dx₊4 Z

BR

v_|∇v^m−1₋2x_|²dx +2¹⁻_m^m

Z B_R

v^m µ°

°

°D²(v^m−1_−B^m−1)

°

°

°

2−(1₋m)

¯

¯

¯∆(v^m−1_−B^m−1)

¯

¯

¯ 2¶

dx

= Z

∂B_R

v^m³ω· ∇|(v^m−1_−B^m−1)_|^2´dσ≤0(by Grisvard’s lemma)

Improvement:∃φsuch thatφ⁰⁰>0,φ(0)₌0andφ⁰(0)₌4[Toscani, JD]

I[v_|Bσ]_≥φ(_F[v_|Bσ]) _⇐= idea: d_I

dt ⁺4I.−I F²

Spectral gap: sharp asymptotic rates of convergence

Assumptions on the initial datumv₀ (H₁)¡

C_{0+ |}x_|^2¢−

1

1−m≤v_0≤^¡C_{1+ |}x_|^2¢−

1 1−m

(H₂)ifd_≥3andm_≤m_∗:₌^d_d⁻₋⁴₂, then(v_0−B)is integrable

Theorem

[Blanchet, Bonforte, JD, Grillo, Vázquez, 2009]Ifm_<1andm₆₌m_∗, then F[v(t,·)]_≤C e^−2γ(m)t _∀t_≥0, γ(m) :₌(1₋m)Λα,d

whereΛα,d>0 is the best constant in the Hardy–Poincaré inequality

Λ_α,d Z

R^d|f_|²dµα−1≤ Z

R^d|∇f_|²dµα ∀f _∈H¹(dµα), Z

R^dfdµα−1=0 withα:_=m¹_−1<0,dµα:₌h_αdx, h_α(x) :₌(1_{+ |}x_|²)^α

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

Spectral gap and the asymptotic time layer

mc=^d−2_d 0

m1=^d−1_d m2=^d+1_d+2

! m2:=^d+4_d+6

m 1

2 4

Case 1 Case 2 Case 3

γ(m)

(d= 5)

! m1:=_d+2^d

F[v(t,·)]_≤C e^−2γ(m)t _∀t_≥0

Spectral gap and improvements... the details

BAsymptotic time layer[BBDGV, 2009] [BDGV, 2010] [JD, Toscani, 2015]

Corollary

Assume thatv solves (4): ∂tv_{+ ∇ ·}^hv^¡_∇v^m−1₋2x^¢i₌0 with initial datumv_0≥0 such that^R_Rdv_0dx₌^R_RdBdx

(i) there is a constantC1>0such that F[v(t,·)]_≤C1e^−2γ(m)t with γ(m)₌2 ifm_1≤m_<1

(ii) ifm_1≤m_<1and^R_Rdx v₀dx₌0, there is a constant_C2>0such that F[v(t_,·)]_≤C2e^−2γ(m)t withγ(m)₌4₋2d(1₋m)

(iii) Assume that ^d_d⁺₊¹_2≤m_<1and^R_Rdx v₀dx₌0and let Bσ:₌σ^−d2B(x/^pσ)

be such that ^R_Rd|x_|²u(t,x)dx₌^R_Rd|x_|²_Bσ(x)dx. Then there is a constant C3>0such that F[v(t,·)_|Bσ]_≤C3e^−4t

BInitial time layerI[v_|Bσ]_≥φ(_F[v_|Bσ])_⇒faster decay fort_∼0

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

The asymptotic time layer improvement

Linearized free energyandlinearized Fisher information F[g] :₌m

2 Z

R^d|g_|²_B^2−m_dx _and I[g] :₌m(1₋m) Z

R^d|∇g_|²_Bdx Hardy-Poincaré inequality. Letd_≥1,m_∈(m₁,1)andg_∈L²(_R^d_,B^2−mdx) such that∇g_∈L²(_R^d_,Bdx),R

R^dg_B^2−mdx₌0andR

R^dx g_B^2−mdx₌0 I[g]_≥4αF[g] where α=2₋d(1₋m)

Proposition

Letm_∈(m₁,1)ifd_≥2,m_∈(1/3,1)ifd₌1,η=2d(m₋m₁)and χ=m/(266₊56m). If^R_Rdvdx_=M,^R_Rdx vdx₌0and

(1₋ε)_B≤v_≤(1₊ε)_B

for someε∈(0,χ η), then

Q[v] :₌^I[v] F[v]^≥4₊η

The initial time layer improvement: backward estimate

Rephrasing thecarré du champmethod,Q[v] :₌^I[v]

F[v]is such that

d_Q

dt ^≤Q(_Q−4)

Lemma

Assume thatm_>m₁ andv is a solution to (4)with nonnegative initial datumv₀. If for someη>0 andT_>0, we have_Q[v(T,·)]_≥4₊η, then

Q[v(t_,·)]_≥4₊ 4ηe^−4T

4_+η−ηe^{−4T ∀}t_∈[0,T]

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

The fast diffusion equation

Regularity and stability Back to entropy-entropy production inequalities

Regularity and stability

Our strategy

Regularity and stability

Our strategy

Choose">0, small enough

Get a threshold timet?(")

0 t?(") t

Backward estimate by entropy methods

Forward estimate based on a spectral gap

E s

⇐

# ^↳

Initial time layer Asymptotic time layer

Uniform convergence in relative error: statement

Theorem

Assume thatm_∈(m₁,1)ifd_≥2,m_∈(1/3,1)ifd₌1 and letε∈(0,1/2), small enough,A_>0, andG_>0 be given. There exists an explicit time t_?≥0 such that, ifu is a solution of

∂u

∂t ^=∆u^m (2)

with nonnegative initial datumu_0∈L¹(_R^d)satisfying

sup

r_>0

r

d(m−mc) (1−m)

Z

|x_|>r

u dx_≤A_{< ∞} (H_A)

R

R^du₀dx₌^R_RdBdx_=M andF[u₀]_≤G, then

sup

x_∈R^d

¯

¯

¯

¯ u(t,x) B(t,x)⁻1

¯

¯

¯

¯≤ε ∀t_≥t?

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

The threshold time

Proposition

Letm_∈(m₁,1)ifd_≥2,m_∈(1/3,1)ifd₌1,ε∈(0,εm,d),A_>0andG_>0

t?=c?

1₊A^1−m₊G^α2 ε^a wherea₌^α_ϑ²₁⁻₋^m_m andϑ=ν/(d₊ν)

c?=c?(m,d)₌ sup

ε∈(0,εm,d)

max^©ε κ₁(ε,m),ε^aκ₂(ε,m),ε κ₃(ε,m)^ª

κ₁(ε,m) :₌maxⁿ 8c (1₊ε)^1−m₋1^,

2^3−mκ?

1₋(1₋ε)^1−m o

κ2(ε,m) :₌(4α)^α−1K^αϑ ε^2−m1−mα

and κ3(ε,m) :₌ 8α⁻¹ 1₋(1₋ε)^1−m

Improved entropy-entropy production inequality

Theorem

Letm_∈(m₁,1)ifd_≥2,m_∈(1/2,1)ifd₌1,A_>0andG_>0. Then there is a positive numberζsuch that

I[v]_≥(4₊ζ)_F[v]

for any nonnegative functionv_∈L¹(_R^d)such thatF[v]₌G, R

R^dvdx_=M, ^R_Rdx vdx₌0andv satisfies (HA) We have theasymptotic time layer estimate

ε∈(0,2ε?), ε?:₌1

2min^©εm,d,χ ηª

with T₌1

2logR(t?) (1_−ε)_B≤v(t_,·)_≤(1₊ε)_{B ∀}t_≥T

and, as a consequence, theinitial time layer estimate

I[v(t,.)]_≥(4₊ζ)_F[v(t,.)] _∀t_∈[0,T], where ζ= 4ηe^−4T 4_+η−ηe^−4T

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

Two consequences

ζ=Z^¡A_,F[u₀]^¢, Z(A,G) :₌ ^ζ?

1₊A^(1−m)α2₊G^{, ζ?} :₌ 4η

4₊η µ ε^a_?

2αc_?

¶_α² cα

BImproved decay rate for the fast diffusion equation in rescaled variables Corollary

Letm_∈(m₁,1)ifd_≥2,m_∈(1/2,1)ifd₌1,A_>0andG_>0. Ifv is a solution of (4) with nonnegative initial datumv_0∈L¹(_R^d)such that F[v₀]₌G,^R_Rdv_0dx_=M, ^R_Rdx v_0dx₌0 andv₀ satisfies (HA), then

F[v(t,.)]_≤F[v₀]e^−(4+ζ)t _∀t_≥0 BThe stability in the entropy - entropy production estimate I[v]₋4_F[v]_≥ζF[v]also holds in a stronger sense

I[v]₋4F[v]_≥ ^ζ 4₊ζI[v]