Stability in Gagliardo-Nirenberg inequalities

(1)

Jean Dolbeault

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

Ceremade, Université Paris-Dauphine July 7, 2020

OWPDE seminar

(2)

Outline

Introduction

B(almost) a firstconstructive resulton thestability of Gagliardo-Nirenberg inequalities

Thefast diffusion flowandentropy methods

BRényi entropy powers:a word on thecarré du champmethod Btheentropy-entropy production inequality

Bspectral gap: theasymptotic time layer

B^theinitial time layer, a backward nonlinear estimate B^thethreshold time

Theuniform convergence in relative error

Bfrom local to global estimates: a quantitative version of Harnack’s inequality and Hölder regularity

Bthe stability result in the entropy framework

J. Dolbeault Stability in Gagliardo-Nirenberg inequalities

(3)

Main results of this lecture 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)

(4)

Introduction

A special family of Gagliardo-Nirenberg inequalities Optimal functions

A stability result

(5)

Gagliardo-Nirenberg inequalities

For any smoothf onR^d with compact support

k∇f_k^θ₂_kf_k¹_p⁻₊^θ₁_≥_C_GN_kf_k_2p (1) [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_k²₂_≥S_d_kf_k²₂_∗

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

(6)

Optimal functions and scalings

k∇f_k^θ₂_kf_k¹_p⁻₊^θ₁_≥_C_GN_kf_k_2p (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⁻^2p^d 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⁺₊¹₁_≥_K_GN_kf_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

R^d|∇f_|²dx_≥1 2 Z

R^d|v_|²log Ã|f_|²

kf_k²₂

! dx₊d

4 log(2πe²)_kf_k²₂

(7)

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⁺₊¹₁_−K_GN_kf_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_d_kf_k²₂_∗_≥_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_k²₂_∗_≥ ^C Sdkf_k²⁽²₂_∗ ^∗⁻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

(8)

A stability result

Therelative entropy E[f] :₌ 2p 1₋p

Z

R^d

³|f_|^p⁺¹₋g^p⁺¹₋¹_2p⁺^pg¹⁻^p^³_|f_|²^p₋g²^p^´´dx

Theorem

Letd_≥1,p_∈(1,p^∗),A_>0andG_>0. There is a _C_>0such that δ[f]_≥_{C E}[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_|²^pdx, Z

R^dx_|f_|²^pdx₌0 sup

r_>0

r^d−p(d−

4) p−1

Z

|x_|>r|f_|^2pdx_≤A and E[f]_≤G Reminder:δ[f] :₌a_k∇f_k²₂₊b_kf_k^p_p⁺₊¹₁₋_K_GN_kf_k²₂^p_p^γ

(9)

Some comments

BThe constantC is explicit

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

°

°°|f_|^2p₋g²^p^°^°_°

L¹(R^d)≤Cp

qE[f] if kf_k_L2p(R^d)= kg_k_L2p(R^d)

BLiterature on stability of Sobolev type inequalities is huge:

– Weak L²^∗^/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

(10)

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

(11)

The fast diffusion equation

∂u

∂t ⁼^∆u^m (2)

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)

(12)

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

(13)

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 with σ=2 d

1 1₋m⁻1

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

(14)

The variation of the Fisher information

Lemma

Ifu solves ^∂_∂^u_t ₌∆u^mwith ^d⁻_d¹₌:m₁_≤m_<1, then

I⁰₌ d dt

Z

R^du_|∇P_|²dx₌₋2 Z

R^du^m µ°

°

°D²P₋_d¹∆PId°

°

2+(m₋m₁) (∆P)²

¶ dx

BThis is where the limitationm_≥m₁:₌^d_d⁻¹ appears .... there are no boundary terms in the integrations by parts ?

(15)

The concavity property

Theorem

[Toscani, Savaré, 2014]Assume that m₁_≤m_<1ifd_>1andm_>1/2if d₌1. ThenF(t)is increasing,(1₋m)F⁰⁰(t)_≤0 and

t_→+∞lim 1

tF(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^θ₂_kf_k¹_p⁻₊^θ₁_≥_C_GN_kf_k_2p (1)

u^m⁻^1/2₌_k_f^f_k

2p andp₌_2m¹₋₁_∈(1,p^∗)_⇐⇒max^©¹₂,m₁^ª_<m_<1

(16)

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/^µ^B

Ã 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 ⁼R¹⁻^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

(17)

Free energy and Fisher information

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

∂v

∂t ^{+ ∇ ·} h

v^³_∇v^m⁻¹₋2x^{´ i}₌0 (3) 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

(18)

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^θ₂_kf_k¹_p⁻₊^θ₁_≥_C_GN_kf_k_2p_⇐⇒δ[f]₌_I[v]₋4F[v]_≥0

Corollary

[del Pino, JD, 2002]A solutionv of (3)with initial datav₀_∈L¹₊(_R^d)such that_|x_|²v₀_∈L¹(_R^d),v₀^m_∈L¹(_R^d)satisfies

F[v(t_,·)]_≤_F[v₀]e⁻⁴^t

(19)

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^m^³ω· ∇|(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_|B_σ]_≥φ(_F[v_|B_σ]) _⇐= idea: d_I

dt ⁺4I≤ −I F²

(20)

Spectral gap: sharp asymptotic rates of convergence

Assumptions on the initial datumv₀ (H₁)¡

C₀_{+ |}x_|²^¢⁻

1

1−m≤v₀_≤^¡C₁_{+ |}x_|²^¢⁻

1 1−m

(H₂)ifd_≥3andm_≤m_∗:₌^d_d⁻₋⁴₂, then(v₀₋_B)is integrable

Theorem

[Blanchet, Bonforte, JD, Grillo, Vázquez, 2009]Ifm_<1andm₆₌m_∗, then F[v(t,·)]_≤C e⁻²^γ^(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

(21)

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⁻²^γ^(m)t _∀t_≥0

[BBDGV, 2009] [BDGV, 2010] [JD, Toscani, 2015]

(22)

Spectral gap and improvements... the details

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

Corollary

Assume thatv solves (3): ∂tv_{+ ∇ ·}^hv^¡_∇v^m⁻¹₋2x^¢ⁱ₌0 with initial datumv₀_≥0 such that^R_Rdv₀_dx₌^R_RdBdx

(i) there is a constantC1>0such that F[v(t,·)]_≤_C₁e⁻²^γ^(m)t with γ(m)₌2 ifm₁_≤m_<1

(ii) ifm₁_≤m_<1and^R_Rdx v₀dx₌0, there is a constant_C₂_>0such that F[v(t_,·)]_≤_C₂e⁻²^γ^(m)t withγ(m)₌4₋2d(1₋m)

(iii) Assume that ^d_d⁺₊¹₂_≤m_<1and^R_Rdx v₀dx₌0and let Bσ:₌σ⁻^d²B(x/^pσ)

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

(23)

The asymptotic time layer improvement

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

2 Z

R^d|g_|²_B²⁻^mdx and I[g] :₌m(1₋m) Z

R^d|∇g_|²_Bdx Hardy-Poincaré inequality. Letd_≥1,m_∈(m₁,1)andg_∈L²(_R^d,B²⁻^mdx) such that∇g_∈L²(_R^d_,Bdx),R

R^dg_B²⁻^mdx₌0andR

R^dx g_B²⁻^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]_≥4_+η

(24)

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 (3)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]

(25)

The fast diffusion equation Regularity and stability Proof

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

(26)

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 datumu₀_∈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?

(27)

Interpolation inequalities

Our regularity theory relies on the inequality kf_k²

L^pm(B)≤K ³ k∇f_k²

L²(B)+ kf_k²

L²(B)

´

whereB_⊂_R^dis the unit ball

p_m _K q β

d_≥3 _d²₋^d₂ _π²Γ(^d₂₊1)^2/d ^d₂ α d₌2 4 0.0564922...<2/p²π≈1.12838 2 2(_α₋1) d₌1 _m⁴ 2¹⁺^m² max^³²⁽²_m⁻_π^m)2 ,¹₄´

2 2₋m

2m 2₋m

(28)

Local estimates (1/2): a Herrero-Pierre type lemma

Lemma

For all(t,R)_∈(0,+∞)², any solutionuof (2)satisfies

sup

y_∈B_R/2(x)

u(t,y)_≤κ Ã 1

t^d/^α µZ

B_R(x)

u₀(y)dy

¶2/α

+ µ t

R²

¶₁_−m¹ !

κ=kK ^2q^β k^β₌^¡_β⁴₊^β₂^¢^β^¡_β⁴₊₂^¢²π⁸^(q⁺¹⁾e⁸

P_∞

j=0log(j+1)(_q+1^q )^j2¹^2m⁻^m(1₊aωd)²b ωd:_{= |S}^d⁻¹_{| =} ²^π^d/2

Γ(d/2)

a₌^3(16(d⁺^{1) (3}⁺^m))

1 1−m

(2−m) (1−m)1−m^m +²

d−m(d+1) 1−m

3^dd and b₌ ³⁸^2(q+¹⁾

¡1−(2/3)

β

4(q+1)¢⁴(q+1)

(29)

Local estimates (2/2)

Lemma

For allR_>0, any solutionuof (2)satisfies

|x₋infx0|≤Ru(t,x)_≥κ³ R⁻²t^´

1 1−m

if t_≥κ?ku₀_k¹_L⁻₁_(B^m

R(x0))R^α₌:2t

κ?=2³^α⁺²d^α and κ=α ωd

Ã (1₋m)⁴

2³⁸d⁴π¹⁶⁽¹⁻^m)^ακ^α²⁽¹⁻^m)

! ²

(1−m)2αd

(30)

Global Harnack Principle

Proposition

Any solutionuof (2)satisfies

u(t,x)_≤B^¡t₊t₋_α¹,x;M^¢ _∀(t,x)_∈[t,+∞)_×R^d u(t,x)_≥B^¡t₋t₋¹_α,x;M^¢ _∀(t,x)_∈[2t_,+∞)_×R^d

c:₌max^©1,2⁵⁻^mκ¹⁻^mb^α^ª, t:₌c t₀A¹⁻^m M:₌2

α

2(1−m)κ^α² (1₊c)^d² b⁻^d²^α_M² M:₌min

(

2⁻^d/2^³^κ b^d

´α/2

, κ

(d(1₋m))^d^/2α

α 2(1−m)

) κ

1 1−m

? M²

(31)

The outer estimates

Corollary

For anyε∈(0,ε), any solutionuof (2)satisfies

u(t_,x)_≥(1₋ε)B(t_,x) if _|x_{| ≥}R(t)ρ(ε) and t_≥T(ε) u(t,x)_≤(1₊ε)B(t,x) if _|x_{| ≥}R(t)ρ(ε) and t_≥T(ε)

(32)

The inner estimate

Proposition

There exist a numerical constantK_>0and an exponentϑ∈(0,1)such that, for anyε∈(0,εm,d)and for anyt_≥4T(ε), any solutionuof (2) satisfies

¯

¯ u(t,x) B(t,x)⁻1

¯

¯≤ K ε¹⁻¹^m

Ã1 t⁺

pG R(t)

!ϑ

if _|x_{| ≤}2ρ(ε)R(t)

K:₌2

3d

α+_α(1−m)³⁺^6α +ϑ+10 (α+M)^ϑ m^ϑ(1₋m)²⁽¹⁺^ϑ⁾⁺1−m²

×

"

1₊b^dC_d_,ν,₁ Ãµ

κM^α² 2^ν 2^ν₋1⁺c

¶_d+ν^d +µ^2d

α^d^α M^d+ν^d

!#

bandcare numerical constants,ϑ=ν/(d₊ν)_andνis a Hölder regularity exponent which arises from Harnack’s inequality in J. Moser’s proof

(33)

An L

^p

-C

^ν

interpolation

Letbu_c_Cν(Ω):₌supx,y_∈Ω x₆₌y

|u(x)₋u(y)_|

|x₋y_|^ν

Lemma

Letp_≥1andν∈(0,1)

ku_k_L∞(R^d)≤C_d_,ν,_p_bu_c

d d+pν

C^ν(_R^d)ku_k

pν d+pν

L^p(_R^d) ∀u_∈L^p(_R^d)_∩C^ν(_R^d)

C_d_,ν,_p₌2

(p−1)(d+pν)+dp p(d+pν) ³

1₊_ω^d

d

´¹_p µ

1₊^³_p^d_ν^´

1 p¶_d+p^d_νµ

³d pν

´ ^pν

d+pν

+¡pν d

¢_d+p^d_ν

¶1/p

(34)

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₊A¹⁻^m₊G^α² ε^a wherea₌^α_ϑ²₁⁻₋^m_m andϑ=ν/(d₊ν)

c?=c?(m,d)₌ sup

ε∈(0,εm,d)

κ₁(ε,m) :₌maxⁿ 8c (1₊ε)¹⁻^m₋1^,

2³⁻^mκ?

1₋(1₋ε)¹⁻^m o

κ2(ε,m) :₌(4α)^α⁻¹K^α^ϑ ε^2−m¹⁻^m^α^ϑ

and κ3(ε,m) :₌ 8α⁻¹ 1₋(1₋ε)¹⁻^m

(35)

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

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⁻⁴^T

(36)

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 (3) with nonnegative initial datumv₀_∈_L¹(_R^d)such that F[v₀]₌G,^R_Rdv₀_dx₌_M, ^R_Rdx v₀_dx₌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]

(37)

A general stability result

λ[f] :₌

µ2dκ[f]^p−1 p²₋1

kf_k^p+_p+¹₁ k∇f_k²₂

¶

2p d−p(d−4)

A[f] :₌ ^M

λ[f]

d−p(d−4) p−1 kf_k²₂^p_p

sup_r_>₀r^d−p(d−

4) p−1 R

|x_|>r|f(x₊x_f)_|²^pdx

E[f] :₌₁^2p₋_p^R_Rd

Ã

κ[f]^p+¹ λ[f]^d

p−1

2p |f_|^p⁺¹₋g^p⁺¹₋¹₂⁺_p^pg¹⁻^p µκ[f]^2p

λ[f]² |f_|^2p₋g²^p

¶! dx S[f]:₌^M

p−1 2p

p²₋1 1

C(p,d)Z(A[f],E[f]) Theorem

Letd_≥1andp_∈(1,p^∗). For anyf_∈_W, we have

³k∇f_k^θ₂_kf_k¹_p⁻₊^θ₁^´²^p^γ₋(_C_GN_kf_k_2p)^2p^γ_≥S[f]_kf_k²₂^p_p^γE[f]

(38)

References

M. Bonforte, J. Dolbeault, B. Nazaret, and N. Simonov.Stability in Gagliardo-Nirenberg inequalities.

Preprinthttps://hal.archives-ouvertes.fr/hal-02887010

M. Bonforte, J. Dolbeault, B. Nazaret, and N. Simonov.Explicit constants in Harnack inequalities and regularity estimates, with an application to the fast diffusion equation(supplementary material).

Preprinthttps://hal.archives-ouvertes.fr/hal-02887013

http://www.ceremade.dauphine.fr/∼dolbeaul (temporary links from the home page)

(39)

(40)

These slides can be found at

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

B^Lectures The papers can be found at

http://www.ceremade.dauphine.fr/∼dolbeaul/Preprints/list/

BPreprints and papers

For final versions, useDolbeaultas login andJeanas password

E-mail:dolbeault@ceremade.dauphine.fr

Thank you for your attention !

(41)

Appendix: linear tools and

corresponding constants

(42)

Linear parabolic equations

∂v

∂t ^{= ∇ ·}

¡A(t,x)_∇v^¢ (4)

onΩT:₌(0,T)_×Ω, whereΩis an open domain,A(t,x)is a real symmetric matrix such that

λ₀|ξ|²≤ d X

i,j₌1

A_i_,_j(t_,x)ξiξj≤λ₁|ξ|² ∀(t_,x_,ξ)_∈_R⁺_×ΩT×R^d

for some positive constantsλ0andλ1

D_R⁺(t₀,x₀) :₌(t₀₊³₄R²,t₀₊R²)_×B_R/2(x₀) D_R⁻(t₀,x₀) :₌^³t₀₋³₄R²,t₀₋¹₄R²^´_×B_R/2(x₀) h:₌exp

"

2^d⁺⁴3^dd₊c₀³2^2(d⁺²⁾⁺³ Ã

1₊ 2^d⁺² (^p2₋1)^2(d⁺²⁾

! σ

#

c₀₌3^d²2^{(d+2) (3d}

2+18d+24)+13 2d

Ã (2+d)¹⁺

4 d2

d¹⁺

2 d2

!(d+1)(d₊2) K ^2d+^d⁴ σ=P_∞

j₌0

³3 4

´j¡

(2₊j) (1₊j)^¢²^d⁺⁴

(43)

Harnack inequality (Moser)

Theorem

LetT_>0,R_∈(0,p

T), and take(t₀,x₀)_∈(0,T)_×Ωsuch that

¡t₀₋R²,t₀₊R²^¢_×B_2R(x₀)_⊂ΩT. Ifu is a weak solution of (4), then sup

D_R⁻(t0,x0)

v_≤h inf

D⁺

R(t0,x0)v

withh:₌h^λ¹⁺^1/^λ⁰

This Harnack inequality goes back to[Moser, 1964] [Moser1971]

The dependence of the constant onλ0andλ1is optimal[Moser, 1971]

The dependence ofhondis pointed out in[Gutierrez, Wheeden, 1990]

To our knowledge, this is the first explicit expression ofh

(44)

Harnack inequality implies Hölder continuity

LetΩ₁⊂Ω₂⊂R^dtwo bounded domains and let us consider

Q₁:₌(T₂,T₃)_×Ω₁⊂(T₁,T₄)_×Ω₂=:Q₂, where0_≤T₁_<T₂_<T₃_<T_<4 We define theparabolic distancebetweenQ₁andQ₂as

d(Q₁,Q₂) :₌ inf

(t,x)∈Q1

(s,y)_∈[T1,T4]_×∂Ω2∪{T1,T4}×Ω2

|x₋y_{| + |}t₋s_|¹²

Theorem

Ifv is a nonnegative solution of (4) onQ₂, then

sup

(t,x),(s,y)∈Q1

|v(t,x)₋v(s,y)_|

¡|x₋y_{| + |}t₋s_|^1/2^¢^ν ^≤2

µ 128

d(Q₁,Q₂)

¶ν

kv_k_L∞(Q2)

whereν:₌log₄^³ ^h

h−1

´

(45)

Stability in Gagliardo-Nirenberg inequalities

Outline

Introduction

Gagliardo-Nirenberg inequalities

Optimal functions and scalings

The stability issue

A stability result

Some comments

The fast diffusion equation

The fast diffusion equation in original variables

The Rényi entropy power F

The variation of the Fisher information

The concavity property

Self-similar variables: entropy-entropy production method

Free energy and Fisher information

The entropy - entropy production inequality

A computation on a large ball, with boundary terms

Spectral gap: sharp asymptotic rates of convergence

Spectral gap and the asymptotic time layer

Spectral gap and improvements... the details

The asymptotic time layer improvement

The initial time layer improvement: backward estimate

Regularity and stability

Regularity and stability

E s

# ↳

Uniform convergence in relative error: statement

Interpolation inequalities

Local estimates (1/2): a Herrero-Pierre type lemma

Local estimates (2/2)

Global Harnack Principle

The outer estimates

The inner estimate

An L

-C

interpolation

The threshold time

Improved entropy-entropy production inequality

Two consequences

A general stability result

References

Thank you for your attention !

Appendix: linear tools and

corresponding constants

Linear parabolic equations

Harnack inequality (Moser)

Harnack inequality implies Hölder continuity

Appendix: still another

interpolation inequality

# ^↳