Optimality in critical and subcritical inequalities: ﬂows, linearization and entropy methods

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Optimality in critical and subcritical inequalities:

flows, linearization and entropy methods

Jean Dolbeault

http://www.ceremade.dauphine.fr/∼dolbeaul Ceremade, Universit´e Paris-Dauphine

December 6, 2016

Tsinghua Sanya International Mathematics Forum Nonlinear Partial Differential Equations and Mathematical Physics

Workshop, December 5–9, 2016

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Outline

BSymmetry breaking and linearization

The critical Caffarelli-Kohn-Nirenberg inequality

A family of sub-critical Caffarelli-Kohn-Nirenberg inequalities Linearization and spectrum

BWithout weights: Gagliardo-Nirenberg inequalities and fast diffusion flows

R´enyi entropy powers

Self-similar variables and relative entropies The role of the spectral gap

BWith weights: Caffarelli-Kohn-Nirenberg inequalities and weighted nonlinear flows

Towards a parabolic proof

Large time asymptotics and spectral gaps A discussion of optimality cases

J. Dolbeault Flows, linearization, entropy methods

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Symmetry breaking and linearization Entropy methods without weights Weighted nonlinear flows and CKN inequalities

Collaborations

Collaboration with...

M.J. Esteban and M. Loss (symmetry, critical case) M.J. Esteban, M. Loss and M. Muratori (symmetry, subcritical case) M. Bonforte, M. Muratori and B. Nazaret (linearization and large time asymptotics for the evolution problem) M. del Pino, G. Toscani (nonlinear flows and entropy methods) A. Blanchet, G. Grillo, J.L. V´azquez (large time asymptotics and linearization for the evolution equations) ...and also

S. Filippas, A. Tertikas, G. Tarantello, M. Kowalczyk ...

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Background references (partial)

Rigidity methods, uniqueness in nonlinear elliptic PDE’s:

[B. Gidas, J. Spruck, 1981],[M.-F. Bidaut-V´eron, L. V´eron, 1991]

Probabilistic methods (Markov processes), semi-group theory and carr´e du champmethods (Γ2theory): [D. Bakry, M. Emery, 1984], [Bakry, Ledoux, 1996], [Demange, 2008],[JD, Esteban, Loss, 2014 & 2015] →D. Bakry, I. Gentil, and M. Ledoux.

Analysis and geometry of Markov diffusion operators (2014) Entropy methods in PDEs

BEntropy-entropy production inequalities: Arnold, Carrillo, Desvillettes, JD, Jüngel, Lederman, Markowich, Toscani, Unterreiter, Villani...,[del Pino, JD, 2001],[Blanchet, Bonforte, JD, Grillo, Vázquez]→A. Jüngel, Entropy Methods for Diffusive Partial Differential Equations (2016)

BMass transportation: [Otto]→C. Villani, Optimal transport.

Old and new (2009)

BR´enyi entropy powers (information theory)[Savar´e, Toscani, 2014], [Dolbeault, Toscani]

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Critical Caffarelli-Kohn-Nirenberg inequality Subcritical Caffarelli-Kohn-Nirenberg inequalities Linearization and spectrum

Symmetry and symmetry breaking results

BThe critical Caffarelli-Kohn-Nirenberg inequality

BA family of sub-critical Caffarelli-Kohn-Nirenberg inequalities BLinearization and spectrum

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Critical Caffarelli-Kohn-Nirenberg inequality

LetD^a,b:=n

v ∈L^p R^d,|x|^−bdx

: |x|^−a|∇v| ∈L² R^d,dx o Z

R^d

|v|^p

|x|^{b p} dx 2/p

≤ C_a,b Z

R^d

|∇v|²

|x|²^a dx ∀v ∈ D^a,b holds under conditions ona andb

p= 2d

d−2 + 2 (b−a) (critical case) BAn optimal function among radial functions:

v?(x) =

1 +|x|^{(p−2) (a}^c^−a)−_p−2²

and C^?_a,b= k |x|^−bv_?k²p

k |x|^−a∇v_?k²2

Question: C_a,b= C^?_a,b (symmetry) orC_a,b>C^?_a,b (symmetry breaking) ?

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Critical CKN: range of the parameters

Figure: d = 3 Z

R^d

|v|^p

|x|^{b p} dx 2/p

≤ C_a,b Z

R^d

|∇v|²

|x|²^a dx

a b

0 1

−1

b=a

b=a+ 1

a=^d−2₂ p

a≤b≤a+ 1ifd≥3

a<b≤a+ 1ifd= 2,a+ 1/2<b≤a+ 1ifd= 1 anda<ac := (d−2)/2

p= 2d

d−2 + 2 (b−a)

[Glaser, Martin, Grosse, Thirring (1976)]

[Caffarelli, Kohn, Nirenberg (1984)]

[F. Catrina, Z.-Q. Wang (2001)]

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Linear instability of radial minimizers:

the Felli-Schneider curve

The Felli & Schneider curve b_FS(a) := d(a_c−a)

2p

(ac−a)²+d−1 +a−a_c

a b

0

[Smets],[Smets, Willem],[Catrina, Wang],[Felli, Schneider]

The functional C^?_a,b

Z

R^d

|∇v|²

|x|²^a dx− Z

R^d

|v|^p

|x|^{b p} dx 2/p

is linearly instable atv =v?

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Symmetry versus symmetry breaking:

the sharp result in the critical case

[JD, Esteban, Loss (Inventiones 2016)]

a b

0

Theorem

Let d≥2and p<2^∗. If either a∈[0,ac)and b>0, or a<0 and b≥b_FS(a), then the optimal functions for the critical

Caffarelli-Kohn-Nirenberg inequalities are radially symmetric

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The Emden-Fowler transformation and the cylinder

BWith an Emden-Fowler transformation, critical the

Caffarelli-Kohn-Nirenberg inequality on the Euclidean space are equivalent to Gagliardo-Nirenberg inequalities on a cylinder

v(r, ω) =r^a−a^cϕ(s, ω) with r =|x|, s=−logr and ω= x r With this transformation, the Caffarelli-Kohn-Nirenberg inequalities can be rewritten as thesubcriticalinterpolation inequality

k∂sϕk²L²(C)+k∇^ωϕk²L²(C)+ Λkϕk²L²(C)≥µ(Λ)kϕk²L^p(C) ∀ϕ∈H¹(C) whereΛ := (a_c−a)²,C=R×S^d−1 and the optimal constantµ(Λ)is

µ(Λ) = 1

C_a,b with a=ac±√

Λ and b= d p ±√

Λ

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Linearization around symmetric critical points

Up to a normalization and a scaling

ϕ?(s, ω) = (coshs)⁻^p−2¹

is a critical point of

H¹(C)3ϕ7→ k∂sϕk²L²(C)+k∇^ωϕk²L²(C)+ Λkϕk²L²(C)

under a constraint onkϕk²L^p(C)

ϕ? is notoptimal for (CKN) if the P¨oschl-Teller operator

−∂_s²−∆_ω+ Λ−ϕ^p−2_? =−∂_s²−∆_ω+ Λ− 1 (coshs)² hasa negative eigenvalue

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Subcritical Caffarelli-Kohn-Nirenberg inequalities

Norms: kwk^L^q,γ⁽R^d) := R

R^d|w|^q|x|^−γ dx^1/q

,kwk^L^q⁽R^d) :=kwk^L^q,0⁽R^d)

(some)Caffarelli-Kohn-Nirenberg interpolation inequalities(1984) kwk^L^2p,γ⁽R^d)≤C_β,γ,pk∇wk^ϑL^2,β(R^d)kwk^1−ϑL^p+1,γ(R^d) (CKN) HereCβ,γ,p denotes the optimal constant, the parameters satisfy

d≥2, γ−2< β < ^d−2_d γ , γ∈(−∞,d), p∈(1,p_?] withp_?:= _d−β−2^d−γ and the exponentϑis determined by the scaling invariance, i.e.,

ϑ= (d−γ) (p−1)

p d+β+2−2γ−p(d−β−2)

Is the equality case achieved by the Barenblatt / Aubin-Talenti type function

w_?(x) = 1 +|x|^2+β−γ^−1/(p−1)

∀x ∈R^d ?

Do we know (symmetry) that the equality case is achieved among radial functions?

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Range of the parameters

Herepis given

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Symmetry and symmetry breaking

[JD, Esteban, Loss, Muratori, 2016]

Let us defineβFS(γ) :=d−2−p

(d−γ)²−4 (d−1) Theorem

Symmetry breaking holds in (CKN) if

γ <0 and βFS(γ)< β < d−2

d γ

In the rangeβFS(γ)< β <^d−2_d γ

w_?(x) = 1 +|x|^2+β−γ^−1/(p−1) is notoptimal

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The green area is the region of symmetry, while the red area is the region of symmetry breaking. The threshold is determined by the hyperbola

(d−γ)²−(β−d+ 2)²−4 (d−1) = 0

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A useful change of variables

With

α= 1 + β−γ

2 and n= 2 d−γ β+ 2−γ, (CKN) can be rewritten for a functionv(|x|^α−1x) =w(x)as

kvk^L^2p,d−n⁽R^d)≤K_α,n,pkD_αvk^ϑL^2,d−n(R^d)kvk^1−ϑL^p+1,d−n(R^d)

with the notationss=|x|,Dαv = α^∂v_∂s,¹_s ∇^ωv

. Parameters are in the range

d≥2, α >0, n>d and p∈(1,p_?], p_?:= n n−2 By our change of variables,w? is changed into

v?(x) := 1 +|x|²−_p−1¹

∀x∈R^d

The symmetry breaking condition (Felli-Schneider) now reads α < αFS with αFS:=

rd−1 n−1

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The second variation

J[v] :=ϑlog kDαvkL^2,d−n(R^d)

+ (1−ϑ) log kvkL^p+1,d−n(R^d)

+ log K_α,n,p−log kvk^L^2p,d−n⁽R^d)

Let us definedµδ:=µδ(x)dx, where µδ(x) := (1 +|x|²)^−δ. Sincev? is a critical point ofJ, a Taylor expansion at orderε²shows that

kDαv?k²L^2,d−n(R^d)J

v?+ε µδ/2f

=¹₂ε²ϑQ[f]+o(ε²) withδ=_p−1²^p and

Q[f] = Z

R^d

|Dαf|²|x|^n−ddµδ− 4pα² p−1

Z

R^d

|f|²|x|^n−ddµδ+1

We assume thatR

R^df|x|^n−ddµδ+1= 0(mass conservation)

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Symmetry breaking: the proof

Proposition (Hardy-Poincar´e inequality)

Let d≥2,α∈(0,+∞), n>d andδ≥n. If f has0 average, then Z

R^d

|Dαf|²|x|^n−ddµδ ≥Λ Z

R^d

|f|²|x|^n−ddµδ+1

with optimal constantΛ = min{2α²(2δ−n),2α²δ η} whereη is the unique positive solution toη(η+n−2) = (d−1)/α². The corresponding eigenfunction is not radially symmetric ifα²> (d−1)δ²

n(2δ−n) (δ−1) Q ≥0iff ^4p_p−1^α² ≤Λand symmetry breaking occurs in (CKN) if

2α²δ η < 4pα²

p−1 ⇐⇒ η <1

⇐⇒ d−1

α² =η(η+n−2)<n−1 ⇐⇒ α > αFS

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Inequalities without weights and fast diffusion equations

BR´enyi entropy powers

BSelf-similar variables and relative entropies BThe role of the spectral gap

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R´ enyi entropy powers and fast diffusion

BR´enyi entropy powers, the entropy approach without rescaling:

[Savar´e, Toscani]: scalings, nonlinearity and a concavity property inspired by information theory

BFaster rates of convergence: [Carrillo, Toscani],[JD, Toscani]

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The fast diffusion equation in original variables

Consider the nonlinear diffusion equation inR^d,d ≥1

∂v

∂t = ∆v^m

with initial datumv(x,t = 0) =v0(x)≥0such thatR

R^dv0dx= 1 and R

R^d|x|²v0dx<+∞. The large time behavior of the solutions is governed by the source-type Barenblatt solutions

U^?(t,x) := 1

κt^1/µdB^? x κt^1/µ

where

µ:= 2 +d(m−1), κ:=2µm m−1

1/µ

andB^? is the Barenblatt profile

B^?(x) :=





C?− |x|²1/(m−1)

+ ifm>1 C_?+|x|²1/(m−1)

ifm<1

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The R´ enyi entropy power F

Theentropy is defined by E :=

Z

R^d

v^mdx and theFisher informationby

I :=

Z

R^d

v|∇p|²dx with p = m m−1v^m−1 Ifv solves the fast diffusion equation, then

E⁰ = (1−m) I To computeI⁰, we will use the fact that

∂p

∂t = (m−1) p ∆p +|∇p|² F := E^σ with σ= µ

d(1−m) = 1+ 2 1−m

1

d +m−1

= 2 d

1 1−m−1 has a linear growth asymptotically ast →+∞

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The concavity property

Theorem

[Toscani-Savar´e]Assume that m≥1−¹d if d>1 and m>0 if d= 1.

Then F(t)is increasing,(1−m) F⁰⁰(t)≤0 and

t→+∞lim 1

t F(t) = (1−m)σ lim

t→+∞E^σ−1I = (1−m)σE^σ−1_? I_? [Dolbeault-Toscani]The inequality

E^σ−1I≥E^σ−1_? I_?

is equivalent to the Gagliardo-Nirenberg inequality k∇wk^θL²(R^d)kwk^1−θL^q+1(R^d)≥C_GNkwkL^2q(R^d)

if1−¹d ≤m<1. Hint: v^m−1/2= _kwk^w

L2q(Rd),q=₂_m−1¹

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The proof

Lemma

If v solves ^∂v_∂t = ∆v^m with ¹_d ≤m<1, then I⁰= d

dt Z

R^d

v|∇p|²dx =−2 Z

R^d

v^m

kD²pk²+ (m−1) (∆p)² dx Explicit arithmetic geometric inequality

kD²pk²− 1

d (∆p)²=

D²p− 1 d∆pId

2

There are no boundary terms in the integrations by parts

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Remainder terms

F⁰⁰=−σ(1−m) R[v]. Thepressure variableisP = _1−m^m v^m−1

R[v] := (σ−1) (1−m) E^σ−1 Z

R^d

v^m ∆P−

R

R^dv|∇P|²dx R

R^dv^mdx

2

dx + 2 E^σ−1

Z

R^d

v^mD²P−d¹∆PId²dx Let

G[v] := F[v] σ(1−m) =

Z

R^d

v^mdx σ−1Z

R^d

v|∇P|²dx The Gagliardo-Nirenberg inequality is equivalent toG[v0]≥G[v?] Proposition

G[v₀]≥G[v_?] + Z ∞

0

R[v(t,·)]dt

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Self-similar variables and relative entropies

The large time behavior of the solution of ^∂v_∂t = ∆v^m is governed by the source-typeBarenblatt solutions

v_?(t,x) := 1 κ^d(µt)^d/µB^?

x κ(µt)^1/µ

where µ:= 2 +d(m−1) whereB^?is the Barenblatt profile (with appropriate mass)

B^?(x) := 1 +|x|²^1/(m−1) A time-dependent rescaling: self-similar variables

v(t,x) = 1 κ^dR^d u

τ, x κR

where dR

dt =R^1−µ, τ(t) := ¹₂ log R(t)

R0

Then the functionusolvesa Fokker-Planck type equation

∂u

∂τ +∇ ·h

u ∇u^m−1− 2x i

= 0

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Free energy and Fisher information

The functionusolves a Fokker-Planck type equation

∂u

∂τ +∇ ·h

u ∇u^m−1− 2x i

= 0 [Ralston, Newman, 1984]Lyapunov functional:

Generalized entropyor Free energy E[u] :=

Z

R^d

−u^m

m +|x|²u

dx− E⁰

Entropy production is measured by theGeneralized Fisher information

d

dtE[u] =−I[u], I[u] :=

Z

R^d

u∇u^m−1+ 2x² dx

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Without weights: relative entropy, entropy production

Stationary solution: chooseC such thatku_∞k^L¹=kuk^L¹=M>0 u_∞(x) := C+ |x|²−1/(1−m)

+

Relative entropy: FixE⁰ so thatE[u_∞] = 0

Entropy – entropy production inequality[del Pino, J.D.]

Theorem

d≥3, m∈[^d−1_d ,+∞), m>¹₂, m6= 1 I[u]≥4E[u]

Corollary

[del Pino, J.D.]A solution u with initial data u0∈L¹₊(R^d)such that

|x|²u₀∈L¹(R^d), u₀^m∈L¹(R^d)satisfies

E[u(t,·)]≤ E[u₀]e⁻^4t

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A computation on a large ball, with boundary terms

∂u

∂τ +∇ ·h

u ∇u^m−1− 2x i

= 0 τ >0, x ∈BR

whereB_R is a centered ball in R^d with radiusR>0, and assume that usatisfies zero-flux boundary conditions

∇u^m−1−2x

· x

|x| = 0 τ >0, x ∈∂B_R. Withz(τ,x) :=∇Q(τ,x) :=∇u^m−1− 2x, therelative Fisher informationis such that

d dτ

Z

B_R

u|z|²dx+ 4 Z

B_R

u|z|²dx + 2^1−m_m

Z

BR

u^m

D²Q²− (1−m) (∆Q)² dx

= Z

∂BR

u^m ω· ∇|z|²

dσ≤0 (by Grisvard’s lemma)

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Another improvement of the GN inequalities

Let us define therelative entropy E[u] :=−1

m Z

R^d

u^m− B^m? − mB^m−1? (u− B^?) dx therelative Fisher information

I[u] :=

Z

R^d

u|z|²dx= Z

R^d

u ∇u^m−1−2x²dx

and R[u] := 21−m m

Z

R^d

u^mD²Q²− (1−m) (∆Q)² dx

Proposition

If1−1/d≤m<1and d ≥2, then I[u₀]− 4E[u₀]≥

Z ∞

0 R[u(τ,·)]dτ

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Entropy – entropy production, Gagliardo-Nirenberg ineq.

4E[u]≤ I[u]

Rewrite it withp=_2m−1¹ ,u=w^2p,u^m=w^p+1 as 1

2 2m

2m−1 ²Z

R^d

|∇w|²dx+ 1

1−m −d Z

R^d

|w|^1+pdx−K ≥0 for some γ,K =K0 R

R^du dx =R

R^dw^2pdx^γ w =w_∞=v∞^1/2p is optimal

Theorem

[Del Pino, J.D.]With1<p≤ d−2^d (fast diffusion case) and d ≥3 kwkL^2p(R^d) ≤ Cp,d^GNk∇wk^θL²(R^d)kwk^1−θL^p+1(R^d)

Cp,d^GN=_y(p−1)2

2πd

^θ₂

2y−d 2y

_2p¹ _Γ(y)

Γ(y−^d₂)

^θ_d

,θ= p(d+2−(d−2)p)^d(p−1) , y =_p−1^p+1

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Sharp asymptotic rates of convergence

Assumptions on the initial datumv0

(H1)VD₀ ≤v0≤VD₁ for someD0>D1>0

(H2)if d≥3 andm≤m_∗,(v₀−V_D)is integrable for a suitable D∈[D₁,D₀]

Theorem

[Blanchet, Bonforte, J.D., Grillo, V´azquez]Under Assumptions (H1)-(H2), if m<1 and m6=m_∗:= ^d−4_d−2, the entropy decays according to

E[v(t,·)]≤C e^{−2 (1−m) Λ}^α,d^t ∀t ≥0

whereΛ_α,d >0is the best constant in the Hardy–Poincar´e inequality Λ_α,d

Z

R^d

|f|²dµα−1≤ Z

R^d

|∇f|²dµα ∀f ∈H¹(dµα) withα:= 1/(m−1)<0, dµα:=hαdx , hα(x) := (1 +|x|²)^α

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Plots (d = 5)

λ01=−4α−2d

λ10=−2α λ11=−6α−2 (d+ 2) λ02=−8α−4 (d+ 2)

λ20=−4α λ30

λ21λ12 λ03

λcont α,d:=¹₄(d+ 2α−2)²

α=−d α=−(d+ 2)

α=−^d+22 α=−^d−22 α=−^d+62

α 0 Essential spectrum ofLα,d

α=−√ d−1−^d2 α=−√

d−1−^d+42 α=−√2d−^d+22

(d= 5) Spectrum ofLα,d

mc=^d−2_d

m1=^d−1_d m2=^d+1_d+2

! m1=d+2^d

! m2=^d+4_d+6

m Spectrum of (1−m)L1/(m−1),d

(d= 5)

Essential spectrum of (1

1

−m)L1/(m−1),d

2 4 6

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Improved asymptotic rates

[Bonforte, J.D., Grillo, V´azquez]Assume thatm∈(m1,1),d≥3.

Under Assumption (H1), ifv is a solution of the fast diffusion equation with initial datumv0such thatR

R^dx v0dx= 0, then the asymptotic convergence holds with an improved rate corresponding to the improved spectral gap.

m1=d+2^d

m1=^d−1d m2=^d+4d+6

m2=^d+1d+2

4

2

m 1 mc=d−2

d (d= 5)

γ(m)

0

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Spectral gaps and best constants

m_c=^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

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Comments

The spectral gap corresponding to the red curves relies on a refined notion of relative entropy with respect tobest matching Barenblatt profiles[J.D., Toscani]

A result by[Denzler, Koch, McCann]Higher order time asymptotics of fast diffusion in Euclidean space: a dynamical systems approach

The constantC in

E[v(t,·)]≤Ce^−2γ(m)^t ∀t≥0

can be made explicit, under additional restrictions on the initial data[Bonforte, J.D., Grillo, V´azquez]

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A parabolic proof ?

Large time asymptotics and spectral gaps Linearization and optimality

Weighted nonlinear flows:

Caffarelli-Kohn-Nirenberg inequalities

BEntropy and Caffarelli-Kohn-Nirenberg inequalities BLarge time asymptotics and spectral gaps

BOptimality cases

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CKN and entropy – entropy production inequalities

When symmetry holds, (CKN) can be written as anentropy – entropy productioninequality

1−m

m (2 +β−γ)²E[v]≤ I[v]

and equality is achieved byB_β,γ. Here thefree energyand the relative Fisher informationare defined by

E[v] := 1 m−1

Z

R^d

v^m−B^m_β,γ−mB^m−1_β,γ (v−Bβ,γ) dx

|x|^γ I[v] :=

Z

R^d

v∇v^m−1− ∇B^m−1_β,γ

2 dx

|x|^β. Ifv solves theFokker-Planck type equation

v_t+|x|^γ∇ ·h

|x|^−βv∇ v^m−1− |x|^2+β−γi

= 0 (WFDE-FP)

then d

dtE[v(t,·)] =− m

1−mI[v(t,·)]

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A parabolic proof ?

Proposition

Let m=^p+1_2p and consider a solution to (WFDE-FP) with nonnegative initial datum u₀∈L^1,γ(R^d)such that ku₀^mkL^1,γ(R^d) and

R

R^du₀|x|^2+β−2γdx are finite. Then

E[v(t,·)]≤ E[u0]e^{−(2+β−γ)}²^t ∀t≥0 if one of the following two conditions is satisfied:

(i) either u₀ is a.e. radially symmetric (ii) or symmetry holds in (CKN)

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A useful change of variables (reminder)

With

α= 1 + β−γ

2 and n= 2 d−γ β+ 2−γ, (CKN) can be rewritten for a functionv(|x|^α−1x) =w(x)as

kvkL^2p,d−n(R^d)≤K_α,n,pkD_αvk^ϑL^2,d−n(R^d)kvk^1−ϑL^p+1,d−n(R^d)

with the notationss=|x|,D_αv = α^∂v_∂s,¹_s ∇^ωv

. Parameters are in the range

d≥2, α >0, n>d and p∈(1,p_?], p_?:= n n−2 By our change of variables,w_? is changed into

v_?(x) := 1 +|x|²^−1/(p−1)

∀x∈R^d

The symmetry breaking condition (Felli-Schneider) now reads α > αFS with αFS:=

rd−1 n−1

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A parabolic proof ?

Towards a parabolic proof

For anyα≥1, let D_αW = α ∂rW,r⁻¹∇^ωW

so that D_α =∇+ (α−1) x

|x|²(x· ∇) =∇+ (α−1)ω ∂r

and define the diffusion operatorL_α by L_α=−D^∗_αD_α =α²

∂²_r +n−1 r ∂r

+∆_ω

r² where∆ω denotes the Laplace-Beltrami operator onS^d−1

∂g

∂t = Lαg^m is changed into

∂u

∂τ = D^∗_α(u z), z := D_αq, q :=u^m−1− Bα^m−1, B^α(x) :=

1 +|x|²

α² m−1¹

by the change of variables

g(t,x) = 1 κⁿRⁿu

τ, x κR

where





dR

dt =R^1−µ, R(0) =R0

τ(t) = ¹₂ log_R(t)

R₀

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If the weight does not introduce any singularity atx= 0...

m 1−m

d dτ

Z

B_R

u|z|²dµn

= Z

∂B_R

u^m ω·D_α|z|²

|x|^n−ddσ (≤0by Grisvard’s lemma)

−2^1−m_m m−1 +¹_n Z

B_R

u^m|Lαq|²dµn

− Z

B_R

u^m

α⁴m1

q⁰⁰−^qr⁰ −α²(n−1)^∆^ω^q r²

2

+²_r^α2²

∇^ωq⁰−^∇r^ω^q

2 dµn

−(n−2) α²_FS−α²Z

B_R

|∇^ωq|² r⁴ dµn

A formal computation that still needs to be justified (singularity atx= 0?)

Other potential application: the computation of Bakry, Gentil and Ledoux (chapter 6) for non-integer dimensions; weights on manifolds

[...]

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A parabolic proof ?

Fast diffusion equations with weights: large time asymptotics

Relative uniform convergence Asymptotic rates of convergence From asymptotic to global estimates

Herev solves the Fokker-Planck type equation vt+|x|^γ∇ ·h

|x|^−βv∇ v^m−1− |x|^2+β−γi

= 0 (WFDE-FP)

Joint work with M. Bonforte, M. Muratori and B. Nazaret

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Relative uniform convergence

ζ:= 1− 1−(1−m)^2−mq

1−^2−m1−mθ θ:= (1−m) (2+β−γ)

(1−m) (2+β)+2+β−γ is in the range0< θ < ^1−m_2−m <1 Theorem

For “good” initial data, there exist positive constantsKand t₀such that, for all q∈_2−m

1−m,∞

, the function w=v/Bsatisfies

kw(t)−1kL^q,γ(R^d) ≤ Ke⁻²^(1−m)2^2−m ^Λ^ζ^(t−t⁰⁾ ∀t ≥t0

in the caseγ∈(0,d), and

kw(t)−1kL^q,γ(R^d)≤ Ke⁻²^(1−m)2^2−m ^{Λ (t−t}⁰⁾ ∀t ≥t₀ in the caseγ≤0

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A parabolic proof ?

0

Λ0,1

Λ1,0

Λess

Essential spectrum

δ4 δ

δ₁ δ₂ δ5

Λ0,1

Λ1,0 Λess Essential spectrum

δ4 δ5:=n 2−η

The spectrum ofLas a function ofδ=_1−m¹ , withn= 5. The essential spectrum corresponds to the grey area, and its bottom is determined by the parabolaδ7→Λ_ess(δ). The two eigenvaluesΛ_0,1and Λ_1,0are given by the plain, half-lines, away from the essential

spectrum. The spectral gap determines the asymptotic rate of convergence to the Barenblatt functions

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Main steps of the proof:

Existence of weak solutions, L^1,γ contraction, Comparison Principle, conservation of relative mass

Self-similar variables and the Ornstein-Uhlenbeck equation in relative variables: the ratiow(t,x) :=v(t,x)/B(x)solves





|x|^−γw_t =−B¹ ∇ ·

|x|^−βBw∇ (w^m−1−1)B^m−1 in R⁺×R^d

w(0,·) =w0:=v0/B in R^d

Regularity: [Chiarenza, Serapioni], Harnack inequalities; relative uniform convergence (without rates) and asymptotic rates

(linearization)

The relative free energy and the relative Fisher information:

linearized free energy and linearized Fisher information A Duhamel formula and a bootstrap

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A parabolic proof ?

Asymptotic rates of convergence

Corollary

Assume that m∈(0,1), with m6=m_∗:= ⁿ⁻⁴_n−2. Under the relative mass condition, for any “good solution” v there exists a positive constantC such that

E[v(t)]≤ Ce⁻^{2 (1−m) Λ}^t ∀t ≥0.

With Csisz´ar-Kullback-Pinsker inequalities, these estimates provide a rate of convergence in L^1,γ(R^d)

Improved estimates can be obtained using “best matching techniques”

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From asymptotic to global estimates

When symmetry holds (CKN) can be written as anentropy – entropy productioninequality

(2 +β−γ)²E[v]≤ m 1−mI[v]

so that

E[v(t)]≤ E[v(0)]e⁻^{2 (1−m) Λ}^?^t ∀t ≥0 with Λ_?:= (2 +β−γ)² 2 (1−m)

Let us consider again theentropy – entropy production inequality K(M)E[v]≤ I[v] ∀v ∈L^1,γ(R^d) such that kvkL^1,γ(R^d)=M, whereK(M)is the best constant: withΛ(M) := ^m₂(1−m)⁻²K(M)

E[v(t)]≤ E[v(0)]e⁻2 (1−m) Λ(M)t

∀t≥0

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A parabolic proof ?

Symmetry breaking and global entropy – entropy production inequalities

Proposition

•In the symmetry breaking range of (CKN), for any M >0, we have 0<K(M)≤m² (1−m)²Λ_0,1

•If symmetry holds in (CKN) then

K(M)≥ ^1−mm (2 +β−γ)² Corollary

Assume that m∈[m₁,1)

(i) For any M>0, if Λ(M) = Λ_? thenβ=βFS(γ)

(ii) Ifβ > βFS(γ)thenΛ_0,1<Λ_? andΛ(M)∈(0,Λ_0,1]for any M>0 (iii) For any M>0, if β < βFS(γ)and if symmetry holds in (CKN), then

Λ(M)>Λ_?

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Linearization and optimality

Joint work with M.J. Esteban and M. Loss

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A parabolic proof ?

Linearization and scalar products

Withu_εsuch that

u_ε=B^? 1 +εfB?^1−m

and Z

R^d

u_εdx=M_? at first order inε→0 we obtain thatf solves

∂f

∂t =Lf where Lf := (1−m)B^m−2? |x|^γD^∗_α |x|^−βB^?D_αf Using the scalar products

hf1,f2i= Z

R^d

f1f2B^2−m? |x|^−γ dx and hhf1,f2ii= Z

R^d

Dαf1·Dαf2B^?|x|^−βdx we compute

1 2

d

dthf,fi=hf,Lfi= Z

R^d

f (Lf)B?^2−m|x|^−γ dx=− Z

R^d

|Dαf|²B^?|x|^−β dx=− hhf,fii for anyf smooth enough:

1 2

d

dthhf,fii= Z

R^d

D_αf ·D_α(Lf)B^?|x|^−βdx=− hhf,Lfii

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Linearization of the flow, eigenvalues and spectral gap

Now let us consider an eigenfunction associated with the smallest positive eigenvalueλ1ofL

− Lf₁=λ1f₁

so thatf₁realizes the equality case in theHardy-Poincar´e inequality hhg,gii=− hg,Lgi ≥λ1kg−g¯k², g¯ :=hg,1i/h1,1i

− hhg,Lgii ≥λ1hhg,gii Proof: expansion of the square :

− hh(g−g¯),L(g −g¯)ii=hL(g−g¯),L(g −g¯)i=kL(g −g¯)k² Key observation:

λ1≥4 ⇐⇒ α≤αFS :=

rd−1 n−1

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A parabolic proof ?

Symmetry breaking in CKN inequalities

Symmetry holds in (CKN) ifJ[w]≥ J[w_?]with J[w] :=ϑlog kDαwkL^2,δ(R^d)

+(1−ϑ) log kwkL^p+1,δ(R^d)

−log kwkL^2p,δ(R^d)

withδ:=d−nand

J[w_?+εg] =ε²Q[g]+o(ε²) where

2

ϑkD_αw_?k²L^2,d−n(R^d)Q[g]

=kD_αgk²L^2,d−n(R^d)+^p^(2+β−γ)_(p−1)₂

d−γ−p(d−2−β) Z

R^d

|g|²1+|x|^|x|^n−d² dx

−p(2p−1)^(2+β−γ)_(p−1)₂² Z

R^d

|g|²_(1+|x^|x|^n−d_|2)² dx is a nonnegative quadratic form if and only ifα≤αFS

Symmetry breaking holds ifα > αFS

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Information – production of information inequality

LetK[u]be such that d

dτI[u(τ,·)] =− K[u(τ,·)] =−(sum of squares) Ifα≤αFS, thenλ1≥4and

u7→ K[u]

I[u] −4 is a nonnegative functional

Withu_ε=B^? 1 +εfB^1−m?

, we observe that

4≤ C²:= inf

u

K[u]

I[u] ≤ lim

ε→0inf

f

K[u_ε] I[uε] = inf

f

hhf,Lfii

hhf,fii = hhf₁,Lf₁ii hhf1,f1ii =λ1

ifλ1= 4, that is, if α=αFS, theninfK/I= 4is achieved in the asymptotic regime asu→ B^?and determined by the spectral gap ofL

ifλ1>4, that is, if α < αFS, thenK/I >4

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A parabolic proof ?

Symmetry in Caffarelli-Kohn-Nirenberg inequalities

Ifα≤αFS, the fact thatK/I ≥4has an important consequence.

Indeed we know that d

dτ (I[u(τ,·)]− 4E[u(τ,·)])≤0 so that

I[u]−4E[u]≥ I[B^?]−4E[B^?] = 0

This inequality is equivalent toJ[w]≥ J[w?], which establishes that optimality in (CKN) is achieved among symmetric functions. In other words, the linearized problem shows that forα≤αFS, the function

τ7→ I[u(τ,·)]−4E[u(τ,·)]

is monotone decreasing

This explains why the method based on nonlinear flows provides theoptimal range for symmetry

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Entropy – production of entropy inequality

Using _dτ^d (I[u(τ,·)]− C²E[u(τ,·)])≤0, we know that I[u]− C²E[u]≥ I[B^?]− C²E[B^?] = 0 As a consequence, we have that

C¹:= inf

u

I[u]

E[u] ≥C²= inf

u

K[u]

I[u]

Withuε=B^? 1 +εfB^1−m?

, we observe that

C¹≤ lim

ε→0inf

f

I[uε] E[u_ε] = inf

f

hf,Lfi

hf,fi = hf1,Lf1i hf1,fi1

=λ1= lim

ε→0inf

f

K[uε] I[u_ε] Iflim_ε→0inff K[uε]

I[uε] =C², thenC¹=C²=λ1

This happens ifα=αFS and in particular in the case without weights (Gagliardo-Nirenberg inequalities)

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A parabolic proof ?

These slides can be found at

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

BLectures The papers can be found at

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

BPreprints / papers

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Optimality in critical and subcritical inequalities: ﬂows, linearization and entropy methods