Entropy methods for parabolic and elliptic equations

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equations

Jean Dolbeault

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

February 6, 2018 Politecnico di Milano

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Outline

BSymmetry breaking and linearization

The critical Caffarelli-Kohn-Nirenberg inequality Linearization and spectrum

A family of sub-critical Caffarelli-Kohn-Nirenberg inequalities

BWithout weights: Gagliardo-Nirenberg inequalities and fast diffusion flows

The Bakry-Emery method on the sphere 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

Large time asymptotics and spectral gaps A discussion of optimality cases

J. Dolbeault Entropy methods

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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 (Γ₂ theory): (D. Bakry, M. Emery, 1984), (Bentaleb),(Bakry, Ledoux, 1996),(Demange, 2008),(JD, Esteban, Kolwalczyk, 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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Recent related papers

JD, M.J. Esteban, and M. Loss. Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces. Invent. Math. 2016

JD, M.J. Esteban, and M. Loss. Interpolation inequalities on the sphere: linear vs. nonlinear flows. Annales de la facult´e des sciences de Toulouse 2017

M. Bonforte, JD, M. Muratori, and B. Nazaret. Weighted fast diffusion equations

(Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities.

(Part II): Sharp asymptotic rates of convergence in relative error by entropy methods. Kinetic and Related Models 2017

JD, M. J. Esteban, M. Loss, and M. Muratori. Symmetry for extremal functions in subcritical CKN inequalities. Comptes Rendus Math´ematique 2017

JD, M. J. Esteban, and M. Loss. Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization. Journal of elliptic and parabolic equations 2016

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Symmetry and symmetry breaking results

BThe critical Caffarelli-Kohn-Nirenberg inequality BLinearization and spectrum

BA family of sub-critical Caffarelli-Kohn-Nirenberg inequalities

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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

≤ Ca,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: Ca,b= C^?_a,b (symmetry) orCa,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

≤ Ca,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(ac−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 := (ac−a)²,C=R×S^d−1 and the optimal constantµ(Λ)is

µ(Λ) = 1 Ca,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, i.e., forΛ>Λ1(explicit)

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The variational problem on the cylinder

Λ7→µ(Λ) := min

ϕ∈H¹(C)

k∂sϕk²L²(C)+k∇^ωϕk²L²(C)+ Λkϕk²L²(C)

kϕk²L^p(C)

is a concave increasing function Restricted to symmetric functions, the variational problem becomes

µ?(Λ) := min

ϕ∈H¹(R)

k∂sϕk²L²(R^d)+ Λkϕk²L²(R^d)

kϕk²L^p(R^d)

=µ?(1) Λ^α

Symmetry meansµ(Λ) =µ?(Λ)

Symmetry breaking meansµ(Λ)< µ?(Λ)

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Numerical results

20 40 60 80 100

10 20 30 40 50

symmetric non-symmetric asymptotic

bifurcation µ

Λ^α µ(Λ)

(Λ) =µ

(1) Λ^α

Parametric plot of the branch of optimal functions forp= 2.8, d= 5.

Non-symmetric solutions bifurcate from symmetric ones at a bifurcation pointΛ1computed by V. Felli and M. Schneider. The branch behaves for large values ofΛ as predicted by F. Catrina and Z.-Q. Wang

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what we want to prove / discard...

non-symmetric symmetric

bifurcation J^θ(µ)

Λ^θ(µ) Λ^θ(µ)

J^θ(µ)

symmetric non-symmetric bifurcation

J^θ(µ) bifurcation

symmetric

non-symmetric

symmetric

Λ^θ(µ) KGN

KCKN(ϑ(p,d), ΛFS(p,θ),p) 1/

1/

∗

When the local criterion (linear stability) differs from global results in a larger family of inequalities (center, right)...

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The elliptic problem: rigidity

The symmetry issue can be reformulated as a uniqueness (rigidity) issue. An optimal function for the inequality

Z

R^d

|v|^p

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

≤ Ca,b

Z

R^d

|∇v|²

|x|²^a dx solves the (elliptic) Euler-Lagrange equation

−∇ · |x|^−2a∇v

=|x|^−bpv^p−1

(up to a scaling and a multiplication by a constant). Is any nonnegative solution of such an equation equal to

v_?(x) = 1 +|x|^{(p−2) (a}^c^−a)⁻_p−2² (up to invariances) ? On the cylinder

−∂²_sϕ−∂ωϕ+ Λϕ=ϕ^p−1 Up to a normalization and a scaling

ϕ?(s, ω) = (coshs)⁻^p−2¹

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Subcritical Caffarelli-Kohn-Nirenberg inequalities

Norms: kwk^L^q,γ(R^d) := R

R^d|w|^q|x|^−γ dx1/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)≤Cβ,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

(M. Bonforte, J.D., M. Muratori and B. Nazaret, 2016)Let us define βFS(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

(JD, Esteban, Loss, Muratori, 2016) Theorem

Symmetry holds in (CKN) if

γ≥0, or γ≤0 and γ−2≤β≤βFS(γ)

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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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Weighted nonlinear flows and CKN inequalities The role of the spectral gap

Inequalities without weights and fast diffusion equations

BThe Bakry-Emery method on the sphere: a parabolic method BEuclidean space: self-similar variables and relative entropies BThe role of the spectral gap

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The Bakry-Emery method on the sphere

Entropy functional E^p[ρ] := _p−2¹ hR

S^dρ²^p dµ− R

S^dρdµ²_pi

if p6= 2 E²[ρ] :=R

S^dρlog

ρ kρk_L1(S_d₎

dµ

Fisher informationfunctional I^p[ρ] :=R

S^d|∇ρ¹^p|²dµ

Bakry-Emery (carr´e du champ) method: use the heat flow

∂ρ

∂t = ∆ρ

and compute _dt^dE^p[ρ] =− I^p[ρ]and _dt^dI^p[ρ]≤ −dI^p[ρ]to get d

dt(I^p[ρ]−dE^p[ρ])≤0 =⇒ I^p[ρ]≥dE^p[ρ]

withρ=|u|^p, ifp≤2^#:= ₍²_d−1)^d²⁺¹2 J. Dolbeault Entropy methods

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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

(Demange),(JD, Esteban, Kowalczyk, Loss): for anyp∈[1,2^∗] K^p[ρ] := d

dt

I^p[ρ]−dE^p[ρ]

≤0

1.0 1.5 2.5 3.0

0.0 0.5 1.5 2.0

(p,m)admissible region,d = 5

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The elliptic version of the problem

The interpolation inequality (p6= 2)

dE^p[ρ] = d p−2

"Z

S^d

ρ^p² dµ− Z

S^d

ρdµ ²_p#

≤ I^p[ρ] = Z

S^d

|∇ρ¹^p|²dµ

can be rewritten foru=ρ¹^p as(W. Beckner, 1993) Z

S^d

|u|^pdµ _p²

≤ p−2 d

Z

S^d

|∇u|²dµ+ Z

S^d

u²dµ ∀u∈H¹(S^d,dµ) and amounts to prove that any nonnegative solution of

−∆_Sdu+λu=u^p−1

is a constant ifλ≤_p−2^d : (B. Gidas, J. Spruck, 1981),

(M.-F. Bidaut-V´eron, L. V´eron, 1991),(Bakry, Ledoux, 1996)

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R´ enyi entropy powers and fast diffusion

The Euclidean space without weights

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 variation of the Fisher information

Lemma

If v solves ^∂v_∂t = ∆v^m with1¹

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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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)≥CGNkwkL^2q(R^d)

if1−¹d ≤m<1. Hint: v^m−1/2= _kwk^w

L2q(Rd),q=₂_m−1¹

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Euclidean space: 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|²1/(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 function usolves 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⁻⁴^t

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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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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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Spectral gap: 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

whereΛα,d >0is the best constant in the Hardy–Poincar´e inequality Λα,d

Z

R^d

|f|²dµ_α−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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Spectral gap 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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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 byBβ,γ(x) := 1 +|x|^2+β−γ_m−1¹

Here thefree energyand therelative 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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Proof of symmetry (1/3: changing the dimension)

We rephrase our problem in a space of higher,artificial dimension n>d (herenis a dimension at least from the point of view of the scaling properties), or to be precise we consider a weight|x|^n−d which is the same in all norms. With

v(|x|^α−1x) =w(x), α= 1 +β−γ

2 and n= 2 d−γ β+ 2−γ, we claim that Inequality (CKN) can be rewritten for a function v(|x|^α−1x) =w(x)as

kvkL^2p,d−n(R^d) ≤Kα,n,pkDαvk^ϑL^2,d−n(R^d)kvk^1−ϑL^p+1,d−n(R^d) ∀v ∈H^p_d−n,d−n(R^d) with the notationss=|x|,Dαv= α^∂v_∂s,¹_s∇^ωv

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

v_?(x) := 1 +|x|²−1/(p−1)

∀x∈R^d

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The strategy of the proof (2/3: R´ enyi entropy)

The derivative of the generalizedR´enyi entropy powerfunctional is

G[u] :=

Z

R^d

u^mdµ σ−1Z

R^d

u|DαP|²dµ

whereσ= ²_d _1−m¹ −1. Heredµ=|x|^n−ddx and the pressure is P := m

1−mu^m−1

Looking for an optimal function in (CKN) is equivalent to minimizeG under a mass constraint

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WithLα=−D^∗_αDα=α² u⁰⁰+ⁿ⁻¹_s u⁰

+_s¹2∆ωu, we consider the fast diffusion equation

∂u

∂t = Lαu^m

in the subcritical range1−1/n<m<1. The key computation is the proof that

−dt^d G[u(t,·)] R

R^du^mdµ1−σ

≥(1−m) (σ−1)R

R^du^m LαP−

R

Rdu|DαP|²dµ R

Rdu^mdµ

²dµ + 2R

R^d

α⁴ 1−¹n P⁰⁰−^Ps⁰ −α²(^∆n−1)^ω^Ps²

²+^2α_s2²

∇^ωP⁰−^∇^ωs^P

² u^mdµ + 2R

R^d

(n−2) α²_FS−α²

|∇^ωP|²+c(n,m,d)^|∇_P^ω2^P|⁴

u^mdµ=:H[u]

for some numerical constantc(n,m,d)>0. Hence ifα≤αFS, the r.h.s.H[u] vanishes if and only ifPis an affine function of|x|², which proves the symmetry result. A quantifier elimination problem (Tarski, 1951)?

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(3/3: elliptic regularity, boundary terms)

This method has a hidden difficulty: integrations by parts ! Hints:

use elliptic regularity: Moser iteration scheme, Sobolev regularity, local H¨older regularity, Harnack inequality, and get global regularity using scalings

use the Emden-Fowler transformation, work on a cylinder, truncate, evaluate boundary terms of high order derivatives using Poincar´e inequalities on the sphere

Summary: ifusolves the Euler-Lagrange equation, we test byLαu^m 0 =

Z

R^d

dG[u]·Lαu^mdµ≥ H[u]≥0

H[u]is the integral of a sum of squares (with nonnegative constants in front of each term)... or test by|x|^γdiv |x|^−β∇w^1+p

the equation (p−1)²

p(p+ 1)w^1−3pdiv |x|^−βw^2p∇w^1−p

+|∇w^1−p|²+|x|^−γ c₁w^1−p−c₂

= 0

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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⁻^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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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→Λess(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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Global vs. asymptotic estimates

Estimates on the global rates. When symmetry holds (CKN) can be written as anentropy – entropy production inequality

(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)

Optimal global rates. Let us consider again theentropy – entropy productioninequality

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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Linearization and optimality

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

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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^?Dαf 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 for anyf smooth enough: with hf,Lfi=− hhf,fii

1 2

d

dthhf,fii= Z

R^d

Dαf ·Dα(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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Why is this method optimal ?

The condition λ1<4 is sufficient for symmetry breaking Withλ1≥4, we prove that

H[v] := m

1−mI[v]−(2 +β−γ)²E[v]

is monotone decaying along the flow and that equality is achieved only on the stationary solution, which attracts all solutions. This has to do with the (formal) gradient flow structure of the problem

The condition λ1≥4 is enough to prove that

Bthe Fisher informationI[v] exponentially decays with ratee⁻⁴^t Bthe functionalH[v]is decreasing

The decay of the Fisher informationI[v]“has to be given” by the conditionλ1≥4 because the problem degenerates into a sharp spectral gap problem in the asymptotic regime

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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?

andα≤αFS, we observe that

4≤ C2:= inf

u

K[u]

I[u] ≤ lim

ε→0inf

f

K[uε] 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, ifα < αFS, thenK/I ≥4... in fact,infK/I= 4 !

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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[B^?], 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[uε] = inf

f

hf,Lfi

hf,fi = hf₁,Lf₁i hf₁,fi1

=λ1= lim

ε→0inf

f

K[uε] I[uε] Iflim_ε→0inff K[uε]

I[u_ε] =C2, thenC1=C2=λ₁

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

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A loss of compactness...

Work in progress with N. Simonov]Caseλ1>4,i.e.,α < αFS

free energy, orgeneralized relative entropy E[v] := 1

m−1 Z

R^d

v^m−B^m−mB^m−1(v−B) dx

|x|^γ relative Fisher information

I[v] :=

Z

R^d

v∇v^m−1− ∇B^m−1² dx

|x|^β Proposition

[JD, Simonov]In the symmetry range, for any M >0, inf

I[v]

E[v] : v∈ D+(R^d), Z

R^d

v|x|^−γ dx=M

= 1−m

m (2 +β−γ)² Conjecture: in the symmetry breaking range,inf^I[v]_E[v] is determined by the spectral gap

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...research in progress

[with N. Simonov] Entropy – entropy production inequalities in the symmetry breaking range of CKN

[with A. Zhang] Towards proofs in the weighted parabolic case (sphere: BGL Sobolev inequality and CKN): regularization of the weight ?

[with N. Simonov and A. Zhang] Doubly nonlinear parabolic case (euclidean space)

[with M. Garc´ıa-Huidobro and R. Man´asevich]Doubly nonlinear parabolic case (sphere)

Full (analytic) parabolic proof based on thecarr´e du champ method based on the analysis of the regularity in the neighborhood of degenerating points / singularities of the potentials[collaborators are welcome !]

Hypo-coercive methods and (sharp) decay rates in coupled kinetic equations...

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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

For final versions, useDolbeaultas login andJeanas password

Entropy methods for parabolic and elliptic equations

equations

Outline

Collaborations

Background references (partial)

Recent related papers

Symmetry and symmetry breaking results

Critical Caffarelli-Kohn-Nirenberg inequality

Critical CKN: range of the parameters

Linear instability of radial minimizers:

the Felli-Schneider curve

Symmetry versus symmetry breaking:

the sharp result in the critical case

The Emden-Fowler transformation and the cylinder

Linearization around symmetric critical points

The variational problem on the cylinder

Numerical results

what we want to prove / discard...

The elliptic problem: rigidity

Subcritical Caffarelli-Kohn-Nirenberg inequalities

Range of the parameters

Symmetry and symmetry breaking

Inequalities without weights and fast diffusion equations

The Bakry-Emery method on the sphere

The evolution under the fast diffusion flow

The elliptic version of the problem

R´ enyi entropy powers and fast diffusion

The fast diffusion equation in original variables

The R´ enyi entropy power F

The variation of the Fisher information

The concavity property

Euclidean space: self-similar variables and relative entropies

Free energy and Fisher information

Without weights: relative entropy, entropy production

A computation on a large ball, with boundary terms

Entropy – entropy production, Gagliardo-Nirenberg ineq.

Spectral gap: sharp asymptotic rates of convergence

Spectral gap and best constants

Weighted nonlinear flows:

Caffarelli-Kohn-Nirenberg inequalities

CKN and entropy – entropy production inequalities

Proof of symmetry (1/3: changing the dimension)

The strategy of the proof (2/3: R´ enyi entropy)

(3/3: elliptic regularity, boundary terms)

Fast diffusion equations with weights: large time asymptotics

Relative uniform convergence

Global vs. asymptotic estimates

Linearization and optimality

Linearization and scalar products

Linearization of the flow, eigenvalues and spectral gap

Why is this method optimal ?

Information – production of information inequality

Symmetry in Caffarelli-Kohn-Nirenberg inequalities

Entropy – production of entropy inequality

A loss of compactness...

...research in progress

Thank you for your attention !