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 21, 2018 Young PDE’s @ Roma

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Outline

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

BSymmetry breaking and linearization

The critical Caffarelli-Kohn-Nirenberg inequality Linearization and spectrum

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

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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Inequalities without weights and fast diffusion equations

BThe Bakry-Emery method on the sphere: a parabolic method BEuclidean space: R´enyi entropy powers

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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Cylindrical coordinates, Schwarz symmetrization, stereographic projection...

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... and the ultra-spherical operator

Change of variablesz = cosθ,v(θ) =f(z),dνd :=ν^d²⁻¹dz/Zd, ν(z) := 1−z²

The self-adjointultrasphericaloperator is

Lf := (1−z²)f⁰⁰−d z f⁰=νf⁰⁰+d 2 ν⁰f⁰ which satisfieshf₁,Lf₂i=−R1

−1f₁⁰f₂⁰νdνd

Proposition

Let p∈[1,2)∪(2,2^∗], d ≥1. For any f ∈H¹([−1,1],dνd),

− hf,Lfi= Z 1

−1|f⁰|²ν dνd ≥dkfk²L^p(S^d)− kfk²L²(S^d)

p−2

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The heat equation ^∂g_∂t =Lg forg =f^p can be rewritten in terms off

as ∂f

∂t =Lf + (p−1)|f⁰|² f ν

−1 2

d dt

Z 1

−1|f⁰|²ν dνd =1 2

d

dt hf,Lfi=hLf,Lfi+(p−1) |f⁰|²

f ν,Lf

d

dtI[g(t,·)] + 2dI[g(t,·)]= d dt

Z 1

−1|f⁰|²νdνd+ 2d Z 1

−1|f⁰|²νdνd

=−2 Z 1

−1

|f⁰⁰|²+ (p−1) d d+ 2

|f⁰|⁴

f² −2 (p−1)d−1 d+ 2

|f⁰|²f⁰⁰ f

ν²dνd

is nonpositive if

|f⁰⁰|²+ (p−1) d d+ 2

|f⁰|⁴

f² − 2 (p−1)d−1 d+ 2

|f⁰|²f⁰⁰ f is pointwise nonnegative, which is granted if

(p−1)d−1 d+ 2

2

≤(p−1) d

d+ 2 ⇐⇒ p≤ 2d²+ 1

(d−1)² = 2^#< 2d d−2 = 2^∗

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Bifurcation point of view

2 4 6 8 10

2 4 6 8

The interpolation inequality from the point of view of bifurcations k∇uk²L²(S^d)+λkuk²L²(S^d)≥µ(λ)kuk²L^p(S^d)

Taylor expansion ofu= 1 +ε ϕ₁as ε→0with−∆ϕ₁=dϕ₁ µ(λ)≤λ if (and only if) λ > d

p−2

BImproved inequalities under appropriate orthogonality constraints ?

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Integral constraints

With the heat flow...

Proposition

For any p∈(2,2^#), the inequality Z 1

−1|f⁰|²ν dνd+ λ

p−2kfk²2≥ λ p−2kfk²p

∀f ∈H¹((−1,1),dνd)s.t.

Z 1

−1

z|f|^pdνd = 0 holds with

λ≥d+ (d−1)²

d(d+ 2)(2^#−p) (λ^?−d)

... and with a nonlinear diffusion flow ?

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Antipodal symmetry

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

Theorem

If p∈(1,2)∪(2,2^∗), we have Z

S^d

|∇u|²dµ≥ d p−2

1 + (d²−4) (2^∗−p)

d(d+ 2) +p−1 kuk²L^p(S^d)− kuk²L²(S^d)

for any u∈H¹(S^d,dµ)with antipodal symmetry. The limit case p= 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 kuk²L²(S^d)

! dµ

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The larger picture: branches of antipodal solutions

8 9 10 11 12 13 14

Case d= 5,p= 3: the branch µ(λ) as a function ofλ

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The optimal constant in the antipodal framework

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= 5 and 1≤p≤10/3≈3.33. The limiting value of the constant is numerically found to be equal to λ?= 2^1−2/pd≈6.59754with d= 5 andp= 10/3

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Work in progress and open questions (1)

BAdding weights

fractional dimensions: in the ultraspherical setting, if one adds the weight(1−z²)^n−d² theultrasphericaloperator becomes

Lf := (1−z²)f⁰⁰−n z f⁰=νf⁰⁰+d 2 ν⁰f⁰ The Bakry-Emery method applies only forp≤2^#

The interpolation inequality is true up top= 2^∗[Pearson]

Parabolic flows: [JD, Zhang]using(ε²+ 1−z²)^n−d² , in progress Open: general singular potentials beyond the range covered by linear flows ?

BConstrained problems

The Lin-Ni problem in convex domains [JD, Kowalczyk]

Branches with turning points corresponding to higher Morse index ?

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The elliptic / rigidity point of view (nonlinear flow)

ut =u^2−2β

Lu+κ^|u_u⁰^|²ν

... Which computation do we have to do ?

− L u−(β−1)|u⁰|²

u ν+ λ

p−2u= λ p−2u^κ Multiply byLuand integrate

...

Z 1

−1Lu u^κdνd=−κ Z 1

−1

u^κ|u⁰|² u dνd

Multiply byκ^|u_u⁰^|² and integrate

...= +κ Z 1

−1

u^κ|u⁰|² u dνd

The two terms cancel and we are left only with Z 1

−1

u⁰⁰−p+ 2 6−p

|u⁰|² u

2

ν²dνd= 0 ifp= 2^∗andβ = 4 6−p

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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)

+

Entropy – entropy production inequality (del Pino, JD)

Theorem

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

p= _2m−1¹ ,u=w^2p: (GN) k∇wk^θL²(R^d)kwk^1−θL^q+1(R^d)≥CGNkwkL^2q(R^d)

Corollary

(del Pino, JD)A solution u with initial data u₀∈L¹₊(R^d)such that

|x|²u₀∈L¹(R^d), u₀^m∈L¹(R^d)satisfiesE[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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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, JD, 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), Z

R^d

fdµ_α−1= 0 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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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 (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 have to 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, JD, 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:

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|^−βdx 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 (P1)

− hhg,Lgii ≥λ1hhg,gii (P2) Proof by expansion of the square

− hh(g−g¯),L(g −g¯)ii=hL(g−g¯),L(g −g¯)i=kL(g −g¯)k² (P1) is associated with the symmetry breaking issue (P2) is associated with thecarr´e du champmethod The optimal constants / eigenvalues are the same

Key observation: λ1≥4 ⇐⇒ α≤αFS:=q

d−1 n−1

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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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Work in progress and open questions (2)

[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 of the flow in the neighborhood of degenerate points / singularities of the potentials [collaborators are welcome !]

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

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Lecture notes onSymmetry and nonlinear diffusion flows...

a course on entropy methods (see webpage)

[JD, Maria J. Esteban, and Michael Loss]Symmetry and symmetry breaking: rigidity and flows in elliptic PDEs ... the elliptic point of view: arXiv: 1711.11291 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

Inequalities without weights and fast diffusion equations

The Bakry-Emery method on the sphere

The evolution under the fast diffusion flow

Cylindrical coordinates, Schwarz symmetrization, stereographic projection...

... and the ultra-spherical operator

Bifurcation point of view

Integral constraints

Antipodal symmetry

The larger picture: branches of antipodal solutions

The optimal constant in the antipodal framework

Work in progress and open questions (1)

The elliptic / rigidity point of view (nonlinear flow)

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

Spectral gap: sharp asymptotic rates of convergence

Spectral gap and best constants

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 have to to prove / discard...

The elliptic problem: rigidity

Subcritical Caffarelli-Kohn-Nirenberg inequalities

Range of the parameters

Symmetry and symmetry breaking

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

A loss of compactness...

Work in progress and open questions (2)

Thank you for your attention !