Jean Dolbeault

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

16 mai 2017 Institut Camille Jordan Universit´e Claude Bernard Lyon 1

### 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 Symmetry, entropy methods and nonlinear flows

### 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 ...

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### 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 (Γ_{2} 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, J¨ungel, Lederman, Markowich, Toscani, Unterreiter, Villani...,(del Pino, JD, 2001),(Blanchet, Bonforte, JD, Grillo, V´azquez)→A. J¨ungel, 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)

J. Dolbeault Symmetry, entropy methods and nonlinear flows

## Symmetry and symmetry breaking results

BThe critical Caffarelli-Kohn-Nirenberg inequality BLinearization and spectrum

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

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### Critical Caffarelli-Kohn-Nirenberg inequality

LetD^{a,b}:=n

v ∈L^{p} R^{d},|x|^{−b}dx

: |x|^{−a}|∇v| ∈L^{2} 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−2}^{2}

and C^{?}_{a,b}= k |x|^{−b}v_{?}k^{2}p

k |x|^{−a}∇v_{?}k^{2}2

Question: Ca,b= C^{?}_{a,b} (symmetry) orCa,b>C^{?}_{a,b} (symmetry breaking) ?

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### 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−2}_{2}
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)]

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### 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?

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### 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

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### 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}^{c}ϕ(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∂sϕk^{2}L^{2}(C)+k∇^{ω}ϕk^{2}L^{2}(C)+ Λkϕk^{2}L^{2}(C)≥µ(Λ)kϕk^{2}L^{p}(C) ∀ϕ∈H^{1}(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

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### Linearization around symmetric critical points

Up to a normalization and a scaling

ϕ?(s, ω) = (coshs)^{−}^{p−2}^{1}

is a critical point of

H^{1}(C)3ϕ7→ k∂sϕk^{2}L^{2}(C)+k∇^{ω}ϕk^{2}L^{2}(C)+ Λkϕk^{2}L^{2}(C)

under a constraint onkϕk^{2}L^{p}(C)

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

−∂s^{2}−∆ω+ Λ−ϕ^{p−2}_{?} =−∂s^{2}−∆ω+ Λ− 1
(coshs)^{2}
hasa negative eigenvalue, i.e., forΛ>Λ1(explicit)

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### The variational problem on the cylinder

Λ7→µ(Λ) := min

ϕ∈H^{1}(C)

k∂sϕk^{2}L^{2}(C)+k∇^{ω}ϕk^{2}L^{2}(C)+ Λkϕk^{2}L^{2}(C)

kϕk^{2}L^{p}(C)

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

µ?(Λ) := min

ϕ∈H^{1}(R)

k∂sϕk^{2}L^{2}(R^{d})+ Λkϕk^{2}L^{2}(R^{d})

kϕk^{2}L^{p}(R^{d})

=µ?(1) Λ^{α}

Symmetry meansµ(Λ) =µ?(Λ)

Symmetry breaking meansµ(Λ)< µ?(Λ)

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### 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

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### what we wan to 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)...

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### 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^{ϑ}L^{2,β}(R^{d})kwk^{1−ϑ}L^{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 ?

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### Range of the parameters

Herepis given

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### 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)

J. Dolbeault Symmetry, entropy methods and nonlinear flows

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

J. Dolbeault Symmetry, entropy methods and nonlinear flows

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

BEuclidean space: self-similar variables and relative entropies BThe role of the spectral gap

J. Dolbeault Symmetry, entropy methods and nonlinear flows

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

### The Bakry-Emery method on the sphere

Entropy functional
E^{p}[ρ] := _{p−2}^{1} hR

S^{d}ρ^{2}^{p} dµ− R

S^{d}ρdµ^{2}_{p}i

if p6= 2
E^{2}[ρ] :=R

S^{d}ρlog

ρ
kρk_{L1(S}_{d}_{)}

dµ

Fisher informationfunctional
I^{p}[ρ] :=R

S^{d}|∇ρ^{1}^{p}|^{2}dµ

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

∂ρ

∂t = ∆ρ

and compute _{dt}^{d}E^{p}[ρ] =− I^{p}[ρ]and _{dt}^{d}I^{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)}^{d}^{2}^{+1}2
J. Dolbeault Symmetry, entropy methods and nonlinear flows

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

### The evolution under the fast diffusion flow

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

∂ρ

∂t = ∆ρ^{m}. (1)

(Demange),(JD, Esteban, Kowalczyk, Loss): for anyp∈[1,2^{∗}]
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

J. Dolbeault Symmetry, entropy methods and nonlinear flows

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

### 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)

J. Dolbeault Symmetry, entropy methods and nonlinear flows

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

### 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) =v_{0}(x)≥0such thatR

R^{d}v_{0}dx= 1 and
R

R^{d}|x|^{2}v_{0}dx<+∞. The large time behavior of the solutions is
governed by the source-type Barenblatt solutions

U^{?}(t,x) := 1

κt^{1/µ}dB^{?} 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|^{2}1/(m−1)

+ ifm>1
C_{?}+|x|^{2}1/(m−1)

ifm<1

J. Dolbeault Symmetry, entropy methods and nonlinear flows

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

### The R´ enyi entropy power F

Theentropy is defined by E :=

Z

R^{d}

v^{m}dx
and theFisher informationby

I :=

Z

R^{d}

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

E^{0} = (1−m) I
To computeI^{0}, 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 →+∞

J. Dolbeault Symmetry, entropy methods and nonlinear flows

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

### The concavity property

Theorem

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

Then F(t)is increasing,(1−m) F^{00}(t)≤0 and

t→+∞lim 1

t F(t) = (1−m)σ lim

t→+∞E^{σ−1}I = (1−m)σE^{σ−1}_{?} I?

[Dolbeault-Toscani]The inequality

E^{σ−1}I≥E^{σ−1}_{?} I?

is equivalent to the Gagliardo-Nirenberg inequality
k∇wk^{θ}L^{2}(R^{d})kwk^{1−θ}L^{q+1}(R^{d})≥CGNkwkL^{2q}(R^{d})

if1−^{1}d ≤m<1. Hint: v^{m−1/2}= _{kwk}^{w}

L2q(Rd),q=_{2}_{m−1}^{1}

J. Dolbeault Symmetry, entropy methods and nonlinear flows

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

### The proof

Lemma

If v solves ^{∂v}_{∂t} = ∆v^{m} with ^{1}_{d} ≤m<1, then
I^{0}= d

dt Z

R^{d}

v|∇p|^{2}dx =−2
Z

R^{d}

v^{m}

kD^{2}pk^{2}+ (m−1) (∆p)^{2}
dx
Explicit arithmetic geometric inequality

kD^{2}pk^{2}− 1

d (∆p)^{2}=

D^{2}p− 1
d∆pId

2

There are no boundary terms in the integrations by parts

J. Dolbeault Symmetry, entropy methods and nonlinear flows

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

### Remainder terms

F^{00}=−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^{d}v|∇P|^{2}dx
R

R^{d}v^{m}dx

2

dx
+ 2 E^{σ−1}

Z

R^{d}

v^{m}D^{2}P−d^{1}∆PId^{2}dx
Let

G[v] := F[v] σ(1−m) =

Z

R^{d}

v^{m}dx
σ−1Z

R^{d}

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

G[v_{0}]≥G[v?] +
Z ∞

0

R[v(t,·)]dt

J. Dolbeault Symmetry, entropy methods and nonlinear flows

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

### 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/µ}B^{?}

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^{d}R^{d} u

τ, x κR

where dR

dt =R^{1−µ}, τ(t) := ^{1}_{2} log
R(t)

R_{0}

Then the functionusolvesa Fokker-Planck type equation

∂u

∂τ +∇ ·h

u ∇u^{m−1}− 2x i

= 0

J. Dolbeault Symmetry, entropy methods and nonlinear flows

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

### 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|^{2}u

dx− E0

Entropy production is measured by theGeneralized Fisher information

d

dtE[u] =−I[u], I[u] :=

Z

R^{d}

u∇u^{m−1}+ 2x^{2} dx

J. Dolbeault Symmetry, entropy methods and nonlinear flows

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

### Without weights: relative entropy, entropy production

Stationary solution: chooseC such thatku_{∞}k^{L}^{1}=kuk^{L}^{1}=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>^{1}_{2}, m6= 1
I[u]≥4E[u]

Corollary

(del Pino, J.D.)A solution u with initial data u_{0}∈L^{1}_{+}(R^{d})such that

|x|^{2}u_{0}∈L^{1}(R^{d}), u_{0}^{m}∈L^{1}(R^{d})satisfies

E[u(t,·)]≤ E[u_{0}]e^{−}^{4}^{t}

J. Dolbeault Symmetry, entropy methods and nonlinear flows

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

### 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|^{2}dx+ 4
Z

B_{R}

u|z|^{2}dx
+ 2^{1−m}_{m}

Z

BR

u^{m}

D^{2}Q^{2}− (1−m) (∆Q)^{2}
dx

= Z

∂BR

u^{m} ω· ∇|z|^{2}

dσ≤0 (by Grisvard’s lemma)

J. Dolbeault Symmetry, entropy methods and nonlinear flows

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

### Entropy – entropy production, Gagliardo-Nirenberg ineq.

4E[u]≤ I[u]

Rewrite it withp=_{2m−1}^{1} ,u=w^{2p},u^{m}=w^{p+1} as
1

2 2m

2m−1 2Z

R^{d}

|∇w|^{2}dx+
1

1−m −d Z

R^{d}

|w|^{1+p}dx−K ≥0
for some γ,K =K_{0} R

R^{d}u dx =R

R^{d}w^{2p}dx^{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^{GN}k∇wk^{θ}L^{2}(R^{d})kwk^{1−θ}L^{p+1}(R^{d})

Cp,d^{GN}=_{y(p−1)}2

2πd

^{θ}_{2}_{2y−d}

2y

_{2p}^{1} _{Γ(y)}

Γ(y−^{d}_{2})

^{θ}_{d}

,θ= p(d+2−(d−2)p)^{d(p−1)} , y =_{p−1}^{p+1}

J. Dolbeault Symmetry, entropy methods and nonlinear flows

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

### Spectral gap: sharp asymptotic rates of convergence

Assumptions on the initial datumv_{0}
(H1)VD_{0} ≤v_{0}≤VD_{1} for someD_{0}>D_{1}>0

(H2)if d≥3 andm≤m_{∗},(v_{0}−V_{D})is integrable for a suitable
D∈[D_{1},D_{0}]

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|^{2}dµ_{α−1}≤
Z

R^{d}

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

J. Dolbeault Symmetry, entropy methods and nonlinear flows

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

### Spectral gap and best constants

m_{c}=^{d−2}_{d} 0

m1=^{d−1}_{d}

m_{2}=^{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}

J. Dolbeault Symmetry, entropy methods and nonlinear flows

## Weighted nonlinear flows:

## Caffarelli-Kohn-Nirenberg inequalities

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

BOptimality cases

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### CKN and entropy – entropy production inequalities

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

1−m

m (2 +β−γ)^{2}E[v]≤ I[v]

and equality is achieved byBβ,γ(x) := 1 +|x|^{2+β−γ}_{m−1}^{1}

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|^{−β}v∇ v^{m−1}− |x|^{2+β−γ}i

= 0 (WFDE-FP) then d

dtE[v(t,·)] =− m

1−mI[v(t,·)]

J. Dolbeault Symmetry, entropy methods and nonlinear flows

Proposition

Let m=^{p+1}_{2}_{p} and consider a solution to (WFDE-FP) with nonnegative
initial datum u_{0}∈L^{1,γ}(R^{d})such that ku_{0}^{m}kL^{1,γ}(R^{d}) and

R

R^{d}u_{0}|x|^{2+β−2γ}dx 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 u_{0} is a.e. radially symmetric
(ii) or symmetry holds in (CKN)

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### 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−1}x) =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−1}x) =w(x)as

kvkL^{2p,d−n}(R^{d}) ≤Kα,n,pkDαvk^{ϑ}L^{2,d−n}(R^{d})kvk^{1−ϑ}L^{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}∇^{ω}v

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}

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### 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^{m}dµ
σ−1Z

R^{d}

u|DαP|^{2}dµ

whereσ= ^{2}_{d} _{1−m}^{1} −1. Heredµ=|x|^{n−d}dx 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

J. Dolbeault Symmetry, entropy methods and nonlinear flows

WithLα=−D^{∗}_{α}Dα=α^{2} u^{00}+^{n−1}_{s} u^{0}

+_{s}^{1}2∆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^{d}u^{m}dµ1−σ

≥(1−m) (σ−1)R

R^{d}u^{m}
LαP−

R

Rdu|DαP|^{2}dµ
R

Rdu^{m}dµ

^{2}dµ
+ 2R

R^{d}

α^{4} 1−^{1}n P^{00}−^{P}s^{0} −α^{2}(^{∆}n−1)^{ω}^{P}s^{2}

^{2}+^{2α}_{s}2^{2}

∇^{ω}P^{0}−^{∇}^{ω}s^{P}

^{2}
u^{m}dµ
+ 2R

R^{d}

(n−2) α^{2}_{FS}−α^{2}

|∇^{ω}P|^{2}+c(n,m,d)^{|∇}_{P}^{ω}2^{P|}^{4}

u^{m}dµ=: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)?

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### (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^{m}dµ≥ 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|^{γ}div |x|^{−β}∇w^{1+p}

the equation
(p−1)^{2}

p(p+ 1)w^{1−3p}div |x|^{−β}w^{2p}∇w^{1−p}

+|∇w^{1−p}|^{2}+|x|^{−γ} c_{1}w^{1−p}−c_{2}

= 0

J. Dolbeault Symmetry, entropy methods and nonlinear flows

## 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|^{−β}v∇ v^{m−1}− |x|^{2+β−γ}i

= 0 (WFDE-FP)

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

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### Relative uniform convergence

ζ:= 1− 1−_{(1−m)}^{2−m}q

1−^{2−m}_{1−m}θ
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 t_{0}such 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−t}^{0}^{)} ∀t ≥t_{0}
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 ≥t_{0}
in the caseγ≤0

J. Dolbeault Symmetry, entropy methods and nonlinear flows

0

Λ0,1

Λ1,0

Λess

Essential spectrum

δ4 δ

δ_{1} δ_{2} δ5

Λ0,1

Λ1,0 Λess Essential spectrum

δ4 δ5:=n 2−η

The spectrum ofLas a function ofδ=_{1−m}^{1} , withn= 5. The
essential spectrum corresponds to the grey area, and its bottom is
determined by the parabolaδ7→Λess(δ). The two eigenvaluesΛ_{0,1}and
Λ_{1,0}are given by the plain, half-lines, away from the essential

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

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### Global vs. asymptotic estimates

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

(2 +β−γ)^{2}E[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) := ^{m}_{2}(1−m)^{−2}K(M)

E[v(t)]≤ E[v(0)]e^{−}2 (1−m) Λ(M)t

∀t≥0

J. Dolbeault Symmetry, entropy methods and nonlinear flows

## Linearization and optimality

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

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### Linearization and scalar products

Withu_{ε}such that

u_{ε}=B^{?} 1 +εfB?^{1−m}

and Z

R^{d}

u_{ε}dx=M_{?}
at first order inε→0 we obtain thatf solves

∂f

∂t =Lf where Lf := (1−m)B^{m−2}? |x|^{γ}D^{∗}_{α} |x|^{−β}B^{?}Dαf
Using the scalar products

hf_{1},f_{2}i=
Z

R^{d}

f_{1}f_{2}B^{2−m}? |x|^{−γ} dx and hhf_{1},f_{2}ii=
Z

R^{d}

Dαf_{1}·Dαf_{2}B^{?}|x|^{−β}dx
we compute

1 2

d

dthf,fi=hf,Lfi= Z

R^{d}

f (Lf)B?^{2−m}|x|^{−γ} dx=−
Z

R^{d}

|Dαf|^{2}B^{?}|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|^{−β}dx=− hhf,Lfii

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### Linearization of the flow, eigenvalues and spectral gap

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

− Lf_{1}=λ1f_{1}

so thatf_{1}realizes the equality case in theHardy-Poincar´e inequality
hhg,gii=− hg,Lgi ≥λ_{1}kg−g¯k^{2}, g¯ :=hg,1i/h1,1i

− hhg,Lgii ≥λ_{1}hhg,gii
Proof: expansion of the square :

− hh(g−g¯),L(g −g¯)ii=hL(g−g¯),L(g −g¯)i=kL(g −g¯)k^{2}
Key observation:

λ1≥4 ⇐⇒ α≤αFS :=

rd−1 n−1

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### 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)^{2}E[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

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### 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] =ε^{2}Q[g]+o(ε^{2})
where

2

ϑkDαw_{?}k^{2}L^{2,d−n}(R^{d})Q[g]

=kDαgk^{2}L^{2,d−n}(R^{d})+^{p}^{(2+β−γ)}_{(p−1)}2

d−γ−p(d−2−β) Z

R^{d}

|g|^{2}_{1+|x|}^{|x|}^{n−d}^{2} dx

−p(2p−1)^{(2+β−γ)}_{(}_{p−1)}2^{2}

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

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### Information – production of information inequality

LetK[u]be such that d

dτI[u(τ,·)] =− K[u(τ,·)] =−(sum of squares)
Ifα≤αFS, thenλ_{1}≥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[uε] I[uε] = inf

f

hhf,Lfii

hhf,fii = hhf_{1},Lf_{1}ii
hhf_{1},f_{1}ii =λ1

ifλ_{1}= 4, that is, if α=αFS, theninfK/I= 4is achieved in the
asymptotic regime asu→ B^{?}and determined by the spectral gap ofL

ifλ_{1}>4, that is, if α < αFS, thenK/I >4

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### 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

J. Dolbeault Symmetry, entropy methods and nonlinear flows

### 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

C^{1}≤ lim

ε→0inf

f

I[uε] E[uε] = inf

f

hf,Lfi

hf,fi = hf_{1},Lf_{1}i
hf_{1},fi1

=λ1= lim

ε→0inf

f

K[uε]
I[uε]
Iflim_{ε→0}inff K[uε]

I[u_{ε}] =C2, thenC1=C2=λ_{1}

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

J. Dolbeault Symmetry, entropy methods and nonlinear flows

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

### Thank you for your attention !

J. Dolbeault Symmetry, entropy methods and nonlinear flows