On a result of symmetry based on nonlinear flows

Jean Dolbeault

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

February 24, 2017 ESI, Vienna

Geometric Transport Equations in General Relativity, February 20–24, 2017

The mexican hat potential in Schr¨ odinger equations

Let us consider a nonlinear Schr¨odinger equation in presence of a radial external potential with a minimum which is not at the origin

−∆u+V(x)u−f(u) = 0

-1.5 -1.0 -0.5 0.5 1.0 1.5

-1.0 -0.5 0.5

A one-dimensional potential V(x)

A two-dimensional potential V(x) with mexican hat shape

J. Dolbeault A symmetry result based on nonlinear flows

Radial solutions to−∆u+V(x)u−F⁰(u) = 0

... give rise to a radial density of energy x7→V|u|²+F(u)

symmetry breaking

... but in some cases minimal energy solutions

... give rise to a non-radial density of energy x7→V|u|²+F(u)

J. Dolbeault A symmetry result based on nonlinear flows

Symmetry breaking may occur...

Bin presence ofnon-cooperativepotentials

Bin weighted equations or weighted variational problems Bin phase transition problems

Bin evolution equations (instability of symmetric solutions)

Bin various models of mathematical physics and quantum field theory etc.

BGK-type kinetic equation as a motivation for nonlinear diffusions – polytropes and fast diffusion / porous medium

ε²∂tf^ε+εv· ∇^xf^ε−ε∇^xV(x)· ∇^vf^ε = Gf^ε−f^ε

f^ε(x,v,t= 0) = f_I(x,v), x,v ∈R³ with the Gibbs equilibriumGf :=γ

|v|²

2 +V(x)−µρf(x,t) The Fermi energyµρ_f(x,t)is implicitly defined by

Z

R³

γ |v|²

2 +V(x)−µρf(x,t)

dv= Z

R³

f(x,v,t)dv=:ρf(x,t) f^ε(x,v,t) . . . phase space particle density

V(x) . . . potential ε . . . mean free path

=⇒ µρf = ¯µ(ρf)

J. Dolbeault A symmetry result based on nonlinear flows

Diffusion limits

[J.D., P. Markowich, D. ¨Olz, C. Schmeiser]

Theorem

For anyε >0, the equation has a unique weak solution

f^ε∈C(0,∞;L¹∩L^p(R⁶))for all p<∞. Asε→0, f^εweakly converges to a local Gibbs state f⁰given by

f⁰(x,v,t) =γ 1

2|v|²−µ(ρ(x,¯ t))

whereρis a solution of the nonlinear diffusion equation

∂tρ=∇^x·(∇^xν(ρ)+ρ∇^xV(x)) with initial dataρ(x,0) =ρI(x) :=R

R³fI(x,v)dv ν(ρ) =

Z ρ 0

sµ¯⁰(s)ds

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 A symmetry result based on 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 ...

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, 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 A symmetry result based on nonlinear flows

Symmetry and symmetry breaking results

BThe critical Caffarelli-Kohn-Nirenberg inequality

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

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|^2a 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) (ac−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) ?

J. Dolbeault A symmetry result based on nonlinear flows

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|^2a 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)]

Linear instability of radial minimizers:

the Felli-Schneider curve

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

2p

(ac−a)²+d−1 +a−ac

a b

0

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

The functional C^?_a,b

Z

R^d

|∇v|²

|x|^2a dx− Z

R^d

|v|^p

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

is linearly instable atv =vJ. Dolbeault? A symmetry result based on 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

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−acϕ(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

C_a,b with a=ac±√

Λ and b= d p ±√

Λ

J. Dolbeault A symmetry result based on nonlinear flows

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

Subcritical Caffarelli-Kohn-Nirenberg inequalities

Norms: kwk^Lq,γ(R^d) := R

R^d|w|^q|x|^−γ dx^1/q

,kwk^Lq(R^d) :=kwk^Lq,0(R^d)

(some)Caffarelli-Kohn-Nirenberg interpolation inequalities(1984) kwk^L2p,γ(R^d)≤C_β,γ,pk∇wk^ϑL^2,β(R^d)kwk^1−ϑL^p+1,γ(R^d) (CKN) HereC_β,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 A symmetry result based on nonlinear flows

Range of the parameters

Herepis given

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 A symmetry result based on 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

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^L2p,d−n(R^d)≤Kα,n,pkDαvk^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−_p−1¹

∀x∈R^d

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

rd−1 n−1

J. Dolbeault A symmetry result based on nonlinear flows

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^L2p,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

kD_αv_?k²L^2,d−n(R^d)J

v_?+ε µδ/2f

=¹₂ε²ϑQ[f]+o(ε²) withδ=_p−1^2p and

Q[f] = Z

R^d

|D_αf|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)

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

J. Dolbeault A symmetry result based on nonlinear flows

Inequalities without weights and fast diffusion equations

B[The Bakry-Emery method on the sphere and its extension by non-linear diffusion flows]

BR´enyi entropy powers

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

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]

J. Dolbeault A symmetry result based on nonlinear flows

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

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 →+∞

J. Dolbeault A symmetry result based on nonlinear flows

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_GNkwk^L2q(R^d)

if1−¹d ≤m<1. Hint: v^m−1/2= _kwk^w

L2q(Rd),q=_2m−1¹

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

J. Dolbeault A symmetry result based on nonlinear flows

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

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

J. Dolbeault A symmetry result based on nonlinear flows

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

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

J. Dolbeault A symmetry result based on nonlinear flows

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 ∈∂BR. 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

∂B_R

u^m ω· ∇|z|²

dσ≤0 (by Grisvard’s lemma)

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^m

D²Q²− (1−m) (∆Q)² dx

Proposition

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

Z ∞

0 R[u(τ,·)]dτ

J. Dolbeault A symmetry result based on nonlinear flows

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 kwk^L2p(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

Sharp asymptotic rates of convergence

Assumptions on the initial datumv0

(H1)V_D0 ≤v₀≤V_D1 for someD₀>D₁>0

(H2)if d≥3 andm≤m∗,(v0−VD)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) Λα,dt} ∀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:=h_αdx , h_α(x) := (1 +|x|²)^α

J. Dolbeault A symmetry result based on nonlinear flows

Spectral gaps and best constants

mc=^d−2_d 0

m₁=^d−1_d

m₂=^d+1_d+2

! m2:=^d+4_d+6

m 2 4

Case 1 Case 2 Case 3

γ(m)

(d= 5)

! m₁:=_d+2^d

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]

J. Dolbeault A symmetry result based on nonlinear flows

Linearization and optimality

Weighted nonlinear flows:

Caffarelli-Kohn-Nirenberg inequalities

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

BOptimality cases

Linearization and optimality

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−B_β,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

vt+|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,·)]

J. Dolbeault A symmetry result based on nonlinear flows

Linearization and optimality

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)2t} ∀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)

Linearization and optimality

The strategy of the proof (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) ≤K_α,n,pkD_αvk^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|,D_αv = 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

J. Dolbeault A symmetry result based on nonlinear flows

Linearization and optimality

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|D_αP|²dµ whereσ= ²_{d 1−m}¹ −1. Heredµ=|x|^n−ddx and the pressure is

P := m 1−mu^m−1

Linearization and optimality

WithL^α=−D^∗_αDα=α² u⁰⁰+ⁿ⁻¹_s u⁰

+_s¹2∆ωu, we consider the fast diffusion equation

∂u

∂t =L^αu^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 L^αP−

R

Rdu|DαP|²dµ R

Rdu^mdµ

2

dµ + 2R

R^d

α⁴ 1−¹n P⁰⁰−^Ps⁰ −α²(n−1)^∆ωPs²

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.

J. Dolbeault A symmetry result based on nonlinear flows

Linearization and optimality

(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 byL^αu^m 0 =

Z

R^d

dG[u]· L^αu^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

w^1−3pdiv |x|^−βw^2p∇w^1−p

+|∇w^1−p|²+|x|^−γ c w^1−p−c

= 0

Linearization and optimality

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 Lα=−D^∗_αDα =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

J. Dolbeault A symmetry result based on nonlinear flows

Linearization and optimality

[...]

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

m 1−m

d dτ

Z

B_R

u|z|²dµn

= Z

∂BR

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|L_αq|²dµn

− Z

B_R

u^m

α⁴m₁q⁰⁰−^qr⁰ −α²(n−1)^∆ωq r²

2

+²_r^α₂² ∇^ωq⁰−^∇r^ωq

2 dµn

−(n−2) α²_FS−α²Z

BR

|∇^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

Linearization and optimality

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

J. Dolbeault A symmetry result based on nonlinear flows

Linearization and optimality

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)22−m Λζ(t−t0)} ∀t ≥t0

in the caseγ∈(0,d), and

kw(t)−1kL^q,γ(R^d)≤ Ke^{−2(1−m)22−m Λ (t−t0)} ∀t ≥t0

in the caseγ≤0

Linearization and optimality

0

Λ0,1

Λ1,0

Λess

Essential spectrum

δ δ₄

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

J. Dolbeault A symmetry result based on nonlinear flows

Linearization and optimality

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|^−γwt =−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

Linearization and optimality

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) Λt} ∀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”

J. Dolbeault A symmetry result based on nonlinear flows

Linearization and optimality

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

Linearization and optimality

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_?

J. Dolbeault A symmetry result based on nonlinear flows

Linearization and optimality

Linearization and optimality

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

Linearization and optimality

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

|D_αf|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

J. Dolbeault A symmetry result based on nonlinear flows

Linearization and optimality

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

Linearization and optimality

Symmetry breaking in CKN inequalities

Symmetry holds in (CKN) ifJ[w]≥ J[w_?]with J[w] :=ϑlog kD_αwkL^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

ϑkDαw?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

J. Dolbeault A symmetry result based on nonlinear flows

Linearization and optimality

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

Linearization and optimality

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

J. Dolbeault A symmetry result based on nonlinear flows

Linearization and optimality

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[u_ε] E[u_ε] = inf

f

hf,Lfi

hf,fi = hf₁,Lf₁i hf1,fi1

=λ1= lim

ε→0inf

f

K[u_ε] 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)

Linearization and optimality

These slides can be found at

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

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

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J. Dolbeault A symmetry result based on nonlinear flows