Nonlinear ﬂows and optimality of entropy - entropy production methods

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Nonlinear flows and optimality of entropy - entropy production methods

Jean Dolbeault

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

December 13, 2016 Vienna

Forefront of PDEs: modelling, analysis & numerics, December 12–14, 2016

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Interpolation inequalities on the sphere Symmetry breaking, linearization, flows Weighted nonlinear flows and CKN inequalities

Outline

BInterpolation inequalities on the sphere spectral methods and fractional operators bifurcations and flow methods

BSymmetry breaking and linearization

The sub-critical and critical Caffarelli-Kohn-Nirenberg inequalities Linearization and spectrum

Diffusions without weights: Gagliardo-Nirenberg inequalities and fast diffusion flows: R´enyi entropy powers, self-similar variables and relative entropies, the role of the spectral gap

BDiffusions with weights: Caffarelli-Kohn-Nirenberg inequalities and weighted nonlinear flows

Towards a parabolic proof

Large time asymptotics and spectral gaps A discussion of optimality cases

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Spectral methods Nonlinear flows on the sphere The bifurcation point of view

Interpolation inequalities on the sphere and eigenvalues of the

(fractional) Laplace operator

BA spectral point of view on the inequalities BThebifurcationpoint of view

BFlows on the sphere Carr´e du champ

Can one prove Sobolev’s inequalities with a heat flow ? Some open problems: constraints and improved inequalities [Beckner, 1993],[J.D., Zhang, 2016]

[Bakry, Emery, 1984]

[Bidault-V´eron, V´eron, 1991],[Bakry, Ledoux, 1996]

[Demange, 2008][J.D., Esteban, Loss, 2014 & 2015]

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Non-fractional interpolation inequalities

On thed-dimensional sphere, let us consider the interpolation inequality

k∇uk²L²(S^d)+ d

p−2kuk²L²(S^d) ≥ d

p−2kuk²L^p(S^d) ∀u∈H¹(S^d,dµ) where the measuredµis the uniform probability measure on S^d⊂R^d+1 induced by the Lebesgue measure onR^d+1

1≤p<2 or 2<p≤2^∗:= 2d d−2

ifd ≥3. We adopt the convention that2^∗=∞ifd = 1ord= 2.

The casep= 2 corresponds to the logarithmic Sobolev inequality

k∇uk²L²(S^d) ≥d 2

Z

S^d

|u|²log |u|² kuk²L²(S^d)

!

dµ ∀u∈H¹(S^d,dµ)\ {0}

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Optimal interpolation inequalities for fractional operators

The sharpHardy-Littlewood-Sobolev inequality onSⁿ[Lieb, 1983]

ZZ

Sⁿ×Sⁿ

F(ζ)|ζ−η|^−λF(η)dµ(ζ)dµ(η)≤ Γ(n) Γ ^n−λ₂ 2^λΓ ⁿ₂

Γ ⁿ_pkFk²L^p(Sⁿ)

λ∈(0,n),p=₂_n−λ²ⁿ ∈(1,2),λ=²_qⁿ

? where _p¹+_q¹

? = 1

sharp GNS inequalities on S^d: [Backner 1993],[Bidaut-V´eron, V´eron, 1991]

Asubcritical interpolation inequality dµis the uniform probability measure onSⁿ

L^s is the fractional Laplace operator of orders∈(0,n) q∈[1,2)∪(2,q?],q?= _n−s²ⁿ

kFk²L^q(Sⁿ)− kFk²L²(Sⁿ)

q−2 ≤C_q,s Z

Sⁿ

FL^sF dµ ∀F ∈H^s/2(Sⁿ)

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The sharp constants

Theorem

[J.D., Zhang]Let n≥1. If either s∈(0,n], q∈[1,2)∪(2,q?], or s=n and q ∈[1,2)∪(2,∞), then

C_q,s = ^n−s_2s ^Γ

n−s 2

Γ ^n+s₂

C⁻¹_q,s =λ1(L^s)= inf

F∈H^s/2(Sⁿ)\RQ[F], Q[F] := (q−2)R

SⁿFL^sF dµ kFk²L^q(Sⁿ)− kFk²L²(Sⁿ)

Sharp subcritical fractional logarithmic Sobolev inequalities Corollary

[J.D., Zhang]Let s∈(0,n]

Z

Sⁿ

|F|² log

|F| kFk^L²⁽Sⁿ)

dµ≤C_2,s Z

Sⁿ

FL^sF dµ ∀F ∈H^s/2(Sⁿ)

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From HLS to Sobolev and subcritical inequalities

Lieb’s approach: F =P∞

k=0F_(k) (spherical harmonics), HLS and Funk-Hecke formula

ZZ

Sⁿ×Sⁿ

F(ζ)|ζ−η|^−λF(η)dµ(ζ)dµ(η)

= Γ(n) Γ ^n−λ₂ 2^λΓ ⁿ₂

Γ ⁿ_p X∞ k=0

Γ(ⁿ_p) Γ(_pⁿ0 +k) Γ(_pⁿ0) Γ(ⁿ_p+k) Z

Sⁿ

|F_(k₎|²dµ

Duality: the fractional Sobolev inequality

kFk²L^q?(Sⁿ)≤ Z

Sⁿ

FK^sF dµ:=

X∞ k=0

γk n q_?

Z

Sⁿ

|F_(k)|²dµ

is dual of HLS, whereq_?=_n−s²ⁿ is the critical exponent and q7→γk n

q

is convex, withγk(x) := Γ(x) Γ(n−x+k)

Γ(n−x) Γ(x+k) is enough to establish the result in the subcritical range

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Fractional flows and related functional inequalities

BSphere: generalized fractional heat flow

∂u

∂t −q∇ ·

u¹⁻¹^q∇(−∆)⁻¹L^su^q¹

= 0 The entropy decays exponentially because of

1 q−2

d dt

" Z

Sⁿ

u dµ ²_q

− Z

Sⁿ

u²^qdµ

#

=−2 Z

Sⁿ

u¹^qL^su¹^qdµ BEuclidean space: any smooth nonnegative solution uof

∂u

∂t =∇ ·√

u∇(−∆)^−su^m−¹² is such that

d dt

Z

R^d

u^mdx= ²^m_2m−1^(1−m) Z

R^d

∇^(1−s)u^m−¹²

2

dx where∇^(1−s)w :=∇(−∆)^−s/2w. Rates ?

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The Bakry-Emery method on the sphere (non-fractional case)

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]

2

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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. (1)

[Demange],[J.D., 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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Can one prove Sobolev’s inequalities with a heat flow ?

1.0 1.5 2.0 2.5 3.0 3.5

1 2 3 4 5 6

(p, β)representation,d= 5. In the dark grey area, the functional is not monotone under the action of the heat flow[J.D., Esteban, Loss]

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The bifurcation point of view

µ(λ)is the optimal constant in the functional inequality

k∇uk²L²(S^d)+λkuk²L²(S^d)≥µ(λ)kuk²L^p(S^d) ∀u∈H¹(S^d,dµ)

2 4 6 8 10

2 4 6 8

Here d= 3 andp= 4

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A critical point ofu7→Q^λ[u] := k∇uk²L²(S^d)+λkuk²L²(S^d)

kuk²L^p(S^d)

solves

−∆u+λu=|u|^p−2u (EL) up to a multiplication by a constant (and a conformal transformation ifp= 2^∗)

The best constantµ(λ) = inf

u∈H¹(S^d,dµ)\{0}Q^λ[u]is such that µ(λ)< λifλ > _p−2^d , and µ(λ) =λifλ≤ p−2^d so that

d

p−2 = min{λ >0 : µ(λ)< λ}

Rigidity: the unique positive solution of (EL) isu=λ^1/(p−2)if λ≤ p−2^d

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Constraints and improvements

Taylor expansion:

d = inf

u∈H¹(S^d,dµ)\{0}

(p−2)k∇uk²L²(S^d)

kuk²_L^p(S^d)− kuk²_L²(S^d)

is achieved in the limit asε→0 withu= 1 +ε ϕ1such that

−∆ϕ1=dϕ1

BThis suggest that improved inequalities can be obtained 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: values of the shooting parametera 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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Critical and subcritical Caffarelli-Kohn-Nirenberg inequality Linearization and spectrum

Entropy methods without weights

Symmetry breaking results and tools for proving symmetry

BThe critical Caffarelli-Kohn-Nirenberg inequality

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

BR´enyi entropy powers and fast diffusion: [Savar´e, Toscani]

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

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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 (Γ2theory): [D. Bakry, M. Emery, 1984], [Bakry, Ledoux, 1996], [Demange, 2008],[JD, Esteban, Loss, 2014 & 2015] →D. Bakry, I. Gentil, and M. Ledoux.

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

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

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

Old and new (2009)

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

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Critical Caffarelli-Kohn-Nirenberg inequality

LetD^a,b:=n

v ∈L^p R^d,|x|^−bdx

: |x|^−a|∇v| ∈L² R^d,dx o Z

R^d

|v|^p

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

≤ C_a,b Z

R^d

|∇v|²

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

p= 2d

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

v?(x) =

1 +|x|^{(p−2) (a}^c^−a)−_p−2²

and C^?_a,b= k |x|^−bv_?k²p

k |x|^−a∇v_?k²2

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

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Critical CKN: range of the parameters

Figure: d = 3 Z

R^d

|v|^p

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

≤ C_a,b Z

R^d

|∇v|²

|x|²^a dx

a b

0 1

−1

b=a

b=a+ 1

a=^d−2₂ p

a≤b≤a+ 1ifd≥3

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

p= 2d

d−2 + 2 (b−a)

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

[Caffarelli, Kohn, Nirenberg (1984)]

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

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Linear instability of radial minimizers:

the Felli-Schneider curve

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

2p

(ac−a)²+d−1 +a−a_c

a b

0

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

The functional C^?_a,b

Z

R^d

|∇v|²

|x|²^a dx− Z

R^d

|v|^p

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

is linearly instable atv =v?

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Symmetry versus symmetry breaking:

the sharp result in the critical case

[JD, Esteban, Loss (Inventiones 2016)]

a b

0

Theorem

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

Caffarelli-Kohn-Nirenberg inequalities are radially symmetric

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The Emden-Fowler transformation and the cylinder

BWith an Emden-Fowler transformation, critical the

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

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

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

µ(Λ) = 1

C_a,b with a=ac±√

Λ and b= d p ±√

Λ

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Linearization around symmetric critical points

Up to a normalization and a scaling

ϕ?(s, ω) = (coshs)⁻^p−2¹

is a critical point of

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

under a constraint onkϕk²L^p(C)

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

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

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Subcritical Caffarelli-Kohn-Nirenberg inequalities

Norms: kwk^L^q,γ⁽R^d) := R

R^d|w|^q|x|^−γ dx^1/q

,kwk^L^q⁽R^d) :=kwk^L^q,0⁽R^d)

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

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

ϑ= (d−γ) (p−1)

p d+β+2−2γ−p(d−β−2)

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

w_?(x) = 1 +|x|^2+β−γ^−1/(p−1)

∀x ∈R^d ?

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

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Range of the parameters

Herepis given

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Symmetry and symmetry breaking

[JD, Esteban, Loss, Muratori, 2016]

Let us defineβFS(γ) :=d−2−p

(d−γ)²−4 (d−1) Theorem

Symmetry breaking holds in (CKN) if

γ <0 and βFS(γ)< β < d−2

d γ

In the rangeβFS(γ)< β <^d−2_d γ

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

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

(d−γ)²−(β−d+ 2)²−4 (d−1) = 0

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A useful change of variables

With

α= 1 + β−γ

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

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

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

. Parameters are in the range

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

v?(x) := 1 +|x|²−_p−1¹

∀x∈R^d

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

rd−1 n−1

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The second variation

J[v] :=ϑlog kDαvkL^2,d−n(R^d)

+ (1−ϑ) log kvkL^p+1,d−n(R^d)

+ log K_α,n,p−log kvk^L^2p,d−n⁽R^d)

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

kDαv?k²L^2,d−n(R^d)J

v?+ε µδ/2f

=¹₂ε²ϑQ[f]+o(ε²) withδ=_p−1²^p and

Q[f] = Z

R^d

|Dαf|²|x|^n−ddµδ− 4pα² p−1

Z

R^d

|f|²|x|^n−ddµδ+1

We assume thatR

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

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Symmetry breaking: the proof

Proposition (Hardy-Poincar´e inequality)

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

R^d

|Dαf|²|x|^n−ddµδ ≥Λ Z

R^d

|f|²|x|^n−ddµδ+1

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

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

2α²δ η < 4pα²

p−1 ⇐⇒ η <1

⇐⇒ d−1

α² =η(η+n−2)<n−1 ⇐⇒ α > αFS

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Symmetry in one slide: 3 steps

Achange of variables: v(|x|^α−1x) =w(x),D_αv = α^∂v_∂s,¹_s ∇^ωv 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)

Concavity of the R´enyi entropy power: with L^α=− D^∗αD_α=α² u⁰⁰+ⁿ⁻¹_s u⁰

+_s¹2∆_ωu and ^∂u_∂t =L^αu^m

−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)^∆^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 Elliptic regularity and the Emden-Fowler transformation: justifying theintegrations by parts

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Inequalities without weights and fast diffusion equations

R´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]

Self-similar variables and relative entropies: the issue of the boundary terms[Carrillo, Toscani],[Carrillo, V´azquez],[Carrillo, J¨ungel, Markowich, Toscani, Unterreiter]

Equivalence of the methods ? The role of the spectral gap

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R´ enyi entropy powers: FDE 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 BBakry-Emery method: ComputeI⁰

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The R´enyi entropy power

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 →+∞

Thepressure variableisP = _1−m^m v^m−1

∂P

∂t = (m−1) P ∆P +|∇P|² ThenF⁰⁰ is proportional to

(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

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The concavity property

Theorem

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

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

t→+∞lim 1

t F(t) = (1−m)σ lim

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

E^σ−1I≥E^σ−1_? I_?

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

if1−¹d ≤m<1. Hint: v^m−1/2= _kwk^w

L2q(Rd),q=₂_m−1¹

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Self-similar variables and relative entropies

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 logR(t) Then the functionusolvesa Fokker-Planck type equation

∂u

∂τ +∇ ·h

u ∇u^m−1− 2x i

= 0 E[u] :=R

R^d B^m−u^m−mB^m−1(B−u)

dx/mis such that d

dtE[u] =−I[u], I[u] :=

Z

R^d

u∇u^m−1+ 2x² dx [del Pino, J.D.], [Carrillo, Toscani]d ≥3,m∈[^d−1_d ,+∞),m> ¹₂, m6= 1

I[u]≥4E[u]

Ifu0∈L¹₊(R^d)is such that|x|²u0∈L¹(R^d),u₀^m∈L¹(R^d), then E[u(t,·)]≤ E[u0]e⁻^4t

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A computation on a large ball, with boundary terms

∂u

∂τ +∇ ·h

u ∇u^m−1− 2x i

= 0 τ >0, x ∈BR

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

∇u^m−1−2x

· x

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

d dτ

Z

B_R

u|z|²dx+ 4 Z

B_R

u|z|²dx + 2^1−m_m

Z

BR

u^m

D²Q²− (1−m) (∆Q)² dx

= Z

∂BR

u^m ω· ∇|z|²

dσ≤0 (by Grisvard’s lemma)

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Reintroducing R´ enyi entropy powers

therelative entropy E[u] :=−1

m Z

R^d

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

I[u] :=

Z

R^d

u|z|²dx= Z

R^d

u ∇u^m−1−2x²dx

R^?[u] = 2^1−m_m Z

R^d

u^m D²u^m−1− d¹∆u^m−1Id²dx + 2 (m−m1)^1−m_m R

R^du^m ∆u^m−1−2d²dx Proposition

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

Z ∞

R[u(τ,·)]dτ

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Sharp asymptotic rates of convergence

Assumptions on the initial datumv0

(H1)VD₀ ≤v0≤VD₁ for someD0>D1>0

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

Theorem

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

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

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

Z

R^d

|f|²dµα−1≤ Z

R^d

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

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Spectral gaps, asymptotic rates

mc=^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

[Bonforte, J.D., Grillo, V´azquez],[Bonforte, J.D., Grillo, V´azquez], [J.D., Toscani],[Denzler, Koch, McCann]

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A parabolic proof ?

Large time asymptotics and spectral gaps Linearization and optimality

Weighted nonlinear flows:

Caffarelli-Kohn-Nirenberg inequalities

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

BOptimality cases

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A parabolic proof ?

CKN and entropy – entropy production inequalities

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

1−m

m (2 +β−γ)²E[v]≤ I[v]

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

E[v] := 1 m−1

Z

R^d

v^m−B^m_β,γ−mB^m−1_β,γ (v−Bβ,γ) dx

|x|^γ I[v] :=

Z

R^d

v∇v^m−1− ∇B^m−1_β,γ

2 dx

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

v_t+|x|^γ∇ ·h

|x|^−βv∇ v^m−1− |x|^2+β−γi

= 0 (WFDE-FP)

then d

dtE[v(t,·)] =− m

1−mI[v(t,·)]

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A parabolic proof ?

Proposition

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

R

R^du₀|x|^2+β−2γdx are finite. Then

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

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

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A parabolic proof ?

Towards a parabolic proof

Letv(|x|^α−1x) =w(x),D_αv = α^∂v_∂s,¹_s∇^ωv

and define the diffusion operatorL_αby

L_α=−D^∗_αD_α =α²

∂²_r +n−1 r ∂r

+∆_ω

r² where∆_ω denotes the Laplace-Beltrami operator onS^d−1and consider the equation

∂u

∂τ = D^∗_α(u z) where

z := D_αq, q :=u^m−1− B^m−1α , B^α(x) :=

1 + |x|²

α² m−1¹

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A parabolic proof ?

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

m 1−m

d dτ

Z

BR

u|z|²dµn

= Z

∂B_R

u^m ω·D_α|z|²

|x|^n−ddσ (≤0by Grisvard’s lemma)

−2^1−m_m m−1 +¹_n Z

BR

u^m|L_αq|²dµn

− Z

B_R

u^m

α⁴m1

q⁰⁰−^qr⁰ −α²(n−1)^∆^ω^q r²

2

+²_r^α2²

∇^ωq⁰−^∇r^ω^q

2 dµn

−(n−2) α²_FS−α²Z

B_R

|∇^ωq|² r⁴ dµn

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

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

[...]

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A parabolic proof ?

Fast diffusion equations with weights: large time asymptotics

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

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

|x|^−βv∇ v^m−1− |x|^2+β−γi

= 0 (WFDE-FP)

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