Entropy methods in partial differential equations: fast diffusion equations

Entropy methods in partial differential equations: fast diffusion equations

Jean Dolbeault

[email protected]

CEREMADE

CNRS & Universit´e Paris-Dauphine

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

SLOVAK-AUSTRIAN MATHEMATICAL CONGRESS

September 22, 2007

Entropy methods in partial differential equations: fast diffusion equations – p.1/3

Abstract

Many new results on asymptotic behavior, sharp rates, optimal regulariza- tion effects, etc. have been achieved for the solutions of nonlinear diffu- sion equations over the last few years. By (generalized) entropy, we mean special Lyapunov functionals which have a probabilistic interpretation or a physical meaning. Such entropies also have deep connections with the (nonlinear) structure of the equation. The key underlying estimate is usu- ally a functional inequality which relates the entropy with its time derivative.

In the case of fast diffusion equations, the functional inequality is an inter- polation inequality of Gagliardo-Nirenberg type. The talk will be devoted to

Contents

Linear Diffusions

- intermediate asymptotics

- the entropy - entropy production method (Ornstein-Uhlenbeck) - (spectral approaches)

The fast diffusion equation

- intermediate asymptotics and interpolation - extensions (finite mass regine)

- the infinite mass regime and Hardy-Poincaré inequalities Other entropies and algebraic rates

- Generalized Poincaré inequalities for second and fourth order equations

- L^q Poincaré inequalities for general measures

Entropy methods in partial differential equations: fast diffusion equations – p.3/3

Intermediate asymptotics of linear diffusion equations

Consider the heat equation:







u^t = ∆u x ∈ R^d , t ∈ R⁺ u(·, t = 0) = u0 ≥ 0

Z

R^d

u0 dx = 1 ⁽¹⁾

As t → +∞, u(x, t) ∼ U(x, t) = ^e_(4πt)^−|x|2_d/^/4₂^t

What is the (optimal) rate of convergence of ku(·, t) − U(·, t)k^L1(R^d) ?

Time dependent rescaling: Fokker-Planck equation

u(x, t) = 1

R^d(t) v

ξ = x

R(t), τ = log R(t) + τ(0)

allows to transform this question into that of the convergence to the stationary solution v∞(ξ) = (2π)^−d/2 e^−|ξ|2/2.

• Ansatz: ^dR_dt = ¹

R R(0) = 1 τ(0) = 0:

R(t) = √

1 + 2t , τ(t) = log R(t) As a consequence: v(τ = 0) = u0.

• Fokker-Planck equation:







v^τ = ∆v + ∇(ξ v) ξ ∈ R^d , τ ∈ R⁺ v(·, τ = 0) = u0 ≥ 0

Z

R^d

u0 dx = 1 ⁽²⁾

Entropy methods for inear diffusion equations – p.2/13

Entropy (relative to the stationary solution v

)

Σ[v] :=

Z

R^d

v log

v v∞

dx

If v is a solution of (2), then (I is the Fisher information) d

dτ Σ[v(·, τ)] = − Z

R^d

v

∇log

v v∞

2

dx =: −I[v(·, τ)]

• Euclidean logarithmic Sobolev inequality: If kvk^L1 = 1, then Z

R^d

v logv dx + d

1 + 1

2 log(2π)

≤ 1 2

Z

R^d

|∇v|² v dx

Σ[v(·, τ)] ≤ ¹₂I[v(·, τ)], Equality: v(ξ) = v∞(ξ) = (2π)^−d/2 e^−|ξ|2/2 Σ[v(·, τ)] ≤ e^−2τΣ[u0] = e^−2τ

Z

u0 log

u0

v∞

dx

Csiszár-Kullback inequality

Consider v ≥ 0, v¯ ≥ 0 such that R

R^d v dx = R

R^d v dx¯ =: M > 0 Z

R^d

v logv v¯

dx ≥ 1

4 M kv − v¯k²L¹(R^d)

=⇒ kv − v∞k²L¹(R^d) ≤ 4 M Σ[u0]e^−2τ τ(t) = log √

1 + 2t

ku(·, t) − u∞(·, t)k²L¹(R^d) ≤ 4

1 + 2t Σ[u0] u∞(x, t) = 1

R^d(t) v∞

x

R(t), τ(t)

The proof of the Csiszár-Kullback inequality is given by a Taylor develop- ment at second order.

Entropy methods for inear diffusion equations – p.4/13

Logarithmic Sobolev inequalities

1) independent of the dimension [Gross, 75]: gaussian form Z

R^d

w logw dµ(x) ≤ 1 2

Z

R^d

w |∇ logw|² dµ(x) with w = ^v

M v_∞ , kvk^L1 = M, dµ(x) = v∞(x)dx 2) invariant under scaling [Weissler, 78]

Z

R^d

v² logv² dx ≤ d

2 log

2 π d e

Z

R^d

|∇v|² dx

for any v ∈ H¹(R^d) such that R

v² dx = 1

Proof: optimize for v^λ(x) = λ^d/2v(λ x) w.r.t. λ > 0

Entropy-entropy production / Bakry-Emery method

... a proof of the Euclidean logarithmic Sobolev inequality:

d dτ

I[v(·, τ)] − 2 Σ[v(·, τ)]

= −C

Xd

i,j=1

Z

R^d

w^ij + a wⁱw^j

w + b w δ^ij

2

dx < 0

for some C > 0, a, b ∈ R and w = √ v

I[v(·, τ)] − 2 Σ[v(·, τ)] & I[v∞] − 2 Σ[v∞] = 0

=⇒ ∀ u0 , I[u0] − 2 Σ[u0] ≥ I[v(·, τ)] − 2 Σ[v(·, τ)]≥ 0 ∀ τ > 0

Entropy methods for inear diffusion equations – p.6/13

Entropy-entropy production method

Goal: large time behavior of parabolic equations:







v^t = divx[D(x) (∇^xv + v∇^xA(x))] = div[e^−A∇(ve^A)]

t > 0, x ∈ R^d v(x, t = 0) = v0(x) ∈ L¹₊(R^d)

(3)

A(x). . . given ‘potential’

v∞(x) = e^−A(x) ∈ L¹ . . . (unique) steady state mass conservation: R

R^d v(t) dx = R

R^d v∞ dx = 1

questions: exponential rate ? connection to logarithmic Sobolev inequalities ? [Bakry-Emery ’84, Gross ’75, Toscani ’96, ...]

Entropy-entropy production method

[Bakry, Emery, 84]

[Arnold, Markowich, Toscani, Unterreiter, 01]

Relative entropy of v w.r.t. v∞:

Σ[v|v∞] :=

Z

R^d

ψ

v v∞

v∞ dx ≥ 0 with ψ(w) ≥ 0 for w ≥ 0, convex

ψ(1) = ψ⁰(1) = 0 2 (ψ⁰⁰⁰)² ≤ ψ⁰⁰ ψ^{I V} Examples:

ψ1 = w lnw − w + 1, Σ₁(v|v∞) = R

v ln

v v_∞

dx . . . physical entropy ψp = ^{wp−p(w−1)−1}_p−1 , 1 < p ≤ 2, Σ₂(v|v∞) = R

R^d(v − v∞)²v_∞⁻¹dx

Entropy methods for inear diffusion equations – p.8/13

Exponential decay of entropy production

I(v(t)|v∞) := d

dtΣ[v(t)|v∞] = − Z

ψ⁰⁰( v v∞

)| ∇ v v∞

| {z }

=:u

|²v∞dx≤ 0

Assume: D ≡ 1, ^∂_∂x^2A2 ≥ λ1Id, λ1 > 0 (A(x) . . . unif. convex) entropy production rate:

I⁰ = 2 Z

ψ⁰⁰( v v∞

)u^T · ∂²A

∂x² · uv∞dx + 2 Z

Tr(XY )v∞dx

| {z }

≥0

≥ −2 λ1I

X = ψ⁰⁰(_v^v

∞) ψ⁰⁰⁰(_v^v

∞) ψ⁰⁰⁰( ^v

v_∞ ) ¹₂ψ^{I V} ( ^v

v_∞)

!

≥ 0 , Y =

P

ij(_∂x^∂ui

j )² u^T · ^∂u_∂x · u u^T · ^∂u_∂x · u |u|⁴

!

≥ 0 Exponential decay of relative entropy: [Arnold, Markowich, Toscani,

Convex Sobolev inequalities

[Arnold, Markowich, Toscani, Unterreiter]: Entropy–entropy production estimate for A(x) = −lnv∞ uniformly convex:

Σ[v|v∞] ≤ 1

2λ1 |I(v|v∞)|

Example 1: logarithmic entropy ψ1(w) = w lnw − w + 1 Z

v ln( v v∞

)dx ≤ 1 2λ1

Z

v|∇ln( v v∞

)|²dx

∀v, v∞ ∈ L¹₊(R^d), R

v dx = R

v∞ dx = 1 Set f² = ^v

v_∞ ⇒

Z

f² lnf²dv∞ ≤ 2 λ1

Z

|∇f|²dv∞

∀f ∈ L²(R^d, dv∞), R

f²dv∞ = 1 logarithmic Sobolev inequality–dv∞ measure version [Gross ’75]

Entropy methods for inear diffusion equations – p.10/13

Convex Sobolev inequalities (continued)

Example 2: non-logarithmic entropies:

ψ^p(w) = ^wp−p_p^(w₋₁⁻¹⁾⁻¹ , 1 < p ≤ 2

(B^p) p p − 1

Z

f²dv∞ − Z

|f|^2pdv∞

^p

≤ 2 λ1

Z

|∇f|²dv∞

With _v^v

∞ = ^|f|

2p

R |f|^p2dv_∞

∀f ∈ L^2p(R^d, v∞dx) Poincaré-type inequality [Beckner ’89], (B^p) ⇒ (B2), 1 < p ≤ 2

Refined convex Sobolev inequalities

Estimate of entropy production rate / entropy production:

I⁰ = 2 Z

ψ⁰⁰( v v∞

)u^T · ∂²A

∂x² · uv∞dx + 2 Z

Tr (XY )v∞dx

| {z }

≥0

≥ −2λ1I

[Arnold, JD]: Observe that for ψ^p(w) = ^{wp−p(w−1)−1}_p−1 , 1 < p < 2:

X = ψ⁰⁰(_v^v

∞) ψ⁰⁰⁰(_v^v

∞ ) ψ⁰⁰⁰( ^v

v_∞) ¹₂ψ^{I V} ( ^v

v_∞)

!

> 0

Entropy methods for inear diffusion equations – p.12/13

Refined Beckner / generalized Poincaré inequalities

• Assume ^∂A_∂x2² ≥ λ1Id ⇒ Σ⁰⁰ ≥ −2λ1Σ⁰+κ^|Σ_1+Σ^0|2, κ = ^2−p

p < 1

⇒ k(Σ[v|v∞]) ≤ _2λ¹₁|Σ⁰| = _2λ¹

1

R ψ⁰⁰(_v^v

∞ )|∇_v^v_∞ |²dv∞

“refined convex Sobolev inequality” with x ≤ k(x) = ^1+x−(1+_1−κ ^x)κ

• Set v/v∞ = |f|^p2/R

|f|^2pdv∞

Theorem 1 (Arnold, JD) 1

2

p p − 1

2 "Z

f²dv∞ − Z

|f|^p2dv∞

2(p−1)Z

f²dv∞

^2−p_p #

≤ 2 λ1

Z

|∇f|²dv∞ ∀f ∈ L^p2(R^d, dv∞)

The Bakry-Emery method revisited

[Gianazza, Savaré, Toscani]

[J.D., Nazaret, Savaré]

Linear diffusions: non-local criterion for the Bakry-Emery method – p.1/5

Consider on a domain Ω ⊂ R^d and dγ = g dx, g = e^−F

Generalized Ornstein-Uhlenbeck operator: ∆gv := ∆v − DF · Dv Z

Ω |Dv|² dγ = − Z

Ω

v ∆gv dγ ∀ v ∈ H₀¹(Ω, dγ)

Let s := v^p/2 and α := (2 − p)/p, p ∈ (1,2]

v^t = ∆gv x ∈ Ω , t ∈ R⁺

∇v · n = 0 x ∈ ∂Ω , t ∈ R⁺ E^p(t) := 1

p − 1 Z

Ω

h

v^p − 1 − p(v − 1)i dγ I^p(t) := 4

p Z

Ω |Ds|² dγ

Written in terms of s = v^p/2, the entropy is E^p = 1

p − 1 Z

Ω

h

s² − 1 − p(s^2/p − 1)i dγ and the evolution is governed by

s^t = ∆gs + α |Ds|² s A simple computation shows that

d

dtE^p(t) := −I^p(t) d

dtI^p(t) := −8

p K^p(t)

Linear diffusions: non-local criterion for the Bakry-Emery method – p.3/5

Using the commutation relation [D,∆_g]s = −D²F Ds, we get Z

Ω

(∆_gs)² dγ = Z

Ω |D²s|² dγ+ Z

Ω

D²F Ds · Ds dγ −

Xd

i,j=1

Z

∂Ω

∂ij² s ∂is nj g dH^d−1

| {z }

≥0 if Ω is convex Let z := √

s. Using : 2 D²s · Dz ⊗ Dz = D |Dz|²

: Dz and i.p.p., we get K^p =

Z

Ω |∆_gs|² dγ + 4 α Z

Ω

∆_gs|Dz|² dγ

≥ Z

Ω |D²s|² dγ + Z

Ω

D²F Ds · Ds dγ + 4² α

Z

|Dz|⁴ dγ − 2 · 4α Z

D²s : Dz ⊗ Dz dγ

An extension of the criterion of Bakry-Emery

Let V (x) := inf_ξ∈S^d−1 D²F(x) ξ, ξ

and define

λ1(p) := inf

w∈D(Ω)\{0}

R

Ω

2 ^p−1_p |Dw|² + V |w|² dγ R

Ω |w|² dγ

Theorem 1 Let F ∈ C²(Ω), γ = e^−F ∈ L¹(Ω), and Ω be a convex domain in R^d. If λ1(p) is positive, then

I^p(t) ≤ I^p(0)e^−2λ1(p)t E^p(t) ≤ E^p(0)e^−2λ1(p)t

Linear diffusions: non-local criterion for the Bakry-Emery method – p.5/5

Fast diffusion equations:

entropy methods and functional inequalities

u^t = ∆u^m x ∈ R^d , t > 0

Entropy methods for fast diffusion and porous media equations:

intermediate asymptotics

Entropy methods and functional inequalities

Porous media / fast diffusion equations

Generalized entropies and nonlinear diffusions (EDP, uncomplete):

[Del Pino, J.D.], [Carrillo, Toscani], [Otto], [Juengel, Markowich, Toscani], [Carrillo, Juengel, Markowich, Toscani, Unterreiter], [Biler, J.D., Esteban], [Markowich, Lederman], [Carrillo, Vázquez], [Cordero-Erausquin, Gangbo, Houdré], [Cordero-Erausquin, Nazaret, Villani], [Agueh, Ghoussoub],...

[del Pino, Sáez], [Daskalopulos, Sesum]...

1) [J.D., del Pino] relate entropy and entropy-production by Gagliardo-Nirenberg inequalities

Various approaches:

2) “entropy – entropy-production method”

3) mass transport techniques

4) hypercontractivity for appropriate semi-groups

Fast diffusion equations: entropy methods and functional inequalities – p.2/9

Heat equation, porous media & fast diffusion equation

d−1 d

m u

= ∆ u

x ∈ R

fast diffusion equation

porous media equation heat equation

1

d−2 d

global existence in L

extinction in finite time

Existence theory, critical values of the parameter m

Intermediate asymptotics for fast diffusion & porous media

u^t = ∆u^m in R^d u|t=0 = u0 ≥ 0 u0(1 + |x|²) ∈ L¹ , u^m₀ ∈ L¹

Intermediate asymptotics: u0 ∈ L^∞, R

u0 dx = M > 0

Self-similar (Barenblatt) function: U(t) = O(t^{−d/(2−d(1−m))}) As t → +∞, [Friedmann, Kamin, 1980]

ku(t,·) − U(t,·)k^L∞ = o(t^{−d/(2−d(1−m))})

=⇒ What about ku(t,·) − U(t,·)k^L1 ?

Fast diffusion equations: entropy methods and functional inequalities – p.4/9

Time-dependent rescaling

Take u(t, x) = R^−d(t)v (τ(t), x/R(t)) where

R˙ = R^d(1−m)−1 , R(0) = 1 , τ = log R v^τ = ∆v^m + ∇ · (x v) , v|τ=0 = u0

[Ralston, Newman, 1984] Lyapunov functional: Entropy or Free energy

Σ[v] =

Z v^m

m − 1 + 1

2|x|²v

dx − Σ₀

d

dτ Σ[v] = −I[v] , I[v] = Z

v

∇v^m

v + x

2

dx

Entropy and entropy production

Stationary solution: choose C such that kv∞k^L1 = kuk^L1 = M > 0

v∞(x) =

C + 1 − m

2m |x|²

−1/(1−m)

+

Fix Σ₀ so that Σ[v∞] = 0. The entropy can be put in an m-homogeneous form

Σ[v] = R ψ

v v_∞

v_∞^m dx with ψ(t) = ^tm−1−m_m−1^(t−1)

Theorem 1 d ≥ 3, m ∈ [^d−1_d ,+∞), m > ¹₂, m 6= 1 I[v] ≥ 2 Σ[v]

Fast diffusion equations: entropy methods and functional inequalities – p.6/9

An equivalent formulation

Σ[v] = R _v^m

m−1 + ¹₂|x|²v

dx − Σ₀ ≤ ¹₂ R v

^∇vm

v + x

² dx = ¹₂I[v] p = _2m−1¹ , v = w^2p, v^m = w^p+1

1 2

2m 2m − 1

2 Z

|∇w|²dx +

1

1 − m − d Z

|w|^1+pdx + K ≥ 0

K < 0 if m < 1, K > 0 if m > 1 and, for some γ, K can be written as K = K0

Z

v dx = Z

w^2p dx ^γ

1/2p is optimal

Gagliardo-Nirenberg inequalities

Theorem 2 [Del Pino, J.D.] Assume that 1 < p ≤ _d−2^d and d ≥ 3 kwk^2p ≤ Ak∇wk^θ2 kwk¹⁻p+1^θ

A =

y(p − 1)² 2πd

^θ₂

2y − d 2y

₂¹_p

Γ(y) Γ(y − ^d₂)

!^θ_d

θ = d(p − 1)

p(d + 2 − (d − 2)p) , y = p + 1 p − 1 Similar results for 0 < p < 1

Uses [Serrin-Pucci], [Serrin-Tang]

1 < p = _2m−1¹ ≤ _d−2^d ⇐⇒ Fast diffusion case: ^d−1_d ≤ m < 1 0 < p < 1 ⇐⇒ Porous medium case: m > 1

Fast diffusion equations: entropy methods and functional inequalities – p.8/9

Intermediate asymptotics

Σ[v] ≤ Σ[u0]e^−2τ+ Csiszár-Kullback inequalities Theorem 3 [Del Pino, J.D.]

(i) ^d−1_d < m < 1 if d ≥ 3 lim sup

t→+∞ t¹

−d(1−m)

2−d(1−m) ku^m − u^m_∞k^L1 < +∞

(ii) 1 < m < 2

lim sup

t→+∞ t¹⁺

d(m−1)

2+d(m−1)k [u − u∞] u^m−1_∞ k^L1 < +∞

u∞(t, x) = R^−d(t)v∞ (x/R(t))

Fast diffusion equations: the finite mass regime

If m ≥ 1: porous medium regime or m1 := ^d−1

d ≤ m < 1, the decay of the entropy is governed by Gagliardo-Nirenberg inequalities, and to the limiting case m = 1 corresponds the logarithmic Sobolev

inequality If mc := ^d−2

d ≤ m < m1, solutions globally exist in L¹ and the Barenblatt self-similar solution has finite mass.

Fast diffusion equations: entropy methods and functional inequalities – p.1/4

A remark on the mass transport approach

The fast diffusion equation can be seen as the gradient flow of the generalized entropy with respect to the Wasserstein distance

Displacement convexity holds in the same range of exponents, m ∈ ((d − 1)/d,1), as for the Gagliardo-Nirenberg inequalities

⇒ How to extend to m^c < m < m1 what has been done for m ≥ m1 ?

Fast diffusion: finite mass regime

Inequalities...

d−1 d

m 1

d−2 d

global existence in L

Bakry-Emery method (relative entropy) v

∈ L

, x

∈ L

S obolev

G agliardo- N irenberg logarithmic S obolev

d d+2

v

... existence of solutions of u^t = ∆u^m

Fast diffusion equations: entropy methods and functional inequalities – p.3/4

Extensions and related results

Mass transport methods: inequalities / rates [Cordero-Erausquin, Gangbo, Houdré], [Cordero-Erausquin, Nazaret, Villani], [Agueh, Ghoussoub, Kang]

General nonlinearities [Biler, J.D., Esteban], [Carrillo-DiFrancesco], [Carrillo-Juengel-Markowich-Toscani-Unterreiter] and gradient flows [Jordan-Kinderlehrer-Otto], [Ambrosio-Savaré-Gigli],

[Otto-Westdickenberg], etc + [J.D.-Nazaret-Savaré, in progress]

Non-homogeneous nonlinear diffusion equations [Biler, J.D., Esteban], [Carrillo, DiFrancesco]

Extension to systems and connection with Lieb-Thirring inequalities [J.D.-Felmer-Loss-Paturel, 2006], [J.D.-Felmer-Mayorga]

Drift-diffusion problems with mean-field terms. An example: the

Keller-Segel model [J.D-Perthame, 2004], [Blanchet-J.D-Perthame, 2006], [Biler-Karch-Laurençot-Nadzieja, 2006],

[Blanchet-Carrillo-Masmoudi, 2007], etc

Fast diffusion equations: the infinite mass regime

If m > m^c := ^d−2

d ≤ m < m1, solutions globally exist in L¹ and the Barenblatt self-similar solution has finite mass.

For m ≤ m^c, the Barenblatt self-similar solution has finite mass

⇒ How to extend to m ≤ m^c what has been done for m > m^c ? Work in relative variables !

Fast diffusion equations: Entropy methods and linearization, intermediate asymptotics, vanishing – p.1/18

Fast diffusion: infinite mass regime

d−1 d

m

d−2

1

d

global existence in L

Bakry-Emery method (relative entropy) v

∈ L

, x

∈ L

d d+2

v v

, V

∈ L

v

− V

∗

∈ L

V

1

− V

0

∈ L

Σ[ V

1

V

0

] < ∞ Σ[ V

1

V

0

] = ∞

m

d−4 d−2

V

1

− V

0

6∈ L

G agliardo- N irenberg

Entropy methods and linearization...

... intermediate asymptotics, vanishing

A. Blanchet, M. Bonforte, J.D., G. Grillo, J.L. Vázquez use the properties of the flow

write everything as relative quantities (to the Barenblatt profile) compare the functionals (entropy, Fisher information) to their linearized counterparts

=⇒ Extend the domain of validity of the method to the price of a restriction of the set of admissible solutions

Fast diffusion equations: Entropy methods and linearization, intermediate asymptotics, vanishing – p.3/18

Setting of the problem

We consider the solutions u(τ, y) of







∂^τu = ∆u^m u(0,·) = u0

where m ∈ (0,1) (fast diffusion) and (τ, y) ∈ Q^T = (0, T) × R^d Two parameter ranges: m^c < m < 1 and 0 < m < m^c, where

m^c := d − 2 d

mc < m < 1, T = +∞: intermediate asymptotics, τ → +∞ 0 < m < m^c, T < +∞: vanishing in finite time

Barenblatt solutions

UD,T(τ, y) := 1

R(τ)^d D + 1 − m 2 m

y R(τ)

2!−_1−m¹

with

R(τ) :=

d(m − m^c) (τ + T)_d(m−¹_mc)

if m^c < m < 1 (vanishing in finite time) if 0 < m < m^c

R(τ) :=

d(m^c − m) (T − τ)−_d(mc¹_−m)

Time-dependent rescaling: t := log

R(τ) R(0)

and x := _R(τ^y ₎. The

function v(t, x) := R(τ)^d u(τ, y) solves a nonlinear Fokker-Planck type equation







∂^tv(t, x) = ∆v^m(t, x) + ∇ · (x v(t, x)) (t, x) ∈ (0,+∞) × R^d v(0, x) = v0(x) = R(0)^d u0(R(0)x) x ∈ R^d

Fast diffusion equations: Entropy methods and linearization, intermediate asymptotics, vanishing – p.5/18

Assumptions

(H1) u0 is a non-negative function in L¹_loc(R^d) and that there exist positive constants T and D0 > D1 such that

UD₀,T(0, y) ≤ u0(y) ≤ UD₁,T(0, y) ∀ y ∈ R^d

(H2) If m ∈ (0, m∗], there exist D∗ ∈ [D1, D0] and f ∈ L¹(R^d) such that u0(y) = U^D∗,T(0, y) + f(y) ∀ y ∈ R^d

(H1’) v0 is a non-negative function in L¹_loc(R^d) and there exist positive constants D0 > D1 such that

V^D₀(x) ≤ v0(x) ≤ V^D₁(x) ∀ x ∈ R^d

(H2’) If m ∈ (0, m ], there exist D ∈ [D , D ] and f ∈ L¹(R ) such that

Convergence to the asymptotic profile (without rate)

m∗ := d − 4

d − 2 < m^c := d − 2

2 , p(m) := d(1 − m) 2 (2 − m)

Theorem 1 Let d ≥ 3, m ∈ (0,1). Consider a solution v with initial data satisfying (H1’)-(H2’)

(i) For any m > m∗, there exists a unique D∗ such that R

R^d(v(t) − V^D∗) dx = 0 for any t > 0. Moreover, for any p ∈ (p(m),∞], lim_t→∞ R

R^d |v(t) − VD_∗|^p dx = 0

(ii) For m ≤ m∗, v(t) − V^D∗ is integrable, R

R^d(v(t) − V^D∗) dx = R

R^d f dx and v(t) converges to V^D∗ in L^p(R^d) as t → ∞, for any p ∈ (1,∞] (iii) (Convergence in Relative Error) For any p ∈ (d/2,∞],

t→∞lim kv(t)/ VD_∗ − 1 kp = 0

[Daskalopoulos-Sesum, 06], [Blanchet-Bonforte-Grillo-Vázquez, 06-07]

Fast diffusion equations: Entropy methods and linearization, intermediate asymptotics, vanishing – p.7/18

Convergence with rate

q∗ := 2 d(1 − m)

2 (2 − m) + d(1 − m)

Theorem 2 If m 6= m∗, there exist t0 ≥ 0 and λ^m,d > 0 such that

(i) For any q ∈ (q∗,∞], there exists a positive constant Cq such that kv(t) − V^D∗k^q ≤ C^q e^{−λm,d t} ∀ t ≥ t0

(ii) For any ϑ ∈ [0,(2 − m)/(1 − m)), there exists a positive constant Cϑ

such that

|x|^ϑ(v(t) − V^D∗)

2 ≤ C^ϑ e^−λm,dt ∀ t ≥ t0

(iii) For any j ∈ N, there exists a positive constant H^j such that

Intermediate asymptotics

Corollary 3 Let d ≥ 3, m ∈ (0,1), m 6= m∗. Consider a solution u with initial data satisfying ^(H1)-(H2). For τ large enough, for any q ∈ (q∗,∞], there exists a positive constant C such that

ku(τ) − U^D∗(τ)k^q ≤ C R(τ)^−α

where α = λm,d + d(q − 1)/q and large means T − τ > 0, small, if m < mc, and τ → ∞ if m ≥ m^c

For any p ∈ (d/2,∞], there exists a positive constant C and γ > 0 such that v(t)/ V^D∗ − 1

L^p(R^d) ≤ C e^{−γ t} ∀ t ≥ 0

Fast diffusion equations: Entropy methods and linearization, intermediate asymptotics, vanishing – p.9/18

Rewriting the equation in relative variables

L¹-contraction, Maximum Principle, conservation of relative mass...

Passing to the quotient: the function w(t, x) := ^v(t,x)

V_D∗(x) solves











w^t = 1 VD_∗

∇ ·

w V^D∗∇

m

m − 1(w^m−1 − 1)V_D^m−1_∗

in (0,+∞) × R^d w(0,·) = w0 := v0

V^D∗

in R^d with

0 < inf

x∈R^d

VD₀

V^D∗

≤ w(t, x) ≤ sup

x∈R^d

VD₁

V^D∗

< ∞ ... Harnack Principle

k ( )k ≤ +∞ ∀ ≥

Relative entropy

Relative entropy

F[w] := 1 1 − m

Z

R^d

(w − 1) − 1

m(w^m − 1)

VD^m_∗ dx Relative Fisher information

J[w] := m (m − 1)²

Z

R^d

∇

w^m−1 − 1

V_D^m−1_∗ ² w V^D∗ dx Proposition 1 Under assumptions (H1)-(H2),

d

dtF[w(t)] = − J [w(t)]

Proposition 2 Under assumptions (H1)-(H2), there exists a constant λ > 0 such that

F[w(t)] ≤ λ⁻¹ J[w(t)]

Fast diffusion equations: Entropy methods and linearization, intermediate asymptotics, vanishing – p.11/18

Heuristics: linearization

Take w(t, x) = 1 + ε ^g(t,x)

V_D^m−1

∗ (x) and formally consider the limit ε → 0 in











wt = 1 V^D∗

∇ ·

w VD_∗∇

m

m − 1(w^m−1 − 1)V_D^m−1_∗

in (0,+∞) × R^d w(0,·) = w0 := v0

V^D∗

in R^d Then g solves

gt = m V_D^m−2_∗ (x) ∇ · [VD_∗(x)∇g(t, x)]

and the entropy and Fisher information functionals

F[g] := 1 2

Z

R^d |g|² V_D^2−m_∗ dx and I[g] := m Z

R^d |∇g|² V^D∗ dx d

Comparison of the functionals

Lemma 3 Let m ∈ (0,1) and assume that u0 satisfies ^(H1)-(H2) [Relative entropy]

C1

Z

R^d

|w − 1|² V_D^m_∗ dx ≤ F[w] ≤ C2

Z

R^d

|w − 1|² V_D^m_∗ dx [Fisher information]

I[g] ≤ β1 J [w] + β2 F[g] with g := (w − 1)V_D^m_∗⁻¹

Theorem 4 (Hardy-Poincaré) There exists a positive constant λ^m,d such that for any m 6= m∗ = (d − 4)/(d − 2), m ∈ (0,1), for any g ∈ D(R^d),

Z

R^d

|g − g|² V_D^2−m_∗ dx ≤ C^m,d Z

R^d

|∇g|² V^D∗ dx with g = R

R^d g V_D^2−m_∗ dx if m > m∗, g = 0 otherwise

Fast diffusion equations: Entropy methods and linearization, intermediate asymptotics, vanishing – p.13/18

Hardy-Poincaré inequalities

With α = _m−1¹ , α∗ = ¹

m_∗−1 = 1 − ^d₂

Theorem 5 Assume that d ≥ 3, α ∈ R \ {α^∗}, dµ^α(x) := h^α(x)dx, hα(x) := (1 + |x|²)^α. Then

Z

R^d

|v|²

1 + |x|² dµα ≤ C^α,d Z

R^d

|∇v|² dµα

holds for some positive constant C^α,d, for any v ∈ D(R^d), under the additional condition R

R^d v dµ^α−1 = 0 if α ∈ (−∞, α^∗)

Limit cases

Poincaré inequality: take α = −1/ε² to v^ε(x) := ε^−d/2 v(x/ε) and let ε → 0 Z

R^d

|v|² dν∞ ≤ 1 2

Z

R^d

|∇v|² dν∞ with dν∞(x) := e^−|x|2dx ... under the additional condition R

R^d v e^−|x|2dx = 0

Hardy’s inequality: take v1/ε(x) := ε^d/2 v(ε x) and let ε → 0 Z

R^d

|v|²

|x|² dν0,α ≤ 1

(α − α∗)² Z

R^d |∇v|² dν0,α with dν0,α(x) := |x|^2α dx ... under the additional condition v¯^α := R

R^d v dν0,α = 0 if α < α^∗

Fast diffusion equations: Entropy methods and linearization, intermediate asymptotics, vanishing – p.15/18

Some estimates of C

α −∞ < α ≤ − d − d < α < α

α

< α ≤ 1 C

2|α|

C

≥

Optimality

? ? yes

α 1 ≤ α ≤ α ¯ ( d ) α ¯ ( d ) ≤ α ≤ d d α > d C

d(d+2α−2)

1

α(d+α−2)

1 2d(d−1)

1

d(d+α−2)

Optimality

? ? yes ?

α∗ = −^d−2₂ , α¯(d) ∈ (1, d)

Hardy’s inequality: the “completing the square method”

Let v ∈ D(R^d) with supp(v) ⊂ R^d \ {0} if α < α^∗

0 ≤ Z

R^d

∇v + λ x

|x|² v

2

|x|^2α dx

= Z

R^d

|∇v|² |x|^2α dx + h

λ² − λ (d + 2α − 2)i Z

R^d

|v|²

|x|² |x|^2α dx An optimization of the right hand side with respect to λ gives λ = α − α^∗, that is (d + 2α − 2)²/4 = λ². Such an inequality is optimal, with optimal constant λ², as follows by considering the test functions:

1) if α > α^∗: v^ε(x) = min{ε^−λ,(|x|^−λ − ε^λ)₊} 2) if α < α^∗: vε(x) = |x|^{1−α−d/2+ε} for |x| < 1

v^ε(x) = (2 − |x|)₊ for |x| ≥ 1 and letting ε → 0 in both cases

Fast diffusion equations: Entropy methods and linearization, intermediate asymptotics, vanishing – p.17/18

The optimality case: Davies’ method

Proposition 4 Let d ≥ 3, α ∈ (α^∗,∞). Then the Hardy-Poincaré inequality holds for any v ∈ D(R^d) with C^α,d := 4/(d − 2 + 2α)² if

α ∈ (α^∗,1] and C^α,d := 4/[d(d − 2 + 2α)] if α ≥ 1. The constant C^α,d is optimal for any α ∈ (α^∗,1].

Proof: h^α = (1 + |x|²)^α, ∇h^α = 2α x h^α−1,

∆h^α = 2α hα−2[d + 2(α − α^∗)|x|²] > 0 By Cauchy-Schwarz

Z

R^d |v|² ∆h^α dx

2

≤ 4 Z

R^d |v| |∇v| |∇h^α| dx 2

≤ 4 Z

R^d

|v|² |∆h^α| dx Z

R^d

|∇v|² |∇h^α|² |∆h^α|⁻¹ dx

|∆h | ≥ 2 |α| min{d,(d − 2 + 2α)} ^hα(x)

Generalized Poincaré inequalities

Coll. J. Carrillo, J.D. , I. Gentil, A. Jüngel

Generalized Poincar´e inequalities: algebraic rates – p.1/17

Higher order diffusion equations

The one dimensional porous medium/fast diffusion equation

∂u

∂t = (u^m)xx , x ∈ S¹ , t > 0 The thin film equation

ut = −(u^m uxxx)_x , x ∈ S¹ , t > 0 The Derrida-Lebowitz-Speer-Spohn (DLSS) equation

ut = −(u(logu)_xx)_xx , x ∈ S¹ , t > 0 ... with initial condition u(·,0) = u0 ≥ 0 in S¹ ≡ [0,1)

Entropies and energies

Averages:

µ^p[v] :=

Z

S¹

v^1/p dx ^p

and v¯ :=

Z

S¹

v dx Entropies: p ∈ (0,+∞), q ∈ R, v ∈ H₊¹ (S¹), v 6≡ 0 a.e.

Σp,q[v] := 1

p q (p q − 1) Z

S¹

v^q dx − (µ^p[v])^q

if p q 6= 1 and q 6= 0 ,

Σ_1/q,q[v] :=

Z

S¹

v^q log

v^q R

S¹ v^q dx

dx if p q = 1 and q 6= 0 , Σ_p,0[v] := −1

p Z

S¹

log

v µp[v]

dx if q = 0

Generalized Poincar´e inequalities: algebraic rates – p.3/17

Convexity

Σ_p,q[v] is non-negative by convexity of

u 7→ u^{p q} − 1 − p q (u − 1)

p q (p q − 1) =: σp,q(u) By Jensen’s inequality,

Σp,q[v] = µ^p[v]^q Z

S¹

σ^p,q

v^1/p (µ^p[v])^1/p

dx

≥ µ^p[v]^qσ^p,q Z

S¹

v^1/p

(µp[v])^1/p dx

= µ^p[v]^qσ^p,q(1) = 0

Limit cases

p q = 1:

p→1/qlim Σp,q[v] = Σ_1/q,q[v] for q > 0 q = 0:

q→0lim Σp,q[v] = Σp,0[v] for p > 0 p = q = 0:

Σ_0,0[v] = − Z

S¹

log

v kvk_∞

dx Some references (>2005):

[ M. J. Cáceres, J. A. Carrillo, and G. Toscani]

[ M. Gualdani, A. Jüngel, and G. Toscani]

[ A. Jüngel and D. Matthes]

[ R. Laugesen]

Generalized Poincar´e inequalities: algebraic rates – p.5/17

Global functional inequalities

Theorem 1 For all p ∈ (0,+∞) and q ∈ (0,2), there exists a positive constant κ^p,q such that, for any v ∈ H₊¹ (S¹),

Σ_p,q[v]^2/q ≤ 1

κ^p,q J1[v] := 1 κ^p,q

Z

S¹

|v⁰|² dx

Corollary 1 Let p ∈ (0,+∞) and q ∈ (0,2). Then, for any v ∈ H₊¹ (S¹),

Σp,q[v]^2/q ≤ 1 4π² κp,q

J2[v] := 1 4π² κp,q

Z

S¹

|v⁰⁰|² dx

A minimizing sequence (vn)_n∈N is bounded in H¹(S¹)

vⁿ * v in H¹(S¹) and Σp,q[vⁿ] → Σp,q[v] as n → ∞ If Σp,q[v] = 0, lim_n→∞ J1[vⁿ] = 0. Let εⁿ := J1[vⁿ], wⁿ := ^v√n_ε⁻¹

n and make a Taylor expansion

(1 + √

ε x)^1/p − 1 −

√ε p x

≤ 1

p r(ε0, p) ε ∀ (x, ε) ∈

− 1

√2, 1

√2

×(0, ε0)

εⁿ := J1[vⁿ] , Σp,q[vⁿ] ≤ c(ε0, p, q)εⁿ Hence, since q < 2,

J1[vⁿ]

Σp,q[vⁿ]^2/q = εⁿ J1[wⁿ]

Σp,q[vⁿ]^2/q ≥ [c(ε0, p, q)]^−2/q ε^1−2/qn → ∞ gives a contradiction

Generalized Poincar´e inequalities: algebraic rates – p.7/17

Asymptotic functional inequalities

The regime of small entropies:

Xε^p,q :=

v ∈ H₊¹ (S¹) : Σp,q[v] ≤ ε and µ^p[v] = 1

Theorem 2 For any p > 0, q ∈ R and ε0 > 0, there exists a positive constant C such that, for any ε ∈ (0, ε0],

Σp,q[v] ≤ 1 + C√ ε

8 p² π² J1[v] ∀ v ∈ Xε^p,q

Without the condition µp[v] = 1:

Σp,q[v] ≤ 1 + C√ ε

8 p² π² (µ^p[v])^q−2 J1[v]

If J1[v] ≤ 8 p² π² ε, define w := (v − 1)/(κ^∞_p √

ε): J1[w] ≤ 1.

Σp,q[v] = 1

pq(pq − 1) Z

S¹

(1+κ^∞p

√εw)^qdx− Z

S¹

(1+κ^∞p

√εw)^1/pdx

^pq

= ε (κ^∞p )² 2p²

"Z

S¹

w² dx − Z

S¹

w dx

2#

+ O(ε^3/2)

= ε (κ^∞p )² 2p²

Z

S¹

(w − w¯)² dx + O(ε^3/2)

≤ ε (κ^∞_p )² 2p²

J1[w]

(2π)² + O(ε^3/2) = J1[v]

8 p² π² + O(ε^3/2) using Poincaré’s inequality

Generalized Poincar´e inequalities: algebraic rates – p.9/17

1

application: Porous media

∂u

∂t = (u^m)xx x ∈ S¹, t > 0 A one parameter family of entropies :

Σk[u] :=















1

k (k + 1) Z

S¹

u^k+1 − u¯^k+1

dx if k ∈ R \ {−1,0} Z

S¹

ulogu u¯

dx if k = 0

− Z

S¹

logu u¯

dx if k = −1

With v := u^p, p := ^m+k₂ , q := ^k+1_p = 2 _m+k^k+1 , Σk[u] = Σp,q[v]

Lemma 1 Let k ∈ R. If u is a smooth positive solution d

dtΣk[u(·, t)] + λ Z

S¹

(u^(k+m)/2)x

2

dx = 0 with λ := 4m/(m + k)² whenever k + m 6= 0, and

d

dtΣk[u(·, t)] + λ Z

S¹

|(logu)x|² dx = 0 with λ := m for k + m = 0.

Generalized Poincar´e inequalities: algebraic rates – p.11/17