(2011,Sep28–30) EVOLworkshop ∼ dolbeaulCeremade,Universit´eParis-DauphineSeptember30,2011Toulouse JeanDolbeaulthttp://www.ceremade.dauphine.fr/ ImprovedSobolevinequalitiesusingnonlinearflows

Improved Sobolev inequalities using nonlinear flows

Jean Dolbeault

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

September 30, 2011

Toulouse

EVOL workshop

(2011, Sep 28 – 30)

A question by H. Brezis and E. Lieb

[Brezis, Lieb (1985)] Is there a natural way to bound S d k∇uk ² L

( R

) − kuk ² L

(R

)

from below in terms of the“distance” off from the set of optimal [Aubin-Talenti] functions when d ≥ 3 ?

[Bianchi-Egnell (1990)] There is a positive constant α such that S d k∇uk ² L

( R

) − kuk ² _L

( R

) ≥ α inf

ϕ∈M k∇u − ∇ϕk ² L

( R

)

[Cianchi, Fusco, Maggi, Pratelli (2009)] (also a version for k∇uk ^p _L

( R

) ) There are constants α and κ such that

S d k∇uk ² L

(R

) ≥ (1 + κ λ(u) ^α ) kuk ² _L

( R

)

where λ(u) = inf ϕ∈M

_ku−ϕk

(Rd)

kuk

L2∗ (Rd)

: kuk ² _L

(R

) = kϕk ² _L

(R

)

J. Dolbeault Improved Sobolev inequalities

A – Sobolev and

Hardy-Littlewood-Sobolev inequalities:

duality, flows

Outline

Outline

A result motivated by [Carrillo, Carlen and Loss]

Sobolev and HLS inequalities can be related using a nonlinear flow compatible with Legendre’s duality

The asymptotic behaviour close to the vanishing time is determined by a solution with separation of variables based on the Aubin-Talenti solution

The entropy H (to be defined) is negative, concave, and we can relate H(0) with H ^′ (0) by integrating estimates on (0, T ), which provides a first improvement of Sobolev’s inequality if d ≥ 5

J. Dolbeault Improved Sobolev inequalities

Sobolev and HLS

As it has been noticed by E. Lieb, Sobolev’s inequality in R ^d , d ≥ 3, kuk ² _L

(R

) ≤ S d k∇uk ² _L

( R

) ∀ u ∈ D ^1,2 ( R ^d ) (1) and the Hardy-Littlewood-Sobolev inequality

S d kv k ²

L

( R

) ≥ Z

R

v (−∆) ⁻¹ v dx ∀ v ∈ L

( R ^d ) (2)

are dual of each other. Here S d is the Aubin-Talenti constant and

2 ^∗ = _d−2 ^{2 d}

Using a nonlinear flow to relate Sobolev and HLS

Consider the fast diffusion equation

∂v

∂t = ∆v ^m t > 0 , x ∈ R ^d (3)

If we define H(t) := H d [v(t , ·)], with H d [v ] :=

Z

R

v (−∆) ⁻¹ v dx − S d kvk ²

L

2d d+2

( R

)

then we observe that 1

2 H ^′ = − Z

R

v ^m+1 dx + S d

Z

R

v

dx

Z

R

∇v ^m · ∇v

dx where v = v (t , ·) is a solution of (3). With the choice m = ^d−2 _d+2 , we find that m + 1 = _d+2 ^2d

J. Dolbeault Improved Sobolev inequalities

A first statement

Proposition

[J.D.] Assume that d ≥ 3 and m = ^d−2 _d+2 . If v is a solution of (3) with nonnegative initial datum in L ^2d/(d+2) ( R ^d ), then

1 2

d dt

Z

R

v (−∆) ⁻¹ v dx − S d kvk ²

L

(R

)

= Z

R

v ^m+1 dx

h

S d k∇uk ² L

( R

) − kuk ² _L

( R

)

i ≥ 0

The HLS inequality amounts to H ≤ 0 and appears as a consequence of Sobolev, that is H ^′ ≥ 0 if we show that lim sup _t>0 H(t) = 0

Notice that u = v ^m is an optimal function for (1) if v is optimal for (2)

Improved Sobolev inequality

By integrating along the flow defined by (3), we can actually obtain optimal integral remainder terms which improve on the usual Sobolev inequality (1), but only when d ≥ 5 for integrability reasons

Theorem

[J.D.] Assume that d ≥ 5 and let q = _d−2 ^d+2 . There exists a positive constant C ≤ 1 + ² _d

1 − e ^−d/2

S d such that

S d kw ^q k ²

L

( R

) − Z

R

w ^q (−∆) ⁻¹ w ^q dx

≤ C kw k

8 d−2

L

(R

)

h

k∇w k ² L

( R

) − S d kwk ² L

(R

)

i for any w ∈ D ^1,2 ( R ^d )

J. Dolbeault Improved Sobolev inequalities

Solutions with separation of variables

Consider the solution vanishing at t = T :

v T (t , x ) = c (T − t ) ^α (F (x))

∀ (t, x) ∈ (0, T ) × R ^d where α = (d + 2)/4, c ^1−m = 4 m d , m = ^d−2 _d+2 , p = d/(d − 2) and F is the Aubin-Talenti solution of

−∆F = d (d − 2) F (d+2)/(d−2)

Let kvk ∗ := sup _x∈R

(1 + |x| ² ) ^d+2 |v (x)|

Lemma

[M. delPino, M. Saez], [J. L. V´ azquez, J. R. Esteban, A. Rodr´ıguez]

For any solution v of (3) with initial datum v 0 ∈ L ^2d/(d+2) ( R ^d ), v 0 > 0, there exists T > 0, λ > 0 and x 0 ∈ R ^d such that

t→T lim

(T − t) ⁻

kv (t , ·)/v(t , ·) − 1k ∗ = 0

with v (t, x) = λ ^(d+2)/2 v T (t , (x − x 0 )/λ)

Improved inequality: proof (1/2)

J(t ) := R

R

v (t , x) ^m+1 dx satisfies

J ^′ = −(m + 1) k∇v ^m k ² L

( R

) ≤ − m + 1 S d

J ¹⁻

If d ≥ 5, then we also have

J ^′′ = 2 m (m + 1) Z

R

v ^m−1 (∆v ^m ) ² dx ≥ 0

Such an estimate makes sense if v = v T . This is also true for any solution v as can be seen by rewriting the problem on S ^d :

integrability conditions for v are exactly the same as for v T Notice that

J ^′

J ≤ − m + 1 S d

J ⁻

≤ −κ with κ T = 2 d d + 2

T S d

Z

R

v ₀ ^m+1 dx −

≤ d 2

J. Dolbeault Improved Sobolev inequalities

Improved inequality: proof (2/2)

By the Cauchy-Schwarz inequality, we have J ^{′ 2}

m + 1 = k∇v ^m k ⁴ L

(R

) = Z

R

v ^(m−1)/2 ∆v ^m · v ^(m+1)/2 dx 2

≤ Z

R

v ^m−1 (∆v ^m ) ² dx Z

R

v ^m+1 dx = Cst J ^′′ J so that Q(t) := k∇v ^m (t , ·)k ² _L

( R

)

R

R

v ^m+1 (t, x) dx −(d−2)/d

is monotone decreasing, and

H ^′ = 2 J (S d Q − 1) , H ^′′ = J ^′

J H ^′ + 2 J S d Q ^′ ≤ J ^′ J H ^′ ≤ 0 H ^′′ ≤ −κ H ^′ with κ = 2 d

d + 2 1 S d

Z

R

v ₀ ^m+1 dx −2/d

By writing that −H(0) = H(T ) − H(0) ≤ H ^′ (0) (1 − e ^{−κ T} )/κ and

using the estimate κ T ≤ d /2, the proof is completed

d = 2: Onofri’s and log HLS inequalities

H 2 [v] :=

Z

R

(v − µ) (−∆) ⁻¹ (v − µ) dx − 1 4 π

Z

R

v log v

µ

dx With µ(x) := _π ¹ (1 + |x | ² ) ⁻² . Assume that v is a positive solution of

∂v

∂t = ∆ log v

µ

t > 0 , x ∈ R ²

Proposition

If v = µ e ^u/2 is a solution with nonnegative initial datum v 0 in L ¹ ( R ² ) such that R

R

v 0 dx = 1, v 0 log v 0 ∈ L ¹ ( R ² ) and v 0 log µ ∈ L ¹ ( R ² ), then d

dt H 2 [v (t, ·)] = 1 16 π

Z

R

|∇u| ² dx − Z

R

e

− 1 u d µ

≥ 1 16 π

Z

R

|∇u| ² dx + Z

R

u dµ − log Z

R

e ^u dµ

≥ 0

J. Dolbeault Improved Sobolev inequalities

Fast diffusion equations

1

entropy methods

2

linearization of the entropy

3

improved Gagliardo-Nirenberg inequalities

B1 – Fast diffusion equations:

entropy methods

J. Dolbeault Improved Sobolev inequalities

Existence, classical results

u t = ∆u ^m x ∈ R ^d , t > 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)) )

d − 1 d

m fast diffusion equation

porous media equation heat equation

1

d − 2 d

global existence in L ¹ extinction in finite time

Existence theory, critical values of the parameter m

Time-dependent rescaling, Free energy

Time-dependent rescaling: Take u(τ, y) = R ^−d (t) v (t , y /R(τ)) where

∂R

∂τ = R ^d(1−m)−1 , R(0) = 1 , t = log R The function v solves a Fokker-Planck type equation

∂v

∂t = ∆v ^m + ∇ · (x v) , v |τ=0 = u 0

[Ralston, Newman, 1984] Lyapunov functional:

Generalized entropy or Free energy Σ[v] :=

Z

R

v ^m m − 1 + 1

2 |x| ² v

dx − Σ 0

Entropy production is measured by the Generalized Fisher information

d

dt Σ[v ] = −I [v ] , I [v ] :=

Z

R

v

∇v ^m v + x

2

dx

J. Dolbeault Improved Sobolev inequalities

Relative entropy and entropy production

Stationary solution: choose C such that kv ∞ k L

= kuk L

= M > 0 v ∞ (x ) := C + ^1−m _2m |x| ² −1/(1−m)

+

Relative entropy: Fix Σ 0 so that Σ[v ∞ ] = 0. The entropy can be put in an m-homogeneous form: for m 6= 1,

Σ[v] = R

R

ψ _v ^v

∞

v ∞ ^m dx with ψ(t ) = ^t

^−1−m _m−1 ^(t−1) Entropy – entropy production inequality

Theorem

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

Corollary

A solution v 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 Σ[v(t, ·)] ≤ Σ[u 0 ] e ^{− 2 t}

An equivalent formulation: Gagliardo-Nirenberg inequalities

Σ[v] = Z

R

v ^m m − 1 + 1

2 |x| ² v

dx−Σ 0 ≤ 1 2

Z

R

v

∇v ^m v + x

2

dx = 1 2 I [v ] Rewrite it with p = _2m−1 ¹ , v = w ^2p , v ^m = w ^p+1 as

1 2

2m 2m − 1

2 Z

R

|∇w | ² dx + 1

1 − m − d Z

R

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

R

v dx = R

R

w ^2p dx γ

w = w ∞ = v ∞ ^1/2p is optimal Theorem

[Del Pino, J.D.] With 1 < p ≤ _d−2 ^d (fast diffusion case) and d ≥ 3 kw k _L

_{( R}

₎ ≤ A k∇w k ^θ _L

( R

) kw k ^1−θ _L

( R

)

A =

y(p−1)

2πd

2y−d 2y

Γ(y) Γ(y −

)

, θ = p(d+2−(d−2)p) ^d(p−1) , y = ^p+1 _p−1

J. Dolbeault Improved Sobolev inequalities

...the Bakry-Emery method

Consider the generalized Fisher information I [v] :=

Z

R

v |Z | ² dx with Z := ∇v ^m v + x and compute

d

dt I [v (t , ·)]+2 I [v (t , ·)] = −2 (m−1) Z

R

u ^m (divZ ) ² dx−2 X d i, j=1

Z

R

u ^m (∂ i Z ^j ) ² dx

the Fisher information decays exponentially:

I [v(t, ·)] ≤ I [u 0 ] e ^{− 2t} lim t→∞ I [v(t, ·)] = 0 and lim t→∞ Σ[v(t, ·)] = 0

d dt

I [v (t, ·)] − 2 Σ[v (t, ·)]

≤ 0 means I [v] ≥ 2 Σ[v]

[Carrillo, Toscani], [Juengel, Markowich, Toscani], [Carrillo, Juengel,

Markowich, Toscani, Unterreiter], [Carrillo, V´ azquez]

Fast diffusion: finite mass regime

Inequalities...

d − 1 d

m 1

d − 2 d

global existence in L ¹

Bakry-Emery method (relative entropy) v ^m ∈ L ¹ , x

∈ L ¹ Sobolev

Gagliardo-Nirenberg logarithmic Sobolev

d d+2

v

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

J. Dolbeault Improved Sobolev inequalities

B2 – Fast diffusion equations: the infinite mass regime

Linearization of the entropy

Extension to the infinite mass regime, finite time vanishing

If m > m c := ^d−2 _d ≤ m < m 1 , solutions globally exist in L ¹ ( R ^d ) and the Barenblatt self-similar solution has finite mass.

For m ≤ m c , the Barenblatt self-similar solution has infinite mass Extension to m ≤ m c ? Work in relative variables !

d−1 d

m 1

d−2 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

− V

∈ L

Σ[ V

V

] < ∞ Σ[ V

V

] = ∞

m

d−4 d−2

V

− V

6∈ L

m

m

Gagliardo-Nirenberg

J. Dolbeault Improved Sobolev inequalities

Entropy methods and linearization: intermediate asymptotics, vanishing

[A. Blanchet, M. Bonforte, J.D., G. Grillo, J.L. V´ azquez]

∂u

∂τ = −∇ · (u ∇u ^m−1 ) = 1 − m

m ∆u ^m (4)

m c < m < 1, T = +∞: intermediate asymptotics, τ → +∞

R(τ) := (T + τ)

0 < m < m c , T < +∞: vanishing in finite time lim τրT u(τ, y) = 0 R(τ) := (T − τ) ⁻

Self-similar Barenblatt type solutions exists for any m Rescaling: time-dependent change of variables

t := ^1−m ₂ log _{R (τ)}

R(0)

and x := q

1 2 d |m−m

|

y

R(τ)

Generalized Barenblatt profiles: V D (x) := D + |x | ²

Sharp rates of convergence

Assumptions on the initial datum v 0

(H1) V D

≤ v 0 ≤ V D

for some D 0 > D 1 > 0

(H2) if d ≥ 3 and m ≤ m ∗ , (v 0 − V D ) is integrable for a suitable D ∈ [D 1 , D 0 ]

Theorem

[Bonforte, J.D., Grillo, V´ azquez] Under Assumptions (H1)-(H2), if m < 1 and m 6= m ∗ := ^d−4 _d−2 , the entropy decays according to

E[v(t, ·)] ≤ C e ^{−2 (1−m) Λ}

^t ∀ t ≥ 0

where Λ α,d > 0 is the best constant in the Hardy–Poincar´e inequality Λ α,d

Z

R

|f | ² dµ α−1 ≤ Z

R

|∇f | ² dµ α ∀ f ∈ H ¹ (d µ α ) with α := 1/(m − 1) < 0, dµ α := h α dx, h α (x) := (1 + |x| ² ) ^α

J. Dolbeault Improved Sobolev inequalities

Plots (d = 5)

λ₀₁=−4α−2d

λ₁₀=−2α λ₁₁=−6α−2 (d+ 2) λ₀₂=−8α−4 (d+ 2)

λ20=−4α λ30

λ₂₁λ₁₂ λ03

λcont α,d:=¹₄(d+ 2α−2)²

α=−d α=−(d+ 2)

α=−^d+2₂ α=−d−2

2 α=−^d+6₂

α 0 Essential spectrum ofLα,d

α=−√d−1−^d₂ α=−√d−1−^d+4₂ α=− −^d+22

√2d (d= 5) Spectrum ofLα,d

mc=^d−2d

m1=^d−1d m₂=^d+1_d+2

e m1=_d+2^d

e m2=^d+4d+6

m Spectrum of (1−m)L1/(m−1),d

(d= 5)

Essential spectrum of (1

1

−m)L1/(m−1),d

2 4 6

Improved asymptotic rates

[Bonforte, J.D., Grillo, V´ azquez] Assume that m ∈ (m 1 , 1), d ≥ 3.

Under Assumption (H1), if v is a solution of the fast diffusion equation with initial datum v 0 such that R

R

x v 0 dx = 0, then the asymptotic convergence holds with an improved rate corresponding to the improved spectral gap.

m1=d+2^d

m1=d−1d m2=^d+4d+6

m2=^d+1d+2

4

2

m 1 mc=d−2

d (d= 5)

γ(m)

0

J. Dolbeault Improved Sobolev inequalities

Higher order matching asymptotics

[J.D., G. Toscani] For some m ∈ (m c , 1) with m c := (d − 2)/d, we consider on R ^d the fast diffusion equation

∂u

∂τ + ∇ · u ∇u ^m−1

= 0

The strategy is easy to understand using a time-dependent rescaling and the relative entropy formalism. Define the function v such that

u(τ, y + x 0 ) = R ^−d v (t, x) , R = R(τ) , t = ¹ ₂ log R , x = y R Then v has to be a solution of

∂v

∂t + ∇ · h v

σ

^(m−m

⁾ ∇v ^m−1 − 2 x i

= 0 t > 0 , x ∈ R ^d with (as long as we make no assumption on R)

2 σ ⁻

^(m−m

⁾ = R ^{1−d (1−m)} dR

dτ

Refined relative entropy

Consider the family of the Barenblatt profiles B σ (x) := σ ⁻

C M + ¹ _σ |x | ²

∀ x ∈ R ^d (5) Note that σ is a function of t : as long as ^dσ _dt 6= 0, the Barenblatt profile B σ is not a solution but we may still consider the relative entropy

F σ [v] := 1 m − 1

Z

R

v ^m − B _σ ^m − m B _σ ^m−1 (v − B σ ) dx Let us briefly sketch the strategy of our method before giving all details

The time derivative of this relative entropy is d

dt F σ(t) [v(t , ·)] = dσ dt

d dσ F σ [v ]

|σ=σ(t)

| {z } choose it = 0

⇐⇒ Minimize F σ [v ] w.r.t. σ ⇐⇒ R

R

|x| ² B σ dx = R

R

|x| ² v dx

+ m

m − 1 Z

R

v ^m−1 − B _σ(t) ^m−1 ∂v

∂t dx

J. Dolbeault Improved Sobolev inequalities

The entropy / entropy production estimate

According to the definition of B σ , we know that 2 x = σ

^(m−m

⁾ ∇B _σ ^m−1

Using the new change of variables, we know that d

dt F σ(t) [v(t, ·)] = − m σ(t )

^(m−m

⁾ 1 − m

Z

R

v ∇ h

v ^m−1 − B _σ(t) ^m−1 i

2

dx Let w := v/B σ and observe that the relative entropy can be written as

F σ [v] = m 1 − m

Z

R

h w − 1 − 1

m w ^m − 1 i B _σ ^m dx (Repeating) define the relative Fisher information by

I σ [v] :=

Z

R

1

m − 1 ∇

(w ^m−1 − 1) B _σ ^m−1

2

B σ w dx

so that d

dt F σ(t) [v (t , ·)] = − m (1 − m) σ(t ) I σ(t) [v (t , ·)] ∀ t > 0

When linearizing, one more mode is killed and σ(t) scales out

Improved rates of convergence

Theorem (J.D., G. Toscani)

Let m ∈ ( m e 1 , 1), d ≥ 2, v 0 ∈ L ¹ ₊ ( R ^d ) such that v ₀ ^m , |y | ² v 0 ∈ L ¹ ( R ^d ) E[v (t , ·)] ≤ C e ^{−2γ(m) t} ∀ t ≥ 0

where

γ(m) =

 

 

 



((d−2) m−(d−4))

4 (1−m) if m ∈ ( m e 1 , m e 2 ] 4 (d + 2) m − 4 d if m ∈ [ m e 2 , m 2 ]

4 if m ∈ [m 2 , 1)

J. Dolbeault Improved Sobolev inequalities

Spectral gaps and best constants

m

=

0

m

=

m

=

e m

:=

m 1

2 4

Case 1 Case 2 Case 3

γ(m)

(d= 5)

e

m

:=

Gagliardo-Nirenberg and Sobolev inequalities :

improvements

[J.D., G. Toscani]

J. Dolbeault Improved Sobolev inequalities

Best matching Barenblatt profiles

(Repeating) Consider the fast diffusion equation

∂u

∂t + ∇ · h u

σ

^(m−m

⁾ ∇u ^m−1 − 2 x i

= 0 t > 0 , x ∈ R ^d with a nonlocal, time-dependent diffusion coefficient

σ(t) = 1 K M

Z

R

|x| ² u(x, t ) dx , K M :=

Z

R

|x| ² B 1 (x) dx where

B λ (x) := λ ⁻

C M + _λ ¹ |x| ²

∀ x ∈ R ^d and define the relative entropy

F λ [u] := 1 m − 1

Z

R

u ^m − B _λ ^m − m B _λ ^m−1 (u − B λ )

dx

Three ingredients for global improvements

1

inf λ>0 F λ [u(x , t )] = F σ(t) [u(x, t)] so that d

dt F σ(t) [u(x, t)] = −J σ(t) [u(·, t)]

where the relative Fisher information is J λ [u] := λ

^(m−m

⁾ m

1 − m Z

R

u ∇u ^m−1 − ∇B _λ ^{m−1 2} dx

2

In the Bakry-Emery method, there is an additional (good) term 4

"

1 + 2 C m,d

F σ(t) [u(·, t )]

M ^γ σ ₀

^(1−m)

# d

dt F σ(t) [u(·, t )]

≥ d

dt J σ(t) [u(·, t)]

3

The Csisz´ ar-Kullback inequality is also improved F σ [u] ≥ m

8 R

R

B ₁ ^m dx C _M ² ku − B σ k ² _L

( R

)

J. Dolbeault Improved Sobolev inequalities

improved decay for the relative entropy

0.0 0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

t 7→ f(t) e

f (0)

t (a)

(b) (c)

Figure: Upper bounds on the decay of the relative entropy: t 7→ f ( t ) e

/ f (0) (a): estimate given by the entropy-entropy production method

(b): exact solution of a simplified equation

(c): numerical solution (found by a shooting method)

A Csisz´ar-Kullback(-Pinsker) inequality

Let m ∈ ( m e 1 , 1) with m e 1 = _d+2 ^d and consider the relative entropy F σ [u] := 1

m − 1 Z

R

u ^m − B _σ ^m − m B _σ ^m−1 (u − B σ ) dx

Theorem

Let d ≥ 1, m ∈ ( m e 1 , 1) and assume that u is a nonnegative function in L ¹ ( R ^d ) such that u ^m and x 7→ |x| ² u are both integrable on R ^d . If kuk L

(R

) = M and R

R

|x| ² u dx = R

R

|x| ² B σ dx, then F σ [u]

σ

^(1−m) ≥ m 8 R

R

B ₁ ^m dx

C M ku − B σ k L

(R

) + 1 σ

Z

R

|x| ² |u − B σ | dx 2

J. Dolbeault Improved Sobolev inequalities

Csisz´ar-Kullback(-Pinsker): proof (1/2)

Let v := u/B σ and dµ σ := B _σ ^m dx Z

R

(v − 1) dµ σ = Z

R

B _σ ^m−1 (u − B σ ) dx

= σ

^(1−m) C M

Z

R

(u − B σ ) dx + σ

^(m

^−m) Z

R

|x| ² (u − B σ ) dx = 0 Z

R

(v − 1) dµ σ = Z

v>1

(v − 1) dµ σ − Z

v<1

(1 − v ) dµ σ = 0 Z

R

|v − 1| dµ σ = Z

v>1

(v − 1) dµ σ + Z

v<1

(1 − v) dµ σ

Z

R

|u − B σ | B _σ ^m−1 dx = Z

R

|v − 1| dµ σ = 2 Z

v<1

|v − 1| dµ σ

Csisz´ar-Kullback(-Pinsker): proof (2/2)

A Taylor expansion shows that

F σ [u] = 1 m − 1

Z

R

v ^m −1 −m (v −1)

d µ σ = m 2

Z

R

ξ ^m−2 |v −1| ² dµ σ

≥ m 2

Z

v<1

|v − 1| ² dµ σ

Using the Cauchy-Schwarz inequality, we get R

v<1 |v − 1| dµ σ 2

= R

v<1 |v − 1| B σ

B σ

dx 2

≤ R

v <1 |v −1| ² dµ σ R

R

B _σ ^m dx and finally obtain that

F σ [u] ≥ m 2

R

v<1 |v − 1| dµ σ 2

R

R

B _σ ^m dx = m 8

R

R

|u − B σ | B _σ ^m−1 dx 2

R

R

B _σ ^m dx

J. Dolbeault Improved Sobolev inequalities

An improved Gagliardo-Nirenberg inequality: the setting

The inequality

kf k L

( R

) ≤ C _p,d ^GN k∇f k ^θ L

( R

) kf k ^1−θ _L

( R

)

with θ = θ(p) := ^p−1 _{p d+2−p} ^d _(d−2) , 1 < p ≤ _d−2 ^d if d ≥ 3 and

1 < p < ∞ if d = 2, can be rewritten, in a non-scale invariant form, as Z

R

|∇f | ² dx + Z

R

|f | ^p+1 dx ≥ K p,d

Z

R

|f | ^{2 p} dx γ

with γ = γ(p, d ) := ^d+2−p _{d−p (d−4)} ^(d−2) . Optimal function are given by

f M,y,σ (x ) = 1 σ

C M + |x − y | ² σ

⁻

∀ x ∈ R ^d where C M is determined by R

R

f _M,y ^2p _,σ dx = M M _d :=

f M,y,σ : (M, y, σ) ∈ M d := (0, ∞) × R ^d × (0, ∞)

An improved Gagliardo-Nirenberg inequality (1/2)

Relative entropy functional R ^(p) [f ] := inf

g∈ M

d

Z

R

h g ^1−p |f | ^{2 p} − g ^{2 p}

− _p+1 ^2p |f | ^p+1 − g ^p+1 i dx

Theorem

Let d ≥ 2, p > 1 and assume that p < d /(d − 2) if d ≥ 3. If R

R

|x| ² |f | ^{2 p} dx R

R

|f | ^2p dx γ = ^{d (p−1)} _d+2−p ^σ _(d−2)

^M

, σ ∗ (p) :=

4 ^d+2−p _(p−1)

^(d−2) (p+1)

for any f ∈ L ^p+1 ∩ D ^1,2 ( R ^d ), then we have Z

R

|∇f | ² dx+

Z

R

|f | ^p+1 dx−K p,d

Z

R

|f | ^{2 p} dx γ

≥ C p,d

R ^(p) [f ] ² R

R

|f | ^{2 p} dx γ

J. Dolbeault Improved Sobolev inequalities

An improved Gagliardo-Nirenberg inequality (2/2)

A Csisz´ ar-Kullback inequality

R ^(p) [f ] ≥ C CK kf k ^2p _L

^(γ−2) ( R

) inf

g∈M

k|f | ^{2 p} − g ^{2 p} k ² L

(R

)

with C CK = ^p−1 _p+1 ^d+2−p _{32 p} ^(d−2) σ ^d

p−1 4p

∗ M ∗ ^1−γ . Let C _p,d := C d,p C CK 2

Corollary

Under previous assumptions, we have Z

R

|∇f | ² dx + Z

R

|f | ^p+1 dx − K p,d

Z

R

|f | ^{2 p} dx γ

≥ C _p,d kf k ^2p _L

^(γ−4) (R

) inf

g∈M

(p) k|f | ^{2 p} − g ^{2 p} k ⁴ L

(R

)

Conclusion 1: improved inequalities

We have found an improvement of an optimal Gagliardo-Nirenberg inequality, which provides an explicit measure of the distance to the manifold of optimal functions.

The method is based on the nonlinear flow

The explicit improvement gives (is equivalent to) an improved entropy – entropy production inequality

J. Dolbeault Improved Sobolev inequalities

Conclusion 2: improved rates

If m ∈ (m 1 , 1), with

f (t ) := F σ(t) [u(·, t)]

σ(t ) = 1 K M

Z

R

|x| ² u(x, t) dx j(t) := J σ(t) [u(·, t)]

J σ [u] := m σ

^(m−m

⁾ 1 − m

Z

R

u ∇u ^m−1 − ∇ B ^m−1 _σ ² dx we can write a system of coupled ODEs

 

 

 



f ^′ = −j ≤ 0 σ ^′ = −2 d ^(1−m) _{m K}

M

σ

^(m−m

⁾ f ≤ 0 j ^′ + 4 j = ^d ₂ (m − m c ) h

j

σ + 4 d (1 − m) _σ ^f i σ ^′ − r

(6)

In the rescaled variables, we have found an improved decay (algebraic

rate) of the relative entropy. This is a new nonlinear effect, which

matters for the initial time layer

Conclusion 3: Best matching Barenblatt profiles are delayed

Let u be such that

v(τ, x) = µ ^d R(D τ) ^d u

1

2 log R(D τ), µ x R(D τ)

with τ 7→ R(τ) given as the solution to 1

R dR dτ =

µ ² K M

Z

R

|x| ² v(τ, x) dx

⁻

(m−m

)

, R(0) = 1 Then

1 R

dR dτ =

R ² (τ) σ

1

2 log R(D τ)

−

(m−m

)

that is R(τ) = R 0 (τ) ≤ R 0 (τ) where _R ^{1 dR} _dτ

= R ₀ ² (τ ) σ (0) −

(m−m

)

and asymptotically as τ → ∞, R(τ) = R 0 (τ − δ) for some delay δ > 0

J. Dolbeault Improved Sobolev inequalities

Thank you for your attention !