· icreauab macroscopic models: pks system ideas of the rigorous proof a byproduct: a new proof of...

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Macroscopic Models: PKS system Ideas of the Rigorous Proof A byproduct: A new proof of HLS inequalities Conclusions

Aggregation versus Diffusionin Mathematical Biology

J. A. Carrillo

ICREA - Universitat Autònoma de Barcelona

Lecture 3: Aggregation versus Diffusion Equations

UIMP, Santander, Spain, 2011

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Outline

1 Macroscopic Models: PKS systemModelling Chemotaxis: First PropertiesPKS as Gradient FlowCritical Fast Diffusion as Gradient FlowNew Liapunov Functionals

2 Ideas of the Rigorous ProofConcentration-Control InequalitiesOne-step JKO

3 A byproduct: A new proof of HLS inequalitiesLog HLS via Fast Diffusion FlowsA New Proof of the HLS inequality with Equality Cases

4 Conclusions

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Modelling Chemotaxis: First Properties

Outline

4 Conclusions

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Chemotaxis

Cell movement and aggregation by chemical interaction.

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PKS System

∂t(x, t) = ∆ρ(x, t)−∇·(ρ(x, t)∇c(x, t)) x ∈ R2 , t > 0 ,

c(x, t) = − 12π

log |x− y| ρ(y, t) dy x ∈ R2 , t > 0 ,

ρ(x, t = 0) = ρ0 ≥ 0 x ∈ R2 .

Huge Literature: Horstmann reviews (2003& 2004), Perthame review (2004).

Conservations:

Conservation of mass:

∫R2ρ0(x) dx =

∫R2ρ(x, t) dx

Conservation of center of mass:

x ρ0(x) dx =

x ρ(x, t) dx .

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PKS System

c(x, t) = − 12π

log |x− y| ρ(y, t) dy x ∈ R2 , t > 0 ,

ρ(x, t = 0) = ρ0 ≥ 0 x ∈ R2 .

Conservations:

∫R2ρ0(x) dx =

∫R2ρ(x, t) dx

x ρ0(x) dx =

x ρ(x, t) dx .

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PKS System

c(x, t) = − 12π

log |x− y| ρ(y, t) dy x ∈ R2 , t > 0 ,

ρ(x, t = 0) = ρ0 ≥ 0 x ∈ R2 .

Conservations:

∫R2ρ0(x) dx =

∫R2ρ(x, t) dx

x ρ0(x) dx =

x ρ(x, t) dx .

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Second Moment

Distributional Solution:

We shall say that ρ ∈ C0([0, T); L1weak(R2)

)is a weak solution to the PKS system if

for all test functions ψ ∈ D(R2),

∫R2ψ(x) ρ(x, t) dx =

∆ψ(x) ρ(x, t) dx− 14π

[∇ψ(x)−∇ψ(y)] · x− y|x− y|2 ρ(x, t) ρ(y, t) dx dy

holds in the distributional sense in (0, T) and ρ(0) = ρ0.

Evolution of second moment:

∫R2|x|2 ρ(x, t) dx = 4M − 1

2πM2,

Struggle between diffusion and aggregation. Balance between these twomechanisms happens precisely at the critical mass M = 8π.

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Second Moment

∆ψ(x) ρ(x, t) dx− 14π

∫R2|x|2 ρ(x, t) dx = 4M − 1

2πM2,

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Second Moment

∆ψ(x) ρ(x, t) dx− 14π

∫R2|x|2 ρ(x, t) dx = 4M − 1

2πM2,

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Second Moment

∆ψ(x) ρ(x, t) dx− 14π

∫R2|x|2 ρ(x, t) dx = 4M − 1

2πM2,

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Second Moment

∆ψ(x) ρ(x, t) dx− 14π

∫R2|x|2 ρ(x, t) dx = 4M − 1

2πM2,

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PKS Cases:

Subcritical Case, M < 8π: Global existence: Jägger-Luckhaus (1992),Dolbeault-Perthame (2004), Blanchet-Dolbeault-Perthame (2006),Blanchet-Calvez-C. (2008).

Supercritical Case, M > 8π: Blow-up: Herrero-Velazquez (1996), Velazquez(2002-2004), Dolbeault-Schmeiser (2009).

Critical Case, M = 8π: Infinite-time aggregation, infinitely many stationarystates: Biler-Karch-Laurençot-Nadzieja (2006), Blanchet-C.-Masmoudi (2008),Blanchet-Carlen-C. (2011), Carlen-C.-Loss (2010), Carlen-Figalli (preprint).

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PKS Cases:

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PKS Cases:

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PKS as Gradient Flow

Outline

4 Conclusions

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Log HLS Inequality by Carlen& LossLet f be a non-negative function in L1(R2) with mass M such that f log f andf log(e + |x|2) belong to L1(R2). Then∫

R2f (x) log f (x) dx +

∫∫R2×R2

f (x)f (y) log |x− y| dx dy ≥ − C(M)

with C(M) := M(1 + logπ − log M).

Equality cases:

There is equality if and only if f (x) = ρλ(x− x0) for some λ > 0 and some x0 ∈ R2,where

ρλ(x) :=Mπ

(λ+ |x|2)2

Natural Liapunov Functional:

Free energy:

FPKS[ρ] :=

∫R2ρ(x) log ρ(x) dx +

∫∫R2×R2

ρ(x) ρ(y) log |x− y| dx dy

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∫∫R2×R2

Equality cases:

ρλ(x) :=Mπ

(λ+ |x|2)2

Free energy:

FPKS[ρ] :=

∫∫R2×R2

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∫∫R2×R2

Equality cases:

ρλ(x) :=Mπ

(λ+ |x|2)2

Free energy:

FPKS[ρ] :=

∫∫R2×R2

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Formal Gradient FlowA formal calculation shows that for all u ∈ C∞c (R2) with zero mean,

limε→0

(FPKS[ρ+ εu]−FPKS[ρ]) =

δFPKS(ρ)

δρ(x) u(x) dx

δFPKS(ρ)

δρ(x) := log ρ(x) +

log |x− y|ρ(y) dy = log ρ(x)− G ∗ ρ(x) .

The PKS equation can be rewritten as

∂t(t, x) = div

(ρ(t, x)∇

[δFPKS(ρ(t))

δρ(t, x)

with entropy dissipation:

ddtFPKS[ρ(t)] = −

∫R2ρ(t, x)

∣∣∣∣∇δFPKS(ρ(t))δρ

(t, x)

∣∣∣∣2 dx .

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limε→0

δFPKS(ρ)

δρ(x) u(x) dx

δFPKS(ρ)

∂t(t, x) = div

(ρ(t, x)∇

[δFPKS(ρ(t))

δρ(t, x)

∫R2ρ(t, x)

(t, x)

∣∣∣∣2 dx .

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limε→0

δFPKS(ρ)

δρ(x) u(x) dx

δFPKS(ρ)

∂t(t, x) = div

(ρ(t, x)∇

[δFPKS(ρ(t))

δρ(t, x)

∫R2ρ(t, x)

(t, x)

∣∣∣∣2 dx .

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Critical Case: Stationary States & Main Result

The critical case M = 8π has a family of explicit stationary solutions of the form

ρλ(x) =8λ

(λ+ |x|2)2

with λ > 0.

All of these stationary solutions have critical mass and infinite second moment.

They are the cases of equality functions for the Log HLS inequality.

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ρλ(x) =8λ

(λ+ |x|2)2

with λ > 0.

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ρλ(x) =8λ

(λ+ |x|2)2

with λ > 0.

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Critical Fast Diffusion as Gradient Flow

Outline

4 Conclusions

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Critical Fast DiffusionThe nonlinear Fokker-Planck equation in R2 with exponent 1/2:

∂v∂t

(t, x) = ∆(√

v(t, x))

+1√2λ

div(x v(t, x)) t > 0 , x ∈ R2 ,

v(0, x) = v0(x) ≥ 0 x ∈ R2 ,

corresponding to the fast diffusion equation ∂u∂t = ∆ (

√u) by a self-similar change

of variable.

It is critical for the direct Bakry-Emery method to the fast-diffusion equation,meaning by a direct entropy-entropy dissipation inequality and a rate of convergencecoming only from displacement convexity.

For λ > 0, define the functionalHλ on the non-negative functions in L1(R2) by

Hλ[v] :=

(√v(x)−

√ρλ(x)

)2ρ−1/2λ (x) dx

This functional is the relative entropy of the fast diffusion equation with respect tothe stationary solution ρλ. The unique minimizer ofHλ is ρλ.

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∂v∂t

(t, x) = ∆(√

v(t, x))

+1√2λ

div(x v(t, x)) t > 0 , x ∈ R2 ,

v(0, x) = v0(x) ≥ 0 x ∈ R2 ,

of variable.

Hλ[v] :=

(√v(x)−

√ρλ(x)

)2ρ−1/2λ (x) dx

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∂v∂t

(t, x) = ∆(√

v(t, x))

+1√2λ

div(x v(t, x)) t > 0 , x ∈ R2 ,

v(0, x) = v0(x) ≥ 0 x ∈ R2 ,

of variable.

Hλ[v] :=

(√v(x)−

√ρλ(x)

)2ρ−1/2λ (x) dx

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Formal Gradient FlowA formal calculation as before shows,

δHλ[v]

1√ρλ− 1√

and the critical fast diffusion equation can be rewritten as

∂v∂t

(t, x) = div(

v(t, x)

[∇δHλ[v]

ddtHλ[v(t)] = −

v(t, x)

∣∣∣∣∇δHλ[v]

∣∣∣∣2 dx.

Let us point out that the functionalHλ[v] can be written as

Hλ[u] :=

[Φ(v(x))− Φ(ρλ(x))− Φ′(ρλ)(v(x)− ρλ(x))

with Φ(s) = −2√

s a convex function.

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δHλ[v]

1√ρλ− 1√

∂v∂t

(t, x) = div(

v(t, x)

[∇δHλ[v]

ddtHλ[v(t)] = −

v(t, x)

∣∣∣∣∇δHλ[v]

∣∣∣∣2 dx.

Hλ[u] :=

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δHλ[v]

1√ρλ− 1√

∂v∂t

(t, x) = div(

v(t, x)

[∇δHλ[v]

ddtHλ[v(t)] = −

v(t, x)

∣∣∣∣∇δHλ[v]

∣∣∣∣2 dx.

Hλ[u] :=

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δHλ[v]

1√ρλ− 1√

∂v∂t

(t, x) = div(

v(t, x)

[∇δHλ[v]

ddtHλ[v(t)] = −

v(t, x)

∣∣∣∣∇δHλ[v]

∣∣∣∣2 dx.

Hλ[u] :=

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New Liapunov Functionals

Outline

4 Conclusions

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New Liapunov FunctionalClaim: The critical fast diffusion functional is also a Liapunov functional for thecritical mass PKS. Formal Computation:

ddtHλ[ρ(t)] =

δHλ[ρ]

δρdiv(ρ(t, x)∇

[δFPKS[ρ]

= −∫R2ρ∇[δHλ[ρ]

]· ∇[δFPKS[ρ]

= −∫R2ρ∇[

1√ρλ− 1√ρ

]· ∇ [log ρ− G ∗ ρ] dx

=−∫R2

λMx ρ+∇√ρ

]· ∇ [log ρ− G ∗ ρ] dx

Now, integrating by parts once more on the term involving the Green’s function,∫R2∇√ρ · ∇ [log ρ− G ∗ ρ] dx =

|∇ρ|2

ρ3/2 +

√ρ∆G ∗ ρ

|∇ρ|2

ρ3/2 −∫R2ρ3/2 dx .

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ddtHλ[ρ(t)] =

δHλ[ρ]

δρdiv(ρ(t, x)∇

[δFPKS[ρ]

]· ∇[δFPKS[ρ]

= −∫R2ρ∇[

1√ρλ− 1√ρ

]· ∇ [log ρ− G ∗ ρ] dx

=−∫R2

λMx ρ+∇√ρ

]· ∇ [log ρ− G ∗ ρ] dx

|∇ρ|2

ρ3/2 +

√ρ∆G ∗ ρ

|∇ρ|2

ρ3/2 −∫R2ρ3/2 dx .

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ddtHλ[ρ(t)] =

δHλ[ρ]

δρdiv(ρ(t, x)∇

[δFPKS[ρ]

]· ∇[δFPKS[ρ]

= −∫R2ρ∇[

1√ρλ− 1√ρ

]· ∇ [log ρ− G ∗ ρ] dx

=−∫R2

λMx ρ+∇√ρ

]· ∇ [log ρ− G ∗ ρ] dx

|∇ρ|2

ρ3/2 +

√ρ∆G ∗ ρ

|∇ρ|2

ρ3/2 −∫R2ρ3/2 dx .

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New Liapunov Functional 2Also,

∫R2 x · ∇ρ dx = −2M and, making the same symmetrization that led to the

evolution of the second moment,∫R2ρ(x) x · ∇G ∗ ρ(x) dx =

∫R2×R2

ρ(t, x) (x−y)· x− y|x− y|2 ρ(t, y) dx dy = −M2

Collecting the above computations:

ddtHλ[ρ(t)] = −1

|∇ρ|2

ρ3/2 dx +

∫R2ρ3/2 dx + 4

√M π

(1− M

Notice that the constant term vanishes in critical mass case M = 8π. Thus, in thecritical mass case, formal calculation yields that for all T > 0,

Hλ[ρ(T)] +

|∇ρ|2

ρ3/2 (t, x) dx−∫R2ρ3/2(t, x) dx

]dt ≤ Hλ[ρ0] .

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∫R2×R2

|∇ρ|2

ρ3/2 dx +

∫R2ρ3/2 dx + 4

√M π

(1− M

Hλ[ρ(T)] +

|∇ρ|2

ρ3/2 (t, x) dx−∫R2ρ3/2(t, x) dx

]dt ≤ Hλ[ρ0] .

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∫R2×R2

|∇ρ|2

ρ3/2 dx +

∫R2ρ3/2 dx + 4

√M π

(1− M

Hλ[ρ(T)] +

|∇ρ|2

ρ3/2 (t, x) dx−∫R2ρ3/2(t, x) dx

]dt ≤ Hλ[ρ0] .

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∫R2×R2

|∇ρ|2

ρ3/2 dx +

∫R2ρ3/2 dx + 4

√M π

(1− M

Hλ[ρ(T)] +

|∇ρ|2

ρ3/2 (t, x) dx−∫R2ρ3/2(t, x) dx

]dt ≤ Hλ[ρ0] .

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New Liapunov Functional 3The missing link to exploit the previous relation is a particular case of theGagliardo-Nirenberg-Sobolev inequalities1.

Gagliardo-Nirenberg-Sobolev inequality

For all functions f in R2 with a square integrable distributional gradient∇f ,

∫R2|f |6 dx ≤

∫R2|∇f |2 dx

∫R2|f |4 dx

and there is equality if and only if f is a multiple of a translate of ρ1/4λ , λ > 0.

Dissipation ofHλ

Applying the GNS to f = ρ1/4: For all densities ρ of mass M = 8π,

D[ρ] :=12

|∇ρ(x)|2

ρ3/2(x)dx−

∫R2ρ3/2(x) dx ≥ 0 ,

and moreover, there is equality if and only ρ is a translate of ρλ for some λ > 0.

1Dolbeault-DelPino, JMPA 2002

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∫R2|f |6 dx ≤

∫R2|∇f |2 dx

∫R2|f |4 dx

Dissipation ofHλ

D[ρ] :=12

|∇ρ(x)|2

ρ3/2(x)dx−

∫R2ρ3/2(x) dx ≥ 0 ,

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∫R2|f |6 dx ≤

∫R2|∇f |2 dx

∫R2|f |4 dx

Dissipation ofHλ

D[ρ] :=12

|∇ρ(x)|2

ρ3/2(x)dx−

∫R2ρ3/2(x) dx ≥ 0 ,

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Main Result Blanchet-Carlen-C.

Bassin of attractions in the Critical Mass PKS

Given any density ρ0 in R2 with total mass 8π such that there exists λ > 0 with

Hλ[ρ0] <∞

Then there exists ρ ∈ AC0([0, T],P2(R2)), with ρ(t) ∈ L1(R2) for all t ≥ 0 being aglobal-in-time weak solution of the critical mass PKS. Moreover, the solutionsconstructed satisfy that

Hλ[ρ(t)] ≤ Hλ[ρ(s)] and FPKS[ρ(t)] ≤ FPKS[ρ(s)] ,

for all 0 ≤ s ≤ t. Moreover, we can show that the constructed weak solutions aredissipative solutions, i.e., they satisfy for all T > 0

Hλ[ρ(T)] +

0D[ρ(t)] dt ≤ Hλ[ρ0]

and limt→∞‖ρ(t)− ρλ‖L1(R2) = 0.

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Hλ[ρ0] <∞

Hλ[ρ(T)] +

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Hλ[ρ0] <∞

Hλ[ρ(T)] +

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Hλ[ρ0] <∞

Hλ[ρ(T)] +

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Mathematical Difficulties

For a density of mass 8π, an upper bound on FPKS[ρ] does not provide anyupper bound on the entropy E [ρ].Indeed, FPKS takes its minimum value for ρ = ρµ for all µ > 0, butE [ρµ]→∞ as µ→ 0 since ρµ converges weakly-* as measures towards aDirac delta at the origin with mass 8π.

The dissipation functional D[ρ] is well defined as long as ‖ρ‖3/2 <∞, but anupper bound on D[ρ] does not give an upper bound on either ‖∇(ρ1/4)‖2 or‖ρ‖3/2 since it is the difference of both terms.

Indeed, D[ρ] = 0 for ρ = ρµ for all µ > 0, but ‖∇(ρ1/4µ )‖2 or ‖ρµ‖3/2 both

diverge to infinity as µ tends to zero.

Stability properties ofHλ over FPKS. In fact, not much was known aboutstability for FPKS till very recently (E. Carlen, A. Figalli, preprint).We do know that if some density ρ with mass 8π satisfies FPKS[ρ] = FPKS[ρλ],then, up to translation, ρ = ρµ for some µ > 0. Even knowing a quantitativeestimate for the error in FPKS, it mainly helps to quantify the decay rate.

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Concentration-Control Inequalities

Outline

4 Conclusions

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Thick Tails & Localization by Hλ

Rigorous Proof of Displacement Convexity ofHλ by regularization.

Proof of Talagrand inequality to control W2(ρ, %λ) in terms ofHλ[ρ].

Thick Tails

A bound onHλ[ρ] implies a "matching tail" estimate.- Let ρ be any density of massM such thatHλ[ρ] <∞. Then for η∗ := 1

5 e−1/5 and any s > 1∫|x|2≥λs2

ρ(x) dx ≥ η∗ e−4√πMλHλ[ρ]

∫|x|2≥λs2

%λ(x) dx =Mη∗

1 + s2 e−4√πMλHλ[ρ].

Solid core

A bound onHλ[ρ] implies localization.- Let ρ be a density of mass M such that forsome λ > 0,Hλ[ρ] <∞. Then∫

{|x|≥4√λ+4(λ/Mπ)1/4

√Hλ[ρ]}

ρ dx ≤ M2.

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Thick Tails

5 e−1/5 and any s > 1∫|x|2≥λs2

∫|x|2≥λs2

%λ(x) dx =Mη∗

Solid core

{|x|≥4√λ+4(λ/Mπ)1/4

√Hλ[ρ]}

ρ dx ≤ M2.

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Thick Tails

5 e−1/5 and any s > 1∫|x|2≥λs2

∫|x|2≥λs2

%λ(x) dx =Mη∗

Solid core

{|x|≥4√λ+4(λ/Mπ)1/4

√Hλ[ρ]}

ρ dx ≤ M2.

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Thick Tails

5 e−1/5 and any s > 1∫|x|2≥λs2

∫|x|2≥λs2

%λ(x) dx =Mη∗

Solid core

{|x|≥4√λ+4(λ/Mπ)1/4

√Hλ[ρ]}

ρ dx ≤ M2.

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Thick Tails

5 e−1/5 and any s > 1∫|x|2≥λs2

∫|x|2≥λs2

%λ(x) dx =Mη∗

Solid core

{|x|≥4√λ+4(λ/Mπ)1/4

√Hλ[ρ]}

ρ dx ≤ M2.

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Thick Tails

5 e−1/5 and any s > 1∫|x|2≥λs2

∫|x|2≥λs2

%λ(x) dx =Mη∗

Solid core

{|x|≥4√λ+4(λ/Mπ)1/4

√Hλ[ρ]}

ρ dx ≤ M2.

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Concentration-Control for the Entropy I

In summary, we can show there exists R > 1 so that for some 0 < a < 8π,∫|x|>R−1

ρ dx ≤ 8π − a and∫|x|<R+1

ρ dx ≤ 8π − a .

The densities lying in sublevel sets of FPKS[ρ] andHλ[ρ] are compact.

Entropy bound

Let ρ be any density with mass M = 8π, withHλ[ρ] <∞ for some λ > 0. Thenthere exist positive computable constants γ1 and CCCF depending only on λ andHλ[ρ] such that

∫R2ρ log+ ρ dx ≤ FPKS[ρ] + CCCF .

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ρ dx ≤ 8π − a .

Entropy bound

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ρ dx ≤ 8π − a .

Entropy bound

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Concentration-Control for Entropy-Dissipation

The densities lying in sublevel sets of FPKS[ρ],Hλ[ρ] and D[ρ] are compact.

Concentration control for DLet ρ be any density with mass 8π, FPKS[ρ] finite, andHλ[ρ] finite for some λ > 0.Then there exist positive computable constants γ2 and CCCD depending only on λ,Hλ[ρ] and FPKS[ρ] such that

∫R2|∇ρ1/4|2 dx ≤ πD[ρ] + CCCD .

This allows to control the L3/2-norm of ρ too.

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∫R2|∇ρ1/4|2 dx ≤ πD[ρ] + CCCD .

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∫R2|∇ρ1/4|2 dx ≤ πD[ρ] + CCCD .

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One-step JKO

Outline

4 Conclusions

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One-step JKO

Set-up

Regularization

For all 0 < ε ≤ 1, define

{ρ ∈ L1(R2, log(e + |x|2) dx) :

∫R2ρ(x) log ρ(x) dx <∞,

∫R2ρ(x) dx = 8π

}For any ρ ∈ S define

FεPKS[ρ] =

∫R2ρ(x) log ρ(x) dx− 1

∫∫R2×R2

ρ(x) Gε(x− y) ρ(y) dx dy .

Here, the regularized Green’s function is

Gε = γε ∗ G ∗ γε

where ∗ denotes convolution, G(x) = −1/(2π) log |x|, and γε(x) = ε−2γ (x/ε). withγ a standard gaussian.

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One-step JKO

Set-up

Regularization

{ρ ∈ L1(R2, log(e + |x|2) dx) :

∫R2ρ(x) dx = 8π

FεPKS[ρ] =

∫∫R2×R2

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One-step JKO

Set-up

Regularization

{ρ ∈ L1(R2, log(e + |x|2) dx) :

∫R2ρ(x) dx = 8π

FεPKS[ρ] =

∫∫R2×R2

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One-step JKO

Difficulties in the JKO scheme + Passing to the LimitRegularized Problem: we can show now existence of minimizers in the JKOscheme and have the right regularity to produce estimates for one step:

arg minρ∈S

2(ρ, ρ0)

2τ+ FεPKS[ρ]

However, the regularization destroys the decay of the functionalHλ. Then wehave to cope with error terms in the dissipation inequality forHλ.

Cure: take step-dependent regularization parameter εk such that the errors aresummable and thus negligible, in the limit of small τ → 0.

For the cure and the compactness, the concentration controlled inequalities anda Lp-smoothing effect are crucial.

The dissipation onHλ in the limit identifies the limiting density %λ. Aqualitative stability estimate on FPKS implies the L1-strong convergence.

Rate of Convergence: (Carlen-Figalli, preprint) Stability estimates for GNSimply stability estimates for the Log-HLS via the fast diffusion flow. This canbe applied to decay rates in L1 for the solutions constructed to the PKS.

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One-step JKO

arg minρ∈S

2(ρ, ρ0)

2τ+ FεPKS[ρ]

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One-step JKO

arg minρ∈S

2(ρ, ρ0)

2τ+ FεPKS[ρ]

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One-step JKO

arg minρ∈S

2(ρ, ρ0)

2τ+ FεPKS[ρ]

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One-step JKO

arg minρ∈S

2(ρ, ρ0)

2τ+ FεPKS[ρ]

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One-step JKO

arg minρ∈S

2(ρ, ρ0)

2τ+ FεPKS[ρ]

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One-step JKO

Aggregation Versus Diffusion: Open ProblemsSharp Conditions on Interaction Potential for linear diffusion to win: Severalpartial answers by (Karch-Suzuki, Nonlinearity 2010) and(Cañizo-C.-Schonbek, preprint).

Asymptotics for small linear diffusion: Several partial answers by(Primi-Stevens-Velázquez, CPDE 2009).

Sharp results in any dimension with nonlinear diffusions:Fair competition m = 2− 2

d : (Blanchet-C.-Laurençot, CVPDE 2008).Diffusion-Dominated case m > 2− 2

d : (Calvez-C., JMPA 2006),(Bedrossian-Rodríguez-Bertozzi, Nonlinearity 2011), (Bedrossian, to appear inCMS), (Kim-Yao, preprint).Aggregation-Dominated Case m < 2− 2

d : very few results.

Long time aysmptotics with nonlinear diffusions as repulsion:(Burger-DiFrancesco, Net. Heterog. Media 2008),(Burger-DiFrancesco-Franek, to appear).

Full-Parabolic system: (Calvez-Corrias, CMS 2008), (DiFrancesco-Rosado,Nonlinearity 2009).

Cross Diffusion: (Hittmeir-Jüngel, SIMA 2011).

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One-step JKO

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One-step JKO

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One-step JKO

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One-step JKO

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One-step JKO

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“Duality” between CFD and PKS

Observation of Matthes, McCann & Savaré, CPDE 2009

Consider two smooth functions Φ and Ψ on Rd, and consider the two ordinarydifferential equations describing gradient flow:

x(t) = −∇Φ[x(t)] and y(t) = −∇Ψ[y(t)] .

Then of course Φ[(x(t)] and Ψ[(y(t)] are monotone decreasing. Now differentiateeach function along the others flow:

Φ[y(t)] = −〈∇Φ[y(t)],∇Ψ[y(t)]〉

Ψ[x(t)] = −〈∇Ψ[x(t)],∇Φ[x(t)]〉 .

Thus, Φ is decreasing along the gradient flow of Ψ for any initial data if and only ifΨ is decreasing along the gradient flow of Φ for any initial data.

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Observation of Matthes, McCann & Savaré, CPDE 2009

Consider two smooth functions Φ and Ψ on Rd, and consider the two ordinarydifferential equations describing gradient flow:

x(t) = −∇Φ[x(t)] and y(t) = −∇Ψ[y(t)] .

Then of course Φ[(x(t)] and Ψ[(y(t)] are monotone decreasing. Now differentiateeach function along the others flow:

Φ[y(t)] = −〈∇Φ[y(t)],∇Ψ[y(t)]〉

Ψ[x(t)] = −〈∇Ψ[x(t)],∇Φ[x(t)]〉 .

Thus, Φ is decreasing along the gradient flow of Ψ for any initial data if and only ifΨ is decreasing along the gradient flow of Φ for any initial data.

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An analog of this holds for well-behaved gradient flows in the 2-Wassersteinsense as used in (Matthes, McCann & Savare, CPDE 2009).

Our case: Apply it to the Log-HLS functional in d = 2.- SinceHλ is decreasingalong the 2-Wasserstein gradient flow for FPKS, i.e., the Patlak-Keller-Segelequation, one can expect that FPKS of the Log-HLS functional is decreasingalong the 2-Wasserstein gradient flow forHλ, i.e., the critical fast diffusion.

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An analog of this holds for well-behaved gradient flows in the 2-Wassersteinsense as used in (Matthes, McCann & Savare, CPDE 2009).

Our case: Apply it to the Log-HLS functional in d = 2.- SinceHλ is decreasingalong the 2-Wasserstein gradient flow for FPKS, i.e., the Patlak-Keller-Segelequation, one can expect that FPKS of the Log-HLS functional is decreasingalong the 2-Wasserstein gradient flow forHλ, i.e., the critical fast diffusion.

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Log HLS via Fast Diffusion Flows

Outline

4 Conclusions

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The connection from FD perspectiveThe log HLS functional F is defined by

F [f ] :=

f (x) log f (x) dx +2∫

R2f (x) dx

∫∫R2×R2

f (x) log |x− y|f (y) dx dy .

The critical fast diffusion is

∂tu(x, t) = ∆u1/2(x, t) ,

with associated Fokker-Planck equation

∂tv(x, t) = ∆v1/2(x, t) +∇ · [xv(x, t)] .

which has as stationary solution

h(x) =Mπ

1 + |x|2

with mass M.Scaling: f(a) := a2f (ax). Then, F [f(a)] = F [f ] for all a. v(x, t) := e2tu(etx, et) andthus F [v(·, t)] = F [u(·, et)].

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F [f ] :=

R2f (x) dx

∫∫R2×R2

∂tu(x, t) = ∆u1/2(x, t) ,

∂tv(x, t) = ∆v1/2(x, t) +∇ · [xv(x, t)] .

h(x) =Mπ

1 + |x|2

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F [f ] :=

R2f (x) dx

∫∫R2×R2

∂tu(x, t) = ∆u1/2(x, t) ,

∂tv(x, t) = ∆v1/2(x, t) +∇ · [xv(x, t)] .

h(x) =Mπ

1 + |x|2

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F [f ] :=

R2f (x) dx

∫∫R2×R2

∂tu(x, t) = ∆u1/2(x, t) ,

∂tv(x, t) = ∆v1/2(x, t) +∇ · [xv(x, t)] .

h(x) =Mπ

1 + |x|2

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F [f ] :=

R2f (x) dx

∫∫R2×R2

∂tu(x, t) = ∆u1/2(x, t) ,

∂tv(x, t) = ∆v1/2(x, t) +∇ · [xv(x, t)] .

h(x) =Mπ

1 + |x|2

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The connection from FD perspectiveLog HLS via fast diffusion flow

Let f be a non-negative measurable functions on R2 such that f logf andf log(e + |x|2) belong to L1(R2). Suppose also that∫

R2f (x) dx =

h(x) dx = M

andsup|x|>R

f (x)|x|4 <∞ .

Let u(x, t) be the solution of critical fast diffusion with u(x, 1) = f (x). Then

F [f ] = F [h] +

∫ ∞1D[u1/4(·, et)] dt ≥ F [h] ,

whereD[g] =

∫R2|∇g(x)|2 dx

g4(x) dx− π∫R2

g6(x) dx ,

which is non-negative by the sharp GNS inequality.

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The connection from FD perspectiveLog HLS via fast diffusion flow

Let f be a non-negative measurable functions on R2 such that f logf andf log(e + |x|2) belong to L1(R2). Suppose also that∫

R2f (x) dx =

h(x) dx = M

andsup|x|>R

f (x)|x|4 <∞ .

Let u(x, t) be the solution of critical fast diffusion with u(x, 1) = f (x). Then

F [f ] = F [h] +

∫ ∞1D[u1/4(·, et)] dt ≥ F [h] ,

whereD[g] =

∫R2|∇g(x)|2 dx

g4(x) dx− π∫R2

g6(x) dx ,

which is non-negative by the sharp GNS inequality.

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The connection from FD perspective

Proof: Let v(x, t) be the solution of the nonlinear FP with v(x, 0) = f (x).

Then it is known that limt→∞ v(x, t) = h(x) strongly in L1(R2) and moreover2, wehave the uniform bounds:

C h(x) ≤ v(x, t) ≤ 1C

h(x) ,

under the assumption on f (x) after some time.

Dominated convergence, implies that limt→∞ F [v(·, t)] = F [h]. Thus,

F [h]−F [f ] =

∫ ∞0

ddtF [v(·, t)] dt .

and by scalingddtF [v(·, t)] =

ddtF [u(·, et)] .

2Bonforte-Vazquez, J. Func. Anal. 2006

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C h(x) ≤ v(x, t) ≤ 1C

h(x) ,

F [h]−F [f ] =

∫ ∞0

ddtF [u(·, et)] .

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C h(x) ≤ v(x, t) ≤ 1C

h(x) ,

F [h]−F [f ] =

∫ ∞0

ddtF [u(·, et)] .

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Hence it shall suffice to compute

ddtF(u(·, t)) =

log u ∆u1/2 dx − 8πM〈(−∆)−1u,∆u1/2〉L2(R2)

= − 12

|∇u|2

u3/2 dx +8πM

u3/2 dx.

Making the change of variablesg = u1/4 ,

again the sharp GNS inequality gives us the non-positivity of the rhs.

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Hence it shall suffice to compute

ddtF(u(·, t)) =

log u ∆u1/2 dx − 8πM〈(−∆)−1u,∆u1/2〉L2(R2)

= − 12

|∇u|2

u3/2 dx +8πM

u3/2 dx.

Making the change of variablesg = u1/4 ,

again the sharp GNS inequality gives us the non-positivity of the rhs.

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A New Proof of the HLS inequality with Equality Cases

Outline

4 Conclusions

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Two Inequalities, One Equation

Sharp Hardy-Littlewood-Sobolev inequalitya It states that for all non-negative measurable functions f on Rd, and all 0 < λ < d,∫

f (x)1

|x− y|λ f (y) dx dy

‖f‖2p

|x− y|λ h(y) dx dy

‖h‖2p

h(x) =

1 + |x|2

)(2d−λ)/2

and p = 2d2d−λ .

Moreover, there is equality if and only if for some x0 ∈ Rd and s ∈ R+, f is anon-zero multiple of h(x/s− x0).

aE. Lieb, Ann. Math 1983.

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Sharp Hardy-Littlewood-Sobolev inequalitya It states that for all non-negative measurable functions f on Rd, and all 0 < λ < d,∫

f (x)1

|x− y|λ f (y) dx dy

‖f‖2p

|x− y|λ h(y) dx dy

‖h‖2p

h(x) =

1 + |x|2

)(2d−λ)/2

and p = 2d2d−λ .

Moreover, there is equality if and only if for some x0 ∈ Rd and s ∈ R+, f is anon-zero multiple of h(x/s− x0).

aE. Lieb, Ann. Math 1983.

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Sharp Gagliardo-Nirenberg-Sobolev inequalitya It states that for all locally integrable functions f on Rd, d ≥ 2, with a squareintegrable distributional gradient, and p with 1 < p < d/(d − 2)

‖∇f‖θ2‖f‖1−θp+1

‖f‖2p≥‖∇g‖θ2‖g‖1−θ

‖g‖2p

g(x) =

1 + |x|2

)1/(p−1)

θ =d(p− 1)

p(d + 2− (d − 2)p).

Moreover, there is equality if and only if for some x0 ∈ Rd and s ∈ R+, f is anon-zero multiple of g(x/s− x0).

aDelPino & Dolbeault, JMPA, 2002.

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Sharp Gagliardo-Nirenberg-Sobolev inequalitya It states that for all locally integrable functions f on Rd, d ≥ 2, with a squareintegrable distributional gradient, and p with 1 < p < d/(d − 2)

‖∇f‖θ2‖f‖1−θp+1

‖f‖2p≥‖∇g‖θ2‖g‖1−θ

‖g‖2p

g(x) =

1 + |x|2

)1/(p−1)

θ =d(p− 1)

p(d + 2− (d − 2)p).

Moreover, there is equality if and only if for some x0 ∈ Rd and s ∈ R+, f is anon-zero multiple of g(x/s− x0).

aDelPino & Dolbeault, JMPA, 2002.

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Two Inequalities, One EquationThe Fast Diffusion Equation

The equality cases are steady states for nonlinear Fokker-Planck equations related tothe fast diffusion equation:

∂tu(x, t) = ∆um(x, t) .

u(x, t) solves the fast diffusion equation if and only if

v(x, t) := etdu(etx, eαt)

with α = 2− d(1− m) satisfies

∂tv(x, t) = ∆vm(x, t) +∇ · [xv(x, t)] .

v∞(x) :=

(1− m)

)1/(1−m)( 11 + |x|2

)1/(1−m)

is a stationary solution with certain mass M.

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∂tu(x, t) = ∆um(x, t) .

∂tv(x, t) = ∆vm(x, t) +∇ · [xv(x, t)] .

v∞(x) :=

(1− m)

)1/(1−m)( 11 + |x|2

)1/(1−m)

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∂tu(x, t) = ∆um(x, t) .

∂tv(x, t) = ∆vm(x, t) +∇ · [xv(x, t)] .

v∞(x) :=

(1− m)

)1/(1−m)( 11 + |x|2

)1/(1−m)

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The connection

The relation between the HLS optimizers, the GNS optimizers and the FDequation is more than a superficial coincidence, at least for certain λ, p and m.

For d ≥ 3, the λ = d − 2 case of the sharp HLS inequality can be proved byusing the fast diffusion flow for m = d/(d + 2) to deform any reasonably nicetrial function into a multiple of h.

Motivation: We choose m for the fast diffusion equation and λ for the HLS inequalityin order to have h as stationary solution/equality case. Thus,

11− m

=d + 2

2⇔ m =

dd + 2

The exact value of the exponent for the GNS inequality

1 < p =d + 1d − 1

d − 2

will be dictated by the evolution of the HLS functional along the flow of the FDequation. Note p is in the correct range of the sharp GNS inequalities.

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The connection

11− m

=d + 2

2⇔ m =

dd + 2

1 < p =d + 1d − 1

d − 2

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The connection

11− m

=d + 2

2⇔ m =

dd + 2

1 < p =d + 1d − 1

d − 2

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The connection

11− m

=d + 2

2⇔ m =

dd + 2

1 < p =d + 1d − 1

d − 2

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The proofThe HLS functional

For λ = d − 2, the HLS inequality can be rewritten as F [f ] ≥ 0 for all f ∈ L2d/(d+2)

F [f ] := CS

(∫Rd

f 2d/(d+2)(x)dx)(d+2)/d

−∫Rd

f (x)[(−∆)−1f

](x) dx ,

withCS :=

4d(d − 2)

|Sd|−2/d .

The GNS inequality

The GNS inequality with p = (d + 1)/(d − 1) can be written as D[g] ≥ 0 for all gwith a square integrable distributional gradient on Rd where

D[g] := CSd(d − 2)

(d − 1)2

(∫Rd

g2d/(d−1) dx)2/d ∫

Rd|∇g|2 dx−

∫g(2d+2)/(d−1) dx .

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The proofThe HLS functional

For λ = d − 2, the HLS inequality can be rewritten as F [f ] ≥ 0 for all f ∈ L2d/(d+2)

F [f ] := CS

(∫Rd

f 2d/(d+2)(x)dx)(d+2)/d

−∫Rd

f (x)[(−∆)−1f

](x) dx ,

withCS :=

4d(d − 2)

|Sd|−2/d .

The GNS inequality

The GNS inequality with p = (d + 1)/(d − 1) can be written as D[g] ≥ 0 for all gwith a square integrable distributional gradient on Rd where

D[g] := CSd(d − 2)

(d − 1)2

(∫Rd

g2d/(d−1) dx)2/d ∫

Rd|∇g|2 dx−

∫g(2d+2)/(d−1) dx .

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The proofDissipation of the HLS functional by the FD equation

Let f ∈ L1(Rd) ∩ L1(R2d/(d+2)) be non-negative, and suppose that∫Rd

f (x) dx =

h(x) dx = M

Let u(x, t) be the solution of the FD equation with u(x, 1) = f (x). Then, for allt > 1,

ddtF [u(·, t)] = −2D[u(d−1)/(d+2)(·, t)] .

HLS inequality via FD flow

f (x) dx =

h(x) dx = M and sup|x|>R

f (x)|x|2/(1−m) <∞ .

Then F [f ] ≥ 0.

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The proofDissipation of the HLS functional by the FD equation

f (x) dx =

h(x) dx = M

Let u(x, t) be the solution of the FD equation with u(x, 1) = f (x). Then, for allt > 1,

ddtF [u(·, t)] = −2D[u(d−1)/(d+2)(·, t)] .

HLS inequality via FD flow

f (x) dx =

h(x) dx = M and sup|x|>R

f (x)|x|2/(1−m) <∞ .

Then F [f ] ≥ 0.

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Conclusions

A new Liapunov functional unveiled for the critical mass PKS: a bassin ofattraction determined for each stationary state.

Optimal Transportation Tools are crucially used to construct the solutions.

A technique of Concentration Controlled inequalities developed to cope withthe lack of compactness.

The duality between the PKS in the critical mass case and the critical nonlinearfast diffusion equation in 2D leads to the discovery of new proofs of log-HLSand HLS inequalities.

References:

1 Blanchet-Carlen-C. (to appear in J. Funct. Analysis).2 Carlen-C.-Loss (PNAS 2010).3 Carlen-Figalli (preprint).

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Conclusions

References:

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Conclusions

References:

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Conclusions

References:

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Conclusions

References:

· icreauab macroscopic models: pks system ideas of the rigorous proof a byproduct: a new proof of...

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