Large time asymptotics for evolution equations with mean field couplings

with mean field couplings

Jean Dolbeault

http://www.ceremade.dauphine.fr/∼dolbeaul Ceremade, Université Paris-Dauphine

March 25, 2021

CIRM conference on Kinetic Equations:

From Modeling Computation to Analysis March 22-26, 2021

Outline

Nonlinear diffusion equations Bthe fast diffusion equation

BThe subcritical Keller-Segel model BA simple mean-field model

Vlasov-Fokker-Planck models BHypocoercivity methods

BHypocoercivity for the linear Vlasov-Fokker-Planck equation BThe Vlasov-Poisson-Fokker-Planck system

Nonlinear diffusion equations

BFast diffusion equation

BThe subcritical Keller-Segel model BA simple mean-field model

The fast diffusion equation equation

(Blanchet, Bonforte, JD, Gillo, Vázquez) (Bonforte, JD, Nazaret, Simonov)

The fast diffusion equation

Consider thefast diffusionequation in R^d,d≥1,m <1

∂u

∂t = ∆u^m, u_|t=0=u0≥0 (FDE) Withp=_2m−1¹ ,u=f^2p

d dt

Z

R^d

u dx

| {z }

=kfk^2p_2p

= 0, d dt

Z

R^d

u^mdx

| {z }

=kfk^p+1_p+1

= (p+ 1)² Z

R^d

|∇f|²dx

Gagliardo-Nirenberg-Sobolev inequalities

k∇fk^θ₂ kfk^1−θ_p+1≥CGNS(p)kfk_2p (GNS) t→+∞asymptotics: u(t, x)∼B(t, x) =t^−d/µg(t^−1/µx)^2p

B Barenblatt self-similar solutions,µ= 2−d(1−m)>1 g(x) = 1 +|x|²⁻p−1¹ Aubin-Talenti type function

Self-similar variables, entropy-entropy production inequality

Inself-similar variables(FDE) becomesa Fokker-Planck type equation

∂v

∂t +∇ ·

v ∇v^m−1− 2x

= 0 (1)

with (GNS)⇐⇒ I[v]≥4F[v] and _dt^dF[v] =−I[v]

Generalized entropy (free energy)andFisher information F[v] :=

Z

R^d

B^m−1(v−B)−^vm−_m^Bm

dx , I[v] :=

Z

R^d

v

∇v^m−1+ 2x

2 dx

whereB(x) =g^2p(x) = 1 +|x|²−_1−m¹

(with appropriate normalizations)

Linearized entropy-entropy production inequality

(BBDGV)... Linearization: Letvε=B 1 +εB^1−mf I[vε]

| {z }

∼ε²R

Rd|∇f|²Bdx

≥ 4 F[vε]

| {z }

∼ε²R

Rd|f|²B^2−mdx

Hardy–Poincaré inequality: withB^2−m= _1+|x|^B 2

Λm,d

Z

R^d

f²B^2−mdx≤ Z

R^d

|∇f|²Bdx ∀f ∈H¹(Bdx), Z

R^d

fB^2−mdx= 0 asymptotic decay rates = rates of the linearized FDE equation

0 =∂_tv+∇ ·

v∇ v^m−1−B^m−1

∼εB^2−m

∂_tf −(1−m)B^m−2∇ · B∇f same rate in the nonlinear regime (Bakry-Emery)

much more (stability results)... but the difficulty lies in the justification of the Taylor expansion

The subcritical Keller-Segel model

(Campos, JD) (Dávila, JD, del Pino, Musso, Wei)

The subcritical Keller-Segel model

M =R

R²n₀dx≤8π: global existence (W. Jäger, S. Luckhaus 1992), (JD, B. Perthame 2004)

Ifusolves

∂u

∂t =∇ ·[u(∇(logu)− ∇v)]

thefree energy

F[u] :=

Z

R²

ulogu dx−1 2

Z

R²

u v dx satisfies

d

dtF[u(t,·)] =− Z

R²

u|∇(logu)− ∇v|²dx Thelogarithmic HLS inequality(E. Carlen, M. Loss 1992)

F is bounded from below if and only ifM ≤8π

Time-dependent rescaling

u(x, t) = 1 R²(t)n

x R(t), τ(t)

and v(x, t) =c x

R(t), τ(t)

withR(t) =√

1 + 2tand τ(t) = logR(t)











∂n

∂t = ∆n− ∇ ·(n(∇c−x)) x∈R², t >0 c=− 1

2π log| · | ∗n x∈R², t >0 n(·, t= 0) =n0≥0 x∈R² (A. Blanchet, JD, B. Perthame 2006)

The convergence in self-similar variables

t→∞lim kn(·,·+t)−n_∞k_L1(R^d)= 0 and lim

t→∞k∇c(·,·+t)− ∇c∞k_L2(R^d)= 0 meansintermediate asymptoticsin original variables:

ku(x, t)−_R2¹(t)n_∞

x

R(t), τ(t)

k_L1(R²)&0

The stationary solution in self-similar variables

n_∞=M e^{c∞−|x|2/2} R

R²e^{c∞−|x|2/2}dx =−∆c_∞, c_∞=− 1

2πlog| · | ∗n_∞

0.5 1.0 1.5 2.0

2 4 6

8 fer⁾

ML8K

Mike

r-1×1

Linearization

We can introduce two functionsf andgsuch that

n=n_∞(1 +f) and c=c_∞(1 +g) = (−∆)⁻¹n and rewrite the Keller-Segel model as

∂f

∂t =Lf + 1

n_∞∇(f n∞∇(c∞g)) where the linearized operator is

Lf = 1

n_∞∇ · n_∞∇(f−c_∞g) and

−∆(c_∞g) =n_∞f

Spectrum of L (lowest eigenvalues only)

5 10 15 20 25

1 2 3 4 5 6

7 7,eigonvalues^ofL

M _8rx25

Figure:The lowest eigenvalues of−L=(−∆)⁻¹(n f)

Functional setting...

Lemma (A. Blanchet, JD, B. Perthame)

Sub-critical HLS inequality(A. Blanchet, JD, B. Perthame) F[n] :=

Z

R²

nlog n

n∞

dx−1

2 Z

R²

(n−n_∞) (c−c_∞)dx≥0 achieves its minimum forn=n_∞

Q1[f] = lim

ε→0

1

ε²F[n_∞(1 +ε f)]≥0 ifR

R²f n∞dx= 0. Notice thatf0generates the kernel ofQ1

Lemma (J. Campos, JD)

Poincaré type inequality. For anyf ∈H¹(R², n_∞dx)such that R

R²f n_∞dx= 0, we have Z

R²

|∇(−∆)⁻¹(f n_∞)|²n_∞dx= Z

R²

|∇(g c_∞)|²n_∞dx≤ Z

R²

|f|²n_∞dx

... and eigenvalues

Withg such that−∆(g c∞) =f n_∞,Q₁ determines a scalar product hf1, f2i:=

Z

R²

f1f2n_∞dx− Z

R²

f1n_∞(g2c_∞)dx on the orthogonal space tof₀ inL²(n_∞dx)

Q2[f] :=

Z

R²

|∇(f −g c∞)|²n∞dx with g=− 1 c_∞

1

2π log|·|∗(f n∞) is a positive quadratic form, whose polar operator is theself-adjoint operatorL

hf,Lfi=Q2[f] ∀f ∈D(L2) Lemma (J. Campos, JD)

Lhas pure discrete spectrum and its lowest eigenvalue is1

A simple Cucker-Smale mean-field model

(Xingyu Li)

A simple version of the Cucker-Smale model

(J. Tugaut, 2014),(A. Barbaro, J. Cañizo, J.A. Carrillo, and P. Degond, 2016),(X. Li)

A model for bird flocking (simplified version)

∂f

∂t =D∆_vf +∇_v·(∇_vφ(v)f−u_ff)

-1.5 -1.0 -0.5 0.5 1.0 1.5

-0.5 0.5 1.0

1.5 4=0

-1.5 -1.0 -0.5 0.5 1.0 1.5 2.0

-0.5 0.5 1.0

1.5 µ 0

whereuf =R

v f dv is the average velocity andφ(v) =¹₄|v|⁴−¹₂|v|²

0.1 0.2 0.3 0.4

-1.0 -0.5 0.5

1.0 U O

u = 0

¥ ^D

U

Relative entropy and related quantities

d

dtFu[f(t,·)] =−I[f]

Relative entropy with respect to a stationary solutionfu

Fu[f] =D Z

R^d

f log f

fu

dv−1

2|uf−u|² Relative Fisher information

I[f] :=

Z

R^d

D∇f

f +α v|v|²+ (1−α)v−uf

2

f dv Non-equilibrium Gibbs state

G_f(v) := e^−D1 (¹2|v−uf|²+α 4|v|⁴−α

2|v|²) R

R^de^−D1 (¹₂^|v−uf|²+α 4|v|⁴−α

2|v|²)dv

Stability and coercivity

(X. Li)

Q_1,u[g] := lim

ε→0

2 ε²F

f_u(1 +ε g)

=D Z

R^d

g²f_udv−D²|vg|² wherevg:= _D¹ R

R^dv g fudv Q2,u[g] := lim

ε→0

1 ε²I

fu(1 +ε g)

=D² Z

R^d

|∇g−vg|² fudv Stability: Q1,u≥0?

Coercivity: Q2,u≥λ Q1,u for someλ >0 ?

Q2,u[g]≥CD 1−κ(D) (vg·u)²

|vg|²|u|²Q1,u[g]

κ(D)<1 and as a special case, if u=u[f], then Q2,u[g]≥CD 1−κ(D)

Q1,u[g]

An exponential rate of convergence for partially symmetric solutions in the polarized case

Proposition (X. Li)

Letα >0,D >0 and consider a solutionf ∈C⁰ R⁺,L¹(R^d) with initial datumfin∈L¹₊(R^d) such thatF[fin]<F[f0]and

uf_in= (u,0. . .0)for some u6= 0. We further assume that

fin(v1, v2, . . . vi−1, vi, . . .) =fin(v1, v2, . . . vi−1,−vi, . . .)for any i= 2, 3,. . .d. Then

0≤F[f(t,·)]−F[f_u]≤C e^{−λ t} ∀t≥0 holds withλ=CD 1−κ(D)

>0

The Vlasov-Poisson-Fokker-Planck system

BHypocoercivity methods

BLinearized Vlasov-Poisson-Fokker-Planck system BA result in the non-linear case, d= 1

The Vlasov-Poisson-Fokker-Planck system:

linearization and hypocoercivity

(JD, Mouhot, Schmeiser, 2015)

(Bouin, JD, Mischler, Mouhot, Schmeiser,2020)Hypocoercivity without confinement

(Arnold, JD, Schmeiser, Wöhrer)Sharpening of decay rates in Fourier based hypocoercivity methods

(Addala, JD, Li, Tayeb)L²-Hypocoercivity and large time asymptotics of the linearized Vlasov-Poisson-Fokker-Planck system.

Preprint hal-02299535 and arxiv: 1909.12762

Hypocoercivity methods

H

-hypocoercivity: an example

∂f

∂t +Tf = ∆vf +∇v·(v f), Tf :=v· ∇xf−x· ∇vf (JD, X. Li)take h= (f /f_∗)^2/p,p∈(1,2)

∂h

∂t +Th=Lh+2−p p

|∇vh|²

h , Lh:= ∆vh−v· ∇vh Twisted Fisher information

Jλ[h] = (1−λ)R

R^d|∇_vh|²dµ+(1−λ)R

R^d|∇_xh|²dµ+λR

R^d|∇_xh+∇_vh|²dµ Theorem (JD, Li)

For an appropriate choice oft7→λ(t), there is a functiont7→ρ(t)>1 a.e.

d

dtJλ(t)[h(t,·)]≤ −ρ(t)Jλ(t)[h(t,·)] ∀t≥0 andJλ(t)[h(t,·)]≤J1/2[h₀] exp

−Rt

0ρ(s)ds

L

-hypocoercivity: the strategy

(JD, Mouhot, Schmeiser)Π is the orthogonal projection on Ker(L) εdF

dt +TF = 1 εLF

Fε=F0+ε F1+ε²F2+O(ε³) asε→0+,u=F0= ΠF0

∂tu+ (TΠ)^∗(TΠ)u= 0

BMain assumption: macroscopic coercivity(Poincaré inequality) kTΠFk²≥λMkΠFk²

ε= 1: the estimate ¹_{2 dt}^dkFk²=hLF, Fi ≤ −λ_mk(1−Π)Fk²is not enough to conclude thatkF(t,·)k²decays exponentially

The operatorA:= 1 + (TΠ)^∗TΠ−1

(TΠ)^∗ is such that hATΠF, Fi ≥ λM

1 +λM

kΠFk² and we can use the L² entropy / Lyapunov functional

H[F] := ¹₂kFk²+δRehAF, Fi

Linearized Vlasov-Poisson-Fokker-Planck system

In collaboration with Lanoir Addala, Xingyu Li and Lazhar M. Tayeb L²-Hypocoercivity and large time asymptotics of the linearized Vlasov-Poisson-Fokker-Planck system. Preprint hal-02299535 and arxiv: 1909.12762

(Hérau, Thomann, 2016),(Herda, Rodrigues, 2018)

Linearized Vlasov-Poisson-Fokker-Planck system

TheVlasov-Poisson-Fokker-Planck systemin presence of an external potentialV is

∂_tf+v· ∇xf−(∇xV +∇xφ)· ∇vf = ∆vf +∇v·(v f)

−∆_xφ=ρ_f = Z

R^d

f dv (VPFP)

Linearized problem aroundf?: f =f?(1 +η h),RR

R^d×R^dh f?dx dv= 0

∂th+v· ∇xh−(∇xV +∇xφ?)· ∇vh+v· ∇xψh−∆vh+v· ∇vh=η∇xψh· ∇vh

−∆xψ_h= Z

R^d

h f_?dv

Drop theO(η) term :linearized Vlasov-Poisson-Fokker-Plancksystem

∂_th+v· ∇xh−(∇xV +∇xφ_?)· ∇vh+v· ∇xψ_h−∆vh+v· ∇vh= 0

−∆_xψ_h= Z

R^d

h f_?dv , Z Z

R^d×R^d

h f_?dx dv= 0

(VPFPlin)

Hypocoercivity

Let us define the norm khk²:=

Z Z

R^d×R^d

h²f?dx dv+ Z

R^d

|∇xψh|²dx

Theorem

Let us assume thatd≥1,V(x) =|x|^α for someα >1 andM >0.

Then there exist two positive constantsCandλsuch that any

solutionhof (VPFPlin)with an initial datumh0 of zero average with kh₀k²<∞ is such that

kh(t,·,·)k²≤Ckh0k² e^{−λ t} ∀t≥0

Diffusion limit

Linearized problem in the parabolic scaling

ε ∂_th+v· ∇xh−(∇xV +∇xφ_?)· ∇vh+v· ∇xψ_h−1

ε ∆_vh−v· ∇vh

= 0

−∆xψh= Z

R^d

h f?dv , Z Z

R^d×R^d

h f?dx dv= 0

(VPFPscal) Expandhε=h0+ε h1+ε²h2+O(ε³) asε→0+. WithW?=V +φ?

ε⁻¹: ∆vh0−v· ∇vh0= 0

ε⁰: v· ∇xh0− ∇xW?· ∇vh0+v· ∇xψh₀ = ∆vh1−v· ∇vh1

ε¹: ∂th0+v· ∇xh1− ∇xW?· ∇vh1= ∆vh2−v· ∇vh2

Withu= Πh0,−∆ψ=u ρ?,w=u+ψ,

u=h0, v· ∇xw= ∆vh1−v· ∇vh1

from which we deduce thath₁=−v· ∇_xwand

∂tu−∆w+∇xW?· ∇u= 0

Rates of convergence

Results in the diffusion limit / in the non-linear case

Theorem

Let us assume thatd≥1,V(x) =|x|^α for someα >1 andM >0.

For anyε >0 small enough, there exist two positive constantsC andλ, which do not depend onε, such that any solution h of (VPFPscal)with an initial datum h0 of zero average satisfies

kh(t,·,·)k²≤Ckh0k² e^{−λ t} ∀t≥0 Corollary

Assume thatd= 1,V(x) =|x|^α for someα >1andM >0. Iff solves (VPFP)with initial datumf0= (1 +h0)f? such that h0 has zero average,kh0k²<∞and(1 +h0)≥0, then

kh(t,·,·)k²≤Ckh₀k² e^{−λ t} ∀t≥0 holds withh=f /f?−1for some positive constants Candλ

These slides can be found at

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

BLectures The papers can be found at

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

BPreprints / papers

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