On Boltzmann’s entropy

Jean Dolbeault

http://www.ceremade.dauphine.fr/∼dolbeaul Ceremade, Université Paris-Dauphine First Belgium-Chile-Italy conference in PDEs

Nov. 16, 2017

A short historical introduction

to entropy methods in PDEs

On the notion of entropy

In physics... the notion of entropy goes back to the 19^thcentury:

Gibbs, Clausius, Maxwell, Boltzmann BThermodynamics: the steam engine

BBoltzmann: how irreversibility arises in large systems Fundamental problems in Mathematics

BLinked to the 6^thHilbert problem Information theory

BShannon, Rényi,...

BWhat von Neumann said to Shannon: When Shannon first derived his famous formula for information, he asked von Neumann what he should call it and von Neumann replied:

“You should call it entropy for two reasons: first because that is what the formula is in statistical mechanics but second and more important, as nobody knows what entropy is, whenever you use the term you will always be at an advantage!”

(to be continued)

On Boltzmann’s entropy

Boltzmann’s equation describes the evolution of a gas of particles by

∂f

∂t +v· ∇xf =Q(f, f) tis the time,xthe position andv the velocity

f(t, x, v) is thedistribution function(a probability density on the phase space). In a collision, ifv andv_∗ are theincomingvelocities andv⁰ andv⁰_∗ theoutgoingvelocities,f_∗=f(t, x, v_∗), etc., then

Q(f, f) = Z Z

R³×S²

σ(v−v_∗, ω) f⁰f_∗⁰ −f f_∗ dv_∗dω

is the collision kernel

The cross-sectionσis nonnegative and has symmetry properties

Boltzmann’s H theorem

Boltzmann’s entropy H =

Z Z

R³×R³

f logf dx dv Boltzmann’s H theorem

dH dt =−1

4 Z

(R³)³×S²

σ(v−v_∗, ω)· f⁰f_∗⁰−f f_∗ log

f⁰f_∗⁰ f f∗

dv_∗dv dx dω

BCarleman (<1949): first mathematical theory BDiPerna and Lions (1989): renormalized solutions

BCercignani, Illner, Pulvirenti (1994): derivation of the equation Can we compute a rate of convergence to an equilibrium usingH ?

On the notion of entropy (continued)

Many other domains of application:

linear diffusions, Markov processes, semi-group theory... a long story ! Bakry & Emery (1984): the carré du champ method PDEs and “entropy methods”: around 1998, Toscani et al., del Pino & JD (1999): the entropy for fast diffusion equations (a nonlinear case)

and also (not discussed here):

Hyperbolic conservation laws

Sinai’s entropy for measure-preserving dynamical system topological entropy, Perelman’s entropy in differential geometry etc.

Outline of the lecture

Entropy methods and diffusion equations

Bϕ-entropies and thecarré du champ method of Bakry & Emery Bthe gradient flow point of view

Brigidity results and entropy methods on compact manifolds BRényi entropy powers on the Euclidean space

Bweighted inequalities and results of symmetry

Bother applications: the (Patlak)-Keller-Segel in mathematical biology and the Oseen attractor in 2D Euler equations

Hypocoercivity in kinetic equations BH¹ methods andϕ-hypocoercivity BL²-hypocoercivity

Entropy methods and

diffusion equations

ϕ-entropies: definition

Theϕ-entropyof a nonnegative functionw∈L¹(R^d, dγ) is E[w] :=

Z

R^d

ϕ(w)dγ

whereϕ:R⁺→R⁺ is such that

ϕ⁰⁰≥0, ϕ≥ϕ(1) = 0 and (1/ϕ⁰⁰)⁰⁰≤0 A classical example of a such a functionϕis given by

ϕp(w) := _p−1¹ w^p−1−p(w−1)

p∈(1,2]

BCasep= 2: ϕ2(w) = (w−1)²

BLimit case asp→1+ : ϕ1(w) :=wlogw−(w−1)

dγis a probability measure, which is absolutely continuous with respect to Lebesgue’s measure

dγ=e^−ψdx

ϕ-entropies and diffusions

Ifusolves theFokker-Planck equation

∂u

∂t = ∆u+∇_x·(u∇_xψ).

thenw=u e^ψ solves theOrnstein-Uhlenbeckorbackward Kolmogorov equation

∂w

∂t =Lw:= ∆w− ∇ψ· ∇w

The Ornstein-Uhlenbeck operatorLon L²(R^d, dγ) is such that

− Z

R^d

(Lw1)w2dγ= Z

R^d

∇w1· ∇w2dγ ∀w1, w2∈H¹(R^d, dγ) Bthe mass is conserved: R

R^dw(t,·)dγ= 1, lim_t→+∞w(t,·) = 1 Btheϕ-entropy decays

d

dtE[w] =− Z

R^d

ϕ⁰⁰(w)|∇xw|²dγ=:−I[w]

whereI[w] denotes theϕ-Fisher informationfunctional

ϕ-entropies: entropy – entropy production inequalities

If for some Λ>0 theentropy – entropy production inequality I[w]≥ΛE[w] ∀w∈H¹(R^d, dγ)

holds, then

E[w(t,·)]≤E[w0]e^−Λt ∀t≥0 Example: withe^−ψ= (2π)^−d/2e^−|x|2/2 asnϕ=ϕp

Bp= 2, Gaussian Poincaré inequality : Λ = 1 f −f¯

2

L²(R^d,dγ)≤ Z

R^d

|∇f|²dγ ∀f ∈H¹(R^d, dγ), f¯= Z

R^d

f dγ

Bp= 1, Logarithmic Sobolev inequality : Λ = 2 Z

R^d

f²log f² kfk²_L2(R^d,dγ)

! dγ≤2

Z

R^d

|∇f|²dγ ∀f ∈H¹(R^d, dγ)

ϕ-entropies: key properties

Generalized Csiszár-Kullback-Pinsker inequality:

ifA:= inf_s∈(0,∞)s^2−pϕ⁰⁰(s)>0, then E[w]≥2^−2pAminn

1,kwk^p−2_Lp(R^d,dγ)

okw−1k²_Lp(R^d,dγ)

Sub-additivity

Eγ1⊗γ2[w]≤ Z

R^d2

Eγ1[w]dγ₂+ Z

R^d1

Eγ2[w]dγ₁ Tensorization

Iγ₁⊗γ2[w] = Z

R^d1×R^d2

ϕ⁰⁰(w)|∇w|²dγ1dγ2≥min{Λ1,Λ2}Eγ₁⊗γ2[w]

Generalized Holley-Stroock perturbation lemma:

ife^−bdγ≤dµ≤e^−adγandwe:=R

R^dw dµ /R

R^ddµ, then e^a−bΛ

Z

R^d

ϕ(w)−ϕ(w)e −ϕ⁰(w)(we −w)e dµ≤

Z

R^d

ϕ⁰⁰(w)|∇w|²dµ

ϕ-entropies: the carré du champ method

(Bakry, Emery, 1984): compute the t-derivative of Fisher On a convex domain Ω, withw=z^2/p so thatI[w] =R

Ω|∇z|²dγ 1

2 d

dtI[w] = −2 p(p−1)

Z

Ω

Hessz

2dγ− Z

Ω

Hessψ:∇z⊗ ∇z dγ

−2−p p

Z

Ω

Hessz−∇z⊗ ∇z z

2

dγ

+ Z

∂Ω

Hessz:∇z⊗ν e^−ψdσ

≤ −I[w]

Key observations: [∇, L] =−Hessψ... ifψ(x) =|x|²/2 Z

Ω

Hessψ:∇z⊗ ∇z dγ= Z

Ω

|∇z|²dγ=I[w]

ϕ-entropies: a statement

Letp∈[1,2] and assume that for anyX ∈H¹(R^d, dγ)^d 2

p(p−1) Z

R^d

|∇X|²dγ+ Z

R^d

Hessψ:X⊗X dγ≥Λ(p) Z

R^d

|X|²dγ

Theorem

Assume thatq∈[1,2). IfΛ = Λ(2/q)>0, then kfk²_L2(R^d,dγ)− kfk²_Lq(R^d,dγ)

2−q ≤ 1

Λ Z

R^d

|∇f|²dγ ∀f ∈H¹(R^d, dγ)

ϕ-entropies: improved inequalities

Remainder terms: withκp= (p−1) (2−p)/p d

dtI[w] + 2I[w]≤ −κ_p I[w]² 1 + (p−1)E[w]

Lete(t) := _p−1¹ R

R^df²dγ−1

wheref =w^p/2 e⁰⁰+ 2e⁰ ≥ κp|e⁰|²

1 + (p−1)e ≥κp|e⁰|² 1 +e Proposition

Assume thatq∈(1,2) anddγ= (2π)^−d/2e^−|x|2/2dx WithF(s) := _1−κ¹

p

1 +s−(1 +s)^κp

, for anyf ∈H¹(R^d, dγ)such thatkfk_Lq(R^d,dγ)= 1

1

qF qkfk²_L2(R^d,dγ)−1 2−q

!

≤ k∇fk²_L2(R^d,dγ)

Bakry-Emery

Photo: Nassif Ghoussoub

ϕ-entropies: a summary

BTo prove the decay of the entropy, we need an entropy – entropy production inequality ΛE≤I

BThe best constant in the entropy – entropy production inequality determines the (exponential) rate of decay: E(t)≤E(O)e^−Λt

BBy differentiating this estimate att= 0, we see that an exponential rate of decay is equivalent to an entropy – entropy production

inequality: −I(0) = _dt^dE(0)≤ −ΛE(0)

BThecarré du champmethod: prove that _dt^d (I(t)−ΛE(t))≤0 BWith an improved inequality ΛF(E)≤I whereF⁰⁰>0,F(0) = 0 andF⁰(0) = 0, optimality in the entropy – entropy production inequality can be achieved only in the asymptotic regime and Λ is given by a spectral gap of a linearized problem

BThe Fokker-Planck equation can be seen as a gradient flow of the ϕ-entropy under an appropriate notion of distance

A (short) review of applications to nonlinear equations

BNonlinear interpolation inequalities

BRigidity results for nonlinear elliptic equations BMonotonicity along nonlinear flows

BSymmetry results in weighted inequalities BOther applications

Background references (partial)

Rigidity methods, uniqueness in nonlinear elliptic PDE’s: (B. Gidas, J. Spruck, 1981),(M.-F. Bidaut-Véron, L. Véron, 1991)

Probabilistic methods (Markov processes), semi-group theory and carré du champmethods (Γ₂theory): (D. Bakry, M. Emery, 1984), (Bentaleb),(Bakry, Ledoux, 1996),(Demange, 2008),(JD, Esteban, Kolwalczyk, Loss, 2014 & 2015)→D. Bakry, I. Gentil, and M.

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

BEntropy-entropy production inequalities: Arnold, Carrillo,

Desvillettes, JD, Jüngel, Lederman, Markowich, Toscani, Unterreiter, Villani...,(del Pino, JD, 2001),(Blanchet, Bonforte, JD, Grillo, Vázquez)→A. Jüngel, Entropy Methods for Diffusive Partial Differential Equations (2016)

BMass transportation: (Otto) →C. Villani, Optimal transport. Old and new (2009)

BRényi entropy powers (information theory)(Savaré, Toscani, 2014), (Dolbeault, Toscani)

Collaborations

Collaboration with...

M.J. Esteban and M. Loss(symmetry, critical case) M.J. Esteban, M. Loss and M. Muratori(symmetry, subcritical case) M. Bonforte, M. Muratori and B. Nazaret(linearization and large time asymptotics for the evolution problem) M. del Pino, G. Toscani (nonlinear flows and entropy methods) A. Blanchet, G. Grillo, J.L. Vázquez (large time asymptotics and linearization for the evolution equations) ...and also

S. Filippas, A. Tertikas, G. Tarantello, M. Kowalczyk ...

Rigidity: the Bakry-Emery method on S

Entropy functional Ep[ρ] := _p−2¹ h

R

S^dρ^2p dµ− R

S^dρ dµ²_pi

if p6= 2 E2[ρ] :=R

S^dρlog _ρ

kρk_{L1 (S}d)

dµ

Fisher informationfunctional Ip[ρ] :=R

S^d|∇ρ^1p|²dµ

Bakry-Emery (carré du champ) method: use the heat flow

∂ρ

∂t = ∆ρ

and compute _dt^dEp[ρ] =−Ip[ρ] and _dt^dIp[ρ]≤ −dIp[ρ] to get d

dt(Ip[ρ]−dEp[ρ])≤0 =⇒ Ip[ρ]≥dEp[ρ]

withρ=|u|^p, ifp≤2^#:=_(d−1)^2d2+12

Rigidity: the fast diffusion flow on S

To overcome the limitationp≤2^#, one can consider a nonlinear diffusion of fast diffusion / porous medium type

∂ρ

∂t = ∆ρ^m. (1)

(Demange),(JD, Esteban, Kowalczyk, Loss): for anyp∈[1,2^∗] Kp[ρ] := d

dt

Ip[ρ]−dEp[ρ]

≤0

1.0 1.5 2.5 3.0

0.0 0.5 1.5 2.0

(p, m)admissible region,d= 5

Rigidity: the functional inequality

The entropy – entropy production method establishes the interpolation inequality

k∇uk²_L2(S^d)+ d

p−2kuk²_L2(S^d)≥ d

p−2kuk²_Lp(S^d) ∀u∈H¹(S^d, dµ) wheredµis the uniform probability measure on S^d andp≥1,p6= 2 andp≤2^∗ :=_d−2^2d ifd≥3

The casep= 2 corresponds to the logarithmic Sobolev inequality k∇uk²_L2(S^d)≥d

2 Z

S^d

|u|² log |u|² kuk²_L2(

S^d)

!

dµ ∀u∈H¹(S^d, dµ)\ {0}

(Beckner, 1993)

(Bidaut-Véron, Véron, 1991) (JD, Esteban, Kowalczyk, Loss)

Rényi entropy powers: the fast diffusion equation on R

Consider the nonlinear diffusion equation inR^d,d≥1

∂v

∂t = ∆v^m

with initial datumv(x, t= 0) =v0(x)≥0 such that R

R^dv0dx= 1 and R

R^d|x|²v0dx <+∞. The large time behavior of the solutions is governed by the source-type Barenblatt solutions

U?(t, x) := 1 κ t^1/µd B?

x κ t^1/µ

where

µ:= 2 +d(m−1), κ:=

2µ m m−1

1/µ

andB? is the Barenblatt profile B?(x) :=







C?− |x|^21/(m−1)

+ ifm >1 C?+|x|²1/(m−1)

ifm <1

The Rényi entropy power F

Theentropy is defined by E:=

Z

R^d

v^mdx

and theFisher informationby I:=

Z

R^d

v|∇p|²dx with p= m

m−1v^m−1 Ifv solves the fast diffusion equation, then

E⁰= (1−m)I To computeI⁰, we will use the fact that

∂p

∂t = (m−1)p∆p+|∇p|² F:=E^σ with σ= µ

d(1−m) = 1+ 2 1−m

1

d+m−1

= 2 d

1 1−m−1 has a linear growth asymptotically ast→+∞

Rényi entropy power and Fisher information

Lemma

Ifv solves ^∂v_∂t = ∆v^m with ¹_d ≤m <1, then I⁰= d

dt Z

R^d

v|∇p|²dx=−2 Z

R^d

v^m

kD²pk²+ (m−1) (∆p)² dx

Explicit arithmetic geometric inequality kD²pk²−1

d(∆p)²=

D²p−1 d∆pId

2

Critical case: ifm= 1−¹_d: F⁰=Iand the inequalityI≥I_?=:I[B?] is Sobolev’s inequality

Rényi entropy power: the subcritical case

Theorem

(Toscani-Savaré)Assume that m≥1−_d¹ ifd >1 andm >0 ifd= 1.

Then(1−m)F⁰⁰(t)≤0

(Dolbeault-Toscani)The inequality

E^σ−1I≥E^σ−1_? I_?

is equivalent to the Gagliardo-Nirenberg inequality k∇wk^θ_L2(R^d)kwk^1−θ_Lq+1(

R^d)≥C_GNkwk_L2q(R^d)

if 1−¹_d ≤m <1

Weighted inequalities and results of

symmetry

Symmetry: critical Caffarelli-Kohn-Nirenberg inequality

LetDa,b:=n

v∈L^p R^d,|x|^−bdx

: |x|^−a|∇v| ∈L² R^d, dxo Z

R^d

|v|^p

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

≤ Ca,b

Z

R^d

|∇v|²

|x|^2a dx ∀v∈Da,b

holds under conditions onaandb:

a < ac= (d−2)/2,a≤b≤a+ 1 ifd≥3

p= 2d

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

v?(x) =

1 +|x|^{(p−2) (ac−a)}−_p−2²

and C^?_a,b= k |x|^−bv?k²_p k |x|^−a∇v?k²₂ Question: Ca,b=C^?_a,b (symmetry) or Ca,b>C^?_a,b (symmetry breaking) ?

Symmetry: the sharp result in the critical case

The Felli & Schneider curve bFS(a) := d(ac−a)

2p

(ac−a)²+d−1+a−ac

a b

0

(JD, Esteban, Loss, 2016)

Theorem

Letd≥2 andp <2^∗. If eithera∈[0, ac)andb >0, ora <0 and b≥bFS(a), then the optimal functions for the critical

Caffarelli-Kohn-Nirenberg inequalities are radially symmetric

Symmetry: an approach based on Rényi entropy powers

We compute the derivative of the generalizedRényi entropy power functional

d dtF[u] :=

Z

R^d

u^mdµ ^σ−1Z

R^d

u|DαP|²dµ

whereσ= ²_{d 1−m}¹ −1. Heredµ=|x|^n−ddxand the pressure variable is

P:= m

1−mu^m−1

Symmetry: the formal computation

WithLα=−D^∗_αDα=α² u⁰⁰+ⁿ⁻¹_s u⁰

+_s¹2∆ωu, we consider the fast diffusion equation

∂u

∂t =Lαu^m

in the subcritical range 1−1/n < m <1. The key computation is the proof that

−_dt^d G[u(t,·)] R

R^du^mdµ1−σ

≥(1−m) (σ−1)R

R^du^m

LαP− R

Rdu|DαP|²dµ

R

Rdu^mdµ

2

dµ

+ 2R

R^d

α⁴ 1−_n¹

P⁰⁰−^P_s⁰ −_α2(n−1)s^∆ωP ₂

2

+²_s^α₂²

∇_ωP⁰−^∇ω_s^P

2 u^mdµ

+ 2R

R^d

(n−2) α²_FS−α²

|∇ωP|²+c(n, m, d)^|∇_P^ω2^P|4

u^mdµ=:H[u]

for some numerical constantc(n, m, d)>0. Hence ifα≤α_FS, the r.h.s.H[u] vanishes if and only ifPis an affine function of|x|², which proves the symmetry result

Other applications

The subcritical regime: (JD, Esteban, Loss, Muratori, 2017) Equations with a mean field coupling

Bthe (Patlak)-Keller-Segel in mathematical biology (Campos, JD, 2014)

Bthe Oseen attractor in 2D Euler equations (with positive vorticity)(Gallay)

Hypocoercivity in kinetic equations

H ¹ methods and ϕ-hypocoercivity

Some references

hypoelliptic methods: (Hörmander),(Hérau, Nier), and many others

H¹-hypocoercive methods: (Gallay),(Villani), (Mouhot-Neumann),(Baudoin),etc.

ϕ-hypocoercivity: (Arnold, Erb),(Achleitner, Arnold, Stürzer), (Achleitner, Arnold, Carlen),(Monmarché et al.),(Evans),(JD, Li)

L²-hypocoercive methods: (JD, Mouhot, Schmeiser),(Bouin, JD, Mouhot, Mischler, Schmeiser),(Arnold et al.)

BMotivation: coupling with mean field equations Partial results: (Hérau, Thomann),(Herda, Rodrigues)

The kinetic Fokker-Planck equation

∂f

∂t +v· ∇xf− ∇xψ· ∇vf = ∆vf +∇v·(v∇vf) withψ(x) =|x|²/2. Under the conditionkfk_L1(R^d×R^d)= 1, it has a unique stationary solution

f?(x, v) = (2π)^−d2 e^−ψ(x)e⁻

1

2|v|² = (2π)^−de⁻

1

2(|x|²+|v|²)

∀(x, v)∈R^d×R^d The functiong:=f /f_? solves

∂g

∂t +Tg=Lg

where the transport operatorTand the Ornstein-Uhlenbeck operator L are defined respectively by

Tg:=v· ∇xg−x· ∇vg and Lg:= ∆vg−v· ∇vg Letdµ:=f_?dx dvbe the invariant measure onR^d×R^d

An optimal decay rate

The functionh:=g^p/2 solves

∂h

∂t +Th=Lh+2−p p

|∇vh|² h . At the kinetic level, we consider theϕp-entropy given by

E[g] :=

Z Z

R^d×R^d

ϕp(g)dµ

Proposition

(Arnold, Erb)With the above notations there exists a constant C>0 for which

E[g(t,·,·)]≤Ce^−t ∀t≥0

ϕ-hypocoercivity: the H

approach

The method is based on theFisher information J[h] =¹₂

Z

R^d

|∇vh|²dµ+¹₂ Z

R^d

|∇xh|²dµ+¹₂ Z

R^d

|∇xh+∇vh|²dµ which involves derivatives inxandv

Acarré du champcomputation shows that d

dtJ[h(t,·)]≤ −J[h(t,·)]

The result follows from the entropy – entropy production inequality ΛE[g(t,·,·)] = ΛE[h^2/p]≤J[h]

ϕ-hypocoercivity: improved rate

Generalized (twisted, time-dependent)Fisher informationfunctional Jλ[h] = (1−λ)

Z

R^d

|∇vh|²dµ+(1−λ) Z

R^d

|∇xh|²dµ+λ Z

R^d

|∇xh+∇vh|²dµ

Theorem

(JD, Li)Let p∈(1,2). There exists a functionλ:R⁺→[1/2,1) and a continuous functionρonR⁺ such that ρ >1/2 a.e., for which we

have d

dtJλ(t)[h(t,·)]≤ −2ρ(t)Jλ(t)[h(t,·)]

As a consequence, for anyt≥0we have the global estimate Jλ(t)[h(t,·)]≤Jλ(0)[h₀] exp

−2 Z t

0

ρ(s)ds

L ² -hypocoercivity

BAbstract statement BA toy model BDecay estimates BDiffusion limits

An abstract evolution equation

Let us consider the equation dF

dt +TF =LF (2)

In the framework of kinetic equations,TandLare respectively the transport and the collision operators

We assume thatTandL are respectively anti-Hermitian and Hermitian operators defined on the complex Hilbert space (H,h·,·i)

A:= 1 + (TΠ)^∗TΠ−1

(TΠ)^∗

∗denotes the adjoint with respect toh·,·i

Π is the orthogonal projection onto the null space ofL

Fokker-Planck kernels with general equilibria

We consider the Cauchy problem

∂tf+v· ∇xf =Lf , f(0, x, v) =f0(x, v) (3) for a distribution functionf(t, x, v), withpositionvariablex∈R^d or x∈T^d the flat d-dimensional torus

Fokker-Planckcollision operator with a general equilibriumM Lf =∇v·h

M∇v M⁻¹fi

Anadmissible local equilibriumM is positive, radially symmetric and Z

R^d

M(v)dv= 1, dγ=γ(v)dv:= dv M(v) Typical example: M(v) = (2π)^−d/2e^−12|v|2 + some technical assumptions (H)

Scattering collision operators

Scatteringcollision operator Lf =

Z

R^d

σ(·, v⁰) f(v⁰)M(·)−f(·)M(v⁰) dv⁰

Main assumption on thescattering rateσ: for some positive, finiteσ 1≤σ(v, v⁰)≤σ ∀v, v⁰∈R^d

Example: linear BGK operator

Lf =M ρ_f−f , ρ_f(t, x) = Z

R^d

f(t, x, v)dv Local mass conservation

Z

R^d

Lf dv= 0

The assumptions

λm,λM, andCM are positive constants such that, for anyF∈H Bmicroscopic coercivity:

− hLF, Fi ≥λ_mk(1−Π)Fk² (H1) Bmacroscopic coercivity:

kTΠFk²≥λ_MkΠFk² (H2) Bparabolic macroscopic dynamics:

ΠTΠF = 0 (H3)

Bbounded auxiliary operators:

kAT(1−Π)Fk+kALFk ≤CMk(1−Π)Fk (H4) The estimate

1 2

d

dtkFk²=hLF, Fi ≤ −λmk(1−Π)Fk² is not enough to conclude thatkF(t,·)k² decays exponentially

Equivalence and entropy decay

For someδ >0 to be determined later, the L² entropy / Lyapunov functional is defined by

H[F] := ¹₂kFk²+δRehAF, Fi

as in(Dolbeault-Mouhot-Schmeiser)so thathATΠF, Fi ∼ kΠFk² and

− d

dtH[F] = :D[F]

= − hLF, Fi+δhATΠF, Fi

− δRehTAF, Fi+δRehAT(1−Π)F, Fi −δRehALF, Fi Bfor anyδ >0 small enough andλ=λ(δ)

λH[F]≤D[F]

Bnorm equivalence ofH[F] andkFk² 2−δ

4 kFk²≤H[F]≤2 +δ 4 kFk²

Exponential decay of the entropy

λ= _{3 (1+λ}^λM

M)minn

1, λm,_(1+λ^λmλM

M)C_M²

o

,δ=¹₂ min

1, λm,_(1+λ^λmλM

M)C_M²

h₁(δ, λ) := (δ C_M)²−4

λ_m− δ−2 +δ

4 λ δ λ_M

1 +λM

−2 +δ 4 λ

Theorem

LetL andTbe closed linear operators (respectively Hermitian and anti-Hermitian) onH. Under (H1)–(H4), for any t≥0

H[F(t,·)]≤H[F0]e^−λ?t whereλ? is characterized by

λ?:= sup

λ >0 : ∃δ >0s.t.h1(δ, λ) = 0, λm− δ−¹₄(2 +δ)λ >0

Hypocoercivity

Corollary

For anyδ∈(0,2), ifλ(δ)is the largest positive root ofh1(δ, λ) = 0for whichλm−δ−¹₄(2 +δ)λ >0, then for any solutionF of (2)

kF(t)k²≤ 2 +δ

2− δe^−λ(δ)tkF(0)k² ∀t≥0 From the norm equivalence ofH[F] andkFk²

2−δ

4 kFk²≤H[F]≤2 +δ 4 kFk² We use ²⁻₄^δkF0k²≤H[F0] so thatλ?≥sup_δ∈(0,2)λ(δ)

A toy problem

du

dt = (L−T)u , L=

0 0 0 −1

, T =

0 −k

k 0

, k²≥Λ>0 Non-monotone decay, a well known picture:

see for instance(Filbet, Mouhot, Pareschi, 2006) H-theorem: _dt^d|u|²=−2u²₂

macroscopic limit: ^du_dt¹ =−k²u1

generalized entropy: H(u) =|u|²− _1+k^{δ k}2 u1u2

dH

dt = −

2− δ k² 1 +k²

u²₂− δ k²

1 +k²u²₁+ δ k 1 +k²u1u2

≤ −(2−δ)u²₂− δΛ

1 + Λu²₁+δ 2u₁u₂

Plots for the toy problem

1 2 3 4 5 6

0.2 0.4 0.6 0.8 1 u1²

1 2 3 4 5 6

0.1 0.2 0.3 0.4 0.5 0.6 0.7 u1²+u2²

1 2 3 4 5 6

0.2 0.4 0.6 0.8 1 u1²

1 2 3 4 5 6

0.2 0.4 0.6 0.8 1 H

Further results

(Bouin, JD, Mouhot, Mischler, Schmeiser)

BIt is possible to make a mode by mode analysis in Fourier variables BOn the whole space without confinement potential (ψ≡0), we obtain rates of decay using Nash’s inequality

Consider

∂tf+v· ∇xf =Lf , f(0, x, v) =f0(x, v)

for a distribution functionf(t, x, v), withpositionvariable in the whole space,x∈R^d, and withtimet≥0

(a) Fokker-Planck collision operator:

Lf =∇v·h

M∇v M⁻¹fi

(b) Scattering collision operator:

Lf = Z

R^d

σ(·, v⁰) f(v⁰)M(·)−f(·)M(v⁰) dv⁰

A statement

A typical example of alocal equilibriumM is the Gaussian function M(v) =e^−12|v|2

(2π)^d/2 but our results apply to more general functionsM Theorem

Let us consider an admissibleM and a collision operatorL satisfying some additional technical assumptions. Assume thatx∈R^d, and γ_k(v) = 1 +|v|^2k/2

for somek∈(d,∞]. There exists a constant C >0 such that the solutionf satisfies

kf(t,·,·)k²_L2(dx dγ_k)≤C

kf0k²_L2(dx dγ_k)+kf0k²_L2(dγ_k; L¹(dx))

(1 +t)^−d2

Diffusion limits:

from kinetic equations to diffusions

BGK-type kinetic equation as a motivation for

nonlinear diffusions – polytropes and fast diffusion / porous medium

ε²∂_tf^ε+εv· ∇xf^ε−ε∇xV(x)· ∇vf^ε = G_fε−f^ε

f^ε(x, v, t= 0) = fI(x, v), x, v ∈R³ with the Gibbs equilibriumGf :=γ

|v|²

2 +V(x)−µρ_f(x, t)

The Fermi energyµρ_f(x, t) is implicitly defined by Z

R³

γ |v|²

2 +V(x)−µ_ρf(x, t)

dv= Z

R³

f(x, v, t)dv=:ρ_f(x, t) f^ε(x, v, t) . . . phase space particle density

V(x) . . . potential ε . . . mean free path

=⇒ µρ_f = ¯µ(ρf)

Diffusion limits

(J.D., P. Markowich, D. Ölz, C. Schmeiser) Theorem

For anyε >0, the equation has a unique weak solution f^ε∈C(0,∞;L¹∩L^p(R⁶))for all p <∞. Asε→0,f^ε weakly converges to a local Gibbs statef⁰ given by

f⁰(x, v, t) =γ 1

2|v|²−µ(ρ(x, t))¯

whereρis a solution of the nonlinear diffusion equation

∂tρ=∇x·(∇xν(ρ)+ρ∇xV(x)) with initial dataρ(x,0) =ρI(x) :=R

R³fI(x, v)dv ν(ρ) =

Z ρ

0

sµ¯⁰(s)ds

Some slides can be found at

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

BLectures The papers can be found at

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

BPreprints / papers

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Thank you for your attention !