pour l’équation de Vlasov-Fokker-Planck

Une approche hypocoercive L

pour l’équation de Vlasov-Fokker-Planck

Jean Dolbeault

dolbeaul@ceremade.dauphine.fr

CEREMADE

CNRS & Universit ´e Paris-Dauphine

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

(EN COLLABORATION AVEC C. MOUHOT ET C. SCHMEISER) EVOL, 26 SEPTEMBRE 2009, TOULOUSE

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

Outline

The goal is to understand the rate of relaxation of the solutions of a kinetic equation

∂_tf + v · ∇_xf − ∇_xV · ∇_vf = Lf

towards a global equilibrium when the collision term acts only on the velocity space. Here f = f(t, x, v) is the distribution function. It can be seen as a probability distribution on the phase space, where x is the position and v the velocity. However, since we are in a linear framework, the fact that f has a constant sign plays no role.

A key feature of our approach [J.D., Mouhot, Schmeiser] is that it distinguishes the mechanisms of relaxation at microscopic level (convergence towards a local equilibrium, in velocity space) and macroscopic level (convergence of the spatial density to a steady state), where the rate is given by a spectral gap which has to do with the underlying diffusion equation for the spatial density

A very brief review of the literature

Non constructive decay results: [Ukai (1974)] [Desvillettes (1990)]

Explicit t^−∞-decay, no spectral gap: [Desvillettes, Villani (2001-05)], [Fellner, Miljanovic, Neumann, Schmeiser (2004)], [Cáceres, Carrillo, Goudon (2003)]

hypoelliptic theory:

[Hérau, Nier (2004)]: spectral analysis of the Vlasov-Fokker-Planck equation

[Hérau (2006)]: linear Boltzmann relaxation operator

Hypoelliptic theory vs. hypocoercivity (Gallay) approach and generalized entropies:

[Mouhot, Neumann (2006)], [Villani (2007, 2008)]

Other related approaches: non-linear Boltzmann and Landau equations:

micro-macro decomposition: [Guo]

hydrodynamic limits (fluid-kinetic decomposition): [Yu]

A toy problem

du

dt = (L − T)u , L = 0 0 0 −1

!

, T = 0 −k k 0

!

, k² ≥ Λ > 0

Nonmonotone decay, reminiscent of [Filbet, Mouhot, Pareschi (2006)]

H-theorem: _dt^d |u|² = −2 u²₂

macroscopic limit: ^du_dt¹ = −k² u₁

generalized entropy: H(u) = |u|² − _1+k^{ε k}₂ u₁ u₂

dH

dt = −

2 − ε k² 1 + k²

u²₂ − ε k²

1 + k² u²₁ + ε k

1 + k² u₁ u₂

≤ −(2 − ε)u²₂ − εΛ

1 + Λ u²₁ + ε

2 u₁u₂

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

0.2 0.4 0.6 0.8 1 u1²

0.2 0.4 0.6 0.8 1 H

. . . compared to plots for the Boltzmann equation

Figure 1: [Filbet, Mouhot, Pareschi (2006)]

The kinetic equation

∂_tf + T f = Lf , f = f(t, x, v), t > 0, x ∈ R^d, v ∈ R^{d (1)} L is a linear collision operator

V is a given external potential on R^d, d ≥ 1 T := v · ∇_x − ∇_xV · ∇_v is a transport operator

There exists a scalar product h·,·i, such that L is symmetric and T is antisymmetric

d

dt kf − Fk² = −2 kLfk²

... seems to imply that the decay stops when f ∈ N(L)

but we expect f → F as t → ∞ since F generates N(L) ∩ N(T) Hypocoercivity: prove an H-theorem for a generalized entropy

H(f) := 1

2 kfk² + ε hAf, fi

Examples

L is a linear relaxation operator L

Lf = Πf − f , Πf := ρ

ρ_F F(x, v) ρ = ρ_f :=

Z

R^d

f dv

Maxwellian case: F(x, v) := M(v) e^−V(x) with M(v) := (2π)^−d/2 e^−|v|2/2 =⇒ Πf = ρ_f M(v)

Linearized fast diffusion case: F(x, v) := ω ¹₂ |v|² + V (x)−(k+1)

L is a Fokker-Planck operator: Lf = ∆_vf + ∇ · (v f)

L is a linear scattering operator (including the case of non-elastic collisions)

Some conventions. Cauchy problem

F is a positive probability distribution

Measure: dµ(x, v) = F(x, v)⁻¹ dx dv on R^d × R^d ∋ (x, v) Scalar product and norm hf, gi = RR

R^d×R^d f g dµ and kfk² = hf, fi The equation

∂_tf + Tf = Lf

with initial condition f(t = 0,·,·) = f₀ ∈ L²(dµ) such that Z Z

R^d×R^d

f₀ dx dv = 1

has a unique global solution (under additional technical assumptions):

[Poupaud], [JD, Markowich,Ölz, Schmeiser]. The solution preserves mass Z Z

R^d×R^d

f(t, x, v) dx dv = 1 ∀t ≥ 0

Maxwellian case: Assumptions

We assume that F(x, v) := M(v) e^−V(x) with M(v) := (2π)^−d/2 e^−|v|2/2 where V satisfies the following assumptions

(H1) Regularity: V ∈ W_loc^2,∞(R^d) (H2) Normalization: R

R^d e^−V dx = 1

(H3) Spectral gap condition: there exists a positive constant Λ ^{such that} R

R^d |u|² e^−V dx ≤ ΛR

R^d |∇_xu|² e^−V dx

for any u ∈ H¹(e^−V dx) ^{such that} R

R^d u e^−V dx = 0

(H4) Pointwise condition 1: there exists c₀ > 0 ^and θ ∈ (0,1) ^{such that}

∆V ≤ ^θ₂ |∇_xV (x)|² + c₀ ∀x ∈ R^d

(H5) Pointwise condition 2: there exists c₁ > 0 ^{such that}

|∇²_xV (x)| ≤ c₁ (1 + |∇_xV (x)|) ∀ x ∈ R^d (H6) Growth condition: R

R^d |∇_xV |² e^−V dx < ∞

Maxwellian case: Main result

Theorem 1. If ∂_tf + T f = Lf^{, for} ε > 0, small enough, there exists an explicit, positive constant λ = λ(ε) ^{such that}

kf(t) − Fk ≤ (1 + ε)kf₀ − Fke^−λt ∀ t ≥ 0

The operator L has no regularization property: hypo-coercivity fundamentally differs from hypo-ellipticity

Coercivity due to L is only on velocity variables d

dtkf(t) − Fk² = −k(1 − Π)fk² = − Z Z

R^d×R^d

|f − ρ_f M(v)|² dv dx

T and L do not commute: coercivity in v is transferred to the x variable. In the diffusion limit, ρ solves a Fokker-Planck equation

∂_tρ = ∆ρ + ∇ · (ρ∇V ) t > 0, x ∈ R^d

the goal of the hypo-coercivity theory is to quantify the interaction of T and L and build a norm which controls k · k and decays exponentially

The operators. A Lyapunov functional

On L²(dµ), define

bf := Π (v f), af := b (Tf), ˆaf := −Π (∇_xf) , A := (1+ˆa·aΠ)⁻¹ ˆa·b

b f = F ρ_F

Z

R^d

v f dv= F

ρ_F j_f with j_f :=

Z

R^d

v f dv

a f = F ρ_F

∇_x · Z

R^d

v ⊗ v f dv + ρ_f ∇_xV

, ˆaf = − F

ρ_F ∇_xρ_f A T = (1 + ˆa · a Π)⁻¹ ˆa · a

Define the Lyapunov functional (generalized entropy)

H(f) := 1

2 kfk² + ε hAf, fi

. . . a Lyapunov functional (continued): positivity, equivalence

hˆa · a Πf, fi = 1 d

Z

R^d

∇_{x ρ}^ρf

F

²m_F dx

with m_F := R

R^d |v|² F(·, v) dv. Let g := Af, u = ρ_g/ρ_F, j_f := R

R^d v f dv (1 + ˆa · aΠ) g = ˆa · bf ⇐⇒ g − 1

d ∇_x (m_F ∇_xu) F

ρ_F = − F

ρ_F ∇_x j_f

ρ_F u − 1

d ∇_x (m_F ∇_xu) = −∇_x j_f kAfk² = R

R^d |u|² ρ_F dx and kT Afk² = ¹_d R

R^d |∇_xu|² m_F dx are such that 2 kAfk² + kT Afk² ≤ k(1 − Π) fk²

As a consequence

(1 − ε)kfk² ≤ 2H(f) ≤ (1 + ε)kfk²

. . . a Lyapunov functional (continued): decay term

H(f) := 1

2 kfk² + ε hAf, fi

The operator T is skew-symmetric on L²(dµ). If f is a solution, then d

dtH(f − F) = D(f − F) D(f) := ...

...hf, Lfi

| {z }

micro

−ε hA TΠ f, fi

| {z }

macro

−εhA T (1−Π) f, fi+ε hT Af, fi+ε hLf,(A+A^∗)fi

Lemma 2. For some ε > 0 small enough, there exists an explicit constant λ > 0 ^such

that

D(f − F) + λH(f − F) ≤ 0

Preliminary computations (1/2)

Maxwellian case: Πf = ρ_f M(v) with ρ_f := R

R^d f dv Replace f by f −F: 0 = RR

R^d×R^d f dx dv = hf, Fi, R

R^d(Π f −f) dv = 0 L is a linear relaxation operator: Lf = Πf − f

hLf, fi ≤ −k(1 − Π)fk² The other terms: for any c,

−εhA T(1 − Π) f, fi = −ε hA T(1 − Π) f,Π fi

≤ c

2 kA T(1 − Π) fk² + ε²

2 c kΠfk² εhT A f, fi = ε hT Af,Π fi ≤ c

2 kTA fk² + ε²

2 c kΠfk² εh(A + A^∗)Lf, fi ≤ εk(1 − Π) fk² + ε kAfk²

Reminder: A and T A are bounded operators

A T (1 − Π) = ΠA T(1 − Π) and T A = ΠT A. With m_F := R

R^d |v|² F dv hˆa · a Πf, fi = 1

d Z

R^d

∇_{x ρ}^ρf

F

²m_F dx

Recall that one ca compute Af := (1 + ˆa · aΠ)⁻¹ ˆa · bf as follows: let g := Af u := ρ_g/ρ_F

By definition of A, ˆa · bf = (1 + ˆa · aΠ)g means

−∇_x · Z

R^d

v f dv =: −∇_x · j_f = ρ_F u − 1

d ∇_x · (m_F ∇_xu) kAfk² = R

R^d |u|² ρ_F dx and kT Afk² = ¹_d R

R^d |∇_xu|² m_F dx are such that 2 kAfk² + kT Afk² ≤ k(1 − Π) fk²

Preliminary computations (2/2)

D(f) ≤ −

1 − c

2 − 2ε

k(1 − Π)fk²

| {z }

micro: ≤0

− ε hA TΠ f, fi

| {z }

macro: “first estimate”

+ c

2 kA T(1 − Π) fk²

| {z }

“second estimate...”

+ ε²

c kΠ fk²

...(A T (1 − Π))^∗f = (ˆa · a(1 − Π))^∗g with g = (1 + ˆa · aΠ)⁻¹ f means

ρ_f = ρ_F u − 1

d ∇_x (m_F ∇_xu) where u = ρ_g/ρ_F. Let q_F := R

R^d |v₁|⁴ F dv, u_ij := ∂²u/∂x_i∂x_j

k(A T(1 − Π))^∗fk²=

Xd

i, j=1

Z

R^d

h_2δ

ij+1

3 q_F − ^m_d2^2F_ρ^δij

F

u_ii u_jj + ^2(1−δ₃ ^ij) q_F u²_iji dx

Maxwellian case

With ρ_F = e^−V = _d¹ m_F = q_F, B = ˆa · aΠ, g = (1 + B)⁻¹ f means ρ_f = u e^−V − ∇_x e^−V ∇_xu

if u = ρ_g ρ_F

Spectral gap condition: there exists a positive constant Λ such that R

R^d |u|² e^−V dx ≤ ΛR

R^d |∇_xu|² e^−V dx Since A TΠ = (1 + B)⁻¹ B, we get the “first estimate"

hA T Πf, fi

| {z }

macro

≥ Λ

1 + Λ kΠ fk²

Notice that B = ˆa · a Π = (TΠ)^∗(TΠ) so that hBf, fi = kTΠfk²

Second estimate (1/3)

We have to bound (H² estimate)

k(A T(1 − Π))^∗fk² ≤ 2

Xd

i, j=1

Z

R^d

|u_ij|² e^−V dx

Let kuk²₀ := R

R^d |u|² e^−V dx. Multiply ρ_f = u e^−V − ∇_x e^−V ∇_xu

by u e^−V to get

kuk²₀ + k∇_xuk²₀ ≤ kΠfk²

By expanding the square in |∇_x(u e^{−V /2})|², with κ = (1 − θ)/(2 (2 + Λc₀)), we obtain an improved Poincaré inequality

κkW uk²₀ ≤ k∇_xuk²₀ for any u ∈ H¹(e^−V dx) such that R

R^d u e^−V dx = 0 Here: W := |∇_xV |

Second estimate (2/3)

Multiply ρ_f = u e^−V − ∇_x e^−V ∇_xu

by W² u with W := |∇_xV | and integrate by parts

kW uk²₀ + kW ∇_xuk²₀ − 2 c₁ k∇_xuk₀ + kW ∇_xuk₀

· kW uk₀

≤ κ

8 kW²uk²₀ + 2

κ kΠ fk² Improved Poincaré inequality applied to W u − R

R^d W u e^−V dx gives κ kW²uk²₀ ≤

Z

R^d

|∇_x(W u)|² e^−V dx + 2κ Z

R^d

W u e^−V dx Z

R^d

W³ u e^−V dx (...) κkW² uk²₀ ≤ 4 kW ∇_xuk²₀ + 8c²₁ (kuk²₀ + kW uk²₀) + 4κkWk⁴₀ kuk²₀

kW ∇_xuk₀ ≤ c₅ kΠ fk

Second estimate (3/3)

Multiply ρ_f = u e^−V − ∇_x e^−V ∇_xu

by ∆u and integrating by parts, we get

k∇²_xuk²₀ − kW ∇_xuk₀ + kΠ fk

k∇²_xuk₀ ≤ kW uk₀ k∇_xuk₀ Altogether (...)

k(A T(1 − Π))fk² ≤ c₆ k (1 − Π)fk²

Summarizing, with λ₁ = 1 − ₂^c (1 + c₆) − 2 ε and λ₂ = _1+Λ^Λε − ^ε_c² D(f) ≤ −λ₁ k(1 − Π) fk² − λ₂ kΠfk²

The Vlasov-Fokker-Planck equation

∂_tf + Tf = Lf with Lf = ∆_vf + ∇_v(v f)

Under the same assumptions as in the linear BGK model... same result ! A list of changes

hf, Lfi

| {z }

micro

= −k∇_vfk² ≤ −k(1 − Π) fk² by the Poincaré inequality

Since Π f = ρ_f M(v), where M(v) is the gaussian function, and F(x, v) = M(v)e^−V(x)

hAf, Lfi = Z Z

ρ_Af M(v) (Lf) dx dv

F =

Z Z

ρ_Af (Lf) e^V dx dv= 0 Af = u F means ρ_F u − ¹_d ∇_x (m_F ∇_xu) = −∇_x j_f, j_f := R

R^d v f dv.

Hence j_Lf = −j_f gives

hA Lf, fi = −hAf, fi

Motivation: nonlinear diffusion as a diffusion limit

ε² ∂_tf + εh

v · ∇_xf − ∇_xV (x) · ∇_vfi

= Q(f) with Q(f) = γ

1

2 |v|² − µ(ρ_f)

− f Local mass conservation determines µ(ρ)

Theorem [Dolbeault, Markowich, Oelz, CS, 2007] ρ_f converges as ε → 0 to a solution of

∂_tρ = ∇_x · (∇_xν(ρ) + ρ∇_xV ) with ν^′(ρ) = ρ µ^′(ρ)

γ(s) = (−s)^k₊, ν(ρ) = ρ^m, 0 < m = m(k) < 5/3 (R³ case)

Linearized fast diffusion case

Consider a solution of ∂_tf + Tf = Lf where Lf = Πf − f, Π f := _ρ^ρ

F

F(x, v) := ω 1

2 |v|² + V (x)

−(k+1)

, V (x) = 1 + |x|²β

where ω is a normalization constant chosen such that RR

R^d×R^d F dx dv = 1 and ρ_F = ω₀ V ^d/2−k−1 for some ω₀ > 0

Theorem 3. Let d ≥ 1^, k > d/2 + 1. There exists a constant β₀ > 1 such that, for any

β ∈ (min{1,(d − 4)/(2k − d − 2)}, β₀), there are two positive, explicit constants C

and λ for which the solution satisfies:

∀t ≥ 0, kf(t) − Fk² ≤ C kf₀ − Fk² e^−λt.

Computations are the same as in the Maxwellian case except for the “first estimate" and the “second estimate"

Linearized fast diffusion case: first estimate

For p = 0, 1, 2, let w_p² := ω₀ V ^p−q, where q = k + 1 − d/2, w₀² := ρ_F

kuk²_i = Z

R^d

|u|² w_i² dx

Now g = (1 + ˆa · aΠ)⁻¹ f means

ρ_f = w₀² u − 2

2k − d ∇_x w₁² ∇_xu

Hardy-Poincaré inequality [Blanchet, Bonforte, J.D., Grillo, Vázquez]

kuk²₀ ≤ Λk∇_xuk²₁ under the condition R

R^d u w₀² dx = 0 if β ≥ 1. As a consequence hA T Πf, fi ≥ Λ

1 + Λ kΠ fk²

Linearized fast diffusion case: second estimate

Observe that ρ_f = w₀² u − _2k−d² ∇_x w₁² ∇_xu

multiplied by u gives

kuk²₀ + (q − 1)⁻¹ k∇_xuk²₁ ≤ kΠ fk² By the Hardy-Poincaré inequality (condition β < β₀)

Z

R^d

V ^{α+1−q−β1} |u|² dx −

R

R^d V ^{α+1−q−β1} u dx2

R

R^d V ^{α+1−q−β1} dx

≤ 1

4 (β₀ − 1)² Z

R^d

V ^α+1−q |∇_xu|² dx

By multiplying ρ_f = w₀² u − _2k−d² ∇_x w₁² ∇_xu

by V ^α u with α := 1 − 1/β or by V ∆u we find

k∇²_xuk²₂ ≤ C kΠfk²

Diffusion limits and hypocoercivity

The strategy of the method is to introduce at kinetic level the macroscopic quantities that arise by taking the diffusion limit

kinetic equation diffusion equation functional inequality (macroscopic) Vlasov + BGK / Fokker-Planck Fokker-Planck Poincaré (gaussian weight)

linearized Vlasov-BGK linearized porous media Hardy-Poincaré nonlinear Vlasov-BGK porous media Gagliardo-Nirenberg

from kinetic to diffusive scales: parabolic scaling and diffusion limit heuristics: convergence of the macroscopic part at kinetic level is governed by the functional inequality at macroscopic level

interplay between diffusion limits and hypocoercivity is still work in progress as well as the nonlinear case

Concluding remarks

hypo-coercivity vs. hypoellipticity

diffusion limits, a motivation for the “fast diffusion case”

other collision kernels: scattering operators other functional spaces

nonlinear kinetic models hydrodynamical models

Reference

J. Dolbeault, C. Mouhot, C. Schmeiser, Hypocoercivity for kinetic

equations with linear relaxation terms, CRAS 347 (2009), pp. 511–516 http://www.ceremade.dauphine.fr/∼dolbeaul/Conferences/