Large time behavior of coagulation-fragmentation equations with - - PowerPoint PPT Presentation

Large time behavior of coagulation-fragmentation equations with degenerate diffusion Laurent Desvillettes CMLA, ENS Cachan & IUF In collaboration with Klemens Fellner

– p. 1

A strange decay rate

Convergence towards equilibrium in ||f(t) − feq|| ≤ Cst e−Cst (log t)β, for all β < 2. Better than any power; Worse than any Cst exp(−Cst tγ), with γ > 0.

– p. 2

Ingredients

√ Entropy (Entropy dissipation) method √ Almost exponential convergence to equilibrium √ Tracking explicitly the evolution of moments

– p. 3

Entropy method for large time behavior

Abstract equation : ∂tf = A f. We suppose that there exists a (bounded below) Lyapounov functional H := H(f) (entropy) and a functional D := D(f) (entropy dissipation) such that ∂tH(f) = −D(f) ≤ 0, and D(f) = 0 ⇐ ⇒ A f = 0 ⇐ ⇒ f = feq, where feq is a given function.

– p. 4

De La Salle’s Principle

Then, one can often prove that

• 1. the decreasing function t → H(f(t)) converges toward its minimum H(feq),
• 2. t → D(f(t)) converges toward 0,

3. lim

t→+∞ f(t) = feq

in a convenient topology.

– p. 5

Explicit rate of convergence toward equilibrium

One looks for functional inequalities like D(f) ≥ Cst (H(f) − H(feq)). Then, using the Lyapounov functional, we get the differential inequality ∂t(H(f) − H(feq)) ≤ −Cst (H(f) − H(feq)), and Gronwall’s lemma ensures that H(f) − H(feq) ≤ Cst e−Cst t. Finally, if H has good properties of coercivity, we obtain ||f − feq|| ≤ Cst e−Cst t.

– p. 6

Coagulation-fragmentation

Aizenmann-Bak model ∂tf(t, y) = QAB(f)(t, y).

• 1. Break-ups of clusters of size y′ larger than y contribute to create clusters of size

y: Q+

frag(f)(t, y) := 2

Z ∞

y

f(t, y′) dy′.

• 2. Break-up of polymers of size y reduces its concentration:

Q−

frag(f)(t, y) := y f(t, y).

– p. 7

• 1. Coalescence of clusters of size y′ ≤ y and y − y′ results into clusters of size y:

Q+

coag(f)(t, y) :=

Z y f(t, y − y′)f(t, y′) dy′.

• 2. Polymerization of clusters of size y with other clusters of size y′ produces a

loss in its concentration: Q−

coag(f)(t, y) := 2f(t, y)

Z ∞ f(t, y′) dy′. Aizenmann-Bak kernel : QAB(f)(t, y) = Q+

frag(f)(t, y)−Q− frag(f)(t, y)+ Q+ coag(f)(t, y)−Q− coag(f)(t, y) .

– p. 8

Entropy structure for Aizenmann-Bak model

Entropy: HAB(f)(t) = Z ∞ „ f(t, y) log f(t, y) − f(t, y) « dy , Entropy dissipation: DAB(f)(t) = Z ∞ Z ∞ „ f(t, y + y′) − f(t, y) f(t, y′) « × „ log[f(t, y + y′)] − log[f(t, y) f(t, y′)] « dy′dy ≥ 0 , Entropy relation: d dtHAB(f)(t) = Z ∞ QAB(f, f)(t, y) log f(t, y) dy = −DAB(f)(t).

– p. 9

Case of equality in the entropy structure for Aizenmann-Bak kernel

DAB(f) = 0 ⇐ ⇒ ∀y ≥ 0, QAB(f)(y) = 0 ⇐ ⇒ ∀y, y′ ≥ 0, f(y + y′) = f(y) f(y′) ⇐ ⇒ ∃c > 0, f(y) = exp (−c y) Analogous to the second part (case of equality) of Boltzmann H-theorem.

– p. 10

Entropy/Entropy Dissipation estimate for AB model

Theorem (M. Aizenmann, T. Bak): For any f := f(y) ≥ 0 such that Nf ≥ N∗ > 0, DAB(f) ≥ Cst(N∗) (HAB(f) − H(MAB(f))) , where MAB(f)(y) = e

−

y

√

Nf ,

Nf = Z ∞ f(y) y dy. Main tool : Elementary convexity inequalities Analogous result for a discrete model : P.-E. Jabin, B. Niethammer

– p. 11

Explicit rate of convergence toward equilibrium (AB)

Theorem (M. Aizenmann, T. Bak): Let fin := fin(y) ≥ 0 be an initial datum such that Z ∞ fin(y) (1 + y + | log fin(y)|) dy < +∞. Then there exists a unique solution to the Aizenmann-Bak equation ∂tf(t, y) = QAB(f)(t, y), f(0, y) = fin(y), such that N(f(t, ·)) = Z ∞ f(t, y) y dy = Z ∞ f(0, y) y dy = N(fin). Moreover this solution satisfies (for some explicit C1, C2 > 0) ˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛f(t, y) − exp „ − y p N(fin) «˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛

L1(R+)

≤ C1 e−C2 t.

– p. 12

Inhomogeneous Aizenmann-Bak equation

New variables for the unknown: f(t, y) → f(t, x, y). Equation (for some a(y) ≥ 0): ∂tf(t, x, y) − a(y) ∆xf(t, x, y) = QAB(f)(t, x, y). Homogeneous Neumann boundary conditions: ∀x ∈ ∂Ω, ∇xf(t, x, y) · n(x) = 0. Initial datum: f(0, x, y) = fin(x, y).

– p. 13

Entropy structure for the Inhomogeneous AB equation

Entropy: HIAB(f)(t) = Z

Ω

Z ∞ „ f(t, x, y) log f(t, x, y) − f(t, x, y) « dydx . Entropy relation: ∂tHIAB(f) = −(D1(f) + D2(f)). Entropy dissipation: D1(f)(t) = Z

Ω

Z ∞ a(y) |∇xf(t, x, y)|2 f(t, x, y) dydx, D2(f)(t) = Z

Ω

Z ∞ Z ∞ „ f(t, x, y + y′) − f(t, x, y) f(t, x, y′) « × „ log[f(t, x, y + y′)] − log[f(t, x, y) f(t, x, y′)] « dy′dydx .

– p. 14

Case of equality in the entropy structure of the inhomogeneous AB model

D1(f) + D2(f) = 0 ⇐ ⇒ f(x, y) = e−c(x) y and ∇xc(x) = 0 ⇐ ⇒ f(x, y) = e−c y.

– p. 15

Entropy/entropy dissipation estimate for the inhomogeneous AB equation

Proposition (J. Carrillo, LD, K. Fellner): Let f := f(x, y) ≥ 0 be such that M(x) := Z ∞ f(x, y) dy ≥ M∗ > 0, N(x) := Z ∞ f(x, y) y dy ≥ N∗ > 0. Then D1(f) + D2(f) ≥ Cst(M∗, N∗, inf a, sup a) ||M||L∞(Ω) log(||M||L∞(Ω)) (HIAB(f) − HIAB(feq)), with feq(x, y) = e

−y r

|Ω| R N(x) dx . – p. 16

Large time behavior of the inhomogeneous AB equation

Theorem (J. Carrillo, LD, K. Fellner): Let 0 < a∗ ≤ a(y) ≤ a∗, and fin := fin(x, y) ≥ 0 be an initial datum such that Z 1 Z ∞ fin(x, y) (1 + y + | log fin(x, y)|) dydx < +∞. According to a theorem by Ph. Laurençot, S. Mischler, there exists a unique solution f := f(t, x, y) ≥ 0 to the spatially inhomogeneous Aizenmann-Bak equation with Neumann BC and initial datum fin. Moreover, for c given by the conservation of total mass and for all q ∈ N, Z ∞ (1 + y)q ||f(t, ·, y) − e−c y||L∞(]0,1[) dy ≤ C1 e−C2 t, where C1, C2 > 0 are explictly computable in terms of a∗, a∗, q, fin.

– p. 17

Estimates used in the proof

• 1. Bounds from above using the entropy dissipation

M ∈ (L1 + L∞)([0, +∞[; L∞(]0, 1[)),

• 2. Bounds from below using the heat kernel and the conservation of total mass

M(t, x) ≥ M∗ > 0, N(t, x) ≥ N∗ > 0,

• 3. Cziszar-Kullback-Pinsker inequality,
• 4. Smoothness estimates using the heat kernel : for all q ∈ N,

Z ∞ (1 + y)q ||f(t, ·, y)||H1(]0,1[) dy ≤ Pol(t).

– p. 18

Almost exponential convergence to equilibrium

Sometimes, the functional inequality D(f) ≥ C (H(f) − H(feq)) cannot be proven, and one has instead D(f) ≥ C A−1 (H(f) − H(feq)) − Cp A−(p+1) for some or all p > 0, and all A > 0. Together with “slowly growing a priori coefficients”: G. Toscani, C. Villani

– p. 19

Almost exponential convergence to equilibrium

D(f) ≥ C A−1 (H(f) − H(feq)) − Cp A−(p+1) for some or all p > 0, and all A > 0. Then by interpolation, D(f) ≥ Cp (H(f) − H(feq))(p+1)/p, and we get the differential inequality ∂t(H(f) − H(feq)) ≤ −Cp (H(f) − H(feq))(p+1)/p, so that thanks to Gronwall’s lemma, H(f)(t) − H(feq) ≤ Cp (1 + t)−p.

– p. 20

Strange rate of convergence toward equilibrium

If Cp = 22p2, then 2−2p2 tp [H(f)(t) − H(feq)] ≤ 1. Taking p = (log t)1−ε for some ε > 0, we get e(log t)2−ε e2 log 2 (log t)2−2ε (H(f)(t) − H(feq)) ≤ 1, so that H(f)(t) − H(feq) ≤ Cst e−Cst (log t)2−ε.

– p. 21

Large time behavior of the degenerate inhomogeneous AB equation

Theorem (LD, K. Fellner): Let 0 <

a∗ 1+y ≤ a(y) ≤ a∗, and fin := fin(x, y) ≥ 0

be an initial datum such that Z 1 Z ∞ fin(x, y) (1 + y + | log fin(x, y)|) dydx < +∞. According to a theorem by Ph. Laurençot, S. Mischler, there exists a unique solution f := f(t, x, y) ≥ 0 to the spatially inhomogeneous Aizenmann-Bak equation with Neumann BC and initial datum fin. Moreover, for c given by the conservation of total mass and for all β < 2, ||f(t, ·, ∗) − e−c ∗||L1(]0,1[×R+) ≤ Cβ e−(log t)β. where Cβ > 0 is explictly computable in terms of a∗, a∗, β, fin.

– p. 22

Estimates used in the proof

• 1. Bounds from above using the entropy dissipation

M ∈ (L1 + L∞)([0, +∞[; L∞(]0, 1[),

• 2. Bounds from below using the heat kernel and the conservation of total mass

M(t, x) ≥ M∗ > 0,

• 3. Cziszar-Kullback-Pinsker inequality,
• 4. Estimates on the moments Mp(f)(t) :=

R 1 R ∞ yp f(t, x, y) dydx.

– p. 23

Estimates used in the proof: cutoff for large y

We define MA(t, x) := R A

0 f(t, x, y) dy and M =

R M dx. Then M − M2

L2

x ≤ 2MA − MA2

L2

x +

4 A2p Z 1 „Z ∞ ypf(t, x, y) dy «2 dx ≤ 2MA − MA2

L2

x +

4 A2p ML∞

x M2p ,

MA − MA2

L2

x ≤ P(Ω)

Z 1 ˛ ˛ ˛ ˛∇x Z A f dy ˛ ˛ ˛ ˛

2

dx ≤ P(Ω) Z 1 ˛ ˛ ˛ ˛ Z A a(y) |∇xf|2 f dy ˛ ˛ ˛ ˛ ˛ ˛ ˛ ˛ Z A f a(y)dy ˛ ˛ ˛ ˛ dx ≤ P(Ω) 1 + A a∗ ML∞

x

Z 1 Z ∞ a(y)|∇xf|2 f dydx since we have

1 a(y) ≤ 1+y a∗ ≤ 1+A a∗ for y ∈ [0, A].

– p. 24

Estimates used in the proof: moments

sup

t≥t∗>0

Mp(f)(t) ≤ (22p Cst)p

• Cf. A. Bobylev, I. Gamba, V. Panferov.

Steps of the proof: 1. Mp(f)(t0) < +∞ ⇒ sup

t≥t0

Mp(f)(t) ≤ Cst „ Mp(f)(t0)+(22p Cst)p « . Difficulty: track the p dependence. 2. Mp(f)(1 + t) ≤ Cst (1 + M2(f)(t)). Difficulty: build a sequence of times between t and t + 1 allowing to pass from Mk(f) to Mk+1(f). 3. M2(f)(t) ≤ Cst (1 + t−Cst). Difficulty: Use De la Vallée-Poussin’s meth., Cf. S. Mischler, B. Wennberg.

– p. 25