nonlinear diffusions and applications

Relative entropy methods for

nonlinear diffusions and applications

Jean DOLBEAULT

Ceremade, Universit´e Paris IX-Dauphine, Place de Lattre de Tassigny,

75775 Paris C´edex 16, France

E-mail: dolbeaul@ceremade.dauphine.fr

http://www.ceremade.dauphine.fr/_edolbeaul/

Results in collaboration with:

N. Ben Abdallah, P. Biler, L. Caffarelli, M. Del Pino, L. Desvillettes, M. Esteban, G. Karch, P. Markowich, T. Nadzieja, C. Schmeiser, A.

Unterreiter

LOGARITHMIC SOBOLEV INEQUALITY AND HEAT EQUATION

Heat equation:

u_t = ∆u in IR^d u_|t=0 = u₀ ≥ 0 u₀(1 + |x|²) ∈ L¹ u₀ log u₀ ∈ L¹

R u₀ dx = 1

Time-dependent rescaling:

u(t, x) = 1

R^d(t) v τ(t), x R(t)

!

R(t) = ^p1 + 2t , τ(t) = logR(t) v_τ = ∆v + ∇ · (x v) , v_|τ=0 = u₀ Stationary solution:

v_∞(x) = (2π)^−d/2 e^−|x|2/2

A Lyapunov functional: the free energy L[v] =

Z

v log

v v_∞

dx i.e. L[v] =

Z

v log v + 1

2|x|²v

dx + C d

dτL[v] = −I[v] , I[v] =

Z

v

∇v

v + x

2

dx

Logarithmic Sobolev inequality [Gross, 1975]

R |g|² log |g| dµ ≤ ^R |∇g|² dµ

dµ(x) = v_∞(x) dx and ^R |g|² dµ = 1.

Take |g|² = _v^v

∞:

I[v] ≥ 2L[v]

L[v] ≤ L[u₀] e^−2τ

Csisz´ar-Kullback inequality [1967]

kv − v_∞k²_L1 ≤ 4L[v]

Intermediate asymptotics u_∞(t, x) = 1

R^d(t) v_∞ x R(t)

!

ku − u_∞k_L¹ = kv − v_∞k_L¹ ≤

q2L[u₀] R(t)

ku − u_∞k_L¹ = O

√1 t

as t → +∞

Probability theory: [Bakry, Emery, Ledoux] Op- timality: [Carlen, 1991] A new proof of Log Sobolev: [Toscani, 1997], [AMTU, 1999]

dτd (I[v] − 2L[v])

= −4^{R Pd}_i,j=1

^∂

2w

∂x_i∂x_j − _w¹ _∂x^∂w_i∂x^∂w_j + ^w₂ δ_ij

2

dx

POROUS MEDIUM / FAST DIFFUSION u_t = ∆u^m in IR^d

u_|t=0 = u₀ ≥ 0

u₀(1 + |x|²) ∈ L¹ , u^m₀ ∈ L¹

Intermediate asymptotics: u₀ ∈ L^∞, ^R u₀ dx = 1, the Green function: G(t) = O(t^{−d/(2−d(1−m))}) as t → +∞, [Friedmann, Kamin, 1980]

ku(t,·) − G(t,·)kL^∞ = o(t^{−d/(2−d(1−m))})

Rescaling: Take u(t, x) = R^−d(t) v (τ(t), x/R(t)) where

R˙ = R^d(1−m)−1 , R(0) = 1 , τ = log R

v_τ = ∆v^m + ∇ · (x v) , v_|τ=0 = u₀

[Ralston, Newman, 1984] Lyapunov functional:

Entropy

L[v] = ^R _m^vm

−1 + ¹₂|x|²v dx + C

d

dτL[v] = −I[v] , I[v] =

Z

v

∇v^m−1

v + x

2

dx Stationary solution: C s.t. kv_∞k_L¹ = kuk_L¹ = 1

v_∞(x) =

C + 1 − m

2m |x|²

_−1/(1−m) +

L[v] = ^R σ _v^vm_m

∞

v_∞^m−1 dx

with σ(t) = ^mt₁^1/m_−m⁻¹ + 1 Theorem 1 m ∈ [^d−_d¹,+∞), m > ¹₂, m 6= 1

I[v] ≥ 2L[v]

p = _2m¹₋₁, v = w^2p

K < 0 if m < 1, K > 0 if m > 1

1

2(_2m^2m₋₁)^{2 R} |∇w|²dx + (₁₋¹_m−d) ^R |w|^1+pdx + K ≥ 0 m = ^d−_d¹: Sobolev, m → 1: logarithmic Sobolev

[Del Pino, J.D.], [Carrillo, Toscani], [Otto]

L[v] ≤ L[u₀] e^{−2τ+ Cisz´}ar-Kullback inequalities

⇒ Intermediate asymptotics [Del Pino, J.D.]

(i) ^d−_d¹ < m < 1 if d ≥ 3, and ¹₂ < m < 1 if d = 2 lim sup_t→+∞ t

1−d(1−m)

2−d(1−m)ku^m − u^m_∞k_L¹ < +∞ (ii) 1 < m < 2

lim sup_t→+∞ t

1+d(m−1)

2+d(m−1)k [u − u_∞] u^m_∞⁻¹ k_L¹ < +∞

Optimal constants for Gagliardo-Nirenberg in- equalities [Del Pino, J.D.]

d ≥ 2, p > 1, p ≤ _d−^d₂ for d ≥ 3 kwk2p ≤ Ak∇wk^θ₂ kwk¹_p+1^−θ A =

y(p−1)² 2πd

^θ

2 _2y−d 2y

¹

2p Γ(y)

Γ(y−₂^d)

!^θ

d

θ = ^d(p−1)

p(d+2−(d−2)p), y = ^p+1_p

−1

Similar results for 0< p < 1. Uses [Serrin-Pucci], [Serrin-Tang].

GENERAL NONLINEAR DIFFUSIONS [Carrillo, Juengel, Markowich, Toscani, Unter- reiter, 2000]

u_t = ∆f(u) + ∇ · (u∇V ) with V (x) ∼ α|x|²

Generalized entropy

L[u] = ^R [H(u) + V u] dx − C

= ^R [u h(u) − f(u) + V u] dx − C with H⁰(u) = h(u) = ^f0(u)_u

Minimizer with given mass

h(u_∞) = C − V (x) u_∞(x) = g(C − V (x))

where g is the generalized inverse of h. Examples:

f(u) = u h(u) = log u , g(t) = e^t f(u) = u^m , h(u) = _m^m₋₁ u^m−1

g(t) = ^m_m⁻¹ t^1/(m−1)

+

L[u] = ^R [u h(u) − f(u) + V u] dx + C such that L[u_∞] = 0

dτd L[u] = −I[u] ,

with I[u] = ^R uh⁰(u)∇u + ∇V u² dx

Generalized Sobolev inequality:

assume that f⁰ > 0 + technical conditions

∃K > 0 L[u] ≤ K I[u]

Rate and intermediate asymptotics:

[Biler, J.D., Esteban]

u_t = ∆f(u) , u_|t=0 = u₀ ≥ 0

Take u(t, x) = R^−d(t)v (τ(t), x/R(t)) where R˙ = R^d(1−m)−1 if f(u) ∼ u^m as m → 0

v_τ = e^mdτ∆f(e^−dτv) + ∇ · (x v)

Assumptions:

f(s) = s^mF(s)

F ∈ C⁰(IR⁺) ∩ C¹(0,+∞) F > 0 , F(0) = 1

F⁰(s) = O(s^k) with k > −1 as s → 0₊ (m − 1)s h(s) − m f(s) ≤ 0

Example: f(s) = s^m + s^q, q > m Entropy

L[τ, v] = e^{mdτ R} [H(e^−dτv) − H(e^−dτv_∞^τ )] dx +¹₂ ^R |x|² (v − v_∞^τ ) dx Theorem 2 Decay rate of the entropy

0 ≤ L[τ, v] ≤ Ke^{−β τ}, β = min (d(k + 1),2)

Corollary 3 Intermediate asymptotics

kH(u) − H(u_∞)k_L¹ ≤ C R^{−d(m−1)−β/2}

A DRIFT-DIFFUSION-POISSON MODEL [Gajewski], [Arnold, Markowich, Toscani], [Arnold, Markowich, Toscani, Unterreiter] – use Holley-

Stroock’s perturbation lemma m = 1: [Biler, J.D.]

m 6= 1: [Biler, J.D., Markowich]

Debye-H¨uckel model for electrolytes, Drift-diffusion model for semi-conductors.

u_t = ∇ · (∇u^m + u∇V + u∇φ) v_t = ∇ · (∇v^m + v∇V − v∇φ)

∆φ = v − u

After a time-dependent rescaling (use abusively same notations):

u_τ = ∇ · (∇u^m + u∇V + β(τ)u∇φ) v_τ = ∇ · (∇v^m + v∇V − β(τ)v∇φ)

∆φ = v − u , β(τ) = e^(2−d)τ Entropy

W[u] =

Z

(u h(u) − f(u) + 1

2|x|²u) dx

L[u, v] = W[u] − W[u_∞] + W[v] − W[v_∞] +^β(₂^{τ) R}(u − v)φ dx

Proposition 4 Entropy decay

dLdτ = −2I[u] − 2I[v] − β^{2 R}(u + v)|∇φ|² dx

Theorem 5 Rates, d = 3

1) L has an exponential decay 2) m = 1: L(τ) ≤ C e^−τ

If ^R u₀ dx = ^R v₀ dx, L(τ) ≤ C e^−2τ

More general nonlinearities – Example:

F(σ) := ^R_IR3 v

dv

+exp(|v|²/2−σ)

f(s) = sF⁻¹(s) − ^R₀^s F⁻¹(x) dx

Stationary solutions, singular limit (dielectric constant → 0)

[Caffarelli, J.D., Markowich, Schmeiser]

[J.D., Markowich, Unterreiter]

∆f(u) + ∇ · (u∇V + u∇φ) = 0

∆f(v) + ∇ · (v∇V − v∇φ) = 0

∆φ = v − u − C(x)

Ω bounded domain, zero flux + insulator bound- ary conditions

R v dx = N, ^R u dx = P given

Global electroneutrality: N−P −^R_Ω C(x) dx = 0 Entropy: L = ^R H(u)dx+^R H(v) dx+¹₂ ^R |∇φ|² dx H⁰(u) = h(u) = ^f0(u)_u , g = h⁻¹,

u = g(α[φ] + φ) , v = g(β[φ] − φ)

∆φ = g(α[φ] + φ) − g(β[φ] − φ) − C(x) A convex functional

J[φ] = ¹₂ ^R |∇φ|² dx + ^R G(α[φ] + φ) dx

+ ^R G(β[φ] − φ) dx − N β[φ] − P α[φ]

which plays a crucial role in the study of sin- gular limits.

ENERGY-TRANSPORT:

STREATER’S MODELS























u_t = ∇ · ^hκ(∇u + ^u_θ(∇φ + ζ∇φ₀))ⁱ (uθ)_t = ∇ · (λ∇θ)

+∇ · [κ(θ∇u + u∇φ + ζu∇φ₀)]

+(∇φ + ζ∇φ₀)

·^hκ(∇u + ^u_θ(∇φ + ζ∇φ₀))ⁱ

±∆φ = u

with the boundary conditions

( ∂_nu + ^u_θ(∂_nφ + ζ∂_nφ₀) = 0 (no mass flux)

∂_nθ = 0 (no heat flux)

Mass: M = ^R u dx

Energy: ^R(u θ + φ₀ + ¹₂φ) dx Entropy: ^R u log(^u_θ) dx

dL

dt = −

Z |∇θ|²

θ² dx−

Z

u|∇u

u +1

θ(∇φ+∇φ₀)|² dx

Electrostatic case: [Biler, Esteban, J.D., Karch]

Stationary solutions: existence, uniqueness in some cases

L[u, θ] − L[u_∞, θ_∞]

= ^R s₁(_u^u

∞) u_∞ dx + ^R s₂(_θ^θ

∞) u dx s₁(t) = t logt + 1 − t

s₂(t) = t − 1 − log t

For uniqueness, use [Desvillettes, J.D.]:

Theorem 6 Ω of class C¹, bounded, ψ ∈ H₀¹(Ω) σ(t) = (−1 + ^q1 + χt²)/t², χ = 2E/M²

−σ(k∇ψk_L²) ∆ψ = e^−ψ

R

Ω e^−ψ dx , ψ_|∂Ω = 0 has a unique solution for any M > 0 and E > 0.

Lemma 7 Ω of class C¹, bounded, ψ ∈ H₀¹(Ω) log

Z

Ω e^−ψ dx

≥ C−2 logk∇ψk_L² (1+o(1)) as k∇ψk_L² → ∞.

Gravitational case: [Biler, Esteban, J.D., Markowich, Nadzieja]

Theorem 8 Ω bounded star-shaped in IR^d d ≥ 3, of class C¹

E/M² > `<sub>1</sub>: existence of bounded solutions E/M<sup>2</sup> < `₀, no nontrivial bounded solution If d ≥ 15, E/M² < ˜`₀: no solution in H₀¹(Ω)

1 2 3 4

2 4 6 8

2.5 3 3.5 4 4.5

2 4 6 8 10

Solutions of −∆ψ = m ^{R eψ}

e^ψ dx in the ball with zero Dirichlet boundary conditions:

left: (m,kψk_H¹), right: (m,kψk_L^∞)

FURTHER RESULTS

ON NONLINEAR DIFFUSIONS u_t = ∆_pu^m

Optimal constants for Gagliardo-Nirenberg ineq.

[Del Pino, J.D.]

Theorem 9 1< p < d, 1< a ≤ ^p(d_d−⁻_p¹⁾, b=p ^a_p⁻₋¹₁

kwk_b ≤ S k∇wk^θ_p kwk¹_a^−θ if a > p kwk^a ≤ S k∇wk^θ_p kwk¹_b^−θ if a < p Equality if w(x) = A(1 + B |x|

p

p−1)⁻

p−1 a−p

+

a > p: θ = _(q ^(q−p)d

−1) (d p−(d−p)q)

a < p: θ = _q(d(p ^(p−q)d

−q)+p(q−1))

Proof based on [Serrin, Tang]

A new logarithmic Sobolev inequality, with op- timal constant [Del Pino, J.D.]

Theorem 10 If kukL^p = 1, then

R |u|^p log |u| dx ≤ _p^d2 log [Lp R

|∇u|^p dx]

L^p = ^p_d ^p−_e^1p−1 π^−2p

"

Γ(₂^d+1) Γ(d ^p−_p¹+1)

#^p

d

Equality: u = π^−2dσ^−d

p−1

p Γ(^d₂+1)

Γ(d ^p−_p¹+1) e^{−σ1|x−x¯|}

p p−1

p = 2: Gross’ logaritmic Sobolev inequality p = 1: [Beckner]

[Del Pino, J.D.] Intermediate asymptotics of u_t = ∆_pu^m

Theorem 11 d ≥ 2, 1 < p < d

d−(p−1)

d(p−1) ≤ m ≤ _p−^p₁ and q = 1 + m − _p−¹₁

(i) ku(t, ·) − u_∞(t, ·)kq ≤ K R^{−(α2+d(1−1q))} (ii) ku^q(t,·) − u^q_∞(t,·)k1/q ≤ K R^−α2

(i): _p¹

−1 ≤ m ≤ _p−^p₁, (ii): ^d_d(p^−(p−1)

−1) ≤ m ≤ _p−¹₁ α = (1 − ¹_p (p − 1)

p−1

p ) _p−^p₁, R = (1 + γ t)^1/γ γ = (md + 1)(p − 1) − (d − 1)

u_∞(t, x) = 1

R^d v_∞(logR, x R) v_∞(x) = (C − ^p_mp⁻¹ (q−1)|x|

p

p−1)^1/(q₊ ⁻¹⁾ m 6= _p¹

−1

v_∞(x) = C e^{−(p−1)2|x|p/(p−1)/p} if m = (p − 1)⁻¹.

References

[Manuel DEL PINO, J.D.] The Optimal Euclidean L^p-Sobolev log- arithmic inequality

[Manuel DEL PINO, J.D.] Best constants for Gagliardo-Nirenberg inequalities and application to nonlinear diffusions

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