Entropy dissipation methods for diﬀusion equations

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Entropy dissipation methods for diffusion equations

Ansgar J¨ungel

Vienna University of Technology, Austria

www.jungel.at.vu

Ansgar J¨ungel (TU Wien) Entropy dissipation methods www.jungel.at.vu 1 / 129

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Literature

Main references

A. J¨ungel. Entropy dissipation methods for nonlinear partial differential equations. Lecture Notes, Bielefeld, 2012.

A. J¨ungel. The boundedness-by-entropy method for cross-diffusion systems. Nonlinearity 28 (2015), 1963-2001.

D. Matthes. Entropy methods and related functional inequalities.

Lecture Notes, 2008.

A. Arnold, P. Markowich, G. Toscani, and A. Unterreiter. On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Commun. PDEs26 (2001), 43-100.

J.A. Carrillo, A. J¨ungel, P. Markowich, G. Toscani, and A. Unterreiter.

Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monats. Math. 133 (2001), 1-82.

L. Evans. Entropy & partial differential equations. Lect. Notes, 2001.

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What mathematics skills are needed?

Entropy methods are intradisciplinary!

Partial differential equations: Fokker-Planck equations, parabolic equations, Sobolev spaces

Functional analysis: Lemma of Lax-Milgram, fixed-point theorems, compactness

Stochastics: Markov processes, Markov chain theory

Numerics: Finite-difference methods, finite-volume methods

Differential geometry: Geodesic convexity of entropy (not convered in these lectures)

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Entropy in physics

Entropy = measure of molecular disorder or energy dispersal Introduced by Clausius (1865) in thermodynamics (measure of irreversibility)

Statistical definition by Boltzmann, Gibbs, Maxwell (1870s)

S =−k_BX

i

p_ilogp_i, p_i : probability of ith microstate Von Neumann (1927): Quantum mechanical entropy

Bekenstein, Hawking (1970s): Black hole entropy (to satisfy second law of thermodynamics), entropy∼radius²: description of volume encoded on its boundary

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Entropy in information theory

1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 1 0 1

Shannon 1948: Concept of information entropy (measure of information density)

Information content: I(p) =−log₂p,p: probability of event Rationale: I(1) = 0: no information content of sure events,

I(p₁p₂) =I(p₁) +I(p₂): information of independent events additive Entropy = expected information content

S =X

i∈Σ

piI(pi) =−X

i∈Σ

pilog₂pi

Applications: Redundancy in language structure, data compression (entropy coding, idea: minimize entropy)

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Entropy in mathematics

Mathematical entropy isnonincreasing, i.e. negative physical entropy Hyperbolic conservation laws(Lax 1971):

∂_tu+∂_xf(u) = 0, u ∈Rⁿ

h is an entropy if∃q:q^′(u) =f^′(u)h^′(u) and entropy inequality:

∂th(u) +∂xq(u)≤0

Kinetic equations: entropy h(f) =R

R^df logf dx gives a priori estimates for Boltzmann equation (DiPerna/Lions 1989), large-time behavior of solutions (Desvillettes/Villani 1990, Mouhot 2006)

Large-time behavior for stochastic processes (Bakry/Emery 1985) and parabolic equations (Toscani 1997)

Regularity for parabolic equations (Nash 1958)

Relations to gradient flows in metric spaces (Ambrosio, Otto, Savar´e...), functional inequalities (Gross, Arnold et al., Dolbeault...)

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Entropy in literature

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Entropy and partial differential equations

Generally: Entropy S(E,X₁, . . . ,Xn) is function of internal energyE and state variables X_i (e.g. density, volume) such that

S is concave, _∂E^∂S >0, S homogeneous of order one.

Def. temperature ¹_θ = _∂E^∂S, chem. potentialµ=−θ^∂S_∂ρ (ρ: mass density) Euler equations in thermodynamics:

∂tρ+ div(ρv) = 0,

∂t(ρv) + div(ρv⊗v−T) = 0,

∂_t(ρe) + div(ρve+q) =T :∇v

wherev: velocity,T: stress tensor,e: internal energy, q: heat flux Energy balance:

d dt

Z

R^d

ρ

2|v|²+ρe

dx = 0 Monoatomic ideal gas: energy densityρe = ³₂ρθ, entropy densityρs =−ρlog(ρ/θ^3/2)⇒ ^∂(ρs)_∂(ρe) = ¹_θ >0

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Aims of lecture course

To introduce into several entropy methods for partial differential equations (PDEs)

To use entropy methods to prove the qualitative behavior of solutions to PDEs (large-time asymptotics, existence analysis,L^∞ bounds) To prove functional inequalities (convex Sobolev inequalities) To relate entropy methods to physical principles and the theory of stochastic processes

To introduce into the theory of cross-diffusion systems

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Overview

1 Introduction

2 Entropies

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Example: Heat equation

∂_tu = ∆u, u(0) =u₀ ≥0 inT^d (torus), t>0 Steady state: u_∞=R

T^du₀dx =R

T^du(t)dx, meas(T^d) = 1 Question: u(t)→u_∞ as t→ ∞ in which sense and how fast?

Define the functionalH₂[u] =R

T^d(u−u_∞)²dx Compute time derivative:

dH₂

dt [u] = 2 Z

T^d

(u−u_∞)∂_tudx =−2

entropy production z }| { Z

T^d|∇u|²dx ≤0 Poincar´e inequality: H2[u] =ku−u_∞k²L² ≤CPk∇uk²L²

Combining expressions:

dH₂

dt =−2k∇uk²L² ≤ −2C_P⁻¹H₂[u]

By Gronwall’s inequality, ku(t)−u_∞k²_L² ≤e⁻^2C^P⁻¹^tku₀−u_∞k²_L²

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Example: Heat equation

∂_tu = ∆u, u(0) =u₀ ≥0 inT^d (torus), t>0 Conclusion: ku(t)−u_∞kL²≤e⁻^C^P⁻¹^tku₀−u_∞kL²

Same result with spectral theory: C_P⁻¹= first eigenvalue of −∆ Since spectral analysis gives the same result: What is the benefit?

First answer: Different “distances” admissible Entropy functional H₁[u] =R

T^dulog(u/u_∞)dx ≥0 dH₁

dt [u] = Z

T^d

log u

u_∞+ 1

∂tudx =−4 Z

T^d|∇√ u|²dx Logarithmic Sobolev ineq.: R

T^dulog(u/u_∞)dx ≤CLR

T^d|∇√ u|²dx By Gronwall inequality,

dH1

dt [u]≤ −4C_L⁻¹H1[u] ⇒ H1[u(t)]≤e⁻^4C^L⁻¹^tH1[u0], t≥0

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Example: Heat equation

Second answer: Method applicable to nonlinear equations Quantum diffusion equation: ∂tu=−div(u∇^∆^√^√_u^u) in T^d Occurs in quantum semiconductor modeling,u: electron density Entropy functional: H₁[u] =R

T^dulog(u/u_∞)dx Entropy production:

dH₁

dt [u] =− Z

T^d

div

u∇∆√u

√u

logudx =− Z

T^d

∆√u

√u ∆udx

≤−κ Z

T^d

(∆√

u)²dx ≤ − κ CP

Z

T^d|∇√

u|²dx ≤ − κ CPCL

H₁[u]

Exponential decay of u(t) to u_∞ with explicit rate:

H₁[u(t)]≤e⁻^κt/(C^P^C^L⁾H₁[u₀], t ≥0

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Strategy

∂_tu+A(u) = 0, t >0, u(0) =u₀ Strategy:

Given an entropy H[u], compute entropy production:

−dH/dt =hA(u),H^′[u]i

Find relation between entropy and entropy production:

H[u]≤ChA(u),H^′[u]i ⇒ dH/dt ≤ −CH

By Gronwall’s inequality, conclude exponential decay:

H[u(t)]≤e⁻^CtH[u₀]

Entropy methods can do much more:

Self-similar asymptotics

A priori estimates and global-in-time existence analysis

Proof of functional inequalities (like logarithmic Sobolev ineq.) Positivity of solutions and L^∞ bounds (no maximum principle!) Uniqueness of weak solutions

Stability of numerical discretizations (structure-preservation)

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Definitions

Setting:

A:D(A)⊂X →X^′ operator, consider ∂tu+A(u) = 0,t >0, u(0) =u₀

Steady state: u_∞∈D(A) solves A(u_∞) = 0 Definitions:

Lyapunov functional: H:D(A)→R such that ^dH_dt[u(t)]≤0, t≥0 Entropy: H :D(A)→Rconvex Lyapunov functional such that

∃Φ∈C⁰(R): Φ(0) = 0 and

d(u,u∞)≤Φ(H[u]−H[u∞]) foru∈D(A) and some metricd. Entropy production: EP[u(t)] =−^dH_dt[u(t)]

Entropy of kth order: containskth-order partial derivatives No clear definition of (mathematical) entropy in the literature!

Examples: F₁: Fisher information H_α[u] =

Z

Ω

(u^α−u^α_∞)dx, F_α[u] = Z

Ω|∇u^α/2|²dx, α≥1

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Heat equation revisited

∂_tu = ∆u, u(0) =u₀ ≥0 inT^d (torus), t>0 Claim: H₁[u] =R

T^dulog(u/u_∞)dx is an entropyfor the heat equation Proof:

Lyapunov functional: ^dH_dt¹[u] =−R

T^d|∇√

u|²dx ≤0 Convexity: u 7→H₁[u] is convex

Csisz´ar-Kullback inequality for Φ(s) =C_φ√

s,d(f,g) =kf −gkL¹: d(u,u_∞)≤C_φ(H1[u]−H1[u_∞])^1/2 using H1[u_∞] = 0

Lemma (Csisz´ar-Kullback-Pinsker)

Let φ∈C²(R) be strictly convex, φ(1) = 0, and R

T^dfdx =R

T^dgdx = 1.

Then, for some C_φ>0,

kf −gk²L¹ ≤C_φ Z

T^d

φ f

g

gdx

Proof: Taylor expansion of φaround 1

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Overview

1 Introduction

2 Entropies

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Linear Fokker-Planck equation

∂_tu = div(∇u+u∇V) in R^d, t >0, u(u) =u₀ ≥0 Assumptions: R

R^du₀dx = 1, lim_|_x_|→∞V(x) =∞ (confinement) Steady state: 0 =∇u_∞+u_∞∇V =u_∞∇(logu_∞+V) ⇒ u_∞=ce⁻^V, wherec is such thatR

R^du_∞dx = 1 Entropy: Let φ∈C⁴ be convex

H_φ[u] = Z

R^d

φ u

u_∞

u_∞dx−φ Z

R^d

udx

Theorem (Bakry/Emery ’85, Arnold/Markowich/Toscani/Unterreiter ’01) Let u₀logu₀∈L¹(R^d),∇²V ≥λ >0,1/φ^′′ concave. Then

ku(t)−u_∞kL¹ ≤e⁻^λtC_φ^1/2H_φ[u0]^1/2, t >0

Example➊:φ(s) =s(logs−1) + 1,φ(s) =s^α−1−α(s−1) (1< α≤2) Example ➋:φ(s) =slogs,V(x) = ¹₂|x|² then λ= 1 (optimal!)

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Proof: First time derivative

∂tu= div(∇u+u∇V) = div

u_∞∇ u u_∞

, u_∞=ce⁻^V First time derivative: H_φ[u] =R

R^dφ(u/u_∞)u_∞dx−φ(1), set ρ:= _u^u_∞ dH_φ

dt = Z

R^d

φ^′(ρ)∂tudx =− Z

R^d

φ^′′(ρ)|∇ρ|²u_∞dx ≤0 Second time derivative: (key idea!)

d²H_φ

dt² [u] =− Z

R^d

φ^′′′(ρ)∂tu|∇ρ|²+ 2φ^′′(ρ)∇ρ· ∇∂tρu_∞

dx =−I₁−I₂ First integral:

I₁ =− Z

R^d∇ φ^′′′(ρ)|∇ρ|²

·(u_∞∇ρ)dx

=− Z

R^d

φ^′′′′(ρ)|∇ρ|⁴+ 2φ^′′′(ρ)∇ρ∇²ρ∇ρ u_∞dx

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Proof: Second time derivative

d²H_φ

dt² [u] =−I1−I2, I1 =− Z

R^d

φ^′′′′(ρ)|∇ρ|⁴+ 2φ^′′′(ρ)∇ρ∇²ρ∇ρ u_∞dx Second integral: compute∇∂tρ=∇∆ρ− ∇²ρ· ∇V − ∇²V∇ρ,ρ= _u^u

∞

I₂ = 2 Z

R^d

φ^′′(ρ)∇ρ· ∇∂tρu_∞dx

= 2 Z

R^d

φ^′′(ρ) ∇ρ· ∇∆ρ− ∇ρ∇²ρ∇V −

≥λ|∇ρ|²

z }| {

∇ρ∇²V∇ρ dx

≤2 Z

R^d

φ^′′(ρ) div(∇²ρ∇ρ)− |∇²ρ|²− ∇ρ∇²ρ∇V −λ|∇ρ|² u_∞dx

= 2 Z

R^d−φ^′′′∇ρ∇²ρ∇ρu_∞−φ^′′∇ρ∇²ρ

z =0}| {

(∇u_∞+u_∞∇V)−φ^′′|∇²ρ|²u_∞ dx

−2λ Z

R^d

φ^′′(ρ)|∇ρ|²u_∞dx, note:

Z

R^d

φ^′′(ρ)|∇ρ|²u_∞dx = dH_φ dt

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Proof: Second time derivative

Add both integralsI₁ and I₂ and useφ convex, 1/φ^′′ concave:

d²Hφ

dt² [u] = Z

R^d

φ^′′′′|∇ρ|⁴+ 4φ^′′′∇ρ∇²ρ∇ρ+ 2φ^′′|∇²ρ|²

u_∞dx−2λdHφ

dt

= Z

R^d

2φ^′′∇²ρ+φ^′′′

φ^′′∇ρ⊗ ∇ρ²

| {z }

≥0

+

φ^′′′′−2(φ^′′′)² φ^′′

| {z }

=−(φ^′′)²(1/φ^′′)^′′≥0

|∇ρ|⁴

u_∞dx

−2λdH_φ

dt ⇒ d²H_φ

dt² [u]≥ −2λdH_φ dt Integrate over (t,∞):

slim→∞

dH_φ dt [u(s)]

| {z }

=0

−dH_φ

dt [u(t)]≥ −2λ lim

s→∞H_φ[u(s)]

| {z }

=0

+2λH_φ[u(t)]

Gronwall lemma and Csisz´ar-Kullback inequality:

ku(t)−u_∞k²L¹ ≤C_φH_φ[u(t)]≤C_φe⁻^2λtH[u₀]

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Bakry-Emery: Remarks

Theorem (Bakry/Emery ’85, Arnold/Markowich/Toscani/Unterreiter ’01) Let u₀logu₀∈L¹(R^d),∇²V ≥λ >0,φconvex,1/φ^′′ concave. Then

ku(t)−u_∞kL¹ ≤e⁻^λtC_φ^1/2H_φ[u0]^1/2 Exponential L¹ decay with (optimal) rate λ

Difficult part of proof: justify computations for weak solutions Proof yields convex Sobolev inequality for all (smooth) u (ρ= _u^u_∞):

H_φ[u] = Z

R^d

φ(ρ)u_∞dx−φ(1)≤ − 1 2λ

dH_φ

dt [u] = 1 2λ

Z

R^d

φ^′′(ρ)|∇ρ|²u_∞dx Example: V(x) = ¹₂|x|²,φ(s) =s(logs−1) + 1, thenλ= 1

Z

R^d

ulogudx +d

2 log(2π) +d ≤ 1 2

Z

R^d

|∇u|² u dx,

Z

R^d

udx = 1 Benefit: Simultaneous proof of deacy rate and convex Sobolev ineq.

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Bakry-Emery for Markov processes

Given Markov process (Xt)_t>0, semigroupStf(x) =E[f(Xt)|X₀ =x], infinitesimal generatorLf = limt→0(Stf −f)/t

Example: Lf = ∆f −x· ∇f onR^d (Fokker-Planck-type), S_tf₀ is solution to∂tf =Lf,f(0) =f₀

Assume: ∃ invariant measureπ: R

fdπ=R Stfdπ Carr´e-du-champ operator: Γ(f,g) = ¹₂(L(fg)−fLg−gLf) Example: Γ(f,g) =∇f · ∇g

Gamma-deux operator: Γ₂(f,g) = ¹₂(LΓ(f,g)−Γ(Lf,g)−Γ(f,Lg)) Example: Γ₂(f,f) =|∇²f|²+|∇f|² ⇒ Γ₂(f,f)≥Γ(f,f)

Theorem (Bakry/Emery 1985)

Let φ∈C² be convex,1/φ^′′ concave, and∃λ >0: Γ2(f,f)≥λΓ(f,f) for all f ≥0. Then for probability density functionsρ,

Z

R^d

φ(ρ)dπ−φ Z

R^d

ρdπ

≤ 1 2λ

Z

R^d

φ^′′(ρ)Γ(ρ, ρ)dπ

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Bakry-Emery for Markov processes

Example: Fokker-Planck-type equation

We have Γ(ρ, ρ) =|∇ρ|² anddπ=u_∞dx withu_∞=ce⁻^V Choose ρ=u/u_∞: R

R^dρdπ=R

R^dudx

Relation to previous convex Sobolev inequality:

Z

R^d

φ(ρ) |{z}dπ

=u^∞dx

−φ Z

R^d

ρdπ

|{z}

=udx

≤ 1 λ

Z

R^d

φ^′′(ρ) Γ(ρ, ρ)

| {z }

=|∇ρ|²

dπ

|{z}

=u^∞dx

Example ➊:φ(s) =s(logs−1) gives logarithmic Sobolev inequality Z

R^d

ρlogρdπ− Z

R^d

ρdπlog Z

R^d

ρdπ ≤ 1 2λ

Z

R^d

Γ(ρ, ρ) ρ dπ Example ➋:φ(s) =s² gives Poincar´e inequality

Z

R^d

ρ−

Z

R^d

udx 2

dπ ≤ 1 λ

Z

R^d

Γ(ρ, ρ)dπ Benefit: Abstract framework for convex Sobolev inequalities

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Extensions of the Bakry-Emery method

➊ More on convex Sobolev inequalities: Compare Poincar´e, logarithmic Sobolev, and Beckner inequalities

➋ Isoperimetric inequality for entropy: Relation to information theoretical approach (entropy power)

➌ Relaxation to self-similarity: Analyze intermediate asymptotics of solution of heat equation

➍ Linear Fokker-Planck equations with variable diffusion matrix

➎ Nonlinear Fokker-Planck equations

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➊ More on convex Sobolev inequalities

Z

R^d

φ(ρ)u_∞dx−φ Z

R^d

ρu_∞dx

≤ 1 2λ

Z

R^d

φ^′′(ρ)|∇ρ|²u_∞dx Logarithmic Sobolev inequality: φ(s) =slogs (forR

R^dρu_∞dx = 1) Z

R^d

ρlogρu_∞dx ≤ 2 λ

Z

R^d|∇ρ^1/2|²u_∞dx Poincar´e inequality: φ(s) =s²

Z

R^d

ρ²u_∞dx− Z

R^d

ρu_∞dx 2

≤ 1 λ

Z

R^d|∇ρ|²u_∞dx Beckner inequality: φ(s) =s^α, 1< α <2

1 α−1

Z

R^d

ρ^αu_∞dx − Z

R^d

ρu_∞dx α

≤ 2 αλ

Z

R^d|∇ρ^α/2|²u_∞dx

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Relations between functional inequalities

1 α−1

Z

R^d

ρ^αu_∞dx− Z

R^d

ρu_∞dx α

≤ 2 αλ

Z

R^d|∇ρ^α/2|²u_∞dx α→1 in Beckner gives logarithmic Sobolev inequality since

1 α−1

Z

R^d

(ρ^α⁻¹−1)ρu_∞dx → Z

R^d

ρlogρu_∞dx α→2 in Beckner gives Poincar´e inequality

Logarithmic Sobolev implies Poincar´e (useρ= 1 +εg with R

R^dgu_∞dx and ε→0) and Beckner (Latala/Oleszkiewicz 2000) Beckner (a)

Poincaré logarithmic

Sobolev

a = 2 a g 1

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➋ Isoperimetric inequality for entropy

Aim: Relation between logarithmic Sobolev ineq. and isoperimetric ineq.

Entropy: H[u] =R

R^dulogudx Fisher information: I[u] = 4R

R^d|∇u^1/2|²dx Entropy power: N[u] = exp(−²dH[u]) Theorem (Isoperimetric inequality for entropy)

For all probability density functions u, N[u]I[u]≥2πed.

Equivalent formulation: 4πexp(_d²H[u])≤ _ed⁸ R

R^d|∇u^1/2|²dx Compare with isoperimetric inequality on R²: 4πA≤L² for closed curve with lengthL and enclosed areaA

Approximating e^z ≥z giveslogarithmic Sobolev inequality:

2 d

Z

R^d

ulogudx = 2

dH[u]≤ 2 πed

Z

R^d|∇u^1/2|²dx

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Isoperimetric inequality for entropy

Theorem (Isoperimetric inequality for entropy)

For all probability density functions u, N[u]I[u]≥2πed.

Proof:

N[u] is concave (Costa 1985, Villani 2000) since d²N

dt² = 2

d 2

N dH

dt 2

−d 2

d²H dt²

≤0 Let v solve ∂tv = ∆v,v(0) =u:

d

dt(N[v]I[v]) = 2 dN

I²+d

2 dI dt

= 2 dN

dH dt

2

−d 2

d²H dt²

≤0 N[v(t)]I[v(t)] reaches minimum as t→ ∞ ⇒ N[v(t)]I[v(t)]≥m Scaling argument: m=N[M]I[M], whereM(x) = _(2πt)¹_d/2exp(−^|^x_2t^|²) Conclusion: N[u]I[u] =N[v(0)]I[v(0)]≥N[M]I[M] = 2πed

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➌ Relaxation to self-similarity

Consider heat equation in whole space:

∂tu = ∆u in R^d, t>0, u(0) =u₀, Z

R^d

u₀dx = 1 Explicit solution:

u(x,t) = (4πt)⁻^d/2R

R^dexp(−|x−y|²/(4t))u0(y)dy, thus u(t)→0 inL^∞ ast → ∞

Entropy is decreasing but H1[u(t)] =

Z

R^d

ulogudx ≤ Z

R^d

u(t)dxlogku(t)k^L^∞ → −∞ (t→ ∞) Entropy method fails! Problem: u_∞= 0 has not unit mass

Solution: Analyze u(t)−U(t)→0, where self-similar solution U(x,t) = 1

(2π(2t+ 1))^d/2 exp

− |x|² 2(2t+ 1)

Idea: Transform variables to makeU stationary: y =x/√ 2t+ 1, s = log√

2t+ 1, and v(y,s) =e^dsu(e^sy,¹₂(e^2s−1))

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Relaxation to self-similarity

∂tu = ∆u in R^d, v(y,s) =e^dsu(e^sy,¹₂(e^2s −1)) Function v solves ∂sv = divy(∇^yv+yv) in R^d

Self-similar solution becomes

M(y) = (2t+ 1)^d/2U(x,t) = (2π)⁻^d/2exp(−|y|²/2) Bakry-Emery shows:

kv(s)−Mk²L¹≤2e⁻^2sH₁[u₀], s >0 Back-transformation:

kv(s)−Mk²L¹ =ku(t)−U(t)k²L¹, 2e⁻^2s = 2(2t+ 1)⁻¹ Theorem

Let R

R^du0dx = 1, u solves ∂tu= ∆u in R^d, u(0) =u0, and U(x,t) = (2π(2t+ 1))⁻^d/2exp(−|x|²/(2(2t+ 1))). Then

ku(t)−U(t)kL¹ ≤(2t+ 1)⁻^1/2(2H₁[u(0)])^1/2 ∼t⁻^1/2 (t → ∞)

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➍ Variable diffusion matrix

∂tu = div(D(x)(∇u+u∇V)) = div(D(x)u_∞∇ρ) inR^d, D(x)∈R^d^×^d Steady state: u_∞=ce⁻^V,ρ= _u^u_∞

Assumptions: D(x) pos. definite, lim_|_x_|→∞V(x) =∞,R

R^du_∞dx = 1 Entropy: H[u] =R

R^dφ(ρ)u_∞dx,φ convex,φ(1) = 0, 1/φ^′′ concave Entropy production: dH

dt [u] =− Z

R^d

φ^′′(ρ)∇ρ^⊤D∇ρu_∞dx ≤0 Theorem (Arnold/Markowich/Toscani/Unterreiter 2001)

Assume H[u(0)]<∞ and

D(x) =const., ∇²V ≥λD⁻¹ or D(x) =a(x)I,

1 2 −^d₄

1

a∇a⊗ ∇a+¹₂(∆a− ∇a· ∇V)I +a∇²V + ¹₂(∇V ⊗ ∇a+∇a⊗ ∇D)≥λI Then

H[u(t)]≤e⁻^2λtH[u(0)], t ≥0

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Nonsymmetric Fokker-Planck equations

∂tu = div[D(x)(∇u+u(∇V +F(x)))] in R^d,

Assume: div(DFu_∞) = 0 in R^d ⇒u_∞=ce⁻^V is still steady state Operator div(D(∇u+u∇V)) = div(Du_∞∇(u/u_∞)) symm. inL²(u_∞⁻¹) Operator div(DuF) is skew-symmetric inL²(u⁻_∞¹)

⇒ evolution = symmetric + skew-symmetric Entropy production: (some computations needed)

dH

dt [u] =− Z

R^d

φ(ρ)∇ρ^⊤D∇ρu_∞dx− Z

R^d

φ(ρ) div(DFu_∞)

| {z }

=0

dx

Entropy and entropy production are independent of F Prove as before that ^d_dt²^H₂ + 2λ^dH_dt ≥0

Implies exponential decay for non-symmetric equation

Bolley/Gentil 2010: Assumption div(DFu_∞) = 0 not necessary

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Degenerate Fokker-Planck equations

∂tu = div(D∇u+Cxu) inR^d, u(0) =u₀ Matrix D∈R^d×d constant anddegenerate,C ∈R^d×d

Assumption➊:∀v: C^⊤v=λCv ⇒ v 6∈ker(D)

Consequence: u₀ ∈L¹ ⇒ u∈C^∞(R₊×R^d) (hypoellipticity) Assumption➋:∀λ_C eigenvalues ofC^⊤: Re(λ_C)>0

Consequence: Drift towards x = 0 due to confinement potential Theorem (Erb/Arnold 2014)

Let assumptions hold,µ= min{Re(λC)}. Then ∃c0>0:

H[u(t)]≤c0e⁻^2µtH[u0], t >0

if all λ∈σ(C) with Re(λ) =µare non-defective (i.e. geometric = algebraic multiplicity), otherwise reduced rate 2(µ−ε),ε >0.

Idea of proof: ^dH_dt[u] = 0 foru 6=u_∞ possible, thus use modified functional I[u] =

Z

R^d

φ^′′(ρ)∇ρ^⊤P∇ρu_∞dx, P positive definite

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Generalized Beckner inequalities

1 α−1

Z

R^d

u^αµdx − Z

R^d

uµdx α

≤ 2 αλ

Z

R^d

D(x)|∇u^α/2|²µdx Valid for u^α/2∈H¹(R^d;µ)∩L^2/α(R^d;µ), 1< α≤2

Ifµ(x) =e^−|^x^|²^/2,D(x) = 1 then λ= 1 for all 1< α≤2

Question: Determine λfor D(x)6= const.? (Matthes/A.J./Toscani ’11) Example: Linearized fast-diffusion eq. ∂tu =D(x)∆u−x· ∇u

D(x) =α²+β²|x|²

µ(x) =C(α²+β²|x|²)⁻¹⁻^1/(2β²⁾ β >0: no Sobolev inequality and λ→0 asα→1

Pointwise Bakry-Emery approach (Γ₂ ≥λΓ) does not work

Idea: Use integral expressions Figure: p= 2/α,Cp= 2/(αλ)

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➎ Nonlinear Fokker-Planck equations

Aim: Extend Bakry-Emery method to

∂tu = div(∇f(u) +u∇V) in Ω, t >0, u(0) =u₀ ≥0 where Ω =R^d or Ω bounded (with no-flux boundary cond.). Assume

f ∈C³ strictly increasing, f(0) = 0, f(s)≤ d−^d1sf^′(s),f^′′(0)>0 Ω convex, ∇²V ≥λ>0, inf_ΩV = 0

Example: f(s) =s^m (m≥ d^d−1),V(x) = ^λ₂|x|² for x∈R^d Steady state: u_∞(x) = (N−^m_2m⁻¹|x|²)^1/(m₊ ⁻¹⁾,N >0 Relative entropy: H^∗[u] =H[u]−H[u_∞], where

H[u] = Z

R^d

(Φ(u) +uV(x))dx, Φ^′′(u) = f^′(u) u Theorem (Carrillo/A.J./Markowich/Toscani/Unterreiter 2001) Let H[u₀]<∞. Then, for t >0,ku(t)−u_∞kL¹ ≤e⁻^λtC(H^∗[u₀]).

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Proof

^(f^{(u) =}^u^m^,^V^(x^{) =}^λ2|x|²)

Theorem (Carrillo/A.J./Markowich/Toscani/Unterreiter 2001) Let H[u₀]<∞. Then, for t >0,ku(t)−u_∞kL¹ ≤e⁻^λtC(H^∗[u₀]).

Step 1. First time derivative (entropy production) dH^∗

dt [u] =− Z

R^d

u|∇(h(u) +V)|²dx ≤0, h(u) = m

m−1u^m⁻¹ Step 2. Second time derivative

d²H^∗

dt² [u] =−2λdH^∗

dt [u]−2R(t) R(t) =

Z

R^d

u^m (m−1)(∆(h(u) +V))²+|∇²(h(u) +V)|² dx ≥0

⇒ d²H^∗

dt² [u]≥ −2λdH^∗ dt [u]

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Proof

^(f^{(u) =}^u^m^,^V^(x^{) =}^λ2|x|²) d²H^∗

dt² [u]≥ −2λdH^∗ dt [u]

Step 3. Functional inequality: integrate, use limt→∞ dH^∗

dt [u(t)] = 0 dH^∗

dt [u(t)]≤ −2λH^∗[u₀] ⇒H^∗[u(t)]≤e⁻^2λtH^∗[u₀] Step 4. Csisz´ar-Kullback inequality: introducebu =αu1_{|_x_|≤_R_}

ku−u_∞kL¹ ≤ ku−bukL¹

| {z }

≤H^∗[u]^1/2

+kbu−u_∞kL¹

| {z }

Ce⁻^λt

≤Ce⁻^λt

Question: Does entropy production ineq. relate to functional ineq.? Yes:

Gagliardo-Nirenberg inequality: Let 1<p <2,u ∈H¹(R^d)∩L^p(R^d):

kukL^p/2+1 ≤Ck∇uk^θL¹kuk¹L⁻^p^θ, θ= _(2+p)(d(2^d(2⁻₋^p)_p)+2p) Proof: Show R

R^dv^mdx ≤AR

R^d|∇v^m⁻^1/2|²dx+B(R

R^dvdx)^γ,m= ^p+2_2p

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Summary

Let u(t) solve ∂tu+A(u) = 0, let u_∞ solveA(u_∞) = 0.

Define entropy H[u]. Entropy method:

ComputedH/dt andd²H/dt²

Show that d²H/dt²+κdH/dt ≥0⇒ H[u(t)]≤e⁻^κtH[u(0)]

Csisz´ar-Kullback inequality gives exponentialL¹ decay rate:

ku(t)−u_∞kL¹ ≤e⁻^(κ/2)tC(H[u(0)]), t>0 Also yields convex Sobolev inequalitywith explicit constant:

entropy =H[u]≤κ⁻¹ −^dHdt[u]

=κ⁻¹×entropy production Applies toMarkov processes (see book of Bakry/Gentil/Ledoux ’14) Also yields intermediate asymptoticsof typeku(t)−U(t)kL¹ ≤Ct⁻^γ Very robust for nonsymm./degenerate/nonlineardiffusion equations Problem: Many integration by parts are needed – make them systematic!

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Overview

1 Introduction

2 Entropies

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Systematic integration by parts: Motivation

Second time derivative d²H/dt² requires well chosen integrations by parts.

Aim: Make the integrations by parts systematic.

Motivation: Consider thin-film equation

∂tu =−(u^βuxxx)x in T(torus), t>0, u(0) =u₀≥0 Models the flow of thin liquid along surface with film height u(x,t) Entropy Hα[u] = _α(α¹₋₁₎R

Tu^αdx: For which α >1 isHα an entropy?

dH_α

dt [u] = 1 α−1

Z

T

u^α⁻¹∂tudx = Z

T

u^α+β⁻²uxxxuxdx

=−(α+β−2) Z

T

u^α+β⁻³u²_xuxxdx− Z

T

u^α+β⁻²u_xx² dx, u_x²uxx = 1 3(u_x³)x

=−1

3(α+β−2)(α+β−3) Z

T

u^α⁻^β⁻⁴u_x⁴dx− Z

T

u^α+β⁻²u_xx² dx≤0

if 2≤α+β ≤3 but ³₂ ≤α+β ≤3 is optimal!

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Idea of method

Example: Thin-film equation∂tu =−(u^βuxxx)x on torus T Entropy production forH_α[u] = _α(α¹₋₁₎R

Tu^αdx dHα

dt [u] = 1 α−1

Z

T

u^α⁻¹∂tudx = Z

T

u^α+β⁻²uxuxxxdx =:−EP[u]≤0 ? Standard integration by parts:

EP[u] =− Z

T

u^α+β⁻²uxuxxxdx = Z

T

u^α+β⁻¹

α+β−1uxxxxdx Formalization of integration by parts:

I₃ = Z

T

u^α+β

(α+β−1)ux

u uxxx

u +uxxxx

u

dx

= Z

T

(u^α+β⁻¹uxxx)xdx = 0

⇒ EP[u] =EP[u] +cI3 with c = _α+β¹₋₁

Entropy dissipation methods for diﬀusion equations

Entropy dissipation methods for diffusion equations

Contents

Literature

What mathematics skills are needed?

Entropy in physics

Entropy in information theory

1

1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 1 0 1

Entropy in mathematics

Entropy in literature

Entropy and partial differential equations

Aims of lecture course

Overview

Example: Heat equation

Example: Heat equation

Example: Heat equation

Strategy

Definitions

Heat equation revisited

Overview

Linear Fokker-Planck equation

Proof: First time derivative

Proof: Second time derivative

Proof: Second time derivative

Bakry-Emery: Remarks

Bakry-Emery for Markov processes

Bakry-Emery for Markov processes

Extensions of the Bakry-Emery method

➊ More on convex Sobolev inequalities

Relations between functional inequalities

➋ Isoperimetric inequality for entropy

Isoperimetric inequality for entropy

➌ Relaxation to self-similarity

Relaxation to self-similarity

➍ Variable diffusion matrix

Nonsymmetric Fokker-Planck equations

Degenerate Fokker-Planck equations

Generalized Beckner inequalities

➎ Nonlinear Fokker-Planck equations

Proof

Proof

Summary

Overview

Systematic integration by parts: Motivation

Idea of method