Entropy dissipation methods for diﬀusion equations

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Entropy dissipation methods for diffusion equations

Ansgar J¨ungel

Vienna University of Technology, Austria

Winter 2017/2018

Ansgar J¨ungel (TU Wien) Entropy dissipation methods Winter 2017/2018 1 / 56

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Literature

Main references

A. J¨ungel. Entropy Methods for Diffusive Partial Differential Equations.

BCAM Springer Briefs, Springer, 2016.

D. Matthes. Entropy methods and related functional inequalities.

Lecture Notes, 2008.

A. J¨ungel and D. Matthes. An algorithmic construction of entropies in higher-order nonlinear PDEs. Nonlinearity19 (2006), 633-659.

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

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

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Introduction

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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Introduction

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

pilogpi, pi : 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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Introduction

Entropy in information theory

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

p_iI(p_i) =−X

i∈Σ

p_ilog₂p_i

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

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Introduction

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:∂_iq(u) =P

j∂_u_if_j(u)∂_u_jh(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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Introduction

Entropy and partial differential equations

Generally: Entropy S(E,X1, . . . ,Xn) is function of internal energyE and state variables X_i (e.g. volume, mole number) 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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Introduction

Aims of lecture course

To introduce into several entropy methods for partial differential equations (PDEs) → Arnold, J¨ungel, Schmeiser

To use entropy methods to prove the qualitative behavior of solutions to PDEs→ Arnold, J¨ungel, Schmeiser

To prove functional inequalities (convex Sobolev inequalities)

→ Arnold

To relate entropy methods to physical principles and the theory of stochastic processes →Schmeiser

To introduce into the theory of cross-diffusion systems→ J¨ungel

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Introduction

Overview

1 Introduction

2 Entropies

5 Exercises

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Entropies

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∞k²_L₂ ≤C_Pk∇uk²_L₂

Combining expressions:

dH2

dt =−2k∇uk²_L2 ≤ −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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Entropies

Example: Heat equation

∂_tu = ∆u, u(0) =u₀ ≥0 inT^d (torus), t>0

Conclusion: ku(t)−u∞k_L2≤e^−C^P⁻¹^tku₀−u∞k_L2

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 ≤C_LR

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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Entropies

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 C_P

Z

T^d

|∇√

u|²dx ≤ − κ

C_PC_LH₁[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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Entropies

Strategy

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

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

−dH/dt =hA(u),H⁰[u]i

Find relation between entropy and entropy production:

H[u]≤ChA(u),H⁰[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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Entropies

Definitions

Setting:

A:D(A)⊂X →X⁰ 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 Hα[u] =

Z

Ω

(u^α−u^α_∞)dx, Fα[u] = Z

Ω

|∇u^α/2|²dx, α≥1

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Entropies

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 −gk_L1: d(u,u∞)≤Cφ(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 Cφ>0,

kf −gk²_L1 ≤C_φ Z

T^d

φ f

g

gdx

Proof: Taylor expansion of φaround 1

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Entropies

Overview

1 Introduction

2 Entropies

5 Exercises

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Systematic integration by parts

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^βu_xxx)_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 H_α[u] = _α(α−1)¹ R

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

dH_α

dt [u] = 1 α−1

Z

T

u^α−1∂_tudx = Z

T

u^α+β−2u_xxxu_xdx

=−(α+β−2) Z

T

u^α+β−3u²_xu_xxdx− Z

T

u^α+β−2u_xx² dx, u_x²u_xx = 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−2u²_xxdx≤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 forHα[u] = _α(α−1)¹ R

Tu^αdx dHα

dt = 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+β−1¹

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Integration-by-parts rules

EP[u] =− Z

T

u^α+β−2uxuxxxdx≥0 ?

Question: How many independent rules of integration by parts?

I₁ = Z

T

u^α+β

(α+β−3)u_x u

4

+ 3u_x u

2 u_xx u

dx = 0 I₂ =

Z

T

u^α+β

(α+β−2)u_x u

2 u_xx

u +u_xx u

2

+u_x u

u_xxx u

dx = 0 I3 =

Z

T

u^α+β

(α+β−1)ux

u uxxx

u +uxxxx

u

dx = 0

Aim: Prove that∃c₁,c₂,c₃∈R: EP[u] =EP[u] +c₁I₁+c₂I₂+c₃I₃≥0 New idea: Identify ξ₁ = ^u_u^x,ξ₂ = ^u_u^xx etc. and formulate using polynomials

EP[u] corresponds to S(ξ) =−ξ₁ξ₃

I1 corresponds to T1(ξ) = (α+β−3)ξ₁⁴+ 3ξ²₁ξ2

I2 corresponds to T2(ξ) = (α+β−2)ξ₁²ξ2+ξ1ξ3+ξ₂² I3 corresponds to T3(ξ) = (α+β−1)ξ1ξ3+ξ4

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Integration-by-parts rules

P[u] corresponds to S(ξ) =−ξ₁ξ₃

I₁ corresponds to T₁(ξ) = (α+β−3)ξ₁⁴+ 3ξ₁²ξ₂ I2 corresponds to T2(ξ) = (α+β−2)ξ₁²ξ2+ξ1ξ3+ξ₂² I₃ corresponds to T₃(ξ) = (α+β−1)ξ₁ξ₃+ξ₄ T_i = integration-by-parts polynomials = shift polynomials Nonnegativity of entropy production follows . . .

∃c₁,c2,c3∈R: P[u] =P[u] +c1I1+c2I2+c3I3 ≥0 . . . from solution of decision problem:

∃c₁,c2,c3∈R:∀ξ : (S+c1T1+c2T2+c3T3)(ξ)≥0 CalculateEP[u] =−^dH_dt, gives polynomial S

Determine shift polynomials T_i (depends on differential order of eq.) Solve decision problem

Show that ∃κ >0 : EP[u]−κQ[u]≥0,Q[u] contains |∇²u^γ|² etc.

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Solution of decision problem

∃c₁,c₂,c₃∈R:∀ξ : (S+c₁T₁+c₂T₂+c₃T₃)(ξ)≥0 Tarski 1930: Polynomial decision problems can be reduced to a quantifier-free statement in an algorithmic way

Problem well known in real algebraic geometry

Implementations in Mathematica,QEPCAD(Collins/Hong 1991) available, give complete and exact answer

Algorithms are doubly exponential in number ofc_i,ξ Reductions:

Not all integration-by-parts rules are needed: reduces number of c_i Write polynomial as sum of squares: many algorithms available, quickly solvable, but only numerical results (relation to Hilbert’s 17th problem), and∃ polynomialP ≥0 withP 6= sum of squares

Several dimensions: symmetry reduction, use scalar variables|∇u|,

∆u,|∇²u|etc.

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Entropies for thin-film equation

∂_tu=−(u^βu_xxx)_x, S(ξ) =−ξ₁ξ₃ Shift polynomials:

T₁(ξ) = (α+β−3)ξ₁⁴+ 3ξ²₁ξ₂, T2(ξ) = (α+β−2)ξ₁²ξ2+ξ²₂+ξ1ξ3

T3(ξ) = (α+β−1)ξ1ξ3+ξ4

Decision problem:

∃c₁,c2,c3 ∈R:∀ξ∈R³: (S+c1T1+c2T2+c3T3)(ξ)≥0 Eliminate ξ₄ ⇒ c₃ = 0; eliminateξ₁ξ₃ ⇒ c₂ = 1

Reduced decision problem: ∃c₁ ∈R:∀ξ∈R² :

(α+β−3)c1ξ₁⁴+ (α+β−2 + 3c1)ξ₁²ξ2+ξ₂² ≥0 Solution: 9(c₁+¹₉(α+β))²+⁸₉(α+β−³₂)(α+β−3)≤0 Choose c₁ =−¹₉(α+β) ⇒positive if and only if ³₂ ≤α+β ≤3

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Bakry-Emery revisited

∂_tu = div(∇u+u∇V) in R^d

Aim: Show ^d_dt²^H2^α +κ^dH_dt^α ≥0 with systematic integration by parts Assume: ∇²V ≥λ, one-dimensional case

Multi-dimensional case: see Matthes/A.J./Toscani 2011 Entropy:

Hα[u] = α 4(α−1)

Z

R

u u∞

α

u∞dx− Z

R

udx α

, 1< α≤2 Set w =u^α/2 and compute

d²Hα

dt² = 2 α

Z

R

w²

αwxx

w 2

+ (2−α)wx

w 2wxx

w

−2αwx

w wxx

w V_x−(2−α)wx

w 3

V_x+αwx

w 2

V_x²

u∞dx Integrand formulated as polynomial:

S2(ξ) =αξ₂²+ (2−α)ξ₁²ξ2−2αξ1ξ2Vx−(2−α)ξ₁³Vx+αξ²₁V_x²

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Shift polynomials

S₂(ξ) =αξ₂²+ (2−α)ξ₁²ξ₂−2αξ₁ξ₂V_x −(2−α)ξ₁³V_x +αξ²₁V_x² First time derivative: −^dH_dt^α =R

Rw_x²u∞dx ⇒S1(ξ) =ξ₁² Shift polynomials: (recall thatu∞,x =−u_∞V_x)

0 = Z

R^d

(w_x²Vxu∞)xdx = Z

R^d

(2wxwxxVx+w_x²Vxx −w_x²V_x²)u∞dx T₁(ξ) = 2ξ₁ξ₂V_x +ξ₁²V_xx−ξ²₁V_x²

0 = Z

R^d

(w⁻¹w_x³u∞)_xdx = Z

R^d

w⁻¹(3w_x²w_xx−w⁻¹w_x⁴−w_x³V_x)u∞dx T2(ξ) = 3ξ₁²ξ2−ξ₁⁴−ξ₁³Vx

Decision problem: ∃c₁,c₂ ∈R,c >0 :∀ξ ∈R³:

S^∗(ξ) = (S₂+c₁T₁+c₂T₂−cS₁)(ξ)≥0

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Solution of decision problem

S^∗(ξ) =αξ₂²+ (2−α+ 3c2)ξ₁²ξ2+ 2(−α+c1)ξ1ξ2Vx

−(2−α+c2)ξ₁³Vx+ (α−c1)ξ₁²V_x²−c2ξ₁⁴+(c1Vxx−c)ξ₁² Eliminate ξ₁ξ₂V_x: c₁=α, eliminate ξ₁³V_x: c₂ =−(2−α)

Since Vxx ≥λ: choose c =αλ This gives with x=ξ₁²,y =ξ2:

S^∗(ξ)≥αξ₂²−2(2−α)ξ₁²ξ₂+(2−α)ξ⁴₁ =αy²−2(2−α)xy+(2−α)x² S^∗(ξ)≥0 if and only ifα(2−α)≥(2−α)² or 2(2−α)(α−1)≥0

⇒ 1≤α≤2

We have shown: ^d_dt²^H2^α +αλ^dH_dt^α ≥0 for 1< α≤2 Theorem

Let ∇²V ≥λ. Then the solution of∂_tu = div(∇u+u∇V) in R^d satisfies H_α[u(t)]≤e^−αλtH_α[u(0)], 1< α≤2

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Overview

1 Introduction

2 Entropies

5 Exercises

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Cross-diffusion systems Examples from physics and biology

Cross-diffusion systems

∂tu−div(A(u)∇u) =f(u) in Ω, t>0, u(0) =u⁰, no-flux b.c.

Meaning: div(A(u)∇u)_i =Pn

j=1div(A_ij(u)∇u_j),A∈R^n×n,u ∈Rⁿ Diagonal diffusion matrix: A_ij(u) = 0 fori 6=j

Cross-diffusion matrix: generallyAij(u)6= 0 fori 6=j Why study cross-diffusion systems?

They arise in many applications from physics, biology, chemistry...

Diffusion-induced instabilities may arise

Cross-diffusion may allow for pattern formation

They may exhibit an unexpected gradient-flow/entropy structure

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Example Ê : Cross-diffusion population dynamics

∂_tu−div(A(u)∇u) =f(u) in Ω, t>0, u(0) =u⁰, no-flux b.c.

u = (u1,u2) andu_i models population density ofith species Diffusion matrix:

A(u) =

a10+a11u1+a12u2 a12u1

a₂₁u₂ a₂₀+a₂₁u₁+a₂₂u₂

Suggested by Shigesada- Kawasaki-Teramoto 1979: models population segregation

Lotka-Volterra functions: fi(u) = (bi0−bi1u1−bi2u2)ui

Diffusion matrix is not symmetric, generally not positive definite

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Example Ë : Ion transport through nano-pores

∂_tu−div(A(u)∇u) =f(u) in Ω, t>0, u(0) =u⁰, no-flux b.c.

(u₁, . . . ,u_N) ion concentrations, u_N = 1−PN−1 j=1 u_j Diffusion matrix forN = 4:

A(u) =





D1(1−u2−u3) D1u1 D1u1

D₂u₂ D₂(1−u₁−u₃) D₂u₂ D3u3 D3u3 D3(1−u2−u3)





Derived by Burger-Schlake-Wolfram 2012 from lattice model Electric field neglected to simplify

Diffusion matrix generally not positive definite – expect that 0≤u_i ≤1

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Cross-diffusion systems

∂_tu−div(A(u)∇u) =f(u) in Ω, t >0, u(0) =u₀, no-flux b.c.

Main features:

Diffusion matrixA(u) non-diagonal

Matrix A(u) may be neithersymmetric norpositive definite Variables ui may beboundedfrom below and/or above Objectives:

Global-in-time existence of weak solutions Positivity and boundedness of weak solutions Large-time asymptotics

Mathematical difficulties:

No general theory for diffusion systems available Generally no maximum principle, no regularity theory Lack of positive definiteness→ local existence nontrivial

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Previous results

∂_tu−div(A(u)∇u) =f(u) in Ω, t>0 Global existence if . . .

Growth conditions on nonlinearities (Ladyˇzenskaya ... 1988) Control onL^∞ and H¨older norms (Amann 1989)

Invariance principle holds (Redlinger 1989, K¨ufner 1996) Positivity, mass control, diagonal A(u) (Pierre-Schmitt 1997) Unexpected behavior:

Finite-time blow-up of H¨older solutions (Star´a-John 1995) Weak solutions may exist after L^∞ blow-up (Pierre 2003)

Cross-diffusion may lead to pattern formation (instability) or may avoid finite-time blow-up (Hittmeir/A.J. 2011)

Special structure needed for global existence theory:

gradient-flow orentropy structure

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Cross-diffusion systems Gradient flows

Abstract gradient flows

Definition: Gradient flow if∂tu =−gradH|_u on differential manifold Example: Rⁿ with Euclidean structure,∂_tu =−∇H(u),H:Rⁿ→R

d

dtH(u) =∇H(u)·∂tu=−|∇H(u)|² ⇒ H is Lyapunov functional Can be generalized to∂_tu ∈ ∇H(u) on Hilbert space (Br´ezis 1973) Heat equation is gradient flow for H(u) = ¹₂R

R^d|∇u|²dx in L²(R^d):

gradH(u)ξ= Z

R^d

∇u· ∇ξdx =− Z

R^d

∆uξdx ⇒ ∂_tu = ∆u Otto 2001: Heat eq. is gradient flow forH(u) =R

R^dulogudx in Wasserstein space (= probability measures with Wasserstein metric) Advantage: allows for geometric interpretation

Reference for abstract gradient flows: Ambrosio/Gigli/Savar´e 2005 Our formal definition: Gradient flow if∂tu= div(B∇gradH(u))

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Gradient flows: Cross-diffusion systems

Main assumption

∂_tu−div(A(u)∇u) =f(u) possesses formal gradient-flow structure

∂_tu−div B∇gradH(u)

=f(u), where B is positive semi-definite,H(u) =R

Ωh(u)dx entropy Equivalent formulation: gradH(u)'h⁰(u) =:w (entropy variable)

∂_tu−div(B∇w) =f(u), B =A(u)h⁰⁰(u)⁻¹ Consequences:

H is Lyapunov functional if f = 0:

dH dt =

Z

Ω

∂tu·h⁰(u)

| {z }

=w

dx =− Z

Ω

∇w :B∇wdx ≤0 L^∞ bounds for u: Leth⁰ :D→Rⁿ (D ⊂Rⁿ) be invertible⇒ u = (h⁰)⁻¹(w)∈D (no maximum principle needed!)

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Example Ê : Population-dynamics model

∂_tu−div(A(u)∇u) = 0 in Ω, t >0, u(0) =u⁰, no-flux b.c.

u = (u1,u2) andui models population density ofith species Diffusion matrix:

A(u) =

a10+a11u1+a12u2 a12u1

a₂₁u₂ a₂₀+a₂₁u₁+a₂₂u₂

Entropy:

H[u] = Z

Ω

h(u)dx = Z

Ω

u1

a₁₂(logu₁−1) + u2

a₂₁(logu₂−1)

dx Entropy production:

dH dt [u] =

Z

Ω

logu1

a12

∂tu1+logu2

a21

∂tu2

dx

=−2 Z

Ω

2

a₁₂(a₁₀+a₁₁u₁)|∇√

u₁|²+ 2

a₂₁(a₂₀+a₂₂u₂)|∇√ u₂|² +|∇√

u1u2|²

dx ≤0

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Example Ê : Population-dynamics model

h(u) = u₁

a₁₂(logu₁−1) + u₂

a₂₁(logu₂−1)

Question: Does the model allow for a gradient-flow/entropy structure?

∂tu−div(B(w)∇w) = 0, B(w) =A(u)h⁰⁰(u)⁻¹ Answer: Yes!

Entropy variablew =h⁰(u):

w1 = ∂h

∂u1

= logu₁ a12

, w2= ∂h

∂u2

= logu₂ a21

⇒u2 ∼e^a²¹^w² positive!

New diffusion matrix:

B(w) =

(a₁₀+a₁₁a⁻¹₂₁e^w¹+e^w²)e^w¹ e^w¹^+w²

e^w¹^+w² (a₂₀+a₂₁a⁻¹₁₂e^w²+e^w¹)e^w²

detB(w)≥a10e^w¹+a20e^w² >0

Matrix B(w) is symmetric, positive definite (not uniform in w ∈R²!)

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Example Ë : Ion-transport model

∂_tu−div(A(u)∇u) = 0 in Ω, t >0, u(0) =u⁰, no-flux b.c.

u = (u1,u2,u3) andui models theith ion concentration Diffusion matrix:

A(u) =





D₁(1−u₂−u₃) D₁u₁ D₁u₁ D2u2 D2(1−u1−u3) D2u2

D3u3 D3u3 D3(1−u2−u3)





Entropy: H[u] =R

Ωh(u)dx,u₄= 1−P3 i=1u_i

h(u) =

3

X

i=1

u_i(logu_i −1) +u₄(logu₄−1) +

3

X

i=1

log(D_i)u_i Entropy production:

dH dt [u] =

Z

Ω

3

X

i=1

∂tuilogui −

3

X

i=1

∂tuilogu4+

3

X

i=1

∂tuilogDi

dx

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Example Ë : Ion-transport model

h(u) =

3

X

i=1

u_i(logu_i −1) +u₄(logu₄−1) +

3

X

i=1

log(D_i)u_i Entropy production:

dH dt [u] =

Z

Ω 3

X

i=1

log Diui

u4

∂tuidx

≤ −C Z

Ω

u₄²

3

X

i=1

|∇√

ui|²+|∇√ u4|²

dx

Difficulty: degeneracy atu4 = 0!

New diffusion matrix:

B(w) =u4diag(D1u1,D2u2,D3u3)

Entropy structure: wi =∂h/∂ui = log(ui/u4), back-transformation:

ui = e^wⁱ

1 +e^w¹+e^w²+e^w³ ∈(0,1) ⇒ L^∞ bounds!

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Relation to nonequilibrium thermodynamics

Chemical potential: µ_i =−_∂ρ^∂s

i,s: physical entropy density,ρ_i: mass density of ith species

Entropy variables: w_i = _∂ρ^∂h

i,h =−s: mathematical entropy Mixture of ideal gases: µ_i =µ⁰_i + logρ_i,µ⁰_i = const. ⇒

w_i =−∂s

∂ρi

=µ⁰_i + logρ_i or ρ_i =e^wⁱ^−µ⁰ⁱ Non-ideal gases: µi = logai,ai =γiρi: thermodynamic activity Example: volume-filling case, γi = 1 +Pn−1

j=1 aj

ρ_i = ai

γ_i = ai

1 +Pn−1

j=1 a_i = exp(µi) 1 +Pn−1

j=1 exp(µ_i)

→ exactly the expression for the ion-transport model!

Open problem: Include nonconstant temperature

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Cross-diffusion systems Boundedness-by-entropy method

Boundedness-by-entropy method

∂tu−div(A(u)∇u) =f(u) in Ω, t >0, u(0) =u0, no-flux b.c.

Assumptions:

1 ∃ entropy densityh∈C²(D; [0,∞)), h⁰ invertible on D ⊂Rⁿ Example: h(u) =ulogu for u∈D= (0,∞), (h⁰)⁻¹(w) =e^w ∈D

2 h⁰⁰(u)A(u) is positive semidefinite foru ∈D

impliesz^>h⁰⁰(u)A(u)z = (h⁰⁰(u)z)^>B(w)(h⁰⁰(u)z)≥0 for z ∈R^N

3 Acontinuous on D,∃C >0 :∀u ∈D: f(u)·h⁰(u)≤C(1 +h(u)) needed to control reaction termf(u)

Problem: h⁰⁰(u)A(u) semidefinite not sufficient, need gradient estimate!

Solution: Assume D ⊂(a,b)ⁿ,a^∗_i >0,mi >0, and z^>h⁰⁰(u)A(u)z ≥

n

X

i=1

a_i(u)²z_i² where a_i(u) =a^∗_i(u_i −a)^mⁱ⁻¹ or a_i(u) =a^∗_i(b−u_i)^mⁱ⁻¹

→ Can probably be generalized to arbitrary increasing functions ai

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Boundedness-by-entropy method

∂_tu−div(A(u)∇u) =f(u) in Ω, t >0, u(0) =u₀, no-flux b.c.

Assumptions:

1 ∃ convex entropyh∈C²(D; [0,∞)), h⁰ invertible on D ⊂Rⁿ

2 Assume D⊂(a,b)ⁿ,a^∗_i >0,m_i >0, and z^>h⁰⁰(u)A(u)z ≥

n

X

i=1

ai(u)²z_i², ai(u)∼u_i^mⁱ⁻¹

3 Acontinuous on D,∃C >0 :∀u ∈D: f(u)·h⁰(u)≤C(1 +h(u)) Consequence of Ë:∇u^>h⁰⁰(u)A(u)∇u ≥C(|∇u^m₁¹|²+|∇u₂^m²|²) Theorem (A.J.,Nonlinearity2015)

Let the above assumptions hold, let D ⊂Rⁿ be bounded, u0 ∈L¹(Ω)∩D.

Then ∃ global weak solution such that u(x,t)∈D and

u∈L²_loc(0,∞;H¹(Ω)), ∂tu ∈L²_loc(0,∞;H¹(Ω)⁰)

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Boundedness-by-entropy method

Theorem (A.J.,Nonlinearity 2015)

Let the above assumptions hold, let D ⊂Rⁿ be bounded, u₀ ∈L¹(Ω)∩D.

u∈L²_loc(0,∞;H¹(Ω)), ∂_tu ∈L²_loc(0,∞;H¹(Ω)⁰) Remarks:

Result valid for rather general model class

YieldsL^∞ boundswithout using a maximum principle

Boundedness assumption onD is strong (can be weakened in some cases; see examples below)

Main assumption: existence of entropyh and invertibility ofh⁰ onD How to find entropy functionsh? Physical intuition, trial-and-error Theorem can be generalized for degenerate problems

What’s next? Proof of existence result, concrete examples, extensions

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Proof of existence theorem

∂_tu−div(A(u)∇u) =f(u) or ∂_tu(w)−div(B(w)∇w) =f(u(w)) Key ideas:

Discretize in time: replace ∂tu(w) by ¹_τ(u(w^k)−u(w^k−1)) Benefit: Avoid issues with time regularity

Regularize in space by adding “ε∆^mw^k”

Benefit: Since div(B(w)∇w) is not uniformly elliptic; yields solutions w^k ∈H^m(Ω)⊂L^∞(Ω) ifm>d/2

Solve problem inw^k by fixed-point argument

Benefit: Problem inw-formulation is elliptic (not true for u-formulation)

Perform limit (ε, τ)→0, obtain solutionu(t) = limu(w^k) Benefit: Compactness comes from entropy estimate;L^∞ bounds coming from u(w^k)∈D ⇒ u ∈D

Strategy: Problem inu → Solve inw → Limit gives problem inu

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Proof of existence theorem

∂tu−div(A(u)∇u) =f(u) or ∂tu(w)−div(B(w)∇w) =f(u(w)) More details:

Implicit Euler: Replace∂tu(t_k) by _τ¹(u(w^k)−u(w^k−1)),t_k =kτ to obtain elliptic problems, w: entropy variable

Regularization: Add ε(−1)^mP

|α|=mD^2αw +εw, where H^m(Ω)⊂L^∞(Ω) uniform ellipticity

Solve approximate problem using Leray-Schauder fixed-point theorem Derive estimates uniform in (τ, ε) from entropy production estimate Use compactness to perform the limit (τ, ε)→0

Approximate problem: Given w^k−1∈L^∞(Ω), solve 1

τ Z

Ω

(u(w^k)−u(w^k−1))·φdx+ Z

Ω

∇φ:B(w^k)∇w^kdx +ε

Z

Ω

X

|α|=m

D^αw^k ·D^αφ+w^k·φ

dx = Z

Ω

f(u(w^k))·φdx

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Step Ê : Lax-Milgram argument

DefineS :L^∞(Ω)×[0,1]→L^∞(Ω), S(y, δ) =w^k andw^k solves linearproblem:

a(w^k, φ) = Z

Ω

∇φ:B(y)∇w^kdx+ε Z

Ω

X

|α|=m

D^αw^k ·D^αφ+w^k·φ

dx

=−δ τ

Z

Ω

(u(y)−u(w^k−1))·φdx+δ Z

Ω

f(u(y))·φdx =F(φ) Lax-Milgram lemma gives solution w^k ⇒ S well defined

Properties: S(y,0) = 0,S compact (since H^m,→L^∞ compact) Theorem (Leray-Schauder)

Let B Banach space, S :B×[0,1]→B compact, S(y,0) = 0for y ∈B,

∃C >0 :∀y ∈B, δ ∈[0,1] : S(y, δ) =y ⇒ kyk_B ≤C. Then S(·,1)has a fixed point.

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Step Ë : Leray-Schauder argument

Discrete entropy estimate: choose test fct. w^k,τ 1, use h convex δ

Z

Ω

h(u(w^k))dx+τ Z

Ω

∇w^k :B∇w^kdx+ετCkw^kk²_Hm

≤ Cτ

|{z}

<1

δ Z

Ω

(1 +h(u(w^k)))dx+ δ

|{z}

≤1

Z

Ω

h(u(w^k−1))dx Yieldskw^kk_L^∞ ≤Ckw^kk_H^m ≤C(ε, τ) ⇒ estimate uniform in (w^k, δ) Leray-Schauder: ∃solution w^k ∈H^m(Ω)

Sum discrete entropy estimate (slightly simplified):

Z

Ω

h(u(w^k))dx+Cτ

k

X

j=1 n

X

i=1

Z

Ω

|∇u_i(w^k)^mⁱ|²dx +ετC

k

X

k=1

kw^jk²_Hm ≤C

Idea: Derive estimates for u =u(w), not forw

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Step Ì : Uniform estimates

Estimates uniform in (τ, ε): set u^(τ)(·,t) =u(w^k), t∈((k−1)τ,kτ] k(u_i^(τ))^mⁱk_L2(0,T;H¹)+√

εkw^(τ)k_L2(0,T;H^m)≤C τ⁻¹ku^(τ)(t)−u^(τ)(t−τ)k_L2(τ,T;(H^m)⁰)≤C Theorem (Nonlinear Aubin-Lions lemma, Chen/A.J./Liu 2014) Let (u^(τ)) be piecewise constant in time, k∈N, s ≥ ¹₂, and

τ⁻¹ku^(τ)(t)−u^(τ)(t−τ)k_L1(τ,T;(H^k)⁰)+k(u^(τ))^sk_L2(0,T;H¹)≤C Then exists subsequence u^(τ⁾→u strongly in L^2s(0,T;L^2s)

Remarks:

Generalization of standard Aubin-Lions lemma (s = 1) Result can be generalized to (u^(τ))^s ∈L^p(0,T;W^1,q) and φ(u^(τ))∈L²(0,T;H¹) if (u^(τ)) bounded inL^∞,φmonotone

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Step Í : Limit (τ, ε) → 0

1 τ

Z _T

0

Z

Ω

(u^(τ)(t)−u^(τ)(t−τ))·φdxdt+ Z _T

0

Z

Ω

∇φ:A(u^(τ))∇u^(τ)dxdt +ε

Z T 0

Z

Ω

X

|α|=m

D^αw^(τ)·D^αφ+w^(τ)·φ

dxdt = Z T

0

Z

Ω

f(u^(τ))·φdxdt Nonlinear Aubin-Lions lemma:

u^(τ)→u strongly inL²(0,T;L²) εw^(τ)→0 strongly inL²(0,T;H^m) A(u^(τ))∇u^(τ)*A(u)∇u weakly inL²(0,T;L²) Limit (τ, ε)→0 in weak formulation ⇒u solves diffusion system u satisfies initial datum: Show that linear interpolant of (u^(τ)) is bounded in C⁰([0,T]; (H^m)⁰) ⇒u(·,0) =u₀ defined in H^m(Ω)⁰ Boundary conditions: Contained in weak formulation

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Summary

Theorem (A.J.,Nonlinearity 2015)

Let the above assumptions hold, let D ⊂Rⁿ be bounded, u₀ ∈L¹(Ω)∩D.

u∈L²_loc(0,∞;H¹(Ω)), ∂_tu ∈L²_loc(0,∞;H¹(Ω)⁰) Strategy of the proof:

Implicit Euler discretization and ∆^m regularization

Entropy formulation gives a priori estimates and L^∞ bounds Compactness from nonlinear Aubin-Lions lemma

Benefits:

General global existence theorem

Yields bounded weak solutions without a maximum principle Limitations:

Boundedness of domain D, how to find entropy density h?

Particular positive definiteness condition onh⁰⁰(u)A(u)

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Ê Population model of Shigesada-Kawasaki-Teramoto

∂_tu−div(A(u)∇u) = 0 in Ω, t>0, u(0) =u₀, no-flux b.c.

Entropy defined onunbounded domainD = (0,∞)² Entropy-dissipation inequality:

dH

dt [u] =−2 Z

Ω

2 a12

(a10+a11u1)|∇√ u1|²

+ 2

a₂₁(a20+a22u2)|∇√

u2|²+|∇√ u1u2|²

dx

Yields estimate for (√

ui) in H¹(Ω): Previous proof applies Main difference: We do not have (u_i) bounded inL^∞(Ω) but only (√

u_i) bounded inL⁶(Ω) (if space dimension≤3) Assumption: Transition rates p_i(u) =a_i0+a_i1u₁+a_i2u₂ Previous technique of proof yield global existence of solutions

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

Entropy in mathematics

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

Systematic integration by parts: Motivation

Idea of method

Integration-by-parts rules

Integration-by-parts rules

Solution of decision problem

Entropies for thin-film equation

Bakry-Emery revisited

Shift polynomials

Solution of decision problem

Overview

Cross-diffusion systems

Example Ê : Cross-diffusion population dynamics

Example Ë : Ion transport through nano-pores

Cross-diffusion systems

Previous results

Abstract gradient flows

Gradient flows: Cross-diffusion systems

Example Ê : Population-dynamics model

Example Ê : Population-dynamics model

Example Ë : Ion-transport model

Example Ë : Ion-transport model

Relation to nonequilibrium thermodynamics

Boundedness-by-entropy method

Boundedness-by-entropy method

Boundedness-by-entropy method

Proof of existence theorem

Proof of existence theorem

Step Ê : Lax-Milgram argument

Step Ë : Leray-Schauder argument

Step Ì : Uniform estimates

Step Í : Limit (τ, ε) → 0

Summary

Ê Population model of Shigesada-Kawasaki-Teramoto