Entropy methods and cross-diﬀusion systems

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Entropy methods and cross-diffusion systems

Ansgar J¨ungel

Vienna University of Technology, Austria

asc.tuwien.ac.at/∼juengel

Ansgar J¨ungel (TU Wien) Cross-diffusion systems asc.tuwien.ac.at/∼juengel 1 / 76

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Literature

Main reference

A. J¨ungel. Entropy methods for diffusive partial differential equations. BCAM Springer Briefs, Springer, 2016.

A. J¨ungel. The boundedness-by-entropy method for cross-diffusion systems. Nonlinearity28 (2015), 1963-2001.

A. J¨ungel. Cross-diffusion systems with entropy structure.

Proceedings of Equadiff 2017, Bratislava, pp. 181-190.

N. Zamponi and A. J¨ungel. Analysis of degenerate cross-diffusion population models with volume filling. Ann. Inst. H. Poincar´e – AN 34 (2017), 1-29. (Erratum: 34 (2017), 789-792.)

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Introduction

Multi-species systems

Examples:

Wildlife populations Tumor growth Gas mixtures

Lithium-ion batteries Population herding Nature is composed of multi-species systems

+^oxygen ^graphite –

Li⁺ Li⁺

Al Cu

separator

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Introduction

How to model multi-species systems?

Microscopic models:

Discrete-time Markov chains:

matrix-based models

Continuous-time Markov chains: species move to neighboring cells with transition ratep^±_j (ui)

Particle models: Newton’s laws with interactions for each individual Continuum models:

Stochastic differential equations: Brownian motion represents erratic motion

Kinetic equations: distribution function depends on phase-space variables (and trait parameters like age, size, maturity)

Diffusive equations: deterministic dynamics for particle densities

→ considered here

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Introduction

Parabolic partial differential equations

Heat equation:

∂tu−∆u= 0 in Ω, t>0, initial & boundary conditions Strongly regularizing: u(0)∈L²(Ω)⇒ u(t)∈C^∞(Ω)

Preserves nonnegativity: u(0)≥0 ⇒u(t)≥0 Reaction-diffusion equations:

∂_tu_i −div(D_i∇u_i) =f_i(u) in Ω, t >0, D_i >0

Still regularizing and nonnegativity preserving (if f_i ≤0 atu_i = 0) Global existence of weak solutions if fi at most quadratic growth Global existence of classical solutions not always guaranteed!

Problem:

Flux D_i∇u_i only depends onu_i (Fick’s law)

In multicomponent systems, flux may depend on all∇u_j

⇒ cross-diffusion systems

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Introduction

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

Cross-diffusion matrix: generallyA_ij(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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Introduction

Overview

1 Introduction

2 Examples

3 Derivation

4 Analysis

5 Nonstandard examples

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Examples

Example ➊ : Cross-diffusion population dynamics

u = (u₁,u₂) andui models population density ofith species Diffusion matrix: (aij≥0)

A(u) =

a₁₀+a₁₁u₁+a₁₂u₂ a₁₂u₁ a₂₁u₂ a₂₀+a₂₁u₁+a₂₂u₂

Suggested by Shigesada- Kawasaki-Teramoto 1979 to model population segregation Lotka-Volterra functions:

f_i(u) = (b_i0−b_i₁u₁−b_i2u₂)ui

Diffusion matrix is not symmetric, generally not positive definite

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Examples

Example ➋ : Ion transport through nanopores

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

Central in biological processes such as neural signal transmission and electrical excitability of muscles

(u₁, . . . ,u_N) ion volume fractions, u_N = 1−PN−1 j=1 uj

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 ⁺

+

+ +

+

+ + +

–

– – – – –

– – positive neutral –

avidin

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Examples

Example ➌ : Tumor-growth modeling

Volume fractions of tumor cells u₁, extracellular matrix u₂, nutrients/wateru₃= 1−u₁−u₂

Diffusion matrix: (β,θ: pressure parameters) A(u) =

2u₁(1−u₁)−βθu₁u₂² −2βu₁u₂(1 +θu₁)

−2u1u₂+βθu₂²(1−u₂) 2βu₂(1−u₂)(1 +θu1)

Derived by Jackson-Byrne 2002 from continuum fluid model Describes avascular growth of symmetric tumor

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

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Examples

Example ➍ : Multicomponent gas mixtures

Volume fractions of gas components u₁,u₂,u₃ = 1−u₁−u₂ Diffusion matrix: δ(u) =d₁d₂(1−u₁−u₂) +d₀(d₁u₁+d₂u₂)

A(u) = 1 δ(u)

d₂+ (d₀−d₂)u₁ (d₀−d₁)u₁ (d₀−d₂)u₂ d₁+ (d₀−d₁)u₂

Application: Patients with airway obstruction inhale Heliox to speed up diffusion

Proposed by Maxwell 1866/Stefan 1871 Duncan-Toor 1962: Fick’s law (J_i ∼ ∇u_i) not sufficient, include cross-diffusion terms Boudin-Grec-Salvarani 2015: Derivation from Boltzmann equation for simple mixtures

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Examples

Difficulties and objectives

∂_tu−div(A(u)∇u) =f(u) in Ω, t >0, u(0) =u⁰ Main features:

Diffusion matrixA(u) non-diagonal(cross-diffusion)

Matrix A(u) may be neithersymmetric norpositive definite Variables u_i expected to bebounded from below and/or above Objectives:

Derivation of equations (formal or rigorous)

Global-in-time existence and uniqueness of weak solutions Positivity and boundedness of solution (if physically expected) Large-time behavior, design of stable numerical schemes Mathematical difficulties:

No general theory for diffusion systems

Generally no maximum principle, no regularity theory Lack of positive definiteness⇒ global existence nontrivial

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Examples

Overview

1 Introduction

2 Examples

3 Derivation

4 Analysis

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Derivation

Derivation of cross-diffusion systems

From random-walk lattice models: Taylor expansion of transition rates and cell size h→0

→ population dynamics & ion transport models From fluid models: diffusion scale in balance equations and force proportional to velocity differences

→ tumor-growth & Maxwell-Stefan models

From kinetic transport equationsfor distribution function f(x,v,t):

mean-free path limit in momentsR

f(x,v,t)φ(v)dv,

→ Maxwell-Stefan equations

From stochastic differential equations: large-number limit, Ito formula

→ cross-diffusion models for multi-species systems

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Derivation From kinetic models to cross diffusion

➊ From kinetic models to cross diffusion

Due to Boudin-Grec-Salvarani 2015

Boltzmann transport equation for f_i(x,v,t) in diffusion scaling ε∂tf_i +v· ∇^xf_i = 1

εQ_i(f_i,f_i) +1 ε

X

j6=i

Q_ij(f_i,f_j), i = 1, . . . ,n

Q_i mono-species,Q_ij bi-species collision operators Collisions are elastic, conserve mass: R

R3(Qi+P

j6=iQij)dv = 0 Particle densities and fluxes:

ρi(x,t) = Z

R3

fi(x,v,t)dv, Xn

i=1

ρi(x,t) = 1, ερi(x,t)vi(x,t) =

Z

R3

f_i(x,v,t)vdv

Ansatz: f_i(x,v,t) =M_i := (2π)^−3/2ρi(x,t) exp(−|v−εvi(x,t)|²/2) (justification: fi close to equilibriumMi,fi =Mi+O(ε))

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➊ From kinetic models to cross diffusion

ε∂_tf_i +v· ∇xf_i =ε⁻¹Q_i(f_i,f_i) +ε⁻¹X

j6=i

Q_ij(f_i,f_j)

1 Ansatz: f_i(x,v,t) =M_i := (2π)^−3/2ρ_i(x,t) exp(−|v−v_i(x,t)|²/2)

2 Insert into Boltzmann equation, multiply by (1,v), and integrate:

∂tρi + divx(ρiv_i) = 0, ε∂t(ρiv_i) + divx

Z

R3

f_iv⊗vdx =ε⁻¹X

j6=i

Z

R3

Q_ij(fi,f_j)vdv

3 Compute integrals:

ε∂t(ρiv_i) +εdivx(ρiv_i⊗v_i) +ε⁻¹∇ρi =ε⁻¹X

j6=i

D_ijρiρj(vj −v_i)

4 Limitε→0 gives Maxwell-Stefan system:

∂tρ_i + div(ρ_iv_i) = 0, ∇ρ_i =X

j6=i

D_ijρ_iρ_j(v_j −v_i), i = 1, . . . ,n

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Maxwell-Stefan system

∂tρi −divJi =fi(ρ), ∇ρi =X

j6=i

Dij(ρjJi −ρiJj) =: (CJ)i, i = 1, . . . ,n

Volume fractions of gas components ρi,Pn

i=1ρi = 1 Can we write this as ∂tρi = div(Pn−1

j=1 A_ij∇ρi)? Yes!

Invert ∇ρ=CJ on ker(C)^⊥, ker(C) ={1}:

∂tρi−divJ_i =f_i(ρ), J_i = Xn−1

j=1

A_ij∇ρj, i = 1, . . . ,n−1 Matrix (A_ij) generally not symm. positive

definite →use entropy variables Local existence analysis: Bothe 2011, Herberg-Meyries-Pr¨uss-Wilke 2017

Global existence analysis: Giovangigli 1999, A.J.-Stelzer 2013

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Derivation From lattice random walk to cross diffusion

➋ From lattice random walk to cross diffusion

Single species: one space dimension to simplify

Master equation: time variation = incoming −outgoing

∂tu(x_i) =p(u(x_i−1) +u(x_i+1))−2pu(x_i) Taylor expansion: (h = grid size)

u(x_i±1)−u(xi) =±h∂xu(xi) +¹₂h²∂_x²u(xi) +O(h³) Diffusion scaling: t7→t/h² ⇒ ∂t h²∂t

h²∂tu(x_i) =p(u(x_i−1)−u(x_i)) +p(u(x_i₊₁)−u(x_i))

=ph²∂_x²u(xi) +O(h³)

Limith→0 gives ∂tu(x) =p∂_x²u(x)(heat equation)

Rigorous limit: De Masi, Lebowitz, Sinai, Spohn etc. (from 1980s on)

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➋ From lattice random walk to cross diffusion

Multiple species:

Master equation for particle number u_j(x_i) atith cell:

∂_tu_j(x_i) =p⁺_j,iu_j(x_i₋₁) +p_j⁻_,i₊₁u_j(x_i+1)−(p⁺_j,i+p_j,i⁻)u_j(x_i) Transition rates: p_j^±_,i =pi(u(xj))qi(un(xj±1))

Taylor expansion, diffusion scaling and limith→0 leads to system of diffusion equations

∂tu_j =∂x

n

X

k=1

A_jk(u)∂xu_k

, j = 1, . . . ,n

Multi-dimensional case analogous Examples:

q_i = 1: A_ij(u) = _∂u^∂

j(uip_i(u)) gives population dynamics models pi = 1: Aij(u) =δijqi(un) +ui d

dunq(un) gives volume-filling models

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Population dynamics & ion transport models

Population dynamics model: q_i(u) = 1

u = (u₁, . . . ,un) andui models population density ofith species Diffusion coefficients for p_i(u) =a_i0+a_i1u₁+· · ·+a_inu_n:

Aij(u) = ∂

∂u_j(uipi(u)) =δija_i0+δij

Xn

k=1

a_iku_k+aijui

Global existence: Kim 1984, Amann 1989, Chen-A.J. 2004 Ion transport model: pi(u) = 1

Ion concentration ui, solvent concentration un,Pn

j=1uj = 1 Diffusion coefficients for q_i(u_n) =D_iu_n:

A_ij(u) =δ_ijq_i(u_n) +u_iq^′(u_n) =δ_ijD_iu_n+D_iu_i

Global existence: Burger et al. 2012, A.J. 2015, Zamponi-A.J. 2015

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Derivation From fluid models to cross diffusion

➌ From fluid models to cross diffusion

Mass and force balance equations:

∂tu_i+ div(uiv_i) = 0, i = 1, . . . ,n ε ∂_t(u_iv_i) + div(u_iv_i ⊗v_i)

−divT_i−p∇u_i =f_i Force terms: f_i =Pn

j=1k_ij(vj−v_i)uiu_j Properties: Pn

i=1u_i = 1, Pn

i=1u_iv_i = 0,Pn

i=1f_i = 0

Interphase pressure: p∇ui,p: phase pressure (Drew-Segel 1971) Assumptions:

Inertia approximation: ε= 0 Stress tensor: Ti =−ui(pId +Pi) Pressures: P_i =P_i(u),P_n= 0,k :=k_ij Consequences:

k :=kij implies thatfi =−kuivi

Pressure: −divT_i−p∇u_i =u_i∇p+ div(u_iP_i)

∂tui + div(uivi) = 0, ui∇p+ div(uiPi) =−kuivi

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From fluid models to cross diffusion

∂tui + div(uivi) = 0, ui∇p+ div(uiPi) =−kuivi

Aim: eliminatep andv_i

Add all force balance equations:

0 =−k Xn

i=1

uivi = Xn

i=1

ui∇p+ div(uiPi)

=∇p+

n−1X

i=1

div(uiPi) Replace∇p and expand divPi =P_u−1

j=1 ∂Pi

∂uj∇uj:

∂tui+

n−1X

j=1

div(Aij(u)∇uj) = 0, i = 1, . . . ,n−1 Diffusion coefficients:

Aii = (1−ui)

Pi+ui

∂Pi

∂u_i

−ui

X

j6=i

uj

∂Pj

∂u_i, Aij =ui(1−ui)∂Pi

∂u_j −uiPj−ui

X

k6=i

uk

∂Pk

∂u_j , j 6=i.

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Tumor-growth & Maxwell-Stefan models

Tumor-growth model:

Volume fractions of tumor cells u₁, extracellular matrix u₂,

nutrients/wateru₃ = 1−u₁−u₂, describes avascular growth of tumor Diffusion matrix forn= 3, P₁ =u₁,P₂ =βu2(1 +θu1):

A(u) =

2u₁(1−u₁)−βθu₁u₂² −2βu₁u₂(1 +θu₁)

−2u1u₂+βθu₂²(1−u₂) 2βu₂(1−u₂)(1 +θu1)

Maxwell-Stefan systems:

General limit diffusion model:

∂_tu_i + div(u_iv_i) = 0, u_i∇p+ div(u_iP_i) =− Xn

j=1

k_ij(v_j −v_i)u_iu_j p = 0 (no phase pressure),P_i = 1

∂tui + div(uivi) = 0, ∇ui =− Xn

j=1

kij(vj −vi)uiuj

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Derivation From SDEs to cross diffusion

➍ From SDEs to cross diffusion

dX_i^k =− Xn

j=1

1 N

XN

ℓ=1

∇V_ij^η(X_i^k −X_j^ℓ)dt+√

2σidW_i^k, X_i^k(0) =ξ_i^k Interacting particles with numbers N=N₁, . . . ,N =N_n, trajectories X_i^k (i = 1, . . . ,n,k= 1, . . . ,N)

Potential: V_ij^η(x) =η^−dV_ij(x/η), η >0,

Given i.i.d. random variables ξ^k_i and∇V_ij^η bounded Lipschitz

⇒ ∃! i.i.d. solutionX_i^k with common lawµ_i Aim: limitN → ∞,η→0

Expected result: (L. Chen-Daus-A.J., in progress) µ_i →µ_i in measure as N→ ∞,µ_i with densityu_i

∂tu_i = div

σ_i∇u_i + Xn

j=1

a_iju_i∇u_j

, a_ij = Z

Ω

V_ij(|x|)dx Open question: How to derive, e.g., population model?

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Derivation From SDEs to cross diffusion

Overview

1 Introduction

2 Examples

3 Derivation

4 Analysis

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Analysis

Local existence analysis

∂tu−div(A(u)∇u) =f(u) in Ω⊂R^d, t >0, u(0) =u⁰ Theorem (Amann 1990)

Let aij, fi smooth, A(u) normally elliptic, u⁰ ∈W^1,p(Ω;Rⁿ) with p>d . Then ∃ unique local solution u

u∈C⁰([0,T^∗);W^1,p(Ω)), u ∈C^∞(Ω×[0,T^∗);Rⁿ), 0<T^∗ ≤ ∞ A(u) normally elliptic = all eigenvalues have positive real parts Linear algebra: If H(u) symmetric positive definite such that H(u)A(u) positive definite then A(u) normally elliptic

Application: Let h(u) convex and set H(u) :=h^′′(u). Then, iff = 0, d

dt Z

Ω

h(u)dx = Z

Ω

h^′(u)·∂tudx =− Z

Ω ∇u:h^′′(u)A(u)∇u

| {z }

≥0 ifh^′′(u)A(u) pos. def.

dx Aim: find a Lyapunov functional (entropy) h(u)

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Analysis

State of the art

∂tu−div(A(u)∇u) =f(u) in Ω⊂R^d, t>0 Global existence if . . .

Growth conditions on nonlinearities (Ladyˇzenskaya ... 1988) Control onW^1,p(Ω) norm withp >d (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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Analysis

Entropy and gradient flows

Entropy: Measure of molecular disorder or energy dispersal Introduced by Clausius (1865) in thermodynamics Boltzmann, Gibbs, Maxwell: statistical interpretation Shannon (1948): concept of information entropy Entropy in mathematics: ∼convex Lyapunov functional

Hyperbolic conservation laws (Lax), kinetic theory (Lions) Relations to stochastic processes (Bakry, Emery) and optimal transportation (Carrillo, Otto, Villani)

Gradient flow: ∂_tu=−gradH|u on differential manifold Example: R^d with Euclidean structure ⇒ ∂tu =−H^′(u) H(u) is Lyapunov functional since∂tH(u) =−|H^′(u)|² Gradient flow of entropy w.r.t. Wasserstein distance (Otto) Entropy H(u) =R

ulogudx: ∂tu = div(u∇H^′(u)) = ∆u

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Analysis

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)≃h^′(u) =:w (entropy variable)

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

1 H is Lyapunov functional if f = 0:

dH dt =

Z

Ω

∂tu·h^′(u)

| {z }

=w

dx =− Z

Ω∇w :B∇wdx ≤0

2 L^∞ bounds for u: Leth^′ :D→Rⁿ (D ⊂Rⁿ) be invertible⇒ u = (h^′)⁻¹(w)∈D (no maximum principle needed!)

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Analysis

Example: Maxwell-Stefan systems for n = 3

Volume fractions of gas components u₁,u₂,u₃ = 1−u₁−u₂

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

A(u) = 1 δ(u)

d₂+ (d₀−d₂)u₁ (d₀−d₁)u₁ (d₀−d₂)u₂ d₁+ (d₀−d₁)u₂

Entropy: H(u) =R

Ωh(u)dx, where

h(u) =u₁(logu₁−1) +u₂(logu₂−1) + (1−u₁−u₂)(log(1−u₁−u₂)−1)

Entropy variables: w =h^′(u)∈R² or u= (h^′)⁻¹(w) w_i = ∂h

∂u_i = logui

u₃, u_i = e^wⁱ

1 +e^w¹+e^w²∈(0,1) Entropy production:

dH

dt (u) =− Z

Ω

2

X

i=1

di|∇ui|²

u_i +d₀u₁u₂|∇u₃|² u₃

dx ≤0

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Analysis

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: µi =µ⁰_i + logρi,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 +P_n−1

j=1 aj

ρ_i = a_i γi

= a_i

1 +Pn−1 j=1 ai

= exp(µ_i) 1 +Pn−1

j=1 exp(µi)

→ exactly the expression for the ion-transport model!

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Analysis Boundedness-by-entropy method

Boundedness-by-entropy method

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

d dt

Z

Ω

h(u)dx = Z

Ω

h^′(u)

| {z }

=w

·∂tudx =− Z

Ω∇w :B(w)∇wdx+ Z

Ω

f(u)·wdx Assumptions:

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

u = (h^′)⁻¹(w) =e^w ∈D

2 “Degenerate” positive definiteness: h^′′(u)A(u)≥diag(ai(ui)²)

∇w^⊤B(w)∇w =∇u^⊤h^′′(u)A(u)∇u ≥ Xn

i=1

ai(ui)²|∇ui|² Gives estimate for|∇α_i(u_i)|², where α^′_i(u_i) =a_i(u_i)∼u_i^mⁱ⁻¹

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

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Boundedness-by-entropy method

Assumptions:

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

2 “Degenerate” positive definiteness: for allu ∈D, z^⊤h^′′(u)A(u)z ≥

Xn

i=1

ai(u)²z_i², ai(u)∼u_i^mⁱ⁻¹

3 Acontinuous on D,∃C >0 :∀u ∈D: f(u)·h^′(u)≤C(1 +h(u)) Theorem (A.J.,Nonlinearity 2015)

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

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

u∈L²loc(0,∞;H¹(Ω)), ∂tu ∈L²loc(0,∞;H¹(Ω)^′)

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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∈L²loc(0,∞;H¹(Ω)), ∂_tu ∈L²loc(0,∞;H¹(Ω)^′) Remarks:

Result valid for rather general model class

YieldsL^∞ boundswithout using a maximum principle

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

Main assumptions: existence of entropy h, pos. def. of h^′′(u)A(u) How to find entropy functionsh? Physical intuition, trial and error Yields immediately global existence for Maxwell-Stefan

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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 approximation involvingu(w^k) Benefit: avoid issues with time regularity

Regularize in space by adding “ε(−∆)^mw^k”,ε >0

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 in w-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 solves problem in u

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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(tk) by _τ¹(u(w^k)−u(w^k−1)),t_k =kτ to obtain elliptic problems, w^k: 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 ➊ : 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^m֒→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 ⇒ kykB ≤C. Then S(·,1)has a fixed point.

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Step ➋ : 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^kk²H^m

≤ Cτ

|{z}

<1

δ Z

Ω

(1 +h(u(w^k)))dx+ δ

|{z}

≤1

Z

Ω

h(u(w^k−1))dx Yieldskw^kkL^∞ ≤Ckw^kkH^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 Xk

j=1

Xn

i=1

Z

Ω|∇u_i(w^k)^mⁱ|²dx +ετC

Xk

k=1

kw^jk²H^m ≤C Idea: derive estimates foru =u(w), not forw

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Step ➌ : uniform estimates

Estimates uniform in (τ, ε): setu^(τ)(·,t) =u(w^k),t ∈((k−1)τ,kτ] k(u_i^(τ))^mⁱkL²(0,T;H¹)+√

εkw^(τ)kL²(0,T;H^m)≤C τ⁻¹ku^(τ)(t)−u^(τ)(t−τ)kL²(τ,T;(H^m)^′)≤C Lemma (Aubin-Lions 1963/69)

Let ku^(τ)kL²(0,T;H¹)+k∂tu^(τ)_i kL²(0,T;H^m(Ω)^′)≤C .

Then exists subsequence u^(τ)→u strongly in L²(0,T;L²).

Problem: discrete time derivative and nonlinear estimate Lemma (Discrete Aubin-Lions; Simon 1987)

Let X ֒→B compact and B ֒→Y continuous, 1≤p<∞, and ku^(τ)kL^p(0,T;X)≤C, sup

τ >0

h→0limku^(τ)(t)−u^(τ)(t−h)kL¹(τ,T;Y)= 0 Then (u^(τ)) is relatively compact in L^p(0,T;B).

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Step ➌ : uniform estimates

Lemma (Discrete Aubin-Lions; Dreher-A.J., 2012) If additionally,(u^(τ))piecewise constant in time, and

ku^(τ)kL^p(0,T;X)+τ⁻¹ku^(τ)(t)−u^(τ)(t−τ)kL¹(τ,T;Y) ≤C Then (u^(τ)) is relatively compact in L^p(0,T;B).

Benefit: studyu^(τ)(t)−u^(τ)(t−τ), not allu^(τ)(t)−u^(τ)(t−h) 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)kL¹(τ,T;(H^k)^′)+k(u^(τ))^skL²(0,T;H¹)≤C Then exists subsequence u^(τ⁾→u strongly in L^2s(0,T;L^2s)

Remark: 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 in L^∞,φmonotone

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Step ➍ : 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)^′) ⇒u(·,0) =u₀ defined in H^m(Ω)^′ Boundary conditions: Contained in weak formulation

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Summary of existence analysis

Theorem (A.J. 2014)

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

u∈L²loc(0,∞;H¹(Ω)), ∂tu ∈L²loc(0,∞;H¹(Ω)^′) 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 onh^′′(u)A(u)

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Analysis Examples revisited

➊ Tumor-growth model

Volume fractions of tumor cells u₁, extracellular matrix (ECM) u₂, nutrients/wateru₃= 1−u₁−u₂, one space dimension

Diffusion matrix: (β,θ: pressure parameters) A(u) =

2u₁(1−u₁)−βθu₁u₂² −2βu₁u₂(1 +θu₁)

−2u1u₂+βθu₂²(1−u₂) 2βu₂(1−u₂)(1 +θu1)

Entropy: H(u) =R

Ωh(u)dx, where

h(u) =u₁(logu₁−1)+u₂(logu₂−1)+(1−u₁−u₂)(log(1−u₁−u₂)−1) Entropy production inequality:

dH

dt [u] +C_θ Z

Ω

(∂xu₁)²+ (∂xu₂)²

dx ≤C(f) andC_θ>0if θ <4/√

β

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Tumor-growth model

Theorem (A.J./Stelzer, M3AS 2012) Let θ <4/√

β, H(u⁰)<∞ ⇒ ∃bounded weak solution with 0≤u₁,u₂≤1

Question: What happens forθ >4/√ β?

Partial answer: Numerical results show “peaks” in ECM fraction

0 0.2 0.4 0.6 0.8 1

0 0.05 0.1 0.15 0.2 0.25

Spatial position

Tumor cells

0 0.2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5 0.6

Spatial position

Extracellular matrix

Tumor front spreads from left to right (production ratef(u) = 0 ) Tumor causes increase of ECM (encapsulation of tumor)

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➋ Maxwell-Stefan model

∂tu_i −divJ_i =f_i(u), ∇u_i =X

j6=i

c_ij(ujJ_i −u_iJ_j) =: (CJ)i

ui(0) =u_i⁰, i = 1, . . . ,n, no-flux b.c.

Volume fractions ui, fluxes Ji

Problem: need to invert relation∇ui ↔Ji but not invertible since Pn

i=1u_i = 1 ⇒ Pn

i=1∇u_i = 0

Solution: solve ∇u =CJ on ker(C)^⊥ using Perron-Frobenius theorem

⇒ J^∗ =C₀⁻¹∇u^∗, whereu^∗= (u₁, . . . ,u_n−1),J^∗ = (J₁, . . . ,J_n−1) Entropy structure: h(u^∗) =Pn

i=1ui(logui −1), un= 1−Pn−1 i=1 ui

Equations: ∂_tu^∗−div(B(w)∇w) =f^∗(u^∗(w))

Difficulty: show thatB(w) =C₀⁻¹h^′′(u^∗(w))⁻¹ positive definite Boundedness-by-entropy theorem applies with D= (0,1)ⁿ⁻¹:

∃ global weak solutionu_i^1/2 ∈L²(0,T;H¹), 0≤u_i ≤1,P_n−1

i=1 u_i ≤1

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➌ Population model of Shigesada-Kawasaki-Teramoto

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

A(u) =

a₁₀+a₁₁u₁+a₁₂u₂ a₁₂u₁ a₂₁u₂ a₂₀+a₂₁u₁+a₂₂u₂

Entropy: H(u) =R

Ωh(u)dx =R

Ω

P2

i=1u_i(logu_i −1) defined on unbounded domainD = (0,∞)²

Entropy production: for some C >0, iff(u) = Lotka-Volterra term dH

dt [u]≤ −C X2

i=1

Z

Ω

(a_i0+a_iiu₁)|∇√u₁|²+|∇√u₁u₂|² dx+C Main difficulty: We do not have (ui) bounded in L^∞(Ω) but only (√u_i) bounded in L⁶(Ω) (if space dimension≤3)

Theorem (Chen-A.J.,SIMA 2004-2006)

Let a_i0 >0or a_ii >0. Then ∃nonnegative weak solution(u₁,u₂)

Entropy methods and cross-diﬀusion systems

Entropy methods and cross-diffusion systems

Contents

Literature

Multi-species systems

How to model multi-species systems?

Parabolic partial differential equations

Cross-diffusion systems

Overview

Example ➊ : Cross-diffusion population dynamics

Example ➋ : Ion transport through nanopores

Example ➌ : Tumor-growth modeling

Example ➍ : Multicomponent gas mixtures

Difficulties and objectives

Overview

Derivation of cross-diffusion systems

➊ From kinetic models to cross diffusion

➊ From kinetic models to cross diffusion

Maxwell-Stefan system

➋ From lattice random walk to cross diffusion

➋ From lattice random walk to cross diffusion

Population dynamics & ion transport models

➌ From fluid models to cross diffusion

From fluid models to cross diffusion

Tumor-growth & Maxwell-Stefan models

➍ From SDEs to cross diffusion

Overview

Local existence analysis

State of the art

Entropy and gradient flows

Gradient flows: Cross-diffusion systems

Example: Maxwell-Stefan systems for n = 3

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 ➊ : Lax-Milgram argument

Step ➋ : Leray-Schauder argument

Step ➌ : uniform estimates

Step ➌ : uniform estimates

Step ➍ : limit ( τ, ε ) → 0

Summary of existence analysis

➊ Tumor-growth model

Tumor-growth model

➋ Maxwell-Stefan model

➌ Population model of Shigesada-Kawasaki-Teramoto