Entropy methods for diﬀusion equations

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Entropy methods for diffusion equations

Ansgar J¨ungel

Vienna University of Technology, Austria www.asc.tuwien.ac.at/∼juengel

1 Introduction

2 Derivation

3 Existence analysis

4 Further topics

Version without figures (because of copyright reasons)

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Multi-species systems

Examples:

Animal populations: observing, predicting, harvesting

Fluid mixtures: heliox (diving, asthma), biofilm reactors, air pollution Cell dynamics: tumor growth, ion transport through membranes Electrolysis: lithium-ion batteries, production of hydrogen from water

Nature is generally composed of multi-species systems!

Modeling: diffusion equations

Ansgar J¨ungel (TU Wien) Entropy methods asc.tuwien.ac.at/∼juengel 2 / 47

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Literature

Needed prerequisites:

Partial differential equations (PDEs)

Sobolev spaces, basics of functional analysis

Optional: basics of probability theory, nonlinear PDEs Main reference

A. J¨ungel. Entropy methods for diffusive partial differential equations. Chap. 4, Springer Briefs, 2016.

A. J¨ungel. The boundedness-by-entropy method for cross-diffusion systems. Nonlinearity, 2015.

A. J¨ungel. Cross-diffusion systems with entropy structure. Proceedings of Equadiff 2017.

X. Chen, E. Daus. A. J¨ungel. Global existence analysis of cross-diffusion population systems for multiple species.

Arch. Ration. Mech. Anal., 2018.

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Diffusion 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 on∇u_i (Fick’s law)

In multicomponent systems, flux may depend on all∇u_j

⇒ cross-diffusion systems

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What are cross-diffusion systems?

∂tu_i−div n

X

j=1

A_ij(u)∇u_j

=f_i(u) in Ω, t >0, i = 1, . . . ,n Systems of quasilinear parabolic equations

Initial and (no-flux) boundary conditions What makes these systems special?

Adding (cross-) diffusion, constant equilibria may become unstable even if equilibria of associated ODE system linearly stable

May lead to physically desired pattern formation (Turing 1952) Uphill diffusion: diffusion flux in higher concentration area

Segregation: supp(ui(t))∩supp(uj(t)) =∅ ∀t (Bertsch et al. 1985) Blow-up inL^∞ norm in finite time possible (Star´a-John 1995) Aim: Develop mathematical theory onlyfor systems from applications

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➊ Multicomponent gas mixtures

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

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 Derivation: Boudin-Grec-Salvarani 2015 A(u) not symm., generally not pos. definite

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➋ Segregating populations

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

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 segregation Lotka-Volterra functions:

f_i(u) = (b_i0−b_i₁u₁−b_i2u₂)ui

Diffusion matrix is not symmetric, generally not positive definite

Figure: Black residential segregation in Milwaukee (blue dots) US Census Bureau 2002

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Difficulties and objectives

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

Diffusion matrixA(u) nondiagonal(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 of weak solutions

Positivity and boundedness of solution (if physically expected) Large-time behavior, uniqueness & regularity of solutions 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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Overview

Introduction Derivation

Existence analysis Further topics

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Derivation of cross-diffusion systems

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

→ Shigesada-Kawasaki-Teramoto (SKT) model From stochastic differential equations: many particle & small interaction limit

→ Shigesada-Kawasaki-Teramoto (SKT) model

From fluid models: high-friction limit and forces proportional to velocity differences

→ Maxwell-Stefan model

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

mean-free path limit in momentsR

f(x,v,t)φ(v)dv close to equilib.

→ Maxwell-Stefan model

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➊ 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, xi =ih)

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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➊ From lattice random walk to cross diffusion

Multiple species:

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

∂tu_j(xi) =p⁺_j,iu_j(xi−1) +p_j⁻_,i₊₁u_j(x_i+1)−(p⁺_j,i+p_j,i⁻)uj(xi) Transition rates: p_j^±_,i =p_i(u(x_j))q_i(un(x_j_±1))

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

∂tu_j =∂x

ⁿ X

k=1

A_jk(u)∂xu_k

, j = 1, . . . ,n Multi-dimensional case analogous

Example: qi = 1,pi(u) =ai0+Pn

k=1aikuk

Aij(u) =pi(u) +ui

∂pi

∂u_j(u) =δija_i0+δij

Xn

k=1

aikuk+aijui

→ n-species generalization of SKT population model

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➋ From SDEs to cross diffusion

Aim: Many-particle limits insingle-species particle system in R^d dX^k =−1

N XN

j=1,j6=k

∇B(X^k −X^j)dt+√

2σdW^k(t), k = 1, . . . ,N X^k(t) stochastic processes (“random position”),W^k(t) independent Wiener processes,B interaction potential, σ >0

Expectation for many-particle limit N→ ∞: 1

N X

j

∇B(x−X^j)∼E(∇B) = Z

Rd∇B(x−y)u(y)dy =∇B∗u Limit eq. for probability density u(x,t): ∂tu =σ∆u+ div(u∇B∗u) (Oelschläger 1989, Sznitman 1991, Méléard 1996)

Localization limit B→δ₀: ∂tu =σ∆u+ div(u∇u)

Goal: extend to multi-species case, expect cross-diffusion div(uj∇u_i)

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➋ From SDEs to cross diffusion

First attempt: Stochastic processesX_i^k(t) solve SDE dX_i^k =−

Xn j=1

1 N

XN ℓ=1, ℓ6=k

∇B_ij^η(X_i^k−X_j^ℓ)dt+√

2σidW_i^k(t) inR^d Species index i = 1, . . . ,n, particle index k= 1, . . . ,N

W_i^k independent Wiener processes,σ_i >0 Interaction potential: B_ij^η(x) =η^−dB_ij(|x|/η),R

R^dB_ijdx =a_ij ⇒ kB_ij^ηkL¹(R^d)=kBijkL¹(R^d), B_ij^η →aijδ₀ in D^′ asη→0 Limit: N → ∞ leads to “nonlocal” SDE, η→0 leads to local SDE with probability density u_i satisfying PDE (by Itˆo’s lemma)

Rigorous limit (L. Chen-Daus-A.J. 2019):

∂tu_i = div σi∇u_i +u_i∇p_i(u)

, p_i(u) = Xn k=1

a_iku_k This is notthe SKT population system ∂tu_i = ∆(σ_iu_i +u_ip_i(u))

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➋ From SDEs to cross diffusion

Second attempt: X_i^k(t) stochastic processes, i = 1, . . . ,n,k = 1, . . . ,N dX_i^k =q

2σ_i+ 2F_N^η(X)dW_i^k(t) inR^d F_N^η(X)=

Xn j=1

1 N

XN ℓ=1

B_ij^η(X_i^k−X_j^ℓ) W_i^k independent Wiener processes,σ_i >0 constants Interaction potential: as before, withB_ij^η →a_ijδ₀ asη→0 LimitN → ∞,η→0 with η^−2d−3 ≤δlogN for “small” δ >0 (L. Chen-Daus-Holzinger-A.J. 2020)

Density functionui solves SKT population model

∂tu_i = ∆ σiu_i +u_ip_i(u)

, p_i(u) = Xn k=1

a_ik∇u_k

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➌ From fluid models to cross diffusion

Mass and momentum balance equations: (Huo-A.J. Tzavaras 2019)

∂tu_i + div(uiv_i) = 0, i = 1, . . . ,n

∂t(uiv_i) + div(uiv_i ⊗v_i) +∇u_i = 1 ε

Xn

j=1

b_iju_iu_j(vj −v_i)

Rigorous high-friction limitε→0: expandui =u_i⁰+εu_i¹+O(ε²), etc.

Simplify (for presentation only): Pn

i=1u_i = 1, barycentr. velocity zero Equations up to order O(ε²), i.e. for u⁰_i +εu_i¹, etc.

Maxwell-Stefan equations: set Ji :=uivi,bij symmetric

∂tu_i + divJ_i = 0, ∇u_i =− Xn

j=1

b_ij(ujJ_i−u_iJ_j), Xn

i=1

u_iv_i = 0 Invert relationJ_i ↔ ∇u_i: problemP

iJ_i = 0 ⇒ invert on span{1}^⊥ Result: J_i =−Pn−1

j=1 A_ij(u)∇u_j

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➍ From kinetic models to cross diffusion

Boltzmann equation: fi =fi(x,v,t) (Boudin-Grec-Salvarani 2015) ε∂tfi +v· ∇^xfi = 1

εQi(fi,fi) +1 ε

X

j6=i

Qij(fi,fj), i = 1, . . . ,n Qi mono-species,Qij bi-species collision operators

Collisions are elastic & conserve mass: R

R3(Q_i +P

j6=iQ_ij)dv = 0 Particle density: u_i =R

R3f_idv, flux: εu_iv_i =R

R3f_ivdv Ansatz: f_i close to equilibrium:

f_i(x,v,t) = u_i(x,t) (2π)^3/2exp

−1

2|v−εvi(x,t)|²

Insert into Boltzmann eq., integrate, limitε→0→ Maxwell-Stefan

∂tu_i + div(u_iv_i) = 0, ∇u_i =− Xn j=1

b_iju_iu_j(v_i −v_j) Rigorous derivation for small initial data: Briant-Grec 2020

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Overview

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State of the art

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

Aim: Develop existence theory (uniqueness, regularity)

Ladyˇzenskaya et al. 1968: growth conditions on nonlinearities needed Many results for small cross diffusion (Kim 1984, Deuring 1987,...) Alt-Luckhaus 1983: global solutions if Onsager matrix unif. pos. def.

Kawashima-Shizuta 1988: hyperbolic-parabolic systems, entropies Amann 1990: parabolic in the sense of Petrovskii⇒ ∃! local classical solution; bounds inW^1,p(Ω), p>d ⇒ ∃global classical solution D. Le 2016: BMO bound & condition on A(u) ⇒ classical solution Burger et al. 2010: globalbounded weak solutions for special model Novelty of approach: degeneracies allowed, global L^∞ solutions

Key idea: exploit formal gradient-flow / entropy structure

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Gradient-flow or entropy structure

Main assumption

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

∂tu−div

B∇δH(u) δu

=f(u), where Onsager matrixB is pos. semi-definite,H(u) =R

Ωh(u)dx entropy Equivalent formulation: δH(u)/δu ≃h^′(u) =:w (entropy variable)

∂tu(w)−div(B(w)∇w) =f(u(w)), B(w) =A(u(w))h^′′(u(w))⁻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(x,t) = (h^′)⁻¹(w(x,t))∈D (no maximum principle needed!)

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Gradient-flow and thermodynamic structure

∂tu_i(w)−div n−1

X

j=1

B_ij(w)∇w_j

=f_i(u(w)), i = 1, . . . ,n−1 Gradient-flow structure: write equations as

∂tu_i −div n−1

X

j=1

B_ij∇δH δuj

= 0, w_i = δH δui

Entropy H=R

Ωh(u)dx

Gradient flow: ∂tu =−K[u^∗]gradH|^u on differential manifold, where K[u^∗]w =−div(B∇w) Onsager operator

Thermodynamic structure:

Mathematical entropy densityh =−s physical entropy density Entropy variable = chemical potential w_i =∂h/∂u_i

Onsager reciprocal relations: B is symmetric Entropy production: −^dH_dt =R

Ω∇w :B∇wdx ≥0

→ expresses second law of thermodynamics

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➊ Maxwell-Stefan models

∂_tu−div(A(u)∇u) = 0 in Ω, t >0 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,u ∈D={u:u_i >0,u₁+u₂<1} ⊂R² h(u) =u1(logu1−1) +u2(logu2−1) + (1−u1−u2

| {z }

=u3

)(log(1−u1−u2)−1) Entropy production:

dH dt +c

Z

Ω

X3

i=1

|∇√u_i|²dx ≤0 Entropy variables: w =h^′(u)∈R² or u= (h^′)⁻¹(w)

w_i = ∂h

∂ui

= log u_i

u₃, u_i = e^wⁱ

1 +e^w¹+e^w²∈D Consequences: gradient estimate for √u_i,u_i(x,t) is bounded

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➋ Population model

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

A(u) =

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

, aij ≥0 Lotka-Volterra terms: f_i(u) = (bi0−b_i1u₁−b_i2u₂)ui,i = 1,2 Entropy: h(u) =a₂₁u₁(logu₁−1) +a₁₂u₂(logu₂−1), D= (0,∞)² Entropy production:

dH dt +C

X2

i=1

Z

Ω

ai0|∇√

ui|²+aii|∇ui|²

dx ≤Cf

Entropy variables: w_i =∂h/∂u_i = logu_i ⇒ u_i = exp(w_i)>0

Consequences: gradient estimates for √ui ifa_i0 >0 and ui ifaii >0, nonnegativity foru_i

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Intermediate summary

Multicomponent systems omnipresent in applications, leads to cross-diffusion systems

Diffusion matrix generally neither symmetric nor positive semidefinite Generally, no full regularity, no maximum principle

Derivation from

diffusion limit for random walks on lattices many-particle limit in interacting particle systems high-friction limit in Euler equations

diffusion limit in system of Boltzmann equations

Mathematical theory based on formal gradient-flow / entropy structure

Structure inspired from thermodynamics

Next lecture: global existence analysis, uniqueness of solutions, large-time asymptotics, regularity of solutions

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Existence analysis

∂tu_i −div ⁿ

X

j=1

A_ij(u)∇u_j

=f_i(u), i = 1, . . . ,n

where A(u) = (Aij(u)) generally neither symm. nor positive semidefinite Main assumption

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

∂tu−div(B∇w) =f(u), w =∇h(u) where Onsager matrixB is pos. semi-definite,H(u) =R

Ωh(u)dx entropy Consequences:

1 H is Lyapunov functional if f = 0: dH/dt+R

Ω∇w :B∇wdx = 0

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

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Boundedness-by-entropy method

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

d dt

Z

Ω

h(u)dx+ Z

Ω∇u:h^′′(u)A(u)∇udx = Z

Ω

f(u)·h^′(u)dx 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 ≥c

Xn

i=1

u_i^2mⁱ⁻²z_i², mi ≥ 1

2 ⇒ estimate for|∇u_i^mⁱ|²

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

Let the above assumptions hold, let D ⊂Rⁿ be bounded,R

Ωh(u⁰)<∞, u_i⁰(x)∈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. 2015)

Let the above assumptions hold, let D ⊂Rⁿ be bounded,R

Ωh(u⁰)<∞, u_i⁰(x)∈D. Then ∃ global weak solution such that u(x,t)∈D and

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 SKT model 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 (mi = ¹₂)

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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 (u(w^k)−u(w^k⁻¹))/τ, τ >0 Benefit: avoid issues with time regularity

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

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

Solve problem inw^k by fixed-point argument

Benefit: elliptic problem in w-formulation (not true foru-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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Existence proof: more details

Approximate problem: Given w^k⁻¹∈L^∞(Ω), solve for φ∈H^m(Ω), 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 Linearized system: S :L^∞(Ω)×[0,1]→L^∞(Ω),S(y, δ) =w^k and w^k solves linearproblem (by Lax-Milgram)

Fixed-point argument: show that S compact, entropy estimate for all fixed points ⇒ ∃w^k ∈H^m(Ω): S(w^k,1) =w^k (by Leray-Schauder)

δ Z

Ω

h(u^k)dx +τ Z

Ω∇w^k :B∇w^kdx+ετkw^kk²H^m(Ω)

≤δ Z

Ω

h(u^k−1)dx+ Cτ

|{z}<1

δ Z

Ω

(1 +h(u^k))dx, u^k :=u(w^k) Limit (ε, τ)→0: Aubin-Lions compactness lemma

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Aubin-Lions lemma

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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Aubin-Lions lemma

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

k(u^(τ))^skL²(0,T;H¹)+τ⁻¹ku^(τ)(t)−u^(τ)(t−τ)kL¹(τ,T;(H^k)^′)≤C Then exists subsequence u^(τ⁾→u strongly in L^2s(0,T;L^2s)

Remark: Alt-Luckhaus 1983: s = 1, Maˆıtre 2003: nonlinear version of Simon 1987, Moussa 2016: monotone nonlinearities

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SKT population model

Diffusion matrix: (aij≥0) A(u) =

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

Entropy H(u) =R

Ωh(u)dx,h(u) =P2

i=1u_i(logu_i −1) for u ∈D = (0,∞)² butno L^∞ bound

Positivity: u_i = exp(w_i)>0 and entropy inequality:

dH dt +C₁

X2

i=1

Z

Ω

(a_i0|∇√u₁|²+a_ii|∇u_i|²)dx ≤C₂ a_ii >0: Gagliardo-Nirenbergu_i ∈L^2+2/d_x,t → enough to treatu_i∇u_j a_i0 >0: more sophisticated estimates sinceui ∈L^1+1/d_x,t only Theorem (L. Chen-A.J. 2004/2006)

Let H(u⁰)<∞. Then ∃solution (u₁,u₂)with u₁, u₂ ≥0 in Ωand ai0>0 : √

ui ∈L²loc(0,∞;H¹(Ω)), aii >0 : ui ∈L²loc(0,∞;H¹(Ω))

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Generalization 1: nonlinear coefficients

Macroscopic limit of random walk on lattice:

A(u) = p₁(u) +u₁^∂p_∂u¹

1(u) u₁^∂p_∂u¹

2(u) u₂^∂p_∂u²

1(u) p₂(u) +u₂^∂p_∂u²

2(u)

!

p_i linear: Chen-A.J. 2004

pi sublinear: Desvillettes-Lepoutre-Moussa 2014 p_i superlinear: p_i(u) =a_i0+a_i1u^s₁+a_i2u^s₂ (i = 1,2), entropy density: hs(u) =a₂₁u₁^s+a₁₂u₂^s,s >1 Theorem (A.J. 2015)

Let 1<s <4 and(1−¹_s)a₁₂a₂₁≤a₁₁a₂₂, H(u⁰)<∞. Then ∃ nonnegative weak solution u_i^s/2 ∈L²_loc(0,∞;H¹(Ω)) Idea of proof: use entropy h_s(u) +εP

iu_i(logu_i −1)

p_i superlinear,s >1: Desvillettes-Lepoutre-Moussa-Trescases 2015

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Generalization 2: more than two species

A_ij(u) = (a_i0+a_i₁u₁+· · ·+a_inu_n)δ_ij +a_iju_i Entropy: H(u) =R

Ωh(u)dx =R

Ω

Pn

i=1πiui(logui−1) Key assumption: π_ia_ij =π_ja_ji (detailed balance),π_i >0 Why detailed balance?

Detailed balance⇔ (πi) reversible measure ⇔h^′′(u)A(u) symmetric

⇒ entropyH(u(t)) decreases∀t

Detailed balancenot satisfied: a_ii “large” ⇒H(u(t)) decreases, otherwise ∃u(0) such that H(u(t))increases

Theorem (X. Chen-Daus-A.J. 2018)

Let a_ij >0and detailed balance hold. Then ∃ nonnegative weak solution u_i ∈L²_loc(0,∞;H¹(Ω)), i = 1, . . . ,n

Nonlinear coefficients: Chen-Daus-A.J. 2018, Lepoutre-Moussa 2017

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Overview

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➊ Entropy structure and normal ellipticity

∂tu−div(A(u)∇u) =f(u) (∗)

Definition: A(u) normally elliptic = A(u) positively stable = eigenvalues of A(u) have positive real parts = (∗) parabolic in sense of Petrovskii

Theorem (X. Chen-A.J. 2019)

If(∗)has entropy structure then A(u) normally elliptic

⇒ local existence of smooth solutions by Amann 1990

If A(u) normally elliptic & h^′′(u)A(u) symmetricthen (∗) has an entropy structure and A(u) diagonalizable with positive eigenvalues symmetry of h^′′(u)A(u) corresponds to Onsager relations

If A=A₀ constant: A normally elliptic⇔ (∗)has entropy structure A(u) =A₀+nonlinear perturbation⇒ ∃entropy structure

Proof: Use Lyapunov theorem and matrix factorization

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Entropy structure

Application: Keller-Segel model with additionalcross-diffusion

∂_tu_i = div(∇u_i−u_i∇c), i = 1, . . . ,n

∂tc = ∆c+δ Xn

j=1

∆uj + Xn

j=1

b_iju_j−c, no-flux b.c.

u_i: cell density ofith species,c: concentration of chemical signal δ >0: strength of additional cross-diffusion, avoids blow-up Diffusion matrixA(u) is normally elliptic

Factorization: A(u) =A₁A₂,A₁ symm. pos. def.,A₂ pos. def.

A₁ =







u₁ 0 0

. .. ...

0 u_n 0

0 · · · 0 δ







, A₂ =







1/u₁ 0 −1

. .. ...

0 1/un −1

1 · · · 1 1/δ







Set h^′′(u) =A⁻¹₁ , thenA₂ =A⁻¹₁ A(u) =h^′′(u)A(u) pos. def.

Compute entropy: h(u) =Pn

i=1ui(logui −1) +u₂²/(2δ)

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➋ Uniqueness of weak solutions

Alt-Luckhaus 1983: linear elliptic operator, ∂tu_i ∈L¹

Gajewski 1994: elliptic Onsager operator monotone in special sense Berendsen et al. 2020: weak-strong uniqueness for special system Result based on entropy method:

∂tu_i = div n

X

j=1

A_ij(u)∇u_j

, A_ij(u) =p(u0)δij +a_ju_iq(u0) u₀=

Xn i=1

a_iu_i, initial & no-flux boundary conditions u_i: species’ concentrations,u₀: solvent concentration Example: ion transport in membrane and nanopore Theorem (X. Chen-A.J. 2018)

Let p(s)≥0, p(s) +sq(s)≥0. Then uniqueness in class of functions p(u0)^1/2∇u_i,|q(u0)|^1/2∇u_i ∈L²,∂tu_i ∈L²(0,T;H¹(Ω)^′).

Idea of proof: combineH⁻¹ method and entropy technique of Gajewski

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Uniqueness of weak solutions

∂tu_i = div n

X

j=1

A_ij(u)∇u_j

, A_ij(u) =p(u0)δij +a_ju_iq(u0) Step ➊:H⁻¹(Ω) method

Sum equations fori = 1, . . . ,n, useu₀=Pn j=1a_ju_j

∂tu₀ = div (p(u0) +u₀q(u0))∇u₀

= ∆Q(u0), no-flux b.c.

whereQ^′(z) =p(z) +zq(z)≥0 (assumption)⇒ Q monotone Let u₀,v₀ be two weak solutions, let ξ solve −∆ξ =u₀−v₀ & b.c.

1 2

d dt

Z

Ω|∇ξ|²dx =− Z

Ω

∂t(∆ξ)ξdx = Z

Ω

∆(Q(u0)−Q(v0))ξdx

=− Z

Ω

(Q(u0)−Q(v0))(u0−v₀)dx ≤0, ξ(0) = 0

Implies thatξ(t) = 0 and hence (u₀−v₀)(t) = 0 ⇒uniqueness foru₀

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Uniqueness of weak solutions

∂tu₀ = ∆Q(u₀), ∂tui = div(p(u₀)∇ui +uiq(u₀)∇u₀) Step ➋:Define Gajewski’s semimetric

G(u,v) = Xn

i=1

Z

Ω

h(ui) +h(vi)−2h

u_i +v_i 2

dx, h(s) =s(logs−1) Compute time derivative

dG

dt (u,v) =−4 Xn

i=1

Z

Ω

p(u₀) |∇√u_i|²+|∇√v_i|²−|∇√

u_i +v_i|² dx ≤0 Test function ∂h/∂ui = logu_i requires to regularizeh(u)

G(u(0),v(0)) = 0 implies thatG(u(t),v(t)) = 0 andu(t) =v(t) Theorem (X. Chen-A.J. 2018)

Let p(s)≥0, p(s) +sq(s)≥0. Then uniqueness of weak solutions satisfying p

p(u₀)∇u_i,p

|q(u₀)|∇u_i ∈L²(Ω×(0,T)).

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Weak-strong uniqueness of renormalized solutions

∂tu_i = div Xn

j=1

A_ij(u)∇u_j +f_i(u), i = 1, . . . ,n A_ij(u) = (ai0+a_i1u₁+· · ·+a_inu_n)δij +a_iju_i Theorem (X. Chen-A.J. 2019)

u: renormalized solution, v : strong solution to SKT model. Then u=v . Renormalized solution: Use test function (∂ξ/∂u_i)φ_i, where ξ∈C^∞ with ξ^′ ∈C₀^∞; needed since no growth condition for f_i(u) supposed

Idea of proof: use relative entropyH(u|v) =R

Ωh(u|v)dx with h(u|v) =h(u)−h(v)−h^′(v)·(u−v), h(u) =

Xn

i=1

u_i(logu_i −1) Aim: Show thatdH/dt ≤CH ⇒ H(u(t)|v(t)) = 0 ⇒u(t) =v(t) Several cutoffs required (J. Fischer 2017), very technical

Relative entropy related to Gajewski’s semimetric

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➌ Large-time asymptotics

∂tu+A(u) =f(u), t >0, u(0) =u⁰ Entropy production:

dH

dt +hA(u),H^′(u)i=hf(u),H^′(u)i Assume: hf(u),H^′(u)i ≤0 and hA(u),H^′(u)i ≥λH. Then

dH

dt +λH≤0 ⇒ H(u(t))≤H(u⁰)e^−λt ⇒ u(t)→0 Convex Sobolev inequality: hA(u),H^′(u)i ≥λH

Example: SKT population model dH

dt +C₁ Xn

i=1

Z

Ω|∇√

ui|²dx ≤0, H(u) = Xn

i=1

Z

Ω

ui(logui −1) Use logarithmic Sobolev inequality:

Z

Ω

u_i(logu_i−1)dx ≤C_S Z

Ω|∇√u_i|²dx ⇒ dH dt + C₁

C_SH≤0

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Large-time asymptotics for reactive mixtures

∂tu+A(u) =f(u), t >0, u(0) =u⁰ Question: What happens if we donothave hf(u),H^′(u)i ≤0?

Example: Maxwell-Stefan systems and mass action kinetics fi(u) =

XN a=1

(βâ_i −αâ_i) k_fâu^αâ−k_bâu^βâ

, i = 1, . . . ,n k_fâ: forward reaction rate, k_bâ: backward reaction rate αâ_i,β_iâ: stoichiometric coefficients, u^αâ:=Qn

j=1u^α

a j

j

Conservation of total mass: Pn

i=1f_i(u) = 0

Aim: Show that u(t)→u_∞ as t→ ∞, use relative entropyH[u|u_∞] Entropy inequality: ^dH_dt +P[u]≤0,we needP[u]≥λH[u|u_∞]

P[u] = Z

Ω∇w :B∇wdx+ XN

a=1

Z

Ω

k_fâu^αâ−k_bâu^βâ

logk_fâu^αâ k_bâu^βâ ≥0

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Large-time asymptotics for reactive mixtures

P[u] = Z

Ω∇w :B∇wdx+ Xa a=1

Z

Ω

k_fâu^αâ−k_bâu^βâ

logk_fâu^αâ k_bâu^βâ ≥0 Homogeneous equilibrium: ∇w = 0 ⇒ u_∞=u(w) constant Detailed-balance equilibrium u: k_fâu^α_∞â =k_bâu_∞^βâ (there are many!) Wegscheider matrix: W = (β_iâ−αâ_i)ia,q₁, . . . ,q_m basis of ker(W^⊤), Q = (q₁, . . . ,q_m)^⊤

Conservation laws: ∂tQR

Ωu(t)dx =R

ΩQf(u)dx = 0,t >0 Theorem (Daus-A.J.-Tang 2020)

∃ unique detailed-balance equilibrium u_∞^∗ satisfying conserv. laws

∃ λ >0: P[u]≥λH[u|u^∗_∞]

Exponential convergence to equilibrium for 1≤p<∞: ku(t)−u_∞^∗ kL^p(Ω)≤C(u⁰)e^−λt/(2p), t >0

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➍ Regularity of solutions

∂_tu−div(A(u)∇u) =f(u) in Ω_T = Ω×(0,T), u(0) =u⁰ Negative result:

Star´a-John 1995: ∃ A∈L^∞: u(t) H¨older blows up att = 1 in L^∞ Full regularity:

Amann 1990: u(t) bounded in W^1,p(Ω), p>d ⇒u classical solution D. Le 2017: A(u) has polynomial growth of order ≤5,u(t)∈BMO

⇒ u classical solution (“Bounded Mean Oscillation”,L^∞⊂BMO⊂L^ploc) Partial regularity:

Giaquinta-Struwe 1982 (A(u) pos. def.): u is H¨older continuous in ΩT \S, whereHd−ε(S) = 0 for some ε >0

Braukhoff-Raithel-Zamponi 2020 (h^′′(u)A(u) pos. def.): u bounded

⇒ u is H¨older continuous in Ω_T \S,Hd−ε(S) = 0

Idea: Use relative entropyh(u|v) =h(u)−h(v)−h^′(v)·(u−v) and h(u|v)∼ |u−v|² for ui far from zero, Aij(u) diagonal forui →0

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Summary

∂iu−div(A(u)∇u) =f(u), t >0, u∈Rⁿ Boundedness-by-entropy method:

Gives global existence of boundedweak solutions

Compared to Alt-Luckhaus: degeneracies allowed, bounded solutions Compared to Amann: “easy-to-verify” conditions for global results Main ingredient: ∃ entropyh(u) such thath^′′(u)A(u) pos. semidef.

Relation to thermodynamics: w =h^′(u) are chemical potentials Entropy methods are used to prove:

Global existence of bounded weak solutions: for volume-filling models Uniqueness of weak solutions, weak-strong uniqueness

Large-time asymptotics: exponential decay to equilibrium Regularity of solutions: only partial results, problem mainly open

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Perspectives

∂tu−div(A(u)∇u) =f(u), t>0, u ∈Rⁿ Further topics:

Numerical schemes preserving entropy structure: finite volumes, finite elements, finite differences (A.J.-Zurek 2020)

Cross-diffusion systems with stochastic noise (Dhariwal-Huber-A.J.-Kuehn-Neamtu 2020) Open problems:

Existence of global weak solutions to n-species SKT population model without detailed balance, for all a_ij >0

Size of class of diffusion systems having an entropy structure Generalization of relative entropy for weak-strong uniqueness Analysis of models for nonisothermal, compressible fluid mixtures Derivation of noise terms for cross-diffusion systems