COLLATZ-WOCHE

Pattern formation & Partial Differential Equations

Felix Otto

Max Planck Institute for Mathematics in the Sciences

Leipzig, Germany

Pattern formation for three specific examples

A) crystal growth under deposition

— roughness of crystal surface

B) demixing of polymers

— labyrinth-like pattern

of concentration field

C) Temperature gradient triggers flow

— mushroom-like pattern

of temperature field

Few elementary mechanisms (diffusion, viscosity, ...)

— complex Pattern

Partial differential equations

Ordinary differential equation for a quantity c

dc

dt(t) = F(c(t))

Partial differential equation for a field c(x)

∂c

∂t(t, x) = F(c(t, x),

∂c

∂x(t, x),

∂2c

∂x2(t, x), · · · )

Predictability

c(t = 0, x) known c determined

Prediction

= Solution of partial differential equations

Explicit solutions e. g. c(t, x) = 1td/2

e−|x|2

4t

Asymptotic

approximate solutions

dcdt = F(ǫ, c)c(t) = c0(t) + ǫ c1(t) + ǫ2 c2(t) + · · ·

Numerical

approximate solutions1h(c(t)− c(t− h)) = F(c(t))

Qualitative properties

of ensemble of solutions

∫

c dx, · · ·

Here: Existence & uniqueness & regularity ok

A. Crystal growth and Kuramoto-Sivashinsky

equation

A. Relevant mechanisms

Crystal lattice favors certain

slopes of the surface

Exposed positions

are disfavored

Vertical growth rate

depends on slope

∂h∂t = − ∂

∂x((1− (∂h∂x)2)∂h∂x)︸ ︷︷ ︸

−∂4h∂x4︸ ︷︷ ︸

+f (1 + 12(

∂h∂x)

2)︸ ︷︷ ︸

A. Qualitatively different behavior

for small/large deposition rate f

Initial data h(t = 0) = white noise of small amplitude

Deposition rate f ≪ 1

• slow growth

• facets with

preferred slope ±1

• number of facets

decreases

Deposition rate f ≫ 1

• fast growth

• slope ≪ 1

• number of maxima/minima

≈ constant

A. Regime of strong deposition:

Kuramoto-Sivashinsky equation

For f ≫ 1, expressed in u = −∂h∂x:

∂u

∂t= −

∂

∂x

(12 u2

)

−∂2u

∂x2−

∂4u

∂x4

A. Three terms — three simple mechanisms

∂2u

∂x2

Growth

∂4u

∂x4

Decay

∂

∂x

(12u

2)

Shear

Periodic configurations u(t, x+ L) = u(t, x); large system L ≫ 1

A. Dynamic equilibrium

Initial data: u(t = 0) = white noise of small amplitude

Observations:

Initial phase

• 1. Smoothing ( ∂4u∂x4

)

• 2. Growth ( ∂2u∂x2

)

• 3. Shear ( ∂∂x(

12u

2) )

Dynamic equilibrium

• average amplitude ∼ 1

• average wave length ∼ 1

• chaotic behavior

in space & time

Shear contains exponential growth

A. Butterfly effect

u(t = 0) = u(t = 0)+localized perturbation of small amplitude

Observations:

• 1. Perturbation stays “invisible”

• 2. Perturbation becomes visible at other place

• 3. u and u differ significantly everywhere

• u and u behave “statistically” similar

Effective unpredictability of details

— robust qualitative behavior

A. Energy spectrum

Decomposition of spatial signal into waves of length L, L2 ,L3 , · · · :

(Fu(t, ·))(k) := L−1∫ L

0eikxu(t, x) dx (Fourier series)

Contribution of wave number (k, k+dk) to total energy:

L|(Fu(t, ·))(k)|2 dk

Time average:limt0↑∞

t−10

∫ t0

0L|(Fu(t, ·))(k)|2dt

A. Equipartition of energy

Observations:

• Equipartition of energy over wave numbers |k| ≪ 1

• Energy spectrum independent of L ≫ 1

“Universal” behavior

A. Challenge for mathematics

Observation:

After initial phase, there is a dynamic equilibrium,

with statistics independent of the initial data u(t = 0)

and of the system size L

Challenge for theory of partial differential equations:

Why?

In mathematics: “Why ?” = “How can it be proved?”

A good proof gives insight into “why”

A. Modest state of mathematical insight

Only statements of the following form have been proved:

space-time averages of |u|, |∂u∂x|, |∂2u

∂x2| . 1,

for all initial data u(t = 0), system sizes L

These statements have been proved step-by-step:

space-time averages of |u|, |∂u∂x|, |∂2u

∂x2| . Lp,

for all initial data u(t = 0)

Nicolaenko & Scheurer & Temam ’85, Goodman ’94: . L2

Collet & Eckmann & Epstein & Stubbe ’93: . L11/10

Bronski & Gambill ’06: . L, Giacomelli & O. ’05: ≪ L

O. ’09 . ln5/3L

bounds on dim(Attractor), dim(Inertial Manifold), Fojas et. al.

A. Insight from proof

Three methods have been developed. Insight of last

method:

Shear term ∂∂x(

12u

2) behaves like a coercive term, i. e.

∫ L

0

∂∂x(

12u

2) u dx as∫ L

0

∣∣∣∣

∣∣∣∂∂x

∣∣∣1/3

u∣∣∣∣

3dx

despite actually being conservative, i. e.

∫ L

0

∂∂x(

12u

2) u dx = 0.

A. Insight: Conservative acts as coercive

in forced inviscid Burgers

Consider f(t, x), g(t, x) with ∂u∂t +

∂∂x(

12u

2) = ∂g∂x,

smooth, periodic in x, compactly supported in t. Then

∫ ∫ ∣∣∣∣

∣∣∣∣∂∂x

∣∣∣∣

13 u

∣∣∣∣

3dx dt

.

modlog

∫ ∫ ∣∣∣∣

∣∣∣∣∂∂x

∣∣∣∣

23 g

∣∣∣∣

32 dx dt,

more precisely expressed in interpolation spaces

‖u; [H1∞, L2]1

3,∞‖3 . ‖g; [H1

2 , L1]23,1

‖32.

A. Insight: Connection with Onsager’s conjecture

on level of forced viscous Burgers

On the one hand, for ∂u∂t +

∂∂x(

12u

2)− ν∂2u

∂x2= ∂g

∂x have

uniform estimate in ν ↓ 0

‖u; [H1∞, L2]1

3,∞‖3 + ν‖u; H1

2‖2 . ‖g; [H1

2 , L1]23,1

‖32.

On the other hand, at ν = 0 if u ∈ [H1∞, L2]1

3,pwith

p < ∞, would have conservation of energy

∂

∂t

∫ 1

2u2 dx =

∫

u∂g

∂xdx.

A. Different physics — same mathematics

Magnetization in thin ferromagnetic films

Local minima of energy functional (u gray scale)

ǫ∫∫

( ∂u∂x1

)2 dx1dx2 +∫∫

(

| ∂∂x1

|−12

(

∂u∂x2

− ∂∂x1

(12u2)))2

dx1dx2

−H∫∫

u2 dx1dx2

A. Similar math: Interpolation inequalities

Kuramoto-Sivashinsky:∫∫ ∣

∣∣∣|

∂∂x1

|13u∣∣∣∣

3dx1dx2 .

ǫ∫∫

( ∂∂x1

u)2 dx1dx2 +1ǫ

∫∫ (

| ∂∂x1

|−1(

∂u∂x2

− ∂∂x1

(12u2)))2

dx1dx2

Magnetism:(∫∫

u2 dx1dx2

)32 .

ǫ∫∫

( ∂∂x1

u)2 dx1dx2 + (ln 1ǫ)

∫∫(

| ∂∂x1

|−12

(

∂u∂x2

− ∂∂x1

(12u2)))2

dx1dx2

B. Demixing and Cahn-Hilliard equation

B. Relevant mechanisms

DiffusionShort-range

attraction

∂u

∂t= −( ∂2

∂x2+ ∂2

∂y2)(u− u3)− ( ∂2

∂x2+ ∂2

∂y2)2u

B. Dynamics in initial phase

Periodic: u(t, x+ L) = u(t, x). Large system: L ≫ 1.

Initial data: u(t = 0) = white noise of small amplitude

Observations:

• 1. average wave length ∼ 1, , • 2. average amplitude ∼ 1

B. Dynamics in later phase

Observations:

width of

transition layer ∼ 1

≪ size of domains ∼ R

≪ size of system ∼ L

B. A geometric evolution equation

Curvature Flow

second order

Mullins-Sekerka

third order

Pego,

Alikakos&Bates&Chen,

Roger & Schatzle

Surface Diffusion

fourth order

B. Statistical self-similarity

earlier later later,rescaled,

periodicallyextended

B. Rate of energy decay

After initial phase: Energy E(u) ≈ length of transition layer

Energy E vs. time t,

double logarithmic plot:

L−(d=2)E(u) ∼ t−1/3

10−4

10−2

100

102

104

103

104

105

Slope: −

1

3

B. Modest state of mathematical insight

Only the following statement has been proved:

L−dE(u) & t−1/3 for t ≫ 1

for all initial data u(t = 0) close to u = 0.

Kohn & O. ’02

B. Insight from proof

Dynamics is steepest descent

in energy landscape

energy ↔ heights,

dissipationmechanism

↔ distances

“Flat” energy landscape =⇒ slow energy decay

B. Dissipation mechanism influences dynamics

Energy functional E ≈ length of interfacial layer

flow,viscosity

first order

L−dE(u) . t−1

Brenier&O.&Seis

bulk diffusion,friction

third order

L−dE(u) . t−1/3

surface diffusion,friction

fourth order

L−dE(u) . t−1/4

B. Dissipation mechanism determines geometry

Distance on configuration space

Monge-Kantorowiczdistance withcost function

c(x1, x2)

= ln(1 + |x1 − x2|)

H−1 norm(∫

||∇|−1(u1−u2)|2)12

Monge-Kantorowiczdistance withcost function

c(x1, x2)

= |x1 − x2|

B. Energy landscape via interpolation estimates

Flatness of energy landscape

L−dHd−1(∂{u ≈ 1})︸ ︷︷ ︸

(

L−d∫

||∇|−1u|2dx)12

︸ ︷︷ ︸

& 1 ≈

(

L−d∫

|u|43dx

)32

Energy distance to u = 0

‖∇u‖L1

12 ‖|∇|−1u‖L2

12 & ‖u‖

L43

Cohen & Dahmen & Daubechies & DeVore, Ledoux

B. Different physics — same mathematics

Domain branching in ferromagnets

Same interpolation estimate:

‖u‖L43. ‖∇u‖

12L1 ‖|∇|−1u‖

12L2

B. A universal pattern

Domainbranching inferromagnets

Hubert,Choksi & Kohn

Domainbranching insuperconductors

Landau,

Choksi & Kohn & O.

Twin-splitting inshape memoryalloys

Kohn & Muller,Conti

C. Rayleigh-Benard convection

C. Relevant mechanisms

Diffusion Buoyancy Convection

temperature T flowvelocity u

driving boundary cond. limiting boundary cond..

C. Dynamic equilibrium

Periodic in horizontal direction: T (t, x+ L, z) = T (t, x, z)

large system H ≫ 1, L ≫ 1.

Initial data: T (t = 0) = linear profile + small amplitudewhite noise

Observation:

Initial stage

• 1. large rolls

• 2. boundary layer,

plumes over half height

Dynamic equilibrium

• plumes over entire height

• chaotic behavior

in space & time

C. Nusselt number

Diffusion and convection vertical heat flux

heat flux q = Tu−∇T, vertical heat flux = q ·(01

)

Nusselt number = space-time average of q ·(01

)

Nu = limt0↑∞

1t0

1Ld−1

1H

∫ t0

0

∫

(0, L)d−1 × (0, H)q ·

(01

)

dx dt

C. Nusselt number Nu independent of height H ≫ 1

Experiments: Nu independent of container height H ≫ 1

Simulation:

100 200 300 400 500 600 7000

0.05

0.1

0.15

Zeit

Nu(

t)

NUSSELT−ZAHL

BLAU:ROT:SCHWARZ:

H = 250,H = 500,H = 1000,

Nu = 0.0648, Varianz = 3.61e−04Nu = 0.0584, Varianz = 2.15e−04Nu = 0.0545, Varianz = 1.19e−04

C. Modest state of mathematical insight

Constantin & Doering (’99): Nu . ln2/3H,

Stokes maximal regularity in L∞

Doering & O. & Reznikoff (’06): Nu . ln1/3H,

logarithmic background temperature profile

O. & Seis (in preparation): Nu . ln1/15H,

logarithmic background profile (optimal)

O. & Seis (in preparation): Nu . ln2/3 lnH,

Stokes maximal regularity in L∞ and

logarithmic background profile

C. Insight from proof

Two different methods of proof. Insights:

1) Non monotonicity of background temperature pro-

file improves stability

stable for H0 . log−1/15H

yields Nu . log1/15H

2) Optimal background profile has no physical meaning

Summary:

Scaling laws in spatially extended systems

B) Gradient flows

C) Driven gradient flows

A) Non-gradient dynamics

Summary

Simple mechanisms — complex patterns

Statistical properties of pattern are universal

Only partial mathematical understanding

Different physics — similar mathematics

Opposite bounds for generic initial data?