collatz-woche pattern formation & partial differential equations · 2010-10-26 ·...
TRANSCRIPT
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?