Frontiers of Mathematics and applications
Santander, UIMP, August 2011
Felix Otto, Max Planck Institute Leipzig
1
Summary
Phenomenon in Cahn-Hilliard: Coarsening
Low volume fraction — Ostwald ripening
With viscous flow — Siggia’s growth
Goal: Bounds on coarsening rates
Gradient flow: Characterize energy landscape
Elements of Analysis
2
Interpolation inequalities (Ledoux ’03),
Quantification of Di Perna-Lions (Crippa-De
Lellis ’08)
Phenomena and Heuristics of coarsening:
Cahn-Hilliard
3
Cahn-Hilliard equation: statics
Scalar order parameter: m(x) ∈ R
2)2
Double-well potential: 1
(1
−
m
4
Ginzburg-Landau energy:
E(m) :=
Z
1 |∇m|2
2
2)2
+1
(1
−
m
4
Large d-dimensional torus:
4
x ∈ (0, Λ)d, Λ ≫ 1
Cahn-Hilliard equation: dynamics
Flux j(x) ∈ Rd
in conservation law: ∂tm + ∇ · j = 0
=⇒
d
dt
Z
m=0
∂E
Dissipative dynamics: j = −∇ ∂m
=⇒
dE
dt
= − |j|2
... turns into Cahn-Hilliard equation
∂E = 0
∂tm − △(−△m + (m3 − m)) = ∂tm − △ ∂m
... as opposed to non-conservative Allen-Cahn equation
5
∂E = 0
∂tm − △m + (m3 − m) = ∂tm + ∂m
Z
Dynamics in initial stage
Initial data:
m(t = 0) = white noise of small amplitude
Observations:
• 1. average wave length ∼ 1
linearization, Fourier transform:
• 2. average amplitude ∼ 1
∂tF δm = (|k|2 − |k|4)F δm,
(1 − m2)2 has stable equilibria {−1, +1}
6
Dynamics in later stage
Observations:
width of
transition layer ∼ 1
≪ size of domains ∼ R
≪ size of system ∼ Λ
7
Dynamics in later stage
driven by interfacial energy
E(m) ≈ energy of interfacial profile × area of interface
Energy of optimal interfacial profile:
8
Z ∞
1 dm 2
1
) + (1 − m2)2 dx1
(
4
−∞ 2 dx1
Z ∞
1
dm 1
dm
2
√ (1 − m ) dx1 with equality iff
= √ (1 − m2)
≥
dx1
−∞ dx1 2
2
Z 1
x1 − x∗1
4
1
2
√ (1 − m ) dm = √
with equality iff m = tanh( √
).
=
−1 2
3 2
2
A geometric evolution equation
Curvature Flow
Mullins-Sekerka
Surface Diffusion
second order
third order
fourth order
9
Pego,
Alikakos&Bates&Chen,
Röger & Schätzle
Statistical self-similarity
earlier
later
later,
rescaled,
periodically
extended
10
Rate of energy decay
5
10
Energy E vs. time t,
double logarithmic plot:
1
Slope: −
3
4
10
Λ−(d=2) E(m) ∼ t−1/3
after initial stage
3
10
−4
10
−2
10
0
10
2
10
11
4 area transition layer
After initial phase: E(m) ≈ √
3 2
system volume
1/3
Typical length scale R = area
∼
t
transition layer
4
10
Abstract framework for gradient flow
Points in space: m
Function: E(m)
Tangent vectors: δm,
e. g. δm = dm
dt to a curve t 7→ m(t), “infinitesimal perturbations”
Differential: diffE|m = linear form on tangent vectors δm;
dm
defined by dE
=
diffE
.
|m
dt
dt
Metric tensor: gm = scalar product on tangent vectors δm,
“gives a size to infinitesimal perturbations”
12
Gradient: gradE|m = tangent vector
defined by ∀ δm gm(gradE|m, δm) = diffE|m.δm,
“defines a direction of steepest ascent”
Heuristic identification of gradient flow structure of CahnHilliard
R
d
Points in space = configurations: m : (0, Λ) → R with m = 0.
Function = Energy: E(m).
Tangent vectors = Infinitesimal variations:
R
d
δm : (0, Λ) → R with δm = 0.
13
Metric tensor = dissipation rate:
R
gm(δm, δm) = inf j : (0,Λ)d→Rd { |j|2 | δm + ∇ · j = 0 }.
Gradient flow (abstract): dm
dt + gradE|m = 0
⇔
∀δm gm( dm
dt , δm) + diffE|m.δm = 0
⇔
dm = argmin
1 g (δm, δm) + diffE .δm
δm 2 m
|m
dt
Gradient flow (Cahn-Hilliard): Write δm + ∇ · j = 0, then
Z
Z
∂E
∂E
(−∇ · j) = ∇
· j,
diffE|m.δm =
∂m
∂m
so that abstract variational principle turns into
1
argminj
2
Z
|j|2 +
Z
∂E
∂E
· j = −∇
.
∇
∂m
∂m
14
R
... opposed to metric tensor gm(δm, δm) = (δm)2
∂E = 0.
that turns gradient flow into Allen-Cahn ∂tm + ∂m
Philosophy: Use gradient flow structure for heuristics
• Identification of a singular limit:
Pass to limit in function E and metric tensor g separately
– Here: Sharp interface limit m
Ω
– and Low volume fraction limit Ω
{Ri}i
• Construction of thermodynamically consistent models:
Postulate driving free energy (=E) and
limiting dissipation mechanism (=g)
15
– Here: Introduction of viscous flow into Cahn-Hilliard
Heuristic identification of interfacial dynamics via gradient
flow structure
Points in( space: sets Ω
) of volume fraction
1 in Ω
via m ≈
.
−1 in CΩ
Function: E(m) ≈ c0Hd−1(∂Ω) , c0 =
1
2
4
√
3 2
Tangent vectors:
R
normal velocities V with ∂Ω V dHd−1 = 0
R
R
via δm φ ≈ ∂Ω 2V φdHd−1
∀ potentials φ : (0, Λ)d → R.
16
Identify differential:
R
d−1
diff[c0H
(∂Ω)].V = c0 ∂Ω HV dHd−1,
where H is mean curvature.
energy transition layer
Metric tensor:
gΩ(V, V ) = inf
j
Z
Z
|j|2 ∀ φ
∂Ω
2V φdHd−1 −
Z
j · ∇φ = 0
Identify metric tensor:
gΩ(V, V ) = inf sup
j
φ
Z
|j|2 + 2
Z
j · ∇φ − 2
Z
∂Ω
2V φdHd−1 .
Optimality in j and φ:
j = −∇φ
and
R
(
∇·j =0
−[j · ν] = 2V
17
Thus gΩ(V, V ) = |∇φ|2 ,
where
φ solves Neumann problem
(
)
−△φ = 0
outside ∂Ω
.
[∇φ · ν] = 2V
on
∂Ω
outside ∂Ω
.
on
∂Ω
.
Identify gradient flow from abstract principle:
1
argminV
gΩ(V, V ) + diff[c0Hd−1(∂Ω)].V
2
Z
Z
1 2
(−2φ + c0H)V dHd−1 .
( |j| + j · ∇φ) +
= argminV inf sup
j
2
∂Ω
φ
Optimality in j and φ as above.
Optimality in V : −2φ + c0H = const = 0 on ∂Ω.
This defines an evolution of Ω:
Given Ω solve Dirichlet problem for φ:
(
−△φ = 0
outside ∂Ω
φ = c20 H
on
∂Ω
)
.
18
Then update Ω via V = [∇φ · ν],
Roughly, V = Dirichlet-to-Neumann∂ΩH.
Scale invariance, Statistical self-similarity,
and coarsening exponent
Scale invariance:
Evolution of Ω invariant under scaling x = µx̂ and t = µ3t̂:
On the one hand,
1 φ̂ =⇒ [∇φ · ν] = 1 [∇
ˆ φ̂ · ν̂].
x = µx̂ =⇒ H = 1
Ĥ
=⇒
φ
=
µ
µ
µ2
On the other hand,
x = µx̂ and t = µ3t̂
Statistical self-similarity:
Spatial correlation function:
R
corr(t, z) := 1d m(t, x)m(t, x + z)dx
=⇒
V = µ12 V̂ .
Λ
Weak version:
∀ generic initial data, t, t′ ≫ 1
∃λ
corr(t, z) ≈ corr(t′ , λz ).
19
Strong version:
∃ corruniv(ẑ), γ ∀ generic initial data, t ≫ 1
corr(t, z) ≈ corruniv( tzγ ).
Value of exponent not a surprise...
Scale invariance (sharp interface level)
x̂ = µx,
t̂ = µ3t
If evolution
is
=⇒
−3t, λ−1x)
d
corr(x, t) = corr(µ
strongly statistically self-similar, i. e.
corr(t, z) ≈ corruniv(t−γ z)
for t ≫ 1,
then coarsening exponent γ must be = 1
3:
corruniv(t−γ z) ≈ corr(t, z)
20
3γ−1t−γ z)
−3t, µ−1z) ≈ corr
d
= corr(µ
univ (µ
... but statistical self-similarity a mystery
Phenomena and Heuristics of coarsening:
Cahn-Hilliard with small volume fraction
21
Small volume fractions = “off critical mixtures”
Z
Instead of Λ−d m = 0
Z
now Λ−d m = −1 + 2Φ
with Φ ≪ 1
−1 −1+2Φ
Typical configuration:
Minority phase Ω of volume fraction Φ
= union of approximately round balls
with approximately immobile centers
22
Evolution via “Ostwald ripening”:
Larger balls grow at the expense of small balls
1
m
Heuristic derivation of the Lifshitz-Slyozov-Wagner theory
for Ostwald ripening via gradient flow structure
Points in space: Radii {Ri}i with
S
via Ω ≈ i BRi (Xi ).
Function: E(m) ≈ cd
P
d
i Ri = const
P
d−1
i Ri
P
Tangent vectors: {δRi}i with i Rid−1δRi = 0
23
Differential: diff[c0Hd−1(∂Ω)].V = cd
P
d−2
δRi.
i Ri
Metric tensor: g{Ri}i,{Xi}i ({δRi}i, {δRi}) =
Z
|∇φ|2,
where φ solves
(
−△φ = 0
outside of ∪i∂BRi (Xi )
.
[∇φ · ν] = 2δRi
on
∂Ω
Monopole approximation of metric tensor for Φ ≪ 1:
g{Ri}i,{Xi}i ({δRi}i, {δRi}i) ≈
XZ
i
where φi solve
24
(
−△φi = 0
outside of BRi (Xi)
∇φi · ν = 2δRi
on
∂BRi (Xi )
Rd −BRi (Xi )
)
.
|∇φi |2,
Solution is given by multiple of fundamental solution
so that
Z

d−1

R

1
i

 d−2
|x−Xi|d−2
φi(x) = −2δRi


1

 Ri ln
|x−Xi |
Rd −BRi (Xi )
|∇φi|2 = cd(δRi )2
(
for




d>2 



for d = 2 
Rid
,
for d > 2
+∞ for d = 2
)
.
25
Hence this approximation has to be refined for d = 2:
Dirichlet integral of fundamental solution only logarithmically
divergent.
Small scale cut-off = average particle radius R.
Large scale cut-off = average particle distance L.
volume fraction Φ = number density of particles × volume per particle = cd L−dRd
large scale cut off
L ∼ ≈ ln 1 .
=
ln
Hence replace +∞ by ln small
scale cut-off
R
Φ
Altogether the LSW-metric tensor is given by
26
g{Ri}i ({δRi}i, {δRi}i) = cd
X
i
Rid(δRi )2 ×
(
1 for d > 2
1 for d = 2
ln Φ
)
.
Identify gradient flow by abstract principle. Have
{
dRi
}i
dt
= argmin{δRi}i
1
g
({δRi}i, {δRi}i) + diffE{Ri}i .{δRi}i
2 {Ri}i
= argmin{δR } |P Rd−1δR =0
i
i i
i i


1 for d > 2
1
for d = 2
ln Φ

X d−2
1X d
2
Ri δRi .
Ri (δRi ) + cd
2 i
i
Find Lifshitz-Slyozov-Wagner evolution
1 for d > 2
1
for d = 2
ln Φ
dRi
×
= −cdRi−2+µRi−1,
dt
P
27
where µ = cd P
d−3
R
i|Ri>0 i
d−2
R
i|Ri>0 i
.
Coarsening rates for small volume fraction
In terms of average radius R̄:
R̄ ∼
1 for d > 2
1
ln Φ
for d = 2
− 1
3
1
× t3 ,
because scale invariance of LSW-system under R = λR̂, t = λ3t̂;
because since for d = 2, logarithm acts as change of variable
1 )t̂
t = (ln Φ
In terms of energy density Λ−dE:
Λ
−d
E ∼ Φ×
1 for d > 2
1
ln Φ
for d = 2
1
3
−1
3
×t
,
28
because energy density = number density × energy per particle
volume fraction × energy per particle
= volume
per particle
≈ cd Φd × Rd−1 ∼ ΦR−1.
R
Phenomena and Heuristics of coarsening:
Cahn-Hilliard with flow
29
Derivation of a Cahn-Hilliard model with incompressible
flow and viscous dissipation
Incompressible flow u : (0, Λ)d → Rd next to diffusive flux j:
∂tm + ∇ · j+∇ · (mu) = 0
with
∇ · u = 0.
Viscous dissipation: strain rate e = ∇u + ∇t u
= rate at which length element is distorted
R
R
2
leads to dissipation 2 |e| = |∇u|2.
Metric tensor g:
30
gm(δm, δm)
)
( Z
Z
1
∂ m + ∇ · j+∇ · (mu) = 0,
.
= inf
|j|2 + |∇u|2 t
∇
·
u
=
0
j,u
λ
Metric tensor: Alternative representation of with help of
pressure = Lagrange multiplier p : (0, Λ)d → R for ∇ · u = 0:
gm(δm, δm)
= inf sup
j,u
p
1
λ
Z
|j|2 +
Z
|∇u|2 − 2
Z
p∇ · u ∂tm + ∇ · j + ∇ · (mu) = 0 .
Identification of gradient flow from abstract principle
dm
1
= argminδm
gm(δm, δm) + diffE|m.δm .
dt
2
Diffusive flux j, viscous flow u, and pressure p are determined by
31
Z
Z
Z
Z
1 2
1
∂E
2
inf sup
|j| +
|∇u| − p∇ · u +
(−∇ · j − ∇ · (mu))
j,u p
2λ
2
∂m
Z
Z
∂E
1
∂E
1 2
) + ( (|∇u|2 + u · ∇p) + mu · ∇
)
= inf sup
( |j| + j · ∇
j,u p
2λ
∂m
2
∂m
First variation
in j :
in u :
in p :
∂E
j = −λ∇
,
∂m
∂E
−△u + ∇p = −m∇
,
∂m
∇ · p = 0.
Evolution
∂E
+ ∇ · (mu) = 0,
∂tm − λ△
∂m
∂E
,
−△u + ∇p = −m∇
∂m
∇ · u = 0.
Given configuration m, determine velocity u (and pressure p)
∂E . Then update m according to
via Stokes system driven by ∂m
diffusion-advection equation.
Heuristic derivation of the interfacial limit for Cahn-Hilliard
with flow via gradient flow structure
1
No diffusion, λ = 0 Points in space: sets Ω of volume fraction 2
(
)
1
in Ω
via m ≈
.
−1 outside Ω
Function: E(m) ≈ c0Hd−1(∂Ω) ,
where c0 = energy of 1-d transition layer
R
32
Tangent vectors: normal velocities V with ∂Ω V dHd−1 = 0
R
R
via δm φ ≈ ∂Ω 2V φdHd−1 ∀ potentials φ : (0, Λ)d → R.
Identify differential:
R
d−1
diff[c0H
(∂Ω)].V = c0 ∂Ω HV dHd−1
Metric tensor:
gΩ(V, V ) = inf
u
= inf
u
|∇u|2 u · ν = V on ∂Ω, ∇ · u = 0
Z
sup
(|∇u|2 − 2p∇ · u) u · ν = V on ∂Ω .
p
Z
Identify gradient flow from abstract principle
dΩ
= argminδΩ gΩ(δΩ, δΩ) + diffE|Ω.δΩ .
dt
Using δΩ = V = u · ν get variational problem
inf sup
u
p
First variation
Z
(|∇u|2 − 2p∇ · u) + c0
in u :
(
in p :
∇·u = 0
Z
∂Ω
Hu · νdHd−1
−△u + ∇p = 0
outside ∂Ω
[∇u − p]ν = c0Hν
on ∂Ω
More physical form in terms of stress tensor S := ∇u + ∇tu − pid:
∇·S = 0
[S]ν = c0Hν
outside ∂Ω
on ∂Ω
Evolution. Given configuration Ω, determine u velocity according to Stokes system driven by H. Update Ω according to
V = u · ν.
33
Scale invariance. x = µx̂ and t = µt̂.
On the one hand: x = µx̂
=⇒
H = µ−1Ĥ
=⇒
u = û.
=⇒
S = µ−1Ŝ
On the other hand: (x = µx̂ and t = µt̂)
=⇒
V = V̂ .
Flow dominated: coarsening exponent 1
Numerical Simulation for diffuse interface with λ = 1
0
energy density E(t)
10
slope −1
−1
10
−2
10
0
10
1
10
2
10
time t
34
Λ−(d=2) E(m) ∼ t−1
3
10
4
10
Dissipation mechanism influences dynamics
Same energy
E ≈
4 area
√
3 2
of transition layer
35
mediated by diffusion,
limited by outer friction
mediated by flow,
limited by viscosity
“Evaporation
-Recondensation”
“Siggia’s growth”
Cross-over from t
1
−3
to t−1
Heuristics:
The mechanism that reduces energy faster dominates
1
−3
initially: Diffusion faster (λt)
later: Flow faster t−1
36
... confirmed by experiments
Statement of rigorous results
37
Cahn-Hilliard with small volume fraction
Theorem. (Conti & Niethammer & O. ’06)
For any initial data, Λ ≫ 1, Φ ≪ 1
Z T
E
(
0 Λd
provided

Z T
>
Φ
)2 ∼
0
E(0) <
2
∼
Φ
Λd
38
Optimal lower bound
×
and
1 for d > 2
1
for d = 2
ln Φ
1
T3
1
≫ Φ
3
×t
2
−1
3
D(0)
−1
2
d
Λ2
.
dt
Cahn-Hilliard with viscous flow
Theorem.
(O. & Seis & Slepcev ’11+, Brenier, O. & Seis ’10)
For any initial data, λ ≫ 1, Λ ≫ 1,
Z T
0
1
max{λ 2 (
Z T
1
E 2 E
t − 1 2 −1
>
) , d } dt ∼
min{λ 2 ( 1/2 ) 3 ) , t } dt
d
0
Λ
Λ
λ
provided
E(0) <
∼ 1
d
Λ
and
39
Optimal lower bound


T
1
λ2
1/3

≫
D(0)
1.
Λd λ 2
Why only lower bounds on E?
Upper bounds on E not independent of initial data:
— too many stationary points of E
m
1
x
−1
Lower bounds on E independently of initial data
40
...avoids to make sense of and cope with “generic”
Basic idea for rigorous lower bounds on E
Dynamics is steepest descent
in energy landscape
energy
dissipation
mechanism
↔
heights,
↔ distances
landscape not steep
=⇒
energy decreases not fast
41
Need to understand distances in the large
metric tensor g m(δm, δm)
local
Simple definition:
inf
(
induced distance dist(m0, m1)
global
dist2(m0, m1) :=
R1
dm
dm
0 g m(s)( ds (s), ds (s)) ds [0, 1] ∋ s 7→ m(s),
m(0) = m0
m(1) = m1
)
but no explicit representation.
42
Need explicit proxy D(m)
for distance dist(m, m∗) to well-mixed state m∗ = 0:
|gradD|m|2 ≤ 1
,
The proxies D to d(m, m∗) for Ostwald
For Ostwald ripening, well-mixed state m∗ = −1 + 2Φ
D(m)2 = inf
=
=
Z
Z
j
Z
2 |j| |∇φ|2 dx
m + 1 − 2Φ + ∇ · j = 0
where
− △φ = m + 1 − 2Φ
||k|−1Fm|2 dk =: k|∇|−1(m + 1 − 2Φ)k2
2
... the H −1-distance function. Note that k|∇|−1uk2
43
is the dual norm to k∇uk2 w. r. t. kuk2.
The proxy D to d(m, m∗) for Siggia
For Siggia’s growth, D(m) is defined as a transportation distance between positive and negative part of m
between m+ := max{m, 0}
and
m− := max{−m, 0}
44
Definition of transportation distance
Given m = m+ − m−,
a measure π(dx− dx+) on [0, L]n × [0, L]n
is called admissible transfer plan provided
Z
Z
ζ(x+ ) π(dx+ dx− ) =
ζ(x− ) π(dx+ dx− ) =
45
D(m) := inf
Z
Z
Z
ζ(x) m+ (x) dx,
ζ(x) m− (x) dx.
c(|x−−x+|)π(dx+ dx−)
π
admissible
Choice of cost function c:
cross-over between linear and logarithmic at z =
c(z) :=
46







z
for z ≤
λ1/2


λ1/2 

z
1 + ln 1/2
for z ≥ λ1/2
λ



1
λ2
Abstract result relating geometry to dynamics
Have by definition of gradient flow dm
dt = −gradg E|m:
dm , dm )
− dE
=
g
(
m
dt
dt dt
Have by construction of proxy, i. e. |gradg D| ≤ 1:
2 ≤ g ( dm , dm )
| dD
|
m dt dt
dt
2 ≤ − dE .
Combines to ( dD
)
dt
dt
Proposition (Kohn & O. ’02)
2 ≤ − dE and:
Assume that E(t) and D(t) satisfy ( dD
)
dt
dt
>
− 2d
−d
Λ E ∼ (Λ D)−α
Then for all σ ∈ (1, α+2
α )
47
Z T
0
(Λ
−d
σ
Z T
>
E) dt ∼
0
provided E ≤ 1 for some α > 0
α
− α+2
)σ
(t
dt
for T ≫ (Λ
− 2d
D)α+2
and Λ−dE(t = 0) ≤ 1.
Bounding the steepness of the energy landscape
for Ostwald ripening
Relation with interpolation inequalities
48
Recall energy,
proxy to distance to well-mixed state m∗ = −1 + 2Φ:
E :=
Z
1
2 1
|∇m| + (1−m2)2
2
4
and
D :=
Z
||∇|−1(m + 1 − 2Φ)|2
Need for abstract result: Geometric exponent α = 1, i. e.
− 2d
−1
−1
−d
2
(Φ Λ E(m)) (Φ Λ D(m)) &
(
1
1
for d > 2
)
1 for d = 2
ln 2 Φ
provided E(m) ≪ 1.
1
2
49
Sharp interface limit: Consider χ = characteristic function of Ω ≈
{m ≈= −1}. Energy and proxy to distance to well-mixed state:
E ≈ c0
Z
|∇χ|
and
D ≈
Z
||∇|−1(χ − Φ)|2
1
2
.
Need for abstract result:
−1 −d
Φ
Λ
Z
|∇χ|
−1 −d
Φ
Λ
Z
||∇|
−1
(χ − Φ)|
1
2 2
&
(
1
1
ln 2
for d > 2
1
Φ
for d = 2
)
.
50
Functional form of geometric estimate = interpolation estimate
Proposition(Cohen & Dahmen & Daubechies & DeVore ’03,
[Ledoux ’03], Cinti & O. ’11+)
Consider functions u(x) on the torus x ∈ (0, Λ)d.
For any d and any function u, we have
kuk 4 .
3
1
k∇uk 2
1 k|∇|
−1
1
uk 2 .
2
For d = 2 and any function u≥ −1, we have
51
1
4
kuln
max{u, e}k 4 .
3
1
k∇uk 2
1 k|∇|
−1
1
uk 2 .
2
Interpolation estimate =⇒ geometeric estimate
Take power 2:
Z
|u
1 4
1 |3
ln 4
3
2
.
Z
|∇u|
Z
||∇|−1u|2
1
2
.
3:
Divide by system volume Λd to power 2
Λ−d
Z
|u
1 4
1 |3
ln 4
3
2
. Λ−d
Z
|∇u|
Λ−d
Z
||∇|−1u|2
1
2
.
52
χ
− 1 ≥ −1. Have
Apply to u = Φ
Z
4
|u| 3
3
2
1
3
1
−1
2 2
Λ
≥ ( − 1) Φ ∼ Φ 2 ,
Φ
3
Z
1 1
3
4
1
1 1
1
2
−1
2
−d
3
3
2
2
2
)Φ ∼ (ln
)Φ 2 ,
Λ
|u| ln max{u, e}
≥ ( − 1) (ln
Φ
Φ
Φ
Z
Z
Λ−d |∇u| = Φ−1Λ−d |∇χ|,
Λ
−d
Z
Multiply with
−d
||∇|
1
Φ2 .
−1
u|
2
2
−1
= Φ
Λ
−d
Z
||∇|
−1
(χ − Φ)|
1
2 2
.
Comments on the nature of the estimate
Useful in many applications
Has many applications: coarsening, domain branching in ferromagnets, superconductors, twin branching in shape memory
alloys ...
... because it has natural form:
kuk 4
| {z 3}
non-convexity
.
1
k∇uk 2
| {z 1}
1
k|∇|−1uk22 .
|
{z
}
interfacial energy field energy
53
Not a Sobolev estimate:
well-behaved w. r. t. volume averages
Among the family of scale-invariant estimates of the form
kukp . k∇ukθ1 k|∇|−1uk1−θ
2
1 ),
+
where 1p = θ(1 − 1d ) + (1 − θ)( 1
2
d
1,
θ≥2
R
1
ours (θ = 2 ) is the only one that holds when integrals dx are
R
1
replaced by volume average d dx:
Λ
Λ
−d
Z
4
|u| 3
3
4
.
Λ
−d
Z
|∇u|
1 2
Λ
−d
Z
||∇|
54
— no elements of the Sobolev estimate (θ = 1).
−1
u|
1
2 4
An end-point estimate
Rather, it should be seen as being embedded into the family of
scale invariant interpolation estimates
kukp .
1
2
k∇uk
q k|∇|
−1
1
uk 2
2
1 1 + 1 1,
where 1p = 2
q
22
q ∈ [1, ∞].
Of this family of estimates, q = p = 2, that is,
−1
kuk2
2 ≤ k∇uk2 k|∇| uk2
is immediate by definition of ||∇|−1uk2 as a dual norm:
55
k|∇|−1uk
R
uv
2 = sup
v k∇vk2
Of this family of estimates, the case 1 < p < ∞ can be easily
treated.
1.) Use definition of k|∇|−1uk2 via potential:
k|∇|−1uk2 = k∇φk2
where
− △φ = u.
Rewrite in terms of φ:
k△φkp .
1
k∇△φk 2
q
1
k∇φk 2 .
2
2.) Use Calderon-Zygmund:
k∇△φkq ∼ k∇3φkq .
56
Rewrite in terms of any component of v = ∇φ:
2
k∇vkp . k∇
1
vk 2
q
1
kvk 2
2
or just for any direction, say, the first coordinate direction:
1
1
k∂1vkp . k∂12vkq2 kvk22 .
3.) Integration by parts and Hölder’s inequality:
k∂1vkpp =
Z
∂1v sign∂1v |∂1v|p−1 = −
≤ kvk2 k∂1vkpp−2 k∂12vkq .
Z
The end-point estimate q = 1 is more subtle.
v |∂1v|p−2 ∂12v
Proof of the interpolation estimate
57