Phase transition of an anisotropic
Ginzburg–Landau equation
PDE workshop in South China University of Technology
刘豫宁
上海纽约大学
Yuning LIU (NYU Shanghai)
Introduction of the model
We consider the complex Ginzburg–Landau type equation
∂t uε = µ∇ div uε + ∆uε − ε−2 ∂F (uε ) in Ω × (0, T ),
where ∂F is the gradient of a potential F > 0 satisfying F
instance Chern–Simons–Higgs potential
−1
(1)
2
1
(0) = {0 , S }. For
F (u) = |u|2 (1 − |u|2 )2 =: f (|u|).
2
This equation is the L -gradient flow of the anisotropic Ginzburg-Landau energy
Z ε
ε
1
Aε [u] =
µ| div u|2 + |∇u|2 + F (u) dx.
2
2
ε
Ω
So the following energy dissipation law holds.
Z
d
Aε [uε (·, t)] +
ε|∂t uε |2 dx = 0.
dt
Ω
Aim: co-dimension one ε-limit of (1) for well-prepared initial data.
Major challenge: localization of the concentration of ∇uε .
Technical issues: lack of maximum principle and maximum modulus estimates.
Existing works are in the static case: Golovaty et.al ’19 ’20 (Ginzburg-Landau),
Lin–Wang ’22. (Ericksen’s model)
Yuning LIU (NYU Shanghai)
Scalar Allen-Cahn equation
System (1) is a vectorial and anisotropic generalization of Allen-Cahn equation
∂t cε = ∆cε − ε−2 F 0 (cε ) where F (c) = (c2 − 1)2 .
The ε-asymptotic of the above equation has been widely investigated. To list a few,
Modica-Mortola ’77 (static case)
ε
1
Γ
µεt =
→ σHd−1
|∇cε |2 + F (cε ) Ld −
2
ε
{z
}
|
It
>|∇ψε |
where ψε (x, t) =
R cε (x,t) p
0
2F (z) dz is bounded in BV class.
De Mottoni–Schatzman ’95 (local, asymptotic expansion),
Evans–Soner–Souganidis ’92 (global, convergence to viscosity sol. by
Chen–Giga–Goto ’91 and Evans–Spruck ’91 ),
Ilmanen ’93, Chen ’96, Tonegawa ’03, Röger-Schätzle ’06 (global, convergence
to Brakke’s flow in geometric measure theory, Brakke ’78)
Modulated energy method (local, Ilmanen ’93, Lin ’95, Soner ’97)
The list is by no means complete. The last two approaches relies essentially on
Modica’s MP in order to control the discrepancy 2ε |∇cε |2 − 1ε F (cε ). So it is hard to
generalize them to the anisotropic/vectorial cases.
Yuning LIU (NYU Shanghai)
Generalization to vectorial cases
Such a modulated energy was first used by Fischer–Laux–Simon ’20 (motivated by
Jerrard–Smets ’15 and Fischer–Hensel ’20) to study the convergence of scalar
parabolic Allen-Cahn equation to MCF. Similar energy can be used for the
anisotropic energy
Z ε
1
ε
Aε [u] =
µ| div u|2 + |∇u|2 + F (u) dx,
2
2
ε
Ω
whose integrand will be concentrated on It . To modulate, we introduce
Z
Z |uε (x,t)| p
Eε [uε |I](t) :=Aε [uε ] −
ξ · ∇ψε (·, t) dx with ψε =
2f (s) ds.
Ω
0
where ξ is an extension of the normal of MCF so that |ξ| = 1 − λd2I (x, t) near It .
Ω−
t
It
Ω−
t
Ω+
t
ξ
∂Ω
Yuning LIU (NYU Shanghai)
It
Ω+
t
Main result
R |u (x,t)| p
Recall F (u) = f (|u|) is a double equal-well potential, ψε := 0 ε
2f (s) ds, and
Z
ε
ε
1
Eε [uε |I](t) :=
µ| div uε |2 + |∇uε |2 + F (uε ) − ξ · ∇ψε
2
ε
Ω 2
R1p
By a normalization, we can choose f s.t 0 2f (s) ds = 1.
L. ’21 preprint
For well-prepared initial datum, there holds
Z
sup Eε [uε |I](t) . ε, sup
|ψε − 1Ω+ | dx . ε1/3 ,
t∈[0,T ]
t∈[0,T ]
Z lim
ε→0
Ω
Z
ε| div uε |2 dx = 0
lim
ε→0
1
ε
|∇uε |2 + F (uε )
2
ε
Ω
t
dx = 1 · H1 (It )
∀t ∈ [0, T ],
∀t ∈ [0, T ].
Ω
Moreover, there exists u ∈ H 1 (Ω+ ; S1 ) with u · n = 0 on ∂Ω+
t such that
ε→0
1
0 2
±
uε −−−→ 1Ω+ u, strongly in L∞
t Lx ∩ Ct Lloc (Ωt )
t
up to the extraction of a subsequence ε = εk . Here Ω+ = ∪t>0 Ω+
t × {t} is the
nematic phase (outside the interface).
Yuning LIU (NYU Shanghai)
Modulated energy by Calibration: scalar case
R c (x,t) p
∇ψε
We denote ψε (x, t) = 0 ε
2F (z) dz, nε = |∇ψ
, and ξ is an extension of the
ε|
normal vector of It (whose modulus decays w.r.t d2I ). It satisfies
∂t ξ + (H · ∇)ξ + (∇H)T ξ = 0 in Bδ (It ).
We can write the density of modulated energy by:
ξ·∇ψε
z
}|
{
p
ε
1
∇ψε
|∇cε |2 + F (cε ) − ξ ·
|∇cε | 2F (cε ) > 0
2
ε
|∇ψε |
p
ε
1
2
= |∇cε | + F (cε ) − |∇cε | 2F (cε ) + (1 − ξ · nε ) |∇ψε |
{z
}
ε
|2
{z
} |
&min(d2
,1)
I
discrepancy 2
By
ε|∇cε |2 = |∇ψε | +
√
ε|∇cε |
√
|
1 p
ε|∇cε | − √
2F (cε )
ε
{z
}
discrepancy
R
and the Cauchy–Schwarz inequ, we can prove Eε [cε |I](t) & Ω ε|∇cε |2 min(d2I , 1).
Moreover, Eε [cε |I](t) satisfies a Grönwall inequality
and thus with appropriate
R
initial data we have Eε [cε |I](t) . ε. Thus Ω |∇cε |2 min(d2I , 1) . 1 for any t ∈ [0, T ].
Yuning LIU (NYU Shanghai)
Modulated energy by Calibration: vectorial case
The goal is to prove that Eε [uε |I](t) = O(ε), if it is the case initially.
Z 1
ε
2
|∇uε | + F (uε ) + |∇ψε | (1 − ξ · nε ) dx . Eε [uε |I](t).
{z
}
|
2
ε
(2)
min(d2
,1)6
I
Use radial projection Π∂i uε := ∂i uε ·
uε
|uε |
uε
|uε |
= ∂i |uε |b
uε to decompose
ε
1
∇ψε
|∇uε |2 + F (uε ) − ξ ·
|∇ψε |
2
ε
|∇ψε |
ε
ε
1
|Π∇uε |2 + F (uε ) − |∇ψε | + (1 − ξ · nε )|∇ψε |.
= |∇uε − Π∇uε |2 +
2
2
ε
Let η be a cut-off function which ≡ 1 in Bδ (It ). Then
Z
µ| div uε |2 + |∇uε − Π∇uε |2 . ε−1 Eε [uε |I](t),
Z
η |∇uε (I2 − n ⊗ n)|2 . ε−1 Eε [uε |I](t).
R
This shows | (div uε ) ∇tan uε | . ε−1 Eε [uε |I](t).
Yuning LIU (NYU Shanghai)
(3)
Deriving a differential inequality of the modulated energy
Let H be a constant extension in normal direction of the mean curvature vector of
It in B2δ (It ) followed by an appropriate cut-off, and Hε be its phase-field analogy:
∇uε
∇ψε
Hε = − ε∆uε − ε−1 ∂F (uε ) ·
,
nε =
.
|∇uε |
|∇ψε |
Laux–L. ’21 proved the following differential inequality (generalizing the scalar one
by Fischer–Laux–Simon ’20):
Z
2
d
1
ε∂t uε − (∇ · ξ)∂dF (uε )
Eε [uε |I] +
dt
2ε
Z
Z
2
1
1
+
ε2 |∂t uε |2 − |Hε |2 +
Hε − ε|∇uε |H . Eε [uε |I].
(4)
2ε
2ε
Using the PDE, we can derive
Z
|∂t uε + (H · ∇)uε |2
Z Z
2
1
6 2
ε2 |∂t uε |2 − |Hε |2 + Hε − ε|∇uε |H + 2µ ∇ div uε · (H · ∇)uε
ε
To close (4), it remains to estimate the term with µ.
Yuning LIU (NYU Shanghai)
Control of the anisotropic terms
Denote Hi,j = ∇H = ∇tan H and div uε = uεk,k . Integrating by parts
Z
Z
Z
ε
ε
− ∇ div uε · (H · ∇)uε = uj,j (H · ∇)uk,k + div uε (∂j H · ∇)uεj
Z
Z
Z
1
=−
(div H)(div uε )2 + (div uε )Hj,k uεj,k + (div uε )(Hk,j − Hj,k )uεj,k
2
The first integral is controlled by ε−1 Eε [uε |I]. The second term involves only
∇tan uε and is controlled by ε−1 Eε [uε |I] too. With µ = 1,
Z
last integral above = − (div uε )(rot uε ) rot H
Z
Z
= (div uε )uε · ∇⊥ rot H − ∇ div uε · (uε )⊥ rot H
{z
}
|
(∂t uε −∆uε )·(uε )⊥
Z
=
Z (div uε )uε · ∇⊥ rot H −
∂t uε + (H · ∇)uε · (uε )⊥ rot H
Z
Z
+ (H · ∇)uε · (uε )⊥ rot H + ∆uε · (uε )⊥ rot H
The last two integrals both involve ∂k uε · (uε )⊥ = (∂k uε − Π∂k uε ) · (uε )⊥ .
Yuning LIU (NYU Shanghai)
Convergence rate of ψε
Z
Convergence rate of ψε :
sup
t∈[0,T ]
Recall ψε (x, t) =
R |uε (x,t)| p
0
Ω
|ψε − 1Ω+ | dx . ε1/3
t
2f (s) ds. We define χ = 1Ω+ − 1Ω− and
t
+
−
2ψε − 1 = 2(ψε − 1) + 1 − 2(ψε − 1)
t
≈ 0 + χ.
To establish Grönwall’s inequality of them, and to avoid concentration, we introduce
Z
gε (t) := (ψε − 1)+ ζ(dI ) dx,
Z hε (t) :=
χ − [1 − 2(ψε − 1)− ] ζ(dI ) dx,
for a cut-off of the identity ζ(·) > 0. The integrands of the two weighted energies
> 0. To estimate their time derivatives, we need
ZZ
(∂t + H · ∇) ζ(dI ) = O(dI ),
|∂t uε + (H · ∇)uε |2 dxdt . 1.
Yuning LIU (NYU Shanghai)
Convergence estimate of ψε
Recalling ψε =
R |uε | p
0
2f (s) ds, we have
∂t ψε =(∂t uε + (H · ∇)uε ) ·
uε p
2F (uε ) − H · ∇ψε .
|uε |
R
We calculate the evolution of gε (t) := (ψε − 1)+ ζ(dI ) dx by
Z
1
1
uε p
gε0 (t) =
ε 2 (∂t uε + (H · ∇)uε ) ·
2F (uε )ε− 2 ζ(dI )
|u
|
ε
{ψε >1}
Z
Z
−
H · ∇ψε ζ(dI ) +
(ψε − 1)+ ∂t ζ(dI )
|
{z
}
{ψε >1}
R
| R
{z
}
[∂t ζ(dI )+H·∇ζ(dI )](ψε −1)+ − (ψε −1)+ H·∇ζ(dI )
H·∇(ψε −1)+ ζ(dI )
Z
6
ε
∂t uε + (H · ∇)uε
8
Z
2
+
8
F (uε )ζ 2 (dI ) +
ε
Z
(div H)(ψε − 1)+ ζ(dI ) + Cgε
This implies gε (t) . ε. To get the desired estimate of ψε , one simply employ
standard trick to remove the weight ζ(dI ) at the price of a weaker convergence rate
ε1/3 .
Yuning LIU (NYU Shanghai)
Anchoring boundary condition
bε =
Let u
uε
.
|uε |
We have from (3) that
Z
bε
µ| div uε |2 + ∇uε − ∇|uε | ⊗ u
2
dx . 1.
Taking the trace on the second integrand, we obtain
Z
2
b ε · ∇|uε | + |uε |2 |∇b
u
uε |2 dx . C(µ).
(5)
|∇uε |2 + 1ε F (uε ) − |∇ψε | . ε implies
Z
(b
uε · nε )2 |∇ψε | dx . εC(µ).
(6)
This together with
R
ε
2
ε→0
Formally |∇ψε | L2 −−−→ H1 It . So one can prove u · n = 0 by establishing
Z
Z
(6)
1
(u · n)2 dH1 6 lim inf
(b
uε · nε )2 |∇ψε | dx = 0.
ε→0
2 It
Ωε
t
where Ωεt are appropriate ψε - sub-level sets. This is done by computation of local
integral of calibration, and finally by a blow-up argument. To this end, we need
b ε on the super level set of ψε .
strong convergence of u
Yuning LIU (NYU Shanghai)
Nice level sets of ψε
Constructing transition layer {x ∈ Ω | ψε ≈ 1/2}
1
Let δ = ε 12 . There exists bε ∈ [ 12 − δ, 12 + δ] and bε →
Ωεt := {x ∈ Ω : ψε (x, t) > bε } has finite perimeter and
1
2
s.t the nematic region
1
H1 (∂Ωεt ) − H1 (It ) . ε 12 .
(7)
The proof uses the local integral of calibration on Stε,δ = {x ∈ Ω : |ψε − 1/2| 6 δ}:
Z
Cε >
(|∇ψε | − ξ · ∇ψε ) dx
(> 0)
ε,δ
St
Z
=
1 +δ
2
H1 ({x : ψε = s}) ds −
1 −δ
2
Z
ε,δ
∂St
ξ · νψε dH1 +
Z
ε,δ
(div ξ)ψε dx,
St
R
The last integral . |Stε,δ | . ε1/3 by Ω |ψε − 1Ω+ | dx . ε1/3 and Chebyshev’s inequ.
t
Z
Z
Z
1
1
−δ
div ξ dx −
+δ
div ξ dx,
ξ · νψε dH1 = −
ε,δ
1 +δ}
2
2
{x:ψε < 1
−δ}
{x:ψε > 2
∂St
2
Z
Z
1
1
−2δH1 (It ) =
−δ
div ξ dx
+
+δ
div ξ dx.
2
2
Ω−
Ω+
t
t
Yuning LIU (NYU Shanghai)
b ε on the level set of ψε
Convergence of u
1
1
There exists bε ∈ [ 12 − ε 12 , 12 + ε 12 ] so that Ωεt := {x ∈ Ω : ψε (x, t) > bε } (the
nematic region) has finite perimeter and
1
H1 (∂Ωεt ) − H1 (It ) 6 Cε 12 .
This combined with the SBV compactness theorem implies, for a.e. t ∈ [0, T ],
ε→0
b ε −−−→ 1Ω+ u weakly-star in BV (Ω),
1Ωεt u
t
ε→0
1Ωεt ∇b
uε −−−→ 1Ω+ ∇u weakly in L1 (Ω).
t
Now we discuss the proof of the anchoring condition by establishing the lower
semi-continuity property:
Z
Z
(6)
1
(u · n)2 dH1 6 lim inf
(b
uε · nε )2 |∇ψε | dx = 0.
ε→0
2 It
ε
Ωt
We shall adapt a density argument from Lin–Wang ’22 (Fonseca–Müller ’93). To
simplify the presentation, we assume a uniform-ε estimate kuε kL∞
6 C. Indeed,
x,t
the lack of such an estimate causes major technical issues.
Yuning LIU (NYU Shanghai)
Deriving the anchoring BCs by a density argument
ε→0
Write s2 = supn∈N (an s + cn ), and θε = 1Ωεt |∇ψε | L2 −−−→ 21 H1 It . For p ∈ It ,
Z
Z
b ε · nε + cn ) dθε
Cε >
(b
uε · nε )2 dθε >
(an u
Br (p)
Br (p)
Z
= an
Br (p)
b ε · nε |∇ψε |1Ωεt dx + cn θε (Br (p))
u
Z
= an
Br (p)∩Ωε
t
b ε · ∇ψ ε dx + cn θε (Br (p)),
u
(8)
where ψ ε = ψε − bε so that ψ ε = 0 on the bdy of Ωεt := {x ∈ Ω : ψε (x, t) > bε } and
L1
a.e.
b ε −−* 1Ω+ div u and product limit lemma,
ψ ε −−→ 12 1Ω+ . By 1Ωεt div u
t
t
Z
Z
b ε · νψ ε dx −
b ε ψ ε dx
red term in (8) =
u
1Ωεt div u
∂Br (p)∩Ωε
t
Br (p)
Z
Z
1
1
u · ν dH −
(div u) dx =
u · ν dH1 .
+
+
2
2
∂Br (p)∩Ω+
B
(p)∩Ω
B
(p)∩∂Ω
r
r
t
t
t
R
Combining with (8) yields 0 > 12 Br (p)∩It (an (u · ν) + cn ) dH1 . Divided by
H1 (Br (p) ∩ It ) and choose appropriate r = rj ↓ 0 yields 0 > an (u · ν) + cn for any
m ∈ N. So we obtain u · ν = 0 at p ∈ It up to a null set.
1
−−−→
2
ε→0
Z
Yuning LIU (NYU Shanghai)
1
Thank you !
Thank you !
Yuning LIU (NYU Shanghai)