Generation of interface for solutions of the mass
conserved Allen-Cahn equation
Danielle Hilhorst 1 , Hiroshi Matano 2 ,
Thanh Nam Nguyen 3 , Hendrik Weber 4
1
University of Paris-Sud, 2 University of Tokyo,
3
NIMS, 4 University of Warwick
The mass conserved Allen-Cahn equation
1
Motivation
The singular limit of the nonlocal Allen-Cahn equation
We consider the problem

Z
1


u = ∆u + 2 f (u) − − f (u)
in Ω × R+ ,


 t
ε
Ω
(P ε )
∂ν u = 0
on ∂Ω × R+ ,




x ∈ Ω,
 u(x, 0) = u0 (x)
where Ω is a bounded domain of RN with smooth boundary, ∂ν is
the outer normal derivative to ∂Ω and
Z
Z
1
f (u(x)) dx.
− f (u) :=
|Ω| Ω
Ω
The mass conserved Allen-Cahn equation
2
The singular limit of the nonlocal Allen-Cahn equation
Problem (P ε ) was proposed by Rubinstein and Sternberg as a
model for phase separation in a binary mixture.
We assume for the moment that f (u) = u(1 − u 2 ).
Problem (P ε ) does not possess any comparison principle, which
makes its study very difficult.
The mass conserved Allen-Cahn equation
3
Propagation of interface
Let Γ0 be a smooth hypersurface without boundary. There exist a
time T ∗ > 0 and a smooth family of initial data u ε (x, 0) = u0 (x)
such that for t ∈ [0, T ∗ )
(
−1 in Ω−
t
u ε (x, t) →
+1 in Ω+
t ,
where
Ω = Ω−
t
[
Ω+
t
+
and where the two subdomains Ω−
t and Ωt are separated by a
smooth interface Γt which propagates according to the law
Z
1
κ) on Γt , t ∈ (0, T ∗ ), Γt=0 = Γ0 .
Vn = (N − 1)(κ −
|Γt | Γt
The mass conserved Allen-Cahn equation
4
Propagation of interface results
Chen, Xinfu; Hilhorst, D.; Logak, E. Mass conserving
Allen-Cahn equation and volume preserving mean curvature
flow. Interfaces Free Bound. 12 (2010), no. 4, 527-549.
Okada, Koji Dynamical approximation of internal transition
layers in a bistable nonlocal reaction-diffusion equation via the
averaged mean curvature flow. Hiroshima Math. J. 38
(2008), no. 2, 263-313.
The mass conserved Allen-Cahn equation
5
Generation of interface
Today : In the very early stage, the diffusion term is negligible
compared with the reaction term, so that the solution of Problem
(P ε ) behaves as that of the initial value problem for the
corresponding ordinary differential equation

Z
1

 ut =
f (u) − − f (u)
in Ω × (0, t ε ),

ε2
ε
Ω
(ODE )



u(x, 0) = u0 (x)
x ∈ Ω.
With the change of time scale τ =
t
, Problem (ODE ε ) becomes
ε2

Z
tε


 uτ = f (u) − − f (u) in Ω × (0, 2 ),
ε
Ω
(ODE )



u(x, 0) = u0 (x)
x ∈ Ω.
The mass conserved Allen-Cahn equation
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Generation of interface
When t → t ε , τ → τ ε := t ε /ε2 ∼ ∞, the solution u ε of Problem
(P ε ) is such that u ε (t ε ) ∼ v∞ where v∞ is a stationary solution of
Problem (ODE ). We will see that in general v∞ only takes two
values a− and a+ which are such that
f (a− ) = f (a+ ) = k
and
f 0 (a− ) < 0,
f 0 (a+ ) < 0,
where k is a constant depending on the initial function. Therefore,
the value of u ε quickly becomes close to either a− or a+ with steep
interfaces (transition layers) between the regions {u ε ≈ a− } and
{u ε ≈ a+ }.
The mass conserved Allen-Cahn equation
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Large time behavior for the nonlocal Allen-Cahn equation
S. Boussaı̈d, D. Hilhorst, T.-N. Nguyen, Convergence to steady
state for the solutions of a nonlocal reaction-diffusion equation,
Evolution Equations and Control Theory, Volume 4, Issue 1,
39–59, (2015).
Z


vt = ∆v + f (v ) − − f (v ) in Ω × R+ ,



Ω
∂ν v = 0
on ∂Ω × R+ ,
(AC )



 v (x, 0) = v0 (x)
x ∈ Ω.
We assume here that
f (s) =
n
X
i=1
ai s i
where n ≥ 3 is an odd number, an < 0.
The mass conserved Allen-Cahn equation
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Large time behavior for the nonlocal Allen-Cahn equation
Mass conservation property
Z
Z
v (x, t) dx =
v0 (x) dx.
Ω
Ω
Lyapunov functional
Z
Z
1
2
E(v ) =
|∇v | dx −
F (v ) dx,
2 Ω
Ω
Z s
where F (s) =
f (τ ) dτ .
0
The mass conserved Allen-Cahn equation
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Large time behavior for a nonlocal Allen-Cahn equation
Boussaı̈d, Hilhorst and Nguyen apply the Lojasiewicz inequality to
prove that as t → ∞
v (t) converges to a stationary solution ϕ in H 1 (Ω).
In other words, the omega-limit set of Problem (AC) is a singleton.
The mass conserved Allen-Cahn equation
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The corresponding nonlocal ordinary differential equation
D. Hilhorst, H. Matano, T.-N. Nguyen and H. Weber, On the large
time behaviour of the solutions of a nonlocal ordinary differential
equation, J. Dynam. Differential Equations 28 (2016), 707–731.
We consider the nonlocal ordinary differential equation

Z


v
=
f
(v
)
−
− f (v ) in Ω × R+ ,
 t
Ω
(ODE )



v (x, 0) = u0 (x)
x ∈ Ω.
We assume that the function f ∈ C 1 (R) and has exactly three
zeros α− < α0 < α+ such that
f 0 (α± ) < 0,
f 0 (α0 ) > 0.
We have studied the omega-limit set
ω(u0 ) := {ϕ ∈ L1 (Ω) : ∃tn →∞ such that
u(tn ) → ϕ in L1 (Ω) as n → ∞}.
The mass conserved Allen-Cahn equation
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The nonlocal ordinary differential equation
f
f 0(m) = f 0(M ) = 0
s1
s∗
m
s∗
O M
s2
s
We choose s1 (small enough) and s2 (large enough) such that
f (s2 ) < f (s) < f (s1 )
s∗ and s ∗ satisfy f (s∗ ) = f (M),
The mass conserved Allen-Cahn equation
s ∈ (s1 , s2 ).
for all
f (s ∗ ) = f (m).
12
Mass conservation and Lyapunov functional
Problem (ODE) has the following properties:
Mass conservation
Lyapunov functional
Z
Z
E (u) = − F (u) dx, where F (s) =
Ω
s
f (τ ) dτ.
0
However, the solution of Problem (ODE) is not very smooth
so the method used to study Problem (PDE) can not be
applied to Problem (ODE). Therefore we use different
methods, which are based on studying the profile of u(t) for
each time t.
The mass conserved Allen-Cahn equation
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Assumptions on function f
We always suppose that
(F1 ) f ∈ C 2 (R), and there exist real numbers m < M such that
(
f 0 (s) > 0 on (m, M),
f 0 (s) < 0 on (−∞, m) ∪ (M, ∞).
(F2 ) There exist s∗ < s ∗ satisfying
(
s∗ < m < M < s ∗ ,
f (s∗ ) = f (M), f (s ∗ ) = f (m).
The mass conserved Allen-Cahn equation
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Assumptions on function f
In some cases, we also consider a special case of f since we can
then derive sharper estimates. Such a function f is assumed to
satisfy, together with (F1 ), (F2 ), the assumption (F3 ):
(F3 ) There exist constants m, M such that m < m < M < M that
f 0 (s) = µ for all s ∈ (m, M).
The mass conserved Allen-Cahn equation
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Assumptions on the initial data
We always suppose that s1 ≤ u0 (x) ≤ s2 for x ∈ Ω. Moreover, let
HN−1 be the (N − 1)-Hausdroff measure and set
Z
1
dHN−1 ,
A(s) :=
{u0 (·)=s} |∇u0 |
where for a function w : Ω → R, we define
{w (·) = s} := {x ∈ Ω : w (x) = s}.
The mass conserved Allen-Cahn equation
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We suppose that one of the following sets of hypotheses holds :

(H11 ) u0 ∈ C 2 (Ω) and s∗ ≤ u0 ≤ s ∗ on Ω,



(H1 )
(H12 ) |{u0 (·) = s}| = 0 for all s ∈ (m, M),



(H13 ) |∇u0 | =
6 0 in {u0 (·) ∈ (m, M)} and A ∈ L∞
loc (m, M).

(H21 )



(H2 )
(H22 )



(H23 )
u0 ∈ C 2 (Ω) and s∗ ≤ hu0 i ≤ s ∗ ,
|{u0 (·) = s}| = 0 for all s ∈ R,
|∇u0 | =
6 0 in Ω and A ∈ L∞
loc (R).
The mass conserved Allen-Cahn equation
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An ODE vision
We denote by v = v (x, τ ) the solution of the equation without
diffusion. Let consider the following initial value problem:
vτ = f (v ) − hf (v )i,
v (x, 0) = u0 (x),
x ∈ Ω.
Note that v satisfies the mass conservation property:
Z
Z
v (x, τ ) dx =
u0 (x) dx for all τ ≥ 0.
Ω
Ω
Let Y (τ ; s) be the unique solution of the initial value problem
Ẏ = f (Y ) − λ(τ ),
Y (0; s) = s, with Ẏ :=
dY
.
dτ
Then Y (τ ; s) is strictly increasing in s and v (x, τ ) = Y (τ ; u0 (x))
for x ∈ Ω, τ ≥ 0.
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Key property
We will use the notations for each τ ≥ 0,
Ω− (τ ) := {x ∈ Ω, v (x, τ ) ≤ m},
Ω0 (τ ) := {x ∈ Ω, m < v (x, τ ) < M},
Ω+ (τ ) := {x ∈ Ω, v (x, τ ) ≥ M}.
Let (H11 ) hold. Then
1
2
s∗ ≤ v (x, τ ) ≤ s ∗ for all x ∈ Ω and all τ ≥ 0.
For every τ 0 > τ ≥ 0,
Ω− (τ ) ⊆ Ω− (τ 0 ), Ω+ (τ ) ⊆ Ω+ (τ 0 ) and Ω0 (τ ) ⊇ Ω0 (τ 0 ).
In other words, Ω− (τ ), Ω+ (τ ) are monotonically expanding in
τ while Ω0 (τ ) is monotonically shrinking in τ .
The mass conserved Allen-Cahn equation
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Basic result
We define
Ω− (∞) :=
[
Ω− (τ ),
\
Ω0 (∞) :=
τ ≥0
Ω0 (τ ),
τ ≥0
Ω+ (∞) :=
[
Ω+ (τ ).
τ ≥0
Let (H11 ) and (H12 ) hold. Then there exists a function ϕ ∈ L1 (Ω)
such that
v (·, τ ) → ϕ in L1 (Ω) as τ → ∞.
Here
ϕ = a− χΩ− (∞) + a+ χΩ+ (∞) ,
where Ω− (∞), Ω+ (∞) are defined as above, χA denotes the
characteristic function of a set A ⊆ Ω and a+ , a− are constants
satisfying
s∗ ≤ a− ≤ m,
M ≤ a+ ≤ s ∗ ,
Furthermore, we have
Z
f (a− ) = f (a+ ) = hf (ϕ)i.
Z
ϕ(x) dx =
Ω
The mass conserved Allen-Cahn equation
u0 (x) dx.
Ω
20
The level sets of v
We define a0 ∈ [m, M] as the unique solution of the equation
f (s) = hf (ϕ)i so that
a− ≤ m ≤ a0 ≤ M ≤ a+ .
In order to analyze the formation of interface in large time for v ,
we fix a constant η ∈ (0, M−m
2 ) arbitrarily and consider the sets:
e − (τ ) := {x ∈ Ω : v (x, τ ) ≤ a− + η},
Ω
e 0 (τ ) := {x ∈ Ω : a− + η < v (x, τ ) < a+ − η},
Ω
e + (τ ) := {x ∈ Ω : a+ − η ≤ v (x, τ )}.
Ω
The mass conserved Allen-Cahn equation
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Some estimates for large time
e ± (τ ) in large
Our purpose is to study the asymptotic behavior of Ω
e
time and estimate the decay of |Ω0 (τ )|.
We have that λ(τ ) = hf (v (·, τ )i → hf (ϕ)i = f (a− ) = f (a+ ) as
τ → ∞. We may choose T1 = T1 (η) > 0 be such that
η
η
f (a+ + ) ≤ λ(τ ) ≤ f (a− − ) for all τ ≥ T1 .
2
2
The mass conserved Allen-Cahn equation
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Diffuse interface
Little after little, we narrow down the study of the diffuse interface
to the set where m < v < M. Let ϕ be the limit of v (·, τ ) in
L1 (Ω) as τ → ∞. Then
f (m) < hf (ϕ)i < f (M).
Moreover since a− < m < a0 < M < a+ , it follows that
f 0 (a− ) < 0,
f 0 (a0 ) > 0,
The mass conserved Allen-Cahn equation
f 0 (a+ ) < 0.
23
Narrowing down the diffuse interface
We can choose a constant δ = δ(η) > 0 small enough such that
(
m < a0 − δ < a0 + δ < M
f (a− + η) < f (a0 − 2δ ) < f (a0 + 2δ ) < f (a+ − η).
and we set
µ∗ :=
inf
s∈[a0 −δ,a0 +δ]
f 0 (s) > 0,
Ω10 (τ ) := {x ∈ Ω : v (x, τ ) ∈ (a0 − δ, a0 + δ)}.
The mass conserved Allen-Cahn equation
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Level sets of v
Let u0 ∈ C (Ω) and let τ ≥ 0, s ∈ R. Assume that {v (·, τ ) = s} is
nonempty. Then
{v (·, τ ) = s} = {u0 (·) = Y −1 (τ ; ·)(s)}.
Proof.
Recall that v (x, τ ) = Y (τ, u0 (x)) and that Y (τ, ·) is strictly
increasing. Thus
{x ∈ Ω : v (x, τ ) = s} = {x ∈ Ω : Y (τ ; u0 (x)) = s}
= {x ∈ Ω : u0 (x) = Y −1 (τ ; ·)(s)}.
The mass conserved Allen-Cahn equation
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Narrowing down the diffuse interface
Lemma
Let δ and µ∗ be as discussed above. Choose T3 > 0 such that
δ
δ
f (a0 − ) < λ(τ ) < f (a0 + ) for all τ ≥ T3 ,
2
2
and set
(Z
C0 := sup
{u0 (·)=s}
s∈Y
−1
1
dHN−1 :
|∇u0 (x)|
(T3 ; ·)([a0 − δ, a0 + δ]).
Then C0 < ∞ and
|Ω10 (τ + T3 )| ≤ 2δC0 exp(−µ∗ τ ) for all τ ≥ 0.
The mass conserved Allen-Cahn equation
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shadow
Coming back to the nonlocal Allen-Cahn equation
We must evaluate the difference between the solutions of the
nonlocal PDE and the nonlocal ODE. We set
û(x, τ ) := u(x, t) = u(x, ε2 τ ).
The re-scaled function satisfies

ûτ = ε2 ∆û + f û − hf (û)i in Ω × (0, +∞),





∂ û
ε
(P̂ )
=0
on ∂Ω × (0, +∞),

∂ν




û(0) = u0
on Ω.
To that purpose we must estimate the term ε2 ∆û.
The mass conserved Allen-Cahn equation
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a priori estimates
Recall that
s1 ≤ û, v ≤ s2 .
We also prove, by means of maximum principle arguments that
|∇û(x, τ )|2 ≤ C4 exp(2µτ ) for all x ∈ Ω, and all τ ≥ 0.
and that
|∆û(x, τ )| ≤ C5 exp(2µτ ) for all x ∈ Ω, τ ≥ 0,
which yields
|û(x, τ ) − v (x, τ )| ≤ C6 ε2 exp(2µτ ) for all x ∈ Ω, τ ≥ 0.
The mass conserved Allen-Cahn equation
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Generation of interface
C6
1
1
1
1
1
log( ) and set tε := ε2 ln( ), τε := ln( ).
2µ
η
µ
ε
µ
ε
Choose ε0 > 0 such that τε0 − p ≥ T1 + T2 . Then for all
ε ∈ (0, ε0 ), we have
Let p ≥
e − (τε − p),
|u(x, tε − pε2 ) − a− | ≤ 2η for all x ∈ Ω
e 0 (tε − pε2 )| ≤ C2 exp(µ∗ p)εµ∗ /µ ,
|Ω
e + (τε − p).
|u(x, tε − pε2 ) − a+ | ≤ 2η for all x ∈ Ω
The mass conserved Allen-Cahn equation
29
Optimal thickness of interface
Suppose that the hypothesis (F3 ) holds, namely that:
there exist constants m, M such that m < m < M < M that
C6
1
log( ) and set
f 0 (s) = µ for all s ∈ (m, M), let p ≥
2µ
η
tε :=
1 2 1
ε ln( ),
µ
ε
τε :=
1
1
ln( ).
µ
ε
Choose ε0 > 0 such that τε0 − p ≥ T1 + T2 . Then for all
ε ∈ (0, ε0 ), we have
e − (τε − p),
|u(x, tε − pε2 ) − a− | ≤ 2η for all x ∈ Ω
e 0 (tε − pε2 )| ≤ C3 exp(µ∗ p)ε,
|Ω
e + (τε − p).
|u(x, tε − pε2 ) − a+ | ≤ 2η for all x ∈ Ω
The mass conserved Allen-Cahn equation
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I thank you for your attention!
The mass conserved Allen-Cahn equation
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