Considering now the usual derivative in time (α = 1), the Allen-Cahn equation can be understood as a gradient flow in L2, minimizing the free energy functional
Fs(u) = ε2
2|u|2Hs(Rn)+ ˆ
Ω
W (u), (5.3.1)
with W (u) = (u2−1)4 2 (see for example [7]). It is well known that the size of ε affects the interface width of the minimizers of Fs. That is, interface width tends to zero with ε → 0. This fact can be easily derived from expression (5.3.1), observing that the right term, which penalizes the variation of u, tends to lose relevance as ε goes to zero, forcing the minimizer u to take values into the set of minimizers of W , that is values belonging to {1, −1}. However, since ε > 0, the right term promote the minimization of the interface length (for n ≥ 2), which implies that the limit behavior cannot be understood as the minimization of Fs with ε = 0. In [74], Savin and Valdinoci show, by means of Γ-convergence theory, that the limit behavior of the problem of minimize Fs tends to a minimal surface problem if s ∈ [1/2, 1), and to a non-local version of the minimal surface problem for s ∈ (0, 1/2).
In our case, numerical experiments (see Figure 5.2) show that the interface width tends to become thinner when the parameter s goes to zero, suggesting that (as in the case ε → 0) a minimizer of Fs should approximate a binary function when s → 0. Motivated by the previous observation, the aim of this section is to analyze the asymptotic behavior of the minimizers of Fs with s tending to zero. To this end, we are going to follow the ideas displayed in [74], and study the Γ-convergence of a suitable modification of the functional Fs.
5.3.1 Γ-convergence when s → 0.
Since Γ-convergence may not be a usual concept in numerical analysis, we start this section by giving its definition and basic properties, and we refer to [19] for further details.
Let X be a topological space, and {Fn}∈N, Fn : X → [−∞, +∞], a sequence of functionals. Then, we say that Fn Γ-converge to F : X → [−∞, +∞], if the following conditions holds:
• For every sequence {xn}n∈N ⊂ X such that xn → x, then F (x) ≤ lim inf
n→∞ Fn(xn).
• For every x ∈ X, there exists a sequence xn converging to x such that F (x) ≥ lim sup
n→∞
Fn(xn).
Also, we define a complementary concept. We say that the family {Fn} has the equi-coerciveness property if for all c ∈ R exists a compact set Kc in such a way that {Fn< c} ⊂ Kc for all n ∈ N.
These two concept allow us to say something about the limiting behavior of the minimizers of Fn in terms of the minimizers of F . That is, if xn is a minimizer of Fn, then every cluster point of {xn}n∈N (if exists) is a minimizer of F . This can be summarized as follow
Equi-coerciveness + Γ-convergence ⇒ Convergence of minimizers
In order to study the Γ-convergence of Fs, we must set an appropriate domain X for Fs,
X = {u ∈ L∞(Rn) with |u| ≤ 1, and u ≡ 0 in Ωc}.
And we are going to consider this space furnished with the norm k · kL1(Ω). Note that if u ∈ X but u 6∈ ˜Hs(Ω), then we can define Fs(u) = +∞.
and, denoting F [u](ξ) as the Fourier transform of u, we know from [28] and Plancharel’s identity that
Since we have ε2 < 1, ˜W (s) is a double-well type potential with minimizers ±√ and, for sake of simplicity, we redefine ˜W as
W (s) = W (s) +˜ ε2 discrete set. First, we want to see
lim inf
We first analyze the left term of the right hand side of (5.3.4). In this case we have
lim inf
From the fact that us→ u in L1(Rn) norm, we have F [us] → F [u] point-wise, and we also have (|ξ|2s− 1)/2s → ln |ξ|. Then, using Fatou’s Lemma, we get the estimation
(i) ≥ where in the last inequality we have use the reverse Fatous’s Lemma.
Hence, the first term in the right hand side of (5.3.4) must be a finite number. This implies that a.e., then u must have the form
u =√
Finally we only need the to verify that if u ∈ X, then F0(u) ≥ lim sup
Fs(u) =
Then, using Monotone Convergence Theorem on the integral over |ξ| > 1, and Domi-nated Convergence Theorem over |ξ| ≤ 1, we have
lim sup
s→0
F˜s(u) = F0(u), which proves (5.3.6).
5.3.2 Equi-coerciveness of ˜ F
sTo complete the analysis we prove the equi-coerciveness of { ˜Fs}s.
Theorem 5.3.2. Suppose sn → 0, and {un}n∈N ⊂ X, such that ˜Fsn(un) ≤ C for all n ∈ N. Then there exists u ∈ X and a subsequence {unj}j∈N, such that unj → u in X.
Proof. First we observe that, as before, we have F˜sn(un) = Now we want to show that (5.3.7) implies that F2(un)[ξ] remains distributed not far from zero. That is, given η > 0, there exists R > 0 such that
ˆ
|ξ|>R
F2(un)[ξ] dξ ≤ η, ∀n ∈ N. (5.3.9) By contradiction suppose that there exists η0 > 0 such that for every R > 0 there is a number m = m(R, η0) ∈ N, in such a way that
which, in view of (5.3.7) results in a contradiction. Hence, assertion (5.3.9) must be true.
On the other hand, the fact that {un}n∈N ⊂ X, implies that this sequence is uni-formly bounded in L2(Ω), and then, we can extract a weakly convergent subsequence {unj}j∈N. Let u ∈ L2(Ω) such that unj * u, our goal now is to show that unj → u strong in L2(Ω), which implies strong convergence in L1(Ω) since |Ω| < ∞.
To this end, we only need to verify that kunjkL2(Ω) → kukL2(Ω) or, equivalently, kF (unj)kL2(Rn) → kF (u)kL2(Rn). From (5.3.9), and the fact that u ∈ L2(Ω), we can take R large enough, in such a way that
ˆ
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Figure 5.1: In red the solution of example 1 at t = 50, in black-dashed the values
±√
1 − ε2. It can be seen that the equilibrium states remain far from 1 and −1, unlike the behavior in the classic AC equation, and approach the values predicted in Section 5.3 (see (5.3.2)).
kF (unj)k2L2(Rn)− kF (u)k2L2(Rn)
≤ 3η. (5.3.11)
Since η can be arbitrary small, we have kunjkL2(Ω) → kukL2(Ω), and the statement of the theorem follows.
5.4 Numerical experiments
In this section, three numerical examples are presented in order to explore the behavior of the solution under fractional parameters s and α.
For the first example, we have used Ω = [−1, 1], a uniform mesh consisting of 3000 nodes, s = 0.005, α = 1, ε2 = 0.5, and the function v = −0.5I(−1,0) + 0.5I[0,1) as the initial data. Here, the aim is to obtain some experimental support for the ideas displayed in Section 5.3, that is, the behavior with a small parameter s. Numerical results are summarized in figure 5.1, and equilibrium values far from 1 and −1 can be observed.
Furthermore, equilibrium values seems to be placed near the values predicted in Section 5.3 or, in other words, the solution seems to approximate a minimizer of (5.3.2).
s = 1, t = 2.5 s = 1, t = 5 s = 1, t = 10
s = 0.85, t = 2.5 s = 0.85, t = 5 s = 0.85, t = 10
s = 0.7, t = 2.5 s = 0.7, t = 5 s = 0.7, t = 10
Figure 5.2: In this example we set Ω = B(0, 1), α = 1, and random noise as initial condition. The evolution in time is displayed for several values of s.
Example 2 and 3 (spinodal decomposition) are showed in figure 5.2 and 5.3 respec-tively. Here we have used Ω as the unitary ball, a uniform triangulation consisting of 16554 triangles, ε2 = 0.02, and random noise as initial data. In example 2 (figure 5.2), the parameter α is fixed in 1, and results for several values of s are shown. Can be observed the fact that, as we have mentioned in Section 5.3, the smaller the parameter s, the thinner the interface. Finally, example 3 (figure 5.3) shows the behavior for fractional values of the parameter α, with s = 1.
Resumen del Cap´ıtulo
En este cap´ıtulo se aplican las t´ecnicas desarrolladas en el cap´ıtulo anterior a una versi´on fraccionaria de la ecuaci´on de Allen-Cahn,
C∂tαu + (−∆)su = f (u) in Ω × (0, T ), con s, α ∈ (0, 1], f (u) = u − u3.
En la Secci´on 5.1 se describe el m´etodo num´erico propuesto (basado en el usado
α = 1, t = 1.25 α = 1, t = 3.75 α = 1, t = 10
α = 0.7, t = 1.25 α = 0.7, t = 3.25 α = 0.7, t = 10
α = 0.4, t = 1.25 α = 0.4, t = 3.25 α = 0.4, t = 10
Figure 5.3: In this example we set Ω = B(0, 1), s = 1, and random noise as initial condition. The evolution in time is displayed for several values of α.
para el caso lineal), mientras que en la Secci´on 5.2 se obtienen estimaciones del error.
En la Secci´on 5.3 se analiza el comportamiento asint´otico de las soluciones con α = 1 y s → 0+. Finalmente, en la Secci´on 5.4 se presentan experimentos num´ericos explorando el comportamiento cualitativo de las soluciones.