^{1}

^{2}

^{*}

^{1}

^{2}

Edited by: Jesus Martin-Vaquero, University of Salamanca, Spain

Reviewed by: Bulent Karasozen, Middle East Technical University, Turkey; Andreas Gustavsson, University of Seoul, South Korea

This article was submitted to Mathematical Physics, a section of the journal Frontiers in Applied Mathematics and Statistics

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In this paper we study the attractor of a parabolic semiflow generated by a singularly perturbed PDE with a non-linear term given by a bistable potential, in an oval surface; the Allen-Cahn equation being a prototypical example. An additional constraint motivated by a variational principle for closed geodesics originally proposed by Poincaré arising from geometric considerations is introduced. The existence of a global attractor is established by modifying standard techniques in order to handle the constraint. Based on previous work on the elliptic case, it is known that when the perturbation parameter tends to zero, minimal energy solutions exhibit a sharp interface concentrated on a closed geodesic. We provide numerical simulations using Galerkin's method. Based on the analytical and numerical results we conjecture that, when the perturbation parameter tends to zero and for large times, the transition layers of the solutions of this PDE consists of closed geodesics or a union of arcs of such geodesics, thus characterizing the structure of the attractor.

The qualitative study of dynamical systems in infinite dimensions has been of fundamental importance. In the case of dynamical systems associated with partial differential equations of evolution having variational structure, many of the ideas and methodologies of gradient-like systems can be extended to infinite dimensions. In particular, the study and characterization of attractors is of special interest.

In this paper, we prove the existence of the global attractor of the parabolic equation associated to:

on an oval surface ^{1}

Example of an oval surface in ℝ^{3}.

Equation (1) arises in many contexts among which we may mention materials science, superconductivity, population dynamics, and pattern formation.

An important case for ^{2})^{2}, which has been widely studied both analytically and numerically for example in Hutchinson and Tonewaga [

In a bounded domain Ω ⊂ ℝ^{n},

This fact is obtained using the variational structure of the problem, because (1) is the Euler Lagrange equation of the functional:

in a suitable functional space.

For ϵ → 0, functions _{ϵ}(_{0}, can be proved to be close to ±1 in most of the domain, except for a transition curve. The proof follows from a classical result in differential geometry due to Birkhoff that guarantees the existence of a closed geodesic on a surface diffeomorphic to the sphere (see Poincaré [

This fact suggests a natural constraint for the problem under consideration. The function

where

On the other hand, solutions of (1) correspond to stationary points of the associated gradient flow:

The main goal of this paper is show the existence of the attractor of the associated parabolic equation to (1) (i.e., Equation 4), and conjecture its structure in terms of functions that possess transition layers determined by closed geodesics or arcs of geodesics. In other words, given any initial condition, the corresponding parabolic semiflow determined by (4) approaches a function with transitions in geodesics. This will be done by considering the special case in which ^{2} and ^{2})^{2}. This will simplify both the analysis and the numerics.

From now on we consider solutions of (4) satisfying the constraint (3). Under the above restrictions, it becomes:

which will be incorporated into the equation later on as a Lagrange multiplier. As a first step, we will proof the existence of an attractor for (4) under the constraint (5). We will recall some standard facts in dynamical systems theory, Sobolev spaces on Riemannian manifolds as well as Gronwall's inequality, which are presented in the following section. This is done for the sake of completeness and to introduce notation and may be skipped by readers familiar with dynamical systems and analysis on manifolds.

Having shown the existence of the attractor, some numerical experiments are performed using the Galerkin method. A few words are in order regarding the limitations of our numerical approach. Even when in principle the method should be applicable for any initial condition, we only considered some that already exhibit a relatively well-defined interface. The aim of the numerical simulations is to make plausible our conjecture on the structure of the global attractor and a more detailed study of the method is not carried out. As for the analytical approach, we remark that the problem of establishing the existence of a global attractor for other surfaces or manifolds in similar situations seems to be a reasonable extension of the methods and ideas here presented. In particular for the case of surfaces with non-zero Euler characteristic as is done in Del Río et al. [

The notation and terminology used in this section is adapted or quoted from Temam [

We will consider dynamical systems whose state is described by an element

The evolution of the dynamical system is described by a family of operators

If ϕ is the state of the dynamical system at time

The semigroup

These operators may or may not be one-to-one; the injectivity property is equivalent to the

_{0} ∈ _{0} is the set ⋃_{t ≥ 0}_{0}.

_{0} is the set

_{0} ∈ _{0} (or

or

where closures are taken in

_{0} ∈

or

_{n} → ∞

_{n} converging to ψ in _{n} → ∞, such that

_{0} ∈

We say that a set _{t ≥ 0} if

It is said to be negatively invariant if {_{t ≥ 0} if

When the set is both positively and negatively invariant, we call it an invariant set or a functional invariant set.

_{t ≥ 0} if

The simplest examples of invariant sets are equilibrium points, heteroclinic orbits and limit cycles.

_{0} > 0

_{0} converges to

as

The distance in (2) is understood to be the distance of a point to a set:

as

The convergence in the above definition is equivalent to the following: for every ϵ > 0, there exists _{ϵ} such that for

_{t ≥ 0} if

It is easy to see that such a set is necessarily unique. Also such a set is maximal for the inclusion relation among the bounded attractors and among the bounded functional invariant sets. For this reason it is also called the maximal attractor.

In order to establish the existence of attractors, a useful concept is the related concept of absorbing sets.

We say also that

The existence of global attractor _{t ≥ 0} implies that of an absorbing set. Indeed, for ϵ > 0, let

Conversely, it is a standard result that a semigroup that possesses an absorbing set and enjoys some other properties possesses an attractor.

In order to prove existence of an attractor when the existence of an absorbing set is known, we need further assumptions on the semigroup {_{t ≥ 0}, and we will make one of the two following:

The operators _{0} which may depend on

is relatively compact in

Alternatively, if

If _{1}(_{2}(_{1}(·) are uniformly compact for _{2}(

For every bounded set

as

Of course, if _{2} = 0.

_{0} ^{+} _{0}

The proof of this theorem is carried out through several steps, which can be found in Temam [

The notation and terminology used in this section can be found in Hebey [

Let (

When

We define the Sobolev space

Note here that one can look at these spaces as subspaces of ^{p}(

_{E}) and (_{F}) two normed vector spaces with the property that _{E}) are relatively compact in (_{F}). This fact is written as

This means that bounded sequences in (_{E}) possess corvergent subsequences in (_{F}). Clearly, if the embedding of _{F} ≤ _{E}.

The following theorem is needed in order to prove the existence of the attractor of the equation in consideration.

The first part of the above theorem has the following consequence:

The following inequality is derived from Gronwall's lemma and will be used later on.

^{1/p}, then ^{1/p}, ∀^{1/p}, then there exists _{0} ∈ (0, ∞) such that ^{1/p} for 0 ≤ _{0}, and ^{1/p} for _{0}.

For _{0}] we write ^{1/p} ≥ 0 and since (^{p} ≥ ^{p} + ^{p} for

Hence

and then by integration

This implies the desired result for _{0}], and since, this inequality holds for _{0}, the lemma is proved.

The main result is the following in which the existence of a global attractor is shown for equation (4) subject to constraint (5).

_{t ≥ 0} ^{2}(^{2})^{2}(^{2})^{2}(^{2}).

for all

where ^{2} into

For fixed ϵ > 0, the existence of this minimum is a consequence of this functional satisfies the Palais–Smale condition (see Struwe [

On other hand it should be noted that:

This last statement ensures the existence of a global solution for

Another way to verify the above statement, is to first prove the existence and uniqueness of a solution of (4)–(5) subject to a suitable initial condition; then the backward uniqueness in order to show existence for all

In the usual way, we shall see the existence of an absorbent set in ^{2}(^{2}) and subsequently, the compactness of the mentioned semigroup, according to theorem 1.

The Euler–Lagrange equation associated to (2) with the constraint (3) (for each ϵ_{i}), contain a Lagrange multiplier λ_{i} as follows:

In Del Río et al. [

In order to prove the existence of an absorbing set in ^{2}(^{2}), we multiply (2) by ^{2}. Using Green's formula we obtain:

where ^{2}(^{2}).

By a standard corollary (see for instance 1) _{0} such that _{1} such that:

An estimate of the third integral in (15) is required, for which the following inequality is used:

and by Hölder's inequality, for a

and for certain

Thanks to (15) and the previous relationship, we conclude that there exists a

Thus:

this meaning that:

According to (16) concluded from (17), there exists a

By using the classical Gronwall lemma, we obtain that:

Therefore:

There exists an absorbing set ^{2}(^{2}), namely, any ball of ^{2}(^{2}) centered at 0 of radius _{0}, as if ^{2}(^{2}), included in a ball ^{2}(^{2}), then

In order to prove the uniform compactness of operators, we proceed using by an argument proposed by B. Nicolaenko (see Temam [^{2}(^{2}) whose existence was established in the previous paragraph.

By Holder inequality:

Analogously to (15), we conclude that:

where

Let ρ_{2} be a real number greater than (γ/δ)^{1/2} and

The above relations show that for any set ^{2}(^{2}), bounded or not, _{2}, if _{0}, thus demostrating the existence of an absorbent set in _{0}, that which is bounded in ^{2}(^{2}) (corollary 1). The existence of the global attractor follows from theorem 1.

Having shown the existence of a global attractor, the question of characterizing its structure arises. This question can be answered provided there is a suitable Lyapunov functional.

_{t ≥ 0} on a set

For each _{0}) is non-increasing.

If _{1}) = _{1}) for some τ > 0, then _{1} is a fixed point of {_{t ≥ 0}, i.e., _{1} = _{1}, ∀

The following standard theorem establishes the structure of the attractor.

_{t ≥ 0}

Remember that, _{*}, which belongs to an orbit {

The details of this proof can be found in Temam [

Once the existence of an attractor is proved, in this section we provide a numerical method for its characterization. In this implementation the Galerkin method is used.

^{2} is parametrized with spherical coordinates by (

Spherical coordinates using longitude θ and latitude ϕ.

Then, the Laplacian in these coordinates is given by:

Using ^{2} is obtained:

Then (4) becomes:

By implementing Galerkin's method, we can approximate the attractor. This is done by projecting Equation (18), with a suitable initial condition on a finite dimensional subspace, thus reducing it to a system of ordinary differential equations. The details are provided in the next section.

The idea is to obtain a finite dimensional reduction of (18). One way to do this is using Galerkin method, which will be described below (for more details see Kythe et al. [

We consider the problem:

Assume that the funtions _{k} = _{k}(θ, ϕ), (^{2}(^{2}). For instance, we could take ^{2}.

Fix now a positive integer _{m} of the form

where we will select the coefficients

and

Here (·, ·) denotes the inner product in ^{2}(^{2}),

and

Thus, we look for a function _{m} of the form (21) that satisfies the

By the standard theorem on existence and uniqueness of systems of ordinary differential equations, we have the following result:

_{m}

Functions _{k}, will be selected via the method of separation of variables, applied to the equation Δ^{2}, i.e., we assume that

The corresponding solutions for Θ are of the form sine and cosine, while those corresponding to Φ are solutions to the Legendre equation, in which the substitution

where

According to the above condition (5) we choose the functions _{m} as:

with

If the initial condition (22) is

we obtain

and

_{1} at different times (

Behavior of _{1}(_{1}(0). _{1}(0). _{1}(0.001). _{1}(0.001). _{1}(1). _{1}(1).

If the initial condition is now,

then

and

_{1} at different times (

Behavior of _{1}(_{1}(0). _{1}(0). _{1}(0.01). _{1}(0.01). _{1}(0.02). _{1}(0.02).

As mentioned in the previous section the legendre equation is involved, we can also choose the Legendre polinomial as follows. If the following functions are now chosen,

where

With the initial condition,

we obtain the following expressions for _{2} for the values _{2} for the values mentioned above.

Behavior of _{2}(_{2}(0). _{2}(0). _{2}(0.0055). _{1}(0.0055). _{2}(0.02). _{2}(0.02).

All the numerical simulations show that the graph of the solution on ^{2} approaches values close to 1 and − 1 when

All datasets generated for this study are included in the article/supplementary material.

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

^{1}A closed and compact surface enclosing a strictly convex set in ℝ^{3}.