Steady State Bifurcations for Phase Field Crystal Equations with underlying two Dimensional Kernel

. This paper is concerned with the study of some properties of stationary solutions to Phase Field Crystal Equations bifurcating from a trivial solution. It is assumed that at this trivial solution, the kernel of the underlying linearized operator has dimension two. By means of the multiparameter method, we give a second order approximation of these bifurcating solutions and analyse their stability properties. The main result states that the stability of these solutions can be described by the variation of a certain angle in a two dimensional parameter space. The behaviour of the parameter curve is also investigated.


Introduction
During the last decades, Pattern Formation Equations have attracted much attention from researchers in applied sciences; see for instance [Hoy06, CH93, CGP00, TAR + 13].
In materials sciences, Pattern Formation Equations (as Allen-Cahn or Cahn-Hilliard equations) are obtained by phase field methods. In 2004, K. Elder and M. Grant have extended these methods by introducing the so-called Phase Field Crystal modelling in order to describe liquid/solid phase transitions in pure materials or alloys ( [EG04]). The solid phase, which can be a crystal, is represented by a periodic field whose wavelength accounts for the distance between neighbouring atoms. The liquid state is described by a (spatially) uniform field. We refer the reader to [PE10, EG04, EHP10, SWWV13, PACI + 13] for a more comprehensive exposition of the Phase Field Crystal method.
The simplest Phase Field Crystal model is the following sixth order evolution equation: ∂ t u − ∂ xx ∂ xxxx u + 2∂ xx u + f (u) = 0, t > 0, x ∈ (0, L). (1.1) Here L is the length of the domain and f is the derivative of a double-well potential. This equation can be viewed as a conservative Swift-Hohenberg equation exactly as the Cahn-Hilliard equation is a conservative version of the Allen-Cahn equation. Performing a linear change of variable mapping (0, L) onto (0, 1), Equation (1.1) can be rewritten as ∂ t u − ε∂ xx ε 2 ∂ xxxx u + 2ε∂ xx u + f (u) = 0, t > 0, x ∈ (0, 1), (1.2) with ε = 1/L 2 . This paper focuses on the stationary solutions to (1.2) complemented with initial and boundary conditions (see (2.7)). In order to gain insight properties of stationary solutions, we use a bifurcation approach.
It is well known that bifurcations occur only if the kernel of the underlying linearized operator is non trivial. For the Phase Field Crystal Equation (1.2), the case of a one dimensional kernel has been investigated in [PR11]. In this paper, we focus on two dimensional kernels. We have choosen the multiparameter method (see [Kie04,Chapter I]) in order to obtain a wild set of solutions; that is to say, a one parameter family of branches of solutions. Notice that one or two branches can also be obtained (see [PR11,Subsection 5.3]) but this is indeed a smaller solution set.
Since two dimensional kernels are considered, we need two independent parameters. It turns out that ε which is the inverse of the domain size square, and the mass M of the initial condition (which is conserved by the dynamics) match the independence condition.
The first step is to characterize the parameters values that give rise to two dimensional kernels. The phase diagramm of Figure 1 features a simple geometric criterion for this (see also Proposition 3.1 for a analytic result).
Then we implement the multiparameter method in order to get bifurcation branches and expansions of solutions to (1.2). According to [Kie04], we have to choose a direction ( α β ) in the kernel which will be tangent to a branch of solutions. Let us denote by y → v(y) this branch, where y ∈ R, y ≃ 0. The parameters ε and M are also parametrized by y; this gives a parameter curve y → (ε(y), M (y)) in R 2 . Theorem 4.1 states an existence result for these bifurcation branches and gives second order expansions of ε(·), M (·) and v(·). In an explicit way, for y ≃ 0, the function x → v(y)(x) is solution to ε(y) 2 ∂ xxxx v(y) + 2ε(y)∂ xx v(y) + f (M (y) + v(y)) = Ω f (M (y) + v(y)) dx a.e. in (0, 1).
We are then led to study two curves: the parameter curve y → (ε(y), M (y)) and the function valued curve y → v(y). The former is studied by considering its oriented tangent at y = 0. This tangent will be denoted by T (α). We show how T (α) behaves w.r.t. α: see Propositions 4.5, 4.9 and Figure 3.
In Proposition 4.10, we state a monotonicity result for α → T (α). More precisely, in a well identified region of the parameter space, T (α) turns clockwise when α goes from 0 to 1. In a quiet surprising way, this monotonicity result is related to the stability of the bifurcating solutions what we will introduce now.
The main result of this paper is stated in Theorem 5.3 and concerns the stability of the bifurcating stationary solutions to the Phase Field Crystal Equation. If the wave numbers of the interactive modes (i.e k * and k * * in the sequel) are not consecutive integers then the bifurcating solutions are unstable. This is easily proved. In order to show stability, we use the principle of reduced stability from [Kie04, Section I-18] (see also [Mie95]). It allows us to reduce some infinite dimensional eigenvalue problem to a two dimensional one. As evoked above, it appears that the bifurcating solutions are stable exactly when the tangent T (α) turns clockwise. So we connect the issue of stability in the PDE (2.7) with the variation of a one dimensional object (the angle between T (α) and the horizontal axis).
Finally, we use a truncated bifurcation equation and symmetries to recover a bifurcation diagramm obtained originally in [PR11] by numerical integration: see Figure 4.

Equations and Functional Setting
Let Ω denote the interval (0, 1) ⊂ R and r be a real number. We define The spaceV 4 is equipped with the bilinear form which becomes in turn a Hilbert space since every v ∈V 4 satisfies by Poincaré-Wirtinger and Poincaré's inequalities. Here σ 1 := π 2 denotes the first eigenvalue of the one-dimensional Laplace operator with homogeneous Dirichlet boundary conditions on Ω. Moreover v ′′ belongs toV 2 thus the same estimates give v ′′ 2 ≤ 1 σ 1 v (4) 2 . Then (2.5) follows. In the same way, if (u, v)V 2 := Ω u (2) v (2) dx then (V 2 , (·, ·)V 2 ) is a Hilbert space. Of course, u (2) stands for the second derivative of u.
Given initial data u 0 = u 0 (x) and a positive parameter ε, the Phase Field Crystal Equation with homogeneous Neumann boundary condition reads (2.7) Since every solution u = u(t, x) to (2.7) satisfies the stationary solutions to the problem above solve The bifurcation problem. We will formulate a bifurcation problem in order to get non trivial solutions of (2.9). To this end, we will introduce some notation. Let ε > 0, ε * > 0 and M , M * be real parameters. We put Let also (2.11) In the sequel, δ * = (ε * , M * ) is the bifurcation point and is fixed; the parameter δ will be closed to δ * . Then we define With these notation, we will consider the following bifurcation problem Remark that the equations in (2.9) and (2.13) are equivalent. We Taylor expend F (µ, v) w.r.t. µ and v at (µ, v) = (0, 0). For this, we write is a continuous bilinear symmetric operator and F 2 (µ)v 2 stands for F 2 (µ)(v, v). We proceed in the same way for F 2 (µ)v 2 , so that The last term is Solutions to (2.9) are critical point of E(M +·, ε) inV 4 where the energy E is defined through (2.14)

The Linearised Equation
For δ = (ε, M ) ∈ (0, ∞) × R, we study the eigenvalue problem (see the previous section and in particular (2.10), for notation) The eigenvalues of (3.1) are with corresponding eigenfunctions Then 0 is an eigenvalue of (3.1) iff there exists a positive integer k such that That is to say, the point (ε, 3M 2 ) is on the parabola given by the function Thus the operator L(·, δ) will have a two dimensional kernel iff the point (ε, 3M 2 ) lies at the intersection of two such parabolas: see Figure 1. If we express this geometric property in an analytical language, we obtain the following result whose proof is straightforward and will be omitted.
Stability of the trivial solution. The trivial solution v = 0 of (2.9) is said to be linearly stable if (3.1) has only positive eigenvalues. In Figure 1, this corresponds to the case where the point (ε, 3M 2 ) is above all parabolas of the form (3.3). If (3.1) has at least one negative eigenvalue then v = 0 is linearly unstable.
Besides, The trivial solution is called neutrally stable if 0 is an eigenvalue of (3.1) and all the other eigenvalues of (3.1) are positive. In order to have stability of solutions to (2.12) bifurcating from v = 0, it is necessary that the trivial solution is neutrally stable. The next result gives a simple criterion for neutrally stability of v = 0 in the case of a 2D kernel.
Proof. For every k ≥ 1, we have with (3.2), (3.4) If v = 0 is neutrally stable and k = k * , k * * then λ k > 0 = λ k * . Hence The value of (k − k * )(k − k * * ) at k = k * * + 1 is k * * + 1 − k * . This number is non positive since by assumption k * * < k * . Thus with (3.6) we get k * = k * * + 1. Conversely, if k * = k * * + 1 then (3.5) imply that λ k − λ k * > 0 for k = k * , k * * . Thus v = 0 is neutrally stable. Figure 1 points out two values of the parameter δ for which the kernel of L(·, δ) has dimension two. One of these values corresponds to the case where k * = 2 and k * * = 1 and lies at the intersection of the green and red parabolas. By Proposition 3.2, the trivial solution is neutrally stable for this value of δ. The other value corresponds to the case where k * = 3 and k * * = 1. In this situation, v = 0 is not neutrally stable. Thus bifurcating solutions will be unstable.
Let us notice that f * , f * * and C S appear naturally if we consider the truncated bifurcation equation of (4.19).
It turns out that this truncated equation has a Z 2 ⊕ Z 2 symmetry; unlike the bifurcation equation (since the fact that u 0 is a solution to (4.19) does not implies that −u 0 is a solution too). Thus we derive from (4.29) Subtracting (4.35) from (4.33), we obtain Also we obtain In order to express A, B, C, D more simply, we put Then, in view of (4.4), Similarly, Then we can state a bifurcation result whose proof comes from the above analysis.
4.1. Sign ofε(0) andM (0). For every α ∈ (0, 1), Theorem 4.1 gives us the parameter curve The tangent to this curve at y = 0 is given bÿ provided that this vector do not vanish. In the sequel, we will compute the signs ofε(0) and M (0) in order to have informations on the profile of the above curve near y = 0. Sign ofε(0). Under the assumptions and notation of Theorem 4.1, it follows from (4.31) thaẗ ε(0) and Aα 2 + B have the same sign. So for every (x, M, α) ∈ (1, ∞) \ {4} × R × (−1, 1), we will compute the sign of A(x, M )α 2 + B(x, M ). Taking advantage of the monotonicity of A(x, M )α 2 + B(x, M ) w.r.t. α 2 , we will look at the sign of B(x, M ) and (A + B)(x, M ). For this, we introduce the so-called cancellation functions of B and A + B, namely (4.53) These functions satisfy The sign of B(x, M ) is given in the following result.
The statement of Lemma 4.2 and those of the seven forcoming results are easily proved; thus their proof will be omitted. Regarding the sign of A + B, we have the lemma below.
Moreover the cancellation functions are ordered or are simultaneously negative.
Proposition 4.5. Under the assumptions and notation of Theorem 4.1, the sign of is as follows.
Sign ofM (0). We proceed as above. We define the cancellation functions D 0 and (C + D) 0 of D and C + D, namely . (4.55) The signs of D(x, M ) and (C +D)(x, M ) are easily determinated with the use of the cancellation functions. More precisely, the following lemmas hold.
Proposition 4.9. Under the assumptions and notation of Theorem 4.1, the sign of is as follows.
As we will see later on, the monotonicity of α →M (0) ε(0) is related to the stability of bifurcating solutions given by Theorem 4.1.
Finally we can combine the above results on the signs ofε(0),M (0) and the variation of M (0) ε(0) to obtain a better insight of the behaviour of the curve y → δ(y) w.r.t. α. We will only investigate the cases useful in the sequel.

Properties of bifurcating solutions
5.1. Energy of the bifurcating solutions. In this section, we will compare the energy of the bifurcating solutions u = M (y) + v(y) given by Theorem 4.1 and the energy of the trivial solution u = M (y). Let us recall that, for (u, ε) ∈ V 2 × (0, ∞), the energy of u is given by (2.14). Moreover, δ(y) := (ε(y), M (y)) for y ≃ 0.
By combining the above results, we prove the assertion of the theorem.
* < (f * * ) 0 (x) thenM (0) > 0 for every α ∈ (0, 1), according to Proposition 4.9. So the assertion follows in this case also. The other case can be proved easily by using the above methods together with Proposition 4.9. 5.2. Stability of bifurcation solutions. We refer to the appendix hereafter for the background concerning the one-dimensional Phase Field Crystal Equation (2.7). The main result of this paper is the following.
Theorem 5.3. Under the assumptions and notation of Theorem 4.1, let us suppose that Then the stability of the stationary solution v(y) to the Phase Field Crystal Problem (A.1) is as follows.
• The above result was unexpected since it connects the stability of bifurcating solutions with the variation of the angle of T (α) with the horizontal axis.
Proof of Theorem 5.3. According to Proposition A.1, it is enough to consider the constrained Swift-Hohenberg Equation (A.2). As explain in Section 3, if the trivial solution v = 0 is not neutrally stable then v(y) is unstable. By Proposition 3.2, it follows that v(y) is unstable if k * = k * * + 1.
The principle of reduced stability states that the eigenvalue problem has two critical eigenvalues λ 1 (y), λ 2 (y) (i.e. eigenvalues close to zero for y ≃ 0) with the following expansions Hence it remains to compute the sign of the eigenvaluesλ 1 andλ 2 of A.
We now state and prove the lemmas used in the proof of Theorem 5.3.
Lemma 5.4. Let A : ker L → ker L be the linear operator defined by (5.12). Then the matrix M (A) of A in the basis (ϕ * , ϕ * * ) is Remark 5.2. The simple formula (5.15) can be obtained, at least at a formal level, by differentiation starting from (4.30). Since (4.30) has a Z 2 ⊕ Z 2 symmetry, (5.15) is in accordance with the results of [GS85, Chap X]. These results are obtained by using symmetries and universal unfolding theory. Moreover the analog of v(y) is obtained in [GS85] as a secondary bifurcation.
Here, there is no Z 2 ⊕ Z 2 symmetry in (2.12) and v(y) is a primary bifurcating solution in the sense that it bifurcates from the trivial solution.
Proof. We compute the first column of the matrix.
Lemma 5.5. Let M * ∈ R, k * , k * * be positive integers such that k * * < k * and f * , f * * , C S be defined by (4.27), (4.28). Let A, B, C, D be given by (4.36)−(4.39) and A be the operator defined through (5.12) (whose matrix is given by (5.15)). Then det A = − 9 8 α 2 β 2 (BC − AD). (5.16) Remark 5.3. To our knowledge, the relation (5.16) is new. It explains why the statement of the stability of v(y) is quiet simple in the sense that the stability is linked to quantities relying on the parameter curve y → δ(y). In particular, if is increasing then (5.16) implies that v(y) is unstable.
• δ 1 and R 1 may be choosen independently of α and β provided that α and β remain bounded away from 0.
• In this setting, the bifurcating solution v(y) of Theorem 4.1 will be denoted by v(y, α, β).
Let (α, β) ∈ (−1, 1) 2 satisfy (4.48). For y close to zero, we have four (distinct) solutions µ(y, ±α, ±β), v(y, ±α, ±β) to Equation (2.12). In view of the remark above, we may suppose that the numbers δ 1 (y, ±α, ±β) are equal. So we will denote their common value by δ m . The goal of this subsection is to establish relations between these solutions. This is achieved by using a suitable translation of the space variable. Let us write k * and k * * under the form k * = 2 r 1 ℓ * , k * * = 2 r 2 ℓ * * , (5.17) where r 1 , r 2 are non negative integers and ℓ * , ℓ * * are positive odd integers. Let us denote by r the minimum of r 1 and r 2 .
The above mentioned translation consists, roughly speaking, in the translation x → x + 2 −r . To be more specific, for every v ∈ L 2 (Ω), let us denote by Jv : R → R the 2−periodic and even function satisfying Jv = v a.e. in [0, 1].
However, if k * = 4k * * then we can prove that µ(·, α, β) is not even. Let us notice that (2.12) has the trivial symmetry This symmetry allows to relate solutions in some cases. However, if k * , k * * are even then it turns out that each of the four solutions µ(y, ±α, ±β), v(y, ±α, ±β) is invariant under S. So S is useless in what case unlike T (see (5.18)).
(ε k * , M 0 ) Remark 6.1. Let us make some comments on the bifurcation diagramm of Figure 4.
• The top blue curve is the graph of where the dependence of (y, α, β) w.r.t. ε is given by (6.2), (6.3).

Thus, in
• We can see secondary secondary bifurcations between interactive modes solutions and single modes solutions. At the bifurcation point, we have α = 0 or β = 0.
• Let k * = 4, k * * = 3 and r = −0.5. If y is small enough then v(y, α, β) is an asymptotically stable solution to Indeed, (4.65) implies that the graphs of x → J 1 (x) and x → J 2 (x) lie respectively between the red curves and the blue curves of Figure 2. Thus in view of the remark above M 2 * ∈ (J 1 (x), J 2 (x)). The claim follows then from Theorem 5.3. This stability result is in accordance with the numerical simulations featured in [PR11, Figure 15].

Appendix A. Phase Field Crystal Equation and stability
The aim of this appendix is to show that stability for the Phase Field Crystal Equation (2.7) and for the following constrained Swift-Hohenberg Equation, are essentially the same. Since the linearized operator corresponding to (2.7) is not symmetric, we will consider for this equation, asymptotic stability in the sense of Lyapunov ; see for instance [Har91,Chap 3]. We will first define the semigroup associated to (2.7). For simplicity, the derivatives ∂ x and ∂ xx will be denoted by ∇ and ∆. Moreover, notice that all of the results below still hold if the interval Ω is replaced by a smooth bounded domain of R 2 or R 3 . Recalling the notation (2.3), we puṫ V 5 := u ∈ H 5 (Ω) | u ′ = u ′′′ = 0 on ∂Ω Ḣ −1 (Ω) := dual of H 1 (Ω) ∩L 2 (Ω).
In the same way but referring to the above constrained Swift-Hohenberg Equation, the problem v ∈ L 2 (0, T,V 4 ) ∩ C([0, T ],V 2 ), d dt v ∈ L 2 (0, T,L 2 (Ω)) has a unique (strong) solution and infer also a semigroup denoted by S cSH . Without los of generality, we may assume ε = 1 and M = Ω u 0 dx = 0. It is clear that (A.1) and (A.2) have the same steady states. Moreover, the energy E − see (2.14) − defined through is a Lyapunov functional for S P F C and S cSH . It is clear for the later. For the former, we test the equation of (A.1) with (−∆ −1 ) d dt v where −∆ : H 1 (Ω) ∩L 2 (Ω) →Ḣ −1 (Ω). We find d dt Notice that the linearized operator for (A.2) at any stationary solution v ∞ is self adjoint with compact resolvant. Thus its spectrum consists on an increasing sequence of eigenvalues.
The main result of this appendix is the following.
Proposition A.1. Let v ∞ be a stationary solution to (A.1). If v ∞ is linearly stable for the semigroup S cSH (i.e. the corresponding eigenvalues are positive) then v ∞ is asymptotically stable in the sense of Lyapunov, for the semigroup S P F C . If one of these eigenvalues is negative and 0 is not an eigenvalue then v ∞ is not stable in the sense of Lyapunov, for the semigroup S P F C .
Roughly speaking, the above results states that if 0 is not an eigenvalue then the stationary solution v ∞ has the same stability for the semigroup S P F C and for the semigroup S cSH .
Proof. Let us assume that v ∞ is linearly stable for S cSH . Expending the energy E(v) for v ∈V 2 , v ≃ v ∞ , we get ).
With Lemma A.2 below, we deduce that there exist ε > 0 and r 2 > 0 such that Since v ∞ is linearly stable, we may assume without los of generality, that v ∞ is the only one stationary solution in the ball B(v ∞ , r 2 ) ofV 2 with radius r 2 and center v ∞ . Since (A.1) and (A.2) have the same steady states, v ∞ is also the unique stationary solution to (A.1) in B(v ∞ , r 2 ). Besides, by continuity of E(·), there exists r 1 ∈ (0, r 2 ) such that Since the energy is a Lyapunov function for the semigroup S P F C , there holds Hence by (A.3), S P F C (t)v 0 ∈ B(v ∞ , r 2 ), ∀t ≥ 0.
Moreover, by standard methods (see [Tem88]), we can show that the trajectory {S P F C (t)v 0 | t ≥ 0} is relatively compact inV 2 . Since v ∞ is a isolated stationary solution, we deduce from Lassalle's Invariance Principle (see [Har91]) that Thus v ∞ is asymptotically stable equilibrium of S P F C .
Conversely, since 0 is not in the spectrum of the linearized operator for (A.2) at v ∞ , there exists r 3 > 0 such that v ∞ is the only one stationary solution to (A.1) in B(v ∞ , r 3 ).
Since v ∞ is linearly unstable, there exists v 0 ∈V 2 arbitrary close to v ∞ such that By continuity of E, there exists a positive number r 1 depending on v 0 such that E(v) > E(v 0 ), ∀v ∈ B(v ∞ , r 1 ).
Moreover, E(S P F C (t)v 0 ) ≤ E(v 0 ), ∀t ≥ 0, since E is a Lyapunov function for the semigroup S P F C . Thus S P F C (t)v 0 ∈ B(v ∞ , r 1 ), ∀t ≥ 0.
Moreover, the Lassalle Invariance Principle implies that where E denotes the set of all stationary solutions to (A.1). Thus for some positive time t 1 and w ∞ ∈ E, there holds However, since v ∞ is the only steady states of (A.1) in B(v ∞ , r 3 ), we have v ∞ − w ∞ V 2 ≥ r 3 .
which means that v ∞ is not stable in the sense of Liapunov. This completes the proof of the proposition.