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The Approximate Solutions of the stochastic Generalized Swift-Hohenberg Equation with Neumann Boundary Conditions

E. M Elabbasy, Wael W. Mohammed, Mahmoud A. Nagy*

Department of Mathematics, Faculty of Science, Mansoura University, Egypt

Abstract: We consider the stochastic Generalized Swift-Hohenberg (GSSH) equation with respect to Neumann boundary conditions on the interval [0, π] in this form Our aim of this paper is to approximate the solutions of (GSSH) via the amplitude equation with quintic term.

Keywords: Generalized Swift-Hohenberg equation, Neumann boundary conditions, amplitude equations, time-scales

Received January 20, 2015, Revised February 05, 2015, Accepted February 10, 2015

Citation: E. M Elabbasy, Wael W. Mohammed, Mahmoud A. Nagy, The Approximate Solutions of the stochastic Generalized Swift-Hohenberg Equation with Neumann Boundary Conditions. International Journal of Partial Differential Equations and Applications, 3 (1). 12-19. January 2015. doi:10.12691/ijpdea-3-1-3

1. Introduction

The stochastic Swift-Hohenberg equation (SH, for short) is one of important equations for description localized structures in the modern physics, which was. rst used as a toy model for the convective instability in Rayleigh-Bénard problem [1] or [2]. Blömker et al [3,5] studied the stochastic Swift-Hohenberg equation in this form

[picture] | (1)

near its change of stability. While in [4] they studied the stochastic Swift-Hohenberg Equation (1) on large domains near a change of stability. They approximated the solutions by amplitude equation with cubic and noise terms.

In this paper we consider the generalized stochastic Swift-Hohenberg equation (GSSH, for short) in this form

[picture] | (2)

where b is constant, is a small deterministic (linear) perturbation and W(t) be a finite Wiener process. Mohammed et al [6] studied (GSSH) wtih and , they obtained the amplitude equation with cubic term.

For simplicity of the presentation here we study the Equation (2) with respect to Neumann boundary conditions on [0, π] in the case of

to obtain the amplitude equation with quintic term as follows

where c₀ constant. Also, we show that near a change of stability on a time-scale of order the solution u(t) of (2) is well approximated by

[picture] | (3)

The organization of this paper as follows. In Section 2 we state the assumptions and definition that we need, while in Section 3 we give the formal derivation of the amplitude equation of (GSSH). In Section 4 we give bounds for high modes.

In Section 5 we state and proof the main results. Finally, In Section 6 we give application to (GSSH) in one dimension with respect to Neumann boundary conditions.

2. Preliminaries

Before we state our assumptions and Definitions that we need, let us rewrite the equation (2) in abstract form as follows

[picture] | (4)

where

is a linear operator with finite dimensional kernel {cos(qcx)}, is a small deterministic perturbation, B(u) is a quadratic nonlinearity and F (u) is a cubic nonlinearity. Let be an eigenfunctions of A in H with the corresponding eigenvalues , which satisfies .

Define N := kerA, S = the orthogonal complement of N in H, and Pc for the projection Pc : H→N and Define Ps := I-Pc where I is the identity operator on H. As the dimension of N is finite, it is well known that both Pc and Ps are bounded linear operators on H (cf. Weidmann[7]).

Definition 1 For , we define the space as

where be an orthonormal basis of H and are real numbers.

For the quadratic nonlinearity B we assume that:

Assumption 2 Assume that be a bounded bilinear, symmetric (i.e. B(u, v) = B(v, u)) and satisfies PcB(ek, ek) ≠ 0 for k∈(0,1). We use B(u) = B(u, u), Bs = PsB and Bc = PcB for short.

For the cubic nonlinearity F we assume that:

Assumption 3 Assume that is trilinear, symmetric and satisfies the following conditions, for some C > 0,

[picture] | (5)

[picture] | (6)

and

[picture] | (7)

We use F(u) = F(u, u, u), Fs = PsF and Fc = PcF for short.

Assumption 4 Assume that

and

where

For the noise we suppose:

Assumption 5 Let W be a finite Wiener process on H. Suppose for t ≥ 0,

where are independent, standard Brownian motions in and are real numbers and PcW = 0.

For our result we rely on a cut off argument. We consider only solutions (a, ψ) that are not too large, as given by the next definition.

Definition 6 For the N X S-valued stochastic process (a, ψ) defined later in (10) we define, for some T₀ > 0 and , the stopping time as

[picture] | (8)

Definition 7 For a real-valued family of processes we say , if for every p1 there exists a constant Cp such that

[picture] | (9)

We use also the analogous notation for time-independent random variables.

3. The amplitude Equation

In this section we present a short formal derivation of the amplitude equation.

We interest here the studying behavior of solution of (4) on time-scales of order . So, we split the solution u(t) into

[picture] | (10)

where a ∈ N and ψ ∈ S. After rescaling to the slow time-scale , we obtain the following system of equations:

[picture] | (11)

and

[picture] | (12)

where is a rescaled version of the Brownian motion. Equation (11) reads in integrated form as

[picture] | (13)

First step, we want to remove the terms depending on . Therefore, we apply Itô formula to to obtain

[picture] | (14)

Second step, we want to remove the term depending on . Therefore, we apply Itô formula to and to obtain

[picture] | (15)

and

[picture] | (16)

We use Assumption 4 to obtain

Third step, we apply Itô formula to to get

[picture] | (17)

Substituting from Equations (14), (15), (17) and (16) into Equation (13) to obtain

[picture] | (18)

where

and

[picture] | (19)

4. Bounds for the high modes

Define Ornstein-Uhlenbeck process Z (OU, for short) as follows

[picture] | (20)

where

[picture] | (21)

Equation (12) reads in integrated form as

[picture] | (22)

where Z(T) is Defined in (20) and is the semi group created by the operator (cf. [8] ).

In next lemma, we will approximate by the fast Ornstein-Uhlenbeck process Z as follows.

Lemma 8 There is a constant C > 0 such that, for > 0 from the definition of and p ≥ 1,

[picture] | (23)

where

[picture] | (24)

and Z(T) is defined in (20).

Proof. From Equations (22) and (24) we obtain

[picture] | (25)

Taking the norm of both sides and using triangle inequality to obtain

Where now bound these terms separately. For the. rst term, we obtain

where we used the definition of . For the second term and the third term, we obtain

and

Analogously, we derive for the fourth term

where we used again the definition of . Combining all results, yields (23).

Lemma 9 There is a constant C > 0 such that, for > 0 from the definition of and p≥ 1,

[picture] | (26)

Proof. We obtain for T ≤.

Lemma 10 Under Assumation 5, for every > 0 and p ≥ 1, there is a constant C, depending on p, k, k, and T₀, such that

where is defined in (21).

Proof. See Lemma 4.2 in [6].

Lemma 11 Let (0) = O(1). Then for p ≥ 0, > 0 and from the definition of , there is a constant C > 0 such that

[picture] | (27)

Proof. From Equations (25), by triangle inequality, Lemma 8, Lemma 10 and Lemma 9 we obtain

5. The main Result

Before we state and proof the main Theorem for the approximation of the solution of Equation (18). We define (T) in N as solution of

[picture] | (28)

Lemma 12 Let (T) be a the solution of Equation (28). Assume that the initial condition satisfies for some p > 1, then for all T₀ > 0 there exists another constant C such that

[picture] | (29)

Proof. Taking the scalar product on both sides of Equation (28) yields

Using Cauchy-Schwarz inequality, we obtain

Using Gronwalls lemma, we obtain for 0≤ T that

Taking supremum on both sides on [0, T₀] yields (29).

Lemma 13 Let R(T) is defined Equation (19), then for all T₀ > 0 there exists another constant C such that

[picture] | (30)

Proof. We follow the same steps of Lemma 8.

In the following we are not able to calculate moments of error terms. Thus we restrict ourselves to a sufficiently large subset of , where our estimate go through.

Definition 14 Define the set such that all these estimates

[picture] | (31)

[picture] | (32)

and

[picture] | (33)

hold on .

Proposition 15 The set has approximately probability 1

Proof.

Using Chebychev inequality and Lemmas 8 and 12, we obtain for sufficiently large q

[picture] | (34)

Theorem 16 Assume that Assumption (5) hold and suppose a(0) = O(1) and (0) = O(1). Let (T) be a solution of (28) and a(T) as defined in (18). If the initial condition satisfies a(0) = (0)and for all , then

[picture] | (35)

Proof. Define (T) as

From (18) we get

[picture] | (36)

Define now

[picture] | (37)

Subtracting Equation (36), Equations (28) and using (37) to obtain

[picture] | (38)

From Equation (38) we obtain

[picture] | (39)

Taking the scalar product on both sides (39), yields

Using Young and Cauchy-Schwartz inequalities, we obtain the following linear ordinary differential inequality

Using Gronwalls lemma we obtain (as (0) = 0) for T ≤≤ T₀

Taking the supremum on both sides and using Equations (32) and (33). The expectation yields

Using Equations (29) and (36). We. nish the proof by using Equations (37), (39) and

Now, we state and prove the main result of this paper as follow:

Theorem 17 (Approximation) Under Assumptions 5 let h be a solution of (4) defined in (10) with the initial condition with and where a(0) and (0) are of order one, and (T) is a solution of Equation (28) with (0) = a(0). Then for all p > 1 and T₀ > 0 and all , there exists C > 0 such that

[picture] | (40)

where Q(T)Defined in (24) with Zk(T) Defined in (21).

Proof. For the stopping time, we note that

Now let us turn to the approximation result. Using (10) and triangle inequality, yields

From Equations (31) and (35), we obtain

Thus

Using Equation (34), yields (40).

6. Applications

In this section, consider the stochastic generalized Swift-Hohenberg equation (2) with respect to Neumann boundary conditions on [0, π]. Let

[picture] | (41)

then N:=span{cos(qcx)} and Moreover, all conditions of Assumptions 3 and 2 are satisfied for the operators B (u, u) = u² and F (u, u, u) =u³ with α = 1 and β = 0, it is easy to check that

and

First Case: If we choose qc = 2, then the stochastic generalized Swift-Hohenberg equation (2) takes this form

[picture] | (42)

For this model we note that

If we see that Assumation 4 is hold where

and

For the noise we study two cases:

(1) If , then the amplitude equation takes the form

[picture] | (43)

where comes from the noise term.

(2) If , then the amplitude equation takes the form

[picture] | (44)

In the two Cases near a change of stability on a time-scale of order the solution u(t) of (42) is well approximated by

where is a solution of (43) (or (44)).

Second Case: If we choose qc = 3, then the stochastic generalized Swift-Hohenberg equation (2) takes this form

[picture] | (45)

and we note that

Assumation 4 is hold if where

and

In this case if we choose then the amplitude equation takes the form

[picture] | (46)

and the solution u(t) of (45) is well approximated by

where is a solution of (46).

Finally, we note that from First and Second cases if

then there is influence from the noise only in the case of L is even and k = L/2.

References

[1]   | M. C. Cross and P. C. Hohenberg. Pattern formation outside of equilibrium, Rev. Mod. Phys. 65: 581.1112, (1993).
[2]   | P. C. Hohenberg and J.B. Swift. Effects of additive noise at the onset of Rayleigh-Bénard convection, Physical Review A, 46:4773.4785, (1992).
[3]   | D Blömker, M Hairer, and G A. Pavliotis. Stochastic Swift-Hohenberg Equation Near A Change of Stability, Proceedings of Equadiff-11. pp. 27.37, (2005).
[4]   | D Blömker, M Hairer, and G A. Pavliotis. Proc. Appl. Math. Mech. 5, 611. 612, (2005).
[5]   | D Blömker and W W. Mohammed. Amplitude equations for SPDEs with cubic nonlinearities, Stochastics: An International Journal of Probability and Stochastic Processes. 1.35, (2011).
[6]   | W W. Mohammed, D Blömker and K Klepel. Multi-scale analysis of SPDEs with degenerate additive noise, J. Evol. Equ. 273-298, (2013).
[7]   | J. Weidmann. Linear operators in Hilbert spaces, volume 68 of Graduate Texts in Mathematics. Springer-Verlag, New York, (1980). Translated from the German by Joseph Szücs.
[8]   | A.Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44. Springer, New York (1983).