Received 5 August 2014; revised 2 September 2014; accepted 8 September 2014

ABSTRACT

In this paper, firstly, some priori estimates are obtained for the existence and uniqueness of solutions of a two dimensional generalized anisotropy Kuramoto-Sivashinsky Equation. Then we prove the existence of the global attractor. Finally, we get the upper bound estimation of the Haus-dorff and fractal dimension of attractor.

Keywords:

Kuramoto-Sivashinsky Equation, Existence, Global Attractor, Dimension Estimation

1. Introduction

In recent years, the infinite dimension dynamic system with high dimension has been studied extensively, and the studies have obtained many achievements [1] -[8] . The related questions of its existence and uniqueness of solutions; the existence and dimension of global attractor; the existence and attraction of inertial manifolds; finite dimension, approximate inertial manifolds and time-lag inertial manifolds are still important contents that are studied.

The celebrated Kuramoto-Sivashinsky Equation

(1.1)

where, is an Equation that for nearly half a century has attracted the attention of many researchers from various areas due to its simple but rich dynamics [9] . It first appeared in the mid-1970s by Kuramoto in the study of angularphase turbulence for a system of reaction-diffusion equations modeling the Belousov Zhabotinskii reaction in three spatial dimensions [10] .

In a physical context, Equation (1.1) is used to model continuous media that exhibits chaotic behavior such as weak turbulence on interfaces among complex flows (quasi-planar flame front and the fluctuation of the positions of a flame front, fluctuations in thin viscous fluid films flowing over inclined planes or vertical walls, dendritic phase change fronts in binary alloy mixtures), small perturbations of a metastable planar front or interface (spatially uniform oscillating chemical reaction in a homogeneous medium) and physical systems driven far from the equilibrium due to intrinsic instabilities (instabilities of dissipative trapped ion modes in plasmas and phase dynamics in reaction-diffusion systems).

As a dynamical system the KSE is known for its chaotic solutions and complicated behavior due to the terms that appear. Namely, the term acts as an energy source and has a destabilizing effect at a large scale, the dissipative term provides dumping in small scales and, finally, the nonlinear term provides stabilization by transferring energy between large and small scales. Because of this fact, Equation (1.1) was studied extensively as a paradigm of finite dynamics in a partial differential equation. Its multi-modal, oscillatory and chaotic solutions have been investigated; its non-integrability was established via its Painlev analysis and due to its bifurcation behavior, a connection to low finite-dimensional dynamical systems is established.

The generalization of KSE to two dimensions comes naturally, the two-dimensional KuramotoCSivashinsky Equation

(1.2)

where now and Equation (1.2) has equally attracted much attention because of the same spatiotemporal chaos properties that exhibits and its applications in modeling complex dynamics in hydrodynamics [11] . Nevertheless, due to the additional spatial dimension Equation (1.2) is very challenging and even its well-posedness is still an open problem.

One generalization of Equation (1.2) which is of much interest is the anisotropic two-dimensional KuramotoCSivashinsky Equation

(1.3)

where the two real parameters control the anisotropy of the linear and the nonlinear term, respectively, in other words, the stability of the solutions of Equation (1.3). The anisotropic two-dimensional KuramotoCSivashinsky Equation, due to the fact that it describes linearly unstable surface dynamics in the presence of in-plane anisotropy, has a wide range of applications, for instance, as a model for the nonlinear evolution of sputter-eroded surfaces and describing the epitaxial growth of a vicinal surface destabilized by step edge barriers; for further details, see the references therein, in particular [12] .

This paper focuses on the following generalization of the anisotropic KSE (1.3)

(1.4)

where and are considered as smooth functions of, and its study under the prism of Lie point symmetries and conservation laws [13] .

According to the above information, the paper mainly thinks about the following generalization of the anisotropic KSE (1.4)

(1.5)

(1.6)

(1.7)

Here is bounded set; is the bound of; and are considered as smooth functions

of. Let.

The following is the rest of this paper. In Section 2, we introduce some basic contents concerning global attractor. In Section 3, we obtain the existence of the global attractor, then we get the upper bound estimation of the Hausdorff and fractal dimension of the global attractor.

2. The Priori Estimate of Solution of Questions (1.5) - (1.7)

Lemma 1. Assume so

the smooth solution of Questions (1.5) - (1.7) satisfies

(2.1)

Proof. We multiply with both sides of Equation (1.5) and obtain

(2.2)

Here

According to Nirenberg-Gagliardo and Cauchy inequality, we obtain

From the (2.2) we obtain

(2.3)

Using the Gronwall inequality, the (2.1) is proved.

Lemma 2. Under the condition of Lemma 1, and so the smooth solution of Questions (1.5) - (1.7) satisfies

(2.4)

Proof. We multiply with both sides of Equation (1.5) and obtain

(2.5)

Here

According to the hypothetical condition and Sobolev interpolation inequalities

so

Using the Young inequality obtain

From the (2.5) we obtain

Here According to the Gronwall inequality,we can get the (2.4).

Lemma 3. Under the condition of Lemma 2, and

so the smooth solution of Questions (1.5) - (1.7) satisfies

(2.6)

Proof. We multiply with both sides of Equation (1.5) and obtain

(2.7)

Here

By Sobolev interpolation inequality

Noticing interpolation inequalities

so

According to the Young inequality,we can obtain

From the (2.7) we obtain

By the Gronwall inequality we can get the (2.6).

Lemma 4. Under the condition of Lemma 3, and

so the smooth solution of Questions (1.5) - (1.7) satisfies

(2.8)

Proof. We multiply with both sides of Equation (1.5) and obtain

(2.9)

Here

By using the Sobolev inequality

So

By the Young inequality, we obtain

From the (2.9), we obtain

So we have

3. Global Attractor and Dimension Estimation

Theorem 1. Assume that and so Questions (1.5) - (1.7) exist a unique smooth so-

lution and

Proof. By the method of Galerkin and Lemma 1-Lemma 3, we can easily obtain the existence of solutions. Next, we prove the uniqueness of solutions in detail.

Amusse are two solutions of Questions (1.5) - (1.7), so the difference of them satisfies

and

The two above formulae subtract and obtain

(3.1)

We multiply with both sides of Equation (3.1) and obtain

(3.2)

Here

Since the assume of Lemma 1, we obtain

From the (3.2) we can obtain

According to the Gronwall inequality,we obtain

So we can get the uniqueness is proved.

Theorem 2. Let be a Banach space, and are the semigroup operators on. here is a unit operator. Set satisfy the follow conditions

1) is bounded. Namely, it exists a constant, so that

;

2) It exists a bounded absorbing set namely it exists a constant so that

;

3) When is a completely continuous operator.

Therefor, the semigroup operators exist a compact global attractor.

Theorem 3. Under the assume of Theorem 1, Questions (1.5) - (1.7) have global attractor

is the bounded absorbing set of and satisfies

1)

2) here and it is a bounded set,

Proof. Under the conditions of Lemma 1 - Lemma 4, it exists the solution semigroup of Questions (1.5)- (1.7),

From Lemma 1 - Lemma 3, to is a bounded set that includes in the ball

This shows is uniformly bounded in

Furthermore, when there is therefore,

is the bounded absorbing set of semigroup

From Lemma 4, there are. Since is tightly em-

bedded, which means that the bounded set in is the tight set in, the semigroup operator to is completely continuous. Furthermore we can know, the attractor A is

w-limited set of the absorptive set

In order to estimate the Hausdorff and fractal dimension of the global attractor A of Questions (1.5) - (1.7), let Questions (1.5) - (1.7) linearize, then we obtain

(3.3)

(3.4)

where

So the solutions of Questions (1.5) - (1.7) are fully smooth. It is easy to prove the initial value, appropriate, smooth and linear Questions (3.3) - (3.4) have global and smooth solutions. Let

and are constants, so it exists a constant and so there is

This suggests that is Frechet differential in

Let be the solutions of the linear Equation (3.3) corresponding to the initial value so there is

(3.5)

is linear mapping that is defined in the (3.4); represents the outer product; represents the trace; is the orthogonal projection from to the span So from (3.5), we can turn dimensions volume element into

is secondly exponential, namely

So

Here

Theorem 4. Under the assume of Theorem 3, the global attractor of Questions (1.5) - (1.7) has finite Hausdorff and fractal dimensin, and

Here is a minimal positive integer of the following inequality

Proof. By theorem [8] , we need to estimate the lower bound of Let be the orthogonal basis of subspace of

Here

Under the bounded condition, is the standard eigenfunction of and the corresponding eigenvalues are and

Therefore, we can get

By and

we have

Therefore

Funding

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11161057.

References