Global attractor for the generalized hyperelastic-rod equation
© Bi et al.; licensee Springer 2014
Received: 24 June 2014
Accepted: 26 August 2014
Published: 25 September 2014
In this paper, we investigate the dynamical behavior of the initial boundary value problem for a class of generalized hyperelastic-rod equations. Under certain conditions, the existence of a global solution in is proved by using some prior estimates and the Galerkin method. Moreover, the existence of an absorbing set and a global attractor in is obtained.
The C-H equation (1.1) was obtained by using an asymptotic expansion directly in the Hamiltonian for the Euler equations in the shallow water regime and possessed a bi-Hamiltonian structure and an infinite number of conservation laws in involution. Research on the C-H equation becomes a hot field due to its good properties – since it was proposed in 1993. Some equations also have similar characters to the C-H equation, which are called C-H family equations. Because of the wide applications in applied sciences such as physics, the C-H family equations have attracted much attention in recent years.
where represents the radial stretch relative to a pre-stressed state. The three coefficients , , and are constants determined by the pre-stress and the material parameters, , , .
The constant γ is called the pre-stressed coefficient of the material rod.
There have been many research results as regards the hyperelastic-rod equation (1.3) –, such as traveling-wave solutions, blow-up of solutions, well-posedness of solutions, the existence of weak solutions, the global solutions of Cauchy problem, the periodic boundary value problem, etc.
The existence of a global weak solution to (1.4) for any initial function belonging to was obtained. They showed stability of the solution when a regularizing term vanishes based on a vanishing viscosity argument and presented a ‘weak equals strong’ uniqueness result.
They considered the Cauchy problem of (1.7) and proved the existence of global and conservative solutions. It was shown that the equation was well-posed for initial data in if one included a Radon measure corresponding to the energy of the system with the initial data.
where . We will study the dynamics behavior of (1.8) and discuss the existence of the global solution and the global attractor under the periodic boundary condition when satisfies the particular conditions.
The rest of this paper is organized as follows: Section 2 describes the main definitions used in this paper. The existence of the global solution is discussed in Section 3. The existence of the absorbing set is detailed in Section 4. Section 5 shows the existence of the global attractor.
In this work, stands for the inner product in the usual sense and represents the norm determined by the inner product, . Apparently, this norm is equal to the natural norm in . The following signs are adopted in this paper to express the norms of different spaces: , , .
The notion of bilinear operator is introduced, , where ∇ is called a first order differential operator. Then we can get .
furthermore, , , so we get and .
Suppose is a second order differential operator, , then A is a self-adjoint operator, which possesses the eigenvalues like , where and . represents the smallest eigenvalue of A.
In this work, we assume that , , and , , C is a constant.
3 The existence of global solution
where . Considering the expressions of , , , according to the qualitative theories of ordinary differential equations, (3.1)-(3.2) have a unique solution in . In order to prove the existence of a global solution, we need to do some prior estimates as regards .
where r, , and are nonnegative constants.
Overall, , , , , that is, , .
where h is a constant which depends on C, , , , .
According to the Aubin compactness theorem, we conclude that there is a convergent subsequence , so that , or equivalently . Suppose that and are replaced by and , then we need to prove that u, v satisfy (2.1).
4 The existence of the absorbing set
It is easy to see that and are uniformly bounded from (4.2). In other words, the semi-group is uniformly bounded in and .
If is an open ball in and whose radius is ρ, it is easy to calculate that when , .
where r, , are nonnegative constants. Let , and then . In other words, is the attracting set of in . This completes the proof of Theorem 2. □
5 The existence of global attractor
Based on the proof of Theorem 2, we only need to prove that is a completely continuous operator, thus the existence of global attractor can be proved.
Let , then we can obtain .
Therefore, we can conclude that is equicontinuous. From the Ascoli-Arzela theorem, is a completely continuous operator. Thus, we have proved that has a global attractor in . □
This work was supported by the National Natural Science Foundation of China (61374194), the National Natural Science Foundation of China (61403081), the Natural Science Foundation of Jiangsu Province (BK20140638), the China Postdoctoral Science Foundation (2013M540405) and the Special Program of China Postdoctoral Science Foundation (2014T70454).
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