Averaging of the 3D non-autonomous Benjamin-Bona-Mahony equation with singularly oscillating forces
© Zhao et al.; licensee Springer. 2013
Received: 29 January 2013
Accepted: 16 April 2013
Published: 30 April 2013
For , we investigate the convergence of corresponding uniform attractors of the 3D non-autonomous Benjamin-Bona-Mahony equation with singularly oscillating force contrast with the averaged Benjamin-Bona-Mahony equation (corresponding to the limiting case ). Under suitable assumptions on the external force, we shall obtain the uniform boundedness and convergence of the related uniform attractors as .
MSC:35B40, 35Q99, 80A22.
Here, , , and is the velocity vector field, is the kinematic viscosity, is a nonlinear vector function, is the singularly oscillating force.
formally corresponding to the case in (1.1).
represents the external forces of problem (1.1)-(1.3) for and of problem (1.4)-(1.6) for , respectively.
for some constants .
note that is of the order as .
The BBM equation is a well-known model for long waves in shallow water which was introduced by Benjamin, Bona, and Mahony (, 1972) as an improvement of the Korteweg-de Vries equation (KdV equation) for modeling long waves of small amplitude in two dimensions. Contrasting with the KdV equation, the BBM equation is unstable in high wavenumber components. Further, while the KdV equation has an infinite number of integrals of motion, the BBM equation only has three. For more results on the wellposedness and infinite dimensional dynamical systems for BBM equations, we can refer to [2–7].
In this paper, firstly, we shall study the asymptotic behavior of the non-autonomous BBM equation depending on the small parameter ε, which reflects the rate of fast time oscillations in the term with amplitude of order , then we shall consider the boundedness and convergence of corresponding uniform attractors of (1.1)-(1.3) in contrast to (1.4)-(1.6).
Throughout this paper, () is the generic Lebesgue space, is the Sobolev space. We set , H, V, W is the closure of the set E in the topology of , , respectively. ‘⇀’ stands for the weak convergence of sequences.
holds for all .
Proof See, e.g., . □
Proof See, e.g., . □
for all , where and are positive constants.
Similar to , by the Galerkin method and a priori estimate, we easily derive the existence of a global weak solution and a uniform attractor which shall be stated in the following theorems.
for all and .
Theorem 2.4 Assume that the external force and (2.8)-(2.11) hold, then the processes generated by the global solution possess uniform attractors in for the non-autonomous system (2.5)-(2.7).
3 Some lemmas
Lemma 3.1 The functions and are taken from the space of translational bounded functions in , then the processes generated by system (1.1)-(1.3) have a uniformly (w.r.t. ) compact attractor for any fixed .
Proof As a similar argument in Section 2, we choose in Theorem 2.4, since and are translational bounded in , then for any fixed , is translational bounded in and we can easily deduce the existence of uniformly compact attractors . □
which is bounded in V for any fixed .
Lemma 3.2 If the function in (1.4) is taken from the space of translational bounded functions in , then the processes generated by system (1.4)-(1.6) have a uniformly (w.r.t. ) compact attractor .
Proof Use a similar technique as that in Theorem 2.4, we can easily deduce the existence of a uniformly compact attractor if we choose . □
4 Uniform boundedness of
Firstly, we shall consider the auxiliary linear equation with a non-autonomous external force and give some useful lemmas, and then we shall prove the uniform boundedness of .
we get the following lemma.
hold for every and some constant , independent of the initial time .
Proof Firstly, using the Galerkin approximation method, we can deduce the existence of a global solution for (4.1), here we omit the details.
By the Gronwall inequality and Poincaré inequality, we can easily prove the lemma. □
Setting , , , we have the following lemma.
where is constant independent of K.
The proof is finished. □
holds for some constants .
where λ is the first eigenvalue of −Δ.
is an absorbing set for which is independent of ε. Since , (4.24) follows and hence the proof is complete. □
5 Convergence of to
The main result of the paper reads as follows.
holds for some , as the size of depends on ρ.
for some positive constants and , both independent of .
where is the solution to (4.25).
where R is a positive constant. The proof is finished. □
Next, we want to generalize Lemma 5.2 to derive the convergence of corresponding uniform attractors. Let the external force in equation (3.3) as , then satisfies inequality (5.22).
holds, here D and R are defined in Lemma 5.2.
Proof As the similar discussion in the proof of Lemma 5.2, replacing , and by , and , respectively, noting that (5.1) still holds for , and the family (), is -continuous, using (5.18) in place of (4.23), we can finally complete the proof of the lemma. □
such that .
Since and is arbitrary, taking the limit , we can prove the theorem. □
All authors give their thanks to the reviewer’s suggestions, XY was in part supported by the Innovational Scientists and Technicians Troop Construction Projects of Henan Province (No. 114200510011) and the Young Teacher Research Fund of Henan Normal University (qd12104).
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