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.
Keywords:Benjamin-Bona-Mahony equation; singularly oscillating forces; uniform attractors; translational bounded functions
Let be a fixed parameter, be a bounded domain with sufficiently smooth boundary ∂Ω. We investigate the long-time behavior for the non-autonomous 3D Benjamin-Bona-Mahony (BBM) equation with singularly oscillating forces:
Here, , , and is the velocity vector field, is the kinematic viscosity, is a nonlinear vector function, is the singularly oscillating force.
Along with (1.1)-(1.3), we consider the averaged Benjamin-Bona-Mahony equation
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.
The functions and are taken from the space of translational bounded functions in , namely,
for some constants .
as a straightforward consequence of (1.7), we have
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.
Lemma 2.1For each , every nonnegative locally summable functionϕon and every , we have
holds for all .
Proof See, e.g., . □
Lemma 2.2Let fulfill that for almost every , the differential inequality
where, for every , the scalar functions and satisfy
for some , and . Then
Proof See, e.g., . □
For the non-autonomous general Benjamin-Bona-Mahony (BBM) equation,
Assume that , the nonlinear vector function , , we denote
In addition, ( ) is a smooth function satisfying
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.
Theorem 2.3Assume that (2.8)-(2.11) hold, , (orV) , then there exists a unique global weak solution of the problem (2.5)-(2.7) which satisfies
for all and .
Theorem 2.4Assume 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 3.1The 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 . □
We can briefly describe the structure of the uniform attractor as follows: if the functions and are translational bounded, problem (1.1)-(1.3) generates the dynamical processes acting on V which is defined by , , where is the solution to (1.1)-(1.3). The processes have a uniformly (w.r.t. ) absorbing set
which is bounded in V for any fixed .
On the other hand, is also bounded in V for each fixed ε since . Assuming , the external force appearing in equation (1.1) belongs to also. Moreover, if and , then
for some and . In this case, to describe the structure of the uniform attractor , we consider the family of equations
For every external force , equation (3.3) generates a class of processes on V, which shares similar properties to those of the processes , corresponding to the original equation (1.1) with the external force . Moreover, the map
Lemma 3.2If 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 .
Considering the linear equation
we get the following lemma.
Lemma 4.1Assume that , then problem (4.1) has a unique solution
Moreover, the following inequalities
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.
Then multiplying (4.1) by Y and AY respectively, we get
By the Gronwall inequality and Poincaré inequality, we can easily prove the lemma. □
Setting , , , we have the following lemma.
Lemma 4.2Assume that the formula
holds for some constant , let . Then the solution yields the following problem:
with satisfying the inequality
where is constant independent ofK.
Moreover, we also have
Proof Noting that
we can derive the following estimates from (4.8):
From Lemma 2.1, we have
Hence, from the Poincaré inequality, combining (4.12) and (4.4)-(4.5), we conclude that
we deduce that for any ,
Integrating (4.9) with respect to time variable from τ to t, we see that is a solution to the problem
such that from (4.13) and (4.14), we can derive
By virtue of , , , we have
Hence, we conclude
The proof is finished. □
Now, we shall use the auxiliary linear equation and some estimates to prove the uniform boundedness of in V. For convenience, we set
and assume that
holds for some constants .
Theorem 4.3The attractors of problem (1.1)-(1.3) (or (1.4)-(1.6)) are uniformly (w.r.t. ε) bounded inV, namely,
Proof Let be the solution to (1.1)-(1.3) with the initial data . For , we consider the auxiliary linear equation
From Lemma 4.2, we have the estimate
Setting the function as
which satisfies the problem
Taking the scalar product of (4.28) with w, we obtain
Using the inequality
where λ is the first eigenvalue of −Δ.
Moreover, notice that
and inserting (4.29)-(4.30) into (4.28), we have
which implies that
Therefore using (1.8), we derive from (4.33)-(4.36) that for any ,
Applying Lemma 2.2 with , , , , we have
Recalling that , and using (4.25) and (4.37), we end up with
Thus, for every , the processes have an absorbing set
On the other hand, if , the processes also possess an absorbing set
In conclusion, for every , the set
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.
Theorem 5.1Assume that and (4.23) holds. Then the uniform attractor (for problem (1.1)-(1.3)) converges to (for problem (1.4)-(1.6)) as in the following sense:
Next, we shall study the difference of two solutions for (1.1) with and (1.4) with which share the same initial data. Denote
with belonging to the absorbing set which can be found in Section 4. In particular, since , the formula corresponding to
holds for some , as the size of depends on ρ.
Lemma 5.2For every , , and , the difference
satisfies the estimate
for some positive constants and , both independent of .
Proof Since the difference solves the equation
fulfills the Cauchy problem
where is the solution to (4.25).
Taking an inner product of equation (5.8) with q in H, we obtain
where λ is the first eigenvalue of −Δ, is the upper bound for (by Lemma 3.1) and
thus, it follows from (5.9) and (5.10) that
Noting that , by the Gronwall inequality, we get
Moreover, we can derive the following formulas:
holds for some positive constants . Finally, since , using (4.26) to control , we may obtain
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).
For any , we observe that is a solution to (3.3) with the external force and . For , we investigate the property of the difference
Lemma 5.3The inequality
holds, hereDandRare 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. □
Proof of Theorem 5.1 For , , we obtain that there exists a complete bounded trajectory of equation (3.3), with some external force
such that .
We choose such that
From the equality
applying Lemma 5.3 with , , we obtain
On the other hand, the set attracts all sets uniformly when . Then, for all , there exists some time which is independent of L such that
Choosing and collecting (5.15)-(5.16), we readily get
Since and is arbitrary, taking the limit , we can prove the theorem. □
The authors declare that they have no competing interests.
The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
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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