Averaging of the 3D non-autonomous Benjamin-Bona-Mahony equation with singularly oscillating forces

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R E S E A R C H

Open Access

Averaging of the

D

non-autonomous

Benjamin-Bona-Mahony equation with

singularly oscillating forces

Mingxia Zhao

1

_{, Xinguang Yang}

2*

_{and Lingrui Zhang}

3

*_{Correspondence:}

[email protected]

2_{College of Mathematics and}

Information Science, Henan Normal University, Xinxiang, 453007, P.R. China

Full list of author information is available at the end of the article

Abstract

For

ε

∈(0, 1), we investigate the convergence of corresponding uniform attractors of the 3Dnon-autonomous Benjamin-Bona-Mahony equation with singularly oscillating force contrast with the averaged Benjamin-Bona-Mahony equation (corresponding to the limiting case

ε

= 0). Under suitable assumptions on the external force, we shall obtain the uniform boundedness and convergence of the related uniform attractors as

ε

→0+_.

MSC: 35B40; 35Q99; 80A22

Keywords: Benjamin-Bona-Mahony equation; singularly oscillating forces; uniform attractors; translational bounded functions

1 Introduction

Let ρ ∈[, ) be a ﬁxed parameter, ⊂ R be a bounded domain with suﬃciently smooth boundary∂. We investigate the long-time behavior for the non-autonomous DBenjamin-Bona-Mahony (BBM) equation with singularly oscillating forces:

ut–ut–νu+∇ ·

–

→

F(u) =f(t,x) +ε–ρf(t/ε,x), x∈, (.)

u(t,x)|∂= , (.)

u(τ,x) =uτ(x), τ∈R. (.)

Here,t∈Rτ,Rτ = (τ,∞), andu=u(t,x) = (u(t,x),u(t,x),u(t,x)) is the velocity vector

ﬁeld,ν>  is the kinematic viscosity,→–F is a nonlinear vector function,f(t,x) +ε–ρf(t/ε,x)

is the singularly oscillating force.

Along with (.)-(.), we consider the averaged Benjamin-Bona-Mahony equation

–

→

F(u) =f(t,x), x∈, (.)

u(t,x)|∂= , (.)

u(τ,x) =uτ(x), τ∈R (.)

formally corresponding to the caseε=  in (.).

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The function

fε(x,t) =

f(x,t) +ε–ρf(x,t/ε),  <ε< ,

f(x,t), ε= 

(.)

represents the external forces of problem (.)-(.) forε>  and of problem (.)-(.) for ε= , respectively.

The functions f(x,s) and f(x,s) are taken from the space L_b(R,H) of translational

bounded functions inL

loc(R,H), namely,

f_L b(R,H)

:=sup

t∈R

t+

t

f(s) 

Hds=M



, (.)

f_L b(R,H):=

sup

t∈R

t+

t

f(s) 

Hds=M



, (.)

for some constantsM,M≥.

Deﬁning

Qε=

M+ Mε–ρ,  <ε< ,

M, ε= ,

as a straightforward consequence of (.), we have

fε_L

b(R,H)≤Q ε

, (.)

note thatQε_{is of the order}_ε–ρ_as_ε_→_+_.

The BBM equation is a well-known model for long waves in shallow water which was in-troduced by Benjamin, Bona, and Mahony ([], ) as an improvement of the Korteweg-de Vries equation (KdV equation) for moKorteweg-deling long waves of small amplituKorteweg-de 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 in-tegrals 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 [–].

In this paper, ﬁrstly, we shall study the asymptotic behavior of the non-autonomous BBM equation depending on the small parameterε, which reﬂects the rate of fast time oscil-lations in the termε–ρ_f

(x,t/ε) with amplitude of orderε–ρ, then we shall consider the

boundedness and convergence of corresponding uniform attractors of (.)-(.) in con-trast to (.)-(.).

2 Preliminaries

Throughout this paper,Lp₍_{) (}_≤_p_≤₊_∞_{) is the generic Lebesgue space,}_Hs₍_{) is the}

Sobolev space. We setE:={u|u∈(C_∞())_}_,_H,_V_,_W _{is the closure of the set}_E_{in the}

topology of (L₍₎₎_{, (H}₍₎₎_{, (H}₍₎₎_{respectively. ‘}_{’ stands for the weak convergence}

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Lemma . For eachτ ∈R,every nonnegative locally summable functionφ on Rτ and

everyβ> ,we have

t

τ

φ(s)e–β(t–s)_ds_≤ 

 –e–β sup θ≥τ

θ+

θ

φ(s)ds, (.)

holds for all t≥τ.

Proof See,e.g., [].

Lemma . Letζ:Rτ →R+fulﬁll that for almost every t≥τ,the diﬀerential inequality

d

dtζ(t) +φ(t)ζ(t)≤φ(t), (.)

where,for every t≥τ,the scalar functionsφandφsatisfy

t

τ

φ(s)ds≥β(t–τ) –γ,

t+

t

φ(s)ds≤M, (.)

for someβ> ,γ ≥and M≥.Then

ζ(t)≤eγζ(τ)e–β(t–τ)+ Me

γ

 –e–β, ∀t≥τ. (.)

Proof See,e.g., [].

For the non-autonomous general Benjamin-Bona-Mahony (BBM) equation,

–

→

F(u) =g(t,x), x∈,t∈Rτ, (.)

u(t,x)|∂= , (.)

u(τ,x) =uτ(x), τ∈R. (.)

Assume thatuτ∈H_(), the nonlinear vector function

–

→

F(s) = (F(s),F(s),F(s)),∀s∈R,

we denote

fi(s) =Fi(s), Fi(s) =

s



Fi(r)dr, (.)

where

–

→

f (s) =f(s),f(s),f(s)

, →–F(s) =F(s),F(s),F(s)

. (.)

In addition,Fi(i= , , ) is a smooth function satisfying

Fi() = , Fi(s)≤C|s|+C|s|, (.)

C_+C_|s| ≤fi(s)≤C+C|s|, Fi(s)≤C|s|+C|s| (.)

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Similar to [], by the Galerkin method anda prioriestimate, we easily derive the exis-tence of a global weak solution and a uniform attractor which shall be stated in the follow-ing theorems.

Theorem . Assume that(.)-(.)hold,g∈L

loc(R,H),uτ∈H() (or V) ,then there

exists a unique global weak solution u(x,t)of the problem(.)-(.)which satisﬁes

u∈C(τ,T);V, ut∈L

(τ,T);V (.)

for allτ∈R and T>τ.

Theorem . Assume that the external force g∈L

loc(R,H)and(.)-(.)hold,then the

processes{U(t,τ),t≥τ}generated by the global solution possess uniform attractorsAg(t)

in H_()for the non-autonomous system(.)-(.).

3 Some lemmas

Lemma . The functions f(x,s)and f(x,s)are taken from the space Lb(R,H)of

transla-tional bounded functions in L_loc(R,H),then the processes{Ufε(t,τ),t≥τ,t,τ∈R}

gener-ated by system(.)-(.)have a uniformly(w.r.t.σ=fε_∈₎_{compact attractor}_Aε_{for any}

ﬁxedε∈(, ).

Proof As a similar argument in Section , we chooseg(t,x) =fε_(t,_{x) in Theorem ., since}

f andf are translational bounded inLloc(R,H), then for any ﬁxedε∈(, ],fε(t,x) is

translational bounded inL

loc(R,H) and we can easily deduce the existence of uniformly

compact attractorsAε_.

We can brieﬂy describe the structure of the uniform attractor as follows: if the func-tionsf(t) andf(t) are translational bounded, problem (.)-(.) generates the dynamical

processes{Uε_(t,_τ_),_t_≥_τ_,_τ _∈_R_}_{acting on}_V_{which is deﬁned by}_Uε_(t,_τ_)uε τ =u

ε_(t),_t_≥_τ_,

whereuε_{(t) is the solution to (.)-(.). The processes}_{_Uε_(t,_τ_),_t_≥_τ_,_τ_∈_R_}_{have a}

uni-formly (w.r.t.t∈R) absorbing set Bε_:=_uε_∈_V_|_uε

V≤CQ

ε _, _(.)

which is bounded inV for any ﬁxedε∈(, ).

On the other hand,Aε_{is also bounded in}_V_{for each ﬁxed}_ε_since_Aε_⊆_Bε

. Assuming

f,f∈Ltc(R,H), the external forcefε(t) appearing in equation (.) belongs toLtc(R,H)

also. Moreover, ifε>  andfˆε_∈_H_(fε_{), then}

ˆ

fε(t) =fˆ(t) +ε–ρfˆ

t ε

, (.)

for somefˆ∈H(f) andfˆ∈H(f). In this case, to describe the structure of the uniform

attractorAε_{, we consider the family of equations}

ˆ

ut+Auˆt+νAuˆ+∇ ·F(u) =ˆ fˆε(t), fˆε∈H

fε. (.)

For every external forceˆfε_∈_H_(fε_{), equation (.) generates a class of processes}_{_U ˆ fε(t,τ)}

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to the original equation (.) with the external forcefε_{(t). Moreover, the map}

uτ,fˆε

→Ufˆε(t,τ)uτ (.)

is (V×H(fε_),_V_{)-continuous.}

Lemma . If the function f(t,x)in(.)is taken from the space Lb(R,H)of translational

bounded functions in L

loc(R,H),then the processes{Uf(t,τ),t≥τ,τ∈R}generated by

sys-tem(.)-(.)have a uniformly(w.r.t.σ=f∈)compact attractorA.

Proof Use a similar technique as that in Theorem ., we can easily deduce the existence of a uniformly compact attractorA_{if we choose}_g(t,_{x) =}_f

(t,x).

4 Uniform boundedness ofAε

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 ofAε_.

Considering the linear equation

Yt+AYt+νAY=K(t), Y|t=τ= , (.)

we get the following lemma.

Lemma . Assume that K∈L

loc(R,H),then problem(.)has a unique solution

Y∈L(τ,T);W∩C(τ,T);V, (.)

∂tY∈L

(τ,T);W. (.)

Moreover,the following inequalities

Y(t)_W≤C t

τ

e–Cν(t–s)K(s)_Hds, (.)

t+

t

Y(s)_Vds≤CY(t)_V+ t+

t

K(s)_Hds

(.)

hold for every t≥τ and some constant C> ,independent of the initial timeτ∈R. Proof Firstly, using the Galerkin approximation method, we can deduce the existence of a global solution for (.), here we omit the details.

Then multiplying (.) byYandAYrespectively, we get

 

d dt

Y+∇Y+ν∇Y=K(t),Y≤ νK(t)



+ν Y

 _(.)

and

 

d dt

∇Y+AY+νAY=K(t),AY≤ νK(t)



+ν AY

_. _(.)

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SettingK(t,τ) =_τtk(s)ds,t≥τ,τ∈R, we have the following lemma. Lemma . Assume that the formula

sup

t≥τ,τ∈R

K(t,τ)_H+ t+

t

K(s,τ)_Hds

≤l (.)

holds for some constant l≥,let k∈L

loc(R,H).Then the solution y(t)yields the following

problem:

yt+Ayt+νAy=k(t/ε), y|t=τ= , (.)

withε∈(, )satisfying the inequality y(t)_V+

t+

t

y(s)_Vds≤Clε, ∀t≥τ, (.)

where C> is constant independent of K. Moreover,we also have

t+

t

Kε(s)



Hds≤C. (.)

Proof Noting that

Kε(t) =

t

τ

k(s/ε)ds=ε t/ε

τ/ε

k(s)ds=εK(t/ε,τ/ε), (.)

we can derive the following estimates from (.):

sup

t≥τ

Kε(t)_H≤lε,

t+

t

Kε(s)



Hds=ε



t+

t

K(s/ε,τ/ε)_Hds

≤Cεsup

t≥τ

t+

t

K(s,τ)_Hds

≤Clε.

From Lemma ., we have t

τ

e–Cν(t–s)Kε(s)



Hds

≤

t

t–

eCν(s–t)Kε(s)



ds+ t–

t–

eCν(s–t)Kε(s)



ds+· · ·

≤

t

t–

Kε(s)



ds+e–Cν t–

t–

Kε(s)



ds+e–Cν t–

t–

Kε(s)



ds+· · ·

≤ +e–Cν+e–Cν+· · ·Kε(s)



L_b(R;H)

≤ 

( –e–Cν₎Kε(s)



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≤ 

( –e–Cν₎sup t≥τ

t+

t

_K_ε(s)_Hds

≤Clε. (.)

Hence, from the Poincaré inequality, combining (.) and (.)-(.), we conclude that

Y(t)_W≤Clε, (.)

t+

t

Y(s)_Vds≤CY(t)_V+ t+

t

K(s)_Hds

≤Clε. (.)

Setting

Y(t) = t

τ

y(s)ds, (.)

we deduce that for anyt≥τ,

∂tY(t) =y(t) =

t

τ

∂ty(s)ds, (.)

sincey(τ) = .

Integrating (.) with respect to time variable fromτtot, we see thatY(t) is a solution to the problem

∂tY(t) +∂t

AY(t)+νAY(t) =Kε(t), qY(t)|t=τ= , (.)

such that from (.) and (.), we can derive

Y(t)_H+∇Y(t)_H+ t+

t

Y(s)_Vds≤Clε. (.)

By virtue ofy(t) =∂tY(t), (AY(t),Y(t))∼ Y(t)V,AY(t) ∼ Y(t)W, we have

∂tY(t)+∂tAY(t)=y(t)+Ay(t)≤νY(t)_W+Kε(t)≤Clε. (.)

Hence, we conclude

y(t)_V≤Cy(t)+Ay(t)≤CνY(t)_W+Kε(t)≤Clε (.)

and t+

t

y(s)_Vds≤Clε. (.)

The proof is ﬁnished.

Now, we shall use the auxiliary linear equation and some estimates to prove the uniform boundedness ofAε_in_V_{. For convenience, we set}

F(t,τ) =

t

τ

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and assume that

sup

t≥τ,τ∈R

F(t,τ) 

+ t+

t

F(s,τ) 

Hds

≤l, (.)

holds for some constantsl≥.

Theorem . The attractorsAε_{of problem}_{(.)-(.) (or}_{(.)-(.))}_{are uniformly}_(w.r_.t.

ε)bounded in V,namely,

sup

ε∈[,)

_Aε

V< +∞. (.)

Proof Letuε_{(t) =}_Uε_(t,_τ_)uε

τ be the solution to (.)-(.) with the initial datauετ ∈V. For

ε> , we consider the auxiliary linear equation

vt+Avt+νAv=ε–ρf(t/ε), v|t=τ = . (.)

From Lemma ., we have the estimate

v(t)_V+ t+

t

v(s)_Vds≤Clε(–ρ), ∀t≥τ. (.)

Setting the functionw(t) as

w(t) =u(t) –v(t), (.)

which satisﬁes the problem

wt+Awt+νAw+∇ ·

–

→

F(w+v) =f, w|t=τ=uτ. (.)

Taking the scalar product of (.) withw, we obtain

 

d dt

w+∇w+ν∇w+∇ ·→–F(w+v),w= (f,w). (.)

Using the inequality

v(t)=v(t)_H≤Cv(t)_V≤Clε(–ρ), ∀t≥τ, (.)

we have

∇ ·–→F(w+v),w≤C

 +w+v+ ν λw



≤C

 +w+lε(–ρ)+ ν λw



≤C

 +w+lε(–ρ)+ ν λw

_, _(.)

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Moreover, notice that

(f,w)≤

ν w



V+

 νf

_, _(.)

and inserting (.)-(.) into (.), we have

 

d dt

w+∇w+ν∇w

≤C

 +w+lε(–ρ)+ ν λw

₊ν

w



V+

 νf



≤C

 +w+lε(–ρ)+ν w



V+

ν w



V+

 νf



=C

 +w+lε(–ρ)+ν w



V+

 νf

_, _(.)

which implies that

d dt

w+w_V+φwV≤φ, (.)

where

φ(t)≡

ν –C

 +u+lε(–ρ), (.)

φ(t)≡

 νf(t)



. (.)

Therefore using (.), we derive from (.)-(.) that for anyt≥τ, t

τ

φ(s)ds≥

ν

(t–τ), (.)

t+

t

φ(s)ds≤CM. (.)

Applying Lemma . withζ(t) =w₊_w

V,β= ν

,γ = ,M=CM 

, we have

w+w_V≤Ce–β(t–τ)uτ+uτ_V

+CM_, ∀t≥τ, (.) which gives

w_V≤Ce–β(t–τ)_u

τ+uτ_V

+CM_, ∀t≥τ. (.)

Recalling thatu=w+v, and using (.) and (.), we end up with

u(t)_V≤ w_V +v_V≤Ce–β(t–τ)uτ+uτV

+Cl+M_, ∀t≥τ. (.) Thus, for every  <ε≤ε, the processes{Uε(t,τ)}have an absorbing set

B:=

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On the other hand, ifε<ε< , the processes{Uε(t,τ)}also possess an absorbing set

Bε₌_u∈_V|_u

V≤CQε . (.)

In conclusion, for everyε∈[, ), the set

B*:=B∪Bε (.)

is an absorbing set for{Uε(t,τ)}which is independent ofε. SinceAε⊂B*, (.) follows

and hence the proof is complete.

5 Convergence ofAεtoA0

The main result of the paper reads as follows.

Theorem . Assume that f,f∈Ltc(R,H)⊂Lb(R,H)and(.)holds.Then the uniform

attractorAε_{(for problem}_{(.)-(.))}_{converges to}_A_{(for problem}_{(.)-(.))}_as_ε_→_+_in

the following sense:

lim

ε→+distV

Aε_,_A_{= .} _(.)

Next, we shall study the diﬀerence of two solutions for (.) withε>  and (.) with ε=  which share the same initial data. Denote

uε(t) :=Uε(t,τ)uτ, (.)

withuτ belonging to the absorbing setB*which can be found in Section . In particular,

sinceuτ ∈B*, the formula corresponding toε= 

u(t)_V+ t+

t

u(s)_Vds≤R_, (.)

holds for someR=R(ρ), as the size ofB*depends onρ.

Lemma . For everyε∈(, ),τ ∈R,uτ∈B*and uε() =u() =uτ,the diﬀerence

w(t) =uε_{(t) –}_u_(t) _(.)

satisﬁes the estimate

w(t)_V≤Dε–ρeR(t–τ), ∀t≥τ, (.)

for some positive constants D=D(ρ,l)and R=R(ρ,l),both independent ofε> .

Proof Since the diﬀerencew(t) solves the equation

wt+Awt+νAw+∇ ·

–→

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the diﬀerence

q(t) =w(t) –v(t), (.)

fulﬁlls the Cauchy problem

qt+Aqt+νAq+∇ ·

–→

Fuε–→–Fu= , q|t=τ = , (.)

wherev(t) is the solution to (.).

Taking an inner product of equation (.) withqinH, we obtain

 

d dt

q+∇q+ν∇q+∇ ·–→Fuε–→–Fu,q= . (.)

Noting that

∇ ·–→Fuε–→–Fu,q

≤sup

i

F_iuε₊_F i

u∇ ·uε_–_{∇ ·}_u

+ ν λq



≤C

 +uε+u∇w+ ν λq



≤C

 +uε₊_R 

∇w+ ν λq



≤C

 +uε+R_q+v_V+ ν λq



≤C

 +K_+R_v_V+ν q



V+

ν λq



=f(t) +ν q



V+h(t)q, (.)

whereλis the ﬁrst eigenvalue of –,Kis the upper bound foruε(by Lemma .) and

h(t) = ν λ,

f(t) =C

 +K+R

v_V≤C +K+R

lε(–ρ),

thus, it follows from (.) and (.) that

 

d dt

q+∇q+ν q



V≤Ch(t)q+f(t)

≤Ch(t)q+∇q+f(t). (.)

Noting thatq(τ)=q(τ)V= , by the Gronwall inequality, we get

q+∇q≤exp

C t τ h(s)ds t τ

f(s)ds. (.)

Moreover, we can derive the following formulas: t

τ

h(s)ds≤ ν

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and t

τ

f(s)ds= t

τ

C +K_+R_v_Vds

≤

t

τ

C +K_+R_lε(–ρ)ds

=C +K_+R_lε(–ρ)(t–τ). (.)

Consequently,

_q(t)

V ≤C

q+∇q

≤C +K_+R_lε(–ρ)(t–τ+ )eνλ(t–τ+)

≤CD_ε(–ρ)eνλ(t–τ) (.)

holds for some positive constantsD=D(ρ,l). Finally, sincew=q+v, using (.) to

controlvV, we may obtain

w(t)_V≤Cq_V+v_V

≤CD_ε(–ρ)eνλ(t–τ)+Cl_ε(–ρ)

≤Dε(–ρ)eR(t–τ)_, _(.)

whereRis a positive constant. The proof is ﬁnished.

Next, we want to generalize Lemma . to derive the convergence of corresponding uniform attractors. Let the external force in equation (.) asfˆ=ˆfε_∈_H_(fε_{), then}_fˆ

∈H(f)

satisﬁes inequality (.). Deﬁne

ˆ

G(t,τ) =

t

τ

ˆ

f(s)ds, t≥τ, (.)

we have

sup

t≥τ,τ∈R

_ˆ G(t,τ)



H+

t+

t

G(s,ˆ τ)_Hds

≤l. (.)

For anyε∈[, ], we observe thatuˆε_{(t) =}_U ˆ

fε(t,τ)yτ is a solution to (.) with the external

forcefˆε₌_fˆ

+ε–ρfˆ(·/ε)∈H(fε) andyτ(fε)∈B*. Forε> , we investigate the property of

the diﬀerence

ˆ

w(t) =uˆε_{(t) –}_u_ˆ_(t). _(.)

Lemma . The inequality

w(t)_ˆ ≤Dε–ρeR(t–τ)_, _∀_t_≥_τ_, _(.)

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Proof As the similar discussion in the proof of Lemma ., replacinguˆε_,ˆ_f

 andfˆ by

uε_, _f

 andf, respectively, noting that (.) still holds for uˆ, and the family{Uˆfε(t,τ)}

(fˆε_∈_H_(fε_{)), is (H}_×_Hε_(fε_),_{H)-continuous, using (.) in place of (.), we can ﬁnally}

complete the proof of the lemma.

Proof of Theorem. Forε> ,uε_∈_Aε_{, we obtain that there exists a complete bounded}

trajectoryuˆε_{(t) of equation (.), with some external force}

ˆ

fε=fˆ+ε–ρfˆ(·/ε)∈H

fε, (.)

such thatuˆε_{() =}_uε_.

We chooseL≥ such that

ˆ

uε(–L)∈Aε⊂B*. (.)

From the equality

uε=Uˆf(, –L)uˆε(–L), (.)

applying Lemma . witht= ,τ= –L, we obtain

uε_–_U ˆ

f(, –L)uˆε(–L)_V≤Dε–ρeRL. (.)

On the other hand, the setAattracts all setsUfˆ(t, –L)B*uniformly whenfˆ∈H(f).

Then, for allδ> , there exists some timeT=T(δ)≥ which is independent ofLsuch that

distV

Ufˆ(T–L, –L)uˆε(–L),A

≤δ. (.)

ChoosingL=T and collecting (.)-(.), we readily get

distV

uε,A≤uε–Ufˆ(, –T)uˆε(–T)_V+distV

Uˆf(, –T)uˆε(–T),A

≤Dε–ρeRT+δ. (.)

Sinceuε_∈_Aε_and_δ_{>  is arbitrary, taking the limit}_ε_→_+_{, we can prove the theorem.}

Competing interests

The authors declare that they have no competing interests. Authors’ contributions

The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the ﬁnal manuscript.

Author details

1_{College of Mathematics and Information Science, Pingdingshan University, Pingdingshan, 467009, P.R. China.}2_{College of}

Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, P.R. China.3_{College of Education and}

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Acknowledgements

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).

Received: 29 January 2013 Accepted: 16 April 2013 Published: 30 April 2013

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doi:10.1186/1687-2770-2013-111