ASYMPTOTIC SMOOTHING AND THE GLOBAL ATTRACTOR …rrosa/dvifiles/kdvl2.pdf · ASYMPTOTIC SMOOTHING AND THE GLOBAL ATTRACTOR OF A WEAKLY DAMPED KDV EQUATION ON THE REAL LINE OLIVIER

ASYMPTOTIC SMOOTHING AND THE GLOBAL ATTRACTOR OF AWEAKLY DAMPED KDV EQUATION ON THE REAL LINE

OLIVIER GOUBET AND RICARDO M. S. ROSA

Abstract. The existence of the global attractor of a weakly damped, forced Korteweg-deVries equation in the phase space L2(R) is proved. An optimal asymptotic smoothing effectof the equation is also shown, namely, that for forces in L2(R), the global attractor in thephase space L2(R) is actually a compact set in H3(R). The energy equation method isused in conjunction with a suitable splitting of the solutions; the dispersive regularizationproperties of the equation in the context of Bourgain spaces are extensively exploited.

1. Introduction

Our aim is to study the existence and the regularity of the global attractor in the phasespace L2(R) for the following weakly damped, forced Korteweg-de Vries equation:

ut + uux + uxxx + γu = f, for (x, t) ∈ R× R. (1.1)

This equation is supplemented with the initial condition

u|t=0 = u0 ∈ L2(R). (1.2)

It is assumed that f is time-independent and belongs to L2(R), and that γ > 0 is a constant.We show that the global attractor exists in L2(R) and is a compact set in H3(R), thusproving an asymptotic smoothing effect (in the terminology used by A. Haraux [13]) sincethe solutions, in general, belong only to L2(R).

Equation (1.1) has been derived by E. Ott and R. N. Sudan [23] as a model for ion-sound waves damped by ion-neutral collisions. For γ = 0 and f = 0, equation (1.1) is thewell-known Korteweg-de Vries (KdV) equation [17]. From the mathematical point of view,the extra term with the factor γ accounts for a weak dissipation with no regularization, orsmoothing, property. The asymptotic smoothing of the global attractor comes essentiallyfrom the dispersive regularization property of the equation.

Date: April 3, 2001 (submitted on July 18, 2000).2000 Mathematics Subject Classification. Primary: 35Q53, 35B40, 37L50; Secondary: 35L30, 37L30.Key words and phrases. Korteweg-de Vries equation, weak damping, noncompact system, global attractor,

asymptotic smoothing, dispersive regularization, Bourgain spaces.The second author was partially supported by a fellowship from CNPq, Brasılia, Brazil, by FAPERJ and

FUJB, Rio de Janeiro, Brazil, and by the Research Fund of the Indiana University and of the NSF - DMSgrant 9705229. The second author also acknowledges the hospitality of the Laboratory of Numerical Analysisat the Universite de Paris-Sud at Orsay.

1

2 OLIVIER GOUBET AND RICARDO M. S. ROSA

For the well-posedness of (1.1) in L2(R), we use the so-called Bourgain function spaces.Those spaces were introduced by J. Bourgain [5] for the well-posedness of the KdV itself inL2(R); we follow here the guidelines of C. E. Kenig, G. Ponce, and L. Vega [15] (see, also,[8]). Those spaces were also used by J. Bona and B.-Y. Zhang [4] for the well-posednessin L2(R) of the forced equation without the weak dissipation (i.e., with γ = 0). The well-posedness of the KdV equation was obtained in higher order Sobolev spaces in, for instance,[27, 3, 26], and in lower order Sobolev spaces in, for instance, [15, 16]. For the existence ofthe global attractor, we use essentially the energy equation method introduced by J. Ball [2](for a wave-type equation) together with a splitting of the solutions.

The existence of the global attractor for hyperbolic equations is obtained by means of theasymptotic compactness or the asymptotic smoothing properties of the solution operatortogether with the existence of a bounded absorbing set [12, 28, 18, 1, 24]. Those propertiesare usually proved by splitting the solutions into a decaying part plus a regular part, or byexploiting suitable energy-type equations, or both. For the energy equation method, theweak continuity of the solutions with respect to the initial condition (in the sense that if theinitial conditions u0n converge to u0 weakly, then the corresponding solutions un(t) convergeweakly to u(t) at all times t) is a crucial step. This approach was used, for instance, in theproof of the existence of the global attractor for (1.1) in H1(R) [25], in H2(R) [22, 19], andin the space-periodic case [7, 21, 10], as well as for several other equations (e.g., [30, 9]).

The energy equation method can also be applied to (1.1) in L2(R). Here, however, sincewe also want to prove the regularity of the attractor, we use this energy equation methodtogether with a splitting of the solutions to obtain, at the same time, the existence of theglobal attractor in L2(R) and its boundedness in H3(R). This is achieved by splitting thesolutions into two parts, one which is regular (it belongs to H3(R)) and the other whichdecays to zero (in L2(R)) as time goes to infinity. In this way, the weak continuity of thesolution operator just mentioned is actually not used, and it is replaced by an asymptoticweak continuity property (in the spirit of [22]). Nevertheless, the weak continuity is aninteresting property by itself, so we include its proof at the end. Because of the lack of thecompact Sobolev embeddings in the unbounded case, the splitting alone is not enough toprove the asymptotic compactness of the solution operator, and the energy equation methodis used.

After obtaining the boundedness in H3(R) of the regular part of the solution, the compact-ness of the global attractor in H3(R) is obtained by applying the energy equation methodto the equation for the time derivative of the solutions in the global atttractor. From theequation (1.1) and from the first part of the proof, we see that the time derivatives are inL2(R) since the global attractor is bounded in H3(R). From the energy equation method,we obtain, loosely speaking, the L2(R) compactness for the time derivatives. Then, goingback to the equation (1.1), this gives the compactness of the global attractor in H3(R). Themajor difficulty here comes from the fact that the time derivative of the solutions is justin L2(R), which makes the nonlinear term in the energy equation for the time derivativemore difficult to handle. This term is handled by using the subtle dispersive regularizationproperties of the equation in the context of the Bourgain spaces.

GLOBAL ATTRACTOR OF A WEAKLY DAMPED KDV EQUATION 3

The splitting used is obtaining by writing u = v + w and splitting the nonlinear term as

uux = vvx + PN((vw)x + wwx) +QN((vw)x + wwx),

where PN denotes the Fourier spectral projector associated with a cut-off of the higher modeswhich retains only the modes with spatial frequency |ξ| ≤ N , for N > 0. The operator QN

is the complement QN = I − PN . Then, we obtain

vt + vvx + vxxx + γv = f − PN((vw)x + wwx), (1.3)

and

wt +QN(wwx) + wxxx + γw = −QN(vw)x, (1.4)

with the initial conditions

v|t=0 = PNu0, w|t=0 = QNu0. (1.5)

Due to the cut-off operator PN in the right hand side of the equation for v and in the initialcondition, the solution v is more regular (“three times” more regular than f in the scale ofSobolev spaces). As for w, there is no forcing term, and the presence of the operator QN

makes the additional linear (driving) term in the right hand side of the equation be relativelysmall with respect to the weakly dissipative term. Then, we obtain that w actually decaysin time in the L2(R)-norm. The appropriate estimate of the driving term is obtained thanksto the dispersive regularization properties of the linear part of the equation in the context ofthe Bourgain spaces; this type of estimate has already been exploited by O. Goubet [10] inthe space-periodic case for the same purpose, and it is slightly simplified here. We then showthat as time t goes to infinity, u(t) = v(t) + w(t) converges strongly in L2(R) to a solutionu(t) on the global attractor, with u(t) in H3(R), and with the regular part v(t) convergingstrongly in H3(R) to u(t). The decaying part w(t) goes to zero in L2(R).

The existence of global attractors for the equation (1.1) was first considered by J. M.Ghidaglia [6, 7], in the spaces H2

per(0, L) of L-periodic functions in H2, where it is assumed

that f ∈ H2per(0, L). This result was extended to Hk

per(0, L), with k ∈ N, k ≥ 3, by I.

Moise and R. Rosa [21], with f ∈ Hkper(0, L), and where the global attractor was proved to

be more regular if so is f (with the attractor as regular as f in the scale of Sobolev spacesHm

per(0, L), m ∈ N, m > k ≥ 3). The whole space case in the phase space H2(R), and

assuming f also in H2(R), was treated independently by P. Laurencot [19], using energy-type equations and weighted spaces, and by I. Moise, R. Rosa, and X. Wang [22], using onlyenergy-type equations. The H1(R) case, assuming f in H1(R), was treated by R. Rosa [25].In all those works, the attractor was proved to be as regular as the forcing term.

If one considers the steady states of (1.1), however, one notes that they are “three times”(in the scale of Sobolev spaces) more regular than the forcing term. One could expectthat the same would happen for the global attractor. The breakthrough to obtain such as-ymptotic regularization was obtained recently by O. Goubet [10] by exploiting the dispersiveregularization properties associated with (1.1) in the context of Bourgain spaces. The authorconsidered the space-periodic case in the phase space L2(0, L) and proved that indeed theglobal attractor exists and is actually compact in H3

per(0, L). The method used is a splitting


of the solutions in a way similar to the one used here. The compact Sobolev embeddingson bounded domains is used to prove the asymptotic compactness in L2(0, L), while thecompactness of the global attractor in H3

per(0, L) is obtained by the energy equation methodapplied to the equation for ut. In the present work, we are able to extend this result tothe whole real line. Some delicate issues appear due to the lack of the compact Sobolevembeddings, and the dispersive regularization is exploited further to compensate for that.

Finally, we mention that this result can be easily extended to show that if the forcingterm f belongs to Hk(R), for a given k ∈ N, then the equation (1.1) generates a group ineach phase space Hm(R), for m = 0, 1, . . . , k, and the global attractor exists and is the samefor each group; moreover, the global attractor is a compact set in Hk+3(R). This completesthe study of the existence and regularity of the global attractor for the equation (1.1) inthe scale of Sobolev spaces Hk(R) for integers k ≥ 0. For the existence of the attractor inGevrey spaces when the force is in Gevrey, see [29].

2. Function Spaces and Preliminary Estimates

2.1. Function Spaces. We consider the spaces L2(R) and Hs(R), s ∈ R, with the normsdenoted respectively by ‖ · ‖L2(R) and ‖ · ‖Hs(R). The inner product in L2(R) is denoted by(·, ·)L2(R). Let L2

loc(R) denote the Frechet space of functions which are locally in L2, i.e.,which belong to L2(J) for every compact interval J in R. We will use the following compactembedding:

H3(R) ⊂c L2loc(R). (2.1)

We also consider spaces of the type C∞c , of infinitely differentiable functions with compactsupport and the Schwartz space S(R2) of tempered test functions on R2. More general spacesof the type Hs, s ∈ R, and Lp, 1 ≤ p ≤ ∞, are also considered, and their norm is denotedwith the appropriate subscript. We recall the Agmon inequality:

‖u‖L∞(R) ≤ ‖u‖1/2

L2(R)‖∂xu‖1/2

L2(R). (2.2)

Whence we deduce the following inequality which will be used in the sequel:

‖u2‖H−1(R) ≤ ‖u‖2L2(R). (2.3)

For a given interval I and a given Banach space E we also consider the spaces Lp(I;E), 1 ≤p ≤ ∞, of E-valued functions on I whose norm in E to the p-th power is integrable on I (oris essentially bounded if p = ∞), and the space Cb(I;E) of bounded, continuous functionson I with values in E. Their respective norms are denoted with the appropriate subscriptsas above. When I is an unbounded interval and 1 ≤ p < ∞, it is also useful to considerthe functions which are locally in Lp, i.e., the space of functions which belong to Lp(J ;E)for every bounded subinterval J ⊆ I. Those spaces are Frechet spaces and are denotedLploc(I;E). Similarly, we denote by C(I;E) the space of functions which belong to Cb(J ;E)for each bounded subinterval J ⊆ I.


An important role is played by the Bourgain space-time function spaces Xs,b, for s, b ∈ R,which are defined as the completion of the Schwartz space S(R2) with respect to the norm

‖f‖2Xs,b =

∫R

∫R

〈τ − ξ3〉2b〈ξ〉2s|f(ξ, τ)|2 dξdτ, (2.4)

where f = f(ξ, τ) is the Fourier transform of f = f(x, t), and

〈λ〉 =√

1 + |λ|2. (2.5)

We will sometimes also denote the Fourier transform in Rn by F :

F(f)(ζ) = f(ζ) =1

(2π)n/2

∫Rn

f(z)e−iz·ζ dz. (2.6)

The inverse Fourier transform is denoted F−1.We have the continuous embedding

Xs,b ⊆ Cb(R, Hsx(R)), (2.7)

for any s ∈ R and b > 1/2. On dealing with space-time functions as in (2.7), we will oftenuse a subscript on the function spaces considered to indicate with respect to which variablethat function space refers to. Similarly, we indicate by Fx the Fourier transform with respectto x of a function depending on x and t.

We will also need to localize in time the estimates obtained in the spaces Xs,b. Forthat purpose, consider a function ψ ∈ C∞c (R) which is equal to one on [−1, 1] and to zerooutside (−2, 2). We set ψT (t) = ψ(t/T ). For any given interval I = [a, b], set also ψI(t) =ψ((2t− a− b)/(b− a)). Then, we consider the seminorms in Xs,b defined by

‖u‖Xs,b[−T,T ]

= ‖ψTu‖Xs,b , (2.8)

and

‖u‖Xs,bI

= ‖ψIu‖Xs,b . (2.9)

The function ψ is fixed once and for all, since some of the constants below depend on it.

2.2. Linear Estimates. Consider the free Airy group {W (t)}t∈R, which is given by W (t) =exp(−t∂3

x). The Bourgain spaces are such that u = u(x, t) belongs to Xs,b if and only ifW (−t)u belongs to Hb

tHsx(R

2). We can write

‖u‖Xs,b = ‖W (−·)u‖HbtH

sx(R2) = ‖〈τ − ξ3〉b〈ξ〉su‖L2

τ,ξ(R2). (2.10)

The following estimate is then easy to obtain:

‖ψTW (t)u0‖Xs,b = ‖ψT‖Hb(R)‖u0‖Hs(R), (2.11)

for every u0 ∈ Hs(R) and s, b ∈ R.We borrow from [8, Lemma 3.2] the following result:


Lemma 2.1. Consider s ∈ R, 0 < T ≤ 1, and −1/2 < b′ − 1 ≤ 0 ≤ b ≤ b′. Let g ∈ Xs,b′−1,and define

w(t) = ψT (t)

∫ t

0

W (t− t′)g(t′) dt′. (2.12)

Then, there exists a constant c1 > 0, independent of s, b, b′ and T, such that the followinginequality holds:

‖w‖Xs,b ≤ c1Tb′−b‖g‖Xs,b′−1 . (2.13)

From [15, Lemma 3.2] we borrow the following result:

Lemma 2.2. There exists a constant c2 > 0 such that for s ∈ R, 1/2 < b ≤ 1, and u ∈ Xs,b,we have

‖ψTu‖Xs,b ≤ c2T(1−2b)/2‖u‖Xs,b . (2.14)

Recall now the well-known Strichartz-type inequalities establishing dispersive-type regu-larizations of the free Airy group (see [14]): for any u0 in L2(R),

‖W (t)u0‖L8x,t(R

2) ≤ c3‖u0‖L2x(R), (2.15)

‖∂xW (t)u0‖L∞x L2t (R

2) ≤ c3‖u0‖L2x(R2), (2.16)

‖(∂x)1/6W (t)u0‖L6x,t(R

2) ≤ c3‖u0‖L2x(R). (2.17)

for some constant c3 > 0. Such estimates may be turned into estimates in the Bourgainspaces with the following lemma, which we borrow from [8, Lemma 3.3]:

Lemma 2.3. Let ‖ · ‖Y be a seminorm in Sx,t(R2) which is stable under multiplication byfunctions in L∞t (R) in the sense that

‖ψf‖Y ≤ c‖ψ‖L∞t (R)‖f‖Y , ∀ψ ∈ L∞t (R), ∀f ∈ Sx,t(R2). (2.18)

Assume the following (Strichartz-type) inequality holds:

‖W (·)u0‖Y ≤ c‖u0‖L2x(R), ∀u0 ∈ L2

x(R). (2.19)

Then, for all b > 1/2, the following inequality holds:

‖f‖Y ≤ cb‖f‖X0,b , ∀f ∈ X0,b, (2.20)

with cb = cb1/2(2b− 1)−1/2.

Hence, using Lemma 2.3, we infer from (2.15), (2.16), and (2.17) that for any b > 1/2 andany f in X0,b,

‖f‖L8x,t(R

2) ≤ c′b‖f‖X0,b , (2.21)

‖∂xf‖L∞x L2t (R

2) ≤ c′b‖f‖X0,b , (2.22)

‖(∂x)1/6f‖L6x,t(R

2) ≤ c′b‖f‖X0,b . (2.23)


where c′b depends on b > 1/2. By interpolating (2.21) with L2x,t = X0,0, we obtain

‖f‖L4x,t(R

2) ≤ c′b‖f‖X0, b2. (2.24)

Similarly, interpolating (2.23), we find

‖(∂x)1/8f‖L4x,t(R

2) ≤ c′b‖f‖X0, 3b4. (2.25)

2.3. Bilinear Estimates. We now describe an estimate needed to handle the bilinear termin the KdV equation. We first recall from [16, Theorem 1.1] the following statement:

Proposition 2.1. Let s ∈ (−3/4, 0] be given. Then there exists a numerical constant c′′s anda number b ∈ (1/2, 1) such that for any function u in Xs,b

‖D(u2)‖Xs,b−1 ≤ c′′s‖u‖2Xs,b . (2.26)

We shall also use below a peculiar version of (2.26) that states as follows (we prove it fors ∈ (−1/2, 0] because we do not need it for smaller values of s):

Proposition 2.2. Let s ∈ (−1/2, 0] be given. Then there exists a numerical constant c′′ssuch that for b′ ∈ (1/2, 9/16], for b ∈ [1/2,min{b′, 1 + s}], and for any function u in Xs,b,

‖D(u2)‖Xs,b′−1 ≤ c′′s‖u‖Xs,b‖u‖X−

12 ,b. (2.27)

As a corollary of this result we have

Corollary 2.1. Assume moreover that the Fourier transform u = u(ξ, τ) of u is supportedin {(ξ, τ); |ξ| ≥ N}, N > 0. Then, for 1/2 < b ≤ 9/16,

‖D(u2)‖X0,b−1 ≤ c′′0√N‖u‖2

X0,b . (2.28)

Proof of Proposition 2.2: Set v = 〈ξ〉s〈τ − ξ3〉bu, where u stands for the Fourier transformF(u) of u. By a duality argument, and by setting ρ = −s, (2.27) stems from the followingassertion: there exists C = c′′s such that for any G and v which are of norm 1 in L2(R2),

Q ≡∫D

ξv(ξ1, τ1)v(ξ2, τ2)G(ξ, τ)

〈τ − ξ3〉1−b′〈τ1 − ξ31〉b〈τ2 − ξ3

2〉b〈ξ1〉ρ〈ξ2〉ρ

〈ξ〉ρdτ1dξ1dτdξ ≤ C‖v‖

Xρ− 12 ,0

(2.29)

where

D = {σ = {ξ, τ, ξ1, τ1} ∈ R2 × R2},and

ξ2 = ξ − ξ1, τ2 = τ − τ1.

We divide the majorization of Q into three cases:First case: |ξ1| ≤ |ξ2| ≤ 1/2. Set D1 for the subset of D where these inequalities are valid.

In that case the function 〈ξ1〉ρ〈ξ2〉ρ|ξ|/〈ξ〉ρ is bounded, and we just have to majorize

Q1 ≡∫D1

|v(ξ1, τ1)v(ξ2, τ2)G(ξ, τ)|〈τ − ξ3〉1−b′〈τ1 − ξ3

1〉b〈τ2 − ξ32〉b

dτ1dξ1dτdξ. (2.30)


Set

w(ξ, τ) =v(ξ, τ)

〈τ − ξ3〉bχ(ξ),

where χ is the characteristic function of the interval [−1/2, 1/2]. Then, using in particularthat b′ − 1 ≤ 0 and G has L2-norm one,

Q1 ≤ ‖G(τ, ξ)〈τ − ξ3〉b′−1‖L2ξ,τ (R)‖w ∗ w‖L2

ξ,τ (R) ≤1

2π‖G(τ, ξ)‖L2

ξ,τ (R)‖w2‖L2x,t(R

2)

=1

2π‖F−1(|v(ξ, τ)χ(ξ, τ)|)〈τ − ξ3〉−b‖2

L4x,t(R), (2.31)

where ∗ denotes the convolution operator. Now, we use (2.24), the embeddings Xs,a ⊂ Xs,0

for a ≥ 0, and the following inverse inequality

‖F−1(|v|χ)‖X0,0 ≤ ‖v‖Xρ− 1

2 ,0(2.32)

to obtain Q1 ≤ C‖v‖Xρ−1/2,0 .Second case:|ξ1| ≤ 1

2≤ |ξ2|. Set D2 for the corresponding region. In that case we use

|ξ| ≤ 2|ξ2| and the fact that the function 〈ξ1〉ρ〈ξ2〉ρ/〈ξ〉ρ is bounded to write

Q2 ≤ C‖G(τ, ξ)〈τ − ξ3〉b′−1‖L2ξ,τ‖F−1(|ξ2||v(ξ2, τ2)|)〈τ2 − ξ3

2〉−b‖L∞x L2t×

× ‖F−1(|v(ξ1, τ1)χ(ξ1)|)〈τ1 − ξ31〉−b‖L2

xL∞t

(2.33)

On the one hand, by (2.22), we have

‖F−1(|ξ2||v(ξ2, τ2))〈τ2 − ξ32〉−b‖L∞x L2

t≤ C‖v‖L2

x,t(R2) = C. (2.34)

On the other hand by the embedding X0,b ⊂ L2xL∞t (R2) and by (2.32), we have

‖F−1(|v(ξ1, τ1)χ(ξ1)|)〈τ1 − ξ31〉−b‖L2

xL∞t≤ c‖F−1(|v(ξ1, τ1)|χ(ξ1))‖X0,0 ≤ c‖v‖

Xρ− 12 ,0. (2.35)

Hence, we obtain Q2 ≤ C‖v‖Xρ−1/2,0 .Third case: 1

2≤ |ξ1| ≤ |ξ2|. In this case, the following algebraic inequality will be useful:

3|ξξ1ξ2| = |ξ3 − ξ31 − ξ3

2 | = |(τ − ξ3)− (τ1 − ξ31)− (τ2 − ξ3

2)|≤ |τ − ξ3|+ |τ1 − ξ3

1 |+ |τ2 − ξ32 |. (2.36)

First subcase: |ξξ1ξ2| ≤ |τ1 − ξ31 |. Set D31 for the corresponding region. For b ≤ 1− ρ, we

shall use |ξ| ≤ 2|ξ2| to write

Q31 ≤ C

∫D31

|v(ξ1, τ1)v(ξ2, τ2)G(ξ, τ)|〈τ − ξ3〉1−b′〈τ2 − ξ3

2〉b|ξ1|ρ−b|ξ2|1−2b dτ1dξ1dτdξ. (2.37)

We then have, using (2.24) and the relations 1− 2b ≤ 0 and b′ − 1 ≤ b/2,

Q31 ≤ C‖F−1(G(τ, ξ)〈τ − ξ3〉b′−1)‖L4x,t‖F−1(|v(ξ2, τ2)|ξ2|1−2b〈τ2 − ξ3

2〉−b)‖L4x,t‖v‖Xρ−b,0

≤ C‖v‖Xρ−b,0 . (2.38)


Second subcase: |ξξ1ξ2| ≤ |τ2 − ξ32 |. Set D32 for the corresponding region. Then,

Q32 ≤ C

∫D32

|v(ξ1, τ1)v(ξ2, τ2)G(ξ, τ)|〈τ − ξ3〉1−b′〈τ1 − ξ3

1〉b|ξ1|ρ−b|ξ2|1−2b dτ1dξ1dτdξ. (2.39)

We obtain

Q32 ≤ C‖v‖X1−2b,0‖F−1(|v(ξ1, τ1)|ξ1|ρ−b〈τ1 − ξ31〉−b)‖L4

x,t‖F−1(G)〈τ − ξ3〉b′−1‖L4

x,t,

≤ C‖v‖Xρ−1/2,0 (2.40)

where we used (2.24) and the relations b ≥ 1/2, ρ− b ≤ 1− 2b ≤ 0, and b′ − 1 ≤ b/2.Third subcase: |ξξ1ξ2| ≤ |τ − ξ3|. In that case, using also |ξ| ≤ 2|ξ2|,

Q33 ≤ C

∫D33

|v(ξ1, τ1)‖ξ1|ρ+b′−1

〈τ1 − ξ31〉b

|v(ξ2, τ2)‖ξ2|2b′−1

〈τ2 − ξ32〉b

|G(τ, ξ)| dτ1dξ1dτdξ. (2.41)

We then have,

Q33 ≤ C‖G‖L2τ,ξ‖F−1(|v(ξ1, τ1)|ξ1|ρ+b′−1|)〈τ1 − ξ3

1〉−b‖L4x,t

‖F−1(|v(ξ2, τ2)|ξ2|2b′−1|)〈τ2 − ξ3

2〉−b‖L4x,t. (2.42)

Thanks to (2.25), this is bounded by

Q33 ≤ C‖v‖X−1/8+ρ+b′−1,0‖v‖X−1/8+2b′−1,0 ≤ c‖v‖Xρ− 1

2 ,0(2.43)

provided b′ ≤ 1/2 + 1/16 = 9/16. The proof of the proposition is complete.

3. Well-Posedness and Absorbing Sets

In this section, we use the approaches in [5, 15, 4] to obtain the local well-posedness ofthe equation (1.1). Then, we prove the L2 energy equation for the solutions and obtain theglobal well-posedness and the existence of bounded absorbing sets in L2(R).

3.1. Local Well-Posedness. We consider fixed but arbitrary −1/2 < s ≤ 0 and 1/2 < b <

b′ ≤ 9/16. Assume γ ∈ R and f = f(x, t) ∈ Xs,b−1[−Tf ,Tf ], for some Tf > 0.

For each u0 ∈ Hs(R), we look for a local solution of (1.1) in the mild sense [11] on aninterval [−T, T ], 0 < T ≤ 1 sufficiently small, as the fixed point in Xs,b of the map Σ(u)given by

Σ(u) = ψ1(t)W (t)u0

+1

2ψT (t)

∫ t

0

W (t− s)[2ψTf (s)f − 2γψ1(s)u(s)− ((ψT (s)u(s))2)x

]ds, (3.1)


By using the estimates from the previous section we find, chosing b, b′ such that ε = b′ −3b+ 1 > 0,

‖Σ(u)‖Xs,b ≤ ‖ψ1‖Hbt (R)‖u0‖Hs

x(R) + c1‖f‖Xs,b−1[−Tf ,Tf ]

+ γc1T‖ψ1‖Hbt (R)‖u‖Xs,b +

1

2c1c′′sc

22T

ε‖u‖2Xs,b .

and

‖Σ(u)− Σ(v)‖Xs,b ≤ γc1T‖ψ1‖Hbt (R)‖u− v‖Xs,b +

1

2c1c′′sc

22T

ε‖u+ v‖Xs,b‖u− v‖Xs,b .

Then, for T > 0 sufficiently small, depending in particular on ‖u0‖Hs(R) (and also on‖f‖Xs,b−1

[−Tf ,Tf ], s, b, and b′), the map Σ is a strict contraction on a closed ball of Xs,b cen-

tered at the origin. Hence, there exists a unique fixed point u, which is a mild solution of(1.1) on the interval [−T, T ]. Moreover, the following bound holds:

‖u‖Xs,b[−T,T ]

≤ c(‖u0‖Hs(R) + ‖f‖Xs,b−1[−Tf ,Tf ]

), (3.2)

for some constant c independent of the data of the problem (we bound ‖ψ1‖Hbt (R) by ‖ψ1‖H1

t (R),

which is independent of b).Now, note that if 1/2 < b < 25/48, then it is possible to choose b′ such that b ≤ b′ ≤ 9/16

and b′ − 3b+ 1 > 0, as required in the calculations above. Then, we obtain

Theorem 3.1. Let γ ∈ R and f ∈ Xs,b−1[−Tf ,Tf ], with −1/2 < s ≤ 0, 1/2 < b < 25/48, and

Tf > 0. Let T = T (‖u0‖Hs(R)) be as described above. Then, for each u0 ∈ Hs(R) there

exists a unique solution u in Xs,b[−T,T ] of the equation (1.1). Moreover, t 7→ u(t) belongs

to Cb([−T, T ];L2(R)) and the map which associates (γ, f, u0) to the corresponding unique

solution is continuous from R × Xs,b−1[−Tf ,Tf ] × Hs(R) into Xs,b

[−T ′,T ′] ∩ C([−T ′, T ′];L2(R)), for

any 0 < T ′ < T (‖u0‖Hs(R)).

Theorem (3.1) applies, in particular, to the case where f is time independent and belongsto Hs(R).

3.2. Global solutions and energy-type equations. We want to establish the globalexistence of the solutions obtained in the previous section. This is achieved with the help ofone of the invariants of the KdV, namely,

I0(u) = ‖u‖2L2(R) =

∫R

u(x)2 dx, (3.3)

This is just one of a countable number of invariants for the KdV equation (see R. M. Miura,C. S. Gardner, and M. D. Kruskal [20], for instance). Upon introducing dissipation andexternal forcing those integrals are no longer invariant but lead to energy-type equationswhich are crucial for proving global existence of the solutions. For a smooth initial conditionu0 ∈ C∞c (R), and a smooth forcing term f ∈ C∞c (R), the local solution u ∈ X0,b

[−T,T ] given by

Theorem 3.1, for a small T > 0, coincides with the classical solution, which exists globally


and belongs to C∞(R× R). By multiplying equation (1.1) by 2u we see that u satisfies theL2 energy-type equation

d

dt‖u(t)‖2

L2(R) + 2γ‖u(t)‖2L2(R) = 2(f , u(t))L2(R), (3.4)

for all t ∈ R. We integrate (3.4) to find

‖u(t)‖2L2(R) + 2γ

∫ t

0

‖u(s‖2L2(R) ds = ‖u0‖2

L2(R) + 2

∫ t

0

(f , u(s))L2(R) ds, (3.5)

for t ∈ [−T, T ]. Now, we consider approximations of u0 ∈ L2(R) and f ∈ L2(R) by smooth

functions u0 and f converging to u0 and f in L2(R), respectively. By the continuity withrespect to the data of the local solution given by Theorem 3.1, we have that the solutionsu with initial condition u(0) = u0 and forcing term f converge in X0,b

[−T,T ], for all T > 0, to

the solution u ∈ X0,b[−T,T ] with initial condition u(0) = u0 and forcing term f . By taking the

limit in (3.5) and using the continuity of the solution with respect to the data, in particularusing that

‖u(t)‖2L2(R) = lim ‖u(t)‖2

L2(R), and (f, u(t))L2(R) = lim(f , u(t))L2(R),

for all t ∈ [−T, T ], which follow from the embedding (2.7), we find that

‖u(t)‖2 + 2γ

∫ t

0

‖u(s)‖2L2(R) ds = ‖u0‖2

L2(R) + 2

∫ t

0

(f, u(s))L2(R) ds, (3.6)

for all t ∈ [−T, T ]. From the energy-type equation (3.6) one can extend the solution u indef-

initely and obtain a global solution u = u(t), t ∈ R, with u ∈ X0,b[−T,T ] ∩ Cb([−T, T ], L2(R))

for all T > 0. One can also check that for each T > 0 and each initial condition u0 ∈ L2(R),there exists a constant C = C(‖u0‖L2(R), T ) such that

‖u‖X0,b[−T,T ]

≤ C(‖u0‖L2(R), ‖f‖L2(R), γ, T ). (3.7)

This can be obtained by dividing each interval [−T, T ] into sufficiently small subintervals, asrequired in the proof of local existence, and by using the estimate provided by the energy-type equation (3.6) for the norm of the solution in L2(R) at each instant of time. We omitthe details since this is straightforward and classical. The energy-type equation (3.6) holdsfor all time, and the continuity of the solutions with respect to the data can be extended toall large times, as well. Hence, we have the following result:

Theorem 3.2. Let γ ∈ R, f ∈ L2(R), and u0 ∈ L2(R). Then, there exists a solution

u ∈ C(R, L2(R)) of equation (1.1) which is the unique solution which belongs to X0,b[−T,T ], for

all T > 0 and all 1/2 < b < 25/48. Moreover, the solution t 7→ u(t) satisfies the energyequation

d

dt‖u(t)‖2

L2(R) + 2γ‖u(t)‖2L2(R) = 2(f, u(t))L2(R), (3.8)


for almost every t in R. Furthermore, the map which associates the data (γ, f, u0) to

the corresponding unique solution u is continuous from R × L2(R) × L2(R) into X0,b[−T,T ] ∩

C([−T, T ];L2(R)) for all T > 0, with, in particular,

‖u‖X0,b[−T,T ]

≤ C(γ, ‖f‖L2(R), ‖u0‖L2(R), T ), (3.9)

for some constant C depending monotonically on the data.

Thanks to Theorem 3.2 we can define a group associated with equation (1.1):

Definition 3.1. For γ ∈ R and f ∈ L2(R) fixed, we denote by {S(t)}t∈R the group in L2(R)defined by S(t)u0 = u(t), where u = u(t) is the unique solution of (1.1) which belongs to

X0,b[−T,T ] for all T > 0.

3.3. Bounded absorbing sets. From this section on we are interested in the long timebehavior of equation (1.1) taking the dissipation into account. Therefore, we assume thatγ > 0. We also assume that the forcing term f belongs to L2(R). We want to obtain theexistence of bounded aborbing sets for the solution operator {S(t)}t∈R. This is achieved withthe help of the energy-type equation proved in the previous section.

By applying Cauchy-Schwarz’ and Young’s inequalities to the term on the right hand sideof (3.8), we see that

d

dt‖u(t)‖2

L2(R) + γ‖u(t)‖2L2(R) =

1

γ‖f‖2

L2(R).

Therefore, upon integrating in time,

‖u(t)‖2L2(R) ≤ ‖u(t0)‖2

L2(R)e−γ(t−t0) +

1

γ2‖f‖2

L2(R)(1− e−γ(t−t0)) (3.10)

whence we deduce that

lim supt→∞

‖u(t)‖L2(R) ≤ ρ0 ≡1

γ‖f‖L2(R), (3.11)

uniformly for u0 bounded in L2(R). Thus, we have proved the following result:

Proposition 3.1. Let γ > 0 and f ∈ L2(R). Then, the solution operator associated withequation (1.1) possesses a bounded absorbing set in L2(R), with the radius of absorbing ballgiven according to (3.11).

4. Splitting of the Solutions

We now consider the equations

vt + vvx + vxxx + γv = f − PN((vw)x + wwx), (4.1)

wt +QN(wwx) + wxxx + γw = −QN(vw)x, (4.2)

with the initial conditions

v|t=0 = PNu0, w|t=0 = QNu0. (4.3)


First, we use that v = u−w to write the equation for w without explicit use of v. Whence,we deduce the global existence of w and the decay in time of w(t) in L2(R). Then, the globalexistence of v follows, and we prove the regularity of v in H3(R) and the L2 energy equationfor v.

4.1. Well-posedness and decay of the w part of the solution. Using that v = u−w,we can write the equation (1.4) for w without explicit reference to v:

wt −QN(wwx) + wxxx + γw = −QN(uw)x, (4.4)

with the initial condition

w|t=0 = QNu0. (4.5)

We will use the global estimates for u to show that w(t) decays in L2(R), as t increases, aslong as the solution is defined. Then, we conclude that w(t) is defined for all positive timesand that it decays exponentially to zero in L2(R), as t goes to infinity.

The local well-posedness of the equation for w follows as that for equation (1.1). For thatwe use the fact that the solution u belongs (locally in time) to X0,b. We consider moregeneral initial conditions of the form

w|t=t0 = w0 ∈ L2(R), (4.6)

where t0 ∈ R is arbitrary, and with QNw0 = w0. Then, proceeding with the fixed pointargument, we find a solution w of the equation

w = ψ1(t)W (t)w0 +1

2ψT (t)

∫ t

0

W (t− s) [−2γψ1(s)w(s)

−(QN(ψT (s)u(s)ψT (s)w(s)))x − (QN(ψT (s)w(s))2)x]ds. (4.7)

By applying the estimates from Section 2 we find, for ε = b′ − 3b+ 1 > 0,

‖w‖X0,b ≤ ‖ψ1‖Hbt (R)‖w0‖L2

x(R)

+ γc1T‖ψ1‖Hbt (R)‖w‖X0,b +

1

2c1c′′sc

22T

ε‖w‖2X0,b + c1c

′′sc

22T

ε‖u‖X0,b‖w‖X0,b .

Hence, for T sufficiently small, we obtain the bound

‖w‖X0,b ≤ c‖w0‖L2(R),

for 1/2 < b < 25/48. Since w coincides with the solution w of (4.4) and (4.6) locally in timearound the origin, we see that

‖w‖X0,b[−T,T ]

≤ c‖w0‖L2(R). (4.8)

We may repeat the argument above for an interval centered at a different “initial” time t0to obtain

‖w‖X0,b[t0−T,t0+T ]

≤ c‖w(t0)‖L2(R), (4.9)


for every t0 in the interval of definition of w, and for

T = T1(‖w(t0)‖L2(R), ‖u(t0)‖L2(R), ‖f‖L2(R), γ). (4.10)

The decay of w is then obtained with the help of the bilinear estimate (2.28). By takingthe inner product of the equation (4.4) with 2w in L2(R), we find

d

dt‖w(t)‖2

L2(R) + 2γ‖w(t)‖2L2(R) = (ux(t), w

2(t))L2(R).

We can split the second term in the left hand side above into two equal parts and integratethe equation with one of the terms for the integrating factor. We obtain

‖w(t)‖2L2(R) = ‖w(t0)‖2

L2(R)e−γ(t−t0) +

∫ t

t0

e−γ(t−s)[(ux(s), w

2(s))L2(R) − γ‖w(s)‖2L2(R)

]ds.

Consider, for the moment, the truncation function ψ = ψ[t0,t], defined in Section 2, and thecharacteristic function χ = χ[t0,t] of the interval [t0, t]. Let b and b′′ be such that 1/2 < b <25/48 and 1− b < b′′ < 1/2. Using integration by parts and duality, we find∫ t

t0

e−γ(t−s)(ux(s), w2(s))L2(R) ds =

∫ ∞−∞

e−γ(t−s)χ(s)(ψ(s)u(s), ∂x((ψ(s)w(s))2))L2(R) ds

≤ ‖χψu‖X0,b′′‖∂x(ψw)2‖X0,−b′′ . (4.11)

From the choice of b′′ and by applying (2.28), we have the estimate

‖∂x(ψw)2‖X0,−b′′ ≤ ‖∂x(ψw)2‖X0,b−1 ≤ c√N‖w‖2

X0,b[t0,t]

. (4.12)

On the other hand, since χ belongs to Hb′′t ∩ L∞t and ψu belongs to X0,b, which is included

in L∞t L2x (see (2.7)), one can show, using (2.10), that

‖χψu‖X0,b′′ ≤ ‖χ‖L∞t (R)‖ψu‖X0,b′′ + ‖χ‖X0,b′′‖ψu‖X0,b ≤ C(χ)‖u‖X0,b[t0,t]

, (4.13)

where C(χ) depends on χ, hence on t0 and t, but is independent of N . Inserting (4.12) and(4.13) into (4.11) yields∫ t

t0

e−γ(t−s)(ux(s), w2(s))L2(R) ds ≤

C(χ)

N1/2‖u‖X0,b

[t0,t]‖w‖2

X0,b[t0,t]

.

Then, using (4.9) (with t0 = s there),∫ t

t0

e−γ(t−s)(ux(s), w2(s))L2(R) ds ≤

C(χ)

N1/2‖u‖X0,b

[t0,t]

(1

t− t0

)∫ t

t0

‖w‖2

X0,b[t0,t]

ds

≤ C(χ)

N1/2‖u‖X0,b

[t0,t]

(1

t− t0

)∫ t

t0

‖w(s)‖2L2(R) ds.

Thus,

‖w(t)‖2L2(R) ≤ ‖w(t0)‖2

L2(R)e−γ(t−t0) +

∫ t

t0

(C(χ)

(t− t0)√N‖u‖X0,b

[t0,t]− e−γ(t−s)γ

)‖w(s)‖2

L2(R) ds.


For N large enough, the second term in the right hand side above is negative, and, hence,

‖w(t)‖2L2(R) ≤ ‖w(t0)‖2

L2(R)e−γ(t−t0). (4.14)

This now can be iterated and shown to hold for all t ≥ t0 ≥ 0 in the interval of definitionof w. Then, we see that the solution w = w(t) can be extended to all positive times and,moreover, (4.14) holds for all t ≥ t0 ≥ 0. In particular, with t0 = 0 and w(t0) = QNu0, wefind

‖w(t)‖2L2(R) ≤ ‖QNu0‖2

L2(R)e−γt, ∀t ≥ 0. (4.15)

The analysis above should actually be done for smooth u0 and f , but, by density and bythe continuity of the solutions with respect to the initial condition, the final bounds (4.14)and (4.15) hold for arbitrary u0 and f in L2(R).

4.2. Regularity of the v part of the solution. Since u and w are defined globally intime, so is v = u− w. We now prove an H3

x(R) bound for v.We first observe that y ≡ PNv = PNu is smooth and satisfies

lim supt→∞

‖y(t)‖H3x(R) ≤ N3, lim sup

t→∞‖u(t)‖L2

x(R) ≤ ρ0N3, (4.16)

where ρ0 is the radius of the absorbing ball given in (3.11). We focus on an H3x(R) estimate

for Z ≡ QNv, which is solution to

Zt + Zxxx +QN((y + Z)(y + Z)x) + γZ = QNf (4.17)

with initial condition Z(0) = 0.We know that Z = QNv remains bounded in L2

x(R). Then, from (4.17),

‖Zx‖L2x(R) ≤ K(‖Zx‖H−2

x (R) + ‖Zxxx‖H−2x (R)) ≤ K(‖Zt‖H−2

x (R) + ‖(y + Z)2‖H−1x (R) + 1)

≤ K(1 + ‖Zt‖1/3

L2x(R)), (4.18)

where we used interpolation between L2x(R) and H−3

x (R), the fact that Zt remains boundedin H−3

x (R), and the inequality (2.3). The coefficient K denotes a “constant” which maydepend on γ, ‖f‖L2(R), and ‖u0‖L2(R), and which may increase from inequality to inequality.

Now, to obtain the H3x(R) bound for Z is equivalent to prove an L2

x(R) estimate on Z ′ = Zt,which solves

Z ′t + Z ′xxx + 2QN∂x((y + Z)Z ′) + γZ ′ = −2QN∂x((y + Z)yt), (4.19)

with initial condition Z ′(0) = QNf −QNy(0)yx(0) in L2x(R).

Multiply this equation by 2Z ′ and integrate over R with respect to x to obtain, after usingYoung’s inequality,

d

dt‖Z ′‖2

L2 + γ‖Z ′‖2L2 ≤

4

γ‖∂x(yt(y + Z))‖L2

x(R) + 2

∫R

(y + Z)∂x(Z′)2 dx (4.20)


Using (2.2) and Young’s inequality, and then (4.18) and inverse (Poincare-type) inequali-ties (for functions with bounded spatial frequency), we find

‖∂x(yt(y + Z))‖L2x(R) ≤ ‖yt‖H1

x(R)‖y + Z‖H1x(R) ≤ K(N)(1 + ‖Z ′‖1/3

L2x(R)), (4.21)

where, as in (4.18), K(N) depends on γ, ‖f‖L2(R), ‖u0‖L2(R), and it is now allowed to dependon N , as well.

We integrate over [0, t] to obtain

‖Z ′(t)‖2L2xeγt ≤ ‖Z ′(0)‖2

L2x

+

∣∣∣∣∫R×[0,t]

eγs{(y + Z)∂((Z ′)2) +K(N)} dxds∣∣∣∣ . (4.22)

We proceed as in (4.8)-(4.14), using the bilinear estimate (2.28) and the local well-posednessof (4.19) in L2, to obtain, for t small enough,

‖Z ′(t)‖2L2x≤ K(N) +

1

N2‖Z ′‖2

X0,b[0,t]

, (4.23)

for 1/2 < b < 25/48, and then conclude that

‖Z ′(t)‖2L2x(R) ≤ K(N), ∀t ≥ 0.

Therefore, there exists K = K(N, ‖u0‖L2x(R)) > 0 such that

‖v(t)‖H3x(R) ≤ K(N, ‖u0‖L2

x(R)), ∀t ≥ 0. (4.24)

4.3. Energy equation for v. Since v(t) belongs to H3x(R), we may take directly the inner

product in L2x(R) of the equation (4.1) for v with 2v to find that

d

dt‖v‖2

L2(R) + 2γ‖v‖2L2(R) = 2(f, v)L2(R) + (PN(2vw + w2), vx)L2(R). (4.25)

Integrating this equation in time yields

‖v(t)‖2L2(R) = ‖v(t0)‖2

L2(R)

+

∫ t

t0

e−2γ(t−t0)[2(f, v(s))L2(R) + (PN(2v(s)w(s) + w2(s)), vx(s))L2(R)

]ds. (4.26)

5. Asymptotic Smoothing and the Global Attractor

The first step is to prove the asymptotic compactness of the group in L2(R). This is doneby showing that for a bounded sequence of initial conditions {u0n}n in L2(R) and a sequenceof positive numbers tn →∞, the solutions un(tn) = vn(tn)+wn(tn) are precompact in L2(R),with wn(tn) decaying to zero in L2(R) and vn(tn) being precompact in L2(R) and weaklyprecompact in H3(R). This will give us the existence of the global attractor A in L2(R),and, at the same time, the boundedness of A in H3(R).

Then, we work with the equations for u′n = dun/dt and we show, using the energy equationmethod applied to u′n, that with the initial conditions {u0n} belonging to A (and, hence,bounded in H3(R)), the sequence u′n(tn) is precompact in L2(R). This implies, from theequation for u, that un(tn) is precompact in H3(R). This shows that the flow restricted to


the global attractor is asymptotically compact in H3(R) and, hence, that the global attractoris compact in H3(R).

5.1. Existence of the global attractor in L2(R). Let {u0n}n be bounded in L2(R) andlet {tn}n be a sequence of positive real numbers going to infinity. Let vn and wn be thesolutions associated with each initial condition u0n; they are defined for all time t ≥ 0. FromSection 4.2, we have that

{vn(tn + ·)}n is bounded in C([−T, T ];H3(R)), (5.1)

and, for the time-derivative,{dvndt

(tn + ·)}n

is bounded in C([−T, T ];L2(R)), (5.2)

for each T > 0 (and starting with n sufficiently large so that tn − T ≥ 0). This implies,by the Arzela-Ascoli Theorem, that {vn(tn + ·)} is precompact in C([−T, T ];L2

loc(R)), forevery T > 0. Then, using again (5.1), we see by interpolation that {vn(tn + ·)} is actuallyprecompact in C([−T, T ];Hs

loc(R)), for all 0 ≤ s < 3. Thus, by a diagonalization process, weobtain a subsequence such that

vnj(tnj + ·)→ u(·) strongly in C([−T, T ];Hsloc(R)), ∀s ∈ [0, 3),

weakly star in L∞([−T, T ];H3(R)), ∀T > 0,(5.3)

Moreover, since ut belongs to L∞([−T, T ];L2(R)), then, for any s < 3,

u ∈ C(R;Hs(R)), ‖u(t)‖H3(R) ≤ C(ρ,N), ∀t ∈ R; (5.4)

the uniform bound follows from the boundeness of v in H3(R) (see (4.24)). In fact, u isweakly continuous with values in H3(R) (by the Strauss theorem). We also find that

vnj(tnj + t) ⇀ u(t) weakly in H3(R), for every t ∈ R. (5.5)

From (4.15), we find that

‖wn(tn + t)‖L2(R) → 0, uniformly for t ≥ −T, ∀T > 0. (5.6)

With (5.3) and (5.6), one can pass to the limit in the weak formulation of the equationfor vnj to find that u is a solution of the weakly damped, forced KdV equation (1.1), thatmoreover satisfy the energy equality (3.6).

We now write the integral form (4.26) of the L2 energy equation for vn with t = tn andt0 = tn − T :

‖vn(tn)‖2L2(R) = e−2γT‖vn(tn − T )‖2

L2(R)

+

∫ T

0

e−2γ(T−s) [2(f, vn)L2(R) − (PN(2vnwn − w2n), vnx)L2(R)

]ds, (5.7)


where, for notational simplicity, we omitted the argument tn − T + s of the functions insidethe time integral. By using the uniform bound for v in H3(R), the decay (5.6) of wn, andthe weak-star limit of vnj in (5.3), we find

lim supj→∞

‖vnj(tnj)‖2L2(R) ≤ C(R,N)e−2γT + 2

∫ T

0

e−2γ(T−s)(f, u(−T + s))L2(R) ds. (5.8)

By substituting for the L2 energy equation (3.8) for u (see (3.6)), we obtain

lim supj→∞

‖vnj(tnj)‖2L2(R) ≤ 2C(R,N)e−2γT + ‖u(0)‖2

L2(R). (5.9)

Let T go to infinity to find that

lim supj→∞

‖vnj(tnj)‖2L2(R) ≤ ‖u(0)‖2

L2(R). (5.10)

This, together with the weak convergence (5.5) and the decay (5.6), implies that

unj(tnj) = vnj(tnj) + wnj(tnj)→ u(0) strongly in L2(R)),weakly in H3(R).

(5.11)

This shows that the solution operator is asymptotically compact in L2(R) and, hence, thereexists a global attractor A in L2(R). Moreover, it also follows that A is a bounded set inH3(R). By interpolation, A is compact in any Hs(R), for 0 ≤ s < 3. It remains to showthat A is compact in H3(R).

5.2. Compactness of the global attractor in H3(R). For the compactness in H3(R),we restrict the flow to the global attractor, which is bounded in H3(R), and we show that theflow is asymptotically compact in H3(R). For that purpose, we assume that the sequence ofinitial conditions {u0n}n belongs to A. Since the global attractor is invariant and is boundedin H3(R), as shown above, the corresponding trajectories un(t) = S(t)u0n belong to and areuniformly bounded (w.r.t. n and t) in H3(R), for all t ∈ R. We want to show that unj(tnj)converges to u(0) in H3(R). For that purpose, we use the equation for u′n = dun/dt :

u′nt + (unu′n)x + u′nxxx + γu′n = 0. (5.12)

From the equation for un, we see that proving that unj(tnj) converges to u(0) in H3(R)amounts to proving that u′nj(tnj) converges to u′(0) strongly in L2(R). Since the trajectories

{un(tn + ·)}n are uniformly bounded in H3(R) and converge strongly in C([−T, T ];L2(R)) tou, we see that

u′nj(tnj + ·) ⇀ u′(·) strongly in C([−T, T ];H−s(R)), ∀s ∈ [0, 3),

and weakly star in L∞([−T, T ];L2(R)), ∀T > 0. (5.13)

Now, we consider the L2 energy equation for u′n :

‖u′n(tn)‖2L2(R) = e−2γT‖u′n(tn − T )‖2

L2(R) −∫ T

0

e−2γ(T−s)(unx, u′n

2)L2(R) ds (5.14)


where, for notational simplicity, we omitted the argument tn − T + s of the functions insidethe time integral. We plan to pass to the limit in (5.14). On the one hand, due to (5.13),when j goes to ∞, ∫ T

0

e−2γ(T−s)((unj)x − ux, u′2)L2(R) ds→ 0. (5.15)

On the other hand, for 0 < s ≤ 23/48 and 1/2 < b < 25/48, we take b′′ such that 1 − b <b′′ < 1/2 and proceed as in (4.11)-(4.13) (but with −s, and using (2.27) with b′ = b, insteadof (2.28)) to find∣∣∣∣∫ T

0

e−2γ(T−s)(unjx, u′nj

2 − u2)L2(R) ds

∣∣∣∣ ≤ c‖unj‖Xs,b[0,T ]‖u′nj − u

′‖2

X−s,b[0,T ]

. (5.16)

From (5.13) and the well-posedness of (5.12) in X−s,b[0,T ] (see Theorem 3.1), we can pass to the

limit as j goes to inifity and thus obtain

limj→∞

∫ T

0

e−2γ(T−s)(unjx(tnj − T + s), u′nj(tnj − T + s)2)L2(R) ds

=

∫ T

0

e−2γ(T−s)(ux(−T + s), u′(−T + s)2)L2(R) ds. (5.17)

Then, from (5.14),

lim supj→∞

‖u′nj(tnj)‖2L2(R) ≤ C(ρ)e−2γT

−∫ T

0

e−2γ(T−s)(ux(−T + s), u′(−T + s)2)L2(R) ds. (5.18)

By substituting for the corresponding L2 energy equation for u′, we find

lim supj→∞

‖u′nj(tnj)‖2L2(R) ≤ 2C(ρ)e−2γT + ‖u′(0)‖2

L2(R). (5.19)

We let T go to infinity to find

lim supj→∞

‖u′nj(tnj)‖2L2(R) ≤ ‖u′(0)‖2

L2(R). (5.20)

Therefore,

u′nj(tnj)→ u′(0) strongly in L2(R). (5.21)

As mentioned before, this implies the convergence in H3(R) of unj(tnj) to u(0), which provesthe desired asymptotic compactness in H3(R) and, hence, the compactness of A in H3(R).


5.3. Conclusion. We have shown the following result:

Theorem 5.1. Let γ > 0 and f ∈ L2(R). Then, the solution operator {S(t)}t∈R in L2(R)associated with equation (1.1) possesses a connected global attractor A in L2(R) which iscompact in H3(R). More precisely, A is a connected and compact set in H3(R); it is invariantfor the system; it attracts (in the L2(R)-metric) all the orbits of the system uniformly withrespect to bounded sets (in L2(R)) of initial conditions; and (with respect to the inclusionrelation) A is maximal among the bounded invariant sets and minimal among the globallyattracting sets.

Similarly, one can show that

Theorem 5.2. Let γ > 0 and f ∈ Hk(R), where k ∈ N. Then, for each m = 0, 1, . . . , k, thesolution operator {Sm(t)}t∈R associated with the equation (1.1) in the phase space Hm(R) iswell-defined and possesses a connected global attractor A in Hm(R), which is the same forall m = 0, 1, . . . , k. Moreover, the global attractor A is compact in Hk+3(R).

6. L2-Weak Continuity of the Solution Operator

As we mentioned in the Introduction, the weak continuity of the solution operator isusually a key issue in the proof of the existence of the global attractor in noncompact systemsvia the energy equation method. We also mentioned that, in the present case, since we alsouse a splitting of the semigroup to obtain the regularity of the global attractor, the weakcontinuity property is actually not needed (it is replaced by an asymptotic weak continuityproperty; see Section 5.1). Nevertheless, the weak continuity is an interesting property byitself. Therefore, we present now a sketch of its proof.

In higher order Sobolev spaces, the weak continuity is relatively easy to prove (see, how-ever, [25], where some difficulties already appear in H1(R)). This is not the case in L2(R),where there are some delicate issues to overcome.

Since we are interested in the weak continuity locally in time, we can take for simplicityγ = 0. The proof for γ ∈ R is similar. Moreover, since the solutions are bounded in L2(R)on a finite interval of time (globally in time for γ = 0), we can also consider a suitably smallinterval of time depending on the L2(R)-norm of the initial condition. This result can thenbe iterated to yield the weak continuity on arbitrarily large intervals of time.

Consider uε0 that converges weakly to u0 in L2(R). Consider u(t) that is solution to

ut + uxxx + uux = 0, (6.1)

with u(0) = u0. This solution is understood in the mild sense of Bourgain-KPV solution toKdV.

Consider uε(t) that solves (6.1) with initial data uε0. We prove that for t small enough(without loss of generality) uε(t) converges weakly to u(t) in L2

x(R).First step: Fix N and consider the projector PN defined by F(PNu)(ξ) = u(ξ)χ( ξ

N), where

χ is the characteristic function of the interval [−1, 1]. Consider vε,N(t) that solves

vt + vxxx + vvx = 0, (6.2)


with initial condition v(0) = PNuε0. For N fixed, we know [27] that vε,N(t) is bounded in the

space

C([−T, T ], H2x(R)) ∩ C1([−T, T ], H−1

x (R)),

since PNuε0 is as smooth as we want and since

‖PNuε0‖H2x(R) ≤ KN2, (6.3)

where K is independent of ε.Hence if L2

x,loc(R) denotes the Frechet space endowed with its natural topology (L2x con-

vergence on bounded sets), vε,N(t) strongly converges to vN in C([−T, T ], L2x,loc(R)) . We

can let ε→ 0 in (6.2) (convergence in D′), and vN is solution to (6.2) with initial data PNu0.On the other hand, the limit vN belongs to L∞([−T, T ], H2

x(R)) (weak-star convergence)and vNt belongs to L∞([−T, T ], H−1

x (R)). Therefore, vN belongs to C([−T, T ], Hsx(R)) for

each s < 2 (and is even weakly continuous in t with values in H2x(R)).

By uniqueness of solution of (6.2) in C([−T, T ], Hsx(R)), for s > 3/2, vN is the usual

solution of (6.2) with initial data PNu0.Second step: Now, wε,N(t) = uε(t)− vε,N(t) is solution to

wt + wxxx + wwx + (vw)x = 0, (6.4)

with initial data w(0) = (Id− PN)uε0 (ε = 0 included).We can prove that on bounded intervals of time, for any s ∈ [0, 3/4),

‖w(t)‖H−s(R) ≤ K‖w(0)‖H−s(R). (6.5)

Here K depends on T but is independent of N and ε.

In fact, (6.5) comes from the well-posedness of KdV equation in Bourgain spaces X−s,1/2+

loc

of negative order.Observe also that since uε0 is bounded in L2(R),

‖w(0)‖H−s(R) = ‖(Id− PN)uε0)‖H−s(R) ≤‖uε0‖L2(R)

N s≤ KN−s. (6.6)

Conclusion: We have, N being fixed, for a test function ψ that is smooth, compactly sup-ported and that satisfies ‖ψ‖L2(R) = 1,

|(u(t)− uε(t), ψ)L2(R)| = |(u(t)− vN(t) + vN(t)− vε,N(t) + vε,N(t)− uε, ψ)L2(R)| (6.7)

= |(w0,N(t) + vN(t)− vN,ε(t) + wε,N(t), ψ)L2(R)| ≤ (6.8)

≤ 2KN−s + |(vN(t)− vε,N(t), ψ)L2(R)|, (6.9)

where we used (6.6). Let ε→ 0 with N fixed to obtain, from the first step,

lim supε→0

|(vN(t)− uε(t), ψ)L2(R)| ≤ 2KN−s. (6.10)

Let N → +∞ to conclude that uε(t) converges to u(t) in the distribution sense. We removethe condition “ψ is smooth and compactly supported” by a density argument, since wε,N isa bounded sequence in L∞([−T, T ], L2(R)). This concludes the proof.


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(Olivier Goubet) Universite de Picardie Jules, LAMFA, Faculte de Mathematiques et In-

formatique, 33 rue Saint-Leu, 80039 Amiens Cedex, FRANCE

E-mail address: [email protected]

(Ricardo M. S. Rosa) Departamento de Matematica Aplicada, Instituto de Matematica, Uni-

versidade Federal do Rio de Janeiro, Caixa Postal 68530 Ilha do Fundao, Rio de Janeiro

RJ 21945-970, BRAZIL

E-mail address: [email protected]

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