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

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1 ASYMPTOTIC SMOOTHING AND THE GLOBAL ATTRACTOR OF A WEAKLY 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-de Vries equation in the phase space L 2 (R) is proved. An optimal asymptotic smoothing effect of the equation is also shown, namely, that for forces in L 2 (R), the global attractor in the phase space L 2 (R) is actually a compact set in H 3 (R). The energy equation method is used in conjunction with a suitable splitting of the solutions; the dispersive regularization properties 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 phase space L 2 (R) for the following weakly damped, forced Korteweg-de Vries equation: u t + uu x + u xxx + γu = f, for (x, t) R R. (1.1) This equation is supplemented with the initial condition u t= = u L 2 (R). (1.2) It is assumed that f is time-independent and belongs to L 2 (R), and that γ > is a constant. We show that the global attractor exists in L 2 (R) and is a compact set in H 3 (R), thus proving an asymptotic smoothing effect (in the terminology used by A. Haraux [13]) since the solutions, in general, belong only to L 2 (R). Equation (1.1) has been derived by E. Ott and R. N. Sudan [23] as a model for ionsound waves damped by ion-neutral collisions. For γ = and f =, equation (1.1) is the well-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, or smoothing, property. The asymptotic smoothing of the global attractor comes essentially from the dispersive regularization property of the equation. Date: April 3, 21 (submitted on July 18, 2). 2 Mathematics Subject Classification. Primary: 35Q53, 35B4, 37L5; Secondary: 35L3, 37L3. 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, Brasília, Brazil, by FAPERJ and FUJB, Rio de Janeiro, Brazil, and by the Research Fund of the Indiana University and of the NSF - DMS grant The second author also acknowledges the hospitality of the Laboratory of Numerical Analysis at the Université de Paris-Sud at Orsay. 1

2 2 OLIVIER GOUBET AND RICARDO M. S. ROSA For the well-posedness of (1.1) in L 2 (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 in L 2 (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-posedness in L 2 (R) of the forced equation without the weak dissipation (i.e., with γ = ). The wellposedness 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 of the 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 the asymptotic compactness or the asymptotic smoothing properties of the solution operator together with the existence of a bounded absorbing set [12, 28, 18, 1, 24]. Those properties are usually proved by splitting the solutions into a decaying part plus a regular part, or by exploiting suitable energy-type equations, or both. For the energy equation method, the weak continuity of the solutions with respect to the initial condition (in the sense that if the initial conditions u n converge to u weakly, then the corresponding solutions u n (t) converge weakly to u(t) at all times t) is a crucial step. This approach was used, for instance, in the proof of the existence of the global attractor for (1.1) in H 1 (R) [25], in H 2 (R) [22, 19], and in the space-periodic case [7, 21, 1], as well as for several other equations (e.g., [3, 9]). The energy equation method can also be applied to (1.1) in L 2 (R). Here, however, since we also want to prove the regularity of the attractor, we use this energy equation method together with a splitting of the solutions to obtain, at the same time, the existence of the global attractor in L 2 (R) and its boundedness in H 3 (R). This is achieved by splitting the solutions into two parts, one which is regular (it belongs to H 3 (R)) and the other which decays to zero (in L 2 (R)) as time goes to infinity. In this way, the weak continuity of the solution operator just mentioned is actually not used, and it is replaced by an asymptotic weak continuity property (in the spirit of [22]). Nevertheless, the weak continuity is an interesting property by itself, so we include its proof at the end. Because of the lack of the compact Sobolev embeddings in the unbounded case, the splitting alone is not enough to prove the asymptotic compactness of the solution operator, and the energy equation method is used. After obtaining the boundedness in H 3 (R) of the regular part of the solution, the compactness of the global attractor in H 3 (R) is obtained by applying the energy equation method to the equation for the time derivative of the solutions in the global atttractor. From the equation (1.1) and from the first part of the proof, we see that the time derivatives are in L 2 (R) since the global attractor is bounded in H 3 (R). From the energy equation method, we obtain, loosely speaking, the L 2 (R) compactness for the time derivatives. Then, going back to the equation (1.1), this gives the compactness of the global attractor in H 3 (R). The major difficulty here comes from the fact that the time derivative of the solutions is just in L 2 (R), which makes the nonlinear term in the energy equation for the time derivative more difficult to handle. This term is handled by using the subtle dispersive regularization properties of the equation in the context of the Bourgain spaces.

3 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 uu x = vv x + P N ((vw) x + ww x ) + Q N ((vw) x + ww x ), where P N denotes the Fourier spectral projector associated with a cut-off of the higher modes which retains only the modes with spatial frequency ξ N, for N >. The operator Q N is the complement Q N = I P N. Then, we obtain and with the initial conditions v t + vv x + v xxx + γv = f P N ((vw) x + ww x ), (1.3) w t + Q N (ww x ) + w xxx + γw = Q N (vw) x, (1.4) v t= = P N u, w t= = Q N u. (1.5) Due to the cut-off operator P N in the right hand side of the equation for v and in the initial condition, the solution v is more regular ( three times more regular than f in the scale of Sobolev spaces). As for w, there is no forcing term, and the presence of the operator Q N makes the additional linear (driving) term in the right hand side of the equation be relatively small with respect to the weakly dissipative term. Then, we obtain that w actually decays in time in the L 2 (R)-norm. The appropriate estimate of the driving term is obtained thanks to the dispersive regularization properties of the linear part of the equation in the context of the Bourgain spaces; this type of estimate has already been exploited by O. Goubet [1] in the space-periodic case for the same purpose, and it is slightly simplified here. We then show that as time t goes to infinity, u(t) = v(t) + w(t) converges strongly in L 2 (R) to a solution ū(t) on the global attractor, with ū(t) in H 3 (R), and with the regular part v(t) converging strongly in H 3 (R) to ū(t). The decaying part w(t) goes to zero in L 2 (R). The existence of global attractors for the equation (1.1) was first considered by J. M. Ghidaglia [6, 7], in the spaces Hper(, 2 L) of L-periodic functions in H 2, where it is assumed that f Hper(, 2 L). This result was extended to Hper(, k L), with k N, k 3, by I. Moise and R. Rosa [21], with f Hper(, k 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 spaces Hper(, m L), m N, m > k 3). The whole space case in the phase space H 2 (R), and assuming f also in H 2 (R), was treated independently by P. Laurençot [19], using energytype equations and weighted spaces, and by I. Moise, R. Rosa, and X. Wang [22], using only energy-type equations. The H 1 (R) case, assuming f in H 1 (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 expect that the same would happen for the global attractor. The breakthrough to obtain such asymptotic regularization was obtained recently by O. Goubet [1] by exploiting the dispersive regularization properties associated with (1.1) in the context of Bourgain spaces. The author considered the space-periodic case in the phase space L 2 (, L) and proved that indeed the global attractor exists and is actually compact in Hper(, 3 L). The method used is a splitting

4 4 OLIVIER GOUBET AND RICARDO M. S. ROSA of the solutions in a way similar to the one used here. The compact Sobolev embeddings on bounded domains is used to prove the asymptotic compactness in L 2 (, L), while the compactness of the global attractor in H 3 per(, L) is obtained by the energy equation method applied to the equation for u t. In the present work, we are able to extend this result to the whole real line. Some delicate issues appear due to the lack of the compact Sobolev embeddings, 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 forcing term f belongs to H k (R), for a given k N, then the equation (1.1) generates a group in each phase space H m (R), for m =, 1,..., k, and the global attractor exists and is the same for each group; moreover, the global attractor is a compact set in H k+3 (R). This completes the study of the existence and regularity of the global attractor for the equation (1.1) in the scale of Sobolev spaces H k (R) for integers k. For the existence of the attractor in Gevrey spaces when the force is in Gevrey, see [29]. 2. Function Spaces and Preliminary Estimates 2.1. Function Spaces. We consider the spaces L 2 (R) and H s (R), s R, with the norms denoted respectively by L 2 (R) and H s (R). The inner product in L 2 (R) is denoted by (, ) L 2 (R). Let L 2 loc (R) denote the Fréchet space of functions which are locally in L2, i.e., which belong to L 2 (J) for every compact interval J in R. We will use the following compact embedding: H 3 (R) c L 2 loc(r). (2.1) We also consider spaces of the type Cc, of infinitely differentiable functions with compact support and the Schwartz space S(R 2 ) of tempered test functions on R 2. More general spaces of the type H s, s R, and L p, 1 p, are also considered, and their norm is denoted with the appropriate subscript. We recall the Agmon inequality: u L (R) u 1/2 L 2 (R) xu 1/2 L 2 (R). (2.2) Whence we deduce the following inequality which will be used in the sequel: u 2 H 1 (R) u 2 L 2 (R). (2.3) For a given interval I and a given Banach space E we also consider the spaces L p (I; E), 1 p, of E-valued functions on I whose norm in E to the p-th power is integrable on I (or is essentially bounded if p = ), and the space C b (I; E) of bounded, continuous functions on I with values in E. Their respective norms are denoted with the appropriate subscripts as above. When I is an unbounded interval and 1 p <, it is also useful to consider the functions which are locally in L p, i.e., the space of functions which belong to L p (J; E) for every bounded subinterval J I. Those spaces are Fréchet spaces and are denoted L p loc (I; E). Similarly, we denote by C(I; E) the space of functions which belong to C b(j; E) for each bounded subinterval J I.

5 GLOBAL ATTRACTOR OF A WEAKLY DAMPED KDV EQUATION 5 An important role is played by the Bourgain space-time function spaces X s,b, for s, b R, which are defined as the completion of the Schwartz space S(R 2 ) with respect to the norm f 2 X = τ ξ 3 2b ξ 2s ˆf(ξ, τ) 2 dξdτ, (2.4) s,b where ˆf = ˆf(ξ, τ) is the Fourier transform of f = f(x, t), and R R λ = 1 + λ 2. (2.5) We will sometimes also denote the Fourier transform in R n by F: F(f)(ζ) = ˆf(ζ) 1 = f(z)e iz ζ dz. (2.6) (2π) n/2 R n The inverse Fourier transform is denoted F 1. We have the continuous embedding X s,b C b (R, H s x(r)), (2.7) for any s R and b > 1/2. On dealing with space-time functions as in (2.7), we will often use a subscript on the function spaces considered to indicate with respect to which variable that function space refers to. Similarly, we indicate by F x the Fourier transform with respect to x of a function depending on x and t. We will also need to localize in time the estimates obtained in the spaces X s,b. For that purpose, consider a function ψ Cc (R) which is equal to one on [ 1, 1] and to zero outside ( 2, 2). We set ψ T (t) = ψ(t/t ). For any given interval I = [a, b], set also ψ I (t) = ψ((2t a b)/(b a)). Then, we consider the seminorms in X s,b defined by and u X s,b = ψ T u X s,b, (2.8) [ T,T ] u X s,b I = ψ I u X s,b. (2.9) The function ψ is fixed once and for all, since some of the constants below depend on it 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 X s,b if and only if W ( t)u belongs to H b t H s x(r 2 ). We can write u X s,b = W ( )u H b t H s x (R2 ) = τ ξ 3 b ξ s û L 2 τ,ξ (R 2 ). (2.1) The following estimate is then easy to obtain: ψ T W (t)u X s,b = ψ T H b (R) u H s (R), (2.11) for every u H s (R) and s, b R. We borrow from [8, Lemma 3.2] the following result:

6 6 OLIVIER GOUBET AND RICARDO M. S. ROSA Lemma 2.1. Consider s R, < T 1, and 1/2 < b 1 b b. Let g X s,b 1, and define w(t) = ψ T (t) W (t t )g(t ) dt. (2.12) Then, there exists a constant c 1 >, independent of s, b, b and T, such that the following inequality holds: From [15, Lemma 3.2] we borrow the following result: w X s,b c 1 T b b g X s,b 1. (2.13) Lemma 2.2. There exists a constant c 2 > such that for s R, 1/2 < b 1, and u X s,b, we have ψ T u X s,b c 2 T (1 2b)/2 u X s,b. (2.14) Recall now the well-known Strichartz-type inequalities establishing dispersive-type regularizations of the free Airy group (see [14]): for any u in L 2 (R), W (t)u L 8 x,t (R 2 ) c 3 u L 2 x (R), (2.15) x W (t)u L x L 2 t (R2 ) c 3 u L 2 x (R 2 ), (2.16) ( x ) 1/6 W (t)u L 6 x,t (R 2 ) c 3 u L 2 x (R). (2.17) for some constant c 3 >. Such estimates may be turned into estimates in the Bourgain spaces with the following lemma, which we borrow from [8, Lemma 3.3]: Lemma 2.3. Let Y be a seminorm in S x,t (R 2 ) which is stable under multiplication by functions in L t (R) in the sense that ψf Y c ψ L t (R) f Y, ψ L t (R), f S x,t (R 2 ). (2.18) Assume the following (Strichartz-type) inequality holds: W ( )u Y c u L 2 x (R), u L 2 x(r). (2.19) Then, for all b > 1/2, the following inequality holds: with c b = cb 1/2 (2b 1) 1/2. f Y c b f X,b, f X,b, (2.2) Hence, using Lemma 2.3, we infer from (2.15), (2.16), and (2.17) that for any b > 1/2 and any f in X,b, f L 8 x,t (R 2 ) c b f X,b, (2.21) x f L x L 2 t (R2 ) c b f X,b, (2.22) ( x ) 1/6 f L 6 x,t (R 2 ) c b f X,b. (2.23)

7 GLOBAL ATTRACTOR OF A WEAKLY DAMPED KDV EQUATION 7 where c b depends on b > 1/2. By interpolating (2.21) with L2 x,t = X,, we obtain Similarly, interpolating (2.23), we find f L 4 x,t (R 2 ) c b f X, b 2. (2.24) ( x ) 1/8 f L 4 x,t (R 2 ) c b f X, 3b 4. (2.25) 2.3. Bilinear Estimates. We now describe an estimate needed to handle the bilinear term in the KdV equation. We first recall from [16, Theorem 1.1] the following statement: Proposition 2.1. Let s ( 3/4, ] be given. Then there exists a numerical constant c s and a number b (1/2, 1) such that for any function u in X s,b D(u 2 ) X s,b 1 c s u 2 X s,b. (2.26) We shall also use below a peculiar version of (2.26) that states as follows (we prove it for s ( 1/2, ] because we do not need it for smaller values of s): Proposition 2.2. Let s ( 1/2, ] be given. Then there exists a numerical constant c s such that for b (1/2, 9/16], for b [1/2, min{b, 1 + s}], and for any function u in X s,b, As a corollary of this result we have D(u 2 ) X s,b 1 c s u X s,b u X 1 2,b. (2.27) Corollary 2.1. Assume moreover that the Fourier transform û = û(ξ, τ) of u is supported in {(ξ, τ); ξ N}, N >. Then, for 1/2 < b 9/16, D(u 2 ) X,b 1 c N u 2 X,b. (2.28) Proof of Proposition 2.2: Set ˆv = ξ s τ ξ 3 b û, where û stands for the Fourier transform F(u) of u. By a duality argument, and by setting ρ = s, (2.27) stems from the following assertion: there exists C = c s such that for any G and v which are of norm 1 in L 2 (R 2 ), ξˆv(ξ 1, τ 1 )ˆv(ξ 2, τ 2 )G(ξ, τ) ξ 1 ρ ξ 2 ρ Q dτ τ ξ 3 1 b τ 1 ξ1 3 b τ 2 ξ2 3 b ξ ρ 1 dξ 1 dτdξ C v X ρ 1 (2.29) 2, where and D D = {σ = {ξ, τ, ξ 1, τ 1 } R 2 R 2 }, ξ 2 = ξ ξ 1, τ 2 = τ τ 1. We divide the majorization of Q into three cases: First case: ξ 1 ξ 2 1/2. Set D 1 for the subset of D where these inequalities are valid. In that case the function ξ 1 ρ ξ 2 ρ ξ / ξ ρ is bounded, and we just have to majorize Q 1 D 1 ˆv(ξ 1, τ 1 )ˆv(ξ 2, τ 2 )G(ξ, τ) τ ξ 3 1 b τ 1 ξ 3 1 b τ 2 ξ 3 2 b dτ 1dξ 1 dτdξ. (2.3)

8 8 OLIVIER GOUBET AND RICARDO M. S. ROSA Set ŵ(ξ, τ) = ˆv(ξ, τ) τ ξ 3 b χ(ξ), where χ is the characteristic function of the interval [ 1/2, 1/2]. Then, using in particular that b 1 and G has L 2 -norm one, Q 1 G(τ, ξ) τ ξ 3 b 1 L 2 ξ,τ (R) ŵ ŵ L 2 ξ,τ (R) 1 2π G(τ, ξ) L 2 ξ,τ (R) w 2 L 2 x,t (R 2 ) = 1 2π F 1 ( ˆv(ξ, τ)χ(ξ, τ) ) τ ξ 3 b 2 L 4 x,t (R), (2.31) where denotes the convolution operator. Now, we use (2.24), the embeddings X s,a X s, for a, and the following inverse inequality F 1 ( ˆv χ) X, v X ρ 1 2, (2.32) to obtain Q 1 C v X ρ 1/2,. Second case: ξ 1 1 ξ 2 2. Set D 2 for the corresponding region. In that case we use ξ 2 ξ 2 and the fact that the function ξ 1 ρ ξ 2 ρ / ξ ρ is bounded to write Q 2 C G(τ, ξ) τ ξ 3 b 1 L 2 ξ,τ F 1 ( ξ 2 ˆv(ξ 2, τ 2 ) ) τ 2 ξ 3 2 b L x L 2 t On the one hand, by (2.22), we have F 1 ( ˆv(ξ 1, τ 1 )χ(ξ 1 ) ) τ 1 ξ 3 1 b L 2 x L t (2.33) F 1 ( ξ 2 ˆv(ξ 2, τ 2 )) τ 2 ξ 3 2 b L x L 2 t C v L 2 x,t (R2 ) = C. (2.34) On the other hand by the embedding X,b L 2 xl t (R 2 ) and by (2.32), we have F 1 ( ˆv(ξ 1, τ 1 )χ(ξ 1 ) ) τ 1 ξ 3 1 b L 2 x L t c F 1 ( ˆv(ξ 1, τ 1 ) χ(ξ 1 )) X, c v X ρ 1 2,. (2.35) Hence, we obtain Q 2 C v X ρ 1/2,. Third case: 1 2 ξ 1 ξ 2. In this case, the following algebraic inequality will be useful: 3 ξξ 1 ξ 2 = ξ 3 ξ 3 1 ξ 3 2 = (τ ξ 3 ) (τ 1 ξ 3 1) (τ 2 ξ 3 2) τ ξ 3 + τ 1 ξ τ 2 ξ 3 2. (2.36) First subcase: ξξ 1 ξ 2 τ 1 ξ1. 3 Set D 31 for the corresponding region. For b 1 ρ, we shall use ξ 2 ξ 2 to write ˆv(ξ 1, τ 1 )ˆv(ξ 2, τ 2 )G(ξ, τ) Q 31 C ξ D 31 τ ξ 3 1 b τ 2 ξ2 3 b 1 ρ b ξ 2 1 2b dτ 1 dξ 1 dτdξ. (2.37) We then have, using (2.24) and the relations 1 2b and b 1 b/2, Q 31 C F 1 (G(τ, ξ) τ ξ 3 b 1 ) L 4 x,t F 1 ( ˆv(ξ 2, τ 2 ) ξ 2 1 2b τ 2 ξ 3 2 b ) L 4 x,t v X ρ b, C v X ρ b,. (2.38)

9 GLOBAL ATTRACTOR OF A WEAKLY DAMPED KDV EQUATION 9 Second subcase: ξξ 1 ξ 2 τ 2 ξ2. 3 Set D 32 for the corresponding region. Then, ˆv(ξ 1, τ 1 )ˆv(ξ 2, τ 2 )G(ξ, τ) Q 32 C ξ D 32 τ ξ 3 1 b τ 1 ξ1 3 b 1 ρ b ξ 2 1 2b dτ 1 dξ 1 dτdξ. (2.39) We obtain Q 32 C v X 1 2b, F 1 ( ˆv(ξ 1, τ 1 ) ξ 1 ρ b τ 1 ξ 3 1 b ) L 4 x,t F 1 (G) τ ξ 3 b 1 L 4 x,t, C v X ρ 1/2, where we used (2.24) and the relations b 1/2, ρ b 1 2b, and b 1 b/2. Third subcase: ξξ 1 ξ 2 τ ξ 3. In that case, using also ξ 2 ξ 2, Q 33 C We then have, (2.4) D33 ˆv(ξ 1, τ 1 ) ξ 1 ρ+b 1 τ 1 ξ 3 1 b ˆv(ξ 2, τ 2 ) ξ 2 2b 1 τ 2 ξ 3 2 b G(τ, ξ) dτ 1 dξ 1 dτdξ. (2.41) Q 33 C G L 2 τ,ξ F 1 ( ˆv(ξ 1, τ 1 ) ξ 1 ρ+b 1 ) τ 1 ξ1 3 b L 4 x,t F 1 ( ˆv(ξ 2, τ 2 ) ξ 2 2b 1 ) τ 2 ξ2 3 b L 4 x,t. (2.42) Thanks to (2.25), this is bounded by Q 33 C v X 1/8+ρ+b 1, v X 1/8+2b 1, c v X ρ 2 1 (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 of the equation (1.1). Then, we prove the L 2 energy equation for the solutions and obtain the global well-posedness and the existence of bounded absorbing sets in L 2 (R) Local Well-Posedness. We consider fixed but arbitrary 1/2 < s and 1/2 < b < b 9/16. Assume γ R and f = f(x, t) X s,b 1 [ T f,t f ], for some T f >. For each u H s (R), we look for a local solution of (1.1) in the mild sense [11] on an interval [ T, T ], < T 1 sufficiently small, as the fixed point in X s,b of the map Σ(u) given by Σ(u) = ψ 1 (t)w (t)u ψ T (t) W (t s) [ 2ψ Tf (s)f 2γψ 1 (s)u(s) ((ψ T (s)u(s)) 2 ) x ] ds, (3.1)

10 1 OLIVIER GOUBET AND RICARDO M. S. ROSA By using the estimates from the previous section we find, chosing b, b such that ε = b 3b + 1 >, and Σ(u) X s,b ψ 1 H b t (R) u H s x (R) + c 1 f X s,b 1 [ T f,t f ] + γc 1 T ψ 1 H b t (R) u X s,b c 1c sc 2 2T ε u 2 X s,b. Σ(u) Σ(v) X s,b γc 1 T ψ 1 H b t (R) u v X s,b c 1c sc 2 2T ε u + v X s,b u v X s,b. Then, for T > sufficiently small, depending in particular on u H s (R) (and also on f X s,b 1, s, b, and b ), the map Σ is a strict contraction on a closed ball of X s,b centered at the origin. Hence, there exists a unique fixed point u, which is a mild solution of [ T f,t f ] (1.1) on the interval [ T, T ]. Moreover, the following bound holds: u X s,b c( u H s (R) + f [ T,T ] X s,b 1 ), (3.2) [ T f,t f ] for some constant c independent of the data of the problem (we bound ψ 1 H b t (R) by ψ 1 H 1 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 >, as required in the calculations above. Then, we obtain Theorem 3.1. Let γ R and f X s,b 1 [ T f,t f ], with 1/2 < s, 1/2 < b < 25/48, and >. Let T = T ( u H s (R)) be as described above. Then, for each u H s (R) there T f exists a unique solution u in X s,b [ T,T ] of the equation (1.1). Moreover, t u(t) belongs to C b ([ T, T ]; L 2 (R)) and the map which associates (γ, f, u ) to the corresponding unique solution is continuous from R X s,b 1 [ T f,t f ] Hs (R) into X s,b [ T,T ] C([ T, T ]; L 2 (R)), for any < T < T ( u H s (R)). Theorem (3.1) applies, in particular, to the case where f is time independent and belongs to H s (R) Global solutions and energy-type equations. We want to establish the global existence of the solutions obtained in the previous section. This is achieved with the help of one of the invariants of the KdV, namely, I (u) = u 2 L 2 (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 [2], for instance). Upon introducing dissipation and external forcing those integrals are no longer invariant but lead to energy-type equations which are crucial for proving global existence of the solutions. For a smooth initial condition ũ Cc (R), and a smooth forcing term f Cc (R), the local solution ũ X,b [ T,T ] given by Theorem 3.1, for a small T >, coincides with the classical solution, which exists globally R

11 GLOBAL ATTRACTOR OF A WEAKLY DAMPED KDV EQUATION 11 and belongs to C (R R). By multiplying equation (1.1) by 2ũ we see that ũ satisfies the L 2 energy-type equation d dt ũ(t) 2 L 2 (R) + 2γ ũ(t) 2 L 2 (R) = 2( f, ũ(t)) L 2 (R), (3.4) for all t R. We integrate (3.4) to find ũ(t) 2 L 2 (R) + 2γ ũ(s 2 L 2 (R) ds = ũ 2 L 2 (R) + 2 ( f, ũ(s)) L 2 (R) ds, (3.5) for t [ T, T ]. Now, we consider approximations of u L 2 (R) and f L 2 (R) by smooth functions ũ and f converging to u and f in L 2 (R), respectively. By the continuity with respect to the data of the local solution given by Theorem 3.1, we have that the solutions ũ with initial condition ũ() = ũ and forcing term f converge in X,b [ T,T ], for all T >, to the solution u X,b [ T,T ] with initial condition u() = u and forcing term f. By taking the limit in (3.5) and using the continuity of the solution with respect to the data, in particular using that u(t) 2 L 2 (R) = lim ũ(t) 2 L 2 (R), and (f, u(t)) L 2 (R) = lim( f, ũ(t)) L 2 (R), for all t [ T, T ], which follow from the embedding (2.7), we find that u(t) 2 + 2γ u(s) 2 L 2 (R) ds = u 2 L 2 (R) + 2 (f, u(s)) L 2 (R) ds, (3.6) for all t [ T, T ]. From the energy-type equation (3.6) one can extend the solution u indefinitely and obtain a global solution u = u(t), t R, with u X,b [ T,T ] C b([ T, T ], L 2 (R)) for all T >. One can also check that for each T > and each initial condition u L 2 (R), there exists a constant C = C( u L 2 (R), T ) such that u X,b C( u L 2 (R), f L 2 (R), γ, T ). (3.7) [ T,T ] This can be obtained by dividing each interval [ T, T ] into sufficiently small subintervals, as required in the proof of local existence, and by using the estimate provided by the energytype equation (3.6) for the norm of the solution in L 2 (R) at each instant of time. We omit the details since this is straightforward and classical. The energy-type equation (3.6) holds for all time, and the continuity of the solutions with respect to the data can be extended to all large times, as well. Hence, we have the following result: Theorem 3.2. Let γ R, f L 2 (R), and u L 2 (R). Then, there exists a solution u C(R, L 2 (R)) of equation (1.1) which is the unique solution which belongs to X,b [ T,T ], for all T > and all 1/2 < b < 25/48. Moreover, the solution t u(t) satisfies the energy equation d dt u(t) 2 L 2 (R) + 2γ u(t) 2 L 2 (R) = 2(f, u(t)) L 2 (R), (3.8)

12 12 OLIVIER GOUBET AND RICARDO M. S. ROSA for almost every t in R. Furthermore, the map which associates the data (γ, f, u ) to the corresponding unique solution u is continuous from R L 2 (R) L 2 (R) into X,b [ T,T ] C([ T, T ]; L 2 (R)) for all T >, with, in particular, u X,b C(γ, f L 2 (R), u L 2 (R), T ), (3.9) [ T,T ] 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 L 2 (R) fixed, we denote by {S(t)} t R the group in L 2 (R) defined by S(t)u = u(t), where u = u(t) is the unique solution of (1.1) which belongs to for all T >. X,b [ T,T ] 3.3. Bounded absorbing sets. From this section on we are interested in the long time behavior of equation (1.1) taking the dissipation into account. Therefore, we assume that γ >. We also assume that the forcing term f belongs to L 2 (R). We want to obtain the existence of bounded aborbing sets for the solution operator {S(t)} t R. This is achieved with the 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 side of (3.8), we see that d dt u(t) 2 L 2 (R) + γ u(t) 2 L 2 (R) = 1 γ f 2 L 2 (R). Therefore, upon integrating in time, whence we deduce that u(t) 2 L 2 (R) u(t ) 2 L 2 (R) e γ(t t ) + 1 γ 2 f 2 L 2 (R) (1 e γ(t t ) ) (3.1) lim sup u(t) L 2 (R) ρ 1 t γ f L 2 (R), (3.11) uniformly for u bounded in L 2 (R). Thus, we have proved the following result: Proposition 3.1. Let γ > and f L 2 (R). Then, the solution operator associated with equation (1.1) possesses a bounded absorbing set in L 2 (R), with the radius of absorbing ball given according to (3.11). We now consider the equations with the initial conditions 4. Splitting of the Solutions v t + vv x + v xxx + γv = f P N ((vw) x + ww x ), (4.1) w t + Q N (ww x ) + w xxx + γw = Q N (vw) x, (4.2) v t= = P N u, w t= = Q N u. (4.3)

13 GLOBAL ATTRACTOR OF A WEAKLY DAMPED KDV EQUATION 13 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 L 2 (R). Then, the global existence of v follows, and we prove the regularity of v in H 3 (R) and the L 2 energy equation for v 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: with the initial condition w t Q N (ww x ) + w xxx + γw = Q N (uw) x, (4.4) w t= = Q N u. (4.5) We will use the global estimates for u to show that w(t) decays in L 2 (R), as t increases, as long as the solution is defined. Then, we conclude that w(t) is defined for all positive times and that it decays exponentially to zero in L 2 (R), as t goes to infinity. The local well-posedness of the equation for w follows as that for equation (1.1). For that we use the fact that the solution u belongs (locally in time) to X,b. We consider more general initial conditions of the form w t=t = w L 2 (R), (4.6) where t R is arbitrary, and with Q N w = w. Then, proceeding with the fixed point argument, we find a solution w of the equation w = ψ 1 (t)w (t)w ψ T (t) W (t s) [ 2γψ 1 (s) w(s) (Q N (ψ T (s)u(s)ψ T (s) w(s))) x (Q N (ψ T (s) w(s)) 2 ) x ] ds. (4.7) By applying the estimates from Section 2 we find, for ε = b 3b + 1 >, w X,b ψ 1 H b t (R) w L 2 x (R) + γc 1 T ψ 1 H b t (R) w X,b c 1c sc 2 2T ε w 2 X,b + c 1 c sc 2 2T ε u X,b w X,b. Hence, for T sufficiently small, we obtain the bound w X,b c w L 2 (R), for 1/2 < b < 25/48. Since w coincides with the solution w of (4.4) and (4.6) locally in time around the origin, we see that w X,b c w L 2 (R). (4.8) [ T,T ] We may repeat the argument above for an interval centered at a different initial time t to obtain w X,b c w(t ) L 2 (R), (4.9) [t T,t +T ]

14 14 OLIVIER GOUBET AND RICARDO M. S. ROSA for every t in the interval of definition of w, and for T = T 1 ( w(t ) L 2 (R), u(t ) L 2 (R), f L 2 (R), γ). (4.1) The decay of w is then obtained with the help of the bilinear estimate (2.28). By taking the inner product of the equation (4.4) with 2w in L 2 (R), we find d dt w(t) 2 L 2 (R) + 2γ w(t) 2 L 2 (R) = (u x(t), w 2 (t)) L 2 (R). We can split the second term in the left hand side above into two equal parts and integrate the equation with one of the terms for the integrating factor. We obtain [ ] w(t) 2 L 2 (R) = w(t ) 2 L 2 (R) e γ(t t ) + (u x (s), w 2 (s)) L 2 (R) γ w(s) 2 L 2 (R) ds. t e γ(t s) Consider, for the moment, the truncation function ψ = ψ [t,t], defined in Section 2, and the characteristic function χ = χ [t,t] of the interval [t, 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 e γ(t s) (u x (s), w 2 (s)) L 2 (R) ds = e γ(t s) χ(s)( ψ(s)u(s), x (( ψ(s)w(s)) 2 )) L 2 (R) ds χ ψu X,b x( ψw) 2 X, b. (4.11) From the choice of b and by applying (2.28), we have the estimate x ( ψw) 2 X, b x( ψw) 2 X,b 1 c w 2. (4.12) N X,b [t,t] On the other hand, since χ belongs to H b t L t and ψu belongs to X,b, which is included in L t L 2 x (see (2.7)), one can show, using (2.1), that χ ψu X,b χ L t (R) ψu X,b + χ X,b ψu X,b C(χ) u X,b, (4.13) [t,t] where C(χ) depends on χ, hence on t and t, but is independent of N. Inserting (4.12) and (4.13) into (4.11) yields e γ(t s) (u x (s), w 2 (s)) L 2 (R) ds C(χ) t N u 1/2 X w 2.,b [t,t] X,b [t,t] Then, using (4.9) (with t = s there), e γ(t s) (u x (s), w 2 (s)) L 2 (R) ds C(χ) ( 1 t N u 1/2 X,b [t,t] t t C(χ) ( 1 N u 1/2 X,b [t,t] t t Thus, w(t) 2 L 2 (R) w(t ) 2 L 2 (R) e γ(t t ) + t ) ) w 2 ds X,b t [t,t] t w(s) 2 L 2 (R) ds. ( ) C(χ) (t t ) N u X e γ(t s) γ w(s) 2,b L 2 (R) ds. [t,t]

15 GLOBAL ATTRACTOR OF A WEAKLY DAMPED KDV EQUATION 15 For N large enough, the second term in the right hand side above is negative, and, hence, w(t) 2 L 2 (R) w(t ) 2 L 2 (R) e γ(t t ). (4.14) This now can be iterated and shown to hold for all t t in the interval of definition of 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 t. In particular, with t = and w(t ) = Q N u, we find w(t) 2 L 2 (R) Q Nu 2 L 2 (R) e γt, t. (4.15) The analysis above should actually be done for smooth u and f, but, by density and by the continuity of the solutions with respect to the initial condition, the final bounds (4.14) and (4.15) hold for arbitrary u and f in L 2 (R) Regularity of the v part of the solution. Since u and w are defined globally in time, so is v = u w. We now prove an H 3 x(r) bound for v. We first observe that y P N v = P N u is smooth and satisfies lim sup y(t) H 3 x (R) N 3, t lim sup u(t) L 2 x (R) ρ N 3, (4.16) t where ρ is the radius of the absorbing ball given in (3.11). We focus on an H 3 x(r) estimate for Z Q N v, which is solution to Z t + Z xxx + Q N ((y + Z)(y + Z) x ) + γz = Q N f (4.17) with initial condition Z() =. We know that Z = Q N v remains bounded in L 2 x(r). Then, from (4.17), Z x L 2 x (R) K( Z x H 2 x (R) + Z xxx H 2 x (R) ) K( Z t H 2 x (R) + (y + Z)2 H 1 x (R) + 1) K(1 + Z t 1/3 L ), (4.18) 2 x(r) where we used interpolation between L 2 x(r) and Hx 3 (R), the fact that Z t remains bounded in Hx 3 (R), and the inequality (2.3). The coefficient K denotes a constant which may depend on γ, f L 2 (R), and u L 2 (R), and which may increase from inequality to inequality. Now, to obtain the Hx(R) 3 bound for Z is equivalent to prove an L 2 x(r) estimate on Z = Z t, which solves Z t + Z xxx + 2Q N x ((y + Z)Z ) + γz = 2Q N x ((y + Z)y t ), (4.19) with initial condition Z () = Q N f Q N y()y x () in L 2 x(r). Multiply this equation by 2Z and integrate over R with respect to x to obtain, after using Young s inequality, d dt Z 2 L 2 + γ Z 2 L 2 4 γ x(y t (y + Z)) L 2 x (R) + 2 (y + Z) x (Z ) 2 dx (4.2) R

16 16 OLIVIER GOUBET AND RICARDO M. S. ROSA Using (2.2) and Young s inequality, and then (4.18) and inverse (Poincaré-type) inequalities (for functions with bounded spatial frequency), we find x (y t (y + Z)) L 2 x (R) y t H 1 x (R) y + Z H 1 x (R) K(N)(1 + Z 1/3 L 2 x (R)), (4.21) where, as in (4.18), K(N) depends on γ, f L 2 (R), u L 2 (R), and it is now allowed to depend on N, as well. We integrate over [, t] to obtain Z (t) 2 L 2 eγt Z () 2 x L + e γs {(y + Z) ((Z ) 2 ) + K(N)} dxds 2 x. (4.22) R [,t] We proceed as in (4.8)-(4.14), using the bilinear estimate (2.28) and the local well-posedness of (4.19) in L 2, to obtain, for t small enough, Z (t) 2 L K(N) x N 2 Z 2, (4.23) X,b [,t] for 1/2 < b < 25/48, and then conclude that Z (t) 2 L 2 x (R) K(N), t. Therefore, there exists K = K(N, u L 2 x (R)) > such that v(t) H 3 x (R) K(N, u L 2 x (R)), t. (4.24) 4.3. Energy equation for v. Since v(t) belongs to H 3 x(r), we may take directly the inner product in L 2 x(r) of the equation (4.1) for v with 2v to find that d dt v 2 L 2 (R) + 2γ v 2 L 2 (R) = 2(f, v) L 2 (R) + (P N (2vw + w 2 ), v x ) L 2 (R). (4.25) Integrating this equation in time yields v(t) 2 L 2 (R) = v(t ) 2 L 2 (R) + e [ ] 2γ(t t ) 2(f, v(s)) L 2 (R) + (P N (2v(s)w(s) + w 2 (s)), v x (s)) L 2 (R) ds. (4.26) t 5. Asymptotic Smoothing and the Global Attractor The first step is to prove the asymptotic compactness of the group in L 2 (R). This is done by showing that for a bounded sequence of initial conditions {u n } n in L 2 (R) and a sequence of positive numbers t n, the solutions u n (t n ) = v n (t n )+w n (t n ) are precompact in L 2 (R), with w n (t n ) decaying to zero in L 2 (R) and v n (t n ) being precompact in L 2 (R) and weakly precompact in H 3 (R). This will give us the existence of the global attractor A in L 2 (R), and, at the same time, the boundedness of A in H 3 (R). Then, we work with the equations for u n = du n /dt and we show, using the energy equation method applied to u n, that with the initial conditions {u n } belonging to A (and, hence, bounded in H 3 (R)), the sequence u n(t n ) is precompact in L 2 (R). This implies, from the equation for u, that u n (t n ) is precompact in H 3 (R). This shows that the flow restricted to

17 GLOBAL ATTRACTOR OF A WEAKLY DAMPED KDV EQUATION 17 the global attractor is asymptotically compact in H 3 (R) and, hence, that the global attractor is compact in H 3 (R) Existence of the global attractor in L 2 (R). Let {u n } n be bounded in L 2 (R) and let {t n } n be a sequence of positive real numbers going to infinity. Let v n and w n be the solutions associated with each initial condition u n ; they are defined for all time t. From Section 4.2, we have that and, for the time-derivative, { } dvn dt (t n + ) n {v n (t n + )} n is bounded in C([ T, T ]; H 3 (R)), (5.1) is bounded in C([ T, T ]; L 2 (R)), (5.2) for each T > (and starting with n sufficiently large so that t n T ). This implies, by the Arzela-Ascoli Theorem, that {v n (t n + )} is precompact in C([ T, T ]; L 2 loc (R)), for every T >. Then, using again (5.1), we see by interpolation that {v n (t n + )} is actually precompact in C([ T, T ]; Hloc s (R)), for all s < 3. Thus, by a diagonalization process, we obtain a subsequence such that v nj (t nj + ) ū( ) strongly in C([ T, T ]; Hloc s (R)), s [, 3), weakly star in L ([ T, T ]; H 3 (R)), T >, Moreover, since ū t belongs to L ([ T, T ]; L 2 (R)), then, for any s < 3, (5.3) ū C(R; H s (R)), ū(t) H 3 (R) C(ρ, N), t R; (5.4) the uniform bound follows from the boundeness of v in H 3 (R) (see (4.24)). In fact, ū is weakly continuous with values in H 3 (R) (by the Strauss theorem). We also find that From (4.15), we find that v nj (t nj + t) ū(t) weakly in H 3 (R), for every t R. (5.5) w n (t n + t) L 2 (R), uniformly for t T, T >. (5.6) With (5.3) and (5.6), one can pass to the limit in the weak formulation of the equation for v nj to find that ū is a solution of the weakly damped, forced KdV equation (1.1), that moreover satisfy the energy equality (3.6). We now write the integral form (4.26) of the L 2 energy equation for v n with t = t n and t = t n T : v n (t n ) 2 L 2 (R) = e 2γT v n (t n T ) 2 L 2 (R) + T e 2γ(T s) [ 2(f, v n ) L 2 (R) (P N (2v n w n w 2 n), v nx ) L 2 (R)] ds, (5.7)

18 18 OLIVIER GOUBET AND RICARDO M. S. ROSA where, for notational simplicity, we omitted the argument t n T + s of the functions inside the time integral. By using the uniform bound for v in H 3 (R), the decay (5.6) of w n, and the weak-star limit of v nj in (5.3), we find lim sup v nj (t nj ) 2 L 2 (R) C(R, N)e 2γT + 2 j T e 2γ(T s) (f, ū( T + s)) L 2 (R) ds. (5.8) By substituting for the L 2 energy equation (3.8) for ū (see (3.6)), we obtain Let T go to infinity to find that lim sup v nj (t nj ) 2 L 2 (R) 2C(R, N)e 2γT + ū() 2 L 2 (R). (5.9) j lim sup v nj (t nj ) 2 L 2 (R) ū() 2 L 2 (R). (5.1) j This, together with the weak convergence (5.5) and the decay (5.6), implies that u nj (t nj ) = v nj (t nj ) + w nj (t nj ) ū() strongly in L 2 (R)), weakly in H 3 (R). (5.11) This shows that the solution operator is asymptotically compact in L 2 (R) and, hence, there exists a global attractor A in L 2 (R). Moreover, it also follows that A is a bounded set in H 3 (R). By interpolation, A is compact in any H s (R), for s < 3. It remains to show that A is compact in H 3 (R) Compactness of the global attractor in H 3 (R). For the compactness in H 3 (R), we restrict the flow to the global attractor, which is bounded in H 3 (R), and we show that the flow is asymptotically compact in H 3 (R). For that purpose, we assume that the sequence of initial conditions {u n } n belongs to A. Since the global attractor is invariant and is bounded in H 3 (R), as shown above, the corresponding trajectories u n (t) = S(t)u n belong to and are uniformly bounded (w.r.t. n and t) in H 3 (R), for all t R. We want to show that u nj (t nj ) converges to ū() in H 3 (R). For that purpose, we use the equation for u n = du n /dt : u nt + (u nu n) x + u nxxx + γu n =. (5.12) From the equation for u n, we see that proving that u nj (t nj ) converges to ū() in H 3 (R) amounts to proving that u n j (t nj ) converges to ū () strongly in L 2 (R). Since the trajectories {u n (t n + )} n are uniformly bounded in H 3 (R) and converge strongly in C([ T, T ]; L 2 (R)) to ū, we see that u n j (t nj + ) ū ( ) strongly in C([ T, T ]; H s (R)), s [, 3), Now, we consider the L 2 energy equation for u n : and weakly star in L ([ T, T ]; L 2 (R)), T >. (5.13) T u n(t n ) 2 L 2 (R) = e 2γT u n(t n T ) 2 L 2 (R) e 2γ(T s) (u nx, u n2 ) L 2 (R) ds (5.14)

19 GLOBAL ATTRACTOR OF A WEAKLY DAMPED KDV EQUATION 19 where, for notational simplicity, we omitted the argument t n T + s of the functions inside the 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 e 2γ(T s) ((u nj ) x ū x, ū 2 ) L 2 (R) ds. (5.15) On the other hand, for < 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, instead of (2.28)) to find T e 2γ(T s) (u nj x, u 2 n j ū 2 ) L 2 (R) ds c u n j X s,b [,T ] u n j ū 2. (5.16) X s,b [,T ] From (5.13) and the well-posedness of (5.12) in X s,b [,T ] (see Theorem 3.1), we can pass to the limit as j goes to inifity and thus obtain T lim e 2γ(T s) (u nj j x (t nj T + s), u n j (t nj T + s) 2 ) L 2 (R) ds Then, from (5.14), = lim sup u n j (t nj ) 2 L 2 (R) C(ρ)e 2γT j T T e 2γ(T s) (ū x ( T + s), ū ( T + s) 2 ) L 2 (R) ds. (5.17) e 2γ(T s) (ū x ( T + s), ū ( T + s) 2 ) L 2 (R) ds. (5.18) By substituting for the corresponding L 2 energy equation for ū, we find We let T go to infinity to find Therefore, lim sup u n j (t nj ) 2 L 2 (R) 2C(ρ)e 2γT + ū () 2 L 2 (R). (5.19) j lim sup u n j (t nj ) 2 L 2 (R) ū () 2 L 2 (R). (5.2) j u n j (t nj ) ū () strongly in L 2 (R). (5.21) As mentioned before, this implies the convergence in H 3 (R) of u nj (t nj ) to ū(), which proves the desired asymptotic compactness in H 3 (R) and, hence, the compactness of A in H 3 (R).

20 2 OLIVIER GOUBET AND RICARDO M. S. ROSA 5.3. Conclusion. We have shown the following result: Theorem 5.1. Let γ > and f L 2 (R). Then, the solution operator {S(t)} t R in L 2 (R) associated with equation (1.1) possesses a connected global attractor A in L 2 (R) which is compact in H 3 (R). More precisely, A is a connected and compact set in H 3 (R); it is invariant for the system; it attracts (in the L 2 (R)-metric) all the orbits of the system uniformly with respect to bounded sets (in L 2 (R)) of initial conditions; and (with respect to the inclusion relation) A is maximal among the bounded invariant sets and minimal among the globally attracting sets. Similarly, one can show that Theorem 5.2. Let γ > and f H k (R), where k N. Then, for each m =, 1,..., k, the solution operator {S m (t)} t R associated with the equation (1.1) in the phase space H m (R) is well-defined and possesses a connected global attractor A in H m (R), which is the same for all m =, 1,..., k. Moreover, the global attractor A is compact in H k+3 (R). 6. L 2 -Weak Continuity of the Solution Operator As we mentioned in the Introduction, the weak continuity of the solution operator is usually a key issue in the proof of the existence of the global attractor in noncompact systems via the energy equation method. We also mentioned that, in the present case, since we also use a splitting of the semigroup to obtain the regularity of the global attractor, the weak continuity property is actually not needed (it is replaced by an asymptotic weak continuity property; see Section 5.1). Nevertheless, the weak continuity is an interesting property by itself. Therefore, we present now a sketch of its proof. In higher order Sobolev spaces, the weak continuity is relatively easy to prove (see, however, [25], where some difficulties already appear in H 1 (R)). This is not the case in L 2 (R), where there are some delicate issues to overcome. Since we are interested in the weak continuity locally in time, we can take for simplicity γ =. The proof for γ R is similar. Moreover, since the solutions are bounded in L 2 (R) on a finite interval of time (globally in time for γ = ), we can also consider a suitably small interval of time depending on the L 2 (R)-norm of the initial condition. This result can then be iterated to yield the weak continuity on arbitrarily large intervals of time. Consider u ε that converges weakly to u in L 2 (R). Consider u(t) that is solution to u t + u xxx + uu x =, (6.1) with u() = u. This solution is understood in the mild sense of Bourgain-KPV solution to KdV. Consider u ε (t) that solves (6.1) with initial data u ε. We prove that for t small enough (without loss of generality) u ε (t) converges weakly to u(t) in L 2 x(r). First step: Fix N and consider the projector P N defined by F(P N u)(ξ) = û(ξ)χ( ξ ), where N χ is the characteristic function of the interval [ 1, 1]. Consider v ε,n (t) that solves v t + v xxx + vv x =, (6.2)

21 GLOBAL ATTRACTOR OF A WEAKLY DAMPED KDV EQUATION 21 with initial condition v() = P N u ε. For N fixed, we know [27] that v ε,n (t) is bounded in the space since P N u ε is as smooth as we want and since C([ T, T ], Hx(R)) 2 C 1 ([ T, T ], Hx 1 (R)), P N u ε H 2 x (R) KN 2, (6.3) where K is independent of ε. Hence if L 2 x,loc (R) denotes the Fréchet space endowed with its natural topology (L2 x convergence on bounded sets), v ε,n (t) strongly converges to v N in C([ T, T ], L 2 x,loc (R)). We can let ε in (6.2) (convergence in D ), and v N is solution to (6.2) with initial data P N u. On the other hand, the limit v N belongs to L ([ T, T ], Hx(R)) 2 (weak-star convergence) and vt N belongs to L ([ T, T ], Hx 1 (R)). Therefore, v N belongs to C([ T, T ], Hx(R)) s for each s < 2 (and is even weakly continuous in t with values in Hx(R)). 2 By uniqueness of solution of (6.2) in C([ T, T ], Hx(R)), s for s > 3/2, v N is the usual solution of (6.2) with initial data P N u. Second step: Now, w ε,n (t) = u ε (t) v ε,n (t) is solution to w t + w xxx + ww x + (vw) x =, (6.4) with initial data w() = (Id P N )u ε (ε = included). We can prove that on bounded intervals of time, for any s [, 3/4), w(t) H s (R) K w() 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 u ε is bounded in L 2 (R), w() H s (R) = (Id P N )u ε ) H s (R) uε L 2 (R) N s KN s. (6.6) Conclusion: We have, N being fixed, for a test function ψ that is smooth, compactly supported and that satisfies ψ L 2 (R) = 1, (u(t) u ε (t), ψ) L 2 (R) = (u(t) v N (t) + v N (t) v ε,n (t) + v ε,n (t) u ε, ψ) L 2 (R) (6.7) = (w,n (t) + v N (t) v N,ε (t) + w ε,n (t), ψ) L 2 (R) (6.8) 2KN s + (v N (t) v ε,n (t), ψ) L 2 (R), (6.9) where we used (6.6). Let ε with N fixed to obtain, from the first step, lim sup (v N (t) u ε (t), ψ) L 2 (R) 2KN s. (6.1) ε Let N + to conclude that u ε (t) converges to u(t) in the distribution sense. We remove the condition ψ is smooth and compactly supported by a density argument, since w ε,n is a bounded sequence in L ([ T, T ], L 2 (R)). This concludes the proof.

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