Long time dynamics for forced and weakly damped KdV on the torus

The forced and weakly damped Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. Starting from $L^2$ and mean-zero initial data we prove that the solution decomposes into two parts; a linear one which decays to zero as time goes to infinity and a nonlinear one which always belongs to a smoother space. As a corollary we prove that all solutions are attracted by a ball in $H^s$, $s\in(0,1)$, whose radius depends only on $s$, the $L^2$ norm of the forcing term and the damping parameter. This gives a new proof for the existence of a smooth global attractor and provides quantitative information on the size of the attractor set in $H^s$.


Introduction
In this paper we study forced and weakly damped Korteweg de Vries (KdV) equation on the torus: u(x, 0) = u 0 (x) ∈L 2 (T) := {g ∈ L 2 (T) : where throughout the paper γ > 0 and f ∈L 2 . We also assume that u and f are real valued.
Note that forced and damped KdV does not satisfy momentum conservation. However, well-posed, if there exists T > 0 and a unique solution u ∈ X ∩ C 0 t H s x ([0, T ] × T). One also demands that there is continuity with respect to the initial data in the appropriate topology. If T can be taken to be arbitrarily large then the problem is globally well-posed.
Local well-posedness for the KdV equation on the torus for nonsmooth data was first obtained by Bourgain, [3]. He proved that the KdV equation is locally well-posed in L 2 (T).
Later, in [12], Kenig, Ponce, Vega extended the theory to H s , s > − 1 2 . The local theory at H −1/2 level was established by Colliander, Keel, Staffilani, Takaoka, Tao, [5]. All these results use the contraction mapping principle and the X s,b spaces of Bourgain. These methods also apply to the equation (1) as we describe in Section 2.
The local L 2 solutions of Bourgain are in fact global since for the KdV equation the L 2 norm is conserved. In [5], it was proved that KdV is globally well-posed in H s (T) for any s ≥ − 1 2 . To extend the local solutions globally in time they used the "I-method", developing a theory of almost conserved quantities starting with the energy. Although the initial data have infinite energy they showed that a smoothed out version of the solution cannot increase much in energy going from one local-in-time interval to another. In [11], Kappeler and Topalov extended the latter result using the integrability properties of the equation and proved that KdV admits global solutions in H s (T) for any s ≥ −1. In [1], Babin, Ilyin, ant Titi gave a new proof of the L 2 theorem of Bourgain using normal form methods. Similar ideas were developed by Shatah [13].
For forced and weakly damped KdV, the conservation of energy does not hold. Nevertheless, the energy remains bounded for positive times and in the long run it is less then 2 f /γ, where · := · L 2 (T) . Therefore global well-posedness for (1) in L 2 follows immediately. For the global well-posedness theory below L 2 see [15] and the references therein.
To obtain the energy bound, note that Setting h(t) = e 2γt u 2 , and using Cauchy Schwarz inequality in the integral f udx, we This implies that Therefore, for positive times Thus for t > T = T (γ, u 0 , f ), we have u(t) < 2 f /γ. Also note that this implies that the set {g : g ≤ f /γ} is invariant under the flow.
We now state the main result of this paper: Consider the forced and weakly damped KdV equation (1) on This theorem in the case when γ = f = 0 was obtained in [6].
This implies that allL 2 solutions are attracted by a ball in H s centered at zero of radius depending only on s, γ, f . An upper bound for this radious can be calculated explicitly by keeping track of the constants in our proof. Moreover, the description of the dynamics is explicit in the sense that after time T the evolution can be written as a sum of the linear evolution which decays to zero exponentially and a nonlinear evolution contained by the attracting ball. We now try to put this corollary into context in the general theory of global attractors.
The problem of global attractors for nonlinear PDEs has generated great interest among engineers, physicists and mathematicians in the last several decades. The theory is concerned with the description of the nonlinear dynamics for a given problem as t → ∞. In particular assuming that one has a well-posed problem for all times we can define the semigroup operator U (t) : u 0 ∈ H → u(t) ∈ H where H is the phase space. We want to describe the long time asymptotics of the solution by an invariant set X ⊂ H (a global attractor) to which the orbit converges as t → ∞: For dissipative systems there are many results (see, e.g., [14]) establishing the existence of a compact set that satisfies the above properties. Dissipativity is characterized by the existence of a bounded absorbing set into which all solutions enter eventually. In the case of the equation (1), the damping parameter γ > 0 makes the system dissipative, c.f. (2).
Notice that this is in contrast with conservative Hamiltonian systems where the orbits may fill the whole space or regions of it. In some cases the global attractor is a "thin" set, for example, it may be a finite dimensional set although the phase space is infinite dimensional.
The candidate for the attractor set is the omega limit set of an absorbing set, B, defined by where the closure is taken on H. To describe the history of the problem more accurately we need some definitions. The distance is understood to be the distance of a point to the set d(x, Y ) = inf y∈Y d(x, y).
We say that A attracts the points of U and we call the largest open such set U the basin of attraction.
Definition 1.4. We say that A ⊂ H is a global attractor for the semigroup {U (t)} t≥0 if A is a compact attractor whose basin of attraction is H.
Remark. We should note that the attracting ball in H s that our theorem provideas is not a global attractor since we don't know whether it is an invariant set.
To state a general theorem for the existence of a global attractor we need one more definition: It is not hard to see that the existence of a global attractor A for a semigroup U (t) implies that of an absorbing set. For the converse we cite the following theorem from [14] which gives a general criterion for the existence of a global attractor.
Theorem A. We assume that H is a metric space and that the operator U (t) is a continuous semigroup from H to itself for all t ≥ 0. We also assume that there exists an open set U and a bounded set is asymptotically compact, i.e. for every bounded sequence x k in H and every sequence In [10], Goubet proved the existence of a global attractor onL 2 and concerning its regularity he proved that the global attractor is a compact subset of H 3 . This was achieved by splitting the solution into two parts, high and low frequencies. The low frequencies are regular and thus in H 3 , while the high frequencies decays to zero in L 2 as time goes to infinity. The existence of a global attractor below L 2 was established by Tsugawa in [15].
The difficulty there lies in the fact that there is no conservation low for the KdV equation below L 2 . He bypasses this problem by using the method of almost conserved quantities of Colliander et al [5]. In addition he proves that the global attractor below L 2 is same as the one obtained by Goubet. One can lower the Sobolev index further, see [16]. Also see [4] for numerical results on the properties of the attractor set for the large and small values of the damping parameter.
We will prove in Section 5 below that the hypothesis of Theorem A can be checked using only Theorem 1.1 and Corollary 1.2. In particular, one does not need to utilize weak topology and Ball's argument. Theorem 1.1 implies asymptotic compactness in the strong topology directly, see Section 5. Therefore, we obtain the following: It should be noted that in the resonant terms the waves interact with no oscillation and hence they are always "the enemy". But it turns out that the nonsmooth resonant terms of the KdV cancel out and the gain of the derivative is more than enough to compensate for the remaining nonli! near terms. For the nonresonant terms, we apply the restricted norm method of Bourgain to the reduced nonlinearity to prove the theorem.

Notation.
To avoid the use of multiple constants, we write A B to denote that there is an absolute constant C such that A ≤ CB. We also write A ≈ B to denote both A B and B A. We define · = 1 + | · |.
We define the Fourier sequence of a 2π-periodic L 2 function u as With this normalization we have Note that for aL 2 function u, u H s ≈ u(k)|k| s ℓ 2 . For a sequence u k , with u 0 = 0, we will use u H s notation to denote u k |k| s ℓ 2 .

Well-posedness theory of forced and weakly damped KdV
We define the X s,b spaces for 2π-periodic KdV via the norm We also define the restricted norm The local well-posedness theory for periodic KdV was established in the space X s,1/2 . Unfortunately, this space fails to control the L ∞ t H s x norm of the solution. To remedy this problem and ensure the continuity of KdV flow, the Y s and Z s spaces are defined in [7] and [5], based on the ideas of Bourgain [3] via the norms Theorem 2.1. The initial value problem (1) is locally and globally well-posed in L 2 . In particular, ∃δ = δ( u 0 , γ, f ) and a unique solution u ∈ C([−δ, δ]; L 2 x (T)) ∩ Y 0 δ with The proof of this theorem follows the arguments in [3] which we briefly sketch. The following lemma is a collection of statements originated in [3]: Finally, The only statement in this lemma which is not explicit in [3] is the last one. By the forth estimate of the lemma, it suffices to consider the second part of the Z 0 δ norm, which follows from In the first inequality we used Holder, and in the second one we used Youngs inequality in the τ variable.
To prove Theorem 2.1, using Duhamel, we write Applying the estimates in the lemma one can easily see that Φ is a contraction on Y 0 δ and in addition it belongs to C([0, δ]; L 2 ). Finally the global well-posedness follows from the energy bound (2). We close this section by stating a Strichartz estimate by Bourgain [3] which will be useful in the proof of Theorem 1.1

Proof of Theorem 1.1
Let Using the notation u(x, t) = k u k (t)e ikx , and v(x) = k v k e ikx , we write (3) on the Fourier side: Because of the mean zero assumption on u and f , there are no zero harmonics in this equation. Using the transformations and the identity the equation can be written in the form We start with the following proposition which follows from differention by parts.
Proposition 3.1. The system (4) can be written in the following form: where we define B(u, v) 0 = ρ 0 = R(u) 0 = 0, and for k = 0, we define Proof of Proposition 3.1. Since e −3ikk 1 k 2 t = ∂ t ( i 3kk 1 k 2 e −3ikk 1 k 2 t ), using differentiation by parts we can rewrite (4) as Recalling the definition of B, we can rewrite this equation in the form: Note that since z 0 = y 0 = 0, in the sums above k 1 and k 2 are not zero. Using (4), we have Using the identity and by renaming the variables µ → k 2 , ν → k 3 , we have that Using (7), we can rewrite (6) as Note that the set on which the phase on the trilinear term vanishes is the disjoint union of the following sets Thus, using the definition of R(u), we have The proposition follows if we show that the last sum above is equal to e −2γt ρ k . Note that Here we used z j = z −j and y j = y −j . Using symmetry, the second line vanishes, which yields the assertion of the proposition.
Integrating (5) from 0 to t, we obtain Transforming back to the w, v variables, we have Proof. By symmetry we can assume in the estimate for B(u, v) that |k 1 | ≥ |k 2 |. Thus, for In the last line we used Young's inequality and Cauchy-Schwarz. Now note that for s < 1, Using the estimates in Lemma 3.2 in the equation (9), we obtain (for s < 1) where the implicit constant in the second inequality depends on γ and s, and We also used the facts that Since our nonlinearity after differentiation by parts is not uu x anymore, we will be able to avoid the Y s 1 and Z s 1 spaces. Instead we will use the embedding X s 1 ,b ⊂ L ∞ t H s 1 x for b > 1/2 and the following lemma. Let η be a smooth function supported on [−2, 2] and η(t) = 1 for |t| ≤ 1.
Then for any δ < 1 where the implicit constant depends on γ and b.
Proof. It suffices to prove the statement with X s,b norms. Note that Therefore it suffices to prove that we see that the Fourier transform of the function inside the norm in the left hand side of For the contribution of this to the left hand side of (12), we use the inequalities and Young's inequality to get The forth inequality holds since η(t)e −γt is a Schwarz function. The last inequality follows from the fact that −1 − b ′ < −1/2.
Proposition 3.4. For s ∈ (0, 1), and ε > 0 sufficiently small, we have We will prove this proposition later on. Using (13) and the proposition above in (10), we see that for |t| < δ, we have In the rest of the proof the implicit constants depend on u 0 , f , γ, and s. Fix t large.
For r ≤ t, we have the bound Thus, by the inequality above and the local theory, with δ ≈ u 0 + f /γ −α (for some for any j ≤ t/δ. Here we used the local theory bound Using this we obtain (with J = t/δ) This completes the proof of the global bound stated in Theorem 1.1. Finally the continuity of u(t) − e t(L−γ) g in H s follows as in [6], we omit the details.

Proof of Theorem 1.6
First of all note that the existence of an absorbing set, B 0 , is immediate from (2). Second, we need to verify the assymptotic compactness of the propagator U (t). To see this note that by Theorem 1.1, U (t)u 0 = e −γt+Lt u 0 + N (t)u 0 where N (t)u 0 is in a ball in H s with radius depending on s, γ, u 0 , f . By Rellich's theorem, {N (t)u 0 : t > 0} is precompact in L 2 . Since the first summand is continuous and goes to zero as t → ∞ in L 2 , we conclude that {U (t)u 0 : t > 0} is precompact in L 2 , which is stronger then asymptotic compactness. This and theorem A imply the existence of a global attractor A ⊂ L 2 .
We now prove that the attractor set A is a compact subset of H s for any s ∈ (0, 1).
By Rellich's theorem, it suffices to prove that for any s ∈ (0, 1), there exists a closed ball B s ⊂ H s of radius C(s, γ, f ) such that A ⊂ B s . By definition Let B s be the ball of radius C(s, γ, f ) (as in Corollary 1.2) centered at zero in H s . By Corollary 1.2, for τ > T , U τ is contained in a δ τ neighborhood N τ of B s in L 2 , where δ τ → 0 as τ tends to infinity. Since B s is a compact subset of L 2 , we have