GLOBAL WELL-POSEDNESS AND EXISTENCE OF THE GLOBAL ATTRACTOR FOR THE KADOMTSEV-PETVIASHVILI II EQUATION IN THE ANISOTROPIC SOBOLEV SPACE

. The global well-posedness for the KP-II equation is established in the anisotropic Sobolev space H s, 0 for s > − 38 . Even though conservation laws are invalid in the Sobolev space with negative index, we explore the asymp- totic behavior of the solution by the aid of the I -method in which Colliander, Keel, Staﬃlani, Takaoka, and Tao introduced a series of modiﬁed energy terms. Moreover, a-priori estimate of the solution leads to the existence of global attractor for the weakly damped, forced KP-II equation in the weak topology of the Sobolev space when s > − 18 .


1.
Introduction. In this article, we consider the initial value problem for the Kadomtsev- u(x, y, 0) = u 0 (x, y) ∈ H s,0 (R 2 ), (1) and the global attractor for the weakly damped, forced Kadomtsev-Petviashvili II (dfKP-II) equation where u = u(x, y, t) is a real-valued function, the damping parameter γ is positive and the external forcing term f ∈ L 2 (R 2 ) is independent of t.
The KP-II equation was initially derived by B.B. Kadomtsev and V.I. Petviashvili [8] to describe the propagation of weakly transverse water waves in the long wave regime with small surface tension. It can be seen as the two-dimensional generalization of the Korteweg-de Vries (KdV) equation. The KP-II equation is completely integrable. In fact, there exists an infinite sequence of conserved quantities [17], such as and However, due to the appearance of ∂ −1 x , these conserved quantities besides E(u) seem to be useless for proofs of well-posedness.
In terms of the Cauchy problem for the KP-II equation, J. Bourgain [2] firstly proved global well-posedness in L 2 (T 2 ) and L 2 (R 2 ) by creatively using the Fourier restriction norm method. H. Takaoka [20] obtained local well-posedness in the anisotropic Sobolev spaces H − 1 4 + ,0 and in the anisotropic homogeneous Sobolev space H − 1 2 +4 ,0 ∩Ḣ − 1 2 + ,0 . In [4], M. Hadac not only proved local well-posedness for the KP-II equation in the inhomogeneous Sobolev space H s,0 in full sub-critical range s > − 1 2 without any additional low frequency assumption, but also considered well-posedness for the generalized KP-II equation in the anisotropic Sobolev space H s1,s2 . Local well-posedness for the critical level of regularity s = − 1 2 in both homogeneous and inhomogeneous cases are solved by Hadac, Herr and Koch via using the atomic space. In [5], they acquired small data global well-posedness and scattering result as well. With regard to global well-posedness for large data below L 2 , Isaza and Mejía showed global result in H s,0 for s > − 1 14 by means of the high-low frequency technique and the almost conservation law (see [6,7]).
Despite that KP-II equation possesses remarkable rich structure and advantage, one cannot neglect energy dissipation mechanism and external excitation in reality. Therefore, sometimes we need add a weak dissipation and an external forcing term to the original KP-II equation. That is the dfKP-II equation (2) we consider here. Global attractor is an invariant compact subset which attracts all trajectories when t approaches to +∞. Tsugawa [22] proved the existence of global attractor for the weakly damped, forced KdV equation on the Sobolev space of negative index by applying the I-method. This idea is also effective for the mKdV equation [15] and the Zakharov-Kuznetsov equation [18,19]. But, not like the torus case, we only showed weak global attractor for the damped, forced Zakharov-Kuznetsov equation because the bounded set in R 2 is just a weakly compact set. What's more, for some kinds of reaction diffusion equations, the weighted L 2 space and the uniform Hölder space are used to recover the precompactness of orbits (see [1] and [14]) . Besides, localization estimate plays an important role in the proofs of precompactness in [23] and [10].
In this article, we refine the global well-posedness for the KP-II equation in H s,0 by pushing s from − 1 14 to − 3 8 through utilizing the atomic space and the I-method. The advantage of U 2 , V 2 spaces is to obtain sharp estimates in the time variable, which provide us with better decay in high frequency. Connecting this merit of the atomic space to I-method, one can explore the global behavior of energy. Then the increment of modified energy allows us to prove a a-priori estimate which helps us better understand the global dynamic of the solution below L 2 (see Proposition 12). Due to a lack of compactness in R 2 , we make an attempt to obtain global attractor in the sense of a stronger topology by using localization estimate as we carried it out for the damped, forced Zakharov-Kuznetsov equation in [10]. But the situation for the KP-II equation is very different from the ZK equation. Because the anti-derivative operator ∂ −1 x will cause increment instead of decay for the low frequency part. What's more, we have to work in negative regularities which seems to bring another difficulty at the technical level.
We now state the main results of this paper.
Then, there exists a semi-group A(t) and maps M 1 and M 2 such that A(t)u 0 is the unique solution of (2) satisfying and for t > T 1 where T 1 depends on u 0 H s,0 , f L 2 and γ, the constant K depends only on f L 2 and γ.
According to this global dynamic result, we gain the weak global attractor in H s,0 from Theorem 1.1 of [21]. Corollary 1. The global attractor for (2) exists in the sense of weak topology in H s,0 (R 2 ) for − 1 8 < s < 0 . Organization of the paper. In Section 2, we shortly list some propositions of U p and V p . In Section 3, we prove global well-posedness by using the I-method on atomic space. Section 4 is devoted to a prior estimate and the proof of Theorem 1.2. Finally, in Section 5 we prove well-posedness, the weakly continuous and Corollary 1.
Now we give the notations used throughout this paper. Let c < 1, C > 3, the notation c+ stands for c + for some 0 < 1. We always use a fixed smooth cut-off function χ ∈ C ∞ 0 ([−2, 2]) which satisfies that χ is even, nonnegative, and χ = 1 on [−1, 1]. We denote spatial variables by x, y and their dual Fourier variables by ξ, η. While, τ is the dual variable of the time t. Letf denote the Fourier transform of f in both time and spatial variables and f denote its Fourier transform only in space or in time. We write ζ = (ξ, η), λ = (ξ, η, τ ) and µ = τ −ξ 3 + ξ −1 η 2 for brevity. The capital letters N, M, N 1 , N 2 and N 3 denote dyadic numbers and we write N ≥1 a N = n∈N a 2 n , N ≥M a N = n∈N;2 n ≥M a 2 n for dyadic summations. The non-isotropic Sobolev space H s1,s2 (R 2 ) is a space of complex valued temperate distributions with the norm
and the atomic space is The norm of U p is defined by and for which the norm ∈ V p and use V p rc (V p −,rc ) to denote the closed subspace of all right continuous functions in V p (V p − ). There is a more intimate connection between U p and V p . (see Section 2 of [5]) Proposition 1. Let 1 < p < q < ∞ and 1 p + 1 p = 1. We have
Actually, one can define k-linear functional as above if M is a symbol with respect to (ζ 1 , · · · , ζ k ). We write Λ k (M ) instead of Λ k (M ; u, · · · , u) for convenience. m(ξ) is a smooth, radially symmetric, non-increasing function satisfying where s < 0, N 1. The corresponding Fourier multiplier operator is Let λ > 0 and N = N λ , we also define and rescaled operator I f (ξ) = m (ξ) f (ξ). We start from well-posedness for the KP-II equation. Acting multiplier operator I on both sides of (1), we obtain One can write this as an integral equation In the next place, we will estimate the Duhamel term following the idea of Proposition 3.1 in [5]. The work space is denoted by Y s which can be defined via the norm It's easy to see that Y s ⊂ L ∞ H s,0 . where S . In order to prove (15), we decompose Id = Q S <M + Q S ≥M and divide the integral into eight pieces.
In the low frequency situation, one has |λ j | < M and |ξ j | ≥ N j /2 due to the cut off operators Q S <M and P Nj . By resonance identity Hence, if we choose M = 1 10 N 1 N 2 N 3 , the integral vanishes. Case 2. At least one of Q S j is high frequency, for instance Q S 1 = Q S ≥M .
Using the L 4 Strichartz estimate (11), we have S . The other cases can be dealt with in exactly the same way.
We use the same decomposition as above to prove (16).
When Q S 1 or Q S 2 is high frequency , by using estimate (8) and bilinear Strichartz estimate (13), one has Therefore, it's easy to see that the following trilinear estimates hold true.
Proof. By definition of Y 0 and symmetry, it suffices to consider the following two terms Combining Proposition 1 (iii) with (15), we obtain The second term can be controlled via (16) and Minkowski's inequality, Analogously, one can estimate the nonlinear term without operator I as Proposition 3 with the help of Lemma 3.2 .
Proof. From Duhamel's principle, we get

Applying Proposition 3, it gives
Then, We get the existence of local solution by the contraction mapping principle. A bootstrap argument yields Iu Y 0 0 . Now we turn to the growth of E(Iu)(t).
Proposition 5. Assume u satisfies (1) and M is a symmetric function. Then where Parseval's formula provides It follows from Proposition 5 that ξ j m 2 (ξ j )).
The first term vanishes since ξ 3 We further give a new modified energy where σ 3 will be set to make a cancelation.
Using Proposition 5 again, one has where Then the two trilinear terms in (22) cancel by taking , and we have Here is the estimate for M 3 (see Proposition 5.16 in [24]). Proposition 6. Let N 1 , N 2 and N 3 be dyadic numbers. Then, in the hyperplane Proof. The fact for any function u j (j = 1, 2, 3) with frequency supported on |ξ j | ∼ N j . One can assume N 1 ∼ N 2 N 3 and N 1 N , otherwise the symbol M 3 vanishes. Notice that |h 3 | > 3|ξ 1 ξ 2 ξ 3 |. Moreover, from the definition of atom, we know that if a is a U p −atom then so is F −1 xy | a xy |. Therefore, we also can assume that the Fourier transform of u with respect to spatial variables is nonnegative due to (23) and (11) , we obtain (11) and bilinear Strichartz estimate (12) This completes the proof of (25), and hence (24).
We complete the proof of (27).
By simple calculation, we know that then we can use Proposition 9 to extend the solution u λ from [0, 1] to [1,2]. Iterating this procedure N 1− steps, we achieve the solution u λ on [0, N 1− ]. Choosing N sufficiently large such that
We first give local well-posedness result for the weakly damped forced KP-II equation.
Proposition 10. Let − 1 8 < s < 0. Assume I v 0 ∈ L 2 (R 2 ) and I g ∈ L 2 (R 2 ), then there is a constant δ = δ( I v 0 L 2 (R 2 ) , λ −3 I g L 2 (R 2 ) , γλ −3 ) > 0 so that there exists a unique solution v(x, y, t) ∈ C([0, δ], H s,0 (R 2 )) of (29) satisfying Proof. Acting I on (29) we rewrite (30) as an integral equation where From the definition of Y 0 and the duality of U p , we know that }, by using Proposition 3 and (31), we get Then, it is easy to see that T : B → B is a strict contraction mapping as we assume We consider σ−scaling of v If σ is chosen to be sufficiently large, it holds (σλ) −3 γ 1, Next, we explore the global dynamic of I v. From (30), we obtain and where M 3 , M 4 are given in Section 3 and Integrating (35) over [0, T ], one gets Lemma 4.1. Assume that v is a solution of (29) on [0, T ]. Then, there exists Proof. From (36) and Therefore, Taking sufficiently small, the last term of (38) will be absorbed by the left side, hence we obtain (37).
Proof. On one hand, from Proposition 7, one has  Besides, On the other hand, in order to prove (41), we need to show .
Therefore, it suffices to show for function with frequency supported on |ξ j | ∼ N j . One can assume N 2 ≥ N 3 and N 2 N .
Hence, the proof is completed.
We give an impactful a-priori estimate which will be used to control u H s,0 .

NOBU KISHIMOTO, MINJIE SHAN AND YOSHIO TSUTSUMI
then Proof. The local existence time for the solution of (29) is Set j ∈ N satisfying δj = T . For 0 ≤ k ≤ j, k ∈ N, we will prove by induction. For k = 0, (43) holds true trivially. We assume (43) holds for k = l where 0 ≤ l ≤ j − 1.
From Lemma 4.1, one has Therefore, it remains to prove and which verifies (44) as C 2 1 and C 4 1. Furthermore, we can get (45) and (46) in a similar way.
From Proposition 8, it holds which gives (47) by taking C 4 sufficiently large and C 2 sufficiently small.
Next ,we show the global dynamic of the solution.
Proof of Theorem 1.2. Fixing 0 < < 1 + 8s, we choose T 1 > 0 so that which is possible because −8s 1− < 1. T 1 depends only on u 0 H s,0 , f L 2 and γ. Set and Hence, from Proposition 12, one gains From (48) and (49) , we know This gives the bound where K 1 depends only on f L 2 and γ.
Then, one can fix T 2 > 0 and solve (2) on time interval [T 1 , T 1 + T 2 ] with initial data u(T 1 ). Let K 2 > 0 be sufficiently large such that for any t > 0, where K 2 depends only on f L 2 and γ. Set N −2s = K 2 e 2γT2 , then the assumptions in Proposition 12 are verified by (50). Therefore, we obtain where K 3 depends only on f L 2 and γ.
Define the maps M 1 (t) and M 2 (t) for t > T 1 as where A(t)u 0 = u(t) and N = (K 2 e 2γ(t−T1) ) − 1 2s , then we have Hence, taking K = max{K

5.
Global attractor in weak topology. In order to define an infinite-dimensional dynamical system from the evolution equation (2) , first of all we should make sure that the corresponding initial value problem is well-posed in H s,0 .
Proof. By Duhamel's formulation From the definition of Y s and the duality of U p , one knows that (20) and (51) give and We rescale equation (2) to get (29) for constructing a strict contraction mapping (52) and (53), and Choosing λ sufficiently large, we have Hence there exists a unique solution v on [0, 1] from the fixed point argument. Thus, (2) is locally well-posed on [0, T ] by taking T = λ −3 .
Next, we consider the weakly continuous dependence of solution on initial data in H s,0 for the Cauchy problem of (2). Without loss of generality, we may assume γ = 0 and f = 0.
We put D x u = F −1 |ξ|û(ξ) . The derivative operators D y and D t are similarly defined. We begin with the following smoothing estimate (see [9, Lemma 3.2 b) on page 1126]).
Lemma 5.1 with a = 1/3 and a = 1/2 and the interpolation yield the following lemma.
We now explain how to show that if a sequence of initial data {u 0n } converges to u 0 weakly in H s,0 and a sequence of solutions {u n } with u n (0) = u 0n converges to a solution u with u(0) = u 0 weakly in X s,0,1/2+ T for s > −1/2, then u n (t) converges to u(t) weakly in H s,0 for each t ∈ [0, T ]. Here we use X s,0,1/2+ T instead of Y s to avoid being too lengthy and tedious. The prerequisite "if "part can be proved through the standard local existence theorem of solution (see Remark 1 below).
Let ψ ∈ C ∞ 0 (R 2 ) and let T be a fixed positive constant with 0 < T < T . The Duhamel formula yields where we write u and u n for χ {0≤t≤T } u and χ {0≤t≤T } u n , moreover we put f = χ {0≤t≤T } e tS ∂ x ψ. It suffices to prove that the the last term on the right hand side of (56) converges to zero as n → ∞, since {u n } is bounded in L ∞ (0, T ; H s,0 ). We put σ = τ −Ŝ(ξ, η). By P N , we denote the projection : u ∈ L 2 (R x ) → F −1 χ {|ξ|≤N } (ξ)û(ξ) . We set u N = P N u and u >N = (I − P N )u. We define a cut-off function ϕ(x, y, t) ∈ C ∞ (R 3 ) as follows.
It follows from (57) that the first three terms on the right hand side can be made arbitrarily small as N gets larger and larger. So, we have only to consider the term u N v N . The contribution of the term u N v N can be divided into the following two terms.
Hence, for ε > 0, we have Let R be a mapping which restricts function f on R 3 to f | {|z|<2R} . Since we have the compact embedding x,y,t (|z| < 2R), ε > 0, we can conclude by Lemma 5.2 (i) and (ii) that the mapping : u → u N is compact from X 0,1/4+ to L 2 x,y,t (|z| < 2R) for small ε > 0. Therefore, the weakly continuous dependence in H s,0 , s > −1/2 of solution on initial data follows from the above argument and (56). Remark 1. Let T be a positive constant which denotes the existence time of solution to (2). For the weakly continuous dependence, we consider the integral equation associated with the Cauchy problem (2).
where α is a cut-off function in C ∞ 0 (R) such that α(t) = 1 (|t| < 1) and α(t) = 0 (|t| > 2) and α T (t) = α(t/T ). All what we need to do is to prove that if a sequence of initial data {u 0n } coverges to u 0 weakly in H s,0 for s > −1/2, a solution u of (60) is given by a weak limit of the sequence of solutions u n of (60) with u 0 replaced by u 0n . In fact, we can extract a subsequence {u n } converges to some u weakly in X s,0,1/2+ , since a sequence of solutions {u n } of (60) with u 0 replaced by u 0n is bounded in X s,0,1/2+ for s > −1/2 (see [4, Theorem 1.1 on page 6556]). Furthermore, by the duality argument similar to above, we can easily see that u is a solution of (60), which is unique in X s,0,1/2+ for s > −1/2 (see [4, Theorem 1.1 on page 6556]). Therefore, the whole sequence of {u n } converges to u weakly in X s,0,1/2+ .
Proof of Corollary 1. By Proposition 13 the initial value problem is well-posed in H s,0 , hence one can define an infinite-dimensional dynamical system from the evolution equation (2). Moreover, we get the weakly continuous in H s,0 by Remark 1. From Theorem 1.2, we know that M 1 (t) is a bounded mapping and M 2 (t) converges uniformly to 0 in H s,0 . It means that the semi-group A(t) is asymptotically compact in the sense of weak topology. Therefore we gain the existence of the global attractor in H s,0 by Remark 1.4 and Theorem 1.1 in [21].