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...e Section 7.2). In fact, as mentioned above, for systems defined on a square, we need to use techniques from analytic number theory as in the work of Bourgain [5]; a similar approach was used also in =-=[7]-=-. The necessary number theory techniques come from [4]; see also [19, § 23.1] and [20].DERIVATION OF THE TWO-DIMENSIONAL NONLINEAR SCHRODINGER EQUATION 7 We should immediately remark that if we consi...

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...3]. We will need a refined version thereof, which is shown by elementary geometric considerations in Section 3. Here, we use arguments similar to those of De Silva, Pavlovic, Staffilani, and Tzirakis =-=[8]-=-, Section 4. Concerning the organization of the paper the following should be added: In Section 2 we introduce the relevant function spaces and state all the nonlinear6 A. GRÜNROCK AND S. HERR estima...

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... where we used the Sobolev theorem and the definition ofXs,br . Finally by Young’s inequality, (4.45) and (4.46) we have the desired estimate. It remains to show (4.43). We use an argument similar to =-=[18]-=-. For fixed τ let S := S(τ,M, β) 6= ∅, then there exists n0 ∈ S and hence (4.47) |S| ≤ 1+|{l ∈ Z/|n0+l | ∼M, |τ+(n0+l)2| ≤Mβ}| ≤ 1+|{l ∈ Z / |l| ≤M, |2n0l+l2| .Mβ}|. (4.48) |2n0l+ l2| = |(l+n0)2−n20| ...

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... easily from conservation of energy and standard arguments. In this work, we extend the range of global well-posedness to s > 2 3 . This generalizes, without any loss in regularity, the results in [5]=-=[18]-=-, where the same result is proved for the torus T2. The proof relies on the I-method of Colliander, Keel, Staffilani, Takaoka, and Tao, a semi-classical bilinear Strichartz estimate proved by the auth...

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...y u∗0 = Ae inx (A ∈ C, n ∈ Z2) in the range 0 < s < 1: 5I.e. the intersection of countably many dense open sets. 6For instance, this is the case for the polynomial cubic nonlinearity whenever s > 2/3 =-=[5, 13]-=-. LONG-TIME INSTABILITY AND UNBOUNDED ORBITS 5 Theorem 1.5. Consider the polynomial cubic NLS equation:{ −i∂tu+∆u = |u| 2u u(0, x) = u0(x) ∈ H s(T2) (1.5) For any 0 < s < 1, 0 < δ ≪ 1, K ≫ 1, there ex...

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...nonlinear Schrd̈inger equations. In the context of compact manifolds, some bilinear estimates on the torus were already implicit in the work of Bourgain [2](cf. [7]) and other variants were proved in =-=[12]-=-. In [7] and [8], the authors prove bilinear Strichartz estimates on spheres S2 and S3 (and on the bit wider class of Zoll manifolds) using bilinear eigenfunction cluster estimates. These bilinear Str...

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...1 Proof. If R � r 1 6 , the estimate (14) follows from Lemma 1. If R ≪ r 1 6, there are at most two lattice points on the intersection of B with the circle of radius ≃ r 1 2 around δ, by Lemma 4.4 of =-=[4]-=-. □ In the sequel we will use the following projections: For a subset M ⊂ Z 2 we define PM by FPMu(k, η, τ) = χM(η)Fu(k, η, τ). Especially, if M is a ball of radius 2 l centered at the origin, we will...

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...the long period linearflow, which is available to the rescaled problem. The kind of the bilinear Strichartz estimate for the periodic nonlinear Schrödinger equation was established simultaneously in =-=[4, 9]-=-, and see also [22] for an improvement version to the one in [9]. Inspired from these papers, we obtain the following bilinear Strichartz estimate on Airy equation here,∥∥∥∥∫∫ ξ=ξ1+ξ2 eixξ1+itξ 3 1eix...

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...ee [1]. In addition there have been many results on the well-posedness of (1) for both focusing and defocusing case for rough initial data (in Hs for s < 1) on two dimensional torus, see [3],[5],[6], =-=[7]-=-,[9] and also on more general compact manifolds, see [4],[14]. Date: May 7, 2014. 1 2 SECKIN DEMIRBAS One of the main tools in proving local well-posedness is the Strichartz estimates, i.e. estimates ...