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...: s> 3 4 [21] F: s> 3 4 ,L2 small [21] F: big L 2 blows up [38] k ≥ 6 sk = 1 2 − F: s<sk, [33];D:?? s ≥ sk, [31] D: s ≥ 1, F: small H 2 k 1 F: big H 1 blows up ?? Our results here and elsewhere [16], =-=[14]-=-, [17] suggest that local well-posedness implies global well-posedness in subcritical dispersive initial value problems. In particular, we believe our methods will extend to prove GWP of mKdV in H 1 4...

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...lobal well-posedness of (1.1)-(1.2) in the case 0 ≤ s < 1 is at this point only partially answered. Previous work establishes that in one dimension the problem is globally well-posed for s > 4/9 (see =-=[7]-=-, [19]) and in two dimensions for s ≥ 1/2 (see [8] [13]). In both these cases no scattering has been proved. In higher dimensions a first result on global well-posedness below the energy norm was pres...

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...H s. (7) pb Here we use the approach introduced by Colliander,Keel, Staffilani, Takaoka and Tao in [6], called the I-method. We also explain why the refined approach introduced by the same authors in =-=[7]-=-, [8] and [10] can not be use to improve our global result stated in Theorem 1.1 (see Proposition 3.1 and Remarks 3.1-3.2 below). Note that when u0 ∈ H s (R) with s < 1 in (1), the energy (4) could be...

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...ssion on them). For the recent developments of I-method, we can refer to [4, 5, 6, 10, 12, 14, 15, 17, 18, 19, 31] on the applications in the context of nonlinear Schrödinger equation (NLS), refer to =-=[8, 9]-=- on the applications in the context of Schrödinger equation with derivative (DNLS), refer to [7, 11, 16, 33] on the applications in the context of gKdV equations. By using “Fourier truncation method”,...

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... well-posed in H1(R) under the condition (1.2). In [30], Takaoka used Bourgain’s “Fourier truncation method” ([6, 7]) to obtain the global well-posedness in Hs(R) for s > 3233 , again under (1.2). In =-=[9, 10]-=-, I-team (Colliander-Keel-Staffilani-Takaoka-Tao) made use of the first, second generations of 2000 Mathematics Subject Classification. Primary 35Q55; Secondary 47J35. Key words and phrases. Bourgain ...

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...a leading to spectral singularities and hence is not inconsistent with the Hayashi-Ozawa conjecture. 50 (263)the Cauchy problem associated to the derivative NLS equation (262) (see in addition [10], =-=[11]-=-, [22], [24], [34], and [35]) all differing in detail but all granting existence of a unique solution under either a sufficiently small L 2 (R) norm or a finite lifetime condition. All of these result...

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...sitive on a set of positive Lebesgue measure.22 D. HUNDERTMARK AND Y.-R. LEE The proof of the multi-linear estimates is based on the by now well-known bilinear Strichartz estimate, see, for example, =-=[6, 13, 28, 43]-=-, ‖Ttf1Ttf2‖ L 2 (R×R,dtdx) � 1 √ dist(supp f1, supp f2) ‖f1‖ L 2 (R)‖f2‖ L 2 (R). (B.9) going back to [6]. For a simple explicit proof of (B.9), see for example, [31] or [24]. Now assume that the...

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...nstant in the Gagliardo-Nirenberg inequality 1 6 ‖u‖6 L 6 ≤ C 2 ‖∇u‖2 L 2‖u‖ 4 L 2. We used the “I-method” that was recently introduced by J. Colliander, M. Keel, G. Stafillani, H.Takaoka, and T.Tao, =-=[5, 7, 8, 9]-=-. This method allows us to define a modification of the energy functional, that is “almost conserved” that is, its time derivative decays with respect to a very large parameter. Since an implementatio...