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...l well-posedness in H s x (R2 ) for all s ≥ 0, and if s is strictly positive then a solution can be continued unless the H s x(R 2 ) norm of the solution goes to infinity at the blowup time (see e.g. =-=[8]-=-, [27]). Also, due to the smooth nature of the nonlinearity, any local H s (R 2 ) solution can be expressed as the limit (in C 0 t,loc Hs x) of smooth solutions. For s < 0 the solution map ceases to b...

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...ying to use the energy E(u) to control the kinetic component ‖u‖ 2 ˙ H 1 (R 2 ) , since the potential energy component of the energy is now negative. However, the sharp Gagliardo-Nirenberg inequality =-=[31]-=- allows one to achieve this control (losing a constant, of course) provided that ‖u‖ L 2 x (R 2 ) < ‖Q‖ L 2 x (R 2 ), allowing one to continue the argument without difficulty. As the modifications are...

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...l-posedness in H s x (R2 ) for all s ≥ 0, and if s is strictly positive then a solution can be continued unless the H s x(R 2 ) norm of the solution goes to infinity at the blowup time (see e.g. [8], =-=[27]-=-). Also, due to the smooth nature of the nonlinearity, any local H s (R 2 ) solution can be expressed as the limit (in C 0 t,loc Hs x) of smooth solutions. For s < 0 the solution map ceases to be unif...

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...he global-in-time problem, in which we allow J to be the whole real line R. Both the local and global-in-time Cauchy problems for this NLS equation (1.1) have attracted a substantial literature [30], =-=[10]-=-, [22], [20] [26], [5], [6], [17], [25], [9], [3], [11]. The local well-posedness theory is now well understood; in particular, one has local well-posedness in H s x (R2 ) for all s ≥ 0, and if s is s...

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...em, in which we allow J to be the whole real line R. Both the local and global-in-time Cauchy problems for this NLS equation (1.1) have attracted a substantial literature [30], [10], [22], [20] [26], =-=[5]-=-, [6], [17], [25], [9], [3], [11]. The local well-posedness theory is now well understood; in particular, one has local well-posedness in H s x (R2 ) for all s ≥ 0, and if s is strictly positive then ...

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...n which we allow J to be the whole real line R. Both the local and global-in-time Cauchy problems for this NLS equation (1.1) have attracted a substantial literature [30], [10], [22], [20] [26], [5], =-=[6]-=-, [17], [25], [9], [3], [11]. The local well-posedness theory is now well understood; in particular, one has local well-posedness in H s x (R2 ) for all s ≥ 0, and if s is strictly positive then a sol...

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...ch we allow J to be the whole real line R. Both the local and global-in-time Cauchy problems for this NLS equation (1.1) have attracted a substantial literature [30], [10], [22], [20] [26], [5], [6], =-=[17]-=-, [25], [9], [3], [11]. The local well-posedness theory is now well understood; in particular, one has local well-posedness in H s x (R2 ) for all s ≥ 0, and if s is strictly positive then a solution ...

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...-time problem, in which we allow J to be the whole real line R. Both the local and global-in-time Cauchy problems for this NLS equation (1.1) have attracted a substantial literature [30], [10], [22], =-=[20]-=- [26], [5], [6], [17], [25], [9], [3], [11]. The local well-posedness theory is now well understood; in particular, one has local well-posedness in H s x (R2 ) for all s ≥ 0, and if s is strictly posi...

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...1) is globally well-posed in H s x (R2 ) for all s > 1/2. Our arguments refine our previous analysis in [17] by adding a “correction term” to a certain modified energy functional E(Iu), as in [15] or =-=[16]-=-, in order to damp out some oscillations in that functional; also, we establish some more refined estimates on the multilinear symbols appearing in those integrals. The main new difficulty is that, du...

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...se. Provided the 2In order to establish a global well-posedness result in L2 x (R2), it is instead necessary to obtain an a priori spacetime bound such as ‖u‖L4 t,x ([0,T]×R2) ≤ C(‖u0‖L2 x (R2)). See =-=[24]-=-, [4], [28] for further discussion. 3 The equation considered in [16] was also non-integrable, but because it was one-dimensional there was still enough cancellation to prevent the contribution of the...

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...e whole real line R. Both the local and global-in-time Cauchy problems for this NLS equation (1.1) have attracted a substantial literature [30], [10], [22], [20] [26], [5], [6], [17], [25], [9], [3], =-=[11]-=-. The local well-posedness theory is now well understood; in particular, one has local well-posedness in H s x (R2 ) for all s ≥ 0, and if s is strictly positive then a solution can be continued unles...

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... ) for all s ≥ 0, and in particular (1.3) holds for all s > 0. This conjecture remains open (though in the radial case, the higher dimensional analogue of this conjecture has recently been settled in =-=[29]-=-). However, there has been some progress in improving the s ≥ 1 results mentioned earlier. The first breakthrough was by Bourgain [5], [6], who established (1.3) (and hence global well-posedness in H ...

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... problem, in which we allow J to be the whole real line R. Both the local and global-in-time Cauchy problems for this NLS equation (1.1) have attracted a substantial literature [30], [10], [22], [20] =-=[26]-=-, [5], [6], [17], [25], [9], [3], [11]. The local well-posedness theory is now well understood; in particular, one has local well-posedness in H s x (R2 ) for all s ≥ 0, and if s is strictly positive ...

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...y in the global-in-time problem, in which we allow J to be the whole real line R. Both the local and global-in-time Cauchy problems for this NLS equation (1.1) have attracted a substantial literature =-=[30]-=-, [10], [22], [20] [26], [5], [6], [17], [25], [9], [3], [11]. The local well-posedness theory is now well understood; in particular, one has local well-posedness in H s x (R2 ) for all s ≥ 0, and if ...

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...H s x (R2 )) for all s > 3/5, using what is now referred to as the Fourier truncation method. In [17] the bound (1.3) was established for all s > 4/7, using the “I-method” developed by the authors in =-=[14]-=-, [15] (see also [23]). The main result of this paper is the following improvement: Theorem 1.1 (Main theorem). The bound (1.3) holds for all s > 1/2. In particular, the Cauchy problem (1.1) is global...

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...be the whole real line R. Both the local and global-in-time Cauchy problems for this NLS equation (1.1) have attracted a substantial literature [30], [10], [22], [20] [26], [5], [6], [17], [25], [9], =-=[3]-=-, [11]. The local well-posedness theory is now well understood; in particular, one has local well-posedness in H s x (R2 ) for all s ≥ 0, and if s is strictly positive then a solution can be continued...

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...s > 3/5, using what is now referred to as the Fourier truncation method. In [17] the bound (1.3) was established for all s > 4/7, using the “I-method” developed by the authors in [14], [15] (see also =-=[23]-=-). The main result of this paper is the following improvement: Theorem 1.1 (Main theorem). The bound (1.3) holds for all s > 1/2. In particular, the Cauchy problem (1.1) is globally well-posed in H s ...

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...bal-in-time problem, in which we allow J to be the whole real line R. Both the local and global-in-time Cauchy problems for this NLS equation (1.1) have attracted a substantial literature [30], [10], =-=[22]-=-, [20] [26], [5], [6], [17], [25], [9], [3], [11]. The local well-posedness theory is now well understood; in particular, one has local well-posedness in H s x (R2 ) for all s ≥ 0, and if s is strictl...

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...currently obtained by the “first-generation” I-method (i.e. without correction terms). A resonant decomposition similar to that employed here appeared previously in the work [7], and more recently in =-=[1]-=-. Inserting the above theorem into the results of [4] (which employ the pseudoconformal transform) we conclude that the equation (1.1) is globally well-posed with scattering when the initial data obey...

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...stablishes (1.3) for s = 1 (with bounds uniform in T), and with some additional arguments one can then deduce the same claim for s > 1 (with the best known bounds growing polynomially in T; see [26], =-=[12]-=-). The mass 1Global well-posedness and even scattering is known when the mass ‖u0‖L2 x (R2) is sufficiently small (see e.g. [8], [27]), or if suitable decay conditions (e.g. xu0 ∈ L2 x(R2 ) are also i...

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...nd Grillakis [21] had also obtained Theorem 1.1, in fact for s ≥ 1/2, by a different method based upon a new type of Morawetz inequality. The Fang-Grillakis interaction Morawetz estimate has recently =-=[13]-=- been improved and combined with the I-method (following the general scheme from [18]) to prove that (1.1) is globally well-posed in H s for s > 2/5. The techniques leading to the improved energy incr...

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...d the 2In order to establish a global well-posedness result in L2 x (R2), it is instead necessary to obtain an a priori spacetime bound such as ‖u‖L4 t,x ([0,T]×R2) ≤ C(‖u0‖L2 x (R2)). See [24], [4], =-=[28]-=- for further discussion. 3 The equation considered in [16] was also non-integrable, but because it was one-dimensional there was still enough cancellation to prevent the contribution of the resonant i...

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...ution equations which are currently obtained by the “first-generation” I-method (i.e. without correction terms). A resonant decomposition similar to that employed here appeared previously in the work =-=[7]-=-, and more recently in [1]. Inserting the above theorem into the results of [4] (which employ the pseudoconformal transform) we conclude that the equation (1.1) is globally well-posed with scattering ...

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...equation (1.1) is globally well-posed with scattering when the initial data obeys 〈x〉 s u0 ∈ L 2 x(R 2 ) for any s > 1/2. During the preparation of this manuscript, we learned that Fang and Grillakis =-=[21]-=- had also obtained Theorem 1.1, in fact for s ≥ 1/2, by a different method based upon a new type of Morawetz inequality. The Fang-Grillakis interaction Morawetz estimate has recently [13] been improve...

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...J to be the whole real line R. Both the local and global-in-time Cauchy problems for this NLS equation (1.1) have attracted a substantial literature [30], [10], [22], [20] [26], [5], [6], [17], [25], =-=[9]-=-, [3], [11]. The local well-posedness theory is now well understood; in particular, one has local well-posedness in H s x (R2 ) for all s ≥ 0, and if s is strictly positive then a solution can be cont...

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...allow J to be the whole real line R. Both the local and global-in-time Cauchy problems for this NLS equation (1.1) have attracted a substantial literature [30], [10], [22], [20] [26], [5], [6], [17], =-=[25]-=-, [9], [3], [11]. The local well-posedness theory is now well understood; in particular, one has local well-posedness in H s x (R2 ) for all s ≥ 0, and if s is strictly positive then a solution can be...

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... (i.e. without correction terms). A resonant decomposition similar to that employed here appeared previously in the work [7], and more recently in [1]. Inserting the above theorem into the results of =-=[4]-=- (which employ the pseudoconformal transform) we conclude that the equation (1.1) is globally well-posed with scattering when the initial data obeys 〈x〉 s u0 ∈ L 2 x(R 2 ) for any s > 1/2. During the ...