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...ar, improvements have utilized an almost Morawetz estimate. (See [5], [10].) Currently, the best known result for n = 2 is Theorem 1.1 (1.1) is globally well-posed when n = 2 for s > 1 4 . Proof: See =-=[13]-=-. In [11], the I-method was extended to prove global well-posedness results for (1.1) when n ≥ 3. The chief difficulty with extending to n ≥ 3 is that the nonlinearity |u| 4/n u is no longer ”algebrai...

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...0. (4.19) Therefore, ∫ T 0 ∫ R |u(t, x)|8dxdt . | ∫ T 0 ∫ ∂t(aj(z)Im[ω(t, z)∂jω(t, z)]dzdt| . ‖u‖2 L∞t Ḣ 1/2([0,T ]×R) ‖u0‖ 6 L2(R). (4.20) Next, we prove an almost Morawetz estimate (see [5], [11], =-=[15]-=- for the two dimensional case; [5], [12], [6] for discussion of the one dimensional case). If u(t, x) solves (4.1), then Iu(t, x) solves iIut(t, x) + ∆Iu(t, x) = I(|u(t, x)| 4u(t, x)) = N . (4.21) Spl...

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...zirakis [7] extended it to s > 25 by means of the I-method and the improved interaction Morawetz inequalities. Laterly, Colliander and Roy [18] improved these results to s > 13 . Subsequently, Dodson =-=[19]-=- showed the global well-posedness for s > 14 by improving the almost Morawetz estimates from [7]. The study of a low regularity problem stimulates the development of the scattering in L2(Rd) for the m...

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..., 2. Since we are truncating to low frequencies, our method is very similar to the almost Morawetz estimates that are often used in conjunction with the I-method. (See [1], [8], [9], [10], [12], [6], =-=[17]-=-, [15], [13], and [14] for more information on the I-method.) We will also use the estimates in §§4− 5 in order to preculde the case (1.15) by proving additional regularity. This method is similar to ...

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...se.) Since we are truncating to low frequencies, our method is very similar to the almost Morawetz estimates that are often used in conjunction with the I-method. (See [1], [8], [9], [10], [12], [6], =-=[19]-=-, [15], [13], and [14] for more information on the I-method.) 2 Function Spaces and linear estimates Linear Strichartz Estimates: Definition 2.1 A pair (p, q) will be called an admissible pair for d =...