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...ard’s fixed point method and controls the nonlinearity in the iteration process by using Strichartz’s type inequalities, then the problem can be shown to be locally well-posed for all s > 0. Bourgain =-=[1]-=- adjusted this approach to the periodic case, where there are certain difficulties due mainly to a “lack of dispersion”. In [1] number theoretic methods were used to show that (1.1)-(1.2) is locally w...

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...ǫ � ‖Iu‖ 5 X 1,1/2+. By the invariant lemma we know that to prove such an estimate it suffices to prove ‖|u| 4 u‖ X s,−1/2+2ǫ � ‖u‖ 5 X s,1/2+. By Hölder’s inequality combined with the Leibnitz rule (=-=[22]-=-) we have ‖|u| 4 u‖Xs,−1/2+2ǫ ≤ ‖|u| 4 u‖Xs,0 � ‖J s u‖L4 t L4 x ‖u‖4 L16 t L16 x , where J s is the Bessel potential of order s. Using (2.6) with λ = 1 we obtain ‖u‖L16 t L16 x � ‖u‖X5/16+,1/2+ � ‖u‖...

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...e interval [0,T]. If T = ∞ we say that a Cauchy problem is globally well-posed. In the case when x ∈ Rd local well-posedness for (1.1)-(1.2) in Hs (Rd) has been studied extensively (see, for example, =-=[8, 16, 20]-=-). In particular if one solves the equivalent integral equation by Picard’s fixed point method and controls the nonlinearity in the iteration process by using Strichartz’s type inequalities, then the ...

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...l well posedness for any s > 4/7. In both results, apart from the application of the I-method, a key element in the proof is the existence of a bilinear refined Strichartz’s estimate due to Bourgain, =-=[4]-=- (see also [9] [10]). In general dimensions d ≥ 2 this estimate reads as follows. Let f and g be any two Schwartz functions whose Fourier transforms are supported in |k| ∼ N1 and |k| ∼ N2 respectively...

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...,− 9 20 � ‖Iu‖3 X 1,1/2+ � ‖u‖3 X s,1/2+. Since the rest of the proof is identical to Proposition 3.4 we briefly recall how (4.5) can be obtained. Note that this estimate was first proved by Bourgain =-=[2]-=- and his proof is based upon a local variant of the well known periodic Strichartz estimate ⎛ ⎝ ∑ ∫ dτ(1 + |τ − |k| 2 |) 2b1 ⎞ 1 2 2 |û(k,τ)| ⎠ (4.6) ‖u‖L4 t L4 � Ns1 x k∈Q where b1 > 1−min(1/2,s1) 2 ...

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...e interval [0,T]. If T = ∞ we say that a Cauchy problem is globally well-posed. In the case when x ∈ Rd local well-posedness for (1.1)-(1.2) in Hs (Rd) has been studied extensively (see, for example, =-=[8, 16, 20]-=-). In particular if one solves the equivalent integral equation by Picard’s fixed point method and controls the nonlinearity in the iteration process by using Strichartz’s type inequalities, then the ...

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...e interval [0,T]. If T = ∞ we say that a Cauchy problem is globally well-posed. In the case when x ∈ Rd local well-posedness for (1.1)-(1.2) in Hs (Rd) has been studied extensively (see, for example, =-=[8, 16, 20]-=-). In particular if one solves the equivalent integral equation by Picard’s fixed point method and controls the nonlinearity in the iteration process by using Strichartz’s type inequalities, then the ...

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...tic transformations that, in some sense, reduce the nonlinear part of the equation to its “essential part”. The I-method, introduced by J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao in =-=[10, 13, 14]-=-, is based on the almost conservation of certain modified Hamiltonians. These two methods together yield global well-posedness in H s (T) with s ≥ 1/2. The refined trilinear Strichartz inequality esta...

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...are at most two lattice points. )1 − 3 R 2 Case 2: 2k N2 N1 ≥ ( 3 3 4 M . In this case we approximate the number of lattice points on γn by the number of lattice points on Cλ n (see for example [1] , =-=[7]-=-): (4.28) #C λ n � Rǫ M ∼ (λN1) ǫ � (λN2) 3ǫ for any ǫ > 0. Combining the estimate in (4.25), Case 1 and Case 2 we conclude that N1 #S � λ 2 + λ 2 (λN2) ǫ , for any ǫ > 0. Since λ,N2 ≥ 1, together wit...

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...,2, see, for example, [4, 10, 23]. Such an estimate allows us to use the I-method machinery in an efficient way. More precisely, when d = 1, beside rescaling, we follow the argument in [25] (see also =-=[12]-=-), where one applies the I-operator to the equation on Tλ, and defines a modified second energy functional as the energy corresponding to the new “I-system”. We prove that such modified second energy ...

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...tic transformations that, in some sense, reduce the nonlinear part of the equation to its “essential part”. The I-method, introduced by J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao in =-=[10, 13, 14]-=-, is based on the almost conservation of certain modified Hamiltonians. These two methods together yield global well-posedness in H s (T) with s ≥ 1/2. The refined trilinear Strichartz inequality esta...

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...tic transformations that, in some sense, reduce the nonlinear part of the equation to its “essential part”. The I-method, introduced by J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao in =-=[10, 13, 14]-=-, is based on the almost conservation of certain modified Hamiltonians. These two methods together yield global well-posedness in H s (T) with s ≥ 1/2. The refined trilinear Strichartz inequality esta...

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...6. Its proof is based on some number theoretic facts that we recall in the following three lemmas; see also related estimates in the work of Bourgain [6]. The following lemma is known as Pick’s Lemma =-=[24]-=-: Lemma 4.3. Let Ar be the area of a simply connected lattice polygon. Let E denote the number of lattice points on the polygon edges and I the number of lattice points in the interior of the polygon....

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...al well-posedness results in the periodic context is exactly the absence of a quantitative refinement of the bilinear Strichartz’s estimate. The reader can consult the paper of Kenig, Ponce and Vega, =-=[21]-=-, where the difference between the real and the periodic case is clearly exposed when one tries to prove bilinear estimates in different functional spaces. In order to overcome the non-availability of...

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...ss for any s > 4/7. In both results, apart from the application of the I-method, a key element in the proof is the existence of a bilinear refined Strichartz’s estimate due to Bourgain, [4] (see also =-=[9]-=- [10]). In general dimensions d ≥ 2 this estimate reads as follows. Let f and g be any two Schwartz functions whose Fourier transforms are supported in |k| ∼ N1 and |k| ∼ N2 respectively. Then we have...

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...ated in frequency space. The constant in the inequality is quantified in terms of λ. As λ → ∞, our estimate reduces to the refined bilinear Strichartz’s inequality1 on Rd , d = 1,2, see, for example, =-=[4, 10, 23]-=-. Such an estimate allows us to use the I-method machinery in an efficient way. More precisely, when d = 1, beside rescaling, we follow the argument in [25] (see also [12]), where one applies the I-op...

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...ith periodic boundary data is considered. We show that the problem is globally well posed in H s (T d ), for s > 4/9 and s > 2/3 in 1D and 2D respectively, confirming in 2D a statement of Bourgain in =-=[3]-=-. We use the “I-method”. This method allows one to introduce a modification of the energy functional that is well defined for initial data below the H 1 (T d ) threshold. The main ingredient in the pr...

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...lementary number theoretic techniques. In order to present our method, we briefly review global well-posedness results on R d . Using an approximation of the modified energy in the I-method, Tzirakis =-=[25]-=- showed that the Cauchy problem (1.1)-(1.2) is globally well posed in H s (R) for any s > 4/9. For u0 ∈ H s (R 2 ) the best known global well posedness result for (1.1)-(1.2) is in [14] where the auth...

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...extend the global theory to infinite energy initial data. Bourgain [3] established global well posedness for (1.1) in H s (T) for any s > 1/2−, by combining a “normal form” reduction method (see also =-=[5]-=-), the “I-method”, and a refined trilinear Strichartz type inequality. The normal form reduction is achieved by symplectic transformations that, in some sense, reduce the nonlinear part of the equatio...

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... for λ-periodic functions stated in Proposition 4.6. Its proof is based on some number theoretic facts that we recall in the following three lemmas; see also related estimates in the work of Bourgain =-=[6]-=-. The following lemma is known as Pick’s Lemma [24]: Lemma 4.3. Let Ar be the area of a simply connected lattice polygon. Let E denote the number of lattice points on the polygon edges and I the numbe...

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...2 (|γ|R −1 ) 3 < 1 2 , where to obtain the last inequality we used the assumption that |γ| < ( 3 4 R) 1/3 . Therefore (4.7) is proved. □ Also we recall the following result of Gauss, see, for example =-=[19]-=- Lemma 4.5. Let K be a convex domain in R 2 . If N(λ) = #{Z 2 ∩ λK},18 DANIELA DE SILVA, NATAˇSA PAVLOVIĆ, GIGLIOLA STAFFILANI, AND NIKOLAOS TZIRAKIS then, for λ >> 1 N(λ) = λ 2 |K| + O(λ), where |K|...

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...4 t L 4 x � ‖u‖ X 0,b, for any b > 3 8 , and (2.4) ‖u‖L6 t L6 x � ‖u‖X0+,1/2+. We note that (2.2) remains true for the λ-periodic problem. This is the case also for (2.3). The proof is essentially in =-=[18]-=-. In fact, it is enough to show, by the standard Xs,b method, that 1 λ sup (ξ,τ)∈ 1 λ Z×R ∑ k1∈ 1 λ Z 〈τ + k 2 1 + (k − k1) 2 〉 1−4b � C. This is done in [18], the only difference being that there are...

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... well-posed for any s > 7 20 in [0,δ] ∼ [0,1]. Proof. By Duhamel’s formula as in Proposition 3.4 we have that ‖Iu‖ X 1,1/2+ � ‖Iu0‖ H 1 + δ 1 20 −ǫ ‖I(|u| 2 u)‖ X 1,− 9 20 By the “invariant lemma” in =-=[11]-=- we know that the estimate is implied by ‖I(|u| 2 u)‖ X 1,− 9 20 (4.5) ‖|u| 2 u‖ X s,− 9 20 � ‖Iu‖3 X 1,1/2+ � ‖u‖3 X s,1/2+. Since the rest of the proof is identical to Proposition 3.4 we briefly rec...

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...ut − ∆u = 0, x ∈ [0,λ] d , that is ∫ Uλ(t)u0(x) = e 2πikx−(2πk)2it û0(k)(dk)λ. We denote by X s,b = X s,b (T d λ ×R) the completion of S(Td λ ×R) with respect to the following norm, see, for example, =-=[15]-=-GLOBAL WELL-POSEDNESS FOR A PERIODIC NONLINEAR SCHRÖDINGER EQUATION 5 ‖u‖Xs,b = ‖Uλ(−t)u‖H s xHb t = ‖〈k〉s 〈τ − 4π 2 k 2 〉 b ũ(k,τ)‖ L2 τ L2 , (dk) λ where ũ(k,τ) is the space-time Fourier Transform ...