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## Global well-posedness for a periodic nonlinear Schrödinger equation (2006)

Venue: | in 1D and 2D, arXiv: math.AP/0602660v1 |

Citations: | 11 - 2 self |

### Citations

475 |
Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part I, Geometric and Functional Analysis 3
- Bourgain
- 1993
(Show Context)
Citation Context ...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... |

277 |
Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle,
- Kenig, Ponce, et al.
- 1993
(Show Context)
Citation Context ...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‖... |

189 |
The Cauchy problem for the critical nonlinear Schrodinger equation in Hs ,
- CAZENAVE, WEISSLER
- 1990
(Show Context)
Citation Context ...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 ... |

179 |
Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity,
- Bourgain
- 1998
(Show Context)
Citation Context ...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... |

168 |
Global Solutions of Nonlinear Schrödinger Equations
- Bourgain
(Show Context)
Citation Context ...,− 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 ... |

141 | On the Cauchy problem for the Zakharov system. - Ginibre, Tsutsumi, et al. - 1997 |

128 |
On nonlinear Schrödinger equations,
- Kato
- 1987
(Show Context)
Citation Context ...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 ... |

108 |
The global Cauchy problem for the non linear Schrödinger equation revisited
- Ginibre, Velo
- 1985
(Show Context)
Citation Context ...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 ... |

88 | Almost conservation laws and global rough solutions to a nonlinear Schrodinger equation,
- Colliander, Keel, et al.
- 2002
(Show Context)
Citation Context ...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... |

79 |
The number of integral points on arcs and ovals
- Bombieri, Pila
- 1989
(Show Context)
Citation Context ...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... |

59 | Tao: A refined global well-posedness result for Schrodinger equations with derivative, preprint arXiv:math.AP/0110026,
- Colliander, Keel, et al.
- 2001
(Show Context)
Citation Context ...,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 ... |

52 | A refined global well-posedness for the Schrodinger equations with derivative,
- Colliander, Keel, et al.
- 2002
(Show Context)
Citation Context ...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... |

41 |
global well-posedness for the KDV and modified
- Colliander, Keel, et al.
(Show Context)
Citation Context ...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... |

35 |
Geometrisches zur Zahlenlehre
- PICK
(Show Context)
Citation Context ...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.... |

30 |
Quadratic forms for the 1-D semilinear Schrodinger equation,
- Kenig, L
- 1996
(Show Context)
Citation Context ...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... |

29 |
Bilinear estimates and applications to 2D
- Colliander, Delort, et al.
(Show Context)
Citation Context ...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... |

27 |
Space-time estimates for null gauge forms and nonlinear Schrodinger equations,
- Ozawa, Tsutsumi
- 1998
(Show Context)
Citation Context ...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... |

19 |
A remark on normal forms and the ’I-Method’ for periodic NLS, preprint
- Bourgain
- 2003
(Show Context)
Citation Context ...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... |

18 |
The Cauchy problem for the semilinear quintic Schrodinger equation in one dimension.
- Tzirakis
- 2005
(Show Context)
Citation Context ...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... |

16 |
Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations. Ergodic Theory Dynam
- Bourgain
(Show Context)
Citation Context ...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... |

16 |
On Strichartz’s Inequalities and the Nonlinear Schrödinger Equation on Irrational Tori
- Bourgain
- 2007
(Show Context)
Citation Context ... 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... |

11 | Mean square discrepancy bounds for the number of lattice points in large convex bodies. II: planar domains,
- Iosevich, Sawyer, et al.
- 2004
(Show Context)
Citation Context ...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|... |

10 |
On the Cauchy- and periodic boundary value problem for a certain class of derivative nonlinear Schrödinger equations
- Grünrock
- 2000
(Show Context)
Citation Context ...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... |

7 |
Multilinear periodic KdV estimates, and applications
- Colliander, Keel, et al.
(Show Context)
Citation Context ... 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... |

4 |
Le problme de Cauchy pour des EDP semi-linaires priodiques en variables d’espace (d’aprs Bourgain
- Ginibre
- 1996
(Show Context)
Citation Context ...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 ... |