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## Resonant decompositions and the I-method for cubic nonlinear Schrödinger

Venue: | on R 2 . Disc. Cont. Dynam. Systems A |

Citations: | 28 - 4 self |

### Citations

408 |
Semilinear Schrödinger equations
- Cazenave
- 2003
(Show Context)
Citation Context ...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... |

353 | Interpolation Spaces, - Bergh, L6fstrSm - 1976 |

268 |
Nonlinear Schrödinger equations and sharp interpolation estimates
- Weinstein
- 1983
(Show Context)
Citation Context ...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... |

251 |
Nonlinear Dispersive Equations, Local and global Analysis
- Tao
(Show Context)
Citation Context ...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... |

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

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

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

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

86 |
Smoothing properties and retarded estimates for some dispersive evolution equations,
- Ginibre, Velo
- 1992
(Show Context)
Citation Context ...-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... |

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 ...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... |

59 |
On the blow up phenomenon of the critical nonlinear Schrodinger equation.
- Keraani
- 2006
(Show Context)
Citation Context ...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... |

58 | Ill posedness for nonlinear Schrödinger and wave equation. arXiv:math/0311048 [math.AP
- Christ, Colliander, et al.
- 2003
(Show Context)
Citation Context ...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... |

39 | Global well-posedness and scattering for the mass-critical defocusing NLS with spherical symmetry in higher dimensions
- Tao, Visan, et al.
(Show Context)
Citation Context ... ) 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 ... |

38 | Global wellposedness and scattering in the energy space for the critical nonlinear Schrödinger equation in R3 - Colliander, Keel, et al. |

34 |
On the growth of high Sobolev norms of solutions for KdV and Schrodinger equations,
- Staffilani
- 1997
(Show Context)
Citation Context ... 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 ... |

34 |
Scattering problem for nonlinear Schrödinger equations
- Tsutsumi
- 1985
(Show Context)
Citation Context ...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 ... |

33 |
Global well-posedness for KdV in Sobolev spaces of negative index,
- Colliander, Keel, et al.
- 2001
(Show Context)
Citation Context ...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... |

32 |
On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations
- Birnir, Kenig, et al.
- 1996
(Show Context)
Citation Context ...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... |

31 | Local and global well-posedness of wave maps on R 1+1 for rough data
- Keel, Tao
- 1998
(Show Context)
Citation Context ...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 ... |

30 |
On nonlinear Schrödinger equations, II. H s -solutions and unconditional wellposedness
- Kato
- 1995
(Show Context)
Citation Context ...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... |

29 | Almost global existence for Hamiltonian semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds.
- Bambusi, Delort, et al.
- 2005
(Show Context)
Citation Context ...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... |

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

24 |
Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on R2
- Colliander, Grillakis, et al.
- 2007
(Show Context)
Citation Context ...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... |

24 | A pseudoconformal compactification of the nonlinear Schrodinger equation and applications,
- Tao
- 2009
(Show Context)
Citation Context ...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... |

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

19 |
On the global existence of rough solutions of the cubic defocusing Schrodinger equation in R2+1.
- Fang, Grillakis
- 2007
(Show Context)
Citation Context ...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... |

13 |
A note on the nonlinear Schrödinger equation in weak Lp spaces
- Cazenave, Vega, et al.
(Show Context)
Citation Context ...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... |

12 |
Dispersive estimates and the 2D cubic NLS equation
- Planchon
(Show Context)
Citation Context ...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... |

7 | Global well-posedness in Sobolev space implies global existence for weighted L2 initial data for L2 critical NLS
- Blue, Colliander
- 2010
(Show Context)
Citation Context ... (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 ... |

1 | Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R 3 - Tao |