#### DMCA

## 4 WELL-POSEDNESS AND SCATTERING FOR NLS ON Rd × T IN THE ENERGY SPACE

Citations: | 2 - 0 self |

### Citations

509 | Endpoints Strichartz estimates
- Keel, Tao
- 1998
(Show Context)
Citation Context ...s about the proof of (2.3). Given any q ≥ 2 for d ≥ 3 (or q > 2 for d = 2, q ≥ 4 for d = 1) we fix the unique 2 ≤ r̃ <∞ such that (2.4) 2 q + d r̃ = d 2 . Hence by the usual Strichartz estimates (see =-=[12]-=-) we get: ‖e−it∆xh‖LqtLr̃x ≤ C‖h‖L2x . In turn it implies ‖e−it∆xh‖LqtW s,r̃ x ≤ C‖h‖Hsx . Notice that if s· r̃ < d then we conclude by the sharp Sobolev embeddingW s,r̃x ⊂ L r x (here r is precisely ... |

408 |
Semilinear Schrödinger Equations
- Cazenave
- 2003
(Show Context)
Citation Context .... We conclude by choosing t0 large enough and by recalling Proposition 7.1. 22 NIKOLAY TZVETKOV AND NICOLA VISCIGLIA Proof of Theorem 1.3. It follows by Proposition 7.2 via a standard argument (see =-=[3]-=-). In fact by using the integral equation associated with (1.1) it is sufficient to prove that (7.4) lim t1,t2→∞ ‖ ∫ t2 t1 e−is∆x,y (u|u|α)ds‖H1x,y = 0 By combining Proposition 2.2 with a duality argu... |

187 |
The Cauchy problem for the critical nonlinear Schrödinger equation in H s Nonlinear Anal
- Cazenave, Weissler
- 1990
(Show Context)
Citation Context ...obal solution to (1.1), with 4/d < α < 4/(d− 1), then (7.1) u(t, x, y) ∈ Lqθt L rθ x H 1/2+δ y for some δ > 0. Proof. We will apply a H 1/2+δ y valued version of the analysis Hsx critical analysis of =-=[4]-=-. Notice that in Proposition 6.1 we have 0 < s < 12 and hence by choosing δ > 0 small , we can control ‖.‖ HsxH 1/2+δ y by ‖.‖H1x,y . By combining this fact with Remark 6.2 we get ‖u(t, x, y)‖ L qθ t>... |

138 | Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds
- Burq, Gérard, et al.
(Show Context)
Citation Context ...rtz estimates for eit∆x . In the cases d = 1 for every α > 0 and d = 2, 3 for the H1-critical nonlinearity α = 4/(d− 1), the proof of Theorem 1.1 can also be deduced respectively from the analysis in =-=[2]-=- and [13]. The main point in our approach is that it works in Rd ×T for every d ≥ 1 and moreover it gives some crucial controls of space-time global norms which are of importance for the scattering an... |

115 |
Scattering theory in the energy space for a class of nonlinear Schrödinger equations
- Ginibre, Velo
- 1985
(Show Context)
Citation Context ... the Cauchy problem (1.1) reduces to L2subcritical NLS in Rd, and at the best of our knowledge no H1-scattering result is available in this situation . It is well-known, since the very classical work =-=[8]-=-, that a key tool to prove scattering for NLS in the euclidean setting Rd, with nonlinearities which are both energy subcritical and L2-supercritical, is the proof of the time-decay of the potential e... |

113 | Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R3
- Colliander, Keel, et al.
(Show Context)
Citation Context ...l, see also Remark 1.7 on this point). NLS ON Rd × T 3 A basic tool that we will use is a suitable version in the partially periodic setting of the interaction Morawetz estimates, first introduced in =-=[6]-=- to study the energy critical NLS in the euclidean space R3. Starting from this work the interaction Morawetz estimates have been exploited in several other papers ([5], [9], [15], [16], [20]), in par... |

66 |
Remarks on the energy scattering for nonlinear Klein-Gordon and Schrodinger equations.
- Nakanishi
- 2001
(Show Context)
Citation Context ... estimates have been exploited in several other papers ([5], [9], [15], [16], [20]), in particular they have been used to provide new and simpler proofs of the classical scattering results in [8] and =-=[14]-=-. We emphasize that we make use of the interaction Morawetz estimates from a different point of view compared with the results above. In particular along the proof of Proposition 1.6 below we are able... |

53 | The nonlinear Schrödinger equation with combined power-type nonlinearities
- Tao, Vişan, et al.
(Show Context)
Citation Context ...t introduced in [6] to study the energy critical NLS in the euclidean space R3. Starting from this work the interaction Morawetz estimates have been exploited in several other papers ([5], [9], [15], =-=[16]-=-, [20]), in particular they have been used to provide new and simpler proofs of the classical scattering results in [8] and [14]. We emphasize that we make use of the interaction Morawetz estimates fr... |

38 | Inhomogeneous Strichartz estimates
- Foschi
(Show Context)
Citation Context ...o extend our analysis to higher dimensional y dependence. A key tool in order to perform a fixed point argument in the space (1.2) are the inhomogenous Strichartz estimates associated with eit∆x (see =-=[7]-=-, [19]). The second step is the proof of the unconditional uniqueness in the space Ct(H 1 x,y). We underline that the proof of Theorem 1.1 in the range of nonlinearity 0 < α < 4/d, can be obtained fol... |

32 |
Bilinear virial identities and applications
- Planchon, Vega
(Show Context)
Citation Context ..., first introduced in [6] to study the energy critical NLS in the euclidean space R3. Starting from this work the interaction Morawetz estimates have been exploited in several other papers ([5], [9], =-=[15]-=-, [16], [20]), in particular they have been used to provide new and simpler proofs of the classical scattering results in [8] and [14]. We emphasize that we make use of the interaction Morawetz estima... |

24 | Tensor products and correlation estimates with applications to nonlinear Schrödinger equations
- Colliander, Grillakis, et al.
(Show Context)
Citation Context ... estimates, first introduced in [6] to study the energy critical NLS in the euclidean space R3. Starting from this work the interaction Morawetz estimates have been exploited in several other papers (=-=[5]-=-, [9], [15], [16], [20]), in particular they have been used to provide new and simpler proofs of the classical scattering results in [8] and [14]. We emphasize that we make use of the interaction Mora... |

20 |
Inhomogeneous Strichartz estimates for the Schrödinger equation, preprint
- Vilela
(Show Context)
Citation Context ...end our analysis to higher dimensional y dependence. A key tool in order to perform a fixed point argument in the space (1.2) are the inhomogenous Strichartz estimates associated with eit∆x (see [7], =-=[19]-=-). The second step is the proof of the unconditional uniqueness in the space Ct(H 1 x,y). We underline that the proof of Theorem 1.1 in the range of nonlinearity 0 < α < 4/d, can be obtained following... |

12 | Global well-posedness of the energy-critical defocusing NLS
- Ionescu, Pausader
(Show Context)
Citation Context ...mates for eit∆x . In the cases d = 1 for every α > 0 and d = 2, 3 for the H1-critical nonlinearity α = 4/(d− 1), the proof of Theorem 1.1 can also be deduced respectively from the analysis in [2] and =-=[13]-=-. The main point in our approach is that it works in Rd ×T for every d ≥ 1 and moreover it gives some crucial controls of space-time global norms which are of importance for the scattering analysis. T... |

8 |
Quadratic Morawetz inequalities and asymptotic completeness in the energy space for nonlinear Schrödinger and Hartree equations, Quart
- GINIBRE, VELO
(Show Context)
Citation Context ...mates, first introduced in [6] to study the energy critical NLS in the euclidean space R3. Starting from this work the interaction Morawetz estimates have been exploited in several other papers ([5], =-=[9]-=-, [15], [16], [20]), in particular they have been used to provide new and simpler proofs of the classical scattering results in [8] and [14]. We emphasize that we make use of the interaction Morawetz ... |

8 | On scattering for the quintic defocusing nonlinear Schrödinger equation
- Hani, Pausader
(Show Context)
Citation Context ...e the unique global solution to (1.1). Then there exist f± ∈ H 1 x,y such that lim t→±∞ ‖u(t, x, y)− e−it∆x,yf±‖H1x,y = 0. Remark 1.4. Concerning scattering results for NLS in product spaces we quote =-=[10]-=- where it is studied the quintic NLS on R × T2. We also underline that using the arguments of [13] (see also [10]) one may obtain that for d = 2, 3, the result of Theorem 1.3 also holds for the H1 cri... |

8 | data scattering for the nonlinear Schrödinger equation on product spaces, Comm. P.D.E. 37 (2012) 125–135
- Tzvetkov, Visciglia, et al.
(Show Context)
Citation Context ...RGY SPACE NIKOLAY TZVETKOV AND NICOLA VISCIGLIA Abstract. We study the Cauchy problem and the large data H1 scattering for energy subcritical NLS posed on Rd × T. 1. Introduction In our previous work =-=[18]-=-, we considered the nonlinear Schrödinger equation on a product space Rd ×M , where M is a compact riemannian manifold. We have seen this problem as a kind of vector valued nonlinear Schrödinger equ... |

7 | Modified scattering for the cubic Schrödinger equation on product spaces and applications
- Hani, Pausader, et al.
(Show Context)
Citation Context ... , where M is a compact riemannian manifold. We have seen this problem as a kind of vector valued nonlinear Schrödinger equation on Rd and we were able to get small data scattering results (cf. also =-=[11]-=- for small data modified scattering results). Our goal here is to extend this view point to a large data problem in the very particular case when M is the one dimensional torus. Therefore, our aim in ... |

5 | On the decay of solutions to a class of defocusing NLS
- VISCIGLIA
(Show Context)
Citation Context ...oduced in [6] to study the energy critical NLS in the euclidean space R3. Starting from this work the interaction Morawetz estimates have been exploited in several other papers ([5], [9], [15], [16], =-=[20]-=-), in particular they have been used to provide new and simpler proofs of the classical scattering results in [8] and [14]. We emphasize that we make use of the interaction Morawetz estimates from a d... |

3 | The Nonlinear Schrödinger equation ground states on product spaces, 2012, Analysis & PDE
- Terracini, Tzvetkov, et al.
(Show Context)
Citation Context .... The second step is the proof of the unconditional uniqueness in the space Ct(H 1 x,y). We underline that the proof of Theorem 1.1 in the range of nonlinearity 0 < α < 4/d, can be obtained following =-=[17]-=-, where it is not needed the use of inhomogeneous Strichartz estimates for eit∆x . In the cases d = 1 for every α > 0 and d = 2, 3 for the H1-critical nonlinearity α = 4/(d− 1), the proof of Theorem 1... |

1 |
Drumond Silva, Scattering for nonlinear Schroedinger equation under partial harmonic confinement
- Antonelli, Carles, et al.
(Show Context)
Citation Context ...imated, then we can be interpolated with the bound ‖u‖L∞t H1x , and hence we get the needed space-time integrality necessary to prove scattering for d ≥ 4. Remark 1.9. We underline that arguing as in =-=[1]-=-, where it is studied NLS with a partially confining potential, one can prove the following version of interaction Morawetz estimate: (1.6) ∫ R ∫ Rd ∣∣|Dx| 3−d2 ( ∫ T |u(t, x, y)|2dy) ∣∣2dxdt <∞ provi... |