### Citations

477 | Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, - Bourgain - 1993 |

209 | A bilinear estimate with application to the KdV equation,
- Kenig, Ponce, et al.
- 1996
(Show Context)
Citation Context ...e C2-well-posedness. The surprising part of this result was that the C∞ critical index s∞c (KdV B) = −1 was lower that the one of the KdV equation ut + uxxx + uux = 0 for which s∞c (KdV ) = −1/2 (cf. =-=[13]-=-, [7]) and also lower than the C ∞ index s∞c (dB) = s 0 c(dB) = −1/2 (cf. [1], [8]) of the dissipative Burgers equation ut − uxx + uux = 0 . On the other hand, using the integrability theory, it was r... |

141 |
On the Cauchy problem for the Zakharov system.
- Ginibre, Tsutsumi, et al.
- 1997
(Show Context)
Citation Context ...t (in time) in [−T, T ], it holds ‖∂x(uv)‖N−1ε . T ν‖u‖S0ε ‖v‖S−1ε . (5.1) Proof. It suffices to slightly modify the proof of Proposition 4.1 to make use of the following result that can be found in [=-=[11]-=-, Lemma 3.1] (see also [[15], Lemma 3.6]): For any θ > 0, there exists µ = µ(θ) > 0 such that for any smooth function f with compact support in time in [−T, T ],∥∥∥∥∥F−1t,x ( f̂(τ, k) 〈τ − k3〉θ )∥∥∥∥∥... |

128 | Asymptotics, frequency modulation and low regularity ill-posedness for canonical defocusing equations, - Chirst, Colliander, et al. - 2002 |

83 | problème de Cauchy pour des EDP semi-linéaires périodiques en variables d’espace (d’après Bourgain), Astérisque - Ginibre, Le - 1996 |

72 | On global existence and scattering for the wave maps equation.
- Tataru
- 2001
(Show Context)
Citation Context ...1(R) is not continuous for the topology inducted by Hs, s < −1, with values even in D′(R). To reach the critical Sobolev space H−1(R) they adapted the refinement of Bourgain’s spaces that appeared in =-=[19]-=- and [18] to the framework developed in [15]. The proof of the main bilinear estimate used in a crucial way the Kato smoothing effect that does not hold on the torus. Our aim here is to give the new i... |

51 |
On the Cauchy problem for the Kadomtsev-Petviashvili equation,
- BOURGAIN
- 1993
(Show Context)
Citation Context ...ce-time frequencies region. Note that our resolution space will still be embedded in C([0, T ];H−1(T)) and that, to get the L∞([0, T ];H−1(T))-estimate in this region, we use an idea that appeared in =-=[4]-=-. Finally, once the well-posedness result is proved, the proof of the ill-posedness result follows exactly the same lines as in [14]. It is due to a high to low frequency cascade phenomena that was fi... |

51 |
Global wellposedness of KdV
- Kappeler, Topalov
(Show Context)
Citation Context ...rder to make our result more transparent, let us first introduce different notions of well-posedness (and consequently ill-posedness) related to the smoothness of the flow-map (see in the same spirit =-=[12]-=-, [9]). Throughout this paper we shall say that a Cauchy problem is (locally) C0-well-posed in some normed function space X if, for any initial data u0 ∈ X, there exist a radius R > 0, a time T > 0 an... |

50 |
Multilinear weighted convolution of L2 functions, and applications to non-linear dispersive equations,
- Tao
- 2001
(Show Context)
Citation Context ...e following crucial bilinear estimate. Proposition 4.1. Let 0 < ε < 1/12. Then for all u, v ∈ S−1ε it holds ‖∂x(uv)‖N−1ε . ‖u‖S−1ε ‖v‖S−1ε . (4.1) We will need the following sharp estimates proved in =-=[17]-=-. Lemma 4.1. Let u1 and u2 be two real valued L 2 functions defined on R×Z with the following support properties (τ, k) ∈ suppui ⇒ |k| ∼ Ni, 〈τ − k 3〉 ∼ Li, i = 1, 2. Then the following estimates hold... |

44 | Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation
- Bejenaru, Tao
(Show Context)
Citation Context ...the well-posedness result is proved, the proof of the ill-posedness result follows exactly the same lines as in [14]. It is due to a high to low frequency cascade phenomena that was first observed in =-=[2]-=- for a quadratic Schrödinger equation. In view of the result of Kappeler and Topalov for KdV it thus appears that, at least on the torus, even if the dissipation part of the KdV-Burgers equation1 all... |

28 |
Damping of solitary waves
- Ott, Sudan
- 1970
(Show Context)
Citation Context ...ies-Burgers (KdV-B) equation posed on the one dimensional torus T = R/2πZ: ut + uxxx − uxx + uux = 0 (1.1) where u = u(t, x) is a real valued function. This equation has been derived by Ott and Sudan =-=[16]-=- as an asymptotic model for the propagation of weakly nonlinear dispersive long waves in some physical contexts when dissipative effects occur. In order to make our result more transparent, let us fir... |

25 |
On the low regularity of the Korteweg-de Vries-Burgers equation,
- MOLINET, RIBAUD
- 2002
(Show Context)
Citation Context ...holds ‖∂x(uv)‖N−1ε . T ν‖u‖S0ε ‖v‖S−1ε . (5.1) Proof. It suffices to slightly modify the proof of Proposition 4.1 to make use of the following result that can be found in [[11], Lemma 3.1] (see also [=-=[15]-=-, Lemma 3.6]): For any θ > 0, there exists µ = µ(θ) > 0 such that for any smooth function f with compact support in time in [−T, T ],∥∥∥∥∥F−1t,x ( f̂(τ, k) 〈τ − k3〉θ )∥∥∥∥∥ L2t,x . T µ‖f‖ L2,2t,x . (5... |

22 | Scattering for the quartic generalised Korteweg-de Vries equation
- Tao
(Show Context)
Citation Context ...ot continuous for the topology inducted by Hs, s < −1, with values even in D′(R). To reach the critical Sobolev space H−1(R) they adapted the refinement of Bourgain’s spaces that appeared in [19] and =-=[18]-=- to the framework developed in [15]. The proof of the main bilinear estimate used in a crucial way the Kato smoothing effect that does not hold on the torus. Our aim here is to give the new ingredient... |

20 | Nonuniqueness and uniqueness in the initial-value problem for Burgers’ equation.
- Dix
- 1996
(Show Context)
Citation Context ...dex s∞c (KdV B) = −1 was lower that the one of the KdV equation ut + uxxx + uux = 0 for which s∞c (KdV ) = −1/2 (cf. [13], [7]) and also lower than the C ∞ index s∞c (dB) = s 0 c(dB) = −1/2 (cf. [1], =-=[8]-=-) of the dissipative Burgers equation ut − uxx + uux = 0 . On the other hand, using the integrability theory, it was recently proved in [12] that the flow-map of KdV equation can be uniquely continuou... |

16 |
Sharp global wellposedness results for periodic and non-periodic
- Colliander, Keel, et al.
(Show Context)
Citation Context ...ell-posedness. The surprising part of this result was that the C∞ critical index s∞c (KdV B) = −1 was lower that the one of the KdV equation ut + uxxx + uux = 0 for which s∞c (KdV ) = −1/2 (cf. [13], =-=[7]-=-) and also lower than the C ∞ index s∞c (dB) = s 0 c(dB) = −1/2 (cf. [1], [8]) of the dissipative Burgers equation ut − uxx + uux = 0 . On the other hand, using the integrability theory, it was recent... |

14 | ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case
- Molinet, Vento, et al.
- 2009
(Show Context)
Citation Context ...uniquely continuously extended in H−1(T). Therefore, on the torus, KdV is C0-well-posed in H−1 if one takes as uniqueness class, the class of strong limit in C([0, T ];H−1(T)) of smooth solutions. In =-=[14]-=- the authors completed the result of [15] in the real line case by proving that the KdV-Burgers equation is analytically well-posed in H−1(R) and C0-ill-posed in Hs(R) for s < −1 in the sense that the... |

11 |
The initial-value problem for the generalized Burgers equation, Differential Integral Equations 9
- Bekiranov
- 1996
(Show Context)
Citation Context ...al index s∞c (KdV B) = −1 was lower that the one of the KdV equation ut + uxxx + uux = 0 for which s∞c (KdV ) = −1/2 (cf. [13], [7]) and also lower than the C ∞ index s∞c (dB) = s 0 c(dB) = −1/2 (cf. =-=[1]-=-, [8]) of the dissipative Burgers equation ut − uxx + uux = 0 . On the other hand, using the integrability theory, it was recently proved in [12] that the flow-map of KdV equation can be uniquely cont... |

6 | Periodic Korteveg de Vries equation with measures as initial data - Bourgain - 1993 |

6 | Nonlinear Schrödinger equations in inhomogeneous media: wellposedness and illposedness of the Cauchy problem
- Gérard
(Show Context)
Citation Context ...o make our result more transparent, let us first introduce different notions of well-posedness (and consequently ill-posedness) related to the smoothness of the flow-map (see in the same spirit [12], =-=[9]-=-). Throughout this paper we shall say that a Cauchy problem is (locally) C0-well-posed in some normed function space X if, for any initial data u0 ∈ X, there exist a radius R > 0, a time T > 0 and a u... |