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## On the persistence properties of solutions of nonlinear dispersive equations in weighted Sobolev spaces (2011)

Venue: | RIMS KÔKYÛROKU BESSATSU, B26 |

Citations: | 6 - 2 self |

### Citations

477 |
Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations,
- Bourgain
- 1993
(Show Context)
Citation Context ...m(R) ∩ L2(|x|mdx)). It should be remarked that in the cases k = 1 and k = 3 the proof of the local theory in Theorem E is based on the spaces Xs,b introduced in the context of dispersive equations in =-=[5]-=-. For all the other powers k one has a local existence theory based on a contraction principle in a spaces defined by mixed norms of the type Lp(R : Lq([0, T ])) or Lq([0, T ] : Lp(R)) (see [37]). Thi... |

342 |
Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations,
- Strichartz
- 1977
(Show Context)
Citation Context ...1 < p <∞ and s ∈ R (2.2) Lps(R n) ≡ (1−∆)−s/2Lp(Rn) = J−s/2Lp(Rn), ‖ · ‖s,p ≡ ‖(1−∆) s · ‖p, with L2s(R n) = Hs(Rn), (b) the pair of indices (q, p) in (2.1) are given by the Strichartz estimates (see =-=[57]-=- and [21]): (2.3) ( ∫ ∞ −∞ ‖eit∆u0‖ q pdt) 1/q ≤ c‖u0‖2, where n 2 = 2 q + n p , 2 ≤ p ≤ ∞, if n = 1, 2 ≤ p < 2n/(n− 2), if n ≥ 2. The value sc = n/2 − 2/(a − 1) in Theorem A is determined by a scalin... |

325 |
The Nonlinear Schrödinger Equation, Self-focusing and Wave Collapse
- Sulem, Sulem
- 1999
(Show Context)
Citation Context ...a number of different physical systems. Also they have been studied because of their relation to inverse scattering theory [20]. The NLS arises as a model in several different physical phenomena (see =-=[58]-=- and references therein). In the particular, case n = 1 and a = 3 it has been shown to be completely integrable [63]. The BO equation (1.3) was first deduced in [3] and [51] as a model for long intern... |

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 ...is LWP in Hs(R) for s ≥ s∗k = (k − 4)/2k. The result s > −3/4 for the case k = 1 was established in [38]. The limiting value s = −3/4 was obtained in [11]. The result for the case k = 2 was proven in =-=[37]-=-. The result s > −1/6 for the case k = 3 was given in [22]. The limiting value s = −1/6 was obtained in [60]. The proof of the cases k ≥ 4 was given in [37]. The above local results apply to both real... |

229 |
Weighted norm inequalities for the Hardy maximal function.
- Muckenhoupt
- 1972
(Show Context)
Citation Context ...s = r in (4.2). The proof of Theorems 6 is based on weighted energy estimates and involves several inequalities concerning the Hilbert transform H. Among them one finds the Ap condition introduced in =-=[47]-=-. Definition 1. A non-negative function w ∈ L1loc(R) satisfies the Ap inequality with 1 < p <∞ if (4.7) sup Q interval ( 1 |Q| ∫ Q w )( 1 |Q| ∫ Q w1−p ′ )p−1 = c(w) <∞, where 1/p+ 1/p′ = 1. It was pro... |

209 | A bilinear estimate with application to the KdV equation,
- Kenig, Ponce, et al.
- 1996
(Show Context)
Citation Context ... k = 3 is LWP in Hs(R) for s ≥ s∗3 = −1/6. (IV) The IVP associated to the equation (1.2) with k ≥ 4 is LWP in Hs(R) for s ≥ s∗k = (k − 4)/2k. The result s > −3/4 for the case k = 1 was established in =-=[38]-=-. The limiting value s = −3/4 was obtained in [11]. The result for the case k = 2 was proven in [37]. The result s > −1/6 for the case k = 3 was given in [22]. The limiting value s = −1/6 was obtained... |

189 |
The Cauchy problem for the critical nonlinear Schrodinger equation in Hs ,
- CAZENAVE, WEISSLER
- 1990
(Show Context)
Citation Context ...arbitrarily large one says that the corresponding IVP is globally well posed (GWP) in X . First, we shall study the Schrödinger equation (1.1). 2. The Schrödinger equation (1.1) The results in [9], =-=[10]-=-, [21], [34], and [62] yield the following LWP theory in the classical Sobolev spaces Hs(Rn) for the IVP associated to the NLS equation (1.1). Theorem A. Let sc = n/2− 2/(a− 1). (I) If s > sc, s ≥ 0, ... |

176 |
Oscillatory integrals and regularity of dispersive equations.
- KENIG, P, et al.
- 1991
(Show Context)
Citation Context ... theory one has the following estimates for the solution u = u(x, t) (3.2) sup x∈R ( ∫ T 0 |∂xD m x u(x, t)| 2dt)1/2 < cT ‖J mu0‖2 = cT ‖u0‖m,2, (the sharp form of the local smoothing effect found in =-=[36]-=--[37]), and (3.3) ‖Dmx u‖L2xL2T = ( ∫ ∞ −∞ ∫ T 0 |Dmx u(x, t)| 2dtdx)1/2 ≤ T 1/2 sup t∈[0,T ] ‖Dmx u(t)‖2 < cT ‖D mu0‖2 ≤ cT ‖u0‖m,2. Now, we consider the extensions of the estimates in (3.2)-(3.3) to... |

168 |
Global solutions of nonlinear Schrödinger equations.
- Bourgain
- 1999
(Show Context)
Citation Context ...s: ‖u(·, t)‖2 = ‖u0‖2, and E(t) = ∫ Rn (|∇xu(x, t)| 2 + 2µ a+ 1 |u(x, t)|a+1)dx = E(0). Using these conservation laws one can extend the LWP results in Theorem A to a GWP one, for details we refer to =-=[6]-=-, [61], and references therein. ON THE PERSISTENCE PROPERTIES OF SOLUTIONS OF DISPERSIVE EQUATIONS 3 Concerning the persistence properties in weighted Sobolev spaces of solutions of the IVP associated... |

152 |
On the Cauchy problem for the (generalized) Korteweg-de Vries equations,”
- Kato
- 1983
(Show Context)
Citation Context ...g internal gravity waves in deep stratified fluids. It was also shown that it is a completely integrable system (see [2], [12] and references therein). We recall the notion of well posedness given in =-=[33]-=- : the IVP is said to be locally well posed (LWP) in the function space X if for each u0 ∈ X there exist T > 0 and a unique solution u ∈ C([−T, T ] : X)∩ .... = YT of the equation, with the map data →... |

138 |
Weighted norm inequalities for the conjugate function and Hilbert transform,”
- Hunt, Muckenhoupt, et al.
- 1973
(Show Context)
Citation Context ...ition 1. A non-negative function w ∈ L1loc(R) satisfies the Ap inequality with 1 < p <∞ if (4.7) sup Q interval ( 1 |Q| ∫ Q w )( 1 |Q| ∫ Q w1−p ′ )p−1 = c(w) <∞, where 1/p+ 1/p′ = 1. It was proven in =-=[29]-=- that this is a necessary and sufficient condition for the Hilbert transform H to be bounded in Lp(w(x)dx) (see [29], ), i.e. w ∈ Ap, 1 < p <∞ if and only if (4.8) ( ∫ ∞ −∞ |Hf |pw(x)dx)1/p ≤ c∗ ( ∫ ∞... |

128 | Asymptotics, frequency modulation and low regularity ill-posedness for canonical defocusing equations,
- Chirst, Colliander, et al.
- 2002
(Show Context)
Citation Context ... and (II) to the critical one. In the latter, one has that if ‖Dscu0‖2 is sufficiently small, then the local solution extends globally in time. For the optimality of the results in Theorem A see [4], =-=[11]-=-, and [39]. Formally, solutions of the NLS equation (1.1) satisfies the following conservation laws: ‖u(·, t)‖2 = ‖u0‖2, and E(t) = ∫ Rn (|∇xu(x, t)| 2 + 2µ a+ 1 |u(x, t)|a+1)dx = E(0). Using these co... |

128 |
On nonlinear Schrödinger equations,
- Kato
- 1987
(Show Context)
Citation Context ...large one says that the corresponding IVP is globally well posed (GWP) in X . First, we shall study the Schrödinger equation (1.1). 2. The Schrödinger equation (1.1) The results in [9], [10], [21], =-=[34]-=-, and [62] yield the following LWP theory in the classical Sobolev spaces Hs(Rn) for the IVP associated to the NLS equation (1.1). Theorem A. Let sc = n/2− 2/(a− 1). (I) If s > sc, s ≥ 0, with [s] ≤ a... |

128 | On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long - Korteweg, Vries - 1985 |

119 |
Internal waves of permanent form in fluids of great depth,
- Benjamin
- 1967
(Show Context)
Citation Context ... different physical phenomena (see [58] and references therein). In the particular, case n = 1 and a = 3 it has been shown to be completely integrable [63]. The BO equation (1.3) was first deduced in =-=[3]-=- and [51] as a model for long internal gravity waves in deep stratified fluids. It was also shown that it is a completely integrable system (see [2], [12] and references therein). We recall the notion... |

99 |
Inequalities for strongly singular convolution operators
- Fefferman
- 1970
(Show Context)
Citation Context ...b/2f‖p ≃ ‖f‖p + ‖D bf‖p ≃ ‖f‖p + ‖D bf‖p. For the proof of Theorem D we refer to [55], where the optimality of the lower bound 2n/(n + 2b) was also established. The case p = 2n/(n + 2b) was proven in =-=[18]-=-. For a detailed discussion on the different characterizations of the Lps(R n) spaces we refer to [55] and [56]. It is easy to see that for p = 2 and b ∈ (0, 1) one has (2.17) ‖Dbf‖2 ≃ ‖D bf‖2, (2.18)... |

99 | On the ill-posedness of some canonical dispersive equations,
- Kenig, Ponce, et al.
- 2001
(Show Context)
Citation Context ...to the critical one. In the latter, one has that if ‖Dscu0‖2 is sufficiently small, then the local solution extends globally in time. For the optimality of the results in Theorem A see [4], [11], and =-=[39]-=-. Formally, solutions of the NLS equation (1.1) satisfies the following conservation laws: ‖u(·, t)‖2 = ‖u0‖2, and E(t) = ∫ Rn (|∇xu(x, t)| 2 + 2µ a+ 1 |u(x, t)|a+1)dx = E(0). Using these conservation... |

93 |
On a class of nonlinear Schrödinger equations
- Ginibre, Velo
- 1978
(Show Context)
Citation Context ...arily large one says that the corresponding IVP is globally well posed (GWP) in X . First, we shall study the Schrödinger equation (1.1). 2. The Schrödinger equation (1.1) The results in [9], [10], =-=[21]-=-, [34], and [62] yield the following LWP theory in the classical Sobolev spaces Hs(Rn) for the IVP associated to the NLS equation (1.1). Theorem A. Let sc = n/2− 2/(a− 1). (I) If s > sc, s ≥ 0, with [... |

90 |
Algebraic solitary waves in stratified fluids,
- Ono
- 1975
(Show Context)
Citation Context ...nt physical phenomena (see [58] and references therein). In the particular, case n = 1 and a = 3 it has been shown to be completely integrable [63]. The BO equation (1.3) was first deduced in [3] and =-=[51]-=- as a model for long internal gravity waves in deep stratified fluids. It was also shown that it is a completely integrable system (see [2], [12] and references therein). We recall the notion of well ... |

79 |
Multipliers on fractional Sobolev spaces
- Strichartz
- 1967
(Show Context)
Citation Context ...the lower bound 2n/(n + 2b) was also established. The case p = 2n/(n + 2b) was proven in [18]. For a detailed discussion on the different characterizations of the Lps(R n) spaces we refer to [55] and =-=[56]-=-. It is easy to see that for p = 2 and b ∈ (0, 1) one has (2.17) ‖Dbf‖2 ≃ ‖D bf‖2, (2.18) ‖Db(fg)‖2 ≤ c(‖f D bg‖2 + ‖gD bf‖2), and for p > 2n/(n+ 2b) (2.19) Db(fg)(x) ≤ ‖f‖∞D bg(x) + |g(x)| Dbf(x). We... |

71 | Global well-posedness of the Benjamin-Ono equation inH1(R),”
- Tao
- 2004
(Show Context)
Citation Context ...s Hs(R) of the IVP associated to the BO equation (1.3) has been largely considered : in [1] and [31] LWP was established for s > 3/2, in [53] for s ≥ 3/2, in [42] for s > 5/4, in [35] for s > 9/8, in =-=[59]-=- for s ≥ 1, in [7] for s > 1/4, and in [30] LWP was proven in Hs(R) for s ≥ 0. Real valued solutions of the IVP (1.3) satisfy infinitely many conservation laws (time invariant quantities), the first t... |

68 | Global well-posedness of the Benjamin-Ono equation with initial data in H1/2
- Kenig, Takaoka
(Show Context)
Citation Context ...ation (1.3) has been largely considered : in [1] and [31] LWP was established for s > 3/2, in [53] for s ≥ 3/2, in [42] for s > 5/4, in [35] for s > 9/8, in [59] for s ≥ 1, in [7] for s > 1/4, and in =-=[30]-=- LWP was proven in Hs(R) for s ≥ 0. Real valued solutions of the IVP (1.3) satisfy infinitely many conservation laws (time invariant quantities), the first three are the following: (4.1) I1(u) = ∫ ∞ −... |

62 | The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical Ap characteristic,
- Petermichl
- 2007
(Show Context)
Citation Context ...er continuity properties of the Hilbert transform. More precisely, the proof requires the constant c∗ in (4.8) to depend only on c(w) the constant describing the Ap condition (see (4.7)) and on p. In =-=[52]-=- precise bounds for the constant c∗ in (4.7) were given which are sharp in the case p = 2 and sufficient for the purpose in [19]. It will be essential in the arguments in [19] that some commutator ope... |

62 |
L2-solutions for nonlinear Schrodinger equations and nonlinear groups,
- Tsutsumi
- 1987
(Show Context)
Citation Context ...says that the corresponding IVP is globally well posed (GWP) in X . First, we shall study the Schrödinger equation (1.1). 2. The Schrödinger equation (1.1) The results in [9], [10], [21], [34], and =-=[62]-=- yield the following LWP theory in the classical Sobolev spaces Hs(Rn) for the IVP associated to the NLS equation (1.1). Theorem A. Let sc = n/2− 2/(a− 1). (I) If s > sc, s ≥ 0, with [s] ≤ a − 1 if a ... |

59 |
Algebras of singular integral operators,”
- Calderon
- 1967
(Show Context)
Citation Context ...lowing estimate: ∀ p ∈ (1,∞), l, m ∈ Z+∪{0}, l+m ≥ 1 ∃ c = c(p; l;m) > 0 such that (4.10) ‖∂lx[H; a]∂ m x f‖p ≤ c‖∂ l+m x a‖∞‖f‖p. In the case l+m = 1, (4.10) is Calderón’s first commutator estimate =-=[8]-=-. The case l+m ≥ 2 of the estimate (4.10) was proved in [14]. ACKNOWLEDGMENT: J. N. was supported by the EAPSI NSF and JSPS program. G.P. was supported by NSF grant DMS-0800967. Part of this work was ... |

57 |
Method for solving the Korteweg–de Vries equation Phys.
- Gardner, Greene, et al.
- 1967
(Show Context)
Citation Context ... the KdV and its modified form (k = 2 in (1.2)) were found to be relevant in a number of different physical systems. Also they have been studied because of their relation to inverse scattering theory =-=[20]-=-. The NLS arises as a model in several different physical phenomena (see [58] and references therein). In the particular, case n = 1 and a = 3 it has been shown to be completely integrable [63]. The B... |

53 |
Nonlocal models for nonlinear, dispersive waves,
- Abdelouhab, Bona, et al.
- 1989
(Show Context)
Citation Context ...T ‖D mv0‖2, ‖Dm+iyx v‖L2xL2T ≤ cT ‖D mv0‖2, for (3.5) v(x, t) = U(t)v0(x) = c ∫ ∞ −∞ eixξeitξ 3 v̂0(ξ)dξ. To apply the three line theorem we consider the function F (z) defined on S = {z ∈ C : ℜ(z) ∈ =-=[0, 1]-=-} F (z) = ∫ ∞ −∞ ∫ T 0 Ds(z)x v(x, t)φ(x, z) f(t) dtdx, 8 J. NAHAS AND G. PONCE where s(z) = (1 − z)(1 +m) + zm, 1/q(z) = (1− z) + z/2, q = 2/(2−m), φ(x, z) = |g(x)|q/q(z) g(x) |g(x)| , with ‖g‖ L 2/(... |

50 |
On the local well-posedness of the Benjamin-Ono equation in Hs
- Koch, Tzvetkov
(Show Context)
Citation Context ...quation (1.3) The LWP in the Sobolev spaces Hs(R) of the IVP associated to the BO equation (1.3) has been largely considered : in [1] and [31] LWP was established for s > 3/2, in [53] for s ≥ 3/2, in =-=[42]-=- for s > 5/4, in [35] for s > 9/8, in [59] for s ≥ 1, in [7] for s > 1/4, and in [30] LWP was proven in Hs(R) for s ≥ 0. Real valued solutions of the IVP (1.3) satisfy infinitely many conservation law... |

49 |
F.: Blow up in finite time and dynamics of blow up solutions for the L2-critical generalized KdV equation.
- Martel, Merle
- 2002
(Show Context)
Citation Context ...1, s = −3/4 and k = 2, s = 1/4 were proven in [24] and [41]. For the case k = 3 the global well posedness is known for s > −1/42, see [23]. For k = 4 blow up of “large” enough solutions was proven in =-=[45]-=-. Similar results for k ≥ 5 remain an open problem. Concerning the persistence of these solutions in weighted Sobolev spaces one has the following result found in [33]. Theorem F. Let m ∈ Z+. Let u ∈ ... |

49 |
On the global well-posedness of the Benjamin-Ono equation
- Ponce
- 1991
(Show Context)
Citation Context ...4. The Benjamin-Ono equation (1.3) The LWP in the Sobolev spaces Hs(R) of the IVP associated to the BO equation (1.3) has been largely considered : in [1] and [31] LWP was established for s > 3/2, in =-=[53]-=- for s ≥ 3/2, in [42] for s > 5/4, in [35] for s > 9/8, in [59] for s ≥ 1, in [7] for s > 1/4, and in [30] LWP was proven in Hs(R) for s ≥ 0. Real valued solutions of the IVP (1.3) satisfy infinitely ... |

41 |
global well-posedness for the KDV and modified
- Colliander, Keel, et al.
(Show Context)
Citation Context ...uation (1.2) formally satisfy at least three conservation laws: I1(u) = ∫ ∞ −∞ u(x, t)dx, I2(u) = ∫ ∞ −∞ (u(x, t))2dx, I3(u) = ∫ ∞ −∞ ((∂xu(x, t)) 2 − 2 (k + 1)(k + 2) u(x, t)k+2)dx. It was proven in =-=[13]-=- that for k = 1 and k = 2 one has global well posedness for s > −3/4 and s > 1/4, respectively. The global cases for k = 1, s = −3/4 and k = 2, s = 1/4 were proven in [24] and [41]. For the case k = 3... |

41 |
On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations
- Kenig, Koenig
(Show Context)
Citation Context ... in the Sobolev spaces Hs(R) of the IVP associated to the BO equation (1.3) has been largely considered : in [1] and [31] LWP was established for s > 3/2, in [53] for s ≥ 3/2, in [42] for s > 5/4, in =-=[35]-=- for s > 9/8, in [59] for s ≥ 1, in [7] for s > 1/4, and in [30] LWP was proven in Hs(R) for s ≥ 0. Real valued solutions of the IVP (1.3) satisfy infinitely many conservation laws (time invariant qua... |

40 |
On the Cauchy problem for the Benjamin-Ono equation
- Iorio
- 1986
(Show Context)
Citation Context ...we shall consider the BO equation (1.3). 4. The Benjamin-Ono equation (1.3) The LWP in the Sobolev spaces Hs(R) of the IVP associated to the BO equation (1.3) has been largely considered : in [1] and =-=[31]-=- LWP was established for s > 3/2, in [53] for s ≥ 3/2, in [42] for s > 5/4, in [35] for s > 9/8, in [59] for s ≥ 1, in [7] for s > 1/4, and in [30] LWP was proven in Hs(R) for s ≥ 0. Real valued solut... |

37 | Nonlinear wave interactions for the Benjamin-Ono equation
- Koch, Tzvetkov
- 2005
(Show Context)
Citation Context ...hibited by the Korteweg-de Vries (KdV) equation, i.e. k = 1 in (1.2). Indeed, it was proven in [46] that for any s ∈ R the map data-solution from Hs(R) to C([0, T ] : Hs(R)) is not locally C2, and in =-=[43]-=- that it is not locally uniformly continuous. In particular, this implies that no LWP results can be obtained by an argument based only on a contraction method. Consider the weighted Sobolev spaces (4... |

35 |
On the well-posedness of the Benjamin-Ono equation
- Burq, Planchon
- 2008
(Show Context)
Citation Context ...associated to the BO equation (1.3) has been largely considered : in [1] and [31] LWP was established for s > 3/2, in [53] for s ≥ 3/2, in [42] for s > 5/4, in [35] for s > 9/8, in [59] for s ≥ 1, in =-=[7]-=- for s > 1/4, and in [30] LWP was proven in Hs(R) for s ≥ 0. Real valued solutions of the IVP (1.3) satisfy infinitely many conservation laws (time invariant quantities), the first three are the follo... |

34 |
A problem in prediction theory,”
- Helson, Szego
- 1960
(Show Context)
Citation Context ... Lp(w(x)dx) (see [29], ), i.e. w ∈ Ap, 1 < p <∞ if and only if (4.8) ( ∫ ∞ −∞ |Hf |pw(x)dx)1/p ≤ c∗ ( ∫ ∞ −∞ |f |pw(x)dx)1/p, In the case p = 2, a previous characterization of w in (4.7) was found in =-=[28]-=-. However, even though the main case is for p = 2, the characterization (4.7) will be the one used in the proof. In particular, one has that in R (4.9) |x|α ∈ Ap ⇔ α ∈ (−1, p− 1). In order to justify ... |

33 | Zihua Global Well-posedness of Korteweg-de Vries equation in H−3/4
- Guo
- 2009
(Show Context)
Citation Context ..., t)k+2)dx. It was proven in [13] that for k = 1 and k = 2 one has global well posedness for s > −3/4 and s > 1/4, respectively. The global cases for k = 1, s = −3/4 and k = 2, s = 1/4 were proven in =-=[24]-=- and [41]. For the case k = 3 the global well posedness is known for s > −1/42, see [23]. For k = 4 blow up of “large” enough solutions was proven in [45]. Similar results for k ≥ 5 remain an open pro... |

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 ... case and (II) to the critical one. In the latter, one has that if ‖Dscu0‖2 is sufficiently small, then the local solution extends globally in time. For the optimality of the results in Theorem A see =-=[4]-=-, [11], and [39]. Formally, solutions of the NLS equation (1.1) satisfies the following conservation laws: ‖u(·, t)‖2 = ‖u0‖2, and E(t) = ∫ Rn (|∇xu(x, t)| 2 + 2µ a+ 1 |u(x, t)|a+1)dx = E(0). Using th... |

32 |
The Characterization of functions arising as potentials
- Stein
(Show Context)
Citation Context ...eibnitz rule for homogeneous fractional derivatives of order b ∈ R (2.14) Dbf(x) ≡ ((2π|ξ|)bf̂)∨(x) deduced as a direct consequence of the characterization of the Lps(R n) spaces (see (2.2)) given in =-=[55]-=-. Theorem D. Let b ∈ (0, 1) and 2n/(n+ 2b) ≤ p < ∞. Then f ∈ Lpb(R n) if and only if (2.15) (a) f ∈ Lp(Rn), (b) Dbf(x) = ( ∫ Rn |f(x)− f(y)|2 |x− y|n+2b dy)1/2 ∈ Lp(Rn), with (2.16) ‖f‖b,p = ‖(1−∆) b/... |

31 |
The inverse scattering transform for the Benjamin– Ono equation — a pivot to multidimensional problems
- Fokas, Ablowitz
- 1983
(Show Context)
Citation Context ...le [63]. The BO equation (1.3) was first deduced in [3] and [51] as a model for long internal gravity waves in deep stratified fluids. It was also shown that it is a completely integrable system (see =-=[2]-=-, [12] and references therein). We recall the notion of well posedness given in [33] : the IVP is said to be locally well posed (LWP) in the function space X if for each u0 ∈ X there exist T > 0 and a... |

20 |
Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity. Differential Integral Equations
- Kishimoto
- 2009
(Show Context)
Citation Context ...x. It was proven in [13] that for k = 1 and k = 2 one has global well posedness for s > −3/4 and s > 1/4, respectively. The global cases for k = 1, s = −3/4 and k = 2, s = 1/4 were proven in [24] and =-=[41]-=-. For the case k = 3 the global well posedness is known for s > −1/42, see [23]. For k = 4 blow up of “large” enough solutions was proven in [45]. Similar results for k ≥ 5 remain an open problem. Con... |

19 |
On solutions of the initial value problem for the nonlinear Schr\"odinger equations
- Hayashi, Nakamitsu, et al.
- 1987
(Show Context)
Citation Context ...ONS OF DISPERSIVE EQUATIONS 3 Concerning the persistence properties in weighted Sobolev spaces of solutions of the IVP associated to the NLS equation (1.1) one has the following result established in =-=[25]-=-, [26], and [27]. Theorem B. In addition to the hypothesis in Theorem A assume u0 ∈ L 2(|x|2mdx), m ∈ Z+ with m ≤ a− 1 if a is not an odd integer. (I) If s ≥ m, then (2.5) u ∈ C([−T, T ] : Hs ∩ L2(|x|... |

18 | On uniqueness properties of solutions of the k-generalized KdV equations
- Escauriaza, Kenig, et al.
(Show Context)
Citation Context ...stablish the main estimate in the proof. For the study of persistence properties of the solution to the IVP associated to the k-gKdV equation (1.2) in exponential weighted spaces we refer to [40] and =-=[15]-=- and references therein. Finally, we shall consider the BO equation (1.3). 4. The Benjamin-Ono equation (1.3) The LWP in the Sobolev spaces Hs(R) of the IVP associated to the BO equation (1.3) has bee... |

18 | The Sharp Hardy Uncertainty Principle for Schrodinger Evolutions.
- Escauriaza, Kenig, et al.
- 2010
(Show Context)
Citation Context ...) ‖Jθa(〈x〉(1−θ)bf)‖2 ≤ c‖〈x〉 bf‖1−θ2 ‖J af‖θ2. For the study of persistence properties of the solution to the IVP associated to the NLS equation (1.1) in exponential weighted spaces we refer to [16], =-=[17]-=-, and references therein. Next, we shall consider the k-gKdV equation (1.2). 3. The k-generalized Korteweg-de Vries equation (1.2) The following theorem describes the LWP theory in the classical Sobol... |

15 |
The scattering transform for the Benjamin-Ono equation, Inverse Problems 6
- Coifman, Wickerhauser
- 1990
(Show Context)
Citation Context ...3]. The BO equation (1.3) was first deduced in [3] and [51] as a model for long internal gravity waves in deep stratified fluids. It was also shown that it is a completely integrable system (see [2], =-=[12]-=- and references therein). We recall the notion of well posedness given in [33] : the IVP is said to be locally well posed (LWP) in the function space X if for each u0 ∈ X there exist T > 0 and a uniqu... |

15 | On the unique continuation of solutions to the generalized KdV equation
- Kenig, Ponce, et al.
(Show Context)
Citation Context ... <∞, to establish the main estimate in the proof. For the study of persistence properties of the solution to the IVP associated to the k-gKdV equation (1.2) in exponential weighted spaces we refer to =-=[40]-=- and [15] and references therein. Finally, we shall consider the BO equation (1.3). 4. The Benjamin-Ono equation (1.3) The LWP in the Sobolev spaces Hs(R) of the IVP associated to the BO equation (1.3... |

14 | On the persistent properties of solutions to semi-linear Schrödinger equation
- Nahas, Ponce
(Show Context)
Citation Context ... regularity”. In particular, (2.6) tells us that tβ∂βxu ∈ L 2 loc(R n), for |β| ≤ m and t ∈ [−T, T ]− {0}. Also one notices that the power of the weight m in Theorem B is assumed to be an integer. In =-=[50]-=- we were able to remove this restriction. Theorem 1. In addition to the hypothesis in Theorem A assume u0 ∈ L 2(|x|2mdx), m > 0 with [m] ≤ a− 1 if a is not an odd integer. (I) If s ≥ m, (2.9) u ∈ C([−... |

11 | The I.V.P for the Benjamin-Ono equation in weighted Sobolev spaces
- Fonseca, Linares, et al.
(Show Context)
Citation Context ...s, the goal was to extend the results in Theorem G and Theorem I from integer values to the continuum optimal range of indices (s, r). In this direction one finds the following results established in =-=[19]-=-: Theorem 4. (I) Let s ≥ 1, r ∈ [0, s], and r < 5/2. If u0 ∈ Zs,r, then the solution u(x, t) of the IVP associated to the BO equation (1.3) satisfies that u ∈ C([0,∞) : Zs,r). (II) For s > 9/8 (s ≥ 3/... |

11 |
Local and Global Analysis of Nonlinear Dispersive
- Tao
- 2006
(Show Context)
Citation Context ...(·, t)‖2 = ‖u0‖2, and E(t) = ∫ Rn (|∇xu(x, t)| 2 + 2µ a+ 1 |u(x, t)|a+1)dx = E(0). Using these conservation laws one can extend the LWP results in Theorem A to a GWP one, for details we refer to [6], =-=[61]-=-, and references therein. ON THE PERSISTENCE PROPERTIES OF SOLUTIONS OF DISPERSIVE EQUATIONS 3 Concerning the persistence properties in weighted Sobolev spaces of solutions of the IVP associated to th... |

10 | quelques généralisations de l’ équations de Korteweg-de Vries - Saut, Sur - 1979 |

9 |
Nonlinear Schrödinger equations in weighted Sobolev spaces
- Hayashi, Nakamitsu, et al.
- 1988
(Show Context)
Citation Context ...E EQUATIONS 3 Concerning the persistence properties in weighted Sobolev spaces of solutions of the IVP associated to the NLS equation (1.1) one has the following result established in [25], [26], and =-=[27]-=-. Theorem B. In addition to the hypothesis in Theorem A assume u0 ∈ L 2(|x|2mdx), m ∈ Z+ with m ≤ a− 1 if a is not an odd integer. (I) If s ≥ m, then (2.5) u ∈ C([−T, T ] : Hs ∩ L2(|x|2mdx)) ∩ Lq([−T,... |

8 | On the decay properties of solutions to a class of Schrödinger equations
- Dawson, McGahagan, et al.
- 2008
(Show Context)
Citation Context ...c(p; l;m) > 0 such that (4.10) ‖∂lx[H; a]∂ m x f‖p ≤ c‖∂ l+m x a‖∞‖f‖p. In the case l+m = 1, (4.10) is Calderón’s first commutator estimate [8]. The case l+m ≥ 2 of the estimate (4.10) was proved in =-=[14]-=-. ACKNOWLEDGMENT: J. N. was supported by the EAPSI NSF and JSPS program. G.P. was supported by NSF grant DMS-0800967. Part of this work was done while J. N. was visiting Prof. Y. Tsutsumi at the Depar... |

7 |
Unique continuation principle for the Benjamin-Ono equation
- Iorio
(Show Context)
Citation Context ... = e −itH∂2xv0(x) = (e −itξ|ξ| v̂0) ∨(x), satisfies that v(·, t) ∈ L2(|x|2kdx), t ∈ [0, T ], when v0 ∈ Zk,k, k ∈ Z + for k = 1, 2, ...... and ∫ ∞ −∞ xj v0(x)dx = 0, j = 0, 1, ..., k − 3, if k ≥ 3. In =-=[32]-=- the unique continuation result in Z4,4 in Theorem G was improved: Theorem I. Let u ∈ C([0, T ] : H2(R)) be a solution of the IVP (1.3). If there exist three different times t1, t2, t3 ∈ [0, T ] such ... |

7 | A decay property of solutions to the k-generalized kdv equation. preprint arxiv:1010.5001
- Nahas
- 2010
(Show Context)
Citation Context ...ded by Theorem E. If u(x, 0) = u0(x) ∈ L 2(|x|mdx), then (I) If m < 1, then for any ǫ > 0 u ∈ C([−T, T ] : Hm(R) ∩ L2(|x|m−ǫdx)). (II) If m ≥ 1, then u ∈ C([−T, T ] : Hm(R) ∩ L2(|x|mdx)). In [48] and =-=[49]-=- the loss of power ǫ > 0 in the weight when m < 1 was removed for the equation (1.2) with non-linearity k = 2, 4, 5, ..... More precisely, the following optimal result was established in [49]: ON THE ... |

6 | Convexity properties of solutions to the free Schrödinger equation with Gaussian decay
- Escauriaza, Kenig, et al.
(Show Context)
Citation Context ... (2.27) ‖Jθa(〈x〉(1−θ)bf)‖2 ≤ c‖〈x〉 bf‖1−θ2 ‖J af‖θ2. For the study of persistence properties of the solution to the IVP associated to the NLS equation (1.1) in exponential weighted spaces we refer to =-=[16]-=-, [17], and references therein. Next, we shall consider the k-gKdV equation (1.2). 3. The k-generalized Korteweg-de Vries equation (1.2) The following theorem describes the LWP theory in the classical... |

5 |
posedness issues for the Benjamin-Ono and related equations
- Molinet, Saut, et al.
(Show Context)
Citation Context ...ation the dispersive effect is described by a non-local operator and is significantly weaker than that exhibited by the Korteweg-de Vries (KdV) equation, i.e. k = 1 in (1.2). Indeed, it was proven in =-=[46]-=- that for any s ∈ R the map data-solution from Hs(R) to C([0, T ] : Hs(R)) is not locally C2, and in [43] that it is not locally uniformly continuous. In particular, this implies that no LWP results c... |

5 |
Scattering for the quartic generalized Korteweg-de Vries equation, J. Differential Equations 232 (2007), 623–651
- Tao
(Show Context)
Citation Context ...e limiting value s = −3/4 was obtained in [11]. The result for the case k = 2 was proven in [37]. The result s > −1/6 for the case k = 3 was given in [22]. The limiting value s = −1/6 was obtained in =-=[60]-=-. The proof of the cases k ≥ 4 was given in [37]. The above local results apply to both real and complex valued functions. The scaling argument described in (2.4) affirms that LWP should hold for s ≥ ... |

4 |
A remark on global well-posedness below L2 for the gKdV-3 equation, Diff
- Grünrock, Panthee, et al.
(Show Context)
Citation Context ... for s > −3/4 and s > 1/4, respectively. The global cases for k = 1, s = −3/4 and k = 2, s = 1/4 were proven in [24] and [41]. For the case k = 3 the global well posedness is known for s > −1/42, see =-=[23]-=-. For k = 4 blow up of “large” enough solutions was proven in [45]. Similar results for k ≥ 5 remain an open problem. Concerning the persistence of these solutions in weighted Sobolev spaces one has t... |

3 |
A decay property of solutions to the mKdV equation
- Nahas
- 2010
(Show Context)
Citation Context ....2) provided by Theorem E. If u(x, 0) = u0(x) ∈ L 2(|x|mdx), then (I) If m < 1, then for any ǫ > 0 u ∈ C([−T, T ] : Hm(R) ∩ L2(|x|m−ǫdx)). (II) If m ≥ 1, then u ∈ C([−T, T ] : Hm(R) ∩ L2(|x|mdx)). In =-=[48]-=- and [49] the loss of power ǫ > 0 in the weight when m < 1 was removed for the equation (1.2) with non-linearity k = 2, 4, 5, ..... More precisely, the following optimal result was established in [49]... |

2 |
Some remarks on the critical nonlinear Schrödinger equation
- Cazenave, Weissler
- 1989
(Show Context)
Citation Context ...aken arbitrarily large one says that the corresponding IVP is globally well posed (GWP) in X . First, we shall study the Schrödinger equation (1.1). 2. The Schrödinger equation (1.1) The results in =-=[9]-=-, [10], [21], [34], and [62] yield the following LWP theory in the classical Sobolev spaces Hs(Rn) for the IVP associated to the NLS equation (1.1). Theorem A. Let sc = n/2− 2/(a− 1). (I) If s > sc, s... |

2 |
A bilinear Airy estimate with application to the 3-gKdV equation Diff
- Grünrock
(Show Context)
Citation Context ...3/4 for the case k = 1 was established in [38]. The limiting value s = −3/4 was obtained in [11]. The result for the case k = 2 was proven in [37]. The result s > −1/6 for the case k = 3 was given in =-=[22]-=-. The limiting value s = −1/6 was obtained in [60]. The proof of the cases k ≥ 4 was given in [37]. The above local results apply to both real and complex valued functions. The scaling argument descri... |

1 |
Exact theory of two dimensional seff-focusing and onedimensional self-modulation of waves in non-linear media, Soviev Physics JETP 34
- Zakharov, Shabat
- 1972
(Show Context)
Citation Context ...theory [20]. The NLS arises as a model in several different physical phenomena (see [58] and references therein). In the particular, case n = 1 and a = 3 it has been shown to be completely integrable =-=[63]-=-. The BO equation (1.3) was first deduced in [3] and [51] as a model for long internal gravity waves in deep stratified fluids. It was also shown that it is a completely integrable system (see [2], [1... |