### Citations

2325 |
Singular Integrals and Differentiability Properties of Functions
- Stein
- 1970
(Show Context)
Citation Context ...ollows Ψ(t, x) = sup s≥0 ∣∣∣∣∣ ∫ |ξ|≤pi/2h |ξ|1/2e−itph(ξ)e−sph(ξ)eixξP̂h1ϕ(ξ)dξ ∣∣∣∣∣ the following |(−∆)1/4(I1ϕ)(t, x)| ≤ Ψ(t, x) holds for any t and x. Classical properties of Poisson’s integrals (=-=[35]-=-, Th. 1, p. 62, Ch. III) shows that the function Ψ satisfies ‖Ψ(·, x)‖L2(Rt) . ‖(−∆)1/4Jϕ(·, x)‖L2(Rt). It remains to prove that Jϕ satisfies (4.52) ∫ |x|<R ∫ ∞ −∞ |(−∆)1/4Jϕ(t, x)|2dtdx . C(R)‖Ph1ϕ‖L... |

513 | Endpoint Strichartz estimates,
- Keel, Tao
- 1998
(Show Context)
Citation Context ....5) ‖S(·)ϕ‖Lq(R, Lr(Rd)) ≤ C(q, r)‖ϕ‖L2(R) for the so-called d/2-admissible pairs (q, r), excepting the limit case (q, r) = (∞, 2) in dimension d = 2. We recall that a σ-admisible pair satisfies (cf. =-=[23]-=-): 2 ≤ q, r ≤ ∞ and (1.6) 1 q = σ ( 1 2 − 1 r ) . The extension to the inhomogeneous linear Schrödinger equation is due to Yajima [43] and Cazenave and Weissler [7]. The estimates presented before pl... |

437 |
Compact sets in the space Lp(0
- SIMON
- 1987
(Show Context)
Citation Context ...,H−2(Rd)) + ‖Ph1(|uh|puh)‖L1(I,H−2(Rd)) ≤ ‖Ph1uh‖L1(I, L2(Rd)) + ‖Ph1(|uh|puh)‖L1(I, L(p+2)′ (Rd)) ≤ C(I, ‖ϕ‖L2(Rd)). Using the embeddings Hs(Ω) ↪→ comp L2(Ω) ↪→ H−2(Ω) and the compactness results of =-=[31]-=- we obtain the existence of a function v such that Ph1u h → v in L2loc(R× Rd). In the following we prove that Ph1u h −Ph0uh → 0 in L2(Rd+1). Classical result on interpolation ([30], Th. 3.1.5, p. 122)... |

436 |
Numerical Approximation of Partial Differential equations
- Quarteroni, Valli
- 1997
(Show Context)
Citation Context ...pactness results of [31] we obtain the existence of a function v such that Ph1u h → v in L2loc(R× Rd). In the following we prove that Ph1u h −Ph0uh → 0 in L2(Rd+1). Classical result on interpolation (=-=[30]-=-, Th. 3.1.5, p. 122) give us that∫ Rd |Ph1uh(t)−Ph0uh(t)|2dx ≤ h2‖∇huh(t)‖2l2(hZd) = h2 ∫ [−pi/h,pi/h]d ph(ξ)e−2|t|ph(ξ)a(h)|ϕ̂h(ξ)|2dξ. Integrating the last inequality on time and using that α(h)→ 1/... |

342 |
Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations
- Strichartz
- 1977
(Show Context)
Citation Context ... a dispersive estimate of the form: (1.3) |S(t)ϕ(x)| = |u(t, x)| ≤ 1 (4pi|t|)d/2 ‖ϕ‖L1(Rd), x ∈ R d, t 6= 0. The Space-Time Estimate (1.4) ‖S(·)ϕ‖L2+4/d(R, L2+4/d(Rd)) ≤ C‖ϕ‖L2(Rd), due to Strichartz =-=[37]-=-, is deeper. It guarantees that the solutions decay in some sense as t becomes large and that they gain some spatial integrability. Inequality (1.4) was generalized by Ginibre and Velo [16]. They prov... |

268 |
Heat kernels and spectral theory, Cambridge Tracts
- Davies
- 1989
(Show Context)
Citation Context ...ph(ξ)eij·ξhdξ, j ∈ Zd, i.e. the conservative semidiscrete kernel restricted to the frequency set [−pi/4h, pi/4h]d. We recall that the operator exp(|t|∆h) is a contraction in l1(hZd) (see for instance =-=[12]-=-, Theorem 1.3.3, p. 14): (4.45) ‖ exp (|t|∆h)‖l1(hZd)−l1(hZd) ≤ 1. Applying Theorem 3.3 and (4.45) we obtain that ‖Kh,1(t)‖l∞(hZd) ≤ c(d)|t|−d/2, which finishes the proof. 4.2. Strichartz like Estim... |

176 |
Oscillatory integrals and regularity of dispersive equations.
- KENIG, P, et al.
- 1991
(Show Context)
Citation Context ... [5], and, in a more general context, by Kato [21, 22]. The Schrödinger equation has another remarkable property guaranteeing the gain of one half space derivative in L2x,t (cf. [32], [10], [11] and =-=[24]-=-): (1.7) sup x0,R 1 R ∫ B(x0,R) ∫ ∞ −∞ |(−∆)1/4eit∆ϕ|2dtdx ≤ C‖ϕ‖2L2(Rd). It has played a crucial role in the study of the nonlinear Schrödinger equation with nonlinearities involving derivatives (se... |

163 |
The Nonlinear Schrödinger Equation
- Sulem, Sulem
- 1999
(Show Context)
Citation Context ...It has played a crucial role in the study of the nonlinear Schrödinger equation with nonlinearities involving derivatives (see [25]). For other deep results on the Schrödinger equations we refer to =-=[38]-=-, [6] and the bibliography therein. NUMERICAL SCHEMES FOR NSE 3 In this paper we analyze whether semidiscrete schemes for LSE have dispersive properties similar to (1.3), (1.5) and (1.7), uniform with... |

158 |
Nonlinear differential-difference equations and Fourier analysis
- Ablowitz, Ladik
- 1976
(Show Context)
Citation Context ... any Lqloc(R, l r(hZ))-norm with r > 2. To explain the necessity of analyzing the dispersive properties at numerical level let us consider the following discretization of the NSE that was proposed in =-=[1]-=- and is accordingly often referred to as the Ablowitz-Ladik NUMERICAL SCHEMES FOR NSE 5 NSE of the form: (2.9) i∂tuhn + (∆hu h)n = |uhn|2(uhn+1 + uhn−1), with initial condition uh(0) = ϕh, ϕh being an... |

139 |
Regularity of solutions to the schrödinger equation.
- SJÖLIN
- 1987
(Show Context)
Citation Context ...lo [14, 15], Cazenave [5], and, in a more general context, by Kato [21, 22]. The Schrödinger equation has another remarkable property guaranteeing the gain of one half space derivative in L2x,t (cf. =-=[32]-=-, [10], [11] and [24]): (1.7) sup x0,R 1 R ∫ B(x0,R) ∫ ∞ −∞ |(−∆)1/4eit∆ϕ|2dtdx ≤ C‖ϕ‖2L2(Rd). It has played a crucial role in the study of the nonlinear Schrödinger equation with nonlinearities invo... |

128 |
On nonlinear Schrödinger equations,
- Kato
- 1987
(Show Context)
Citation Context ... case p = 4/d. The case of H1-solutions has been analyzed by Baillon, Cazenave and Figueira [3], Lin and Strauss [26], Ginibre and Velo [14, 15], Cazenave [5], and, in a more general context, by Kato =-=[21, 22]-=-. The Schrödinger equation has another remarkable property guaranteeing the gain of one half space derivative in L2x,t (cf. [32], [10], [11] and [24]): (1.7) sup x0,R 1 R ∫ B(x0,R) ∫ ∞ −∞ |(−∆)1/4eit... |

120 |
Existence of solutions for Schrodinger evolution equations,
- YAJIMA
- 1987
(Show Context)
Citation Context ...imension d = 2. We recall that a σ-admisible pair satisfies (cf. [23]): 2 ≤ q, r ≤ ∞ and (1.6) 1 q = σ ( 1 2 − 1 r ) . The extension to the inhomogeneous linear Schrödinger equation is due to Yajima =-=[43]-=- and Cazenave and Weissler [7]. The estimates presented before play an important role in the proof of the well-posedness of the nonlinear Schrödnger equation (NSE). Typically the dispersive estimates... |

110 |
Local smoothing properties of dispersive equations.
- CONSTANTIN, SAUT
- 1988
(Show Context)
Citation Context ..., 15], Cazenave [5], and, in a more general context, by Kato [21, 22]. The Schrödinger equation has another remarkable property guaranteeing the gain of one half space derivative in L2x,t (cf. [32], =-=[10]-=-, [11] and [24]): (1.7) sup x0,R 1 R ∫ B(x0,R) ∫ ∞ −∞ |(−∆)1/4eit∆ϕ|2dtdx ≤ C‖ϕ‖2L2(Rd). It has played a crucial role in the study of the nonlinear Schrödinger equation with nonlinearities involving ... |

93 | Maximal functions associated to filtrations
- Christ, Kiselev
(Show Context)
Citation Context ...mate (4.51) holds for s(r) = 1/4. The homogenous case has been proved by Kenig, Ponce and Vega [24]. The inhomogeneous case is reduced to the homogenous one by using the results of Christ and Kiselev =-=[9]-=- and Strichartz estimates. In our case the arguments of [9] can not be applied. The key point in their proof is that the Schrödinger semigroup satisfies S(t− s) = S(t)S(s)∗ for all reals t and s, ide... |

93 |
On a class of nonlinear Schrödinger equations
- Ginibre, Velo
- 1978
(Show Context)
Citation Context ...nave and Weissler [8] proved the local existence in the critical case p = 4/d. The case of H1-solutions has been analyzed by Baillon, Cazenave and Figueira [3], Lin and Strauss [26], Ginibre and Velo =-=[14, 15]-=-, Cazenave [5], and, in a more general context, by Kato [21, 22]. The Schrödinger equation has another remarkable property guaranteeing the gain of one half space derivative in L2x,t (cf. [32], [10],... |

79 | Strichartz estimates for a Schrödinger operator with nonsmooth coefficients
- Staffilani, Tataru
(Show Context)
Citation Context ...p II. Regularity of the inhomogeneous part. In the following we prove (5.77). This estimate will be reduced to the homogenous one (5.76) by using the argument of Christ and Kiselev [9] (see also [4], =-=[33]-=- in the context of PDE). A simplified version, useful in PDE application is given in [33] : Lemma 5.4. Let X and Y be Banach spaces and assume that K(t, s) is a continuous function taking its values i... |

72 | Spectral Methods
- Trefethen
- 2000
(Show Context)
Citation Context ...s for scheme (1.8) for which at the discrete level, nor (1.5) or 1.7 hold uniformly with respect to parameter h. In our analysis, we make use of the semidiscrete Fourier transform (SDFT) (we refer to =-=[39]-=- for the mains properties of the SDTF). For any v ∈ l2(hZd) we define its SDFT at the scale h by: (3.10) v̂(ξ) = (Fhv)(ξ) = hd ∑ j∈Zd e−iξ·jhvj, ξ ∈ [−pi/h, pi/h]d. To avoid the presence of constants,... |

71 |
analysis: real-variable methods, orthogonality, and oscillatory integrals
- Harmonic
- 1993
(Show Context)
Citation Context ... sufficient to prove (3.31) in the one-dimensional case. Using that the second derivative of the function sin2(ξ/2) is positive on Ω1,1 we obtain by the Van der Corput Lemma (Prop. 2, Ch. 8, p. 332, =-=[36]-=-) that ‖K1,d (t)‖l∞ ≤ c()|t|−1/2 which finishes the proof. A similar result for the local smoothing effect can be stated. For a positive , let us define the set Ah of all points situated at a di... |

65 |
Fourier Analysis of Numerical Approximations of Hyperbolic Equations,
- Vichnevetsky, Bowles
- 1982
(Show Context)
Citation Context ...he nonlinear term. Another difficulty comes from the fact that the interpolator E has no compact support in the Fourier space. To simplify the proof we consider the band-limited interpolator Ph∗ (cf. =-=[41]-=-, Ch. II) and prove the compactness for Ph∗ . Once this is obtained we transfer the L2-strong convergence of Ph∗uh to Ph0uh. This is a consequence of the following property of the piecewise constant i... |

62 |
L2-solutions for nonlinear Schrodinger equations and nonlinear groups,
- Tsutsumi
- 1987
(Show Context)
Citation Context ...o that the L∞-norm of the fundamental solution behaves like t−d/2 [23]. The nonlinear problem with nonlinearity F (u) = |u|p−1u, p < 4/d and initial data in L2(Rd) has been first analyzed by Tsutsumi =-=[40]-=-. The author proved that, in this case, NSE is globally well posed in L∞(R, L2(Rd)) ∩ Lqloc(R, L r(Rd)), where (q, r) is an d/2-admissible pair depending on the nonlinearity F . Also, Cazenave and Wei... |

58 |
Decay and scattering of solutions of a nonlinear Schrodinger equation,
- Lin, Strauss
- 1978
(Show Context)
Citation Context ...inearity F . Also, Cazenave and Weissler [8] proved the local existence in the critical case p = 4/d. The case of H1-solutions has been analyzed by Baillon, Cazenave and Figueira [3], Lin and Strauss =-=[26]-=-, Ginibre and Velo [14, 15], Cazenave [5], and, in a more general context, by Kato [21, 22]. The Schrödinger equation has another remarkable property guaranteeing the gain of one half space derivativ... |

54 |
The Cauchy problem for the nonlinear Schrödinger Equation
- Cazenave, Weissler
- 1990
(Show Context)
Citation Context ...a σ-admisible pair satisfies (cf. [23]): 2 ≤ q, r ≤ ∞ and (1.6) 1 q = σ ( 1 2 − 1 r ) . The extension to the inhomogeneous linear Schrödinger equation is due to Yajima [43] and Cazenave and Weissler =-=[7]-=-. The estimates presented before play an important role in the proof of the well-posedness of the nonlinear Schrödnger equation (NSE). Typically the dispersive estimates are used when the energy meth... |

45 |
Fonctions entières et intégrales de fourier multiples
- Plancherel, Pólya
- 1937
(Show Context)
Citation Context ...Z, S1(t)ϕ as in (3.19), which is defined for all x ∈ R, is in fact the band-limited interpolator of the semi-discrete function S1(t)ϕ. The results of Magyar et al. [27] (see also Plancherel and Polya =-=[29]-=-) on band-limited functions show that the following inequality holds for any q > q0 ≥ 1 and for all continuous, 2pi-periodic functions ϕ̂: (3.20) ‖S1(t)ϕ‖lq(Z) ‖ϕ‖lq0 (Z) ≥ c(q, q0) ‖S1(t)ϕ‖Lq(R) ‖ϕ‖L... |

41 |
On nonlinear Schrödinger equations in exterior domains
- Burq, Gérard, et al.
(Show Context)
Citation Context .... Step II. Regularity of the inhomogeneous part. In the following we prove (5.77). This estimate will be reduced to the homogenous one (5.76) by using the argument of Christ and Kiselev [9] (see also =-=[4]-=-, [33] in the context of PDE). A simplified version, useful in PDE application is given in [33] : Lemma 5.4. Let X and Y be Banach spaces and assume that K(t, s) is a continuous function taking its va... |

35 |
Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems
- Ablowitz, Prinari, et al.
- 2004
(Show Context)
Citation Context ... solutions. We remark that any solution uh satisfies uhn(t) = 1 h u1n ( t h2 ) , n ∈ Z, t ≥ 0, where u1 is the solution on the mesh size h = 1. In this case there are explicit solutions of (2.9) (cf. =-=[2]-=-, p. 84) of the form: u1n(t) = A exp(i(an− bt)) sech(cn− dt) for suitable constants A, a, b, c, d (for the explicit values we refer to [2]). The solutions of (2.9) obtained by scaling on t of this one... |

28 |
Ensuring well-posedness by analogy; stokes problem and boundary control for the wave equation,”
- Glowinski
- 1992
(Show Context)
Citation Context ...Rd)) where u is the unique weak solution of critical NSE. 5. A two-grid algorithm In this section to compensate the lack of dispersion proved in Section 3 we propose a two-grid algorithm (inspired by =-=[17]-=-) and that, to some extent, acts as a filter for those unwanted high frequency components. The method is roughly as follows. We consider two meshes: the coarse one of size 4h, h > 0, 4hZd, and the fin... |

27 |
An introduction to multigrid methods, Pure and Applied Mathematics
- Wesseling
- 1992
(Show Context)
Citation Context ...vergence of the method. This particular structure of the data cancels the two pathologies of the discrete symbol mentioned above. Indeed, a careful Fourier analysis of those initial data (we refer to =-=[42]-=- for the theory of multi-grid methods) shows that their discrete Fourier transform vanishes quadratically in each variable at the points ξ = (±pi/2h)d and ξ = (±pi/h)d. As we shall see, this suffices ... |

19 | Discrete analogues in harmonic analysis: spherical averages
- Magyar, Stein, et al.
- 2002
(Show Context)
Citation Context ...int out that for any sequence {ϕj}j∈Z, S1(t)ϕ as in (3.19), which is defined for all x ∈ R, is in fact the band-limited interpolator of the semi-discrete function S1(t)ϕ. The results of Magyar et al. =-=[27]-=- (see also Plancherel and Polya [29]) on band-limited functions show that the following inequality holds for any q > q0 ≥ 1 and for all continuous, 2pi-periodic functions ϕ̂: (3.20) ‖S1(t)ϕ‖lq(Z) ‖ϕ‖l... |

13 |
Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and KleinGordon equations
- Stefanov, Kevrekidis
- 2005
(Show Context)
Citation Context ...ating wave packets at these pathological points it is possible to prove the lack of any uniform estimate of the type (1.3), (1.5) or (1.7). For the semidiscrete Schrödinger equation we also refer to =-=[34]-=-. In that paper the authors analyze the Schrödinger equation on the lattice hZd without concentrating on parameter h. They obtain Strichartz-like estimates in a class of exponents q and r larger than... |

7 | Dispersive properties of a viscous numerical scheme for the Schrödinger equation
- Ignat, Zuazua
(Show Context)
Citation Context ...roximation of ∆: (∆huh)j = h−2 d∑ k=1 (uhj+ek + u h j−ek − 2uhj ). In the one-dimensional case, the lack of uniform dispersive estimates for the solutions of (1.8) has been observed by the authors in =-=[19, 20]-=-. In that case the symbol of the Laplacian, ξ2, is replaced by a discrete one 4/h2 sin2(ξh/2) which vanishes its first and second derivative at the points ±pi/h and ±pi/2h of the spectrum. By concentr... |

6 |
Equations de Schrödinger non linéaires en dimension deux
- Cazenave
- 1979
(Show Context)
Citation Context ...8] proved the local existence in the critical case p = 4/d. The case of H1-solutions has been analyzed by Baillon, Cazenave and Figueira [3], Lin and Strauss [26], Ginibre and Velo [14, 15], Cazenave =-=[5]-=-, and, in a more general context, by Kato [21, 22]. The Schrödinger equation has another remarkable property guaranteeing the gain of one half space derivative in L2x,t (cf. [32], [10], [11] and [24]... |

3 |
On a sharp estimate for oscillatory integrals associated with the Schrödinger equation
- Gigante, Soria
(Show Context)
Citation Context ...xd) we have the right decay τ−(d−k)/2 and the bad one τ−k/3 on the hyperplane (x1, . . . , xk). 10 L. I. IGNAT AND E. ZUAZUA Proof of Lemma 3.1. The techniques used below are similar to those used in =-=[13]-=- to get lower bounds on oscillatory integrals. We define the relevant initial data through its Fourier transform. Let us first fix a positive function ϕ̂ supported on (−1, 1) such that ∫ pi−pi ϕ̂ = 1.... |

3 |
The discretized generalized Korteweg-de Vries equation with fourth order nonlinearity
- Nixon
(Show Context)
Citation Context ...ter h. They obtain Strichartz-like estimates in a class of exponents q and r larger than in the continuous one, none being independent of the parameter h. In the case of fully discrete schemes, Nixon =-=[28]-=- considers an approximation of the one-dimensional KdV equation based on the backward Euler approximation of the linear semigroup and proves space time estimates for that approximation. For the Schrö... |

2 |
Équation de Schrödinger non linéaire
- Baillon, Cazenave, et al.
- 1977
(Show Context)
Citation Context ...depending on the nonlinearity F . Also, Cazenave and Weissler [8] proved the local existence in the critical case p = 4/d. The case of H1-solutions has been analyzed by Baillon, Cazenave and Figueira =-=[3]-=-, Lin and Strauss [26], Ginibre and Velo [14, 15], Cazenave [5], and, in a more general context, by Kato [21, 22]. The Schrödinger equation has another remarkable property guaranteeing the gain of on... |

2 | elliptic equations
- Semilinear
- 1994
(Show Context)
Citation Context ... played a crucial role in the study of the nonlinear Schrödinger equation with nonlinearities involving derivatives (see [25]). For other deep results on the Schrödinger equations we refer to [38], =-=[6]-=- and the bibliography therein. NUMERICAL SCHEMES FOR NSE 3 In this paper we analyze whether semidiscrete schemes for LSE have dispersive properties similar to (1.3), (1.5) and (1.7), uniform with resp... |

2 |
American Mathematical Society (AMS
- Providence
- 2003
(Show Context)
Citation Context ..., 15], Cazenave [5], and, in a more general context, by Kato [21, 22]. The Schrödinger equation has another remarkable property guaranteeing the gain of one half space derivative in L2x,t (cf. [32], =-=[10]-=-, [11] and [24]): (1.7) sup x0,R 1 R ∫ B(x0,R) ∫ ∞ −∞ |(−∆)1/4eit∆ϕ|2dtdx ≤ C‖ϕ‖2L2(Rd). It has played a crucial role in the study of the nonlinear Schrödinger equation with nonlinearities involving ... |

2 | Fully discrete schemes for the Schrödinger equation. Dispersive properties.Mathematical Models and
- Ignat
(Show Context)
Citation Context ...ximation of the one-dimensional KdV equation based on the backward Euler approximation of the linear semigroup and proves space time estimates for that approximation. For the Schrödinger equation in =-=[18]-=- necessary and sufficient conditions to guarantee the existence, at the discrete level, of dispersive properties for the Schrödinger equation are given. 4 L. I. IGNAT AND E. ZUAZUA The paper is organ... |

1 |
remarks on the nonlinear Schrödinger equation in the critical case, Nonlinear semigroups, partial differential equations and attractors
- Some
- 1987
(Show Context)
Citation Context ...author proved that, in this case, NSE is globally well posed in L∞(R, L2(Rd)) ∩ Lqloc(R, L r(Rd)), where (q, r) is an d/2-admissible pair depending on the nonlinearity F . Also, Cazenave and Weissler =-=[8]-=- proved the local existence in the critical case p = 4/d. The case of H1-solutions has been analyzed by Baillon, Cazenave and Figueira [3], Lin and Strauss [26], Ginibre and Velo [14, 15], Cazenave [5... |

1 |
smoothing properties of Schrödinger equations
- Local
- 1989
(Show Context)
Citation Context ... Cazenave [5], and, in a more general context, by Kato [21, 22]. The Schrödinger equation has another remarkable property guaranteeing the gain of one half space derivative in L2x,t (cf. [32], [10], =-=[11]-=- and [24]): (1.7) sup x0,R 1 R ∫ B(x0,R) ∫ ∞ −∞ |(−∆)1/4eit∆ϕ|2dtdx ≤ C‖ϕ‖2L2(Rd). It has played a crucial role in the study of the nonlinear Schrödinger equation with nonlinearities involving deriva... |

1 |
Schrödinger equations, Schrödinger operators (Sønderborg
- Nonlinear
- 1988
(Show Context)
Citation Context ... case p = 4/d. The case of H1-solutions has been analyzed by Baillon, Cazenave and Figueira [3], Lin and Strauss [26], Ginibre and Velo [14, 15], Cazenave [5], and, in a more general context, by Kato =-=[21, 22]-=-. The Schrödinger equation has another remarkable property guaranteeing the gain of one half space derivative in L2x,t (cf. [32], [10], [11] and [24]): (1.7) sup x0,R 1 R ∫ B(x0,R) ∫ ∞ −∞ |(−∆)1/4eit... |

1 | solutions to nonlinear Schrödinger equations, Ann - Small - 1993 |