Nonlinear Schrödinger evolution equations

Brezis, Haı̈m; Gallouët, Thierry

doi:10.1016/0362-546x(80)90068-1

Cited by 489 publications

(415 citation statements)

References 3 publications

Supporting

Mentioning

403

Contrasting

Unclassified

Order By: Relevance

“…The proof relies on some Brezis-Gallouet inequality similar to those that appeared in [2]. After this work was completed, we learned that P. Gérard (see [5]) proved the global well-posedness for these equations, in the cases D = 2 or D = 3, in the energy space [4] for where the author uses similar framework to handle more general nonlinearities and problem in exterior domains.…”

Section: Gallo Has Proved the Following Resultsmentioning

confidence: 92%

Two Remarks on Solutions of Gross-Pitaevskii Equations on Zhidkov Spaces

Goubet

2007

Mh Math

View full text Add to dashboard Cite

show abstract

Section: Gallo Has Proved the Following Resultsmentioning

confidence: 92%

Two Remarks on Solutions of Gross-Pitaevskii Equations on Zhidkov Spaces

Goubet

2007

Mh Math

View full text Add to dashboard Cite

show abstract

“…We prove that by the Brézis-Gallouët technique applied to higher order energy with integration by parts, the standard theory developed in [5] and [18] for NLS and half-wave equation with cubic nonlinearity, has an improvement to quartic nonlinearity.…”

mentioning

confidence: 90%

“…In this paper in fact the authors treat, beside very interesting blow-up results, the Cauchy theory for the half-wave equation with cubic nonlinearity via the classical approach in [5]. We should also notice that in [18] the authors work in H 1 2 (R), while in Theorem 0.2 we work in H 1 (R).A basic tool along the proof of Theorem 0.2 will be the following version of (0.2):Its proof follows by a straightforward adaptation of the argument in [5]. Hence we skip it and we shall make an extensive use of (0.10) without any further comment.…”

mentioning

confidence: 99%

See 1 more Smart Citation

An improvement on the Brézis–Gallouët technique for 2D NLS and 1D half-wave equation

Ozawa

Visciglia

2016

Ann. Inst. H. Poincaré C Anal. Non Linéaire

View full text Add to dashboard Cite

Abstract. We revise the classical approach by Brézis-Gallouët to prove global well posedness for nonlinear evolution equations. In particular we prove global well-posedness for the quartic NLS posed on general domains Ω in R 2 with initial data in H 2 (Ω)∩H 1 0 (Ω), and for the quartic nonlinear half-wave equation on R with initial data in H 1 (R).The main aim of this paper is to revise the technique developed by Brézis-Gallouët to study the global well-posedness of Cauchy problems associated with some nonlinear evolution equations. We prove that by the Brézis-Gallouët technique applied to higher order energy with integration by parts, the standard theory developed in [5] and [18] for NLS and half-wave equation with cubic nonlinearity, has an improvement to quartic nonlinearity.Our first result concerns an extension to higher order nonlinearities of the very classical result in [5]. More precisely the first family of problems that we shall address is the following one:where λ = ±1, Ω ⊂ R 2 is open and satisfies the following hypothesis:is self-adjoint . By the celebrate Brézis-Gallouët inequality it follows that if Ω satisfies (H1), then the following logarithmic Sobolev embedding occurs:There has been a growing interest in the last decades on the Cauchy problem associated with NLS on domains, starting from the pioneering paper [5]. In this paper the authors can deduce global well-posedness for the defocusing cubic NLS on domains Ω ⊂ R 2 , by combining (0.2) with the conservation of the energy. A first extension of the result by Brézis-Gallouët, up to the fourth order nonlinearity, was obtained in [21] under some restrictive conditions on the initial data ϕ. More precisely it is assumed ϕ|ϕ| ∈ H 3 (Ω) ∩ H 1 0 (Ω), ∆ϕ ∈ H 1 0 (Ω). A fundamental tool to treat NLS on domains, with higher order nonlinearities, are the so called Strichartz inequalities (see [8] and the bibliography therein for the case Ω = R 2 ). In [6] it is proved a suitable version of Strichartz inequalities with loss, on general compact manifolds. Beside other results in this paper it is studied the Cauchy problem Due to the huge literature devoted to NLS on 2D domains, Theorem 0.1 below could be considered somewhat weaker compared with the known results, however we prefer to keep its statement along this paper for three reasons. First of all our argument is exclusively based on integration by parts and energy estimates, and hence it is independent on the use of Strichartz estimates. The second reason is that the proof of Theorem 0.1 can help to understand the idea behind the more involved proof of our second result concerning the nonlinear half-wave equation, where as far as we know our result is a novelty in the literature. The third reason is that as far as we know it is unclear whether or not the aforementioned Strichartz estimates are available under the rather general assumptions (H1), (H2).Let us recall that by the usual energy estimates, in conjunction with the classical Sobolev embedding H 2 (Ω) ֒→ L ∞ (Ω), one can prove that the Cauchy ...

show abstract

“…, for any T > 0 cf., e.g., [4,10,13]. In contrast, for the critical exponent p * = 2 d , global existence of a solution u of (1.1) is guaranteed only under a smallness condition, elaborated in equation (1.3).…”

Section: Introductionmentioning

confidence: 99%

A Posteriori Error Analysis for Evolution Nonlinear Schrödinger Equations up to the Critical Exponent

Katsaounis¹,

Kyza²

2018

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

Abstract. We provide a posteriori error estimates in the L ∞ ([0, T ]; L 2 (Ω))−norm for relaxation time discrete and fully discrete schemes for a class of evolution nonlinear Schrödinger equations up to the critical exponent. In particular for the discretization in time we use the relaxation Crank-Nicolson-type scheme introduced by Besse in [9]. The space discretization consists of finite element spaces that are allowed to change between time steps. The estimates are obtained using the reconstruction technique. Through this technique the problem is converted to a perturbation of the original partial differential equation and this makes it possible to use nonlinear stability arguments as in the continuous problem. Our analysis includes as special cases the cubic and quintic nonlinear Schrödinger equations in one spatial dimension and the cubic nonlinear Schrödinger equation in two spatial dimensions. Numerical results illustrate that the estimates are indeed of optimal order of convergence.

show abstract

Nonlinear Schrödinger evolution equations

Cited by 489 publications

References 3 publications

Two Remarks on Solutions of Gross-Pitaevskii Equations on Zhidkov Spaces

Two Remarks on Solutions of Gross-Pitaevskii Equations on Zhidkov Spaces

An improvement on the Brézis–Gallouët technique for 2D NLS and 1D half-wave equation

A Posteriori Error Analysis for Evolution Nonlinear Schrödinger Equations up to the Critical Exponent

Contact Info

Product

Resources

About