Bilinear eigenfunction estimates and the nonlinear Schr�dinger equation on surfaces

Burq, Nicolas; Gérard, Patrick; Tzvetkov, Nikolay

doi:10.1007/s00222-004-0388-x

Cited by 145 publications

(242 citation statements)

References 51 publications

Supporting

Mentioning

233

Contrasting

Unclassified

Order By: Relevance

“…It is in fact possible to prove a bilinear version of this result (see [7], [8], [9]), and, combining with the clustering properties of the spectrum, one can prove that (5.1) holds on S 2 for every s 0 > 1 4 , which proves that…”

Section: Notice That Estimate (51) Implies the Following Strichartz mentioning

confidence: 84%

“…This estimate was proved in [7] to be a criterion for the existence of a local-in-time smooth flow map for (1.1) on…”

Section: Notice That Estimate (51) Implies the Following Strichartz mentioning

confidence: 94%

“…Notice that s c (R/aZ × R/bZ) is still unknown for arbitrary values of a, b, though it has been proved to be ≤ 1/3 by Catoire-Wang in [10]. The part of Theorem 4.3 concerning the sphere is deduced from arguments by Burq-Gérard-Tzvetkov in [6] and [7] (see also [13]). A generalization of these arguments shows that, if M is a surface of revolution with a nondegenerate equator, s c (M ) ≥ 1/4, but the upper bound is still open.…”

Section: Nicolas Burq Patrick Gérard and Nikolay Tzvetkovmentioning

confidence: 99%

See 2 more Smart Citations

High frequency solutions of the nonlinear Schrödinger equation on surfaces

Burq

Gérard²,

Tzvetkov

2009

Quart. Appl. Math.

View full text Add to dashboard Cite

Abstract. We address the problem of describing solutions of the nonlinear Schrödin-ger equation on a compact surface in the high frequency regime. In this context, we introduce a nonnegative threshold, depending on the geometry of the surface, which can be seen as a measurement of the nonlinear character of the equation, and we compute this number for the torus and for the sphere, as a consequence of earlier arguments. The last part is devoted to the study, on the sphere, of the critical regime associated to this threshold. We prove that the effective dynamics are described by a new evolution equation, the Resonant Hermite-Schrödinger equation.

show abstract

Section: Notice That Estimate (51) Implies the Following Strichartz mentioning

confidence: 84%

“…This estimate was proved in [7] to be a criterion for the existence of a local-in-time smooth flow map for (1.1) on…”

Section: Notice That Estimate (51) Implies the Following Strichartz mentioning

confidence: 94%

Section: Nicolas Burq Patrick Gérard and Nikolay Tzvetkovmentioning

confidence: 99%

See 1 more Smart Citation

High frequency solutions of the nonlinear Schrödinger equation on surfaces

Burq

Gérard²,

Tzvetkov

2009

Quart. Appl. Math.

View full text Add to dashboard Cite

show abstract

“…Interestingly, many tools in [8]- [9] resemble the Birkhoff normal form techniques developed by Bambusi, Delort, Grebért, Szeftel [2] for PDEs on spheres and Zoll manifolds, and seem deeply related with the methods of Burq, Gérard, Tzvetkov [15] for the initial value problem.…”

Section: Introductionmentioning

confidence: 99%

Quasi-periodic solutions of Hamiltonian PDEs

Berti¹

2012

Journées Équations Aux Dérivées Partielles

View full text Add to dashboard Cite

“…If the Euclidean space is replaced by a manifold, we refer to [Burq et al 2005] and [Hani 2010]. The case of the wave equation is treated by Klainerman, Machedon, Bourgain, and Tataru [Klainerman and Machedon 1996], and Foschi and Klainerman [2000].…”

mentioning

confidence: 99%

Untitled

2013

Anal. PDE

View full text Add to dashboard Cite

See inside back cover or msp.org/apde for submission instructions.The subscription price for 2013 is US $160/year for the electronic version, and $310/year (+$35, if shipping outside the US) for print and electronic. Subscriptions, requests for back issues from the last three years and changes of subscribers address should be sent to MSP.Analysis & PDE (ISSN 1948(ISSN -206X electronic, 2157 We consider solutions to the linear wave equation on higher dimensional Schwarzschild black hole spacetimes and prove robust nondegenerate energy decay estimates that are in principle required in a nonlinear stability problem. More precisely, it is shown that for solutions to the wave equation g D 0 on the domain of outer communications of the Schwarzschild spacetime manifold .M n m ; g/ (where n 3 is the spatial dimension, and m > 0 is the mass of the black hole) the associated energy flux EOE . † / through a foliation of hypersurfaces † (terminating at future null infinity and to the future of the bifurcation sphere) decays, EOE . † / Ä CD= 2 , where C is a constant depending on n and m, and D < 1 is a suitable higher-order initial energy on † 0 ; moreover we improve the decay rate for the first-order energy to EOE@ t . † R / Ä CD ı = 4 2ı for any ı > 0, where † R denotes the hypersurface † truncated at an arbitrarily large fixed radius R < 1 provided the higher-order energy D ı on † 0 is finite. We conclude our paper by interpolating between these two results to obtain the pointwise estimate j j † R Ä CD 0 ı = 3 2 ı . In this work we follow the new physical-space approach to decay for the wave equation of Dafermos and Rodnianski (2010).

show abstract

Bilinear eigenfunction estimates and the nonlinear Schr�dinger equation on surfaces

Cited by 145 publications

References 51 publications

High frequency solutions of the nonlinear Schrödinger equation on surfaces

High frequency solutions of the nonlinear Schrödinger equation on surfaces

Quasi-periodic solutions of Hamiltonian PDEs

Untitled

Contact Info

Product

Resources

About