Inventiones mathematicae

, Volume 159, Issue 1, pp 187–223

Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces

Article

Abstract

We study the cubic non linear Schrödinger equation (NLS) on compact surfaces. On the sphere \(\mathbb{S}^2\) and more generally on Zoll surfaces, we prove that, for s>1/4, NLS is uniformly well-posed in Hs, which is sharp on the sphere. The main ingredient in our proof is a sharp bilinear estimate for Laplace spectral projectors on compact surfaces.

Résumé

On étudie l’équation de Schrödinger non linéaire (NLS) sur une surface compacte. Sur la sphère \(\mathbb{S}^2\) et plus généralement sur toute surface de Zoll, on démontre que pour s>1/4, NLS est uniformément bien posée dans Hs, ce qui est optimal sur la sphère. Le principal ingrédient de notre démonstration est une estimation bilinéaire pour les projecteurs spectraux du laplacien sur une surface compacte.

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Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  1. 1.MathématiquesUniversité Paris SudOrsayFrance

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