Bilinear Eigenfunction Estimates and the Nonlinear Schroedinger Equation on Surfaces

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 $H^s$, which is sharp on the sphere. The main ingredient in our proof is a sharp bilinear estimate for Laplace spectral projectors on compact surfaces. 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 $H^s$, ce qui est optimalsur la sphère. Le principal ingrédient de notre démonstration est une estimation bilinéaire pour les projecteurs spectraux du laplacien sur une surface compacte.

34 pages, 2 figures

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