Bilinear Eigenfunction Estimates and the Nonlinear Schroedinger Equation on Surfaces

Details Bilinear Eigenfunction Estimates and the Nonlinear Schroedinger Equation on Surfaces Bilinear Eigenfunction Estimates and the Nonlinear Schroedinger Equation on Surfaces

2003-08-22

arxiv.org/abs/math/0308214v1

Mathematics

Analysis of PDEs

34 pages, 2 figures

Scientific paper

10.1007/s00222-004-0388-x

We study the cubic non linear Schr\"odinger 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 \'etudie l'\'equation de Schr\"odinger non lin\'eaire (NLS) sur une surface compacte.Sur la sph\`ere $\mathbb{S}^2$ et plus g\'en\'eralement sur toute surface de Zoll, on d\'emontre que pour $s>1/4$, NLS est uniform\'ement bien pos\'ee dans $H^s$, ce qui est optimalsur la sph\`ere. Le principal ingr\'edient de notre d\'emonstration est une estimation bilin\'eaire pour les projecteurs spectraux du laplacien sur une surface compacte.

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