2 2 A ug 2 00 3 BILINEAR EIGENFUNCTION ESTIMATES AND THE NONLINEAR SCHRÖDINGER EQUATION ON SURFACES
نویسندگان
چکیده
— We study the cubic non linear Schrödinger equation (NLS) on compact surfaces. On the sphere 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. Résumé. — Onétudie l'´ equation de Schrödinger non linéaire (NLS) sur une surface com-pacte. Sur la sphère 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 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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