Global existence of solutions to Schrödinger equations on compact riemannian manifolds below $H^1$

Zhong, Sijia

Bulletin de la Société Mathématique de France, Tome 138 (2010) no. 4, pp. 583-613.

Nous nous intéressons dans cet article au caractère bien posé des équations de Schrödinger non-linéaires cubiques défocalisantes sur les variétés riemanniennes compactes sans bord, en régularité $H^s$, $s<1$, sous certaines conditions bilinéaires de Strichartz. Nous trouvons un $\tilde{s}<1$ tel que la solution est globale pour $s>\tilde{s}$.

In this paper, we will study global well-posedness for the cubic defocusing nonlinear Schrödinger equations on the compact Riemannian manifold without boundary, below the energy space, i.e. $s<1$, under some bilinear Strichartz assumption. We will find some $\tilde{s}<1$, such that the solution is global for $s>\tilde{s}$.

Classification : 35Q55, 37K05, 37L50, 81Q20
Keywords: schrödinger equation, compact riemannian manifold, global, I-method
Mot clés : Équation de schrödinger, variété riemanienne compacte, globalité, I-méthode

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