Local Uniform Well-posedness for Nonlinear Schrödinger Equation with General Nonlinearity on Product Manifolds

Abstract

We study the Cauchy problem of the nonlinear Schrödinger equation posed on a (d + 1)-dimensional product manifolds $M={Z^{d}} \times {{\mathbb S}^1}$, where Z^d denotes the d-dimensional compact manifold under the eigenvalue assumption. We prove the uniform local well-posedness of H^s solution with the defocusing subcritical nonlinearity and extend the results of [Amer. J. Math., 2004, 126(3): 569–605], [Ann. Sci. École Norm. Sup. (4), 2005, 38(2): 255–301] and [Sci. China Math., 2015, 58(5): 1023–1046] to the more general geometry. The main tools in our argument are multilinear estimates and Bony’s linearized technique.

Keywords

Multilinear spectral estimates / nonlinear Schrödinger equation / Bourgain space