Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces. (English) Zbl 1092.35099

The authors prove that the nonlinear Schrödinger equation is uniformly well-posed on the two-dimensional sphere (or more generally on Zoll surfaces) in the Sobolev spaces of order \(s\) bigger than \(1/4\); the assertion is sharp on the sphere. The main tool for proving this are bilinear estimates for Laplace spectral projectors on compact surfaces.


35Q55 NLS equations (nonlinear Schrödinger equations)
58J32 Boundary value problems on manifolds
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35P15 Estimates of eigenvalues in context of PDEs
35Q40 PDEs in connection with quantum mechanics
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