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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.

MSC:
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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