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Bilinear Strichartz estimates for Schrödinger operators in two-dimensional compact manifolds with boundary and cubic NLS. (English) Zbl 1240.35556

Summary: In this paper we establish bilinear and gradient bilinear Strichartz estimates for Schrödinger operators in two-dimensional compact manifolds with boundary. Using these estimates, we can infer the local well-posedness of the cubic nonlinear Schrödinger equation in \(H^{s}\) for every \(s>2/3\) on such manifolds.

MSC:

35R01 PDEs on manifolds
35Q41 Time-dependent Schrödinger equations and Dirac equations
35Q40 PDEs in connection with quantum mechanics
58Jxx Partial differential equations on manifolds; differential operators
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