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

Yilin Song , Ruixiao Zhang

Frontiers of Mathematics ›› : 1 -37.

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Frontiers of Mathematics ›› : 1 -37. DOI: 10.1007/s11464-023-0122-x
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Local Uniform Well-posedness for Nonlinear Schrödinger Equation with General Nonlinearity on Product Manifolds

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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 Zd denotes the d-dimensional compact manifold under the eigenvalue assumption. We prove the uniform local well-posedness of Hs 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.

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Multilinear spectral estimates / nonlinear Schrödinger equation / Bourgain space

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Yilin Song,Ruixiao Zhang. Local Uniform Well-posedness for Nonlinear Schrödinger Equation with General Nonlinearity on Product Manifolds. Frontiers of Mathematics 1-37 DOI:10.1007/s11464-023-0122-x

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