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arXiv · 1905.04352

Optimal local well-posedness for the periodic derivative nonlinear Schrodinger equation

Yu Deng·Andrea R. Nahmod·Haitian Yue
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Abstract

We prove local well-posedness for the periodic derivative nonlinear Schrodinger's equation, which is L^2 critical, in Fourier-Lebesgue spaces which scale like H^s(T) for s>0. In particular we close the existing gap in the subcritical theory by improving the result of Grunrock and Herr [25], which established local well-posedness in Fourier-Lebesgue spaces which scale like H^s(T) for s>1 . We achieve this result by a delicate analysis of the structure of the solution and the construction of an adapted nonlinear submanifold of a suitable function space. Together these allow us to construct the unique solution to the given subcritical data. This constructive procedure is inspired by the theory of para-controlled distributions developed by Gubinelli-Imkeller-Perkowski [26] and Cantellier-Chouk [10] in the context of stochastic PDE. Our proof and results however, are purely deterministic.

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Yu Deng, Andrea R. Nahmod, Haitian Yue. 2019-05-10. Optimal local well-posedness for the periodic derivative nonlinear Schrodinger equation. https://doi.org/10.1007/s00220-020-03898-8

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