Optimal Local Well-Posedness for the Periodic Derivative Nonlinear Schrödinger Equation

We prove local well-posedness for the periodic derivative nonlinear Schrödinger equation, which is L 2 critical, in Fourier-Lebesgue spaces which scale like H s ( T ) for s > 0 . Our result is optimal in the sense that it covers the full subcritical regime. In particular we close the existing gap in the subcritical theory by improving the result of Grünrock and Herr (SIAM J Math Anal 39(6):1890–1920, 2008), which established local well-posedness in Fourier-Lebesgue spaces which scale like H s ( T ) for s > 1 4 . 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 et al. (Forum Math Pi, 3:75, 2015) and Catellier and Chouk (Ann Probab 46(5):2621–2679, 2018) in the context of stochastic PDE. Our proof and results however, are purely deterministic.