Quasilinear Schrödinger Equations III: Large Data and Short Time

In this article we prove short time local well-posedness in low-regularity Sobolev spaces for large data general quasilinear Schrödinger equations with a nontrapping assumption. These results represent improvements over the small data regime considered by the authors in Marzuola et al. (Adv Math 231(2):1151–1172, 2012; Kyoto J Math 54(3):529–546, 2014), as well as the pioneering works by Kenig et al. (Invent Math 158:343–388, 2004; Adv Math 196(2), 402–433, 2005; Adv Math 206(2):373–486, 2006), where viscosity methods were used to prove the existence of solutions for localized data in high regularity spaces. Our arguments here are purely dispersive. The function spaces in which we show existence are constructed in ways motivated by the results of Mizohata, Ichinose, Doi, and others, including the authors.