Quasilinear Schrödinger equations III: Large Data and Short Time

Abstract

In this article we prove short time local well-posedness in low-regularity Sobolev spaces for large data general quasilinear Schrödinger equations with a non-trapping assumption. These results represent improvements over the small data regime considered by the authors in previous works, as well as the pioneering works by Kenig-Ponce-Vega and Kenig-Ponce-Rolvung-Vega, where viscosity methods were used to prove 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.

47 pages, no figures. Updates made from referee reports. Comments welcome!!

References in corpus (2)

• The General Quasilinear Ultrahyperbolic Schrödinger Equation
• Quasilinear Schrödinger Equations

Cited by in corpus (8)

• Smoothing effect for time-degenerate Schrödinger operators
• Long time solutions for quasi-linear Hamiltonian perturbations of Schrödinger and Klein-Gordon equations on tori
• Low regularity solutions for the general quasilinear ultrahyperbolic Schrödinger equation
• Local well-posedness of skew mean curvature flow for small data in $d\ge 4$ dimensions
• Local well-posedness for a quasilinear Schrödinger equation with degenerate dispersion
• Local well-posedness for the quasi-linear Hamiltonian Schrödinger equation on tori
• Illposedness for dispersive equations: Degenerate dispersion and Takeuchi--Mizohata condition
• Exotic traveling waves for a quasilinear Schrödinger equation with nonzero background