Quasilinear Schrö;dinger equations, II: Small data and cubic nonlinearities

In part I of this project we examined low-regularity local well-posedness for generic quasilinear Schrodinger equations with small data. This improved, in the small data regime, the preceding results of Kenig, Ponce, and Vega as well as Kenig, Ponce, Rolvung, and Vega. In the setting of quadratic interactions, the (translation invariant) function spaces which were utilized incorporated an l1-summability over cubes in order to account for Mizohata's integrability condition, which is a necessary condition for the L2 well-posedness for the linearized equation. For cubic interactions, this integrability condition meshes better with the inherent L2-nature of the Schrodinger equation, and such summability is not required. Thus we are able to prove small data well-posedness in Hs-spaces. © 2014 by Kyoto University