Globally well and ill posedness for non-elliptic derivative Schrödinger equations with small rough data

Abstract

We show that there exists sc>0 such that the cubic (quartic) non-elliptic derivative Schrödinger equations with small data in modulation spaces M2,1s(Rn) for n≥3 (n=2) are globally well-posed if s≥sc, and ill-posed if s<sc. In 2D cubic case, using the Gabor frame, we get some time-global dispersive estimates for the Schrödinger semi-group in anisotropic Lebesgue spaces, which include a time-global maximal function estimate in the space Lx12Lx2,t∞. By resorting to the smooth effect estimate together with the dispersive estimates in anisotropic Lebesgue spaces, we show that the cubic hyperbolic derivative NLS in 2D has a unique global solution if the initial data in Feichtinger–Segal algebra or in weighted Sobolev spaces are sufficiently small.