Sharp global well-posedness for non-elliptic derivative Schrödinger equations with small rough data

Abstract

We show the sharp global well posedness for the Cauchy problem for the cubic (quartic) non-elliptic derivative Schrödinger equations with small rough data in modulation spaces $M_{2,1}^s({\mathbb{R}}^n)$ for $n\ge 3$ ($n=2$). 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 $L_{x_1}^2{L}_{x_2,t}^{\infty }$. 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.

42 Pages

References in corpus (3)

• On Schrödinger Maps
• Global results for Schrödinger Maps in dimensions $n\ge 3$
• Global well posedness and scattering for the elliptic and non-elliptic derivative nonlinear Schrodinger equations with small data

Cited by in corpus (4)

• Schrödinger equations with rough Hamiltonians
• Spherically averaged maximal function and scattering for the 2D cubic derivative Schrödinger equation
• Gabor representations of evolution operators
• The inviscid limit for the Landau-Lifshitz-Gilbert equation in the critical Besov space