RANDOM DATA CAUCHY PROBLEM FOR THE NONLINEAR SCHRÖDINGER EQUATION WITH DERIVATIVE NONLINEARITY

摘要

We consider the Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity (i∂t + △)u = ±∂(u~m) on R~d, d ≥ 1, with random initial data, where d is a first order derivative with respect to the spatial variable, for example a linear combination of ∂/∂x_1, • • •, ∂/∂x_d or |▽| = F~(-1) [lξlF]. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in H~s(R~d) with s > mex(d-1/ds_c,s_c/2t, s_c - d/2(d+1)) for d + m ≥ 5, where s is below the scaling critical regularity s_c :=d/2-1/m-1.