Cauchy problem of nonlinear Schrödinger equation with initial data in Sobolev space W s , p for p < 2

In this paper, we consider in R the Cauchy problem for nonlinear Schrödinger equation with initial data in Sobolev space W s,p for p < 2. It is well known that this problem is ill posed. However, We show that after a linear transformation by the linear semigroup the problem becomes locally well posed in W s,p for 2n n+1 < p < 2 and s > n(1− 1 p ). Moreover, we show that in one space dimension, the problem is locally well posed in L for any 1 < p < 2. Keyword:Cauchy problem, nonlinear Schrödinger equation, locally well-posedness, scaling limit.