We consider the cubic nonlinear Schrödinger equation (NLS) on ℝ3 with randomized initial data. In particular, we study an iterative approach based on a partial power series expansion in terms of the random initial data. By performing a fixed point argument around the second order expansion, we improve the regularity threshold for almost sure local well-posedness from our previous work. We further investigate a limitation of this iterative procedure. Finally, we introduce an alternative iterative approach, based on a modified expansion of arbitrary length, and prove almost sure local well-posedness of the cubic NLS in an almost optimal regularity range with respect to the original iterative approach based on a power series expansion.
|Number of pages||47|
|Journal||Transactions of the American Mathematical Society: Series B|
|Publication status||Published - 4 Mar 2019|