Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on ℝ³

Abstract

We consider the cubic nonlinear Schrödinger equation (NLS) on R 3 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.

Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation

“…From the proof of Theorem 1.3, it is clear that our methods easily generalize to other space dimensions d ≥ 3 and to other power-type nonlinearities. We also expect that our functional framework is compatible with the iterative procedure put forth in [5] and that these ideas can be combined to further lower the regularity threshold.…”

“…We also refer to Brereton [13] for analogous results for the defocusing quintic NLS. In the context of the cubic NLS on R 3 , Bényi-Oh-Pocovnicu [5] recently introduced an iterative procedure based on a partial power series expansion in terms of the free evolution of the random data, which allows to lower the regularity threshold for almost sure local well-posedness obtained in their previous work [6].…”

Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation

“…The literature on random dispersive partial differential equations is vast. We refer the interested reader to the survey [6], and mention the related works [2,4,8,9,11,12,14,15,16,38,39,41,42,44]. In the following discussion, we focus on the Wiener randomization [3,38] of a function f P H s pR d q.…”

On the Probabilistic Cauchy Theory for Nonlinear Dispersive PDEs

“…While it is theoretically possible to repeat this procedure based on partial power series expansions, it seems virtually impossible to keep track of all the terms as the order of the expansion grows. In [7], we introduced modified higher order expansions to avoid this combinatorial nightmare and established improved almost sure local well-posedness based on the modified higher order expansions of arbitrary length.…”