Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on ${\mathbb{R}}^3$

Abstract

We consider the cubic nonlinear Schrödinger equation (NLS) on ${\mathbb{R}}^3$ 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 [2]. 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.

48 pages. To appear in Trans. Amer. Math. Soc

References in corpus (6)

• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation
• Ill-posedness for nonlinear Schrodinger and wave equations
• Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity
• Almost sure local well-posedness for the supercritical quintic NLS
• On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities
• Almost sure scattering for the energy-critical NLS with radial data below $H^1({\mathbb{R}}^4)$

Cited by in corpus (4)

• Almost sure scattering for the radial energy critical nonlinear wave equation in three dimensions
• Random initial conditions for semi-linear PDEs
• Pointwise convergence problem of Ostrovsky equation with rough data and random data
• Probabilistic pointwise convergence problem of some dispersive equations