Almost sure scattering for the energy-critical NLS with radial data below $H^1({\mathbb{R}}^4)$

Abstract

We prove almost sure global existence and scattering for the energy-critical nonlinear Schrödinger equation with randomized spherically symmetric initial data in $H^s({\mathbb{R}}^4)$ with $\frac{5}{6}<s<1$. We were inspired to consider this problem by the recent work of Dodson--Lührmann--Mendelson, which treated the analogous problem for the energy-critical wave equation.

19 pages

References in corpus (4)

• Ill-posedness for nonlinear Schrodinger and wave equations
• Almost sure local well-posedness for the supercritical quintic NLS
• Solutions globales pour l'équation de Schrödinger cubique en dimension 3
• Solutions globales pour des équations de Schrödinger sur-critiques en toutes dimensions

Cited by in corpus (6)

• On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities
• Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on ${\mathbb{R}}^3$
• Almost sure well-posedness for the cubic nonlinear Schrödinger equation in the super-critical regime on $\TTT^d$, $d\ge 3$
• Pointwise convergence problem of Ostrovsky equation with rough data and random data
• Probabilistic pointwise convergence problem of some dispersive equations
• Large global solutions for nonlinear Schrödinger equations II, mass-supercritical, energy-subcritical cases