Almost sure scattering for the nonradial energy-critical NLS with arbitrary regularity in 3D and 4D cases

Abstract

In this paper, we study the defocusing energy-critical nonlinear Schrödinger equations i∂ t u + ∆u = |u| 4 d−2 u. When d = 3, 4, we prove the almost sure scattering for the equations with nonradial data in H s x for any s ∈ R. In particular, our result does not rely on any spherical symmetry, size or regularity restrictions. Contents 1. Introduction 1.1. Definition of randomization 1.2. Main result 1.3. Sketch of the proof. 1.4. Organization of the paper 2. Preliminary 2.1. Notation 2.2. Useful lemmas 2.3. Probabilistic theory 3. Almost sure Strichartz estimates 3.1. Strichartz estimates 3.2. Linear estimates in 3D case 3.3. Linear estimates in 4D case 4. Global well-posedness and scattering in 3D case 4.1. Reduction to the deterministic problem 4.2. Local theory 4.3. Modified Interaction Morawetz 4.4. Almost conservation law 4.5. Perturbations 4.6. Proof of Proposition 4.1 5. Global well-posedness and scattering in 4D case 5.1. Reduction to the deterministic problem 5.

Almost sure local well-posedness for cubic nonlinear Schrodinger equation with higher order operators

“…Hence, we recover the range of regularities of [DLM19] for d = 4, almost recover [SSW21] when d = 3 (we do not prove the borderline case) and improve that of [BOP15] for any dimension. We believe our methods can be refined to recover probabilistic estimates for the time of existence of solutions.…”

“…Later, Brereton [Bre19] obtained analogous results for P = −∆ and quintic nonlinearity. When d = 3, Shen, Soffer, and Wu [SSW21], very recently, obtained the local well-posedness of (1.1) (with the Laplacian) for S ≥ 1 6 improving [BOP15]. All the previous results rely on a fixed point argument for operators on variants of the X s,b spaces adapted to the variation spaces V p and U p introduced by Koch, Tataru, and collaborators [HHK09,HTT11,KTV14].…”

On the almost sure scattering for the energy-critical cubic wave equation with supercritical data

“…We note that in [10] a different randomization than in [22,21] for radially symmetric data was used. In fact, most of the aforementioned results (except [9,40,38]) relied on the so-called Wiener randomization, which is based on a unit-scale decomposition of frequency space (see Subsection 1.2 below for details).…”

Remarks on the almost sure well-posedness for incompressible Navier-Stokes equations with arbitrary regularity

“…Furthermore, other kinds of randomization were also introduced in the Euclidean space setting R d , such as the ones based on the wave packet [8], annuli [7], or angular variable [10] decompositions. Very recently, Shen, Soffer and Wu [41] constructed a "narrowed" Wiener randomization, and gave the first probabilistic result without any regularity restrictions, in the context of nonlinear Schrödinger equations. Furthermore, there are some initial data with very low regularity in practical use, such as the white noise in Ḣ−d/2− and the Brownian motion in Ḣ−d/2+1− .…”

“…Furthermore, there are some initial data with very low regularity in practical use, such as the white noise in Ḣ−d/2− and the Brownian motion in Ḣ−d/2+1− . This motivates us to study the problem with arbitrarily rough data, as what was done in [41]. Therefore, in this paper, we aim to apply the "narrowed" Wiener randomization to study the almost sure well-posedness for the incompressible Navier-Stokes equations.…”