Almost sure well-posedness and scattering of the 3D cubic nonlinear Schrödinger equation

2021/10/22 by Jia Shen, Avy Soffer, Shen, Jia +3
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #FOS: Mathematics #Quantum Chromodynamics and Particle Interactions

paper · pdf · doi:10.48550/arxiv.2110.11648

openalex publication_date 2021/10/22 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

We study the random data problem for 3D, defocusing, cubic nonlinear Schrödinger equation in H_x^s(ℝ³) with s<\frac 12. First, we prove that the almost sure local well-posedness holds when (1)/(6)\leqslant s<\frac 12 in the sense that the Duhamel term belongs to H_x^1/2(ℝ³). Furthermore, we prove that the global well-posedness and scattering hold for randomized, radial, large data f∈ H_x^s(ℝ³) when (17)/(40)< s<\frac 12. The key ingredient is to control the energy increment including the terms where the first order derivative acts on the linear flow, and our argument can lower down the order of derivative more than \frac12. To our best knowledge, this is the first almost sure large data global result for this model.

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