Growth of Sobolev Norms for 2d NLS with harmonic potential

2021/10/28 by F. Planchon, N. Tzvetkov, Planchon, F. +3 · 3 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · doi:10.48550/arxiv.2110.14912

openalex publication_date 2021/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove polynomial upper bounds on the growth of solutions to 2d cubic NLS where the Laplacian is confined by the harmonic potential. Due to better bilinear effects our bounds improve on those available for the 2d cubic NLS in the periodic setting: our growth rate for a Sobolev norm of order s=2k, k∈ ℕ, is t^2(s-1)/3+ε. In the appendix we provide an direct proof, based on integration by parts, of bilinear estimates associated with the harmonic oscillator.

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