2025/07/15 by Zexing Li, Li, Zexing · 1 citation
Engineering · Physics and Astronomy · #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Ionosphere and magnetosphere dynamics #Magnetic confinement fusion research #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2507.11248
openalex publication_date 2025/07/15 · openalex created_date 2025/10/08 · openalex updated_date 2026/07/28
We consider self-similar blowup for (NLS) i∂t u + Δu + u|u|p-1 = 0 in d ≥ 1 and slightly mass-supercritical range 0 < sc := \frac d2 - (2)/(p-1) ≪ 1. The existence and stability of such dynamics [Merle-Raphaël-Szeftel, 2010] and construction of suitable profiles [Bahri-Martel-Raphaël, 2021] lead to the question of asymptotic stability. Based on our previous work [Li, 2023], this nonlinear problem is reduced to linear mode stability of the matrix linearized operator. In this work, we prove mode stability for the low-energy spectrum in d ≥ 1 as a perturbation of the linearized operator around ground state for mass-critical NLS. The main difficulty of this spectral bifurcation problem arises from the non-self-adjoint, relatively unbounded and high-dimensional nature, for which we exploit the Jost function argument from [Perelman, 2001], qualitative WKB analysis generalized from [Bahri-Martel-Raphaël, 2021], matched asymptotics method and uniform estimates for high spherical classes based on special functions.