Blow-up dynamics and spectral property in the $L^2$-critical nonlinear Schrödinger equation in high dimensions

Abstract

We study stable blow-up dynamics in the $L^2$-critical nonlinear Schrödinger equation in high dimensions. First, we show that in dimensions $d=4$ to $d=12$ generic blow-up behavior confirms the "log-log" regime in our numerical simulations, including the log-log rate and the convergence of the blow-up profiles to the rescaled ground state; this matches the description of the stable blow-up regime in the dimension $d=2$ (for the 2d cubic NLS equation). Next, we address the question of rigorous justification of the "log-log" dynamics in higher dimensions ($d\ge 5)$, at least for the initial data with the mass slightly larger than the mass of the ground state, for which the spectral conjecture has yet to be proved, see [34] and [10]. We give a numerically-assisted proof of the spectral property for the dimensions from $d=5$ to $d=12$, and a modification of it in dimensions $2\le d\le 12$. This, combined with previous results of Merle-Raphaël, proves the "log-log" stable blow-up regime in dimensions $d\le 10$ and radially stable for $d\le 12$.

Cited by in corpus (8)

• Blow-up dynamics in the mass super-critical NLS equations
• Threshold for Blowup and Stability for Nonlinear Schrödinger Equation with Rotation
• Behavior of solutions to the 1D focusing stochastic $L^2$-critical and supercritical nonlinear Schrödinger equation with space-time white noise
• A Spectral Analysis of the Nonlinear Schroedinger Equation in the Co-Exploding Frame
• Low regularity blowup solutions for the mass-critical NLS in higher dimensions
• Behavior of solutions to the 1D focusing stochastic nonlinear Schrödinger equation with spatially correlated noise
• A Normal Form for the Onset of Collapse: the Prototypical Example of the Nonlinear Schrodinger Equation
• Spectral property for the 2D Zakharov-Kuznetsov equation