Blow-up dynamics and spectral property in the $L^2$-critical nonlinear Schr\"odinger equation in high dimensions

TL;DR

This paper investigates the stable blow-up behavior of the $L^2$-critical nonlinear Schrödinger equation in high dimensions, confirming the 'log-log' regime through numerical simulations and providing a numerically-assisted proof of spectral properties.

Contribution

It extends the understanding of blow-up dynamics and spectral properties of the NLS in high dimensions, including numerical verification and a numerically-assisted proof of the 'log-log' regime.

Findings

Confirmation of 'log-log' blow-up regime in dimensions 4 to 12.

Numerically-assisted proof of spectral property for dimensions 5 to 12.

Stable blow-up regime proven for dimensions up to 10, radially stable up to 12.

Abstract

We study stable blow-up dynamics in the $L^2$-critical nonlinear Schr\"{o}dinger 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 ($d \geq5)$, 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 $2 \leq d \leq 12$. This, combined with previous results of Merle-Rapha\"el, proves the "log-log" stable blow-up regime in dimensions $d \leq 10$ and radially stable for $d \leq 12$.