Low regularity blowup solutions for the mass-critical NLS in higher dimensions

TL;DR

This paper investigates the stability of log-log blowup solutions for the focusing mass-critical nonlinear Schrödinger equation in dimensions three and higher, extending previous two-dimensional results using bootstrap and commutator techniques.

Contribution

It extends the $H^s$-stability results of log-log blowup regimes from 2D to higher dimensions $d \\geq 3$ for the mass-critical NLS.

Findings

Established $H^s$-stability of blowup solutions in higher dimensions.

Extended previous 2D results to $d \\geq 3$.

Utilized bootstrap and commutator estimates in the analysis.

Abstract

In this paper, we study the $H^s$-stability of the log-log blowup regime (which has been completely described in a series of recent works by Merle and Raphael) for the focusing mass-critical nonlinear Schr\"odinger equations $i\partial_tu+\Delta u+|u|^\frac4du=0$ in $\mathbb{R}^d$ with $d\geq3$. We aim to extend the result in [Colliander and Raphael, Rough blowup solutions to the $L^2$ critical NLS, Math. Anna., 345(2009), 307-366.] for dimension two to the higher dimensions cases $d\geq3$, where we use the bootstrap argument in the above paper and the commutator estimates in [M. Visan and X. Zhang, On the blowup for the $L^2$-critical focusing nonlinear Schr\"odinger equation in higher dimensions below the energy class. SIAM J. Math. Anal., 39(2007), 34-56.].