[Paper Review] Log-log blow up solutions blow up at exactly m points

Chenjie Fan|arXiv (Cornell University)|Oct 4, 2015

Advanced Mathematical Physics Problems23 references3 citations

TL;DR

This paper constructs finite-time blow-up solutions to the mass-critical focusing nonlinear Schrödinger equation in dimensions d=1,2 that blow up at exactly m distinct points, using log-log blow-up solutions—derived from the log-log law—as building blocks. The construction relies on a compactness argument and perturbation theory, proving that such solutions exhibit stable m-point blow-up dynamics with precise asymptotic behavior near the blow-up time.

ABSTRACT

We study the focusing mass-critical nonlinear Schrödinger equation, and construct certain solutions which blow up at exactly $m$ points according to the log-log law.

Motivation & Objective

• To construct solutions of the mass-critical focusing nonlinear Schrödinger equation that blow up at exactly m distinct spatial points in finite time.
• To extend the known theory of single-point log-log blow-up solutions to multi-bubble configurations with m ≥ 2.
• To establish the asymptotic behavior of such solutions near the blow-up time, showing convergence to a sum of m decoupled log-log blow-up profiles.
• To prove the stability and persistence of m-point blow-up dynamics under small perturbations of initial data.

Proposed method

• Uses the log-log blow-up solutions—solutions with blow-up rate $ \|\nabla u(t)\|_{L^2} \sim \left( \frac{\ln|\ln(T-t)|}{T-t} \right)^{1/2} $—as fundamental building blocks.
• Applies a compactness argument based on the concentration-compactness method to extract a limit profile from a sequence of approximate solutions.
• Employs perturbation theory and energy estimates to control the interaction between m distinct blow-up bubbles.
• Relies on the stability of the log-log dynamics under small perturbations, as established in prior works by Merle and Raphaël.
• Uses the pseudo-conformal transformation and scaling symmetry to relate the log-log solutions to the explicit soliton solution $ Q(x)e^{it} $.
• Implements a refined analysis of the evolution of parameters $ \lambda_j(t) $ and $ x_j(t) $ for each bubble to ensure they remain separated and evolve independently near blow-up time.

Experimental results

Research questions

• RQ1 Can solutions of the mass-critical focusing NLS blow up at exactly m distinct spatial points for any m ≥ 1?
• RQ2 What is the precise asymptotic behavior of such m-point blow-up solutions near the blow-up time?
• RQ3 How do the parameters $ \lambda_j(t) $ and $ x_j(t) $ of each blow-up bubble evolve, and can they be controlled independently?
• RQ4 Is the m-point blow-up dynamics stable under small perturbations of the initial data?

Key findings

• The paper constructs solutions to the mass-critical NLS that blow up at exactly m distinct points $ x_1, \dots, x_m \in \mathbb{R}^d $, with $ x_i \neq x_j $ for $ i \neq j $.
• Near the blow-up time $ T $, the solution asymptotically behaves as $ u(t,x) \sim \sum_{j=1}^m S\left( \frac{t-T}{\lambda_j^2}, \frac{x-x_j}{\lambda_j} \right) $, where $ S $ is the log-log blow-up solution.
• Each bubble evolves independently with $ \lambda_j(t) \to 0 $ as $ t \to T $, and the blow-up rate is $ \|\nabla u(t)\|_{L^2} \sim \left( \frac{\ln|\ln(T-t)|}{T-t} \right)^{1/2} $.
• The stability of the m-point blow-up is established via a perturbation argument: small changes in initial data lead to small changes in the blow-up parameters $ \lambda_j $ and $ x_j $.
• The construction relies on the fact that the log-log dynamics are stable and that the blow-up time of each bubble is controlled by the slowest-decaying $ \lambda_j $, ensuring no premature collapse.