On the stability of blowup solutions to the complex Ginzburg-Landau equation in R^d

TL;DR

This paper proves the full stability of a specific blowup solution in the complex Ginzburg-Landau equation by introducing a novel dynamic rescaling method that controls unstable modes and captures the blowup rate with log correction.

Contribution

It develops a generalized dynamic rescaling formulation with new modulation parameters, enabling stability analysis without spectrum analysis or topological arguments.

Findings

Established stability of type-I blowup with log correction.

Introduced a new modulation framework controlling all unstable modes.

Achieved stability proof using energy estimates and normalization conditions.

Abstract

Building upon the idea in \cite{HNWarXiv24}, we establish stability of the type-I blowup with log correction for the complex Ginzburg-Landau equation. In the amplitude-phase representation, a generalized dynamic rescaling formulation is introduced, with modulation parameters capturing the spatial translation and rotation symmetries of the equation and novel additional modulation parameters perturbing the scaling symmetry. This new formulation provides enough degrees of freedom to impose normalization conditions on the rescaled solution, completely eliminating the unstable and neutrally stable modes of the linearized operator around the blowup profile. It enables us to establish the full stability of the blowup by enforcing vanishing conditions via the choice of normalization and using weighted energy estimates, without relying on a topological argument or a spectrum analysis. The log correction for the blowup rate is captured by the energy estimates and refined estimates of the modulation parameters.