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<p>Building upon the idea in&#xa0;[Hou, arXiv:2404.09410 2024], we establish the 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 anisotropic 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, for a non-variational problem. No topological argument or spectrum analysis is needed, opening up the possibility to tackle a wide range of type-I singularities. The log correction for the blowup rate is automatically inferred via the local normalization conditions, captured by the energy estimates and refined estimates of the modulation parameters.</p>

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On the Stability of Blowup Solutions to the Complex Ginzburg-Landau Equation in \(\mathbb{R}^d\)

  • 期刊论文

  • 发表日期：

    2025年11月17日

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  • Jiajie Chen,
  • Thomas Y. Hou,
  • Van Tien Nguyen,
  • Yixuan Wang

摘要

Building upon the idea in [Hou, arXiv:2404.09410 2024], we establish the 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 anisotropic 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, for a non-variational problem. No topological argument or spectrum analysis is needed, opening up the possibility to tackle a wide range of type-I singularities. The log correction for the blowup rate is automatically inferred via the local normalization conditions, captured by the energy estimates and refined estimates of the modulation parameters.