Singular limits for general energy-critical complex Ginzburg-Landau equations
Abstract
Motivated by dissipative approximations of the three-dimensional focusing energy-critical nonlinear Schrödinger equation, we study the inviscid and zero-dispersion limits for general energy-critical complex Ginzburg-Landau equations in dimensions 3≤ d≤6. We develop a new framework for these limits that does not require a global theory for the limiting equation. For limiting data in H1, we prove strong convergence in the critical space, in particular in Lt∞ Hx1, on every compact subinterval of the maximal lifespan of the limiting solution. This includes strong H1 inviscid convergence to the 3D focusing energy-critical NLS without higher regularity of the limiting data. The diffusion and nonlinear coefficients may vary independently, and both focusing and defocusing cases are treated. Both limits use coefficient-uniform critical stability. The inviscid argument uses low-frequency truncation to overcome the derivative loss at H1, whereas uniform parabolic smoothing yields a quantitative H1 estimate in the zero-dispersion limit. Moreover, for the inviscid limit, we prove the required homogeneous Strichartz estimates and obtain the retarded estimates by establishing a uniform retarded double-endpoint bound for the critical forcing space.
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BibTeX
@misc{gao2026singular,
title = {{Singular limits for general energy-critical complex Ginzburg-Landau equations}},
author = {Lingbang Gao and Jie Xin and Yunrui Zheng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.24088},
eprint = {2608.24088},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24088},
}
RIS
TY - UNPB
TI - Singular limits for general energy-critical complex Ginzburg-Landau equations
AU - Lingbang Gao
AU - Jie Xin
AU - Yunrui Zheng
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2608.24088
UR - https://arxiv.org/abs/2608.24088
AB - Motivated by dissipative approximations of the three-dimensional focusing energy-critical nonlinear Schrödinger equation, we study the inviscid and zero-dispersion limits for general energy-critical complex Ginzburg-Landau equations in dimensions 3≤ d≤6. We develop a new framework for these limits that does not require a global theory for the limiting equation. For limiting data in H^1, we prove strong convergence in the critical space, in particular in L_t^∞ H_x^1, on every compact subinterval of the maximal lifespan of the limiting solution. This includes strong H^1 inviscid convergence to the 3D focusing energy-critical NLS without higher regularity of the limiting data. The diffusion and nonlinear coefficients may vary independently, and both focusing and defocusing cases are treated. Both limits use coefficient-uniform critical stability. The inviscid argument uses low-frequency truncation to overcome the derivative loss at H^1, whereas uniform parabolic smoothing yields a quantitative H^1 estimate in the zero-dispersion limit. Moreover, for the inviscid limit, we prove the required homogeneous Strichartz estimates and obtain the retarded estimates by establishing a uniform retarded double-endpoint bound for the critical forcing space.
ER -