发表机构
Hohai University; Nanjing University of Aeronautics and Astronautics; CNRS, Université Paris-Est Créteil; Shandong University（河海大学; 南京航空航天大学; 法国国家科学研究中心，巴黎东克雷泰伊大学; 山东大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对三维黎曼流形上能量次临界与临界的复金兹堡-朗道方程，通过纯能量方法分析解的渐近行为、二分性与收敛性，还推导了相关非线性薛定谔方程的全局弱解及特定流形上解的衰减性。
AI 中文摘要
我们研究三维黎曼流形上能量次临界与能量临界的复金兹堡-朗道方程，通过纯能量方法建立解的渐近行为，进而证明解的快/慢二分性。对于慢解，我们证明其收敛到显式渐近剖面，特别地，精确描述了主导阶动力学的振幅与相位。作为副产品，复金兹堡-朗道方程近似给出紧三维黎曼流形上能量次临界与能量临界散焦非线性薛定谔方程的全局弱解。我们还研究一类完备非紧三维黎曼流形上能量次临界与能量临界复金兹堡-朗道方程解的长时间行为，在适当几何假设下证明全局解的衰减性。
英文摘要
We study the energy-subcritical and energy-critical complex Ginzburg-Landau equations on three-dimensional Riemannian manifolds. We establish the asymptotic behavior of solutions via a purely energy-based approach and subsequently prove a fast/slow dichotomy of the solutions. Moreover, for the slow solution, we show that the solution converges to an explicit asymptotic profile. In particular, we obtain a precise description of both the amplitude and the phase of the leading-order dynamics. As a by-product, the complex Ginzburg-Landau equation approximation yields global weak solutions to the energy-subcritical and energy-critical defocusing nonlinear Schrödinger equations on compact three-dimensional Riemannian manifolds. We also investigate the long time behavior of the solutions to the energy-subcritical and energy-critical complex Ginzburg-Landau equations on a class of complete noncompact three-dimensional Riemannian manifolds and prove decay of global solutions under suitable geometric assumptions.
