论文标题
非线性演变的绝热近似框架的通用框架II
A generic framework of adiabatic approximation for nonlinear evolutions II
论文作者
论文摘要
在本文中，我们继续开发非线性发展的通用绝热方案。我们考虑了某些能量功能的抽象梯度流，以及由损坏的对称性产生的静态溶液的给定层次。首先，我们在能量功能上列出了许多明确和通用条件，以确保我们的绝热方案的有效性。然后，我们构建了一些显式的低能，但不一定是静态配置，这形成了稳定的歧管，并具有有限的编织，用于给定的梯度流。第三，我们表明梯度流在全球范围内具有良好的配置，并具有稳定歧管的初始配置。最后，我们表明，从稳定的歧管开始的所有梯度流动的解决方案基本上都由静态溶液的多种方程式控制，直至均匀且耗散的误差项。
In this paper, we continue the development of a generic adiabatic scheme for nonlinear evolutions. We consider an abstract gradient flow of some energy functional, together with a given manifold of static solutions arising from broken symmetries. First, we list a number of explicit and generic conditions on the energy functional that ensures the validity of our adiabatic scheme. Then, we construct some explicit low-energy but no necessarily static configurations, which form a stable manifold with finite codimensions for the given gradient flow. Thirdly, we show that the gradient flow is globally well-posed with initial configuration from the stable manifold. Finally, we show that any solution to the full gradient flow starting from the stable manifold is essentially governed by an effective equation on the manifold of static solutions, up to a uniformly small and dissipating error term.
