非线性演化绝热近似的一般框架II
A generic framework of adiabatic approximation for nonlinear evolutions II
摘要：
在本文中，我们继续发展的非线性演化的一般绝热计划。我们考虑一个抽象的梯度流的一些能量泛函，连同一个给定的流形的静态解决方案所产生的对称性破缺。首先，我们列出了一些明确的和一般的能量泛函的条件，确保我们的绝热计划的有效性。然后，我们构造了一些明确的低能量，但不一定是静态的配置，形成一个稳定的流形有限余维给定的梯度流。第三，我们证明了梯度流是全局适定的，初始配置来自稳定流形。最后，我们表明，任何解决方案的完整的梯度流从稳定的流形基本上是由一个有效的方程的静态解决方案的流形上，到一个统一的小和耗散的误差项。
英文摘要：
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.
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