带有守恒律的耗散系统中 Whitham 调制方程的有效性——广义 Ginzburg-Landau 系统中的相动力学——
Validity of Whitham's modulation equations for dissipative systems with a conservation law -- Phase dynamics in a generalized Ginzburg-Landau system --
AI总结：
本文研究了带有守恒律的耗散系统中 Whitham 调制方程的有效性，证明了其在广义 Ginzburg-Landau 系统中逼近真实解的严格误差估计，核心方法结合了 Cauchy-Kovalevskaya 理论、Gevrey 空间能量估计与 Fourier 空间局部分解。
摘要：
众所周知，Whitham 调制方程可用于逼近色散系统中缓慢变化的周期波列动力学。我们关注其在带有守恒律的耗散系统中的有效性。此类系统的典型例子是广义 Ginzburg-Landau 系统，它作为一个普适振幅系统，用于描述具有中性稳定长模的空间扩展斑图形成系统中的 Turing-Hopf 分岔。本文证明了通过 Whitham 调制方程获得的逼近与该 Ginzburg-Landau 系统真实解之间的严格误差估计。证明依赖于解析平滑、Cauchy-Kovalevskaya 理论、Gevrey 空间中的能量估计，以及 Fourier 空间中的局部分解，该分解将中心模与稳定模分离，并揭示了相关非线性项前的（半）导数。
英文摘要：
It is well-established that Whitham's modulation equations approximate the dynamics of slowly varying periodic wave trains in dispersive systems. We are interested in its validity in dissipative systems with a conservation law. The prototype example for such a system is the generalized Ginzburg-Landau system that arises as a universal amplitude system for the description of a Turing-Hopf bifurcation in spatially extended pattern-forming systems with neutrally stable long modes. In this paper we prove rigorous error estimates between the approximation obtained through Whitham's modulation equations and true solutions to this Ginzburg-Landau system. Our proof relies on analytic smoothing, Cauchy-Kovalevskaya theory, energy estimates in Gevrey spaces, and a local decomposition in Fourier space, which separates center from stable modes and uncovers a (semi)derivative in front of the relevant nonlinear terms.
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