On the Domain of Analyticity for Solutions of Second Order Analytic Nonlinear Differential Equations
On the Domain of Analyticity for Solutions of Second Order Analytic Nonlinear Differential Equations
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论二阶解析非线性微分方程解的解析性域
DOI：
10.1006/jdeq.2000.3927
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发表时间：
2001
影响因子：
2.4
通讯作者：
E. Titi
中科院分区：
文献类型：
--
作者：
M. Oliver;E. Titi
摘要周期解析函数的解析性半径可用其傅里叶系数的衰减来表征。这一观察结果导致了所谓的Gevrey规范作为一种简单的方法来估计抛物型和非抛物型偏微分方程解的解析性的空间半径的时间演化。在本文中，我们证明，使用一个简单的，明确可解的模型方程，通常的Gevrey类方法得到的解析性的半径上的估计不跨一个家庭的解决方案的最佳规模，也不最佳规模作为方程的物理参数的函数。我们的属性观察到的缺乏锋利的一个特定的嵌入不等式，并给出了一个修改后的定义的Gevrey规范，这表明最终产生一个尖锐的估计半径的解析。
Abstract The radius of analyticity of periodic analytic functions can be characterized by the decay of their Fourier coefficients. This observation has led to the use of so-called Gevrey norms as a simple way of estimating the time evolution of the spatial radius of analyticity of solutions to parabolic as well as non-parabolic partial differential equations. In this paper we demonstrate, using a simple, explicitly solvable model equation, that estimates on the radius of analyticity obtained by the usual Gevrey class approach do not scale optimally across a family of solutions, nor do they scale optimally as a function of the physical parameters of the equation. We attribute the observed lack of sharpness to a specific embedding inequality, and give a modified definition of the Gevrey norms which is shown to finally yield a sharp estimate on the radius of analyticity.