Well-posedness and Ill-posedness for Linear Fifth-Order Dispersive Equations in the Presence of Backwards Diffusion
Well-posedness and Ill-posedness for Linear Fifth-Order Dispersive Equations in the Presence of Backwards Diffusion
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存在向后扩散的线性五阶色散方程的适定性和不适定性
DOI：
10.1007/s10884-020-09905-9
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发表时间：
2022
影响因子：
1.3
通讯作者：
Woods, Jacob
中科院分区：
文献类型：
--
作者：
Ambrose, David M.;Woods, Jacob
五阶色散方程出现在诸如水波等现象的高阶模型的上下文中。对于五阶变系数线性色散方程，给出了初值问题适定或不适定的条件。对于适定性，必须在首阶色散和四阶导数项可能的向后扩散之间取得平衡。这概括了第一作者和赖特的工作为三阶方程。除了对五阶色散方程的固有兴趣之外，这项工作还受到数值分析的一个问题的启发：三阶数值方程的有限差分格式可以产生有效满足五阶方程的近似解。我们发现这样的五阶方程是适定的当且仅当基本的三阶方程是不适定的。
Fifth-order dispersive equations arise in the context of higher-order models for phenomena such as water waves. For fifth-order variable-coefficient linear dispersive equations, we provide conditions under which the intitial value problem is either well-posed or ill-posed. For well-posedness, a balance must be struck between the leading-order dispersion and possible backwards diffusion from the fourth-derivative term. This generalizes work by the first author and Wright for third-order equations. In addition to inherent interest in fifth-order dispersive equations, this work is also motivated by a question from numerical analysis: finite difference schemes for third-order numerical equations can yield approximate solutions which effectively satisfy fifth-order equations. We find that such a fifth-order equation is well-posed if and only if the underlying third-order equation is ill-posed.
周期五阶KdV型方程的局部适定性
DOI：
10.1007/s12220-013-9443-4
发表时间：
2011
期刊：
The Journal of Geometric Analysis
影响因子：
--
作者：
Y. Hu
通讯作者：
Y. Hu
关于周期性KDV型方程的柯西问题
DOI：
10.1201/9780429332838-3
发表时间：
2020
期刊：
影响因子：
--
作者：
J. Bourgain
通讯作者：
J. Bourgain
关于色散差分方案。
影响因子：
3
作者：
J. Goodman
通讯作者：
J. Goodman