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Well-posedness and Ill-posedness for Linear Fifth-Order Dispersive Equations in the Presence of Backwards Diffusion
Journal of Dynamics and Differential Equations ( IF 1.4 ) Pub Date : 2020-10-31 , DOI: 10.1007/s10884-020-09905-9
David M. Ambrose , Jacob Woods

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.



中文翻译:

存在向后扩散的线性五阶色散方程的适定性和不适定性

在水波等现象的高阶模型中,出现了五阶色散方程。对于五阶变系数线性色散方程,我们提供了条件条件下初始值问题是适定的还是不适定的条件。为了具有良好的适度性,必须在前导色散与四阶导数项可能的向后扩散之间取得平衡。这归纳了第一作者和赖特对三阶方程的研究。除了对五阶色散方程固有的兴趣外,这项工作还受到数值分析问题的推动:三阶数值方程的有限差分方案可以产生有效满足五阶方程的近似解。

更新日期:2020-11-02
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