Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II. The nonlinear theory
Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II. The nonlinear theory
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DOI：
10.1088/0951-7715/17/3/010
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发表时间：
2004-05
期刊：
影响因子：
1.7
通讯作者：
J. Bona;Min Chen;J. Saut
中科院分区：
文献类型：
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作者：
J. Bona;Min Chen;J. Saut
在这项工作的第一部分(博纳J L，陈明和Saut J-C 2002 Boussinesq方程和其他系统的非线性色散介质中的小振幅长波I：推导和线性理论J.非线性科学J.12283-318)，建立了描述表层水波传播的四参数Boussinesq系统族。在传播的主要方面是对流的非线性效应和频率色散的线性效应之间的平衡的其他物理环境中，预计也会出现类似的系统。除了得到这些系统外，我们在第一部分中准确地确定了它们中的哪些在各种自然函数类中是线性适定的。有人认为，线性适定性是所讨论的模型可能的物理相关性的自然必然要求。本文证明了线性适定的一阶修正模型实际上是局部非线性适定的。此外，在某些特定情况下，建立了物理上相关的初始数据的全局适定性。在第一部分中，还推导了高阶修正模型。对这些模型中一个很有前途的子类的初步分析表明，它们是很好的适定性的。
In part I of this work (Bona J L, Chen M and Saut J-C 2002 Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media I: Derivation and the linear theory J. Nonlinear Sci. 12 283–318), a four-parameter family of Boussinesq systems was derived to describe the propagation of surface water waves. Similar systems are expected to arise in other physical settings where the dominant aspects of propagation are a balance between the nonlinear effects of convection and the linear effects of frequency dispersion. In addition to deriving these systems, we determined in part I exactly which of them are linearly well posed in various natural function classes. It was argued that linear well-posedness is a natural necessary requirement for the possible physical relevance of the model in question. In this paper, it is shown that the first-order correct models that are linearly well posed are in fact locally nonlinearly well posed. Moreover, in certain specific cases, global well-posedness is established for physically relevant initial data. In part I, higher-order correct models were also derived. A preliminary analysis of a promising subclass of these models shows them to be well posed.