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Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II: The nonlinear theory. (English) Zbl 1059.35103

Summary: In part I of this work [the authors, J. Nonlinear Sci. 12, 283–318 (2002; Zbl 1022.35044)], 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.

MSC:

35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35B35 Stability in context of PDEs
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

Citations:

Zbl 1022.35044
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