Boussinesq equations and other
systems
for small-amplitude long waves in nonlinear dispersive media. Part
II:
the nonlinear theory.
In Part I of this work, 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 the present article, 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.
Min Chen (chen@math.purdue.edu)