  There are many subtle issues associated with solving the Navier-Stokes
equations. In this paper, several of these issues, which have been observed
previously in research involving the Navier-Stokes equations, are studied
within the framework of the one-dimensional Kuramoto-Sivashinsky (KS) equation,
a model nonlinear partial-differential equation. This alternative approach is
expected to more easily expose major points and hopefully identify open
questions that are related to the Navier-Stokes equations. In particular, four
interesting issues are discussed. The first is related to the difficulty in
defining regions of linear stability and instability for a time-dependent
governing parameter; this is equivalent to a time-dependent base flow for the
Navier-Stokes equations. The next two issues are consequences of nonlinear
interactions. These include the evolution of the solution by exciting its
harmonics or sub-harmonics (Fourier components) simultaneously in the presence
of a constant or a time-dependent governing parameter; and the sensitivity of
the long-time solution to initial conditions. The final issue is concerned with
the lack of convergent numerical chaotic solutions, an issue that has not been
previously studied for the Navier-Stokes equations. The last two issues,
consequences of nonlinear interactions, are not commonly known. Conclusions and
questions uncovered by the numerical results are discussed. The reasons behind
each issue are provided with the expectation that they will stimulate interest
in further study.
