  There are a number of mathematical theorems in the literature on the dynamics
of cosmological models with accelerated expansion driven by a positive
cosmological constant $\Lambda$ or a nonlinear scalar field with potential $V$
(quintessence) which do not assume homogeneity and isotropy from the beginning.
The aim of this paper is to generalize these results to the case of $k$-essence
models which are defined by a Lagrangian having a nonlinear dependence on the
kinetic energy. In particular, Lagrangians are included where late time
acceleration is driven by the kinetic energy, an effect which is qualitatively
different from anything seen in quintessence models. A general criterion for
isotropization is derived and used to strengthen known results in the case of
quintessence.
