  Scalar-tensor theories with first-derivative self interactions, known as
$k$-essence, may provide interesting phenomenology on cosmological scales. On
smaller scales, however, initial value evolutions (which are crucial for
predicting the behavior of astrophysical systems, such as binaries of compact
objects) may run into instabilities related to the Cauchy problem becoming
potentially ill-posed. Moreover, on local scales the dynamics may enter in the
nonlinear regime, which may lie beyond the range of validity of the infrared
theory. Completions of $k$-essence in the ultraviolet, when they are known to
exist, mitigate these problems, as they both render Cauchy evolutions
well-posed at all times, and allow for checking the relation between
nonlinearities and the low energy theory's range of validity. Here, we explore
these issues explicitly by considering an ultraviolet completion to $k$-essence
and performing vacuum 1+1 dynamical evolutions within it. The results are
compared to those obtained with the low-energy theory, and with the low-energy
theory suitably deformed with a phenomenological "fixing the equations"
approach. We confirm that the ultraviolet completion does not incur in any
breakdown of the Cauchy problem's well-posedness, and we find that evolutions
agree with the results of the low-energy theory, when the system is within the
regime of validity of the latter. However, we also find that the nonlinear
behavior of $k$-essence lies (for the most part) outside this regime.
