  In this paper, we consider the $L^2$-gradient flow for the modified
$p$-elastic energy defined on planar closed curves. We formulate a notion of
weak solution for the flow and prove the existence of global-in-time weak
solutions with $p \ge 2$ for initial curves in the energy space via minimizing
movements. Moreover, we prove the existence of unique global-in-time solutions
to the flow with $p=2$ and obtain their subconvergence to an elastica as $t \to
\infty$.
