  If $\psi:M^n\to \mathbb{R}^{n+1}$ is a smooth immersed closed hypersurface,
we consider the functional $\mathcal{F}_m(\psi) = \int_M 1 + |\nabla^m \nu |^2
\, d\mu$, where $\nu$ is a local unit normal vector along $\psi$, $\nabla$ is
the Levi-Civita connection of the Riemannian manifold $(M,g)$, with $g$ the
pull-back metric induced by the immersion and $\mu$ the associated volume
measure. We prove that if $m>\lfloor n/2 \rfloor$ then the unique globally
defined smooth solution to the $L^2$-gradient flow of $\mathcal{F}_m$, for
every initial hypersurface, smoothly converges asymptotically to a critical
point of $\mathcal{F}_m$, up to diffeomorphisms. The proof is based on the
application of a Lojasiewicz-Simon gradient inequality for the functional
$\mathcal{F}_m$.
