  We prove long-time existence for the negative $L^2$-gradient flow of the
$p$-elastic energy, $p\geq 2$, with an additive positive multiple of the length
of the curve. To achieve this result we regularize the energy by adding a small
multiple of a higher order energy, namely the square of the $L^2$-norm of the
normal gradient of the curvature $\kappa$. Long-time existence is proved for
the gradient flow of these new energies together with the smooth
sub-convergence of the evolution equation's solutions to critical points of the
regularized energy in $W^{2,p}$. We then show that the solutions to the
regularized evolution equations converge to a weak solution of the negative
gradient flow of the $p$-elastic energies. These latter weak solutions also
sub-converge to critical points of the $p$-elastic energy.
