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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 ofweak solution for the flow and prove the existence of global-in-time weaksolutions with $p \ge 2$ for initial curves in the energy space via minimizingmovements. Moreover, we prove the existence of unique global-in-time solutionsto the flow with $p=2$ and obtain their subconvergence to an elastica as $t \to\infty$.

The $p$-elastic flow for planar closed curves with constant parametrization