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Register a new userThe $p$-elastic flow for planar closed curves with constant parametrization
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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 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 $t to infty$.
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In this paper we consider the steepest descent $L^2$-gradient flow of the entropy functional. The flow expands convex curves, with the radius of an initial circle growing like the square root of time. Our main result is that, for any initial curve (either immersed locally convex of class $C^2$ or embedded of class $W^{2,2}$ bounding a convex domain), the flow converges smoothly to a round expanding multiply-covered circle.
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