We prove long-time existence for the negative L 2 {L}^{2} -gradient flow of the p-elastic energy, p ≥ 2 p\ge 2 , with an additive positive multiple of the length of the curve. To achieve this result, we regularize the energy by cutting off the degeneracy at points with vanishing curvature and add a small multiple of a higher order energy, namely, the square of the L 2 {L}^{2} -norm of the normal gradient of the curvature κ \kappa . Long-time existence is proved for the gradient flow of these new energies together with the smooth subconvergence of the evolution equation’s solutions to critical points of the regularized energy in W 2 , p {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 subconverge to critical points of the p-elastic energy.

中文翻译：

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p-弹性能量的正则化梯度流

我们证明负数的长期存在 大号 2 {L}^{2} -梯度流p- 弹性能量， p ≥ 2 p\ge 2 ，具有曲线长度的加性正倍数。为了达到这个结果，我们通过在曲率消失的点处切断退化并添加高阶能量的小倍数，即 大号 2 {L}^{2} - 曲率法线梯度的范数 κ \卡帕 . 证明了这些新能量的梯度流的长期存在性以及演化方程对正则化能量临界点的解的平滑次收敛性 W 2 , p {W}^{2,p} . 然后我们证明正则化进化方程的解收敛于负梯度流的弱解p- 弹性能量。后面的这些弱解也子收敛到p- 弹性能量。