Existence and convergence of the length-preserving elastic flow of clamped curves (Q6592841)

The Euler-Bernoulli energy of an immersed curve \(f:I=[0,1]\rightarrow \mathbb{R}^{d}\), \(d\geq 2\), is defined as \[\mathcal{E}(f)=\frac{1}{2} \int_{I}\left\vert \overrightarrow{\kappa }\right\vert ^{2}ds,\] where \( ds=\gamma dx=\left\vert \partial _{x}f\right\vert dx\) is the arc-length element, and \(\overrightarrow{\kappa }=\partial _{s}^{2}f\) the curvature vector field, \(\partial _{s}=\gamma ^{-1}\partial _{x}\) being the arc-length derivative. \N\NThe authors consider the deformation of an initial curve \(f_{0}\) in such a way that its elastic energy decreases as fast as possible, while keeping the (total) length \(\mathcal{L}(f)=\int_{I}ds\) fixed, which yields the geometric evolution equation \[\partial ^{\perp }_{t}f=-\nabla _{s}^{2} \overrightarrow{\kappa }-\frac{1}{2}\left\vert \overrightarrow{\kappa } \right\vert ^{2}\overrightarrow{\kappa }+\lambda \overrightarrow{\kappa },\] where \(\nabla _{s}\) denotes the connection on the normal bundle along \(f\) and \(\lambda \) is the Lagrange multiplier that depends on the solution \(f\). They finally get the initial boundary value problem \[\partial _{t}f=-\nabla _{s}^{2}\overrightarrow{\kappa }-\frac{1}{2}\left\vert \overrightarrow{ \kappa }\right\vert ^{2}\overrightarrow{\kappa }+\lambda \overrightarrow{ \kappa }+\theta \partial _{s}f\] on \((0,T)\times I\), \(f(0,x)=f_{0}(x)\), \( x\in I\), \(f(t,y)=p_{y}\), and \(\partial _{s}f(t,y)=\tau _{y}\), \(0\leq t<T\), \( y\in \partial I\), where the unknown \(\theta :[0,T)\times I\rightarrow \mathbb{R}\) is the tangential velocity \(\theta =\left\langle \partial _{t}f,\partial _{s}f\right\rangle \). The boundary data \(p_{y}\in \mathbb{R} ^{d}\), \(\tau _{y}\in \mathbb{S}^{d-1}\subset \mathbb{R}^{d}\) satisfy the compatibility conditions \(f_{0}(y)=p_{y}\) and \(\partial _{s}f_{0}(y)=\tau _{y}\), \(y\in \partial I\). \N\NThe main result of the paper proves that if \( f_{0}\in W^{2,2}(I;\mathbb{R}^{d})\) is immersed, \(p_{0},p_{1}\in \mathbb{R} ^{d}\) and \(\tau _{0},\tau _{1}\in \mathbb{S}^{d-1}\) satisfy the compatibility conditions, there exist \(T>0\) and a solution \[f\in W^{1,2}(0,T;L^{2}(I;\mathbb{R}^{d}))\cap L^{2}(0,T;W^{4,2}(I;\mathbb{R}^{d})) \] to the initial boundary value problem. \N\NFor the proof, the authors build an explicit tangential motion to transform the initial boundary value problem into a quasilinear parabolic system.\ They then introduce a linearization, apply the theory of maximal \(L^{p}\)-regularity and prove suitable contraction estimates to prove the result using a fixed point argument. They prove the existence of \(0<T_{1}\leq T\) such that the solution \(f\) belongs to \(C^{\infty }((0,T_{1})\times \ I;\mathbb{R}^{d})\). The second main result is a convergence result obtained under the same hypotheses as above: there exists a smooth family of curves \(f:(0,\infty )\times I\rightarrow \mathbb{R} ^{d}\) solving the initial boundary value problem, such that \(f(t)\rightarrow f_{0}\) in \(W^{2,2}(I;\mathbb{R}^{d})\) as \(t\rightarrow 0\), and \( f(t)\rightarrow f_{\infty }\) smoothly after reparametrization as \( t\rightarrow \infty \), where \(f_{\infty }\) is a constrained clamped elastica, i.e., a solution of \[-\nabla _{s}^{2}\overrightarrow{\kappa }-\frac{ 1}{2}\left\vert \overrightarrow{\kappa }\right\vert ^{2}\overrightarrow{ \kappa }+\lambda \overrightarrow{\kappa }=0\] on \(I\), \(f(y)=p_{y}\), and \( \partial _{s}f(y)=\tau _{y}\), \(y\in \partial I\), for some \(\lambda \in \mathbb{R}\). The authors prove the existence of a global solution to the initial boundary value problem, and they prove a refined Łojasiewicz-Simon gradient inequality to obtain the convergence result after reparametrization.