Global analysis of the generalised Helfrich flow of closed curves immersed in $\mathbb {R}^n$

Abstract:

In this paper we consider the evolution of regular closed elastic curves $\gamma$ immersed in $\mathbb {R}^n$. Equipping the ambient Euclidean space with a vector field $\mathbb {c}:\mathbb {R}^n\rightarrow \mathbb {R}^n$ and a function $f:\mathbb {R}^n\rightarrow \mathbb {R}$, we assume the energy of $\gamma$ is smallest when the curvature $\vec {\kappa }$ of $\gamma$ is parallel to $\vec {c}_0 = (\mathbb {c}\circ \gamma ) + (f\circ \gamma )\tau$, where $\tau$ is the unit vector field spanning the tangent bundle of $\gamma$. This leads us to consider a generalisation of the Helfrich functional $\mathcal {H}^{\vec {c}_0}_{\lambda }$, defined as the sum of the integral of $|\vec {\kappa }-\vec {c}_0|^2$ and $\lambda$-weighted length. We primarily consider the case where $f:\mathbb {R}^n\rightarrow \mathbb {R}$ is uniformly bounded in $C^\infty (\mathbb {R}^n)$ and $\mathbb {c}:\mathbb {R}^n\rightarrow \mathbb {R}^n$ is an affine transformation. Our first theorem is that the steepest descent $L^2$-gradient flow of $\mathcal {H}^{\vec {c}_0}_{\lambda }$ with smooth initial data exists for all time and subconverges to a smooth solution of the Euler-Lagrange equation for a limiting functional $\mathcal {H}^{\vec {c}_\infty }_{\lambda }$. We additionally perform some asymptotic analysis. In the broad class of gradient flows for which we obtain global existence and subconvergence, there exist many examples for which full convergence of the flow does not hold. This may manifest in its simplest form as solutions translating or spiralling off to infinity. We prove that if either $\mathbb {c}$ and $f$ are constant, the derivative of $\mathbb {c}$ is invertible and non-vanishing, or $(f,\gamma _0)$ satisfy a ‘properness’ condition, then one obtains full convergence of the flow and uniqueness of the limit. This last result strengthens a well-known theorem of Kuwert, Schätzle and Dziuk on the elastic flow of closed curves in $\mathbb {R}^n$ where $f$ is constant and $\mathbb {c}$ vanishes.