# Well-posedness and applications of classical elastohydrodynamics for a
swimming filament

Yoichiro Mori  Department of Mathematics, University of Pennsylvania,
Philadelphia, PA 19104<EMAIL_ADDRESS>and  Laurel Ohm  Department of
Mathematics, Princeton University, Princeton, NJ 08540
<EMAIL_ADDRESS>
###### Abstract.

We consider a classical elastohydrodynamic model of an inextensible filament
undergoing planar motion in $\mathbb{R}^{3}$. The hydrodynamics are described
by resistive force theory, and the fiber elasticity is governed by Euler-
Bernoulli beam theory. Our aim is twofold: (1) Serve as a starting point for
developing the mathematical analysis of filament elastohydrodynamics,
particularly the analytical treatment of an inextensibility constraint, and
(2) As an application, prove conditions on internal fiber forcing that allow a
free-ended filament to swim. Our analysis of fiber swimming speed is
supplemented with a numerical optimization of the internal fiber forcing, as
well as a novel numerical method for simulating an inextensible swimmer.

###### Contents

  1. 1 Introduction
    1. 1.1 Setup and statement of results
  2. 2 Well-posedness: local for large data, global for small data
    1. 2.1 Preliminaries
    2. 2.2 Tension equation
    3. 2.3 Evolution equation
  3. 3 Periodic active forcing and swimming
    1. 3.1 Existence of solution with force
    2. 3.2 Existence of unique periodic solution
    3. 3.3 Swimming
  4. 4 Observations and numerics
    1. 4.1 Observations
    2. 4.2 Numerical method
    3. 4.3 Swimmer numerics
  5. A Validation of numerical method

##  1\. Introduction

We consider the planar motion of an inextensible, free-ended filament immersed
in a Stokes fluid in $\mathbb{R}^{3}$. The hydrodynamic effects on the
filament are described using resistive force theory (or local slender body
theory), one of the most fundamental tools for modeling flagellar locomotion
in viscous fluids [16, 21, 22, 45]. The structure of the filament itself is
described by the classical Euler-Bernoulli beam theory, coupled with internal
forcing corresponding to a (possibly nonzero) preferred curvature [19, 61, 62,
3, 2, 48, 18, 49, 55]. Resistive force theory coupled with the elastic
response of the fiber has been shown to be remarkably effective at capturing
actual filament dynamics, as demonstrated by [8, 64].

_Aim 1: Analysis of inextensibility._ A key feature of this model is the
inextensibility constraint which prohibits the fiber from growing or shrinking
in length over time. Inextensibility features prominently in related models
of, for example, vesicle dynamics [57, 56, 47], but its treatment is
relatively underdeveloped from the perspective of mathematical analysis,
particularly in the dynamical setting.

We begin by considering the relaxation of the filament with zero preferred
curvature and show global-in-time existence and uniqueness of solutions for
small initial data, as well as local well-posedness for large initial data.
Global well-posedness for large data is hindered by the behavior of the
filament tension. However, we show that if the filament’s initial bending
energy is small, then a solution exists globally. As a byproduct, we obtain
nonlinear stability of the straight filament. The filament evolution in this
setting may be regarded as a simpler version of the Peskin problem (immersed
boundary method), which has been the subject of many recent PDE works [13, 14,
29, 40, 54]. This work also follows a recent program by the authors to place
nonlocal slender body theory on firm theoretical footing [38, 39, 36, 37, 44].
These previous papers have all considered the static boundary value problem
for Stokes flow about a curve at a single instant in time. The form of the
integral operator in nonlocal slender body renders the dynamic problem for the
curve evolution much more difficult. Using resistive force theory for the
force-to-velocity map along the fiber simplifies the curve evolution problem
while still remaining relevant from a current modeling perspective (see, for
example, [1, 4, 5, 6, 30, 43, 46, 52, 60, 63]).

_Aim 2: Analysis of swimming and applications._ Resistive force theory coupled
with fiber elastodynamics has been especially useful for understanding
undulatory swimming at low Reynolds number [7, 9, 10, 20, 34, 26, 24, 51, 27].
This brings us to our second aim: understanding conditions on a filament’s
internal forcing which allow it to swim. On the analysis side, we consider the
filament dynamics under small-amplitude periodic forcing, corresponding to a
nonzero preferred curvature along the fiber. We show existence of a unique
periodic solution and derive an expression for the swimming speed of the
filament. From the swimming expression, we obtain conditions on the periodic
forcing that give rise to propulsion. Our swimming analysis may also be
compared to other analysis works on swimming, including [11, 31, 12, 42].

The swimming expression we obtain allows us to perform a small numerical
optimization to determine the internal fiber forcing which gives rise to the
fastest swimming for a fixed amount of work and bending energy. The optimal
forcing that we find is a type of traveling wave; notably, we do not _a
priori_ assume any conditions on the forcing other than time periodicity. Our
optimization can be compared with related optimizations of the actual filament
shape [28, 51, 26, 41, 45], although we emphasize that our optimization is for
the active forcing along the filament, and the resulting fiber deformation is
an emergent property.

Finally, we develop a novel numerical method based on two recent approaches to
inextensibility, due to Moreau et al. [35] and Maxian et al. [32], which rely
on different but related methods of avoiding the need to solve for the
filament tension. These reformulations of filament elastohydrodynamics are
interesting from an analysis perspective because, as seen in the well-
posedness results of this paper, estimates for the filament tension are a
limiting factor in the solution theory which hinder us in showing global
existence for large initial data. A weaker notion of inextensibility – perhaps
related to these reformulations – may overcome this difficulty. Moreover, a
broader aim of this paper is to serve as a starting point for a full numerical
analysis of the various formulations of fiber inextensibility used in
numerical simulations, ranging from penalization methods [55] to variants of
the more recent approaches which avoid solving for the fiber tension
altogether [35, 17, 59, 58, 32, 33]. Our own numerical method is validated
against a direct implementation of the classical formulation of filament
elastohydrodynamics. We use the method to verify numerically the observations
from our swimming analysis.

###  1.1. Setup and statement of results

We consider an inextensible elastic filament in $\mathbb{R}^{3}$ with
centerline $\bm{X}:[0,L]\times[0,T]\to\mathbb{R}^{3}$ parameterized by
arclength $s$. Throughout, we will use the subscript $\cdot_{s}$ to denote
differentiation $\frac{\partial}{\partial_{s}}\cdot$ with respect to
arclength. The filament is assumed to undergo planar deformations only; in
particular, we can define a unique in-plane unit normal vector $\bm{e}_{\rm
n}(s,t)$ along the filament centerline. In the presence of active forcing in
the form of a preferred curvature $\kappa_{0}(s,t)$, the classical resistive
force theory formulation of the evolution of $\bm{X}(s,t)$ is given by

$\displaystyle\frac{\partial\bm{X}}{\partial t}(s,t)$
$\displaystyle=\big{(}c_{\perp}{\bf
I}+(c_{\parallel}-c_{\perp})\bm{X}_{s}\bm{X}_{s}^{\rm
T}\big{)}\big{(}E(\bm{X}_{sss}-(\kappa_{0})_{s}\bm{e}_{\rm
n})-\tau(s,t)\bm{X}_{s}\big{)}_{s}$ (1)
$\displaystyle\left\lvert\bm{X}_{s}\right\rvert^{2}$ $\displaystyle=1$

along with the force-free and torque-free boundary conditions
$\big{(}E(\bm{X}_{sss}-(\kappa_{0})_{s}\bm{e}_{\rm
n})-\tau\bm{X}_{s}\big{)}\big{|}_{s=0,L}=0$,
$E(\bm{X}_{ss}-\kappa_{0}\bm{e}_{\rm n})\big{|}_{s=0,L}=0$. For a filament of
radius $\varepsilon$, the constants
$c_{\perp}\approx\frac{\log(\varepsilon/L)}{4\pi}$ and
$c_{\parallel}\approx\frac{\log(\varepsilon/L)}{2\pi}$ for $\varepsilon/L\ll
1$; in particular, $c_{\parallel}\approx 2c_{\perp}<0$. (Note that some
sources define these constants as $1/c_{\perp}$ and $1/c_{\parallel}$; see,
for example, [25, Chapter 6.3]). The hydrodynamic force on the right hand side
consists of an elastic term with constant bending stiffness $E>0$, the unknown
tension $\tau(s,t)$ which serves as a Lagrange multiplier to enforce
$\left\lvert\bm{X}_{s}\right\rvert^{2}=1$, and active forcing due to a time-
varying preferred curvature $\kappa_{0}(s,t)$ along the filament (see [2, 53]
or the derivation in Section 3). We will consider both the evolution of a
passive filament ($\kappa_{0}(s,t)\equiv 0$) and an active filament with time-
periodic forcing.

Rescaling arclength as $s/L$, $\bm{X}$ as $\bm{X}/L$, tension as
$\frac{L^{2}}{E}\tau$, forcing as $L\kappa_{0}$, time as $\frac{E\left\lvert
c_{\perp}\right\rvert}{L^{4}}t$, and defining
$\gamma=\frac{c_{\parallel}}{c_{\perp}}-1$, we may rewrite (1) as

$\displaystyle\frac{\partial\bm{X}}{\partial t}(s,t)$
$\displaystyle=-\big{(}{\bf I}+\gamma\bm{X}_{s}\bm{X}_{s}^{\rm
T}\big{)}\big{(}\bm{X}_{sss}-\tau(s,t)\bm{X}_{s}-(\kappa_{0})_{s}\bm{e}_{\rm
n}\big{)}_{s}$ (2) $\displaystyle\left\lvert\bm{X}_{s}\right\rvert^{2}$
$\displaystyle=1$ $\displaystyle(\bm{X}_{ss}-\kappa_{0}\bm{e}_{\rm
n})\big{|}_{s=0,1}$
$\displaystyle=0\,,\quad(\bm{X}_{sss}-\tau\bm{X}_{s}-(\kappa_{0})_{s}\bm{e}_{\rm
n})\big{|}_{s=0,1}=0\,.$

Throughout we will use the notation $I=[0,1]$ to denote the unit interval. We
also note that the constant $\gamma\approx 1>0$. Since the filament is planar
and inextensible, the tangent vector $\bm{e}_{\rm t}(s,t)=\bm{X}_{s}(s,t)$ at
each point may be determined with respect to the angle $\theta(s,t)$ between
$\bm{e}_{\rm t}(s,t)$ and a fixed axis, which we take to be $\bm{e}_{\rm
t}(0,0)$. In particular, we may write

$\displaystyle\bm{X}_{s}$ $\displaystyle=\begin{pmatrix}\cos\theta\\\
\sin\theta\end{pmatrix}=\bm{e}_{\rm
t}\,,\quad\bm{X}_{ss}=\begin{pmatrix}-\sin\theta\\\
\cos\theta\end{pmatrix}\theta_{s}=\theta_{s}\bm{e}_{\rm n}\,.$

Differentiating the formulation (2) with respect to $s$ gives the evolution of
the tangent vector $\dot{\bm{e}}_{\rm t}=\dot{\bm{X}}_{s}$. The normal and
tangential components of this evolution yield two equations, one for the
evolution of the tangent angle $\theta$ and one for the tension $\tau$ at each
time:

$\displaystyle\dot{\theta}$
$\displaystyle=-\theta_{ssss}+(2+\gamma)(\theta_{s}^{3})_{s}+(2+\gamma)\tau_{s}\theta_{s}+\tau\theta_{ss}+(\kappa_{0})_{sss}-(1+\gamma)\theta_{s}^{2}(\kappa_{0})_{s}$
(3) $\displaystyle(1+\gamma)\tau_{ss}-(\theta_{s})^{2}\tau$
$\displaystyle=(\theta_{s})^{4}+\theta_{ss}^{2}-(4+3\gamma)(\theta_{ss}\theta_{s})_{s}+(2+\gamma)(\kappa_{0})_{ss}\theta_{s}+(1+\gamma)\theta_{ss}(\kappa_{0})_{s}$
(4) $\displaystyle(\theta_{s}-\kappa_{0})\big{|}_{s=0,1}$
$\displaystyle=0\,,\quad(\theta_{ss}-(\kappa_{0})_{s})\big{|}_{s=0,1}=0\,,\quad(\tau+\kappa_{0}^{2})\big{|}_{s=0,1}=0\,.$
(5)

Note that the boundary conditions for $\tau$ are due to the torque-free
condition $\bm{X}_{ss}\big{|}_{s=0,1}=\theta_{s}\bm{e}_{\rm
n}\big{|}_{s=0,1}=\kappa_{0}\bm{e}_{\rm n}\big{|}_{s=0,1}$ along with the form
of $\bm{X}_{sss}=-\theta_{s}^{2}\bm{e}_{\rm t}+\theta_{ss}\bm{e}_{\rm n}$ in
the force-free condition in (2). The tangent angle formulation (3)-(4), with
or without a forcing $\kappa_{0}$, is a popular choice for numerical
simulations [3, 2, 18, 19, 20, 35, 49, 61, 64]. Given the boundary conditions
(5) and the fact that only derivatives of $\theta$ appear on the right hand
side of (3) and in the tension equation (4), we find it more convenient to
work with the evolution of the filament curvature $\kappa=\theta_{s}$ rather
than $\theta$, as is done in [15, 53]. In particular, given an initial tangent
angle $\theta\big{|}_{t=0}=\theta_{\rm in}(s)$, once the curvature evolution
$\kappa(s,t)$ is known we may uniquely recover $\theta(s,t)$ by

$\dot{\theta}=-\kappa_{sss}+(2+\gamma)(\kappa^{3})_{ss}+(2+\gamma)\tau_{s}\kappa+\tau\kappa_{s}+(\kappa_{0})_{sss}-(1+\gamma)\kappa^{2}(\kappa_{0})_{s}\,.$
(6)

Furthermore, it will then be convenient to consider the evolution of the
difference $\overline{\kappa}=\kappa-\kappa_{0}$ as well as
$\overline{\tau}=\tau+\kappa_{0}^{2}$. Using equations (3)–(5), we have that
$\overline{\kappa}$ and $\overline{\tau}$ satisfy the equations

$\displaystyle\dot{\overline{\kappa}}$
$\displaystyle=-\overline{\kappa}_{ssss}-\dot{\kappa}_{0}+\big{[}3(2+\gamma)\overline{\kappa}(\overline{\kappa}+2\kappa_{0})\overline{\kappa}_{s}+(5+3\gamma)\kappa_{0}^{2}\overline{\kappa}_{s}+(5+2\gamma)\overline{\kappa}^{2}(\kappa_{0})_{s}$
$\displaystyle\hskip
85.35826pt+2(3+\gamma)\overline{\kappa}\kappa_{0}(\kappa_{0})_{s}+(2+\gamma)\overline{\tau}_{s}(\overline{\kappa}+\kappa_{0})+\overline{\tau}(\overline{\kappa}+\kappa_{0})_{s}\big{]}_{s}$
(7)
$\displaystyle\overline{\tau}_{ss}-\frac{(\overline{\kappa}+\kappa_{0})^{2}}{1+\gamma}\overline{\tau}$
$\displaystyle=\frac{1}{1+\gamma}\big{[}\overline{\kappa}(\overline{\kappa}+\kappa_{0})^{2}(\overline{\kappa}+2\kappa_{0})+(\overline{\kappa}+\kappa_{0})_{s}\overline{\kappa}_{s}$
$\displaystyle\hskip
56.9055pt-(1+\gamma)\big{(}\overline{\kappa}(\overline{\kappa}+2\kappa_{0})\big{)}_{ss}-(2+\gamma)\big{(}\overline{\kappa}_{s}(\overline{\kappa}+\kappa_{0})\big{)}_{s}\big{]}$
(8) $\displaystyle\overline{\kappa}\big{|}_{s=0,1}$
$\displaystyle=0\,,\quad\overline{\kappa}_{s}\big{|}_{s=0,1}=0\,,\quad\overline{\tau}\big{|}_{s=0,1}=0\,.$
(9)

The formulation (7)–(9) serves as the basis of our analysis. We will begin by
considering the passive filament $\kappa_{0}(s,t)\equiv 0$, for which the
dynamics reduce to the simpler system

$\displaystyle\dot{\kappa}$
$\displaystyle=-\kappa_{ssss}+(2+\gamma)(\kappa^{3})_{ss}+(2+\gamma)(\tau_{s}\kappa)_{s}+(\tau\kappa_{s})_{s}$
(10) $\displaystyle\tau_{ss}-\frac{\kappa^{2}}{1+\gamma}\tau$
$\displaystyle=\frac{1}{1+\gamma}\bigg{(}\kappa^{4}+(\kappa_{s})^{2}-(4+3\gamma)(\kappa_{s}\kappa)_{s}\bigg{)}$
(11) $\displaystyle\kappa\big{|}_{s=0,1}$
$\displaystyle=0\,,\quad\kappa_{s}\big{|}_{s=0,1}=0\,,\quad\tau\big{|}_{s=0,1}=0\,.$
(12)

We define the operator $\mathcal{L}$ to be the linear evolution of (10) near
$\kappa\equiv 0$:

$\mathcal{L}[\psi]:=\partial_{ssss}\psi\,,\qquad\psi(0)=\psi(1)=0\,,\;\psi_{s}(0)=\psi_{s}(1)=0\,.$
(13)

Since the remaining nonlinear terms of (10) are a perfect derivative in $s$,
we write the remainder as $(\mathcal{R}[\kappa])_{s}$, where

$\mathcal{R}[\kappa]=3(2+\gamma)\kappa^{2}\kappa_{s}+(2+\gamma)\tau_{s}\kappa+\tau\kappa_{s}\,.$
(14)

Letting $\kappa_{\rm in}(s)=\kappa(s,0)$ denote the initial curvature of the
fiber, we may define a mild solution $\kappa(s,t)$ to (10) by the Duhamel
formula

$\kappa(s,t)=e^{-t\mathcal{L}}\kappa_{\rm
in}(s)+\int_{0}^{t}e^{-(t-t^{\prime})\mathcal{L}}(\mathcal{R}[\kappa(s,t^{\prime})])_{s}\,dt^{\prime}\,.$
(15)

The fiber curvature in (10)–(12) obeys the energy identity

$\frac{1}{2}\frac{d}{dt}\int_{0}^{1}\kappa^{2}\,ds+\int_{0}^{1}\left(\kappa_{ss}-\kappa^{3}-\tau\kappa\right)^{2}\,ds+\int_{0}^{1}(1+\gamma)\left(3\kappa\kappa_{s}+\tau_{s}\right)^{2}\,ds=0\,.$
(16)

In particular, $\left\lVert\kappa\right\rVert_{L^{2}}$ is nonincreasing in
time. The identity (16) is more easily seen at the level of (2), where
sufficiently regular curves $\bm{X}(s,t)$ satisfy

$\frac{1}{2}\frac{d}{dt}\int_{0}^{1}\left\lvert\bm{X}_{ss}\right\rvert^{2}\,ds=-\int_{0}^{1}\bigg{(}(\bm{X}_{sss}-\tau\bm{X}_{s})_{s}^{2}+\gamma\big{(}\bm{X}_{s}\cdot(\bm{X}_{sss}-\tau\bm{X}_{s})_{s}\big{)}^{2}\bigg{)}\,ds\leq
0\,.$ (17)

This may be shown by multiplying both sides of (2) by
$(\bm{X}_{sss}-\tau\bm{X}_{s})_{s}$ and integrating in $s$. Integration by
parts on the left hand side yields

$\displaystyle\int_{0}^{1}\frac{\partial\bm{X}}{\partial
t}\cdot(\bm{X}_{sss}-\tau\bm{X}_{s})_{s}\,ds$
$\displaystyle=-\int_{0}^{1}\dot{\bm{X}}_{s}(s,t)\cdot(\bm{X}_{sss}-\tau\bm{X}_{s})\,ds$
$\displaystyle=-\int_{0}^{1}\dot{\bm{X}}_{s}\cdot\bm{X}_{sss}\,ds=\int_{0}^{1}\dot{\bm{X}}_{ss}\cdot\bm{X}_{ss}\,ds$
$\displaystyle=\frac{1}{2}\frac{d}{dt}\int_{0}^{1}\left\lvert\bm{X}_{ss}\right\rvert^{2}\,,$

where we have used the inextensibility constraint
$\dot{\bm{X}}_{s}\cdot\bm{X}_{s}=0$.

The system (10)–(12) is invariant under the scaling

$\displaystyle(\kappa(s,t),\tau(s,t))\mapsto(\lambda\kappa(\lambda
s,\lambda^{4}t)\,,\lambda^{2}\tau(\lambda s,\lambda^{4}t))\,.$

Norms which are invariant under the above scaling symmetry are known as
critical norms and play an important role in the well-posedness theory. Norms
which decay upon ‘zooming in’ via the above scaling are known as subcritical.
The natural expectation is that the PDE is (locally) well-posed in subcritical
spaces and solutions belonging to subcritical cases are in fact smooth.
Typically, if we have _a priori_ control on a subcritical norm for all time,
then the solution should be globally well-posed. This is the case for the 1D
viscous Burgers’ equation, for example. In our case, any $H^{s}$ norm with
$s>-1/2$ is subcritical. Furthermore, by (16) we have that
$\left\lVert\kappa\right\rVert_{L^{2}}$ is monotone decreasing. However, we
have found the regularity theory to be much more subtle due to the behavior of
the tension $\tau$ (see Remark 2.4 and Section 2.2).111In particular, the
local solution to (10)–(12) described in Theorem 1.1 may be extended up to
some maximal time $T^{*}(\kappa_{\rm in})$ depending on the initial data. If
the energy identity (16) can be used (in conjunction with a Grönwall argument)
to obtain a maximal existence time which depends only on the norm
$\left\lVert\kappa_{\rm in}\right\rVert_{L^{2}}$ rather than the full initial
data, or if it can be shown that, given $T$, the local solution $\kappa$
remains in $C([0,T];L^{2}(I))$ up to time $T$, then the solution can be
extended globally. The main difficulty in proving these arises in bounding the
tension $\tau$ (see e.g. bound (37)), which seems to necessarily depend on
$\left\lVert\kappa\right\rVert_{\dot{H}^{1}}^{2}$ – i.e. a rather high power
of a high regularity norm for $\kappa$.  This makes the system (10)–(12)
somewhat surprising from a regularity point of view. Nevertheless, the energy
inequality (16) motivates our choice of function space for the following well-
posedness result.

######  Theorem 1.1 (Well-posedness).

_Local-in-time for large data:_ For any $\kappa_{\rm in}\in L^{2}(I)$ there
exists a time $0<T_{0}<1$ such that the system (10)–(12) admits a unique mild
solution $\kappa\in C([0,T_{0}];L^{2}(I))\cap C((0,T_{0}];H^{1}(I))$.

_Global-in-time for small data:_ There exists a constant $0<c_{0}<1$ such
that, given $\kappa_{\rm in}\in L^{2}(I)$ with $\left\lVert\kappa_{\rm
in}\right\rVert_{L^{2}(I)}\leq c_{0}$, the system (10)–(12) admits a unique
mild solution $\kappa\in C([0,T];L^{2}(I))\cap C((0,T];H^{1}(I))$ for any
$T>0$. The solution $\kappa$ satisfies

$\left\lVert\kappa\right\rVert_{L^{2}(I)}+\min\\{t^{1/4},1\\}\left\lVert\kappa\right\rVert_{\dot{H}^{1}(I)}\leq
ce^{-t\lambda_{1}}\left\lVert\kappa_{\rm in}\right\rVert_{L^{2}(I)}\,.$ (18)

Here $\lambda_{1}$ is the first eigenvalue of the linearized operator
$\mathcal{L}$ (see (30)).

The time weight in (18) allows us to consider $\kappa_{\rm in}\in L^{2}(I)$
rather than $H^{1}(I)$. The norm $\left\lVert\kappa_{\rm
in}\right\rVert_{L^{2}(I)}$ has physical significance, as it is exactly the
initial bending energy of the filament. The proof of Theorem 1.1 is given in
Section 2. In principle, one can further show that $\kappa(s,t)$ belongs to
$C^{\infty}(I)$ for all $t>0$ using a bootstrapping argument, although this
requires a bit of additional care (see Remark 2.6).

Noting that $\kappa\equiv 0$ corresponds to a straight filament, as an
immediate corollary of the exponentially decaying bound (18), we obtain:

######  Corollary 1.2 (Stability of straight fiber).

The straight filament $\kappa\equiv 0$ is nonlinearly stable to small
perturbations.

We now turn to the filament with active internal forcing $\kappa_{0}\neq 0$ in
(7)-(9). We take $\kappa_{0}(s,t)$ to be time-periodic with period $T>0$ to
simulate a waving flagellum. Note that existence and uniqueness of solutions
to (7)–(9) with sufficiently small $\kappa_{0}$ and $\overline{\kappa}_{\rm
in}$ follow by similar arguments to Theorem 1.1 (see Lemma 3.1). Moreover, we
show the existence of a unique $T$-periodic solution $\overline{\kappa}$:

######  Theorem 1.3 (Existence of unique periodic solution).

There exist constants $0<c_{0}<1$ and $c_{1}\geq 0$ such that, given a
$T$-periodic preferred curvature $\kappa_{0}(s,t)\in C^{1}([0,T];H^{1}(I))$
satisfying

$\displaystyle T\bigg{(}\sup_{0\leq t\leq
T}\left\lVert\dot{\kappa}_{0}\right\rVert_{L^{2}(I)}\bigg{)}=\varepsilon\leq
c_{0}\,,\quad\sup_{0\leq t\leq
T}\left\lVert\kappa_{0}\right\rVert_{H^{1}}=c_{1}\varepsilon\leq c_{0}\,,$

there exists a unique $T$-periodic mild solution $\overline{\kappa}\in
C([0,T];H^{1}(I))$ to the system (7)–(9) satisfying the bound

$\sup_{0\leq t\leq T}\left\lVert\overline{\kappa}\right\rVert_{H^{1}(I)}\leq
c(T)\varepsilon\,.$ (19)

Furthermore, there exists $c_{2}\geq 0$ such that, for any
$\overline{\varphi}_{\rm in}\in L^{2}(I)$ satisfying
$\left\lVert\overline{\varphi}_{\rm in}\right\rVert_{L^{2}}\leq
c_{2}\varepsilon$, the corresponding unique solution to (7)-(9) satisfies

$\left\lVert\overline{\kappa}(\cdot,nT)-\overline{\varphi}(\cdot,nT)\right\rVert_{L^{2}(I)}\leq
ce^{-nT\lambda_{1}}\left\lVert\overline{\kappa}_{\rm
in}-\overline{\varphi}_{\rm in}\right\rVert_{L^{2}(I)},\quad n=1,2,\dots\,,$
(20)

where $\lambda_{1}$ is the first eigenvalue of the linearized operator
$\mathcal{L}$ (see (30)).

The proof of Theorem 1.3 appears in Section 3.2. Note that by (20), nearby
solutions converge very rapidly to the periodic solution. We consider this
periodic solution in all further analysis and use it to calculate the fiber
swimming speed. To do so, we first must recover the equations for the actual
translational motion of the fiber.

Given the unique periodic solution
$(\overline{\kappa},\overline{\tau})=(\kappa-\kappa_{0},\tau+\kappa_{0}^{2})$
to (7)–(9) and the initial tangent angle $\theta_{\rm in}$, we may uniquely
calculate $\theta$ via the equation (6) for $\dot{\theta}$. From (6) we may
also solve for the frame evolution:

$\dot{\bm{e}}_{\rm t}(s,t)=\dot{\theta}(s,t)\bm{e}_{\rm
n}(s,t)\,,\qquad\dot{\bm{e}}_{\rm n}(s,t)=-\dot{\theta}(s,t)\bm{e}_{\rm
t}(s,t)\,.$ (21)

We may then obtain the full evolution of the fiber via

$\frac{\partial\bm{X}}{\partial
t}(s,t)=\big{(}-(\kappa-\kappa_{0})_{ss}+\kappa^{3}+\tau\kappa\big{)}\bm{e}_{\rm
n}+(1+\gamma)\big{(}\tau_{s}+2\kappa\kappa_{s}+\kappa(\kappa-\kappa_{0})_{s}\big{)}\bm{e}_{\rm
t}\,.$ (22)

The swimming velocity of the fiber
$\bm{V}(t)=\int_{0}^{1}\frac{\partial\bm{X}}{\partial t}(s,t)\,ds$ may be
calculated as

$\bm{V}(t)=-\gamma\int_{0}^{1}\bigg{(}\big{(}\overline{\kappa}\overline{\kappa}_{s}-\overline{\tau}_{s}-\kappa_{0}\overline{\kappa}_{s}\big{)}\bm{e}_{\rm
t}+2\overline{\kappa}\kappa^{2}\bm{e}_{\rm n}\bigg{)}\,ds\,.$ (23)

(See Section 3.3, equation (72) for the full calculation).

For suitably small $\kappa_{0}$, we seek an expression for the average
swimming speed of the fiber in the direction $\bm{e}_{\rm t}(0,0)$ over the
course of one period $0\leq t\leq T$. In what follows, for any function
$h(t)$, we will use the notation

$\langle h\rangle:=\frac{1}{T}\int_{0}^{T}h(t)\,dt$ (24)

to denote the time average over one period. We show the following.

######  Theorem 1.4 (Small amplitude swimming).

Suppose that $\kappa_{0}(s,t)\in C^{1}([0,T];H^{3}(I))$ is $T$-periodic in
time and satisfies

$\displaystyle T\bigg{(}\sup_{0\leq t\leq
T}\left\lVert\dot{\kappa}_{0}\right\rVert_{L^{2}(I)}\bigg{)}=\varepsilon\leq
c_{0}\,,\quad\sup_{0\leq t\leq
T}\left\lVert\kappa_{0}\right\rVert_{H^{1}}=c_{1}\varepsilon\leq c_{0}\,.$

Then a filament whose curvature evolution satisfies (7)-(9) swims with
velocity

$\bm{V}(t)=U(t)\,\bm{e}_{\rm t}(0,0)+\bm{r}_{\rm v}(t)\,,$ (25)

where $\sup_{0\leq t\leq T}\left\lvert U\right\rvert\leq c\varepsilon^{2}$ and
$\sup_{0\leq t\leq T}\left\lvert\bm{r}_{\rm v}\right\rvert\leq
c\varepsilon^{3}$, and $U$ is given by

$U(t)=-\gamma\int_{0}^{1}(\kappa_{0})_{s}\overline{\kappa}\,ds\,.$ (26)

Writing $\kappa_{0}(s,t)=\sum_{m,k=1}^{\infty}\big{(}a_{m,k}\cos(\omega
mt)-b_{m,k}\sin(\omega mt)\big{)}\psi_{k}(s)$ where $\psi_{k}(s)$ are the
eigenfunctions of the linearized operator $\mathcal{L}$ (see (30)) and
$\omega=\frac{2\pi}{T}$, we have that the time-averaged speed $\langle
U\rangle$ may be written

$\displaystyle\langle U\rangle$
$\displaystyle=\frac{\gamma}{2}\sum_{m,k,\ell=1}^{\infty}\frac{\omega^{2}m^{2}}{\omega^{2}m^{2}+\lambda_{k}^{2}}\bigg{(}\frac{\lambda_{k}}{\omega
m}\big{(}a_{m,k}b_{m,\ell}-b_{m,k}a_{m,\ell}\big{)}$ (27) $\displaystyle\hskip
142.26378pt+a_{m,k}a_{m,\ell}+b_{m,k}b_{m,\ell}\bigg{)}\int_{0}^{1}\psi_{k}(\psi_{\ell})_{s}\,ds\,+r_{\rm
u}\,,$

where $\lambda_{k}$ are the eigenvalues of $\mathcal{L}$ and $\left\lvert
r_{\rm u}\right\rvert\leq c\varepsilon^{3}$.

The proof of Theorem 1.4 is given in Section 3.3. We note that the additional
$H^{3}(I)$ regularity on $\kappa_{0}$ is used to ensure that the fiber frame
$(\bm{e}_{\rm t},\bm{e}_{\rm n})$ is changing very little over time (see
equation (21)). As in previous works (see, e.g., [25]), we obtain that an
$O(\varepsilon)$ forcing results in propulsion at most at
$O(\varepsilon^{2})$. From (26), we see that propulsion occurs due to a lag
between the actual and preferred curvature of the fiber, which in turn depends
on the rate of change of the preferred curvature via (7). However, from (27),
we see that further conditions on $\kappa_{0}(s,t)$ are needed to actually
generate net movement over the course of one full period. Observations about
the swimming speed (27) are made more precise in Section 4.1, but may be
summarized as follows:

######  Observations 1.5.

  1. (1)

Forcing only the lowest nonzero temporal frequency appears to yield the
furthest filament displacement for a given amount of work. Thus in further
applications we consider the filament’s preferred curvature to be of the form

$\kappa_{0}(s,t)=F_{1}(s)\cos(\omega\,t)+F_{2}(s)\sin(\omega\,t)\,.$ (28)

  2. (2)

If $F_{1}$ and $F_{2}$ are both even or both odd about $s=\frac{1}{2}$, the
integral $\int_{0}^{1}\psi_{k}(\psi_{\ell})_{s}\,ds$ in (27) vanishes and the
filament does not swim to leading order.

  3. (3)

If either $F_{1}=0$ or $F_{2}=0$, or if $F_{1}=\pm F_{2}$, then the term
proportional to $\big{(}a_{m,k}b_{m,\ell}-b_{m,k}a_{m,\ell}\big{)}$ in (27)
vanishes. The filament may still swim at leading order, but its displacement
will be very small, due to the size of $\lambda_{k}$ (see (30)).

Observation (1), which was also noted in [24], comes from numerically solving
a simple constrained optimization problem (see (79)). The requirements on
$F_{1}$ and $F_{2}$ may be considered in terms of the Scallop Theorem for a
swimmer in a linear, time-independent flow: violating observation (2) can be
seen numerically to lead to a time-reversible filament deformation (see
Section 4.3), while violating (3) gives rise to a nearly time-reversible fiber
deformation.

A waveform fitting the above criteria is a traveling wave propagating along
the filament; for example, $F_{1}=\sin(\omega s)$ and $F_{2}=\cos(\omega s)$.
This is consistent with classical calculations by Pironneau and Katz [45]
which find that a filament should deform as a traveling wave to reach a given
swimming speed while minimizing energy expenditure.

A further small numerical optimization problem (see (80)) where
$\left\lVert\kappa_{0}\right\rVert_{L^{2}}$ is constrained as well yields a
$\kappa_{0}$ (28) in the form of a traveling wave which outperforms the simple
prescription $F_{1}=\sin(\omega s)$ and $F_{2}=\cos(\omega s)$. A notable
feature of this optimal $\kappa_{0}$ is that $F_{1}$ and $F_{2}$ look like
$\sin(\omega s)$ and $\cos(\omega s)$ away from the filament ends, but
$F_{1}=F_{2}=0$ and $F_{1}^{\prime}=F_{2}^{\prime}=0$ at $s=0,1$. This is
reminiscent of the results of [41], which find that an inactive flagellar
endpiece enhances swimming movement.

The above observations are explored in more detail in Section 4.1, and are
further supplemented with numerical simulations in Section 4.2. We implement a
reformulated version of equation (2) using a novel combination of the methods
proposed by Moreau et al. [35] and Maxian et al. [32], which we validate
against a direct discretization of (2) (see Appendix A). Using the
observations surrounding the form (28) of $\kappa_{0}$, we consider
combinations of $F_{1}(s)$ and $F_{2}(s)$ that give rise to non-swimmers, bad
swimmers, and good swimmers, and create visualizations of their respective
positions over time.

##  2\. Well-posedness: local for large data, global for small data

###  2.1. Preliminaries

We begin by stating a few useful lemmas.

######  Lemma 2.1 (Gagliardo–Nirenberg inequality).

For any function $u\in H^{1}(I)$, the following inequality holds for each
$2\leq p\leq\infty$:

$\displaystyle\left\lVert u\right\rVert_{L^{p}(I)}$ $\displaystyle\leq
c\left\lVert u\right\rVert_{H^{1}(I)}^{\frac{p-2}{2p}}\left\lVert
u\right\rVert_{L^{2}(I)}^{\frac{p+2}{2p}}\,.$

If $u\in H^{1}_{0}(I)$, then we may replace $\left\lVert
u\right\rVert_{H^{1}(I)}$ in the above estimate with $\left\lVert
u\right\rVert_{\dot{H}^{1}(I)}$.

In addition to Lemma 2.1, we will need estimates for the semigroup generated
by the operator $\mathcal{L}$ (13). We first state some properties of the
well-studied Dirichlet problem for the bilaplace equation in 1D (see [23, 61,
62]). In particular, we have explicit expressions for the eigenfunctions and
eigenvalues of $\mathcal{L}$. First, define an increasing sequence
$\\{\xi_{k}\\}$ satisfying

$\cos(\xi_{k})\cosh(\xi_{k})=1\,,\qquad\xi_{0}=0\,.$ (29)

The first few values of $\xi_{k}$ are given by $\xi_{1}\approx 4.730$,
$\xi_{2}\approx 7.853$, $\xi_{3}\approx 10.996$. Note that since $\cosh(\xi)$
increases exponentially as $\left\lvert\xi\right\rvert$ increases, we have
$\xi_{k}\to\frac{(2k+1)\pi}{2}$ as $k\to\infty$. The eigenfunctions $\psi_{k}$
and eigenvalues $\lambda_{k}$ of $\mathcal{L}$ are given by

$\displaystyle\psi_{k}(s)$
$\displaystyle=\frac{\widehat{\psi}_{k}(s)}{\|\widehat{\psi}_{k}\|_{L^{2}}}\,,\qquad\lambda_{k}=\xi_{k}^{4}\,,\qquad
k=1,2,\dots$ (30) $\displaystyle\text{where }\widehat{\psi}_{k}(s)$
$\displaystyle=\left(\cos(\xi_{k})-\cosh(\xi_{k})\right)\left(\cos(\xi_{k}s)-\cosh(\xi_{k}s)\right)$
$\displaystyle\qquad+\left(\sin(\xi_{k})+\sinh(\xi_{k})\right)\left(\sin(\xi_{k}s)-\sinh(\xi_{k}s)\right)\,.$

We note in particular that the smallest eigenvalue of $\mathcal{L}$ is given
by $\lambda_{1}\approx(4.73)^{4}\approx 500$. The first six eigenfunctions
$\psi_{k}$ are plotted in Figure 1. Note that from (30), for odd $k$,
$\psi_{k}(s)$ is even about $s=\frac{1}{2}$, while for even $k$, $\psi_{k}(s)$
is odd about $s=\frac{1}{2}$.

Figure 1. The first six eigenfunctions of $\mathcal{L}$.

Using (30), we may define the domain of $\mathcal{L}$ by

$D(\mathcal{L})=\bigg{\\{}u\in
L^{2}(I)\;:\;u=\sum_{k}a_{k}\psi_{k},\;a_{k}=\int_{0}^{1}u(s)\psi_{k}(s)ds;\;\sum_{k}\lambda_{k}^{2}a_{k}^{2}<\infty\bigg{\\}}\,.$
(31)

Furthermore, following [50, Chapter 3.7], we may define the domain of
fractional powers $\mathcal{L}^{r}$, $0\leq r\leq 1$, by

$D(\mathcal{L}^{r})=\bigg{\\{}u\in
L^{2}(I)\;:\;u=\sum_{k}a_{k}\psi_{k},\;a_{k}=\int_{0}^{1}u(s)\psi_{k}(s)ds;\;\sum_{k}\lambda_{k}^{2r}a_{k}^{2}<\infty\bigg{\\}}\,.$
(32)

Noting the quartic growth of $\lambda_{k}=\xi_{k}^{4}$ as $k\to\infty$, we
have the following inclusions:

$D(\mathcal{L}^{r})\subseteq H^{4r}(I)\,,\quad 0\leq r\leq 1\,;\quad
D(\mathcal{L}^{0})=L^{2}(I)\,.$ (33)

We will repeatedly use the following lemma.

######  Lemma 2.2 (Semigroup estimates).

For $0\leq m+j\leq 4$ we have

$\left\lVert
e^{-t\mathcal{L}}(\partial_{s}^{j}u)\right\rVert_{\dot{H}^{m}}\leq
c\max\\{t^{-(m+j)/4},1\\}e^{-t\lambda_{1}}\left\lVert
u\right\rVert_{L^{2}}\,,$ (34)

where $\lambda_{1}=\xi_{1}^{4}$ is the first eigenvalue of the operator
$\mathcal{L}$ (30).

###### Proof.

For $u\in L^{2}(I)$, write $u=\sum_{k=1}^{\infty}a_{k}\psi_{k}(s)$ where
$a_{k}=\int_{0}^{1}u(s)\psi_{k}(s)\,ds$. For $0\leq r\leq 1$, we have

$\displaystyle\left\lVert\mathcal{L}^{r}e^{-t\mathcal{L}}u\right\rVert_{L^{2}(I)}$
$\displaystyle=\bigg{\|}\sum_{k=1}^{\infty}\lambda_{k}^{r}e^{-t\lambda_{k}}a_{k}\psi_{k}\bigg{\|}_{L^{2}}\leq\sup_{\lambda_{k}}\bigg{(}\lambda_{k}^{r}e^{-t(\lambda_{k}-\lambda_{1})}\bigg{)}e^{-t\lambda_{1}}\bigg{\|}\sum_{k=1}^{\infty}a_{k}\psi_{k}\bigg{\|}_{L^{2}}$
$\displaystyle\leq c_{r}\max\\{t^{-r},1\\}e^{-t\lambda_{1}}\left\lVert
u\right\rVert_{L^{2}(I)}\,.$

Using (33), we then have

$\left\lVert e^{-t\mathcal{L}}u\right\rVert_{\dot{H}^{4r}}\leq
c_{r}\max\\{t^{-r},1\\}e^{-t\lambda_{1}}\left\lVert
u\right\rVert_{L^{2}(I)}\,,\qquad 0\leq r\leq 1\,.$ (35)

It remains to show the smoothing estimate, for which we use a duality
argument. For $u,w\in C_{c}^{\infty}(I)$, we have

$\displaystyle\sup_{\left\lVert u\right\rVert_{L^{2}}=1}\left\lVert
e^{-t\mathcal{L}}(\partial_{s}^{j}u)\right\rVert_{L^{2}}$
$\displaystyle=\sup_{\left\lVert u\right\rVert_{L^{2}}=1}\sup_{\left\lVert
w\right\rVert_{L^{2}}=1}\big{(}w,e^{-t\mathcal{L}}\partial_{s}^{j}u\big{)}_{L^{2}}$
$\displaystyle=\sup_{\left\lVert u\right\rVert_{L^{2}}=1}\sup_{\left\lVert
w\right\rVert_{L^{2}}=1}\big{(}\partial_{s}^{j}e^{-t\mathcal{L}}w,u\big{)}_{L^{2}}$

since $\mathcal{L}$ is self-adjoint. Using (35), we then have

$\displaystyle\sup_{\left\lVert u\right\rVert_{L^{2}}=1}\left\lVert
e^{-t\mathcal{L}}(\partial_{s}^{j}u)\right\rVert_{L^{2}}\leq\sup_{\left\lVert
u\right\rVert_{L^{2}}=1}\sup_{\left\lVert
w\right\rVert_{L^{2}}=1}\left\lVert\partial_{s}^{j}e^{-t\mathcal{L}}w\right\rVert_{L^{2}}\left\lVert
u\right\rVert_{L^{2}}\leq c_{j}\max\\{t^{-j/4},1\\}e^{-t\lambda_{1}}\,.$

The same result holds for $u\in L^{2}(I)$ by density. ∎

Finally, we will require the following lemma to show local existence for large
initial data.

######  Lemma 2.3.

Fix $u\in L^{2}(I)$ and let $0<r\leq 1$, $\varepsilon>0$. There exists
$T_{\varepsilon}>0$ which depends on $u$ and satisfies

$\sup_{t\in[0,T_{\varepsilon}]}\,\min\\{t^{r},1\\}e^{t\lambda_{1}}\left\lVert
e^{-t\mathcal{L}}u\right\rVert_{\dot{H}^{4r}(I)}\leq\varepsilon\,.$ (36)

The smallness in Lemma 2.3 is achieved by approximating $u\in L^{2}(I)$ by
slightly smoother functions.

###### Proof.

Consider a sequence of functions $\phi_{j}\in D(\mathcal{L}^{r})$ with
$\phi_{j}\to u$ in $L^{2}$. Since $\phi_{j}\in D(\mathcal{L}^{r})$, by (32) we
have

$\displaystyle\left\lVert
e^{-t\mathcal{L}}\phi_{j}\right\rVert_{\dot{H}^{4r}}\leq\left\lVert\mathcal{L}^{r}e^{-t\mathcal{L}}\phi_{j}\right\rVert_{L^{2}}\leq\left\lVert\sum_{k=1}^{\infty}\lambda_{k}^{r}e^{-t\lambda_{k}}(\phi_{j},\psi_{k})\psi_{k}\right\rVert_{L^{2}}\leq
e^{-t\lambda_{1}}\bigg{(}\sum_{k=1}^{\infty}\lambda_{k}^{2r}(\phi_{j},\psi_{k})^{2}\bigg{)}^{1/2}\leq
e^{-t\lambda_{1}}c_{j}$

for some constant $0<c_{j}<\infty$. Then, using Lemma 2.2, we have

$\displaystyle\min\\{t^{r},1\\}e^{t\lambda_{1}}\left\lVert
e^{-t\mathcal{L}}u\right\rVert_{\dot{H}^{4r}}$
$\displaystyle\leq\min\\{t^{r},1\\}e^{t\lambda_{1}}\left\lVert
e^{-t\mathcal{L}}\phi_{j}\right\rVert_{\dot{H}^{4r}}+\min\\{t^{r},1\\}e^{t\lambda_{1}}\left\lVert
e^{-t\mathcal{L}}(u-\phi_{j})\right\rVert_{\dot{H}^{4r}}$
$\displaystyle\leq\min\\{t^{r},1\\}\,c_{j}+c\left\lVert
u-\phi_{j}\right\rVert_{L^{2}}\,.$

Taking $j$ large enough that $c\left\lVert
u-\phi_{j}\right\rVert_{L^{2}}\leq\frac{\varepsilon}{2}$ and taking $t$ small
enough that $t^{r}\,c_{j}\leq\frac{\varepsilon}{2}$, we obtain the bound (36).
∎

With Lemmas 2.1, 2.2, and 2.3, we may now turn to the proof of Theorem 1.1:
well-posedness for the unforced system (10)–(12). First, a remark:

######  Remark 2.4.

To highlight the subtlety of the well-posedness theory for the system
(10)–(12), we may consider the following reduced PDE system consisting of the
most problematic terms of (10)–(12):

$\displaystyle\dot{\kappa}=-\kappa_{ssss}+(\tau\kappa_{s})_{s}\,,\quad\tau_{ss}=\kappa_{s}^{2}\,.$

We may rewrite this simplified system as a single equation:

$\displaystyle\dot{\kappa}=-\kappa_{ssss}+\big{(}\partial_{ss}^{-1}(\kappa_{s}^{2})\kappa_{s}\big{)}_{s}\,,$

where $\partial_{ss}^{-1}$ is the inverse second derivative with homogeneous
Dirichlet conditions at $s=0,1$. This equation is invariant under the same
scaling symmetry as the full system: $\kappa(s,t)\mapsto\lambda\kappa(\lambda
s,\lambda^{4}t)$. Again, $\left\lVert\kappa\right\rVert_{L^{2}}$ is
subcritical for this equation. However, to actually make sense of the tension
$\tau$, we need $\kappa_{s}\in L^{2}$. Taking initial data $\kappa_{\rm in}\in
L^{2}(I)$ only, we expect that
$\sup_{t}\,t^{1/4}\left\lVert\kappa_{s}\right\rVert_{L^{2}(I)}<\infty$ due to
the smoothing effect of $\mathcal{L}$. Thus every $\kappa_{s}$ which appears
in the nonlinear term ‘costs’ $t^{-1/4}$. From Lemma 2.2, using the smoothing
estimate on the semigroup to account for the outside derivative in the
nonlinear term also costs $t^{-1/4}$. The nonlinearity then contains three
additional powers of $\left\lVert\kappa_{s}\right\rVert_{L^{2}}$, costing a
total of $t^{-3/4}$. Altogether, we must integrate $t^{-1}$ in the Duhamel
formula, so the full power of the smoothing of $\mathcal{L}$ is required to
close the contraction.

###  2.2. Tension equation

We first consider the elliptic equation (11) and derive the following
estimates for the tension $\tau$ in terms of $\kappa$.

######  Lemma 2.5.

Given $\kappa\in H^{1}(I)$, there exists a unique weak solution $\tau$ to
equation (11) satisfying

$\left\lVert\tau\right\rVert_{H^{1}(I)}\leq
c\left\lVert\kappa\right\rVert_{\dot{H}^{1}(I)}^{2}\left(\left\lVert\kappa\right\rVert_{L^{2}(I)}+1\right)\,.$
(37)

Given $\kappa,\varphi\in H^{1}(I)$, define $\tau^{\kappa}$, $\tau^{\varphi}$
to be the $H^{1}_{0}$ solution to (11) using $\kappa$, $\varphi$ respectively
on the right hand side. The difference $\tau^{\kappa}-\tau^{\varphi}$ then
satisfies

$\displaystyle\|\tau^{\kappa}-\tau^{\varphi}\|_{H^{1}(I)}$ $\displaystyle\leq
c\left\lVert\kappa-\varphi\right\rVert_{L^{2}}\bigg{(}\left\lVert\kappa\right\rVert_{\dot{H}^{1}}^{2}(\left\lVert\kappa\right\rVert_{L^{2}}^{2}+1)+\left\lVert\varphi\right\rVert_{\dot{H}^{1}}^{2}(\left\lVert\varphi\right\rVert_{L^{2}}^{2}+1)\bigg{)}$
(38)
$\displaystyle\qquad+c\left\lVert\kappa-\varphi\right\rVert_{\dot{H}^{1}}\big{(}\left\lVert\kappa\right\rVert_{\dot{H}^{1}}+\left\lVert\varphi\right\rVert_{\dot{H}^{1}}\big{)}\,.$

Note that the right hand sides of both (37) and (38) involve two powers of an
$\dot{H}^{1}$ norm, and it is not clear that a lower power is possible. This
factor of $\left\lVert\kappa\right\rVert_{\dot{H}^{1}}^{2}$ presents an
obstacle to showing global existence for large data, as demonstrated by Remark
2.4.

###### Proof.

Given $\kappa\in H^{1}(I)$, for $\phi\in H^{1}_{0}(I)$, we may define

$\mathcal{B}(\tau,\phi):=\int_{0}^{1}\bigg{(}\tau_{s}\phi_{s}+\frac{\kappa^{2}}{1+\gamma}\tau\phi\bigg{)}\,ds\,,$
(39)

which, due to the sign of the potential term (note that $\kappa\in
L^{\infty}(I)$) is always bounded and coercive on $H^{1}_{0}(I)$. We may
define a weak solution of the tension equation as $\tau$ satisfying

$\mathcal{B}(\tau,\phi)=-\int_{0}^{1}\frac{1}{1+\gamma}\bigg{(}\kappa^{4}\phi+(\kappa_{s})^{2}\phi+(4+3\gamma)\kappa_{s}\kappa\phi_{s}\bigg{)}\,ds\quad\text{for
all }\phi\in H^{1}_{0}(I)\,.$ (40)

By the Lax–Milgram lemma, there exists a unique weak solution $\tau\in
H^{1}_{0}(I)$, and, using Lemma 2.1, $\tau$ satisfies

$\displaystyle\mathcal{B}(\tau,\tau)$ $\displaystyle\leq
c\bigg{(}\left\lVert\kappa\right\rVert_{L^{6}}^{3}\left\lVert\kappa\tau\right\rVert_{L^{2}}+\left\lVert\kappa_{s}\right\rVert_{L^{2}}^{2}\left\lVert\tau\right\rVert_{L^{\infty}}+\left\lVert\kappa_{s}\right\rVert_{L^{2}}\left\lVert\tau_{s}\right\rVert_{L^{2}}\left\lVert\kappa\right\rVert_{L^{\infty}}\bigg{)}$
$\displaystyle\leq
c\bigg{(}\left\lVert\kappa\right\rVert_{\dot{H}^{1}}\left\lVert\kappa\right\rVert_{L^{2}}^{2}\left\lVert\kappa\tau\right\rVert_{L^{2}}+\left\lVert\kappa\right\rVert_{\dot{H}^{1}}^{2}\left\lVert\tau\right\rVert_{H^{1}}+\left\lVert\kappa\right\rVert_{\dot{H}^{1}}^{3/2}\left\lVert\kappa\right\rVert_{L^{2}}^{1/2}\left\lVert\tau\right\rVert_{H^{1}}\bigg{)}\,.$

Note that, in terms of regularity, the limiting factor here is the term
$\kappa_{s}^{2}\tau$, which requires 2 powers of
$\left\lVert\kappa\right\rVert_{\dot{H}^{1}}$ to estimate, while the
additional regularity of $\tau$ is wasted.

Using Young’s inequality, we then have

$\mathcal{B}(\tau,\tau)\leq
c\bigg{(}\left\lVert\kappa\right\rVert_{\dot{H}^{1}}^{2}\left\lVert\kappa\right\rVert_{L^{2}}^{4}+\left\lVert\kappa\right\rVert_{\dot{H}^{1}}^{4}+\left\lVert\kappa\right\rVert_{\dot{H}^{1}}^{3}\left\lVert\kappa\right\rVert_{L^{2}}\bigg{)}\,,$
(41)

and therefore, using that $\left\lVert\tau\right\rVert_{H^{1}}^{2}\leq
c\left\lVert\tau_{s}\right\rVert_{L^{2}}^{2}\leq c\mathcal{B}(\tau,\tau)$, we
obtain the bound (37).

We next show the Lipschitz estimate (38) for $\tau$. Let $\tau^{\kappa}$,
$\tau^{\varphi}$ be as defined in Lemma 2.5. The difference
$\widehat{\tau}=\tau^{\kappa}-\tau^{\varphi}$ then satisfies

$\displaystyle\widehat{\tau}_{ss}$
$\displaystyle=\frac{\kappa^{2}}{1+\gamma}\tau^{\kappa}-\frac{\varphi^{2}}{1+\gamma}\tau^{\varphi}+\frac{1}{1+\gamma}\bigg{(}\kappa^{4}-\varphi^{4}+(\kappa_{s})^{2}-(\varphi_{s})^{2}-\frac{4+3\gamma}{2}\big{(}\kappa^{2}-\varphi^{2}\big{)}_{ss}\bigg{)}$
$\displaystyle=\frac{1}{2(1+\gamma)}\bigg{(}(\kappa^{2}+\varphi^{2})\widehat{\tau}+(\kappa-\varphi)(\kappa+\varphi)(\tau^{\kappa}+\tau^{\varphi})\bigg{)}$
$\displaystyle\quad+\frac{1}{1+\gamma}\bigg{(}(\kappa-\varphi)(\kappa+\varphi)(\kappa^{2}+\varphi^{2})+(\kappa_{s}-\varphi_{s})(\kappa_{s}+\varphi_{s})-\frac{4+3\gamma}{2}\big{(}(\kappa-\varphi)(\kappa+\varphi)\big{)}_{ss}\bigg{)}\,.$

Upon multiplying by $\widehat{\tau}$ and integrating by parts in $s$, we
obtain

$\displaystyle\int_{0}^{1}\bigg{(}\widehat{\tau}_{s}^{2}+\frac{\kappa^{2}+\varphi^{2}}{2(1+\gamma)}\widehat{\tau}^{2}\bigg{)}ds$
$\displaystyle\leq
c\bigg{(}\left\lVert\kappa-\varphi\right\rVert_{L^{2}}\left\lVert\kappa+\varphi\right\rVert_{L^{2}}\|\tau^{\kappa}+\tau^{\varphi}\|_{L^{\infty}}\bigg{)}\|\widehat{\tau}\|_{L^{\infty}}$
$\displaystyle\quad+c\bigg{(}\left\lVert\kappa-\varphi\right\rVert_{L^{\infty}}\left\lVert\kappa+\varphi\right\rVert_{L^{\infty}}\|(\kappa^{2}+\varphi^{2})\widehat{\tau}\|_{L^{2}}+\left\lVert\kappa_{s}-\varphi_{s}\right\rVert_{L^{2}}\left\lVert\kappa_{s}+\varphi_{s}\right\rVert_{L^{2}}\|\widehat{\tau}\|_{L^{\infty}}\bigg{)}$
$\displaystyle\quad+c\bigg{(}\left\lVert\kappa-\varphi\right\rVert_{L^{\infty}}\left\lVert\kappa_{s}+\varphi_{s}\right\rVert_{L^{2}}+\left\lVert\kappa_{s}-\varphi_{s}\right\rVert_{L^{2}}\left\lVert\kappa+\varphi\right\rVert_{L^{\infty}}\bigg{)}\|\widehat{\tau}_{s}\|_{L^{2}}$
$\displaystyle\leq
c\left\lVert\kappa-\varphi\right\rVert_{L^{2}}(\left\lVert\kappa\right\rVert_{L^{2}}+\left\lVert\varphi\right\rVert_{L^{2}})\bigg{(}\left\lVert\kappa\right\rVert_{\dot{H}^{1}}^{2}(\left\lVert\kappa\right\rVert_{L^{2}}+1)+\left\lVert\varphi\right\rVert_{\dot{H}^{1}}^{2}(\left\lVert\varphi\right\rVert_{L^{2}}+1)\bigg{)}\|\widehat{\tau}\|_{L^{\infty}}$
$\displaystyle\quad+c\left\lVert\kappa-\varphi\right\rVert_{\dot{H}^{1}}^{1/2}\left\lVert\kappa-\varphi\right\rVert_{L^{2}}^{1/2}(\left\lVert\kappa\right\rVert_{\dot{H}^{1}}^{1/2}\left\lVert\kappa\right\rVert_{L^{2}}^{1/2}+\left\lVert\varphi\right\rVert_{\dot{H}^{1}}^{1/2}\left\lVert\varphi\right\rVert_{L^{2}}^{1/2})\|(\kappa^{2}+\varphi^{2})\widehat{\tau}\|_{L^{2}}$
$\displaystyle\quad+c\left\lVert\kappa-\varphi\right\rVert_{\dot{H}^{1}}(\left\lVert\kappa\right\rVert_{\dot{H}^{1}}+\left\lVert\varphi\right\rVert_{\dot{H}^{1}})\|\widehat{\tau}\|_{H^{1}}\,.$

Here we have used Lemma 2.1 as well as the bound (37). Using Young’s
inequality, we then have

$\displaystyle\|\widehat{\tau}\|_{H^{1}}^{2}$
$\displaystyle\leq\int_{0}^{1}\bigg{(}\widehat{\tau}_{s}^{2}+\frac{\kappa^{2}+\varphi^{2}}{2(1+\gamma)}\widehat{\tau}^{2}\bigg{)}ds$
$\displaystyle\leq
c\left\lVert\kappa-\varphi\right\rVert_{L^{2}}^{2}(\left\lVert\kappa\right\rVert_{L^{2}}+\left\lVert\varphi\right\rVert_{L^{2}})^{2}\bigg{(}\left\lVert\kappa\right\rVert_{\dot{H}^{1}}^{2}(\left\lVert\kappa\right\rVert_{L^{2}}+1)+\left\lVert\varphi\right\rVert_{\dot{H}^{1}}^{2}(\left\lVert\varphi\right\rVert_{L^{2}}+1)\bigg{)}^{2}$
$\displaystyle\quad+c\left\lVert\kappa-\varphi\right\rVert_{\dot{H}^{1}}^{2}\big{(}\left\lVert\kappa\right\rVert_{\dot{H}^{1}}+\left\lVert\varphi\right\rVert_{\dot{H}^{1}}\big{)}^{2}\,.$

In particular, we have

$\displaystyle\|\tau^{\kappa}-\tau^{\varphi}\|_{H^{1}(I)}$ $\displaystyle\leq
c\left\lVert\kappa-\varphi\right\rVert_{L^{2}}(\left\lVert\kappa\right\rVert_{L^{2}}+\left\lVert\varphi\right\rVert_{L^{2}})\bigg{(}\left\lVert\kappa\right\rVert_{\dot{H}^{1}}^{2}(\left\lVert\kappa\right\rVert_{L^{2}}+1)+\left\lVert\varphi\right\rVert_{\dot{H}^{1}}^{2}(\left\lVert\varphi\right\rVert_{L^{2}}+1)\bigg{)}$
$\displaystyle\qquad+c\left\lVert\kappa-\varphi\right\rVert_{\dot{H}^{1}}\big{(}\left\lVert\kappa\right\rVert_{\dot{H}^{1}}+\left\lVert\varphi\right\rVert_{\dot{H}^{1}}\big{)}\,,$

from which we obtain the bound (38). ∎

###  2.3. Evolution equation

We now consider the evolution equation (10) for the curvature $\kappa(s,t)$.
To prove the existence of a unique mild solution (15) in both the large data
and small data settings, we will rely on a contraction mapping argument that
makes use of the semigroup estimates of Lemma 2.2. We will also require Lemma
2.3 in the large data setting.

###### Proof of Theorem 1.1.

We begin by defining the function spaces $\mathcal{Y}_{0}$, $\mathcal{Y}_{1}$
as

$\displaystyle\mathcal{Y}_{0}$ $\displaystyle=\big{\\{}u\in
C([0,T];L^{2}(I))\,:\,\left\lVert
u\right\rVert_{\mathcal{Y}_{0}}<\infty\big{\\}}\,,$
$\displaystyle\mathcal{Y}_{1}$ $\displaystyle=\big{\\{}u\in
C((0,T];\dot{H}^{1}(I))\,:\,\left\lVert
u\right\rVert_{\mathcal{Y}_{1}}<\infty\big{\\}}\,,$

where the norms $\left\lVert\cdot\right\rVert_{\mathcal{Y}_{0}}$,
$\left\lVert\cdot\right\rVert_{\mathcal{Y}_{1}}$, respectively, are given by

$\displaystyle\left\lVert\cdot\right\rVert_{\mathcal{Y}_{0}}:=\sup_{t\in[0,T]}e^{t\lambda_{1}}\left\lVert\cdot\right\rVert_{L^{2}(I)}\,,\qquad\left\lVert\cdot\right\rVert_{\mathcal{Y}_{1}}:=\sup_{t\in[0,T]}\min\\{t^{1/4},1\\}\,e^{t\lambda_{1}}\left\lVert\cdot\right\rVert_{\dot{H}^{1}(I)}\,.$

We will close a contraction mapping argument in
$\mathcal{Y}_{0}\cap\mathcal{Y}_{1}$. We consider the intersection of two
closed balls in $\mathcal{Y}_{0}$ and $\mathcal{Y}_{1}$:

$\displaystyle B_{M_{0}}(\mathcal{Y}_{0})\cap
B_{M_{1}}(\mathcal{Y}_{1})=\big{\\{}u\in\mathcal{Y}_{0}\cap\mathcal{Y}_{1}\,:\,\left\lVert
u\right\rVert_{\mathcal{Y}_{0}}\leq M_{0},\;\left\lVert
u\right\rVert_{\mathcal{Y}_{1}}\leq M_{1}\big{\\}}\,.$

Given either sufficiently small $T$ or sufficiently small initial data, we
show that the mild solution formula (15) maps $B_{M_{0}}(\mathcal{Y}_{0})\cap
B_{M_{1}}(\mathcal{Y}_{1})$ into itself for some choice of $M_{0}>0$ and
$0<M_{1}<1$.

For any $\varphi\in B_{M_{0}}(\mathcal{Y}_{0})\cap
B_{M_{1}}(\mathcal{Y}_{1})$, let $\Psi$ denote the mapping

$\displaystyle\Psi[\varphi]:=e^{-t\mathcal{L}}\kappa_{\rm
in}+\int_{0}^{t}e^{-(t-t^{\prime})\mathcal{L}}\big{(}\mathcal{R}[\varphi(s,t^{\prime})]\big{)}_{s}dt^{\prime}\,.$

We start by estimating the nonlinear terms of (15). First, using (37) and
Lemma 2.1, we have

$\displaystyle\left\lVert\mathcal{R}[\kappa(\cdot,t)]\right\rVert_{L^{2}}$
$\displaystyle\leq
c\bigg{(}\left\lVert\kappa\right\rVert_{L^{\infty}}^{2}\left\lVert\kappa_{s}\right\rVert_{L^{2}}+\left\lVert\tau\right\rVert_{L^{2}}\left\lVert\kappa\right\rVert_{L^{\infty}}+\left\lVert\tau\right\rVert_{L^{\infty}}\left\lVert\kappa_{s}\right\rVert_{L^{2}}\bigg{)}$
(42) $\displaystyle\leq
c\bigg{(}\left\lVert\kappa\right\rVert_{L^{2}}\left\lVert\kappa\right\rVert_{\dot{H}^{1}}^{2}+\left\lVert\kappa\right\rVert_{\dot{H}^{1}}^{5/2}(\left\lVert\kappa\right\rVert_{L^{2}}^{3/2}+\left\lVert\kappa\right\rVert_{L^{2}}^{1/2})+\left\lVert\kappa\right\rVert_{\dot{H}^{1}}^{3}(\left\lVert\kappa\right\rVert_{L^{2}}+1)\bigg{)}$
$\displaystyle\leq
c\left\lVert\kappa\right\rVert_{\dot{H}^{1}}^{3}(\left\lVert\kappa\right\rVert_{L^{2}}+1)\,.$

Here we note that the term $\tau\kappa_{s}$ from (14) is the limiting factor
in terms of regularity (and note that writing
$\tau_{s}\kappa+\tau\kappa_{s}=(\tau\kappa)_{s}$ and using the smoothing of
the semigroup in Lemma 2.2 will still lead to the same result). In particular,
since we only have the $H^{1}$ bound (37) for $\tau$, the regularity of $\tau$
in this estimate ends up getting wasted, and we end up with 3 powers of
$\left\lVert\kappa\right\rVert_{\dot{H}^{1}}$ on the right hand side of (42).

Using Lemma 2.2 and (42), we then have

$\displaystyle\left\lVert\int_{0}^{t}e^{-(t-t^{\prime})\mathcal{L}}\big{(}\mathcal{R}[\kappa(\cdot,t^{\prime})]\big{)}_{s}\,dt^{\prime}\right\rVert_{\dot{H}^{1}(I)}\leq
c\int_{0}^{t}\max\\{(t-t^{\prime})^{-1/2},1\\}e^{-(t-t^{\prime})\lambda_{1}}\left\lVert\mathcal{R}[\kappa]\right\rVert_{L^{2}(I)}dt^{\prime}$
$\displaystyle\hskip 28.45274pt\leq
c\int_{0}^{t}\max\\{(t-t^{\prime})^{-1/2},1\\}e^{-(t-t^{\prime})\lambda_{1}}\left\lVert\kappa\right\rVert_{\dot{H}^{1}}^{3}(\left\lVert\kappa\right\rVert_{L^{2}}+1)dt^{\prime}$
$\displaystyle\hskip 28.45274pt\leq
c\int_{0}^{t}\max\\{(t-t^{\prime})^{-1/2},1\\}e^{-(t-t^{\prime})\lambda_{1}}\left(e^{-3t^{\prime}\lambda_{1}}\max\\{(t^{\prime})^{-3/4},1\\}\left\lVert\kappa\right\rVert_{\mathcal{Y}_{1}}^{3}\left(e^{-t^{\prime}\lambda_{1}}\left\lVert\kappa\right\rVert_{\mathcal{Y}_{0}}+1\right)\right)\,dt^{\prime}$
$\displaystyle\hskip 28.45274pt\leq
ce^{-t\lambda_{1}}\int_{0}^{t}\max\\{(t-t^{\prime})^{-1/2},1\\}\max\\{(t^{\prime})^{-3/4},1\\}\,e^{-2t^{\prime}\lambda_{1}}\,dt^{\prime}\,\left(\left\lVert\kappa\right\rVert_{\mathcal{Y}_{1}}^{3}\left(\left\lVert\kappa\right\rVert_{\mathcal{Y}_{0}}+1\right)\right)$
$\displaystyle\hskip 28.45274pt\leq
c\,\max\\{t^{-1/4},1\\}e^{-t\lambda_{1}}\left(\left\lVert\kappa\right\rVert_{\mathcal{Y}_{1}}^{3}\left(\left\lVert\kappa\right\rVert_{\mathcal{Y}_{0}}+1\right)\right)\,.$

By an analogous series of estimates, we also have

$\displaystyle\left\lVert\int_{0}^{t}e^{-(t-t^{\prime})\mathcal{L}}\big{(}\mathcal{R}[\kappa(\cdot,t^{\prime})]\big{)}_{s}\,dt^{\prime}\right\rVert_{L^{2}(I)}$
$\displaystyle\leq
ce^{-t\lambda_{1}}\left(\left\lVert\kappa\right\rVert_{\mathcal{Y}_{1}}^{3}\left(\left\lVert\kappa\right\rVert_{\mathcal{Y}_{0}}+1\right)\right)\,.$

Then, using Lemma 2.2, for $\varphi\in B_{M_{0}}(\mathcal{Y}_{0})\cap
B_{M_{1}}(\mathcal{Y}_{1})$ we obtain the following $\mathcal{Y}_{0}$ bound:

$\displaystyle\left\lVert\Psi[\varphi]\right\rVert_{\mathcal{Y}_{0}}$
$\displaystyle\leq\sup_{t\in[0,T]}e^{t\lambda_{1}}\left\lVert
e^{-t\mathcal{L}}\kappa_{\rm
in}\right\rVert_{L^{2}}+c\left\lVert\varphi\right\rVert_{\mathcal{Y}_{1}}^{3}\left(\left\lVert\varphi\right\rVert_{\mathcal{Y}_{0}}+1\right)$
(43) $\displaystyle\leq\left\lVert\kappa_{\rm
in}\right\rVert_{L^{2}(I)}+cM_{1}^{3}(M_{0}+1)\leq M_{0}\,,$

where we have taken $M_{0}=2\left\lVert\kappa_{\rm in}\right\rVert_{L^{2}(I)}$
and $M_{1}$ small enough that $cM_{1}^{3}(M_{0}+1)\leq\frac{M_{0}}{2}$.

Furthermore, to obtain a $\mathcal{Y}_{1}$ bound, we use that

$\displaystyle\left\lVert\Psi[\varphi]\right\rVert_{\mathcal{Y}_{1}}$
$\displaystyle\leq\sup_{t\in[0,T]}\,\min\\{t^{1/4},1\\}\,e^{t\lambda_{1}}\left\lVert
e^{-t\mathcal{L}}\kappa_{\rm
in}\right\rVert_{\dot{H}^{1}}+c\left\lVert\varphi\right\rVert_{\mathcal{Y}_{1}}^{3}\left(\left\lVert\varphi\right\rVert_{\mathcal{Y}_{0}}+1\right)$
(44)
$\displaystyle\leq\sup_{t\in[0,T]}\,\min\\{t^{1/4},1\\}\,e^{t\lambda_{1}}\left\lVert
e^{-t\mathcal{L}}\kappa_{\rm
in}\right\rVert_{\dot{H}^{1}}+cM_{1}^{3}(M_{0}+1)$
$\displaystyle\leq\sup_{t\in[0,T]}\,\min\\{t^{1/4},1\\}\,e^{t\lambda_{1}}\left\lVert
e^{-t\mathcal{L}}\kappa_{\rm in}\right\rVert_{\dot{H}^{1}}+\frac{M_{1}}{2}$

for $M_{1}$ sufficiently small. It remains to show that

$\displaystyle\sup_{t\in[0,T]}\,\min\\{t^{1/4},1\\}\,e^{t\lambda_{1}}\left\lVert
e^{-t\mathcal{L}}\kappa_{\rm
in}\right\rVert_{\dot{H}^{1}}\leq\frac{M_{1}}{2}\,,$

for which we have two options:

  1. (1)

_Small time:_ Using Lemma 2.3, we may choose a time
$T_{M_{1}}=T_{M_{1}}(\kappa_{\rm in})$ such that

$\displaystyle\sup_{t\in[0,T_{M_{1}}]}\,\min\\{t^{1/4},1\\}\,e^{t\lambda_{1}}\left\lVert
e^{-t\mathcal{L}}\kappa_{\rm
in}\right\rVert_{\dot{H}^{1}}\leq\frac{M_{1}}{2}\,.$

Note that $T_{M_{1}}$ depends on $\kappa_{\rm in}$, particularly on how well
$\kappa_{\rm in}\in L^{2}(I)$ can be approximated by slightly smoother
functions.

  2. (2)

_Small data:_ Using Lemma 2.2, we have

$\displaystyle\sup_{t\in[0,T]}\,\min\\{t^{1/4},1\\}\,e^{t\lambda_{1}}\left\lVert
e^{-t\mathcal{L}}\kappa_{\rm in}\right\rVert_{\dot{H}^{1}}\leq
c\left\lVert\kappa_{\rm in}\right\rVert_{L^{2}}\,.$

For $\left\lVert\kappa_{\rm in}\right\rVert_{L^{2}}$ sufficiently small, we
may take $M_{1}=2c\left\lVert\kappa_{\rm in}\right\rVert_{L^{2}}$ to obtain
the desired bound.

We now show that $\Psi$ is a contraction on $B_{M_{0}}(\mathcal{Y}_{0})\cap
B_{M_{1}}(\mathcal{Y}_{1})$. For $\kappa,\varphi\in H^{1}(I)$ we first note
that

$\displaystyle\left\lVert\mathcal{R}[\kappa(\cdot,t)]-\mathcal{R}[\varphi(\cdot,t)]\right\rVert_{L^{2}(I)}$
(45) $\displaystyle\leq
c\left(\left\lVert\kappa^{2}\kappa_{s}-\varphi^{2}\varphi_{s}\right\rVert_{L^{2}}+\left\lVert\tau^{\kappa}_{s}\kappa-\tau^{\varphi}_{s}\varphi\right\rVert_{L^{2}}+\left\lVert\tau^{\kappa}\kappa_{s}-\tau^{\varphi}\varphi_{s}\right\rVert_{L^{2}}\right)$
$\displaystyle\leq
c\bigg{(}\left\lVert(\kappa^{2}+\varphi^{2}-\varphi\kappa)(\kappa_{s}-\varphi_{s})\right\rVert_{L^{2}}+\left\lVert(\kappa-\varphi)(\kappa\varphi_{s}+\varphi\kappa_{s})\right\rVert_{L^{2}}+\|(\tau^{\kappa}-\tau^{\varphi})_{s}(\kappa+\varphi)\|_{L^{2}}$
$\displaystyle\quad+\|(\tau^{\kappa}+\tau^{\varphi})_{s}(\kappa-\varphi)\|_{L^{2}}+\|(\tau^{\kappa}-\tau^{\varphi})(\kappa_{s}+\varphi_{s})\|_{L^{2}}+\|(\tau^{\kappa}+\tau^{\varphi})(\kappa_{s}-\varphi_{s})\|_{L^{2}}\bigg{)}$
$\displaystyle\leq
c\bigg{(}\left(\left\lVert\kappa\right\rVert_{L^{\infty}}^{2}+\left\lVert\varphi\right\rVert_{L^{\infty}}^{2}\right)\left\lVert\kappa_{s}-\varphi_{s}\right\rVert_{L^{2}}+\left\lVert\kappa-\varphi\right\rVert_{L^{\infty}}\big{(}\left\lVert\kappa\right\rVert_{L^{\infty}}\left\lVert\varphi_{s}\right\rVert_{L^{2}}+\left\lVert\varphi\right\rVert_{L^{\infty}}\left\lVert\kappa_{s}\right\rVert_{L^{2}}\big{)}$
$\displaystyle\quad+\|(\tau^{\kappa}-\tau^{\varphi})_{s}\|_{L^{2}}(\left\lVert\kappa\right\rVert_{L^{\infty}}+\left\lVert\varphi\right\rVert_{L^{\infty}})+(\|\tau^{\kappa}_{s}\|_{L^{2}}+\|\tau^{\varphi}_{s}\|_{L^{2}})\left\lVert\kappa-\varphi\right\rVert_{L^{\infty}}$
$\displaystyle\quad+\|\tau^{\kappa}-\tau^{\varphi}\|_{L^{\infty}}(\left\lVert\kappa_{s}\right\rVert_{L^{2}}+\left\lVert\varphi_{s}\right\rVert_{L^{2}})+(\|\tau^{\kappa}\|_{L^{\infty}}+\|\tau^{\varphi}\|_{L^{\infty}})\left\lVert\kappa_{s}-\varphi_{s}\right\rVert_{L^{2}}\bigg{)}$
$\displaystyle\leq
c\bigg{(}\left\lVert\kappa-\varphi\right\rVert_{L^{2}}\left(\left\lVert\kappa\right\rVert_{\dot{H}^{1}}^{3}(\left\lVert\kappa\right\rVert_{L^{2}}^{2}+1)+\left\lVert\varphi\right\rVert_{\dot{H}^{1}}^{3}(\left\lVert\varphi\right\rVert_{L^{2}}^{2}+1)\right)$
$\displaystyle\quad+\left\lVert\kappa-\varphi\right\rVert_{\dot{H}^{1}}\left(\left\lVert\kappa\right\rVert_{\dot{H}^{1}}^{2}\left(\left\lVert\kappa\right\rVert_{L^{2}}+1\right)+\left\lVert\varphi\right\rVert_{\dot{H}^{1}}^{2}\left(\left\lVert\varphi\right\rVert_{L^{2}}+1\right)\right)\bigg{)}\,.$

Here we have used Lemma 2.1 in the third inequality, and in the fourth we used
the estimates (38) and (37) to bound $\tau^{\kappa}$, $\tau^{\varphi}$ in
terms of $\kappa$, $\varphi$.

Using (45), for $\kappa,\varphi\in B_{M_{0}}(\mathcal{Y}_{0})\cap
B_{M_{1}}(\mathcal{Y}_{1})$ we have

$\displaystyle\left\lVert\Psi[\kappa]-\Psi[\varphi]\right\rVert_{\mathcal{Y}_{1}}$
$\displaystyle\quad\leq\sup_{t\in[0,T]}\min\\{t^{1/4},1\\}\,e^{t\lambda_{1}}\int_{0}^{t}\left\lVert
e^{-(t-t^{\prime})\mathcal{L}}\big{(}\mathcal{R}[\kappa]-\mathcal{R}[\varphi]\big{)}_{s}\right\rVert_{\dot{H}^{1}}\,dt^{\prime}$
$\displaystyle\quad\leq\sup_{t\in[0,T]}\min\\{t^{1/4},1\\}\,e^{t\lambda_{1}}\int_{0}^{t}\max\\{(t-t^{\prime})^{-1/2},1\\}\,e^{-(t-t^{\prime})\lambda_{1}}\left\lVert\mathcal{R}[\kappa]-\mathcal{R}[\varphi]\right\rVert_{L^{2}}\,dt^{\prime}$
$\displaystyle\quad\leq
c\sup_{t\in[0,T]}\min\\{t^{1/4},1\\}\int_{0}^{t}\max\\{(t-t^{\prime})^{-1/2},1\\}\max\\{(t^{\prime})^{-3/4},1\\}\,e^{-2t^{\prime}\lambda_{1}}\,dt^{\prime}\;\bigg{(}\left\lVert\kappa-\varphi\right\rVert_{\mathcal{Y}_{0}}\big{(}\left\lVert\kappa\right\rVert_{\mathcal{Y}_{1}}^{3}(\left\lVert\kappa\right\rVert_{\mathcal{Y}_{0}}^{2}+1)$
$\displaystyle\hskip
56.9055pt+\left\lVert\varphi\right\rVert_{\mathcal{Y}_{1}}^{3}(\left\lVert\varphi\right\rVert_{\mathcal{Y}_{0}}^{2}+1)\big{)}+\left\lVert\kappa-\varphi\right\rVert_{\mathcal{Y}_{1}}\big{(}\left\lVert\kappa\right\rVert_{\mathcal{Y}_{1}}^{2}(\left\lVert\kappa\right\rVert_{\mathcal{Y}_{0}}+1)+\left\lVert\varphi\right\rVert_{\mathcal{Y}_{1}}^{2}(\left\lVert\varphi\right\rVert_{\mathcal{Y}_{0}}+1)\big{)}\bigg{)}$
$\displaystyle\quad\leq
c\,M_{1}^{3}(M_{0}^{2}+1)\left\lVert\kappa-\varphi\right\rVert_{\mathcal{Y}_{0}}+c\,M_{1}^{2}(M_{0}+1)\left\lVert\kappa-\varphi\right\rVert_{\mathcal{Y}_{1}}\,,$

and, by a similar series of estimates,

$\displaystyle\left\lVert\Psi[\kappa]-\Psi[\varphi]\right\rVert_{\mathcal{Y}_{0}}$
$\displaystyle\leq
c\,M_{1}^{3}(M_{0}^{2}+1)\left\lVert\kappa-\varphi\right\rVert_{\mathcal{Y}_{0}}+c\,M_{1}^{2}(M_{0}+1)\left\lVert\kappa-\varphi\right\rVert_{\mathcal{Y}_{1}}\,.$

For fixed $M_{0}$, taking $M_{1}$ small enough that both
$c\,M_{1}^{3}(M_{0}^{2}+1)<\frac{1}{2}$ and
$c\,M_{1}^{2}(M_{0}+1)<\frac{1}{2}$, by the contraction mapping theorem there
exists a unique fixed point of the map $\Psi$ in
$B_{M_{0}}(\mathcal{Y}_{0})\cap B_{M_{1}}(\mathcal{Y}_{1})$. Furthermore, in
the case of small data, combining the bounds (43) and (44), and using that
$M_{0}=2\left\lVert\kappa_{\rm in}\right\rVert_{L^{2}}$ and
$M_{1}=2c\left\lVert\kappa_{\rm in}\right\rVert_{L^{2}}$, we obtain the
estimate (18). ∎

######  Remark 2.6.

Here we sketch a bootstrapping argument for showing that $\kappa$ is in fact
smooth for all positive times. By a calculation analogous to that above
equation (43), it can be shown that $\kappa(s,t)$ in fact belongs to
$H^{2}(I)$ for any positive time. Using this additional regularity on the
right hand side of the equation (11) for $\tau$, we may show that $\tau\in
H^{2}(I)$ as well. Then, applying $\mathcal{L}^{1-\epsilon/4}$ to the Duhamel
formula (15) for $\kappa$:

$\displaystyle\mathcal{L}^{1-\epsilon/4}\kappa(s,t)=\mathcal{L}^{1-\epsilon/4}e^{-(t-t_{0})\mathcal{L}}\kappa\big{|}_{t=t_{0}}+\int_{t_{0}}^{t}\mathcal{L}^{1-\epsilon/4}e^{-(t-t^{\prime})\mathcal{L}}(\mathcal{R}[\kappa(s,t^{\prime})])_{s}\,dt^{\prime}\,,$

since $(\mathcal{R}[\kappa(s,t^{\prime})])_{s}\in L^{2}(I)$ (see (14)), we may
use Lemma 2.2 to obtain that $\kappa\in C([t_{0},T];H^{4-\epsilon})$ for any
$\epsilon,t_{0}>0$. Note that we can also show that $\kappa\in
L^{2}([t_{0},T];H^{4}(I))$ by applying the full operator $\mathcal{L}$ above
and estimating the integral term in $L^{2}$ in time.

Passing to higher regularity becomes a bit tricky with the boundary conditions
(12): if we were instead working on the torus, we could simply apply
$\mathcal{L}^{k}$, $k\geq 1$, to the Duhamel formula, and use that
$\mathcal{L}$ commutes with the semigroup $e^{-t\mathcal{L}}$. However, on the
interval $I$, this requires $(\mathcal{R}[\kappa])_{s}\in D(\mathcal{L})$,
which is not necessarily the case since, e.g., $\tau_{s}\kappa_{s}$ does not
satisfy the boundary conditions (12).

Instead, we can use that $\dot{\kappa}\in L^{2}([t_{0},T];L^{2}(I))$ since the
right hand side of the $\dot{\kappa}$ equation (10) now belongs to
$L^{2}([t_{0},T];L^{2}(I))$. Taking a time derivative of the equation (10) for
$\kappa$, we have

$\displaystyle\partial_{t}(\dot{\kappa})$
$\displaystyle=-\mathcal{L}\dot{\kappa}+(2+\gamma)(3\kappa^{2}\dot{\kappa})_{ss}+(\tau\dot{\kappa})_{ss}+(2+\gamma)(\dot{\tau}\kappa)_{ss}+(1+\gamma)(\tau_{s}\dot{\kappa}-\dot{\tau}\kappa_{s})_{s}\,.$

Here $\dot{\tau}$ satisfies the elliptic equation

$\displaystyle\dot{\tau}_{ss}-\frac{\kappa^{2}}{1+\gamma}\dot{\tau}$
$\displaystyle=\frac{1}{1+\gamma}\bigg{(}2\kappa\dot{\kappa}\tau+4\kappa^{3}\dot{\kappa}+2((\kappa_{s}\dot{\kappa})_{s}-\kappa_{ss}\dot{\kappa})-(4+3\gamma)(\dot{\kappa}_{s}\kappa+\kappa_{s}\dot{\kappa})_{s}\bigg{)}\,,$

and we have that at each time $t>t_{0}$,
$\left\lVert\dot{\tau}\right\rVert_{H^{1}(I)}\leq
c\big{(}\left\lVert\kappa\right\rVert_{H^{2}}^{3}\left\lVert\dot{\kappa}\right\rVert_{L^{2}}+\left\lVert\kappa\right\rVert_{H^{1}}\left\lVert\dot{\kappa}\right\rVert_{\dot{H}^{1}}\big{)}$.
Using the Duhamel formula for $\dot{\kappa}$, we have

$\displaystyle\dot{\kappa}$
$\displaystyle=e^{-(t-t_{0})\mathcal{L}}\dot{\kappa}\big{|}_{t=t_{0}}+\int_{t_{0}}^{t}e^{-(t-t^{\prime})\mathcal{L}}\bigg{(}(\mathcal{R}_{1}[\kappa(s,t^{\prime})])_{ss}+(\mathcal{R}_{2}[\kappa(s,t^{\prime})])_{s}\bigg{)}\,dt^{\prime}\,,$

where
$\mathcal{R}_{1}=(2+\gamma)3\kappa^{2}\dot{\kappa}+\tau\dot{\kappa}+(2+\gamma)\dot{\tau}\kappa$
and $\mathcal{R}_{2}=(1+\gamma)(\tau_{s}\dot{\kappa}-\dot{\tau}\kappa_{s})$.
Since we need $\dot{\kappa}\in H^{1}(I)$ to make sense of $\dot{\tau}$, we can
again obtain estimates for $\dot{\kappa}$ in a time-weighted space analogous
to $\mathcal{Y}_{0}\cap\mathcal{Y}_{1}$. In particular, we can estimate

$\displaystyle\sup_{t\in[t_{0},T]}\bigg{(}\left\lVert\dot{\kappa}\right\rVert_{L^{2}(I)}+\min\\{(t-t_{0})^{1/4},1\\}\left\lVert\dot{\kappa}\right\rVert_{\dot{H}^{1}(I)}\bigg{)}$

in a similar way as in estimates (43) and (44), and the above argument begins
to repeat. Repeating this, we can show that $\partial_{t}^{k}\kappa\in
L^{2}([kt_{0},T];H^{4}(I))$. To obtain spatial regularity from the time
regularity, we may use that, once $\dot{\kappa}\in
L^{2}([2t_{0},T];H^{4}(I))$, we have $\mathcal{L}\dot{\kappa}\in
L^{2}([2t_{0},T];L^{2}(I))$, and, using the equation (10) for $\dot{\kappa}$,
$\mathcal{L}(-\mathcal{L}\kappa+(2+\gamma)(\kappa^{3})_{ss}+(2+\gamma)(\tau_{s}\kappa)_{s}+(\tau\kappa_{s})_{s})\in
L^{2}([2t_{0},T];L^{2}(I))$, i.e. $\kappa\in L^{2}([2t_{0},T];H^{8}(I))$.

##  3\. Periodic active forcing and swimming

We now consider the system (7)–(9) with nonzero $T$-periodic active forcing
$\kappa_{0}(s,t)$ along the filament, corresponding to a nonzero preferred
curvature, as derived in [2, 53]. In particular, at a fixed time $t$, let
$\kappa_{0}(s)$ denote the preferred curvature of the fiber. For a planar
fiber, recalling that $\bm{X}_{ss}=\kappa(s)\bm{e}_{\rm n}(s)$ and
$(\bm{e}_{\rm n})_{s}=-\kappa(s)\bm{e}_{\rm t}(s)$, we may compute the elastic
energy of the fiber as

$\displaystyle G[\bm{X}]$
$\displaystyle=\frac{1}{2}\int_{0}^{1}\bigg{(}\big{(}\kappa(s)-\kappa_{0}(s)\big{)}^{2}+\lambda(s)(\left\lvert\bm{X}_{s}\right\rvert^{2}-1)\bigg{)}\,ds$
(46)
$\displaystyle=\frac{1}{2}\int_{0}^{1}\bigg{(}\left\lvert\bm{X}_{ss}-\kappa_{0}\bm{e}_{\rm
n}\right\rvert^{2}+\lambda(\left\lvert\bm{X}_{s}\right\rvert^{2}-1)\bigg{)}\,ds\,,$

along with the boundary conditions $(\bm{X}_{ss}-\kappa_{0}\bm{e}_{\rm
n})\big{|}_{s=0,1}=0$, $(\bm{X}_{ss}-\kappa_{0}\bm{e}_{\rm
n})_{s}\big{|}_{s=0,1}=0$, $\lambda|_{s=0,1}=0$. The Lagrange multiplier
$\lambda(s)$ is equal to the tension $\tau$ when $\kappa_{0}\equiv 0$. Taking
the variation and integrating by parts yields

$\displaystyle\frac{d}{d\varepsilon}\bigg{|}_{\varepsilon=0}G[\bm{X}+\varepsilon\bm{Y}]$
$\displaystyle=-\int_{0}^{1}{\bm{Y}}_{s}\cdot\bigg{(}\bm{X}_{sss}-(\kappa_{0})_{s}\bm{e}_{\rm
n}-\kappa_{0}(\bm{e}_{\rm n})_{s}-\lambda\bm{X}_{s}\bigg{)}\,ds\,.$

Defining the tension of the fiber to be $\tau=\lambda-\kappa\kappa_{0}$, we
obtain the form of forcing appearing on the right hand side of (2).

###  3.1. Existence of solution with force

Given the complicated way that the forcing terms due to $\kappa_{0}$ enter the
system (7)–(9), we will first need to establish existence and uniqueness of
solutions for $t\in[0,T]$ for sufficiently small $\kappa_{0}$. In particular,
we show the following lemma.

######  Lemma 3.1 (Well-posedness with forcing).

There exists a constant $0<M<1$ such that, given a preferred curvature
$\kappa_{0}(s,t)\in C^{1}([0,T];H^{1}(I))$ and an initial condition
$\overline{\kappa}_{\rm in}\in L^{2}(I)$ satisfying

$\displaystyle cT\bigg{(}\sup_{0\leq t\leq
T}\left\lVert\dot{\kappa}_{0}\right\rVert_{L^{2}(I)}\bigg{)}=c_{1}M\,,\quad\sup_{0\leq
t\leq
T}\left\lVert\kappa_{0}\right\rVert_{H^{1}}=c_{2}M\,,\qquad\left\lVert\overline{\kappa}_{\rm
in}\right\rVert_{L^{2}}=c_{3}M\,,$

for $c_{1}>0$, $c_{2}>0$, $c_{3}\geq 0$, and $c_{1}+c_{3}\leq\frac{1}{2}$,
there exists a unique mild solution $\overline{\kappa}\in
C([0,T];L^{2}(I))\cap C((0,T];H^{1}(I))$ to the forced system (7)–(9)
satisfying

$\sup_{0\leq t\leq
T}\bigg{(}\left\lVert\overline{\kappa}\right\rVert_{L^{2}(I)}+\min\\{t^{1/4},1\\}\left\lVert\overline{\kappa}\right\rVert_{\dot{H}^{1}(I)}\bigg{)}\leq
cM\,.$ (47)

###### Proof.

We begin by deriving estimates for the forced tension equation (8). Recalling
the definition of the bilinear form $\mathcal{B}$ (39), where now we write
$\kappa=\overline{\kappa}+\kappa_{0}$, we have that a weak solution of (8)
satisfies

$\displaystyle\mathcal{B}(\overline{\tau},\phi)$
$\displaystyle=-\int_{0}^{1}\frac{1}{1+\gamma}\bigg{(}\overline{\kappa}(\overline{\kappa}+\kappa_{0})^{2}(\overline{\kappa}+2\kappa_{0})\phi+\overline{\kappa}_{s}(\overline{\kappa}+\kappa_{0})_{s}\phi$
(48)
$\displaystyle\qquad+(1+\gamma)\big{(}\overline{\kappa}(\overline{\kappa}+2\kappa_{0})\big{)}_{s}\phi_{s}+(2+\gamma)\overline{\kappa}_{s}(\overline{\kappa}+\kappa_{0})\phi_{s}\bigg{)}\,ds\qquad\forall\,\phi\in
H^{1}_{0}(I)\,.$

Existence of a weak solution $\overline{\tau}\in H^{1}_{0}(I)$ again follows
by Lax-Milgram, and, using Lemma 2.1 as in the proof of the unforced estimate
(37), we have that $\overline{\tau}$ satisfies

$\displaystyle\mathcal{B}(\overline{\tau},\overline{\tau})$ $\displaystyle\leq
c\bigg{(}\left\lVert\overline{\kappa}\right\rVert_{\dot{H}^{1}}\big{(}\left\lVert\overline{\kappa}\right\rVert_{\dot{H}^{1}}\left\lVert\overline{\kappa}\right\rVert_{L^{2}}+\left\lVert\kappa_{0}\right\rVert_{H^{1}}\left\lVert\kappa_{0}\right\rVert_{L^{2}}\big{)}\left\lVert(\overline{\kappa}+\kappa_{0})\overline{\tau}\right\rVert_{L^{2}}$
$\displaystyle\qquad+\left\lVert\overline{\kappa}\right\rVert_{\dot{H}^{1}}\big{(}\left\lVert\overline{\kappa}\right\rVert_{\dot{H}^{1}}+\left\lVert\kappa_{0}\right\rVert_{H^{1}}\big{)}\left\lVert\overline{\tau}\right\rVert_{H^{1}}\bigg{)}\,.$

Applying Young’s inequality and again using that
$\left\lVert\overline{\tau}\right\rVert_{H^{1}}^{2}\leq
c\left\lVert\overline{\tau}_{s}\right\rVert_{L^{2}}^{2}\leq
c\mathcal{B}(\overline{\tau},\overline{\tau})$, we obtain

$\left\lVert\overline{\tau}\right\rVert_{H^{1}}\leq
c\big{(}\left\lVert\overline{\kappa}\right\rVert_{\dot{H}^{1}}^{2}+\left\lVert\kappa_{0}\right\rVert_{H^{1}}^{2}\big{)}\big{(}\left\lVert\overline{\kappa}\right\rVert_{L^{2}}+\left\lVert\kappa_{0}\right\rVert_{L^{2}}+1\big{)}\,.$
(49)

We also need to amend the Lipschitz estimate of Lemma 2.5 to account for the
presence of the forcing terms. As before, given $\overline{\kappa}$,
$\overline{\varphi}\in H^{1}(I)$, we define $\overline{\tau}^{\kappa}$ and
$\overline{\tau}^{\varphi}$ to be the $H^{1}_{0}$ solutions to the forced
tension equation (8) using $\overline{\kappa}$ and $\overline{\varphi}$,
respectively, on the right hand side. We have that the difference
$\widehat{\tau}:=\overline{\tau}^{\kappa}-\overline{\tau}^{\varphi}$ now
satisfies

$\displaystyle\widehat{\tau}_{ss}-\frac{(\overline{\kappa}+\kappa_{0})^{2}+(\overline{\varphi}+\kappa_{0})^{2}}{2(1+\gamma)}\widehat{\tau}$
$\displaystyle=\frac{1}{1+\gamma}\bigg{(}\frac{1}{2}(\overline{\kappa}-\overline{\varphi})(\overline{\kappa}+\overline{\varphi}+2\kappa_{0})(\overline{\tau}^{\kappa}+\overline{\tau}^{\varphi})$
$\displaystyle\qquad+(\overline{\kappa}-\overline{\varphi})\big{(}\overline{\kappa}^{3}+\overline{\varphi}^{3}+(\overline{\kappa}\overline{\varphi}+5\kappa_{0}^{2})(\overline{\kappa}+\overline{\varphi})+4\kappa_{0}(\overline{\kappa}^{2}+\overline{\kappa}\overline{\varphi}+\overline{\varphi}^{2})+2\kappa_{0}^{3}\big{)}$
$\displaystyle\qquad+(\overline{\kappa}-\overline{\varphi})_{s}(\overline{\kappa}+\overline{\varphi}+\kappa_{0})_{s}-(1+\gamma)\big{(}(\overline{\kappa}-\overline{\varphi})(\overline{\kappa}+\overline{\varphi}+2\kappa_{0})\big{)}_{ss}$
$\displaystyle\qquad-(2+\gamma)\big{(}(\overline{\kappa}-\overline{\varphi})_{s}(\overline{\kappa}+\kappa_{0})+(\overline{\kappa}-\overline{\varphi})\overline{\varphi}_{s}\big{)}_{s}\bigg{)}\,.$

Proceeding similarly to the proof of Lemma 2.5 and using the bound (49) to
estimate the first term involving
$\overline{\tau}^{\kappa}+\overline{\tau}^{\varphi}$, we have

$\displaystyle\left\lVert\widehat{\tau}\right\rVert_{H^{1}}^{2}$
$\displaystyle\leq
c\int_{0}^{1}\bigg{(}\widehat{\tau}_{s}^{2}+\frac{(\overline{\kappa}+\kappa_{0})^{2}+(\overline{\varphi}+\kappa_{0})^{2}}{2(1+\gamma)}\widehat{\tau}^{2}\bigg{)}\,ds$
$\displaystyle\leq
c\left\lVert\overline{\kappa}-\overline{\varphi}\right\rVert_{L^{2}}\big{(}\left\lVert\overline{\kappa}\right\rVert_{\dot{H}^{1}}^{2}+\left\lVert\overline{\varphi}\right\rVert_{\dot{H}^{1}}^{2}+\left\lVert\kappa_{0}\right\rVert_{H^{1}}^{2}\big{)}\big{(}\left\lVert\overline{\kappa}\right\rVert_{L^{2}}+\left\lVert\overline{\varphi}\right\rVert_{L^{2}}+\left\lVert\kappa_{0}\right\rVert_{L^{2}}+1\big{)}^{2}\left\lVert\widehat{\tau}\right\rVert_{L^{\infty}}$
$\displaystyle\qquad+c\left\lVert\overline{\kappa}-\overline{\varphi}\right\rVert_{L^{2}}\big{(}\left\lVert\overline{\kappa}\right\rVert_{L^{6}}^{3}+\left\lVert\overline{\varphi}\right\rVert_{L^{6}}^{3}+\left\lVert\kappa_{0}\right\rVert_{L^{6}}^{3}\big{)}\left\lVert\widehat{\tau}\right\rVert_{L^{\infty}}$
$\displaystyle\qquad+c\left\lVert\overline{\kappa}-\overline{\varphi}\right\rVert_{\dot{H}^{1}}(\left\lVert\overline{\kappa}\right\rVert_{\dot{H}^{1}}+\left\lVert\overline{\varphi}\right\rVert_{\dot{H}^{1}}+\left\lVert\kappa_{0}\right\rVert_{H^{1}})\left\lVert\widehat{\tau}\right\rVert_{H^{1}}\,.$

Using Lemma 2.1 to absorb the $L^{6}$ terms into the first expression and
applying Young’s inequality, we obtain the Lipschitz bound

$\displaystyle\left\lVert\overline{\tau}^{\kappa}-\overline{\tau}^{\varphi}\right\rVert_{H^{1}}$
$\displaystyle\leq
c\left\lVert\overline{\kappa}-\overline{\varphi}\right\rVert_{L^{2}}\big{(}\left\lVert\overline{\kappa}\right\rVert_{\dot{H}^{1}}^{2}+\left\lVert\overline{\varphi}\right\rVert_{\dot{H}^{1}}^{2}+\left\lVert\kappa_{0}\right\rVert_{H^{1}}^{2}\big{)}\big{(}\left\lVert\overline{\kappa}\right\rVert_{L^{2}}+\left\lVert\overline{\varphi}\right\rVert_{L^{2}}+\left\lVert\kappa_{0}\right\rVert_{L^{2}}+1\big{)}^{2}$
(50)
$\displaystyle\qquad+c\left\lVert\overline{\kappa}-\overline{\varphi}\right\rVert_{\dot{H}^{1}}\big{(}\left\lVert\overline{\kappa}\right\rVert_{\dot{H}^{1}}+\left\lVert\overline{\varphi}\right\rVert_{\dot{H}^{1}}+\left\lVert\kappa_{0}\right\rVert_{H^{1}}\big{)}\,.$

Equipped with estimates (49) and (50), we now consider the forced evolution
equation (7). We take $\overline{\kappa}_{\rm in}(s)=\overline{\kappa}(s,0)$
to be the initial difference from the fiber’s preferred curvature and recall
the definition (13) of the operator $\mathcal{L}$. We may then write the mild
solution formula for (7)–(9), which now consists of three terms:

$\overline{\Psi}[\overline{\kappa}](s,t)=e^{-t\mathcal{L}}\overline{\kappa}_{\rm
in}(s)-\int_{0}^{t}e^{-(t-t^{\prime})\mathcal{L}}\,\dot{\kappa}_{0}(s,t^{\prime})\,dt^{\prime}+\int_{0}^{t}e^{-(t-t^{\prime})\mathcal{L}}(\overline{\mathcal{R}}[\overline{\kappa}(s,t^{\prime})])_{s}\,dt^{\prime}\,.$
(51)

Here $(\overline{\mathcal{R}}[\overline{\kappa}(s,t^{\prime})])_{s}$ contains
all nonlinear terms in equation (7); in particular,

$\displaystyle\overline{\mathcal{R}}[\overline{\kappa}]$
$\displaystyle=3(2+\gamma)\overline{\kappa}(\overline{\kappa}+2\kappa_{0})\overline{\kappa}_{s}+(5+3\gamma)\kappa_{0}^{2}\overline{\kappa}_{s}+(5+2\gamma)\overline{\kappa}^{2}(\kappa_{0})_{s}$
$\displaystyle\hskip
85.35826pt+2(3+\gamma)\overline{\kappa}\kappa_{0}(\kappa_{0})_{s}+(2+\gamma)\overline{\tau}_{s}(\overline{\kappa}+\kappa_{0})+\overline{\tau}(\overline{\kappa}+\kappa_{0})_{s}\,.$

We show that (51) admits a unique fixed point. We define $\mathcal{Y}_{T}$ as
the space of functions for which the following norm is finite:

$\left\lVert\cdot\right\rVert_{\mathcal{Y}_{T}}:=\sup_{0\leq t\leq
T}\bigg{(}\left\lVert\cdot\right\rVert_{L^{2}(I)}+\min\\{t^{1/4},1\\}\left\lVert\cdot\right\rVert_{\dot{H}^{1}(I)}\bigg{)}\,.$
(52)

Considering a ball of radius $M$ in $\mathcal{Y}_{T}$,

$\displaystyle B_{M}(\mathcal{Y}_{T})=\\{u\in\mathcal{Y}_{T}\,:\,\left\lVert
u\right\rVert_{\mathcal{Y}_{T}}\leq M\\}\,,$

we have that $\overline{\Psi}$ is bounded from $B_{M}(\mathcal{Y}_{T})$ to
$B_{M}(\mathcal{Y}_{T})$ for some $M>0$. In particular, by Lemma 2.2, the
second term of $\overline{\Psi}[\overline{\kappa}]$ involving
$\dot{\kappa}_{0}$ satisfies

$\displaystyle\left\lVert\int_{0}^{t}e^{-(t-t^{\prime})\mathcal{L}}\,\dot{\kappa}_{0}\,dt^{\prime}\right\rVert_{\dot{H}^{m}(I)}$
$\displaystyle\leq
c\int_{0}^{t}\max\\{(t-t^{\prime})^{-m/4},1\\}\,e^{-(t-t^{\prime})\lambda_{1}}\left\lVert\dot{\kappa}_{0}\right\rVert_{L^{2}(I)}dt^{\prime}$
(53) $\displaystyle\leq c\,\max\\{t^{1-m/4},1\\}\bigg{(}\sup_{0\leq t\leq
T}\left\lVert\dot{\kappa}_{0}\right\rVert_{L^{2}(I)}\bigg{)}\,.$

Furthermore, we may bound the nonlinear terms from
$(\overline{\mathcal{R}}[\overline{\kappa}])_{s}$ following similar steps to
the unforced estimate (42) for $\mathcal{R}[\kappa]$. Using Lemma (2.1) and
the tension estimate (49), we have

$\displaystyle\left\lVert\overline{\mathcal{R}}[\overline{\kappa}(\cdot,t)]\right\rVert_{L^{2}}$
$\displaystyle\leq
c\big{(}\left\lVert\overline{\kappa}\right\rVert_{L^{\infty}}^{2}+\left\lVert\kappa_{0}\right\rVert_{L^{\infty}}^{2}+\left\lVert\overline{\tau}\right\rVert_{H^{1}}\big{)}\big{(}\left\lVert\overline{\kappa}\right\rVert_{\dot{H}^{1}}+\left\lVert\kappa_{0}\right\rVert_{H^{1}}\big{)}$
(54) $\displaystyle\leq
c\big{(}\left\lVert\overline{\kappa}\right\rVert_{\dot{H}^{1}}^{3}+\left\lVert\kappa_{0}\right\rVert_{H^{1}}^{3}\big{)}\big{(}\left\lVert\overline{\kappa}\right\rVert_{L^{2}}+\left\lVert\kappa_{0}\right\rVert_{L^{2}}+1\big{)}\,.$

By Lemma 2.2, we then have

$\displaystyle\left\lVert\int_{0}^{t}e^{-(t-t^{\prime})\mathcal{L}}\big{(}\overline{\mathcal{R}}[\overline{\kappa}(\cdot,t^{\prime})]\big{)}_{s}dt^{\prime}\right\rVert_{\dot{H}^{m}(I)}$
(55) $\displaystyle\quad\leq
c\int_{0}^{t}\max\\{(t-t^{\prime})^{-(m+1)/4},1\\}\,e^{-(t-t^{\prime})\lambda_{1}}\left\lVert\overline{\mathcal{R}}[\overline{\kappa}]\right\rVert_{L^{2}}dt^{\prime}$
$\displaystyle\quad\leq
c\int_{0}^{t}\max\\{(t-t^{\prime})^{-(m+1)/4},1\\}\,e^{-(t-t^{\prime})\lambda_{1}}\big{(}\left\lVert\overline{\kappa}\right\rVert_{\dot{H}^{1}}^{3}+\left\lVert\kappa_{0}\right\rVert_{H^{1}}^{3}\big{)}\big{(}\left\lVert\overline{\kappa}\right\rVert_{L^{2}}+\left\lVert\kappa_{0}\right\rVert_{L^{2}}+1\big{)}\,dt^{\prime}$
$\displaystyle\quad\leq
c\int_{0}^{t}\max\\{(t-t^{\prime})^{-(m+1)/4},1\\}\,\max\\{(t^{\prime})^{-3/4},1\\}\,e^{-(t-t^{\prime})\lambda_{1}}\,dt^{\prime}\,\big{(}\left\lVert\overline{\kappa}\right\rVert_{\mathcal{Y}_{T}}^{3}$
$\displaystyle\hskip
256.0748pt+\left\lVert\kappa_{0}\right\rVert_{\mathcal{Y}_{T}}^{3}\big{)}\left(\left\lVert\overline{\kappa}\right\rVert_{\mathcal{Y}_{T}}+\left\lVert\kappa_{0}\right\rVert_{\mathcal{Y}_{T}}+1\right)$
$\displaystyle\quad\leq
c\,\max\\{t^{-m/4},1\\}\big{(}\left\lVert\overline{\kappa}\right\rVert_{\mathcal{Y}_{T}}^{3}+\left\lVert\kappa_{0}\right\rVert_{\mathcal{Y}_{T}}^{3}\big{)}\left(\left\lVert\overline{\kappa}\right\rVert_{\mathcal{Y}_{T}}+\left\lVert\kappa_{0}\right\rVert_{\mathcal{Y}_{T}}+1\right)\,.$

Combining the estimates (53) and (55) for $\dot{\kappa}_{0}$ and
$\overline{\mathcal{R}}[\overline{\kappa}]$, we then have

$\displaystyle\left\lVert\overline{\Psi}[\overline{\kappa}]\right\rVert_{\mathcal{Y}_{T}}$
$\displaystyle\leq\left\lVert\overline{\kappa}_{\rm
in}\right\rVert_{L^{2}}+cT\bigg{(}\sup_{0\leq t\leq
T}\left\lVert\dot{\kappa}_{0}\right\rVert_{L^{2}}\bigg{)}+c\big{(}\left\lVert\overline{\kappa}\right\rVert_{\mathcal{Y}_{T}}^{3}+\left\lVert\kappa_{0}\right\rVert_{\mathcal{Y}_{T}}^{3}\big{)}\left(\left\lVert\overline{\kappa}\right\rVert_{\mathcal{Y}_{T}}+\left\lVert\kappa_{0}\right\rVert_{\mathcal{Y}_{T}}+1\right)$
(56)

for $\overline{\kappa}\in B_{M}(\mathcal{Y}_{T})$. Noting that
$\left\lVert\kappa_{0}\right\rVert_{\mathcal{Y}_{T}}\leq
c(T)\big{(}\sup_{0\leq t\leq
T}\left\lVert\kappa_{0}\right\rVert_{H^{1}}\big{)}$, we take the preferred
curvature $\kappa_{0}(s,t)$ and initial condition $\overline{\kappa}_{\rm
in}(s)$ to satisfy

$cT\bigg{(}\sup_{0\leq t\leq
T}\left\lVert\dot{\kappa}_{0}\right\rVert_{L^{2}(I)}\bigg{)}=c_{1}M,\quad\sup_{0\leq
t\leq
T}\left\lVert\kappa_{0}\right\rVert_{H^{1}}=c_{2}M,\qquad\left\lVert\overline{\kappa}_{\rm
in}\right\rVert_{L^{2}}=c_{3}M,$ (57)

where $c_{1}>0$, $c_{2}>0$, $c_{3}\geq 0$, and $c_{1}+c_{3}\leq\frac{1}{2}$.
We then have

$\left\lVert\overline{\Psi}[\overline{\kappa}]\right\rVert_{\mathcal{Y}_{T}}\leq\left(c_{1}+c_{3}+c(M^{3}+M^{2})\right)M\leq
M$ (58)

for $M$ sufficiently small.

We now show that $\overline{\Psi}$ is a contraction on
$B_{M}(\mathcal{Y}_{T})$. Given $\overline{\kappa}$, $\overline{\varphi}\in
B_{M}(\mathcal{Y}_{T})$, we first amend the unforced Lipschitz estimate (45)
for $\mathcal{R}[\kappa]$ to account for the forcing terms in
$\overline{\mathcal{R}}[\overline{\kappa}]$. Using the tension bound (49) and
tension Lipschitz estimate (50), as well as Lemma 2.1, we have

$\displaystyle\left\lVert\overline{\mathcal{R}}[\overline{\kappa}(\cdot,t)]-\overline{\mathcal{R}}[\overline{\varphi}(\cdot,t)]\right\rVert_{L^{2}(I)}$
$\displaystyle\qquad\leq
c\bigg{(}\left\lVert\overline{\kappa}-\overline{\varphi}\right\rVert_{L^{\infty}}\big{(}\left\lVert\overline{\kappa}\right\rVert_{L^{\infty}}+\left\lVert\overline{\varphi}\right\rVert_{L^{\infty}}+\left\lVert\kappa_{0}\right\rVert_{L^{\infty}}\big{)}\big{(}\left\lVert\overline{\kappa}\right\rVert_{\dot{H}^{1}}+\left\lVert\kappa_{0}\right\rVert_{H^{1}}\big{)}$
$\displaystyle\qquad\quad+\left\lVert\overline{\kappa}-\overline{\varphi}\right\rVert_{\dot{H}^{1}}\big{(}\left\lVert\overline{\varphi}\right\rVert_{L^{\infty}}^{2}+\left\lVert\kappa_{0}\right\rVert_{L^{\infty}}^{2}\big{)}+\left\lVert\overline{\tau}^{\kappa}-\overline{\tau}^{\varphi}\right\rVert_{H^{1}}\big{(}\left\lVert\overline{\kappa}\right\rVert_{\dot{H}^{1}}+\left\lVert\kappa_{0}\right\rVert_{H^{1}}\big{)}+\left\lVert\overline{\kappa}-\overline{\varphi}\right\rVert_{\dot{H}^{1}}\left\lVert\overline{\tau}^{\varphi}\right\rVert_{H^{1}}\bigg{)}$
$\displaystyle\qquad\leq
c\bigg{(}\left\lVert\overline{\kappa}-\overline{\varphi}\right\rVert_{L^{2}}\big{(}\left\lVert\overline{\kappa}\right\rVert_{\dot{H}^{1}}^{3}+\left\lVert\overline{\varphi}\right\rVert_{\dot{H}^{1}}^{3}+\left\lVert\kappa_{0}\right\rVert_{H^{1}}^{3}\big{)}\big{(}\left\lVert\overline{\kappa}\right\rVert_{L^{2}}+\left\lVert\overline{\varphi}\right\rVert_{L^{2}}+\left\lVert\kappa_{0}\right\rVert_{L^{2}}+1\big{)}^{2}$
$\displaystyle\qquad\quad+\left\lVert\overline{\kappa}-\overline{\varphi}\right\rVert_{\dot{H}^{1}}\big{(}\left\lVert\overline{\kappa}\right\rVert_{\dot{H}^{1}}^{2}+\left\lVert\overline{\varphi}\right\rVert_{\dot{H}^{1}}^{2}+\left\lVert\kappa_{0}\right\rVert_{H^{1}}^{2}\big{)}\big{(}\left\lVert\overline{\varphi}\right\rVert_{L^{2}}+\left\lVert\kappa_{0}\right\rVert_{L^{2}}+1\big{)}\bigg{)}\,.$

Then, using the above estimate along with Lemma 2.2, we have

$\displaystyle\left\lVert\int_{0}^{t}e^{-(t-t^{\prime})\mathcal{L}}\left(\overline{\mathcal{R}}[\overline{\kappa}]-\overline{\mathcal{R}}[\overline{\varphi}]\right)_{s}\,dt^{\prime}\right\rVert_{\mathcal{Y}_{T}}$
$\displaystyle\quad\leq\sup_{t\in[0,T]}\int_{0}^{t}\bigg{(}\max\\{(t-t^{\prime})^{-1/4},1\\}$
$\displaystyle\hskip
56.9055pt+\min\\{t^{1/4},1\\}\max\\{(t-t^{\prime})^{-1/2},1\\}\bigg{)}e^{-(t-t^{\prime})\lambda_{1}}\left\lVert\overline{\mathcal{R}}[\overline{\kappa}]-\overline{\mathcal{R}}[\overline{\varphi}]\right\rVert_{L^{2}(I)}\,dt^{\prime}$
$\displaystyle\quad\leq
c\sup_{t\in[0,T]}\int_{0}^{t}e^{-(t-t^{\prime})\lambda_{1}}\bigg{(}\max\\{(t-t^{\prime})^{-1/4},1\\}$
$\displaystyle\hskip
56.9055pt+\min\\{t^{1/4},1\\}\max\\{(t-t^{\prime})^{-1/2},1\\}\bigg{)}\max\\{(t^{\prime})^{-3/4},1\\}\,dt^{\prime}\,\bigg{(}\left\lVert\overline{\kappa}\right\rVert_{\mathcal{Y}_{T}}^{5}+\left\lVert\overline{\varphi}\right\rVert_{\mathcal{Y}_{T}}^{5}+\left\lVert\kappa_{0}\right\rVert_{\mathcal{Y}_{T}}^{5}$
$\displaystyle\hskip
142.26378pt+\left\lVert\overline{\kappa}\right\rVert_{\mathcal{Y}_{T}}^{2}+\left\lVert\overline{\varphi}\right\rVert_{\mathcal{Y}_{T}}^{2}+\left\lVert\kappa_{0}\right\rVert_{\mathcal{Y}_{T}}^{2}\bigg{)}\left\lVert\overline{\kappa}-\overline{\varphi}\right\rVert_{\mathcal{Y}_{T}}$
$\displaystyle\quad\leq
c\left\lVert\overline{\kappa}-\overline{\varphi}\right\rVert_{\mathcal{Y}_{T}}\bigg{(}M^{5}+M^{2}+\left\lVert\kappa_{0}\right\rVert_{\mathcal{Y}_{T}}^{5}+\left\lVert\kappa_{0}\right\rVert_{\mathcal{Y}_{T}}^{2}\bigg{)}\,.$

Using the bound (57) on $\kappa_{0}$, we therefore have

$\left\lVert\overline{\Psi}[\overline{\kappa}]-\overline{\Psi}[\overline{\varphi}]\right\rVert_{\mathcal{Y}_{T}}\leq
c(M^{5}+M^{2})\left\lVert\overline{\kappa}-\overline{\varphi}\right\rVert_{\mathcal{Y}_{T}}\,.$
(59)

Taking $M$ small enough that $c(M^{5}+M^{2})\leq\frac{1}{2}$, we obtain a
contraction on $B_{M}(\mathcal{Y}_{T})$, yielding Lemma 3.1. ∎

###  3.2. Existence of unique periodic solution

We now show that given a $T$-periodic preferred curvature $\kappa_{0}$
satisfying the smallness assumptions of Lemma 3.1, there exists a unique
$\overline{\kappa}_{\rm in}$ such that the solution $\overline{\kappa}$ to
(7)–(9) is $T$-periodic in time, and solutions starting from nearby initial
data converge to this $T$-periodic $\overline{\kappa}$ over time.

###### Proof of Theorem 1.3.

Given an initial condition $\overline{\kappa}_{\rm in}(s)$ and $T$-periodic
preferred curvature $\kappa_{0}(s,t)$ satisfying the conditions of Lemma 3.1,
we consider the unique solution $\overline{\kappa}(s,t)$ to the system (7)–(9)
at time $T$. Let $\Phi^{T}[\overline{\kappa}_{\rm in}]$ denote the time $T$
map $\overline{\kappa}_{\rm in}\mapsto\overline{\kappa}(s,T)$:

$\Phi^{T}[\overline{\kappa}_{\rm in}]=e^{-T\mathcal{L}}\overline{\kappa}_{\rm
in}-\int_{0}^{T}e^{-(T-t^{\prime})\mathcal{L}}\,\dot{\kappa}_{0}\,dt^{\prime}+\int_{0}^{T}e^{-(T-t^{\prime})\mathcal{L}}\big{(}\overline{\mathcal{R}}[\overline{\kappa}]\big{)}_{s}\,dt^{\prime}\,.$
(60)

We consider the map $\Phi^{T}$ from $B_{M^{\prime}}(L^{2}(I))$ to
$B_{M^{\prime}}(L^{2}(I))$ and show that there exists a unique fixed point.
Here $B_{M^{\prime}}(L^{2}(I))$ is the ball

$\displaystyle B_{M^{\prime}}(L^{2}(I))=\\{u\in L^{2}\,:\,\left\lVert
u\right\rVert_{L^{2}}\leq M^{\prime}\\}$

where the constant $M^{\prime}$ is given by

$M^{\prime}=\min\bigg{(}2c_{1}M,c_{2}M,\frac{M}{4}\bigg{)}$ (61)

for $c_{1}$, $c_{2}$, and $M$ as in Lemma 3.1.

We will again make use of the space $\mathcal{Y}_{T}$ (52), and note that for
$u(s,t)\in\mathcal{Y}_{T}$, we have

$\displaystyle\left\lVert u(\cdot,T)\right\rVert_{L^{2}(I)}\leq\left\lVert
u\right\rVert_{\mathcal{Y}_{T}}\,.$

Taking $\overline{\kappa}_{\rm in}\in B_{M^{\prime}}(L^{2}(I))$, we first note
that, using the bounds (57) and the definition (61) of $M^{\prime}$, the
estimate (58) on $\overline{\Psi}$ implies that the second and third terms on
the right hand side of (60) satisfy

$\displaystyle\left\lVert\int_{0}^{T}e^{-(T-t^{\prime})\mathcal{L}}\,\dot{\kappa}_{0}\,dt^{\prime}\right\rVert_{L^{2}}$
$\displaystyle\leq\frac{M^{\prime}}{2}\,,\qquad\left\lVert\int_{0}^{T}e^{-(T-t^{\prime})\mathcal{L}}(\overline{\mathcal{R}}[\overline{\kappa}])_{s}\,dt^{\prime}\right\rVert_{L^{2}}\leq\frac{M^{\prime}}{4}\,.$

Furthermore, by Lemma 2.2, we have

$\displaystyle\left\lVert e^{-T\mathcal{L}}\overline{\kappa}_{\rm
in}\right\rVert_{L^{2}}$ $\displaystyle\leq
e^{-T\lambda_{1}}\left\lVert\overline{\kappa}_{\rm in}\right\rVert_{L^{2}}\leq
e^{-T\lambda_{1}}M^{\prime}\,.$

Taking the period $T$ large enough that $e^{-T\lambda_{1}}<\frac{1}{4}$, the
time $T$ map satisfies

$\displaystyle\left\lVert\Phi^{T}[\overline{\kappa}_{\rm
in}]\right\rVert_{L^{2}(I)}\leq M^{\prime}\,.$

Next we need a Lipschitz estimate for $\Phi^{T}$. Given two initial conditions
$\overline{\kappa}_{\rm in}$, $\overline{\varphi}_{\rm in}\in L^{2}(I)$, we
begin by noting that, by Lemma 2.2, the difference in the linear evolution
alone satisfies

$\displaystyle\left\lVert
e^{t\lambda_{1}}\,e^{-t\mathcal{L}}(\overline{\kappa}_{\rm
in}-\overline{\varphi}_{\rm in})\right\rVert_{\mathcal{Y}_{T}}$
$\displaystyle=\sup_{0\leq t\leq T}e^{t\lambda_{1}}\bigg{(}\left\lVert
e^{-t\mathcal{L}}(\overline{\kappa}_{\rm in}-\overline{\varphi}_{\rm
in})\right\rVert_{L^{2}}+\min\\{t^{1/4},1\\}\left\lVert
e^{-t\mathcal{L}}(\overline{\kappa}_{\rm in}-\overline{\varphi}_{\rm
in})\right\rVert_{\dot{H}^{1}}\bigg{)}$ $\displaystyle\leq\sup_{0\leq t\leq
T}e^{t\lambda_{1}}\bigg{(}ce^{-t\lambda_{1}}\left\lVert\overline{\kappa}_{\rm
in}-\overline{\varphi}_{\rm in}\right\rVert_{L^{2}}\bigg{)}$
$\displaystyle=c\left\lVert\overline{\kappa}_{\rm in}-\overline{\varphi}_{\rm
in}\right\rVert_{L^{2}(I)}\,.$

Furthermore, using the estimate (59), we may obtain the following bound on the
difference in the evolution of the nonlinear terms:

$\displaystyle\left\lVert
e^{t\lambda_{1}}\int_{0}^{t}e^{-(t-t^{\prime})\mathcal{L}}\big{(}\overline{\mathcal{R}}[\overline{\kappa}]-\overline{\mathcal{R}}[\overline{\varphi}]\big{)}_{s}\,dt^{\prime}\right\rVert_{\mathcal{Y}_{T}}$
$\displaystyle\leq\frac{1}{2}\left\lVert
e^{t\lambda_{1}}\big{(}\overline{\kappa}-\overline{\varphi}\big{)}\right\rVert_{\mathcal{Y}_{T}}\,.$

We then have

$\displaystyle\left\lVert
e^{t\lambda_{1}}\big{(}\Phi^{t}[\overline{\kappa}_{\rm
in}]-\Phi^{t}[\overline{\varphi}_{\rm
in}]\big{)}\right\rVert_{\mathcal{Y}_{T}}$ $\displaystyle\leq
c\left\lVert\overline{\kappa}_{\rm in}-\overline{\varphi}_{\rm
in}\right\rVert_{L^{2}}+\frac{1}{2}\left\lVert
e^{t\lambda_{1}}(\overline{\kappa}-\overline{\varphi})\right\rVert_{\mathcal{Y}_{T}}\,.$

Noting that $\Phi^{t}[\overline{\kappa}_{\rm in}](s)=\overline{\kappa}(s,t)$,
we may subtract $\frac{1}{2}\left\lVert
e^{t\lambda_{1}}(\overline{\kappa}-\overline{\varphi})\right\rVert_{\mathcal{Y}_{T}}$
from both sides to obtain

$\left\lVert e^{t\lambda_{1}}\big{(}\Phi^{t}[\overline{\kappa}_{\rm
in}]-\Phi^{t}[\overline{\varphi}_{\rm
in}]\big{)}\right\rVert_{\mathcal{Y}_{T}}\leq
c\left\lVert\overline{\kappa}_{\rm in}-\overline{\varphi}_{\rm
in}\right\rVert_{L^{2}}\,.$ (62)

In particular, at time $T$ we have

$\displaystyle\left\lVert
e^{T\lambda_{1}}\big{(}\Phi^{T}[\overline{\kappa}_{\rm
in}]-\Phi^{T}[\overline{\varphi}_{\rm
in}]\big{)}\right\rVert_{L^{2}}\leq\left\lVert
e^{t\lambda_{1}}\big{(}\Phi^{t}[\overline{\kappa}_{\rm
in}]-\Phi^{t}[\overline{\varphi}_{\rm
in}]\big{)}\right\rVert_{\mathcal{Y}_{T}}\leq
c\left\lVert\overline{\kappa}_{\rm in}-\overline{\varphi}_{\rm
in}\right\rVert_{L^{2}}\,,$

which we may rewrite as

$\left\lVert\Phi^{T}[\overline{\kappa}_{\rm
in}]-\Phi^{T}[\overline{\varphi}_{\rm in}]\right\rVert_{L^{2}(I)}\leq
ce^{-T\lambda_{1}}\left\lVert\overline{\kappa}_{\rm
in}-\overline{\varphi}_{\rm in}\right\rVert_{L^{2}(I)}\,.$ (63)

Taking the period $T$ of the preferred curvature $\kappa_{0}(s,t)$ to be large
enough that $ce^{-T\lambda_{1}}<1$, equation (63) is a contraction on
$L^{2}(I)$, yielding the unique periodic solution of Theorem 1.3.

As a slight abuse of notation, we now let $\overline{\kappa}_{\rm in}$ denote
the unique initial condition such that $\Phi^{T}[\overline{\kappa}_{\rm
in}]=\overline{\kappa}_{\rm in}$. The estimate (19) then follows from estimate
(47) of Lemma 3.1 since we must have $\overline{\kappa}_{\rm in}\in H^{1}(I)$.
In particular, using the periodicity of
$\overline{\kappa}=\Phi^{t}[\overline{\kappa}_{\rm in}]$, we have

$\displaystyle\left\lVert\overline{\kappa}_{\rm in}\right\rVert_{H^{1}(I)}$
$\displaystyle=\left\lVert\overline{\kappa}(\cdot,T)\right\rVert_{H^{1}(I)}\leq
c(T)\left(\left\lVert\overline{\kappa}(\cdot,T)\right\rVert_{L^{2}(I)}+\min\\{T^{1/4},1\\}\left\lVert\overline{\kappa}(\cdot,T)\right\rVert_{\dot{H}^{1}(I)}\right)$
$\displaystyle\leq
c(T)\left\lVert\overline{\kappa}\right\rVert_{\mathcal{Y}_{T}}\leq c(T)M\,.$

Furthermore, the bound (62) implies that for any initial condition
$\overline{\varphi}_{\rm in}\in L^{2}(I)$ satisfying
$\left\lVert\overline{\varphi}_{\rm in}\right\rVert_{L^{2}}\leq M^{\prime}$,
the iterated time $T$ map $\Phi^{nT}[\overline{\varphi}_{\rm in}]$ satisfies

$\displaystyle\left\lVert\overline{\kappa}_{\rm
in}-\Phi^{nT}[\overline{\varphi}_{\rm in}]\right\rVert_{L^{2}(I)}\leq
ce^{-nT\lambda_{1}}\left\lVert\overline{\kappa}_{\rm
in}-\overline{\varphi}_{\rm in}\right\rVert_{L^{2}(I)},\quad n=1,2,\dots\,,$

which yields the convergence estimate (20). ∎

###  3.3. Swimming

Finally, we calculate the small-amplitude fiber swimming speed stated in
Theorem 1.4. The proof will rely on two auxiliary lemmas, which we introduce
below.

To show the swimming expression (26), we will need to use the smallness and
additional $H^{3}(I)$ regularity of $\kappa_{0}$ to show that the fiber
tangent vector $\bm{e}_{\rm t}(s,t)$ is not varying greatly in space and time.
In particular:

######  Lemma 3.2.

Given a $T$-periodic preferred curvature $\kappa_{0}(s,t)\in
C^{1}([0,T];H^{3}(I))$ satisfying

$\displaystyle T\bigg{(}\sup_{0\leq t\leq
T}\left\lVert\dot{\kappa}_{0}\right\rVert_{L^{2}(I)}\bigg{)}=\varepsilon\,,\quad\sup_{0\leq
t\leq T}\left\lVert\kappa_{0}\right\rVert_{H^{1}}=c_{1}\varepsilon$

for some $0<\varepsilon<1$, let
$(\overline{\kappa},\overline{\tau})=(\kappa-\kappa_{0},\tau+\kappa_{0}^{2})$
denote the unique $T$-periodic mild solution to (7)-(9). The corresponding
evolution of the frame vector $\bm{e}_{\rm t}(s,t)$, given by (21) and (6),
then satisfies

$\displaystyle\sup_{0\leq t\leq T}\left\lVert\bm{e}_{\rm
t}(\cdot,t)-\bm{e}_{\rm t}(0,0)\right\rVert_{L^{2}(I)}\leq c\varepsilon\,.$
(64)

In addition to Lemma 3.2, in order to show the swimming expansion (27), we
will need to consider a nearby simpler problem. We define
$(\overline{\kappa}^{\rm lin},\overline{\tau}^{\rm lin})\in
C([0,T];H^{1}(I)\times H^{1}(I))$ to be the unique periodic solution to the
linear evolution equation

$\displaystyle\dot{\overline{\kappa}}^{\rm lin}$
$\displaystyle=-\overline{\kappa}^{\rm lin}_{ssss}-\dot{\kappa}_{0}$ (65)
$\displaystyle(1+\gamma)\bigg{(}\overline{\tau}^{\rm
lin}_{ss}+\big{(}\overline{\kappa}^{\rm lin}(\overline{\kappa}^{\rm
lin}+2\kappa_{0})\big{)}_{ss}+\big{(}\overline{\kappa}^{\rm
lin}_{s}(\overline{\kappa}^{\rm lin}+\kappa_{0})\big{)}_{s}\bigg{)}$
$\displaystyle=-\overline{\kappa}^{\rm lin}_{ss}(\overline{\kappa}^{\rm
lin}+\kappa_{0})$ (66) $\displaystyle\overline{\kappa}^{\rm
lin}\big{|}_{s=0,1}=0,\;\overline{\kappa}^{\rm lin}_{s}\big{|}_{s=0,1}$
$\displaystyle=0,\;\overline{\tau}^{\rm lin}\big{|}_{s=0,1}=0\,,$ (67)

where $\overline{\kappa}^{\rm lin}(s,0):=\overline{\kappa}^{\rm lin}_{\rm
in}$. The existence of a unique periodic mild solution
$(\overline{\kappa}^{\rm lin},\overline{\tau}^{\rm lin})$ follows by a simple
modification of Section 3.2, and can be shown to satisfy analogous estimates
to (49) and (19) for the full system (7)-(9).

For sufficiently small $\kappa_{0}$, we have that the unique periodic solution
$(\overline{\kappa},\overline{\tau})$ to (7)-(9) is close to the unique
periodic $(\overline{\kappa}^{\rm lin},\overline{\tau}^{\rm lin})$:

######  Lemma 3.3.

Given a $T$-periodic preferred curvature $\kappa_{0}(s,t)$ satisfying

$\displaystyle T\bigg{(}\sup_{0\leq t\leq
T}\left\lVert\dot{\kappa}_{0}\right\rVert_{L^{2}(I)}\bigg{)}=\varepsilon\,,\quad\sup_{0\leq
t\leq T}\left\lVert\kappa_{0}\right\rVert_{H^{1}}=c_{1}\varepsilon$

for some $0<\varepsilon\ll 1$, the unique $T$-periodic mild solutions
$(\overline{\kappa},\overline{\tau})$ and $(\overline{\kappa}^{\rm
lin},\overline{\tau}^{\rm lin})$ to (7)-(9) and (65)-(67), respectively,
satisfy the following:

$\displaystyle\sup_{0\leq t\leq T}\|\overline{\kappa}-\overline{\kappa}^{\rm
lin}\|_{H^{1}}$ $\displaystyle\leq c\varepsilon^{3}\,;$ (68)
$\displaystyle\sup_{0\leq t\leq T}\|\overline{\tau}-\overline{\tau}^{\rm
lin}\|_{H^{1}}$ $\displaystyle\leq c\varepsilon^{4}\,.$ (69)

The proofs of Lemmas 3.2 and 3.3 are given below.

###### Proof of Lemma 3.2.

First, using the Duhamel formula (51) for $\overline{\kappa}(s,t)$, by Lemma
2.2 and estimate (55), we may show that

$\displaystyle\min\\{t^{m/4},1\\}\left\lVert\overline{\kappa}(\cdot,t)\right\rVert_{\dot{H}^{m}(I)}\leq
c\left\lVert\overline{\kappa}_{\rm in}\right\rVert_{L^{2}(I)}$

for $0\leq m\leq 3$, due to the additional regularity of $\kappa_{0}$. By the
periodicity of $\overline{\kappa}$, we in fact have

$\displaystyle\left\lVert\overline{\kappa}\right\rVert_{\dot{H}^{m}(I)}\leq
c(T,m)\left\lVert\overline{\kappa}_{\rm in}\right\rVert_{L^{2}(I)}\leq
c(T,m)\varepsilon\,.$

Then, using the equation (6) for $\dot{\theta}$, we have

$\displaystyle\sup_{0\leq t\leq T}\left\lVert\bm{e}_{\rm
t}(\cdot,t)-\bm{e}_{\rm t}(\cdot,0)\right\rVert_{L^{2}(I)}$ $\displaystyle\leq
c\sup_{0\leq t\leq T}\|\dot{\theta}\|_{L^{2}(I)}$ $\displaystyle\leq
c\sup_{0\leq t\leq
T}\big{(}\left\lVert\overline{\kappa}_{sss}\right\rVert_{L^{2}(I)}+\left\lVert\dot{\kappa}_{0}\right\rVert_{L^{2}(I)}\big{)}+c\varepsilon^{2}$
$\displaystyle\leq c(T)\varepsilon\,.$

Furthermore, using the frame relation $(\bm{e}_{\rm t})_{s}=\kappa\bm{e}_{\rm
n}$, we have that

$\displaystyle\left\lVert\bm{e}_{\rm t}(s,0)-\bm{e}_{\rm
t}(0,0)\right\rVert_{L^{2}(I)}\leq
c\left\lVert\kappa\right\rVert_{L^{2}(I)}\leq c\varepsilon\,.$

Combining the above two estimates yields Lemma 3.2. ∎

###### Proof of Lemma 3.3.

We may use Duhamel’s formula to write the difference
$\overline{\kappa}-\overline{\kappa}^{\rm lin}$ as

$\displaystyle\overline{\kappa}-\overline{\kappa}^{\rm
lin}=e^{-t\mathcal{L}}(\overline{\kappa}_{\rm in}-\overline{\kappa}_{\rm
in}^{\rm
lin})+\int_{0}^{t}e^{-(t-t^{\prime})\mathcal{L}}\big{(}\overline{\mathcal{R}}[\overline{\kappa}(s,t^{\prime})]\big{)}_{s}\,dt^{\prime},$

where $\overline{\mathcal{R}}[\overline{\kappa}]$ is as in (3.1). Recalling
that $\overline{\kappa},\overline{\kappa}^{\rm lin}\in C([0,T];H^{1}(I))$ by
periodicity, we may replace the space $\mathcal{Y}_{T}$ of (52) by
$C([0,T];H^{1}(I))$ in the estimate (55) on the remainder terms
$\overline{\mathcal{R}}[\overline{\kappa}]$. We may use this estimate along
with Lemma 2.2 to obtain

$\sup_{0\leq t\leq T}\|\overline{\kappa}-\overline{\kappa}^{\rm
lin}\|_{H^{1}(I)}\leq\|\overline{\kappa}_{\rm in}-\overline{\kappa}_{\rm
in}^{\rm lin}\|_{L^{2}(I)}+c\varepsilon^{3}\,.$ (70)

Furthermore, using the $T$-periodicity of both $\overline{\kappa}$ and
$\overline{\kappa}^{\rm lin}$, we have

$\displaystyle\|\overline{\kappa}_{\rm in}-\overline{\kappa}_{\rm in}^{\rm
lin}\|_{L^{2}(I)}$ $\displaystyle=\left\lVert
e^{-T\mathcal{L}}(\overline{\kappa}_{\rm in}-\overline{\kappa}_{\rm in}^{\rm
lin})+\int_{0}^{T}e^{-(T-t^{\prime})\mathcal{L}}\big{(}\overline{\mathcal{R}}[\overline{\kappa}(s,t^{\prime})]\big{)}_{s}\,dt^{\prime}\right\rVert_{L^{2}(I)}$
$\displaystyle\leq ce^{-T\lambda_{1}}\|\overline{\kappa}_{\rm
in}-\overline{\kappa}_{\rm in}^{\rm lin}\|_{L^{2}(I)}+c\varepsilon^{3},$

where we have again used Lemma 2.2. In particular, provided that $T$ is large
enough that $ce^{-T\lambda_{1}}\leq\frac{1}{2}$, we have

$\|\overline{\kappa}_{\rm in}-\overline{\kappa}_{\rm in}^{\rm
lin}\|_{L^{2}}\leq c\varepsilon^{3}.$ (71)

Combining (70) and (71), we obtain the desired estimate for
$\overline{\kappa}-\overline{\kappa}^{\rm lin}$.

For the tension estimate (69), letting
$\widetilde{\kappa}=\overline{\kappa}-\overline{\kappa}^{\rm lin}$, the
difference $\widetilde{\tau}:=\overline{\tau}-\overline{\tau}^{\rm lin}$
satisfies

$\displaystyle(1+\gamma)\bigg{(}\widetilde{\tau}_{ss}$
$\displaystyle+\big{(}\widetilde{\kappa}(\overline{\kappa}+2\kappa_{0})\big{)}_{ss}+\big{(}\overline{\kappa}^{\rm
lin}\widetilde{\kappa}\big{)}_{ss}+\big{(}\widetilde{\kappa}_{s}(\overline{\kappa}+\kappa_{0})\big{)}_{s}+\big{(}\overline{\kappa}^{\rm
lin}_{s}\widetilde{\kappa}\big{)}_{s}\bigg{)}$
$\displaystyle=-\widetilde{\kappa}_{ss}(\overline{\kappa}+\kappa_{0})-\overline{\kappa}^{\rm
lin}_{ss}\widetilde{\kappa}+(\overline{\kappa}+\kappa_{0})^{2}\overline{\tau}+\overline{\kappa}(\overline{\kappa}+\kappa_{0})^{2}(\overline{\kappa}+2\kappa_{0})\,.$

Multiplying by $\widetilde{\tau}$ and integrating by parts, we obtain

$\displaystyle\left\lVert\widetilde{\tau}_{s}\right\rVert_{L^{2}}^{2}$
$\displaystyle\leq
c\bigg{(}\left\lVert\big{(}\widetilde{\kappa}(\overline{\kappa}+2\kappa_{0})\big{)}_{s}\right\rVert_{L^{2}}+\|\big{(}\overline{\kappa}^{\rm
lin}\widetilde{\kappa}\big{)}_{s}\|_{L^{2}}+\left\lVert\widetilde{\kappa}_{s}(\overline{\kappa}+\kappa_{0})\right\rVert_{L^{2}}+\|\overline{\kappa}^{\rm
lin}_{s}\widetilde{\kappa}\|_{L^{2}}\bigg{)}\left\lVert\widetilde{\tau}_{s}\right\rVert_{L^{2}}$
$\displaystyle+c\bigg{(}\left\lVert\widetilde{\kappa}_{s}\right\rVert_{L^{2}}\left\lVert(\overline{\kappa}+\kappa_{0})_{s}\right\rVert_{L^{2}}+\|\overline{\kappa}^{\rm
lin}_{s}\|_{L^{2}}\left\lVert\widetilde{\kappa}\right\rVert_{L^{2}}+\left\lVert\overline{\kappa}+\kappa_{0}\right\rVert_{L^{2}}^{2}\left\lVert\overline{\tau}+\overline{\kappa}(\overline{\kappa}+2\kappa_{0})\right\rVert_{L^{\infty}}\bigg{)}\left\lVert\widetilde{\tau}\right\rVert_{L^{\infty}}\,.$

Using Young’s inequality as well as Lemma 2.1, we have

$\displaystyle\left\lVert\widetilde{\tau}\right\rVert_{H^{1}}^{2}$
$\displaystyle\leq\bigg{(}\left\lVert\widetilde{\kappa}\right\rVert_{H^{1}}^{2}\big{(}\left\lVert\overline{\kappa}\right\rVert_{H^{1}}^{2}+\|\overline{\kappa}^{\rm
lin}\|_{H^{1}}^{2}+\left\lVert\kappa_{0}\right\rVert_{H^{1}}^{2}\big{)}$
$\displaystyle\qquad+(\left\lVert\overline{\kappa}\right\rVert_{H^{1}}^{4}+\left\lVert\kappa_{0}\right\rVert_{H^{1}}^{4})\left\lVert\overline{\tau}\right\rVert_{H^{1}}^{2}+(\left\lVert\overline{\kappa}\right\rVert_{H^{1}}^{6}+\left\lVert\kappa_{0}\right\rVert_{H^{1}}^{6})\left\lVert\overline{\kappa}\right\rVert_{H^{1}}^{2}\bigg{)}\leq
c\varepsilon^{8}\,,$

where we have used (49), (19), and (68) to obtain the final inequality. ∎

Equipped with Lemmas 3.2 and 3.3, we may now prove Theorem 1.4.

###### Proof of Theorem 1.4.

We begin by calculating the form of the swimming velocity
$\bm{V}(t)=\int_{0}^{1}\frac{\partial\bm{X}}{\partial t}(s,t)\,ds$ stated in
equation (23). Using the formulation (2) and recalling that
$\bm{X}_{sss}=-\kappa^{2}\bm{e}_{\rm t}+\kappa_{s}\bm{e}_{\rm n}$,
$(\bm{e}_{\rm t})_{s}=\kappa\bm{e}_{\rm n}$, and $(\bm{e}_{\rm
n})_{s}=-\kappa\bm{e}_{\rm t}$, we have

$\displaystyle\int_{0}^{1}\frac{\partial\bm{X}}{\partial t}(s,t)\,ds$
$\displaystyle=-\int_{0}^{1}\big{(}{\bf I}+\gamma\bm{e}_{\rm t}\bm{e}_{\rm
t}^{\rm T}\big{)}\big{(}-\kappa^{2}\bm{e}_{\rm t}+\kappa_{s}\bm{e}_{\rm
n}-\tau\bm{e}_{\rm t}-(\kappa_{0})_{s}\bm{e}_{\rm n}\big{)}_{s}\,ds$ (72)
$\displaystyle=-\gamma\int_{0}^{1}\big{(}-2\kappa\kappa_{s}-\tau_{s}-\kappa\kappa_{s}+\kappa(\kappa_{0})_{s}\big{)}\bm{e}_{\rm
t}\,ds$
$\displaystyle=-\gamma\int_{0}^{1}\big{(}-2(\kappa-\kappa_{0})\kappa_{s}-2\kappa_{0}(\kappa-\kappa_{0})_{s}-\overline{\tau}_{s}-\kappa(\kappa-\kappa_{0})_{s}\big{)}\bm{e}_{\rm
t}\,ds$
$\displaystyle=-\gamma\int_{0}^{1}\bigg{(}\big{(}2(\kappa-\kappa_{0})_{s}(\kappa-\kappa_{0})-\overline{\tau}_{s}-\kappa(\kappa-\kappa_{0})_{s}\big{)}\bm{e}_{\rm
t}+2(\kappa-\kappa_{0})\kappa^{2}\bm{e}_{\rm n}\bigg{)}\,ds$
$\displaystyle=-\gamma\int_{0}^{1}\bigg{(}\big{(}(\kappa-\kappa_{0})_{s}(\kappa-\kappa_{0})-\overline{\tau}_{s}-\kappa_{0}(\kappa-\kappa_{0})_{s}\big{)}\bm{e}_{\rm
t}+2(\kappa-\kappa_{0})\kappa^{2}\bm{e}_{\rm n}\bigg{)}\,ds\,.$

Here in the second line we have used the boundary conditions in (2), in third
line we have replaced $\tau$ with $\overline{\tau}=\tau+\kappa_{0}^{2}$, in
the fourth line we have integrated $-2(\kappa-\kappa_{0})\kappa_{s}\bm{e}_{\rm
t}$ by parts, and in the final line we have added and subtracted
$\kappa_{0}(\kappa-\kappa_{0})_{s}\bm{e}_{\rm t}$.

Then we have that the swimming velocity $\bm{V}(t)$ satisfies

$\displaystyle\bm{V}(t)$
$\displaystyle=-\gamma\int_{0}^{1}\bigg{(}\big{(}\overline{\kappa}\overline{\kappa}_{s}-\overline{\tau}_{s}-\kappa_{0}\overline{\kappa}_{s}\big{)}\bm{e}_{\rm
t}(s,t)+2\overline{\kappa}\kappa^{2}\bm{e}_{\rm n}(s,t)\bigg{)}\,ds$
$\displaystyle=-\gamma\int_{0}^{1}\big{(}\overline{\kappa}\overline{\kappa}_{s}-\overline{\tau}_{s}-\kappa_{0}\overline{\kappa}_{s}\big{)}\bm{e}_{\rm
t}(0,0)\,ds+\bm{r}_{\rm v}(t)$
$\displaystyle=\gamma\int_{0}^{1}\kappa_{0}\overline{\kappa}_{s}\,ds\,\bm{e}_{\rm
t}(0,0)+\bm{r}_{\rm v}(t)$

where, using Lemma 3.2, we have

$\displaystyle\left\lvert\bm{r}_{\rm v}(t)\right\rvert$
$\displaystyle=\left\lvert\gamma\int_{0}^{1}\bigg{(}\big{(}\overline{\kappa}\overline{\kappa}_{s}-\overline{\tau}_{s}-\kappa_{0}\overline{\kappa}_{s}\big{)}(\bm{e}_{\rm
t}(s,t)-\bm{e}_{\rm t}(0,0))+2\overline{\kappa}\kappa^{2}\bm{e}_{\rm
n}(s,t)\bigg{)}\,ds\right\rvert$ $\displaystyle\leq
c\,\gamma\bigg{(}\big{(}\left\lVert\overline{\kappa}\right\rVert_{H^{1}}^{2}+\left\lVert\overline{\tau}\right\rVert_{H^{1}}+\left\lVert\kappa_{0}\right\rVert_{H^{1}}\left\lVert\overline{\kappa}\right\rVert_{H^{1}}\big{)}\left\lVert\bm{e}_{\rm
t}(\cdot,t)-\bm{e}_{\rm
t}(0,0)\right\rVert_{L^{2}}+2\left\lVert\overline{\kappa}\right\rVert_{H^{1}}\left\lVert\kappa\right\rVert_{H^{1}}^{2}\bigg{)}\leq
c\varepsilon^{3}\,.$

Integration by parts yields the swimming expression (26).

We now consider the equation (65) satisfied by $\overline{\kappa}^{\rm lin}$.
We aim to use $\overline{\kappa}^{\rm lin}$, along with Lemma 3.3, to better
understand when expression (26) leads to nonnegligible net motion over one
period in time.

Defining $\omega=\frac{2\pi}{T}$, we may first expand $\overline{\kappa}^{\rm
lin}(s,t)$ and $\kappa_{0}(s,t)$ as a Fourier series in time:

$\kappa_{0}=\sum_{m=1}^{\infty}A_{m}(s)\cos(\omega m\,t)-B_{m}(s)\sin(\omega
m\,t)\,,\quad\overline{\kappa}^{\rm
lin}=\sum_{m=1}^{\infty}C_{m}(s)\cos(\omega m\,t)-D_{m}(s)\sin(\omega
m\,t)\,.$ (73)

Then, using equation (65), the coefficients $A_{m}$, $B_{m}$, $C_{m}$, and
$D_{m}$ satisfy

$\displaystyle C_{m}$ $\displaystyle=-\frac{1}{\omega
m}(D_{m})_{ssss}-A_{m}\,,\qquad D_{m}=\frac{1}{\omega
m}(C_{m})_{ssss}-B_{m}\,.$

Further expanding $A_{m}$, $B_{m}$, $C_{m}$, and $D_{m}$ in eigenfunctions of
$\mathcal{L}$ (30), i.e.

$\displaystyle A_{m}$
$\displaystyle=\sum_{k=1}^{\infty}a_{m,k}\psi_{k}(s)\,,\;B_{m}=\sum_{k=1}^{\infty}b_{m,k}\psi_{k}(s)\,,\;C_{m}=\sum_{k=1}^{\infty}c_{m,k}\psi_{k}(s)\,,\;D_{m}=\sum_{k=1}^{\infty}d_{m,k}\psi_{k}(s)\,,$

we may solve for the coefficients $c_{m,k}$, $d_{m,k}$ in terms of $a_{m,k}$,
$b_{m,k}$ as

$c_{m,k}=\frac{\omega^{2}m^{2}}{\omega^{2}m^{2}+\lambda_{k}^{2}}\bigg{(}\frac{\lambda_{k}}{\omega
m}b_{m,k}-a_{m,k}\bigg{)}\,,\quad
d_{m,k}=\frac{\omega^{2}m^{2}}{\omega^{2}m^{2}+\lambda_{k}^{2}}\bigg{(}-\frac{\lambda_{k}}{\omega
m}a_{m,k}-b_{m,k}\bigg{)}\,.$ (74)

Here $\lambda_{k}=\xi_{k}^{4}$ are the eigenvalues of $\mathcal{L}$, given by
(29). Inserting the above expansions into the swimming expression (26) (and
replacing $\overline{\kappa}$ with $\overline{\kappa}^{\rm lin}$, by Lemma
3.3), we may write

$\displaystyle-\gamma$
$\displaystyle\int_{0}^{1}\langle(\kappa_{0})_{s}\overline{\kappa}^{\rm
lin}\rangle\,ds=-\gamma\sum_{m,k,\ell=1}^{\infty}\bigg{(}\frac{1}{2}a_{m,\ell}c_{m,k}+\frac{1}{2}b_{m,\ell}d_{m,k}\bigg{)}\int_{0}^{1}\psi_{k}(\psi_{\ell})_{s}\,ds$
$\displaystyle=\frac{\gamma}{2}\sum_{m,k,\ell=1}^{\infty}\frac{\omega^{2}m^{2}}{\omega^{2}m^{2}+\lambda_{k}^{2}}\bigg{(}\frac{\lambda_{k}}{\omega
m}\big{(}a_{m,k}b_{m,\ell}-b_{m,k}a_{m,\ell}\big{)}+a_{m,k}a_{m,\ell}+b_{m,k}b_{m,\ell}\bigg{)}\int_{0}^{1}\psi_{k}(\psi_{\ell})_{s}\,ds\,.$

∎

##  4\. Observations and numerics

In this section, we begin by exploring the observations 1.5 about the swimming
speed predicted by the expression (27). We proceed to perform a small
numerical optimization of the swimming speed subject to a fixed amount of work
performed by the filament. Finally, we outline a numerical method for a
reformulation of equation (2) and verify the observations numerically.

###  4.1. Observations

We first consider the form of the eigenfunctions $\psi_{k}$ of $\mathcal{L}$
(see (30)). Since $\psi_{2k}(s)$ is odd and $\psi_{2k-1}(s)$ is even about
$s=\frac{1}{2}$ for each $k=1,2,3,\dots$, we have that eigenfunctions with the
same parity do not contribute to net motion:

$\int_{0}^{1}\psi_{2k}(\psi_{2\ell})_{s}\,ds=0\,,\quad\int_{0}^{1}\psi_{2k-1}(\psi_{2\ell-1})_{s}\,ds=0\,,\qquad
k,\ell=1,2,3,\dots\,.$ (75)

If $\kappa_{0}(s,t)$ is always odd about $s=\frac{1}{2}$, we may write

$\displaystyle\kappa_{0}=\sum_{m=1}^{\infty}\sum_{k=1}^{\infty}\big{(}a_{m,2k}\cos(\omega
m\,t)+b_{m,2k}\sin(\omega m\,t)\big{)}\psi_{2k}(s)\,.$

Likewise, if $\kappa_{0}(s,t)$ is always even about $s=\frac{1}{2}$, we may
write

$\displaystyle\kappa_{0}=\sum_{m=1}^{\infty}\sum_{k=1}^{\infty}\big{(}a_{m,2k-1}\cos(\omega
m\,t)+b_{m,2k-1}\sin(\omega m\,t)\big{)}\psi_{2k-1}(s)\,.$

In either of the above cases, due to (75), we have that the leading order term
of the swimming expression (27) satisfies

$\displaystyle-\gamma\int_{0}^{1}\langle(\kappa_{0})_{s}\overline{\kappa}^{\rm
lin}\rangle\,ds=0\,.$

Furthermore, writing

$\displaystyle\kappa_{0}=\sum_{m=1}^{\infty}A_{m}(s)\cos(\omega
m\,t)+B_{m}(s)\sin(\omega m\,t)\,,$

we see that if $A_{m}=\pm B_{m}$ or either $A_{m}=0$ or $B_{m}=0$ for all $m$,
then the first term $b_{m,k}a_{m,\ell}-a_{m,k}b_{m,\ell}$ in the swimming
expression (27) vanishes. Recalling that the eigenvalues
$\lambda_{k}=\xi_{k}^{4}$ of $\mathcal{L}$ satisfy
$\xi_{k}\to\frac{(2k+1)\pi}{2}$ as $k\to\infty$ and that the smallest
eigenvalue $\lambda_{1}\approx(4.73)^{4}\approx 500$ (see (30)), we note that
this initial term has the greatest relative contribution to filament
propulsion. (The remaining terms $a_{m,k}a_{m,\ell}+b_{m,k}b_{m,\ell}$ become
relatively more important for large $m$, but the optimimzation results below
prompt us to consider only $m$ small). In particular, if this initial term
vanishes, the fiber may still have nonzero net motion over the course of one
period, but we expect its displacement to be very small.

We may also consider optimizing the swimming speed (27) given certain
constraints on the filament deformation, such as a fixed amount of work [51,
25]. We begin by calculating the average work done by the swimmer over one
period. Starting with the formulation (2), we have

$\displaystyle W$
$\displaystyle=\int_{0}^{1}\bigg{\langle}\frac{\partial\bm{X}}{\partial
t}\cdot\big{(}\bm{X}_{sss}-\tau\bm{X}_{s}-(\kappa_{0})_{s}\bm{e}_{\rm
n}\big{)}_{s}\bigg{\rangle}\,ds=-\int_{0}^{1}\bigg{\langle}\frac{\partial\bm{X}_{s}}{\partial
t}\cdot\big{(}\bm{X}_{sss}-(\kappa_{0})_{s}\bm{e}_{\rm
n}\big{)}\bigg{\rangle}\,ds$
$\displaystyle=-\int_{0}^{1}\langle\dot{\theta}(\kappa-\kappa_{0})_{s}\rangle\,ds=\int_{0}^{1}\langle\dot{\kappa}\overline{\kappa}\rangle\,ds=\int_{0}^{1}\langle\dot{\kappa}_{0}\overline{\kappa}\rangle\,ds\,.$

Here we have used that $\overline{\kappa}=\kappa-\kappa_{0}$ is $T$-periodic.
Using Lemma 3.3, we have that

$\displaystyle|W-W^{\rm lin}|\leq c\varepsilon^{3}\,,$

where, using the expansion (73) and the form of the coefficients (74), we may
write

$\displaystyle W^{\rm lin}$
$\displaystyle=\int_{0}^{1}\langle\dot{\kappa}_{0}\overline{\kappa}^{\rm
lin}\rangle\,ds\,dt=\sum_{m,k,\ell=1}^{\infty}\frac{\omega
m}{2}\bigg{(}b_{m,\ell}c_{m,k}-a_{m,\ell}d_{m,k}\bigg{)}\int_{0}^{1}\psi_{k}\psi_{\ell}\,ds$
(76)
$\displaystyle=\sum_{m,k=1}^{\infty}\frac{\lambda_{k}}{2}\frac{\omega^{2}m^{2}}{\omega^{2}m^{2}+\lambda_{k}^{2}}\big{(}a_{m,k}^{2}+b_{m,k}^{2}\big{)}\,.$

Here we seek to numerically optimize the leading order swimming speed (27)
over a (small) finite number of spatial and temporal modes $a_{m,k}$,
$b_{m,k}$, subject to a fixed amount of work (76). For a given $m_{\max}$,
$k_{\max}$, we aim to construct the preferred curvature

$\displaystyle\kappa_{0}(s)=\sum_{m=1}^{m_{\max}}\sum_{k=1}^{k_{\max}}\big{(}a_{m,k}\cos(\omega
m\,t)-b_{m,k}\sin(\omega m\,t)\big{)}\psi_{k}(s)$

which results in the fastest swimming speed. Note that we do not necessarily
expect the true optimal preferred curvature to be representable by a finite
number of spatial modes $k$, as we cannot construct functions with
nonhomogeneous boundary conditions using finitely many eigenfunctions
$\psi_{k}$ of $\mathcal{L}$. We instead aim to visualize general trends in the
emerging optimal $\kappa_{0}$.

We define the finite-mode swimming speed and work

$\displaystyle U_{m_{\max},k_{\max}}$
$\displaystyle:=\frac{\gamma}{2}\sum_{m=1}^{m_{\max}}\sum_{k,\ell=1}^{k_{\max}}\frac{\omega^{2}m^{2}}{\omega^{2}m^{2}+\lambda_{k}^{2}}\bigg{(}\frac{\lambda_{k}}{\omega
m}\big{(}a_{m,k}b_{m,\ell}-b_{m,k}a_{m,\ell}\big{)}$ $\displaystyle\hskip
142.26378pt+a_{m,k}a_{m,\ell}+b_{m,k}b_{m,\ell}\bigg{)}\int_{0}^{1}\psi_{k}(\psi_{\ell})_{s}\,ds$
(77) $\displaystyle W_{m_{\max},k_{\max}}$
$\displaystyle:=\sum_{m=1}^{m_{\max}}\sum_{k=1}^{k_{\max}}\frac{\lambda_{k}}{2}\frac{\omega^{2}m^{2}}{\omega^{2}m^{2}+\lambda_{k}^{2}}\big{(}a_{m,k}^{2}+b_{m,k}^{2}\big{)}$
(78)

and solve the following constrained optimization problem for the coefficients
$a_{m,k}$, $b_{m,k}$:

$\displaystyle\min_{a_{m,k},b_{m,k}}$ $\displaystyle U_{m_{\max},k_{\max}}$
(79) subject to $\displaystyle W_{m_{\max},k_{\max}}={\rm const.}$

Note that we are seeking the fastest swimming speed in the negative direction
with respect to the unit tangent vector $\bm{e}_{\rm t}(0,0)=\bm{e}_{x}$, i.e.
leftward. The optimization is performed using Matlab’s fmincon starting from
random initial $a_{m,k}$, $b_{m,k}$ satisfying the constraint, where
$\text{const.}=1$. We use $m_{\max}=3$ and $k_{\max}=10$. For each $m=1,2,3$,
the optimal temporal eigenmodes
$A_{m}(s)=\sum_{k=1}^{k_{\max}}a_{m,k}\psi_{k}(s)$ and
$B_{m}(s)=\sum_{k=1}^{k_{\max}}b_{m,k}\psi_{k}(s)$ are plotted in Figure 2(a).
As in [24, Section B], we find that forcing only the lowest temporal mode
$m=1$ leads to the optimal swimming speed for a given amount of work. Notably,
the optimal $A_{1}(s)$ is perfectly odd about $s=\frac{1}{2}$ while $B_{1}(s)$
is perfectly even. The phase difference between $A_{1}(s)$ and $B_{1}(s)$ is
consistent with the classical optimization paper [45] which finds that, to
reach a given swimming speed while minimizing energy expenditure, the filament
should deform as a traveling wave.

(a)

(b)

(c)

Figure 2. (A) For the optimization problem (79), the optimal temporal
eigenmodes $A_{m}(s)=\sum_{k=1}^{k_{\max}}a_{m,k}\psi_{k}(s)$ and
$B_{m}(s)=\sum_{k=1}^{k_{\max}}b_{m,k}\psi_{k}(s)$ for $k_{\max}=10$ are
plotted for each $m=1,2,3$. Forcing only the lowest temporal mode $m=1$ leads
to the fastest swimming speed for a given amount of work. (B)–(C) When the
bending energy $\left\lVert\kappa_{0}\right\rVert_{L^{2}}$ is also constrained
(see (80)), the optimal eigenmodes $A_{1}$ and $B_{1}$ are far less
oscillatory. Here we plot two different optima for $k_{\max}=12$ which both
yield the same swimming speed of $-0.1008$.

Note that for small $m$, the expression for average work (76) is essentially
the $H^{-4}$ norm of $\kappa_{0}$ along the length of the filament, due to the
$\sim\lambda_{k}^{-1}$ coefficient of the summand. Thus, given a fixed amount
of work, we expect the optimal $\kappa_{0}$ to oscillate at the highest
allowable frequency in $s$, as seen in Figure 2(a). Constraining only the work
done by the filament thus may not lead to the most physically realistic
optimum. As noted in [28, 51], the bending energy of the filament can also be
taken into account.

Thus we also consider the additional constraint
$\left\lVert\kappa_{0}\right\rVert_{L^{2}}=$ const., and solve the following
optimization problem for the coefficients $a_{m,k}$, $b_{m,k}$:

$\displaystyle\min_{a_{m,k},b_{m,k}}$ $\displaystyle U_{m_{\max},k_{\max}}$
(80) subject to $\displaystyle W_{m_{\max},k_{\max}}=1$
$\displaystyle\sum_{m,k}a_{m,k}^{2}=\sum_{m,k}b_{m,k}^{2}=1\,.$

Due to the results of optimization problem (79), we take $m_{\max}=1$ and
optimize only over spatial modes $k$. Two different results of the
optimization for $k_{\max}=12$ are plotted in Figures 2(b) and 2(c). Both
yield the same swimming speed $U_{m_{\max},k_{\max}}=-0.1008$.

Case (B) is similar to a simple sine and cosine, i.e. perhaps the most
familiar traveling wave example. However, since $A_{1}$ and $B_{1}$ are
constructed of finitely many eigenfunctions $\psi_{k}(s)$ (see (30)), they are
constrained to have homogeneous boundary conditions. We also note the striking
symmetry in the solutions of case (C).

In the next section we try swimming using both of the optimal curvatures
constructed in Figures 2(b) and 2(c) in order to visualize their performance.
We also explore other forms of $\kappa_{0}(s,t)$ to verify the observations
about the swimming expression (27) noted above.

###  4.2. Numerical method

Here we numerically verify the swimming predictions of the previous section.
We develop a simple numerical method for simulating the motion of an
inextensible filament via resistive force theory based on a combination of the
methods of [32] and [35]. We apply the method to the planar swimmer studied in
the previous sections, but note that the formulation itself is applicable more
generally.

While the curvature evolution equations (7)-(9) in terms of $\kappa$ and
$\tau$ are convenient for analysis, a direct numerical implementation can be
cumbersome due in part to complicated dependence on the forcing $\kappa_{0}$
as well as the need to solve for the actual fiber movement separately (22).
Numerically, however, we can make use of reformulations that may simplify the
computation. As noted, our future goal is to do numerical analysis comparing
the many existing formulations of filament inextensibility.

To introduce the numerical method, we return to the original formulation of
inextensible fiber dynamics (2), which we reiterate here for convenience:

$\displaystyle\frac{\partial\bm{X}}{\partial t}(s,t)$
$\displaystyle=-\big{(}{\bf I}+\gamma\bm{X}_{s}\bm{X}_{s}^{\rm
T}\big{)}\big{(}\bm{X}_{sss}-\tau\bm{X}_{s}-(\kappa_{0})_{s}\bm{e}_{\rm
n}\big{)}_{s}$ (81) $\displaystyle\left\lvert\bm{X}_{s}\right\rvert^{2}$
$\displaystyle=1$ $\displaystyle(\bm{X}_{ss}-\kappa_{0}\bm{e}_{\rm
n})\big{|}_{s=0,1}$
$\displaystyle=0\,,\quad(\bm{X}_{sss}-\tau\bm{X}_{s}-(\kappa_{0})_{s}\bm{e}_{\rm
n})\big{|}_{s=0,1}=0\,.$

We enforce inextensibility directly by parameterizing $\bm{X}(s,t)$ as

$\bm{X}(s,t)=\bm{X}_{0}(t)+\int_{0}^{s}\bm{e}_{\rm
t}(s^{\prime},t)ds^{\prime}\,,$ (82)

where $\bm{e}_{\rm t}\in S^{2}$; in particular, $\dot{\bm{e}}_{\rm
t}\cdot\bm{e}_{\rm t}=0$.

Along the fiber we may rewrite (81) as the coupled system

$\displaystyle\dot{\bm{X}}_{0}+\int_{0}^{s}\dot{\bm{e}}_{\rm
t}(s^{\prime})\,ds^{\prime}$ $\displaystyle=-({\bf I}+\gamma\bm{e}_{\rm
t}\bm{e}_{\rm t}^{\rm T})\bm{h}(s)$ (83) $\displaystyle\big{(}{\bf
I}-\bm{e}_{\rm t}(s)\bm{e}_{\rm t}(s)^{\rm
T}\big{)}\int_{0}^{s}\bm{h}(s^{\prime})ds^{\prime}$ $\displaystyle=\big{(}{\bf
I}-\bm{e}_{\rm t}(s)\bm{e}_{\rm t}(s)^{\rm
T}\big{)}\big{(}\bm{X}_{sss}-(\kappa_{0})_{s}\bm{e}_{\rm n}\big{)}\,,$ (84)

where $\bm{h}(s)$ is the hydrodynamic force density along the filament, and
only the normal components of the integrated force density (84) are specified.
To satisfy the boundary conditions of (81), we must also ensure that the total
hydrodynamic force and torque along the fiber vanish:

$\int_{0}^{1}\bm{h}(s)\,ds=0,\qquad\int_{0}^{1}(\bm{X}(s)-\bm{X}(0))\times\bm{h}(s)\,ds=0\,.$
(85)

The formulation (83)-(84), combined with total force vanishing (85), is
equivalent to the formulation in [32], which uses slender body theory rather
than resistive force theory in equation (83). As noted in [32], equations
(83)-(84) along with (85) are enough to recover the rate of work exerted by
the filament on the fluid in (81). In particular, the rate of work may be
calculated using (83)-(84) as

$\displaystyle\dot{W}=\int_{0}^{1}\bm{h}(s)\cdot\bigg{(}\dot{\bm{X}}_{0}+\int_{0}^{s}\dot{\bm{e}}_{\rm
t}(s^{\prime})\,ds^{\prime}\bigg{)}ds$
$\displaystyle=-\int_{0}^{1}\bigg{(}\int_{0}^{s}\bm{h}(s^{\prime})ds^{\prime}\bigg{)}\cdot\dot{\bm{e}}_{\rm
t}(s)\,ds$
$\displaystyle=-\int_{0}^{1}(\bm{X}_{sss}-(\kappa_{0})_{s}\bm{e}_{\rm
n})\cdot\dot{\bm{e}}_{\rm t}(s)\,ds\,,$

where we have integrated by parts and used that $\bm{e}_{\rm
t}\cdot\dot{\bm{e}}_{\rm t}=0$. Starting instead from (81) and again
integrating by parts and using $\bm{X}_{s}=\bm{e}_{\rm
t}\perp\dot{\bm{e}}_{\rm t}$, we also obtain

$\displaystyle\dot{W}=\int_{0}^{1}\big{(}\bm{X}_{sss}-\tau\bm{X}_{s}-(\kappa_{0})_{s}\bm{e}_{\rm
n}\big{)}_{s}\cdot\frac{\partial\bm{X}}{\partial
t}\,ds=-\int_{0}^{1}\big{(}\bm{X}_{sss}-(\kappa_{0})_{s}\bm{e}_{\rm
n}\big{)}\cdot\dot{\bm{e}}_{\rm t}\,ds\,.$

Since we are using resistive force theory, equation (83) may be directly
inverted to solve for $\bm{h}$:

$\bm{h}=-({\bf I}-\frac{\gamma}{1+\gamma}\bm{e}_{\rm t}\bm{e}_{\rm t}^{\rm
T})\bigg{(}\dot{\bm{X}}_{0}+\int_{0}^{s}\dot{\bm{e}}_{\rm
t}(s^{\prime})\,ds^{\prime}\bigg{)}\,,$ (86)

and this expression may be inserted into (84) to condense (83) and (84) into a
single equation along the filament. For a planar fiber, recalling that we may
parameterize $\bm{e}_{\rm t}=(\cos\theta,\sin\theta)^{\rm T}$, $\bm{e}_{\rm
n}=(-\sin\theta,\cos\theta)^{\rm T}$, and that
$\bm{X}_{sss}=-(\theta_{s})^{2}\bm{e}_{\rm t}+\theta_{ss}\bm{e}_{\rm n}$, we
may rewrite (83) and (84) as

$\bm{e}_{\rm n}(s,t)\cdot\int_{0}^{s}({\bf
I}-\frac{\gamma}{1+\gamma}\bm{e}_{\rm t}\bm{e}_{\rm t}^{\rm
T})\bigg{(}\dot{\bm{X}}_{0}+\int_{0}^{s^{\prime}}\dot{\bm{e}}_{\rm
t}(\overline{s})\,d\overline{s}\bigg{)}ds^{\prime}=-\theta_{ss}+(\kappa_{0})_{s}\,.$
(87)

Equation (87) will be the main evolution equation for the fiber, for which the
unknowns are the basepoint $\bm{X}_{0}$ and the tangent angle $\theta(s)$. We
combine (87) with the total force balance (85) to enforce the boundary
condition $(-\theta_{ss}+(\kappa_{0})_{s})\big{|}_{s=1}=0$:

$\int_{0}^{1}({\bf I}-\frac{\gamma}{1+\gamma}\bm{e}_{\rm t}\bm{e}_{\rm t}^{\rm
T})\bigg{(}\dot{\bm{X}}_{0}+\int_{0}^{s}\dot{\bm{e}}_{\rm
t}(s^{\prime})\,ds^{\prime}\bigg{)}\,ds=0.$ (88)

The torque condition in (85) leads to the boundary condition
$(\theta_{s}-\kappa_{0})\big{|}_{s=0,1}=0$, which will be enforced directly in
the discretization of $\theta_{ss}$ on the right hand side of (87).

The fiber is discretized into $N$ segments between $N+1$ points $\bm{X}_{i}$
along the fiber, and $\theta_{i}$, $i=1,\dots,N$, is taken to be the angle
between segment $i$ and the $x$-axis (see Figure 3). The equation (87) is
enforced at the midpoint of each segment at
$\bm{X}_{i-\frac{1}{2}}:=\frac{\bm{X}_{i-1}+\bm{X}_{i}}{2}$, $i=1,\dots,N$.

Figure 3.  Filament discretization used to implement (90)–(93).

As mentioned, the boundary condition
$(\theta_{s}-\kappa_{0})\big{|}_{s=0,1}=0$ is enforced in the discretization
of $\theta_{ss}$ on the right hand side of (87) via the approximation

$\theta_{ss}\big{|}_{s=0}\approx
N^{2}(2\theta_{2}-2\theta_{1})-2N\kappa_{0,1}\,,\qquad\theta_{ss}\big{|}_{s=1}\approx
N^{2}(2\theta_{N-1}-2\theta_{N})+2N\kappa_{0,N}\,.$ (89)

At the $s=1$ endpoint, equations (87) and (88) coincide to yield the boundary
condition $\theta_{ss}\big{|}_{s=1}=(\kappa_{0})_{s}\big{|}_{s=1}$, which,
using (89), yields an equation for $\theta_{N}$:

$\theta_{N}=\theta_{N-1}+\frac{1}{N}\kappa_{0,N}-\frac{1}{2N^{2}}(\kappa_{0})_{s,N}\,.$
(90)

The full discretized system is then given by

$\displaystyle\frac{1}{N}\begin{pmatrix}-\sin\theta_{j}\\\
\cos\theta_{j}\end{pmatrix}\cdot\sum_{i=1}^{j}\bm{M}_{\rm
RFT}(\theta_{i})\dot{\bm{X}}_{i-\frac{1}{2}}$
$\displaystyle=-N^{2}(\theta_{j-1}-2\theta_{j}+\theta_{j+1})+(\kappa_{0})_{s,j}\,,\quad
j=2,\dots,N-1$ (91) $\displaystyle\frac{1}{N}\begin{pmatrix}-\sin\theta_{1}\\\
\cos\theta_{1}\end{pmatrix}\cdot\bm{M}_{\rm
RFT}(\theta_{1})\dot{\bm{X}}_{\frac{1}{2}}$
$\displaystyle=-N^{2}(2\theta_{2}-2\theta_{1})+2N\kappa_{0,1}$ (92)
$\displaystyle\frac{1}{N}\sum_{i=1}^{N-1}\bm{M}_{\rm
RFT}(\theta_{i})\dot{\bm{X}}_{i-\frac{1}{2}}$ $\displaystyle=0\,,$ (93)

along with equation (90). Here the evolution $\dot{\bm{X}}_{i-\frac{1}{2}}$ of
each segment midpoint is parameterized as

$\displaystyle\dot{\bm{X}}_{i-\frac{1}{2}}=\begin{pmatrix}\dot{x}_{0}\\\
\dot{y}_{0}\end{pmatrix}+\frac{1}{2N}\begin{pmatrix}-\sin\theta_{i}\\\
\cos\theta_{i}\end{pmatrix}\dot{\theta}_{i}+\frac{1}{N}\sum_{k=1}^{i}\begin{pmatrix}-\sin\theta_{k}\\\
\cos\theta_{k}\end{pmatrix}\dot{\theta}_{k}\,,\quad i=1,\dots,N\,,$

and the $2N\times 2N$ matrix $\bm{M}_{\rm RFT}(\theta_{i})$ is given by

$\displaystyle\bm{M}_{\rm
RFT}(\theta_{i})=\begin{pmatrix}1-\frac{\gamma}{1+\gamma}\cos^{2}\theta_{i}&-\frac{\gamma}{1+\gamma}\cos\theta_{i}\sin\theta_{i}\\\
-\frac{\gamma}{1+\gamma}\cos\theta_{i}\sin\theta_{i}&1-\frac{\gamma}{1+\gamma}\sin^{2}\theta_{i}\end{pmatrix}\,.$

The expressions (91) and (92) include $N-1$ equations enforced at
$\bm{X}_{\frac{1}{2}},\dots,\bm{X}_{N-\frac{3}{2}}$; the expression (93)
involves two equations enforced at $\bm{X}_{N-\frac{1}{2}}$; and the equation
(90) gives one additional equation to uniquely determine the $N+2$ unknowns
$x_{0}$, $y_{0}$, $\theta_{1}$, …, $\theta_{N}$. We use a standard ODE solver
to evolve (90)–(93) in time.

The formulation (90)–(93) is very similar to that of Moreau et al. [35], which
also avoids the need to solve for the unknown tension by projecting away from
the tangential direction along the fiber. At the continuous level, the
formulation in [35] is essentially based on the identity

$\bm{X}_{s}\times(\bm{X}_{sss}-\tau(s)\bm{X}_{s}-(\kappa_{0})_{s}\bm{e}_{\rm
n})=\big{(}\bm{X}_{s}\times(\bm{X}_{ss}-\kappa_{0}\bm{e}_{\rm
n})\big{)}_{s}\,.$ (94)

Integrating (94) with respect to arclength from $s$ to 1 ($0\leq s\leq 1$) and
using the boundary conditions in (81), the identity (94) yields

$\displaystyle\bm{X}_{s}(s)\times(\bm{X}_{ss}(s)-\kappa_{0}(s)\bm{e}_{\rm n})$
$\displaystyle=-\int_{s}^{1}\bm{X}_{s^{\prime}}(s^{\prime})\times\big{(}\bm{X}_{sss}(s^{\prime})-\tau(s^{\prime})\bm{X}_{s}(s^{\prime})-\kappa_{0}(s^{\prime})\bm{e}_{\rm
n}\big{)}\,ds^{\prime}$ (95)
$\displaystyle=\int_{s}^{1}\big{(}\bm{X}(s^{\prime})-\bm{X}(s)\big{)}\times\big{(}\bm{X}_{sss}-\tau\bm{X}_{s}-\kappa_{0}\bm{e}_{\rm
n}\big{)}_{s^{\prime}}\,ds^{\prime}\,.$

In the case of a planar filament parameterized by the tangent angle $\theta$,
we have $\bm{X}_{s}\times\bm{X}_{ss}=\theta_{s}\bm{e}_{z}$. Combining (95)
with the definition of $\bm{h}$ (86), we obtain (in the $\bm{e}_{z}$
direction):

$\displaystyle\theta_{s}(s)-\kappa_{0}(s)$
$\displaystyle=\int_{s}^{1}\big{(}\bm{X}(s^{\prime})-\bm{X}(s)\big{)}\times\bm{h}(s^{\prime})\,ds^{\prime}$
(96)
$\displaystyle=-\int_{s}^{1}\big{(}\bm{X}(s^{\prime})-\bm{X}(s)\big{)}\times\bigg{(}({\bf
I}-\frac{\gamma}{1+\gamma}\bm{e}_{\rm t}\bm{e}_{\rm t}^{\rm
T})\bigg{(}\dot{\bm{X}}_{0}+\int_{0}^{s^{\prime}}\dot{\bm{e}}_{\rm
t}(\overline{s})\,d\overline{s}\bigg{)}\bigg{)}\,ds^{\prime}\,.$

The equation (96) is also enforced at each of the $N$ midpoints
$\bm{X}_{i-\frac{1}{2}}$ along the discretized fiber (see Figure 3). To close
the discretized system, equation (96) is accompanied by the total force
constraint

$\int_{0}^{1}({\bf I}-\frac{\gamma}{1+\gamma}\bm{e}_{\rm t}\bm{e}_{\rm t}^{\rm
T})\bigg{(}\dot{\bm{X}}_{0}+\int_{0}^{s}\dot{\bm{e}}_{\rm
t}(s^{\prime})\,ds^{\prime}\bigg{)}\,ds=0\,,$ (97)

which gives two additional equations.

In Appendix A, we validate the method (90)–(93) against a direct
discretization of the classical expression (81). A more thorough comparison of
different numerical implementations of fiber inextensibility is the intended
subject of future work.

###  4.3. Swimmer numerics

We use the numerical method (90)–(93) to test the swimming predictions from
Section 4.1. In each simulation, the initial condition is a straight fiber
lying on the $x$-axis from $x=0$ to $x=1$. We discretize the fiber into
$N=100$ segments and prescribe a preferred curvature of the form

$\displaystyle\kappa_{0}(s,t)=F_{1}(s)\cos(\omega\,t)+F_{2}(s)\sin(\omega\,t)\,,\qquad\left\lVert
F_{1}\right\rVert_{L^{2}(I)}=\left\lVert F_{2}\right\rVert_{L^{2}(I)}=1\,.$

Temporal forcing is only at the lowest mode, in accordance with the
conclusions of the previous section. We take $\gamma=1$ and $\omega=2\pi$, and
consider different waveforms $F_{1}(s)$ and $F_{2}(s)$.

Non-swimmers: We begin by considering combinations of $F_{1}$ and $F_{2}$
which are predicted to produce no net motion over each time period. In
accordance with the observations of Section 4.1, we consider examples where
$F_{1}$ and $F_{2}$ are either both even or both odd about $s=\frac{1}{2}$.
Specifically, we take

  1. (1)

$F_{1}=\cos(4\pi s)$, $F_{2}=\cos(2\pi s)$ (both even about $s=\frac{1}{2}$)

  2. (2)

$F_{1}=\sin(4\pi s)$, $F_{2}=\sin(2\pi s)$ (both odd about $s=\frac{1}{2}$)

and normalize both such that $\left\lVert
F_{1}\right\rVert_{L^{2}}=\left\lVert F_{2}\right\rVert_{L^{2}}=1$. The two
combinations of $F_{1}$ and $F_{2}$ are plotted in Figure 4, along with
snapshots of the spatial positions of both swimmer (1) and swimmer (2) over
the course of 50 time units. As predicted, neither swimmer exhibits net
displacement in the $x$-direction.

(a)

(b)

(c)

Figure 4. (A) Plot of the two combinations of non-swimming $F_{1}$ and $F_{2}$
considered here. (B) Snapshots of the swimmer position in case (1) over the
course of 50 time units. (C) Snapshots of the swimmer position in case (2)
over the course of 50 time units.

Bad swimmers: We next consider swimmers predicted in Section 4.1 to possibly
swim, but very poorly. We choose $F_{2}(s)$ to be neither even nor odd about
$s=\frac{1}{2}$, and consider $F_{1}=\pm F_{2}$ and $F_{1}=0$:

  1. (3)

$F_{1}=\cos(2\pi s)+\sin(2\pi s)$, $F_{2}=F_{1}$

  2. (4)

$F_{1}=0$, $F_{2}=\cos(2\pi s)+\sin(2\pi s)$

  3. (5)

$F_{1}=\cos(2\pi s)+\sin(2\pi s)$, $F_{2}=-F_{1}$ .