Solvability of the Stokes Immersed Boundary Problem in Two Dimensions

TL;DR

This paper establishes local and global existence and uniqueness results for the two-dimensional Stokes immersed boundary problem, showing convergence to equilibrium for near-circular initial configurations.

Contribution

It introduces a contour dynamic formulation for the problem and proves well-posedness and exponential convergence to equilibrium, extending potential methods to three dimensions.

Findings

Local-in-time existence and uniqueness for arbitrary initial configurations.

Global-in-time convergence to equilibrium near circular configurations.

Method potentially applicable to three-dimensional problems.

Abstract

We study coupled motion of a 1-D closed elastic string immersed in a 2-D Stokes flow, known as the Stokes immersed boundary problem in two dimensions. Using the fundamental solution of the Stokes equation and the Lagrangian coordinate of the string, we write the problem into a contour dynamic formulation, which is a nonlinear non-local equation solely keeping track of evolution of the string configuration. We prove existence and uniqueness of local-in-time solution starting from an arbitrary initial configuration that is an $H^{5/2}$-function in the Lagrangian coordinate satisfying the so-called well-stretched assumption. We also prove that when the initial string configuration is sufficiently close to an equilibrium, which is an evenly parameterized circular configuration, then global-in-time solution uniquely exists and it will converge to an equilibrium configuration exponentially as $t\rightarrow +\infty$. The technique in this paper may also apply to the Stokes immersed boundary problem in three dimensions.

Equations

$$\begin{array}{rl}& -{\mu }_0\Delta u+\nabla p=f(x,t),x\in {\mathbb{R}}^2,t>0,\\ & \mathrm{div}u=0,\\ & |u|,|p|\to 0\text{mbox}as|x|\to \infty .\end{array}$$

$$f(x,t)={\int }_{\mathbb{T}}F(s,t)\delta (x-X(s,t))ds.$$

$$F(s,t)=\frac{\partial }{\partial s}\left(\mathcal{T}(|X_s|)\frac{X_s}{|X_s|}\right),\mathcal{T}(|v|)={\mathcal{E}}^{\prime }(|v|).$$

$$\mathcal{E}(|v|)=k_0|v{|}^2/2.$$

$$\frac{\partial X}{\partial t}(s,t)=u(X(s,t),t),$$

$$|X(s_1,t)-X(s_2,t)|\ge \lambda |s_1-s_2|,$$

$$X_t(s,t)=\mathcal{L}X(s,t)+g_X(s,t),X(s,0)=X_0(s),$$

$$g_X(s,t)=$$

$${\Gamma }_0(s,s^{\prime },t)=$$

$$(-\Delta {)}^{1/2}Y(s)\triangleq -\frac{1}{\pi }p.v.{\int }_{\mathbb{T}}\frac{Y(s^{\prime })-Y(s)}{4{\sin }^2\left(\frac{s^{\prime }-s}{2}\right)}d{s}^{\prime },$$

$$G(x)=\frac{1}{4\pi }\left(-\ln |x|Id+\frac{x\otimes x}{|x{|}^2}\right)$$

$${\Omega }_T=\left\{Y(s,t)\in L_T^{\infty }{H}^{5/2}\cap L_T^2{H}^3(\mathbb{T}):Y_t(s,t)\in L_T^2{H}^2(\mathbb{T})\right\}.$$

$$\Vert Y(s,t){\Vert }_{{\Omega }_T}\triangleq \Vert Y{\Vert }_{L_T^{\infty }{H}^{5/2}(\mathbb{T})}+\Vert Y{\Vert }_{L_T^2{H}^3(\mathbb{T})}+\Vert Y_t{\Vert }_{L_T^2{H}^2(\mathbb{T})}.$$

$$|X_0(s_1)-X_0(s_2)|\ge \lambda |s_1-s_2|,\forall s_1,s_2\in \mathbb{T}.$$

$$\Vert X{\Vert }_{L_{T_0}^{\infty }{\dot{H}}^{5/2}\cap L_{T_0}^2{\dot{H}}^3(\mathbb{T})}\le 4\Vert X_0{\Vert }_{{\dot{H}}^{5/2}(\mathbb{T})},\Vert X_t{\Vert }_{L_{T_0}^2{\dot{H}}^2(\mathbb{T})}\le \Vert X_0{\Vert }_{{\dot{H}}^{5/2}(\mathbb{T})},$$

$$\left|X(s_1,t)-X(s_2,t)\right|\ge \frac{\lambda }{2}|s_1-s_2|.$$

$$|X(s_1,t)-X(s_2,t)|\ge c\lambda |s_1-s_2|.$$

$$\frac{1}{2}{\int }_{\mathbb{T}}Y(s)\times Y^{\prime }(s)ds=\pi R_Y^2.$$

$$Y_{\theta ,x}(s)=(R_Y\cos (s+\theta ),R_Y\sin (s+\theta ){)}^T+x$$

$$({\theta }_{\ast },x_{\ast })=\underset{\theta \in [0,2\pi ),x\in {\mathbb{R}}^2}{\mathrm{argmin}}{\int }_{\mathbb{T}}|Y(s)-Y_{\theta ,x}(s){|}^{2}ds.$$

$$\Vert X_0(s)-X_{0\ast }(s){\Vert }_{{\dot{H}}^{5/2}(\mathbb{T})}\le$$

$$\Vert X_0(s)-X_{0\ast }(s){\Vert }_{{\dot{H}}^1(\mathbb{T})}\le$$

$$\Vert X-X_{\ast }{\Vert }_{L_{[0,+\infty )}^{\infty }{\dot{H}}^{5/2}(\mathbb{T})}\le$$

$$\left|X(s_1,t)-X(s_2,t)\right|\ge$$

$$\Vert X{\Vert }_{L_{[0,+\infty )}^{\infty }{\dot{H}}^{5/2}(\mathbb{T})}\le C{R}_{X_0}$$

$$\Vert X_0(s)-X_{0\ast }(s){\Vert }_{{\dot{H}}^1(\mathbb{T})}\le {\xi }_{\ast \ast }{R}_{X_0},$$

$$\begin{array}{rl}& \Vert X-X_{\ast }{\Vert }_{{\dot{H}}^{5/2}(\mathbb{T})}(t)\\ & \le C{e}^{-{\alpha }_{\ast }t}\max \{\Vert X_0-X_{0\ast }{\Vert }_{{\dot{H}}^{5/2}(\mathbb{T})},\Vert X_0-X_{0\ast }{\Vert }_{{\dot{H}}^1(\mathbb{T})}(|\ln \Vert X_0-X_{0\ast }{\Vert }_{{\dot{H}}^1(\mathbb{T})}|+1{)}^2\}\\ & \triangleq C{e}^{-{\alpha }_{\ast }t}B(X_0).\end{array}$$

$$\Vert X(t)-X_{\infty }{\Vert }_{{\dot{H}}^{5/2}(\mathbb{T})}\le CB(X_0)e^{-{\alpha }_{\ast }t},$$

$$G(x)=\frac{1}{4\pi }\left(-\ln |x|Id+\frac{x\otimes x}{|x{|}^2}\right),Q(x)=\frac{x}{2\pi |x{|}^2},$$

$$\begin{array}{rl}{u}_X(x,t)=& {\int }_{{\mathbb{R}}^2}G(x-y)f(y,t)dy={\int }_{{\mathbb{R}}^2}{\int }_{\mathbb{T}}G(x-y)\delta (x-X(s^{\prime },t))F(s^{\prime },t)d{s}^{\prime }dy\\ =& {\int }_{\mathbb{T}}G(x-X(s^{\prime },t))X_{ss}(s^{\prime },t)d{s}^{\prime },\end{array}$$

$$|G(X(s)-X(s^{\prime }))|\le C(\lambda )(1+|\ln |s-s^{\prime }||).$$

$$u_X^i(x)={\int }_{\mathbb{T}}-{\partial }_{s^{\prime }}[G^{ij}(x-X(s^{\prime }))][X^{\prime }(s^{\prime })-C_x{]}^{j}d{s}^{\prime },$$

Full text

Keywords. Immersed boundary problem, Stokes flow, fractional Laplacian, solvability, stability. AMS subject classifications. 35C15, 35Q35, 35R11, 76D07.

1 Introduction

The immersed boundary method was initially formulated by Peskin [1, 2] in early 1970s to study flow patterns around heart valves, and later it develops into a generally effective method to solve fluid-structure interaction problems [3]. It gives birth to numerous studies of the numerical methods, along with applications in physics, biology and medical sciences. See [3, 4] and the references therein. Various mathematical analysis have also been performed based on the model formulation itself, e.g. [5, 6, 7]. From the analysis point of view, the immersed boundary problem is intriguing on its own right. It is nonlinear by nature, featuring free moving boundary and singular forcing, which are not well-studied in the classic mathematical theory of hydrodynamics [8].

In this paper, we shall consider Stokes immersed boundary problem in two dimensions. It models the scenario where there is a 1-D closed elastic string (or fibre) immersed and moving in the 2-D Stokes flow: the string exerts force on the fluid and generates the flow, while the flow in turn moves the string and changes its configuration. The mathematical formulation will be given below. We will prove solvability of the string motion and its asymptotic behavior near equilibrium. Much of the analysis in this paper also applies to immersed boundary problems in three dimensions.

A similar type of problems on one- [9, 10, 11] or two-phase [12, 13, 14, 15, 16, 17, 18, 19] incompressible fluid motion has been extensively studied. In these settings, the space is occupied by one incompressible viscous fluid and the vacuum, or by two immiscible incompressible viscous fluids; the fluids move with or without surface tension on their interface. Solvability results have been established in various function spaces. The main difference between these problems and ours is that only the geometry (such as length, area and curvature) of the interface is involved there in determining the force balance at the interface. In particular, it does not depend on how the immersed string or membrane is parametrized. Consequently, one can use either Eulerian or Lagrangian approach to study the evolution of interfaces. However, in the immersed boundary problems, elastic strings or membranes have their internal structures and their dynamics also depends on constitutive law of elasticity, which varies from case to case. In other words, intrinsic parametrization of the immersed boundary and its elastic deformation should play a role. Indeed, immersed boundaries with identical overall shape can generate force differently. One can easily construct a 1-D closed string with a circular shape, yet far more stretched at some point than somewhere else. In this case, we shall see that the force on the string is not everywhere pointing inward normal to the string. This suggests that a pure Eulerian approach employed in many mathematical studies of free boundary problems in hydrodynamics (e.g. [20]) would not suffice. One needs to keep track of the configuration of the immersed boundary, which is typical in the nonlinear elasticity problems, and different techniques need to be used.

1.1 The Stokes immersed boundary problem in two dimensions

Consider a 1-D neutrally buoyant massless elastic closed string immersed in 2-D Stokes flow. The string is modeled as a Jordan curve $\Gamma_{t}$ parameterized by $X(s,t)$, where $s\in\mathbb{T}$ is the Lagrangian coordinate (or the material coordinate) and $t\geq 0$ is the time variable. Here, $\mathbb{T}\triangleq\mathbb{R}/2\pi\mathbb{Z}$ is the 1-D torus equipped with the induced metric. We always assume that at least $X(\cdot,t)\in H^{2}(\mathbb{T})$ for all $t$. The flow field in the immersed boundary problem is determined by

[TABLE]

Here $u(x,t)$ is the velocity field in $\mathbb{R}^{2}$ and $p$ is the pressure; $\mu_{0}>0$ is the dynamic viscosity; $f(x,t)$ is the elastic force exerted on the fluid generated by the string, given by [3]

[TABLE]

Here $\delta$ is the 2-D delta measure, which means the force is only supported on the string. $F(s,t)$ is the force in the Lagrangian formulation; it is given by

[TABLE]

where $X_{s}=\partial X/\partial s$, $\mathcal{T}$ is the tension in the string and $\mathcal{E}$ is the elastic energy density. In the following discussion, we shall take

[TABLE]

In this case, each infinitesimal segment of the string behaves like a Hookean spring with elasticity coefficient $k_{0}>0$, and thus $F(s,t)=k_{0}X_{ss}(s,t)$. It will be clear below that most of the discussion in this paper can also apply to more general elastic energy of other forms. The model is closed by the kinematic equation of the string,

[TABLE]

which means the string moves with the flow.

For simplicity, we shall take $\mu_{0}=k_{0}=1$ in the rest of the paper. Indeed, one can easily normalize both coefficients simultaneously by properly redefining $u$, $p$ and the time variable $t$. We shall always omit the $t$-dependence whenever it is convenient; and we shall also write $X^{\prime}(s^{\prime})$ and $X^{\prime\prime}(s^{\prime})$ in the places of $X_{s}(s^{\prime},t)$ and $X_{ss}(s^{\prime},t)$ respectively.

1.2 Contour dynamic formulation

The starting point of the analysis in this paper is the following proposition. It rewrites the original immersed boundary problem (1)-(5) that is in mixed Eulerian and Lagrangian formulation into a pure Lagrangian formulation, which we will call contour dynamic formulation.

Proposition 1**.**

Under the assumptions that $X(\cdot,t)\in H^{2}(\mathbb{T})$ for all $t$, and that there $\exists\,\lambda>0$, s.t. $\forall\,s_{1},s_{2}\in\mathbb{T}$,

[TABLE]

where $|s_{1}-s_{2}|$ is the distance between $s_{1}$ and $s_{2}$ on $\mathbb{T}$, the evolution of $X(s,t)$ in the 2-D Stokes immersed boundary problem (1)-(5) is equivalently given by

[TABLE]

where $\mathcal{L}\triangleq-\frac{1}{4}(-\Delta)^{1/2}$, and

[TABLE]

Here $(-\Delta)^{1/2}$ on $\mathbb{T}$ is understood as a Fourier multiplier or equivalently the following singular integral

[TABLE]

and

[TABLE]

is the fundamental solution of the 2-D Stokes equation for the velocity field [21].

We call (6) well-stretched assumption; (7) is called the contour dynamic formulation of the immersed boundary problem. The proof of Proposition 1 is left to Section 2. In the sequel, we shall focus on (7) and prove existence and uniqueness of its solutions and their properties. Estimates of the velocity field $u_{X}(x,t)$ can be easily obtained based on that; see Lemma 1 below. Note that the subscript of $u_{X}$ stresses that it is determined by $X(s,t)$; see Section 2 for more details.

1.3 Main results

Let us introduce a notation before we state the main results of the paper. With $T>0$, define

[TABLE]

It is equipped with the norm

[TABLE]

Here $L^{\infty}_{T}{H}^{5/2}(\mathbb{T})=L^{\infty}([0,T];{H}^{5/2}(\mathbb{T}))$, and $L^{2}_{T}{H}^{3}(\mathbb{T})$ and $L^{2}_{T}{H}^{2}(\mathbb{T})$ have similar meanings. Then we are able to prove the local well-posedness of the immersed boundary problem (7).

Theorem 1** (Existence of the local-in-time solution).**

Suppose $X_{0}(s)\in H^{5/2}(\mathbb{T})$, s.t. there exists some $\lambda>0$,

[TABLE]

Then there exists $T_{0}=T_{0}(\lambda,\|X_{0}\|_{\dot{H}^{5/2}})\in(0,+\infty]$ and a solution $X(s,t)\in\Omega_{T_{0}}\cap C_{[0,T_{0}]}H^{5/2}(\mathbb{T})$ of the immersed boundary problem (7), satisfying that

[TABLE]

and that for $\forall\,s_{1},s_{2}\in\mathbb{T}$ and $t\in[0,T_{0}]$,

[TABLE]

We write $C_{[0,T_{0}]}H^{5/2}(\mathbb{T})$ instead of $C_{T_{0}}H^{5/2}(\mathbb{T})$ to stress continuity up to the end points of the time interval.

Theorem 2** (Uniqueness of the local-in-time solution).**

Suppose $X_{0}(s)\in H^{5/2}(\mathbb{T})$ satisfies (13) with some $\lambda>0$. Given an arbitrary $c\in(0,1)$, the immersed boundary problem (7) has at most one solution $X\in\Omega_{T}$ satisfying that $\forall\,s_{1},s_{2}\in\mathbb{T}$ and $\forall\,t\in[0,T]$,

[TABLE]

In particular, the local-in-time solution obtained in Theorem 1 is unique in $\Omega_{T_{0}}$.

To state the results on the global existence of solutions near equilibrium configurations and its exponential convergence, we need the following definition.

Definition 1**.**

Assume $Y(s)\in H^{5/2}(\mathbb{T})$ defines a Jordan curve in the plane, s.t. the area of domain enclosed by $Y$ is $\pi R_{Y}^{2}$ with $R_{Y}>0$, i.e.,

[TABLE]

We call $R_{Y}$ the effective radius of $Y(s)$. Define

[TABLE]

with $\theta\in[0,2\pi)$ and $x\in\mathbb{R}^{2}$. Let

[TABLE]

Then $Y_{*}(s)\triangleq Y_{\theta_{*},x_{*}}(s)$ is called the closest equilibrium configuration to $Y(s)$.

Properties of the closest equilibrium configuration will be discussed in Section 5. Now we have

Theorem 3** (Existence and uniqueness of global-in-time solution near equilibrium).**

There exist universal $\varepsilon_{*},\xi_{*}>0$, such that for $\forall\,X_{0}(s)\in H^{5/2}(\mathbb{T})$ satisfying

[TABLE]

with $X_{0*}(s)$ being the closest equilibrium configuration to $X_{0}(s)$, there exists a unique solution $X(s,t)\in C_{[0,+\infty)}H^{5/2}\cap L^{2}_{[0,+\infty),loc}H^{3}(\mathbb{T})$ satisfying $X_{t}(s,t)\in L^{2}_{[0,+\infty),loc}H^{2}(\mathbb{T})$ for the immersed boundary problem (7). It satisfies the following estimates

[TABLE]

In particular,

[TABLE]

for some universal $C$.

Theorem 4** (Exponential convergence to the equilibriums).**

Let $X_{0}\in H^{5/2}(\mathbb{T})$ satisfy all the assumptions in Theorem 3 and let $X$ be the unique global solution of (7) starting from $X_{0}$ obtained in Theorem 3. There exist universal constants $\xi_{**},\alpha_{*}>0$, such that if in addition

[TABLE]

then

With some universal constant $C>0$,

[TABLE] 2. 2.

There exists an equilibrium configuration $X_{\infty}\triangleq x_{\infty}+(R_{X_{0}}\cos(s+\theta_{\infty}),R_{X_{0}}\sin(s+\theta_{\infty}))^{T}$, such that

[TABLE]

where $C$ is a universal constant and $B(X_{0})$ is defined in (25).

The rest of the paper is organized as follows. In Section 2, the reformulation in Proposition 1 is justified. We also discuss properties of the flow field and law of energy dissipation in the system; their proofs are left to the Appendix A.1. In Section 3, we will prove a priori estimates necessary for proving the local well-posedness of the contour dynamic formulation (7). In particular, in Section 3.1, we prove some preliminary estimates as building blocks of more complicated bounds in Section 3.2, which is devoted to finding out derivatives of $g_{X}$ and proving its $H^{2}$-estimate. In Section 4, we will establish the local well-posedness of (7). In Section 5, we will show global-in-time existence of solutions of (7) provided that the initial configuration is sufficiently close to an equilibrium configuration. In Section 6, we will first prove a lower bound of the rate of energy dissipation in Section 6.1 when the solution is close to an equilibrium. Based on that, we will show exponential convergence of the solution to an equilibrium configuration in Section 6.2. Some other auxiliary results will be stated and proved in the Appendix A.2 and A.3.

2 Problem Reformulation and the Flow Field

2.1 Proof of Proposition 1

We first justify Proposition 1, which reformulates the original immersed boundary problem (1)-(5) into the contour dynamic formulation (7). Some of the arguments are redundant for proving the proposition itself, but we still derive them here as they will be useful in proving Lemma 1 and Lemma 2 below.

Proof of Proposition 1.

In 2-D stationary Stokes flow, the velocity field $u$ and the pressure $p$ are instantaneously determined by the forcing $f$ through fundamental solutions

[TABLE]

respectively [21], where $Id$ is the $2\times 2$-identity matrix. Hence,

[TABLE]

This is well-defined for $x\not\in\Gamma_{t}$ and $X(\cdot,t)\in H^{2}(\mathbb{T})$. The subscript of $u_{X}$ stresses that it is determined by the configuration $X$. For $x=X(s,t)\in\Gamma_{t}$, by (6),

[TABLE]

Hence, $G(X(s)-X(\cdot))\in L^{2}(\mathbb{T})$ and (28) is well-defined.

For $x\not\in\Gamma_{t}$, we do integration by parts in (28) and find that

[TABLE]

where the superscripts stand for the indices of entries, and $C_{x}$ is any arbitrary constant vector independent of $s^{\prime}$. We may take $C_{x}=X^{\prime}(s_{x})$, where $s_{x}$ is defined by

[TABLE]

Note that $s_{x}$ may not be unique; pick an arbitrary one if it is the case. Hence,

[TABLE]

Similarly, by integration by parts and taking the indetermined constant to be [math], we find for $x\not\in\Gamma_{t}$,

[TABLE]

For $x=X(s,t)\in\Gamma_{t}$, by (28),

[TABLE]

Using (6) and the assumption that $X(\cdot,t)\in H^{2}(\mathbb{T})$, we find

[TABLE]

It goes to [math] as $\varepsilon\rightarrow 0^{+}$. A similar bound holds for $|G(X(s)-X(s+\varepsilon))(X^{\prime}(s+\varepsilon)-X^{\prime}(s))|$. Hence,

[TABLE]

In (9), we denoted the integrand in (33) by $\Gamma_{0}(s,s^{\prime})$. It is trivial to show that

[TABLE]

Hence, $\Gamma_{0}(s,s^{\prime})$ is integrable, and the principal value integral in (33) can be replaced by the usual integral. As a byproduct, we also find a bound for $u_{X}(X(s))$,

[TABLE]

(33) together with (5) gives (7). Once (7) is solved, we can recover $u$ and $p$ by (28) and (32). The original immersed boundary problem is then solved. This completes the proof of Proposition 1. ∎

Remark 1*.*

(33) can be equivalently written as

[TABLE]

Indeed, under the assumptions $X(\cdot,t)\in H^{2}(\mathbb{T})$ and (6),

[TABLE]

To obtain the last convergence, we derive that, since $|X^{\prime}(s)|\geq\lambda$,

[TABLE]

and similarly,

[TABLE]

(35) can be viewed as taking $C_{x}=0$ in (29). ∎

Remark 2*.*

The reason why we single out the term $\mathcal{L}X$ in Proposition 1 comes from the following suggestive calculation starting from (35). Note that the integrals in (33) and (35) give the same value, so we use them interchangeably.

Suppose $X(\cdot,t)$ is sufficiently smooth. There is a singularity in the integrand of (35) as $s^{\prime}\rightarrow s$. Consider $s^{\prime}$ very close to $s$ and we formally use $(s^{\prime}-s)X^{\prime}(s^{\prime})$ to approximate $X(s^{\prime})-X(s)$ in (35). In this way, when $|s^{\prime}-s|$ is sufficiently small, we formally find

[TABLE]

which presumably accounts for the principal part of the singular integral in (35). Recall that the Hilbert transform $\mathcal{H}$ on $\mathbb{T}$ is defined as [22]

[TABLE]

Hence, if we take out $-\frac{1}{4}\mathcal{H}X^{\prime}=\mathcal{L}X$ in (35), what remains is expected to be regular. We shall see that $\mathcal{L}X$ provides nice dissipation property that helps prove well-posedness of (7). See Lemma 14 and Lemma 15 for some relevant estimates.

It should be noted that the very idea has been adopted in early numerical literature to, for example, remove stiffness in computing the evolution of elastic immersed boundary in 2-D Stokes flow or the motion of interface with surface tension in 2-D incompressible irrotational flow. See e.g. [23, 24] and references therein. ∎

2.2 Regularity of the flow field and energy dissipation

As is mentioned above, once (7) is solved, we can obtain the flow field $u_{X}$ by (28). The following lemma characterizes its regularity.

Lemma 1**.**

Let $X(\cdot,t)\in H^{2}(\mathbb{T})$ and satisfy the well-stretched condition (6). Then $u_{X}(\cdot,t)$ defined by (28) (or equivalently (31), (33) and (35)) is continuous in $\mathbb{R}^{2}$. Moreover, $\nabla u_{X}(\cdot,t)\in L^{2}(\mathbb{R}^{2})$.

Remark 3*.*

That $u_{X}(x,t)$ is continuous throughout $\mathbb{R}^{2}$ agrees with the intuition that the string moves with the ambient flow, and there is no jump in velocity across the string. ∎

As a dissipative system, the Stokes immersed boundary problem enjoys a natural law of energy dissipation, which is useful in proving existence and asymptotic behavior of global solution near equilibrium in Section 5 and Section 6.

Lemma 2**.**

Assume $X(s,t)\in C_{T}H^{2}(\mathbb{T})$ with $X_{t}(s,t)\in L^{2}_{T}H^{1}(\mathbb{T})$ is a solution of (7) with some $T>0$ satisfying (6) with constant $\lambda>0$, and $u_{X}(x,t)$ is the corresponding velocity field defined by the Stokes equation (1), with $\nabla u_{X}(x,t)\in L^{\infty}_{T}L^{2}(\mathbb{R}^{2})$ (showed in (179) in the proof of Lemma 1). Then

[TABLE]

holds in the scalar distribution sense, and

[TABLE]

In particular, the total elastic energy of the string $\mathcal{E}_{X}\triangleq\frac{1}{2}\|X(\cdot,t)\|_{\dot{H}^{1}(\mathbb{T})}^{2}$ always decreases in $t$.

The proofs of these lemmas are technical. We leave them to Appendix A.1.

3 A Priori Estimates

In this section, we shall prove a priori estimates that are needed in proving well-posedness of (7).

3.1 Preliminaries

First we introduce some notations that will be heavily used in the rest of the paper. Suppose $X\in H^{3}(\mathbb{T})$. For $s,s^{\prime}\in\mathbb{T}$, let $\tau=s^{\prime}-s\in[-\pi,\pi)$. For $s^{\prime}\not=s$, define

[TABLE]

and

[TABLE]

It is straightforward to calculate that for $s^{\prime}\not=s$,

[TABLE]

In the sequel, we shall omit the arguments in $L(s,s^{\prime})$, $M(s,s^{\prime})$ and $N(s,s^{\prime})$ whenever it is convenient. Without assuming the well-stretched assumption (6), we have the following estimates for $L$, $M$ and $N$, which will be building blocks of more complicated estimates in Section 3.2.

Lemma 3**.**

For $\forall\,1\leq p\leq q\leq\infty$, $q>1$ and any interval $I\subset\mathbb{T}$ satisfying $0\in I$

[TABLE]

where the constants $C>0$ only depend on $p$ and $q$. 2. 2.

For $\forall\,1<p\leq q\leq\infty$ and any interval $I\subset\mathbb{T}$ satisfying $0\in I$

[TABLE]

where the constants $C>0$ only depend on $p$ and $q$. 3. 3.

Let $\mathcal{M}$ be the centered Hardy-Littlewood maximal operator on $\mathbb{T}$. Then for $\forall\,s,s^{\prime}\in\mathbb{T}$,

[TABLE] 4. 4.

If $X\in C^{2}(\mathbb{T})$,

[TABLE] 5. 5.

Moreover, if (6) is satisfied with constant $\lambda>0$,

[TABLE]

and

[TABLE]

Proof.

(54) and (55) are obvious. To prove the $L^{p}$-estimates and the continuity of $L$, $M$ and $N$, we rewrite

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

(53) is immediate by the continuity of $X^{\prime}$ and $X^{\prime\prime}$ at $s$. To prove (52), we use the above representation to derive that

[TABLE]

Now we turn to (42)-(51). When $p=q=\infty$, (42)-(51) immediately follow from the above representations. When $1\leq p\leq q\leq\infty$, $p<\infty$ and $q>1$, we find that

[TABLE]

We applied Minkowski inequality in the second line and H$\mathrm{\ddot{o}}$lder’s inequality in the fourth line; we also used the fact that $s+\theta I\subset s+I$. This proves (42); (43) could be proved in exactly the same way simply by replacing $X^{\prime\prime}$ by $X^{\prime\prime\prime}$. For (44),

[TABLE]

(45) could be proved in exactly the same way simply by replacing $X^{\prime\prime}$ by $X^{\prime\prime\prime}$. For (46),

[TABLE]

For (47), we first consider the case $p=q\in(1,\infty)$. (42) implies that, $\|L(s,s^{\prime})\|_{L^{p}_{s^{\prime}}(s+I)}\leq C\|X^{\prime}\|_{L^{p}(s+I)}$. Hence, by Fubini’s Theorem,

[TABLE]

On the other hand, by (42), $\|L(s,s^{\prime})\|_{L^{\infty}_{s}(\mathbb{T})L^{p}_{s^{\prime}}(s+I)}\leq C\|X^{\prime}\|_{L^{p}(\mathbb{T})}$. Hence, by interpolation between $L^{p}$-spaces, we proved (47). In a similar manner, we can prove (48)-(51). ∎

3.2 $H^{2}$-estimate of $g_{X}$

In Section 4, we will prove well-posedness of (7) via a fixed-point-type argument by making use of dissipation structure of the operator $\mathcal{L}$ (see Lemma 14 and Lemma 15 in the Appendix A.2). In order to do that, in this section, we focus on the term $g_{X}$ in (7) and establish its $H^{2}$-estimate; recall that $g_{X}$ is defined in (8). We are also going to prove an $H^{2}$-estimate of $g_{X_{1}}-g_{X_{2}}$, which will be used in proving the uniqueness of the local solution.

We start from a pointwise estimate of $g_{X}$.

Lemma 4**.**

Suppose $X\in H^{2}(\mathbb{T})$ satisfies (6) with some $\lambda>0$. Then

[TABLE]

where $C>0$ is a universal constant.

Proof.

Recall that $\Gamma_{0}(s,s^{\prime})$ is defined in (9). By (33), and the definitions of $L$ and $M$, we have

[TABLE]

Hence, by (54),

[TABLE]

This implies by H$\mathrm{\ddot{o}}$lder’s inequality and Lemma 3 that

[TABLE]

The other term in $g_{X}(s)$, $(-\Delta)^{1/2}X$, has mean zero on $\mathbb{T}$. By Gagliardo-Nirenberg interpolation inequality,

[TABLE]

Using (54), we find that

[TABLE]

(56) is then proved by (59) and (60). ∎

Remark 4*.*

If we further assume $X\in H^{3}(\mathbb{T})\subset C^{2}(\mathbb{T})$, using the continuity of $L(s,\cdot)$ and $M(s,\cdot)$, it is not difficult to show in (57) that

[TABLE]

This will be useful below in proving Lemma 5. ∎

Corollary 1**.**

Let $X_{1}(s),X_{2}(s)\in H^{2}(\mathbb{T})$ both satisfy (6) with some $\lambda>0$. Then

[TABLE]

where $C>0$ is a universal constant.

Proof.

By the definition of $g_{X}$ in (8) and (57),

[TABLE]

where $L_{i}$, $M_{i}$ and $X_{i}^{\prime}$ denote the corresponding quantities associated with $X_{i}(\cdot)$; see definitions in (39) and (40). To make an $L^{2}$-estimate, for conciseness, we only consider a part of the difference above. By (6) and (54),

[TABLE]

By Lemma 3 and Sobolev inequality,

[TABLE]

Similarly,

[TABLE]

We can estimate the other terms in (63) in a similar fashion and obtain (62). ∎

In order to estimate $H^{2}$-norm of $g_{X}$, we find out its weak derivatives $g_{X}^{\prime}$ and $g_{X}^{\prime\prime}$ in the following two lemmas.

Lemma 5**.**

Suppose $X\in H^{3}(\mathbb{T})$ and satisfies (6) with some $\lambda>0$. Then

[TABLE]

Proof.

We define a cut-off function $\varphi(y)\in C^{\infty}(\mathbb{T})$ such that

$\varphi(y)=\varphi(-y)$, $\forall\,y\in\mathbb{T}$. 2. 2.

$\varphi(y)=1$ for $|y|\leq 1$; $\varphi(y)=0$ for $|y|\geq 2$; and $|\varphi^{\prime}(y)|\leq C$. 3. 3.

$\varphi(y)$ is decreasing on $[0,\pi]$ and increasing on $[-\pi,0]$.

Define $\psi_{\varepsilon}(y)=1-\varphi\left(\frac{y}{\varepsilon}\right)$. Let

[TABLE]

By (58) and Lemma 3,

[TABLE]

which implies that $g_{X,1},g_{X,1}^{\varepsilon}\in L^{\infty}(\mathbb{T})$, and $g_{X,1}^{\varepsilon}\rightarrow g_{X,1}$ in $L^{\infty}(\mathbb{T})$. In particular, for any test function $\eta\in C^{\infty}(\mathbb{T})$,

[TABLE]

where $(\cdot,\cdot)$ is the $L^{2}$-inner product on $\mathbb{T}$. Since there is no singularity in the integral in $g_{X,1}^{\varepsilon}$, we apply integration by parts on the left hand side above and exchange the derivative and the integral. We will obtain

[TABLE]

It is not difficult to show that

[TABLE]

On the other hand, since $\psi_{\varepsilon}^{\prime}(\cdot-s)$ is of mean zero on $\mathbb{T}$ and $\|\psi_{\varepsilon}^{\prime}(\cdot-s)\|_{L^{1}(\mathbb{T})}=2$ due to the monotonicity assumption on $\varphi$, we have that

[TABLE]

where the convergence comes from (61). Combining (66), (67), (68) and (69), we find

[TABLE]

We used (36) in the last line.

For the other term in $g_{X}(s)$, namely $\frac{1}{4}(-\Delta)^{1/2}X$, we note that $(-\Delta)^{1/2}$ and the derivative commute since they are both Fourier multipliers. This gives

[TABLE]

Combining (70) and (71), we proved (64).

∎

Lemma 6**.**

Suppose $X\in H^{3}(\mathbb{T})$ and satisfies (6) with some $\lambda>0$. Then

[TABLE]

Proof.

Denote the integrand of (64) by $\Gamma_{1}(s,s^{\prime})$, i.e.

[TABLE]

What we are going to show in (72) is exactly

[TABLE]

We claim that for $s\not=s^{\prime}$,

[TABLE]

For conciseness, we leave its proof in Lemma 16 in Appendix A.3. With (74) in hand, we use (54) and (55) to derive that,

[TABLE]

By the continuity of $L$, $M$ and $N$, i.e. (53), we also know by (74) that

[TABLE]

To this end, we can prove (72) by arguing as in the proof of Lemma 5. We omit the details. ∎

Similar to Corollary 1, one can prove that

Corollary 2**.**

Let $X_{1}(s),X_{2}(s)\in H^{2}(\mathbb{T})$ both satisfy (6) with some $\lambda>0$. Then

[TABLE]

where $C>0$ is a universal constant.

Proof.

With (73) and (74) in hand, we simply argue as in the proof of Corollary 1 to obtain the desired estimate. We omit the details. ∎

The following lemma is devoted to $H^{2}$-estimate of $g_{X}$.

Lemma 7**.**

Suppose $X\in H^{3}(\mathbb{T})$ and satisfies (6) with some $\lambda>0$. Then for $\forall\,\delta\in(0,\pi)$,

[TABLE]

where $C>0$ is a universal constant.

Proof.

By Lemma 6, we look into $\partial_{s}\Gamma_{1}(s,s^{\prime})$. We take $s$-derivative of (74) and use $\partial_{s}L=N$ in (41) to find that

[TABLE]

The following estimate is also useful by substituting (41) into the above formula

[TABLE]

In order to prove (77), we split $g_{X}^{\prime\prime}$, an integral of $\partial_{s}\Gamma_{1}$ with respect to $s^{\prime}$, into two terms — the integral in a neighborhood of the singularity at $s^{\prime}=s$, and the rest. To be more precise, for $\forall\,\delta\in(0,\pi)$, we have

[TABLE]

For $I_{\delta}$, we use (78). Applying Lemma 3 with $I=B_{\delta}(0)$, we obtain that

[TABLE]

We used (54) and Sobolev inequality in the last line. For $II_{\delta}$, we used (79). Applying Lemma 3 with $I=\mathbb{T}$, we obtain that

[TABLE]

Here we used Lemma 3 and Sobolev inequality. Combining the above two estimates of $I_{\delta}$ and $II_{\delta}$, we proved (77). ∎

Remark 5*.*

It is clear from the proof that, the goal of splitting $g_{X}^{\prime\prime}$ into two parts in (80) is to introduce a small parameter $\delta$ in front of $\|X\|_{\dot{H}^{3}}$ in (77). This will be useful in the proof of local well-posedness. See Section 4. ∎

We can also show that

Corollary 3**.**

Let $X_{1}(s),X_{2}(s)\in H^{3}(\mathbb{T})$ both satisfy (6) with some $\lambda>0$. Then for $\forall\,\delta\in(0,\pi)$, and $\forall\,\mu>0$,

[TABLE]

where $C_{\mu}>0$ is a universal constant depending on $\mu$.

Proof.

We take $s$-derivative in (74) first, and argue as in the proofs of Corollary 1 and Lemma 7. The calculation is unnecessarily long but tedious. We omit the details here. ∎

4 Existence and Uniqueness of the Local-in-time Solution

To this end, we are able to prove the local well-posedness of (7).

4.1 Existence

Proof of Theorem 1 (existence of the local-in-time solution).

For $\forall\,Y\in L^{1}(\mathbb{T})$, we split it into its mean $\bar{Y}$ and its oscillation $\tilde{Y}$, i.e.

[TABLE]

Then (7) could be split into two equations as well, one for $\tilde{X}$ and the other for $\bar{X}$. Namely,

[TABLE]

and

[TABLE]

We first consider the existence of solutions of the $\tilde{X}$-equation (82). Given $X_{0}$, with $T>0$ to be determined, we define

[TABLE]

The subscript [math] stresses that functions in $\Omega_{0,T}(X_{0})$ has mean zero on $\mathbb{T}$. We remark that only the seminorms are used, since the mean of $X_{0}$ is irrelevant in the equation for $\tilde{X}$, which is always this case in the sequel. $\Omega_{0,T}(X_{0})$ is non-empty. Indeed, by Lemma 14 and Lemma 15, $\mathrm{e}^{t\mathcal{L}}\widetilde{X_{0}}\in\Omega_{0,T}(X_{0})$. It is also convex and closed in $\Omega_{T}$. By Aubin-Lions lemma, $\Omega_{0,T}(X_{0})$ is compact in $C_{T}H^{2}(\mathbb{T})$.

By Lemma 15, for $\forall\,Y\in\Omega_{0,T}(X_{0})$, $\|Y\|_{L^{\infty}_{T}\dot{H}^{5/2}\cap L^{2}_{T}\dot{H}^{3}(\mathbb{T})}\leq 4\|X_{0}\|_{\dot{H}^{5/2}(\mathbb{T})}$. Moreover, by taking $T$ sufficiently small, we will have

[TABLE]

In fact, if we assume $C_{1}\|X_{0}\|_{\dot{H}^{5/2}(\mathbb{T})}T^{1/2}\leq\lambda/2$, with $C_{1}$ being a universal constant coming from Sobolev inequality that will be clear below,

[TABLE]

Here we used the assumptions that $\|Y_{t}(s,t)\|_{L^{2}_{T}\dot{H}^{2}(\mathbb{T})}\leq\|X_{0}\|_{\dot{H}^{5/2}(\mathbb{T})}$ and $Y(s,0)=\widetilde{X_{0}}(s)$. Then (85) follows from (13) and the triangle inequality.

Under the above assumption, we define a map $V:\Omega_{0,T}(X_{0})\rightarrow\Omega_{0,T}(X_{0})$ as follows, with $T$ to be determined. For given $Y(s,t)\in\Omega_{0,T}(X_{0})$, let $Z\triangleq VY$ solve

[TABLE]

To show $V$ is well-defined, we first claim that $Z\in\Omega_{T}$. In fact, for $Y\in\Omega_{0,T}(X_{0})$, by Lemma 7,

[TABLE]

In the last line, we used (54) and Sobolev inequality to have that $\lambda\leq C\|X_{0}\|_{\dot{H}^{5/2}(\mathbb{T})}$. Then for $\forall\,T>0$, Lemma 15 gives the existence and uniqueness of the solution $Z\in\Omega_{T}$, which satisfies

[TABLE]

$Z$ obviously has mean zero for all time.

Now consider $W=Z-\mathrm{e}^{t\mathcal{L}}\widetilde{X_{0}}$, which solves

[TABLE]

By Lemma 15 and (87), we find that

[TABLE]

To this end, we first take $\delta\leq\delta_{0}(\lambda,\|X_{0}\|_{\dot{H}^{5/2}})$ sufficiently small, s.t.

[TABLE]

and then assume $T\leq T_{0}(\lambda,\|X_{0}\|_{\dot{H}^{5/2}},\delta)$ sufficiently small as well, s.t.

[TABLE]

This implies $\|Z-\mathrm{e}^{t\mathcal{L}}\widetilde{X_{0}}\|_{L^{\infty}_{T}\dot{H}^{5/2}\cap L^{2}_{T}\dot{H}^{3}(\mathbb{T})}\leq\|X_{0}\|_{\dot{H}^{5/2}(\mathbb{T})}$ by (89). Also by (87) and (88)

[TABLE]

Hence, $V$ is well-defined from $\Omega_{0,T}(X_{0})$ to itself. We note that the upper bound of valid $T$, which is $T_{0}$, essentially only depends on $\lambda$ and $\|X_{0}\|_{\dot{H}^{5/2}(\mathbb{T})}$.

By Aubin-Lions lemma, $V(\Omega_{0,T}(X_{0}))$ is compact in $C_{T}H^{2}(\mathbb{T})$. By Schauder fixed point theorem, there is a fixed point of the map $V$ in $V(\Omega_{0,T}(X_{0}))\subset\Omega_{0,T}(X_{0})$, denoted by $\tilde{X}\in\Omega_{T}$, which is a solution of (82). It satisfies

[TABLE]

and

[TABLE]

To this end, we turn to the ODE (83) for $\bar{X}$. By Lemma 4, for $\forall\,s\in\mathbb{T}$ and $t\in[0,T]$,

[TABLE]

It is then easy to show that (83) admits a unique solution $\bar{X}(t)\in C^{0,1}([0,T])$ once $\tilde{X}$ is given. The solution for (7) is thus given by $X(s,t)=\bar{X}(t)+\tilde{X}(s,t)$. This proves the existence of the local-in-time solutions in $\Omega_{T}$. (14) and (15) follow from (91) and (92) respectively.

That $X\in L^{2}_{T}H^{3}(\mathbb{T})$ together with $X_{t}\in L^{2}_{T}H^{2}(\mathbb{T})$ implies that $X$ is almost everywhere equal to a continuous function valued in $H^{5/2}(\mathbb{T})$, i.e. $X$ could be realized as an element in $C([0,T];H^{5/2}(\mathbb{T}))$. This can be proved by classic arguments (see Temam [8], § 1.4 of Chapter III). ∎

4.2 Uniqueness

Proof of Theorem 2 (uniqueness of the local-in-time solution).

Suppose $X_{1},X_{2}\in\Omega_{T}$ are two solutions of (7), both satisfying the assumption (16). Let

[TABLE]

and $Q(s,t)\triangleq X_{1}-X_{2}$. Then $\tilde{Q}=\widetilde{X_{1}}-\widetilde{X_{2}}$ solves

[TABLE]

By Corollary 3 with $\mu=2$ and Sobolev inequality, with $t\in(0,T]$ to be determined

[TABLE]

Here we repeatedly used (93). By Lemma 15,

[TABLE]

Hence, we first take $\delta=\delta_{*}(\lambda,R,c)$ sufficiently small and then take $t=t_{*}(\lambda,R,c)$ sufficiently small, such that $C(c)[\delta^{1/2}+(|\ln\delta|+1)t^{1/2}]\lambda^{-4}R^{4}\in(0,1)$. This implies that

[TABLE]

i.e. $\tilde{Q}(s,t)=0$ for $t\in[0,t_{*}]$. Since (16) and (93) are uniform throughout $[0,T]$, the above argument is also true for arbitrary initial time, i.e. provided that $\tilde{Q}(s,t_{0})=0$ for some $t_{0}\in[0,T]$, then $\tilde{Q}(s,t)=0$ for $t\in[t_{0},\min\{t_{0}+t_{*},T\}]$. Hence, $\tilde{Q}(s,t)\equiv 0$ for $t\in[0,T]$, i.e. $\widetilde{X_{1}}(s,t)\equiv\widetilde{X_{2}}(s,t)$.

Recall that in (83), the solution $\bar{X}(t)$ is uniquely determined in $C^{0,1}([0,T])$ by $\tilde{X}(s,t)$. This implies that $\overline{X_{1}}(t)\equiv\overline{X_{2}}(t)$, and thus $X_{1}(s,t)\equiv X_{2}(s,t)$ for $(s,t)\in\mathbb{T}\times[0,T]$. This proves the uniqueness under the assumption (16).

The uniqueness of the local-in-time solution obtained in Theorem 1 follows immediately. ∎

5 Existence and Uniqueness of Global-in-time Solutions near Equilibrium Configurations

In this section, we will prove that existence of global solution provided that the initial string configuration is sufficiently close to equilibrium. The closeness is measured using the difference between a string configuration $Y$ and its closet equilibrium configuration $Y_{*}$ (see Definition 1). We start with several remarks on the definition of the closest equilibrium configuration.

Remark 6*.*

In the definition of $Y_{*}$, we have $x_{*}=\frac{1}{2\pi}\int_{\mathbb{T}}Y(s)\,ds$. This can be seen from the Fourier point of view. Assume $Y(s)=\sum_{k\in\mathbb{Z}}\hat{Y}_{k}\mathrm{e}^{iks}$, where $\hat{Y}_{k}$’s are complex-valued 2-vectors. By Parseval’s identity,

[TABLE]

In order to achieve its minimum, we should take $x_{*}=\hat{Y}_{0}=\frac{1}{2\pi}\int_{\mathbb{T}}Y(s)\,ds$. In the sequel, we shall denote $Y_{\theta}(s)\triangleq Y_{\theta,x_{*}}(s)$ and only minimize $\|Y-Y_{\theta}\|_{L^{2}(\mathbb{T})}$ with respect to $\theta$. ∎

Remark 7*.*

Although $Y_{*}$ is defined to be the closest to $Y$ in the $L^{2}$-distance among all $Y_{\theta}$, it is also the closest in the $H^{s}$-sense for all $s\geq 0$. Indeed, by Parseval’s identity,

[TABLE]

The last term in the last line is constant with respect to $\theta$, which implies that $\theta_{*}$ also optimizes $\|Y-Y_{\theta}\|_{\dot{H}^{s}(\mathbb{T})}$. ∎

The following lemma establishes the equivalence of the $H^{1}$-distance and the energy difference between a string configuration $Y$ and its closest equilibrium configuration $Y_{*}$. Recall that the elastic energy of $Y$ is $\|Y\|_{\dot{H}(\mathbb{T})}^{2}/2$ (see Lemma 2). The motivation is that we wish to transform the global coercive bound on the energy difference, which comes from (38), into a bound for more convenient quantity $\|Y-Y_{*}\|_{\dot{H}^{1}}$.

Lemma 8**.**

We have the following estimates for $Y$ and its closest equilibrium configuration $Y_{*}$:

[TABLE]

Proof.

Without loss of generality, we assume that $(\theta_{*},x_{*})=(0,0)$. Otherwise we simply make a translation and rotation of $Y$. Define $D(s)=Y(s)-Y_{*}(s)$. By Remark 6 and the above assumption, $D(s)$ is of mean zero on $\mathbb{T}$.

We first prove the upper bound. By the definition of $\theta_{*}$ and $Y_{*}$, we know that

[TABLE]

and

[TABLE]

Here we used $Y_{*}^{\prime\prime}=-Y_{*}$. Moreover, since $Y$ and $Y_{*}$ have the same effective radius, by (17),

[TABLE]

Since $Y_{*}^{\prime}(s)=(-R_{Y}\sin s,R_{Y}\cos s)=Y_{*}^{\perp}(s)$ and $Y_{*}^{\prime\prime}=-Y_{*}$, it is further simplified to be

[TABLE]

In the sequel, we shall write $Y_{*}$ and $D$ in terms of their Fourier coefficients. With the assumption that $(\theta_{*},x_{*})=(0,0)$, we have

[TABLE]

Assume $D(s)=\sum_{k\in\mathbb{Z}}\hat{D}_{k}\mathrm{e}^{iks}$, where $\hat{D}_{k}$’s are complex-valued $2$-vectors satisfying $\hat{D}_{-k}=\overline{\hat{D}_{k}}$. Hence, (97) could be rewritten as

[TABLE]

where $\hat{D}_{1,1}$ and $\hat{D}_{1,2}$ represent the first and the second component of $\hat{D}_{1}$ respectively. This implies that

[TABLE]

Similarly, the terms in (99) could be rewritten as follows:

[TABLE]

and

[TABLE]

Here we used the fact that $\hat{D}_{-1}=\overline{\hat{D}_{1}}$. By (LABEL:eqn:_second_order_equation_for_the_optimal_approximated_equilibrium) and (101), we know that

[TABLE]

We calculate that

[TABLE]

and

[TABLE]

Case 1*.*

If $\int_{\mathbb{T}}2D\cdot Y_{*}\,ds\geq 0$, we readily proved the upper bound in (96) by comparing (104) and (105).

Case 2*.*

If $\int_{\mathbb{T}}2D\cdot Y_{*}\,ds<0$, by (101), $\mathrm{Re}\,\hat{D}_{1,1}-\mathrm{Im}\,\hat{D}_{1,2}<0$. Then by (99), (101), (102) and (105),

[TABLE]

In the last line, we used the fact that $\hat{D}_{-1}=\overline{\hat{D}_{1}}$ and $\mathrm{Im}\,\hat{D}_{1,1}\mathrm{Re}\,\hat{D}_{1,2}\leq 0$ due to (100).

If $\mathrm{Re}\,\hat{D}_{1,1}$ and $\mathrm{Im}\,\hat{D}_{1,2}$ have the same sign, then $12\pi\mathrm{Re}\,\hat{D}_{1,1}\mathrm{Im}\,\hat{D}_{1,2}-2\pi R_{Y}(\mathrm{Re}\,\hat{D}_{1,1}-\mathrm{Im}\,\hat{D}_{1,2})\geq 0$. Hence,

[TABLE]

Otherwise, if $\mathrm{Re}\,\hat{D}_{1,1}$ and $\mathrm{Im}\,\hat{D}_{1,2}$ have different signs, i.e., $\mathrm{Re}\,\hat{D}_{1,1}\leq 0$ and $-\mathrm{Im}\,\hat{D}_{1,2}\leq 0$ since $\mathrm{Re}\,\hat{D}_{1,1}-\mathrm{Im}\,\hat{D}_{1,2}<0$, we know that

[TABLE]

Also, by (103),

[TABLE]

This implies

[TABLE]

Therefore,

[TABLE]

This proves the upper bound in (96).

Now we turn to the lower bound. By (99) and (105),

[TABLE]

Here we used the fact that $D$ has mean zero on $\mathbb{T}$.

This completes the proof. ∎

Remark 8*.*

As a byproduct, we know that for any Jordan curve $Y(s)\in H^{1}(\mathbb{T})$, $\|Y_{*}\|_{\dot{H}^{1}(\mathbb{T})}\leq\|Y\|_{\dot{H}^{1}(\mathbb{T})}$. The equality holds if and only if $Y=Y_{*}$. Hence, Lemma 8 implies that the string configuration having a circular shape and uniform parameterization has the lowest elastic energy among all the $H^{1}$-configurations that enclose the same area. This can also be showed by isoperimetric inequality and Cauchy-Schwarz inequality. ∎

Let $X$ be a local solution of (7) obtained in Theorem 1. By Lemma 2 and Lemma 8, we readily have global bound on $\|X-X_{*}\|_{\dot{H}^{1}(\mathbb{T})}(t)$. It would be very ideal if we could show that $\|X-X_{*}\|_{\dot{H}^{5/2}(\mathbb{T})}(t)$ can not be (always) big when $\|X-X_{*}\|_{\dot{H}^{1}(\mathbb{T})}(t)$ is small. The following lemma is an effort in this direction, which is crucial in proving Theorem 3.

Lemma 9**.**

Suppose $T\in(0,1]$ and $X_{0}\in H^{5/2}(\mathbb{T})$. Let $X(s,t)\in\Omega_{T}$ be a (local) solution of (7), s.t.

[TABLE]

and for some $\lambda>0$,

[TABLE]

There exists $T_{*}=T_{*}(T,R,\lambda)\in(0,T]$, s.t.

[TABLE]

where $X_{*}(\cdot,t)$ and $X_{0*}$ are the closest equilibrium configuration to $X(\cdot,t)$ and $X_{0}(\cdot)$ respectively. 2. 2.

Given $T^{\prime}\in(0,T_{*}]$, there exist a constant $c_{*}=c_{*}(R,\lambda,T^{\prime})>0$, s.t. if

[TABLE]

then there exists $t_{*}\in[T^{\prime}/4,T^{\prime}]$, s.t.

[TABLE]

In particular,

[TABLE]

Proof.

It is easy to see that $X_{0*}(s,t)\equiv X_{0*}(s)\in H^{5/2}(\mathbb{T})$ is the (unique, by Theorem 2) solution for (7) starting from $X_{0*}$. Consider $\tilde{X}-\widetilde{X_{0*}}$, which satisfies

[TABLE]

Similar to (94), we use the assumptions (108) and (109) and Corollary 3 with $\mu=2$ to find that for $\forall\,t\in[0,T]$ with $T\leq 1$ and $\forall\,\delta\in(0,1]$,

[TABLE]

where $C=C(R,\lambda)$. By Lemma 15 and the interpolation inequality, for $\forall\,t\in[0,T]$ and $\forall\,\delta\in(0,1]$,

[TABLE]

where $C_{3}=C_{3}(R,\lambda)$ is a constant. For simplicity, let us assume $C_{3}(R,\lambda)\geq 1$. Take $\delta=t\leq 1$ with $t\in[0,T_{*}]$ and we find that

[TABLE]

Now we take $T_{*}\leq T\leq 1$ sufficiently small, s.t.

[TABLE]

and $x(|\ln x|+1)^{2}$ is increasing in $[0,T_{*}]$. In this way, $C_{3}t\leq C_{3}T_{*}(|\ln T_{*}|+1)^{2}\leq 1/8$. By (116),

[TABLE]

By taking supremum in $t\in[0,T_{*}]$ on the left hand side, we find that

[TABLE]

In view of Remark 7, (110) immediately follows with $T_{*}$ defined in (117).

Next we shall prove the second part of the Lemma for given $T^{\prime}\in(0,T_{*}]$. Putting (119) back into the third line of (118) and take $t=T^{\prime}$, we find that

[TABLE]

In the last inequality, we introduce the notation $c=\|X_{0}-X_{0*}\|_{\dot{H}^{5/2}(\mathbb{T})}/\|X-X_{0*}\|_{L^{\infty}_{T^{\prime}}\dot{H}^{1}(\mathbb{T})}$. Denote $J(t)=\|\tilde{X}-\widetilde{X_{0*}}\|_{\dot{H}^{5/2}(\mathbb{T})}^{2}(t)$. By interpolation, for $\forall\,t\in[0,T^{\prime}]$,

[TABLE]

We multiply both sides of (120) by $c^{-2/3}J(0)^{1/3}$ and find that

[TABLE]

Now suppose the statement of the Lemma is false. Namely, for $\forall\,c>0$, there exists a solution $X^{(c)}(s,t)$ with $t\in[0,T^{\prime}]$, starting from some $X_{0}^{(c)}(s)\in H^{5/2}(\mathbb{T})$, satisfying (108), (109) and that

[TABLE]

while for $\forall\,t\in[T^{\prime}/4,T^{\prime}]$,

[TABLE]

Since (121) holds with $J$ replaced by $J^{(c)}$, we find that

[TABLE]

which implies that

[TABLE]

Since $T^{\prime}\leq T_{*}\leq 1$,

[TABLE]

Then (122) implies that

[TABLE]

Let $c_{+}$ be the unique positive real number such that the equality is achieved in (123). Then we have

[TABLE]

which implies that

[TABLE]

Here $C_{4}\geq 1$ is some constant depending only on $R$ and $\lambda$; it will be used in the proof of Theorem 3 and Theorem 4. Therefore, if $c\geq C_{4}(R,\lambda)(|\ln T^{\prime}|+1)^{2}$, (123) does not hold, which is a contradiction. Hence, we proved (112) with

[TABLE]

(113) immediately follows from (112) by virtue of Remark 7. This completes the proof. ∎

Remark 9*.*

Taking smaller $\mu$ in (115) can give sharper bound for $c_{*}$ in (125), but that is not necessary for the remaining results. ∎

Using Lemma 9, we are able to prove Theorem 3.

Proof of Theorem 3 (existence and uniqueness of global solution near equilibrium).

With no loss of generality, we assume $R_{X_{0}}=1$; otherwise, simply rescale $X_{0}$ by a factor of $R_{X_{0}}^{-1}$. Note that the contour dynamic formulation (7) is translation and scaling-invariant. Moreover, we note that the effective radius of $X(\cdot,t)$ is invariant in time, since the flow is volume-preserving.

Define

[TABLE]

We claim that there exists a universal constant $\varepsilon_{0}$ which will be clear below, for $\forall\,Z(s)\in S_{\varepsilon_{0}}$, $\|Z\|_{\dot{H}^{5/2}}\leq C$ for some universal constant $C$, and

[TABLE]

In fact,

[TABLE]

where $C_{5}>0$ is a universal constant coming from Sobolev inequality. Hence, it suffices to take $\varepsilon_{0}=\min\{(C_{5}\pi)^{-1},1\}$; that $\|Z\|_{\dot{H}^{5/2}}\leq C$ is obvious.

The above uniform estimates, together with Theorem 1 and Theorem 2, imply that there is a universal constant $T_{0}\in(0,1)$, s.t. for $\forall\,X_{0}\in S_{\varepsilon_{0}}$, there is a unique solution $X(s,t)$ for (7) in $C_{[0,T_{0}]}H^{5/2}\cap L^{2}_{T_{0}}H^{3}(\mathbb{T})$ starting from $X_{0}$, s.t.

[TABLE]

where $C_{6}$ is a universal constant. Moreover, for $\forall\,s_{1},s_{2}\in\mathbb{T}$ and $t\in[0,T_{0}]$,

[TABLE]

That is, $X(s,t)$ satisfies the assumption of Lemma 9 with $T=T_{0}$, $R=C_{6}$, and $\lambda=(2\pi)^{-1}$, which are all universal constants. Hence, by Lemma 9, there exists a universal constant $T_{*}=T_{*}(T_{0},C_{6},1/(2\pi))\in(0,T_{0}]$ such that

[TABLE]

To this end, we shall first investigate $\|\tilde{X}-\widetilde{X_{0*}}\|_{L^{\infty}_{[0,t]}\dot{H}^{1}(\mathbb{T})}$. Using the equation for $\tilde{X}$ (see (82)), we find that for $\forall\,t\in[0,T_{0}]$,

[TABLE]

In order to give an estimate for $\|\widetilde{g_{\tilde{X}}}\|_{\dot{H}^{1}}$, we should go back to (73) and (75) and apply Lemma 3. Indeed, with (128) and (129), we have

[TABLE]

where $C$ is a universal constant. Hence, by (128), (130) and Lemma 8,

[TABLE]

where $C_{7}$ is a universal constant. Here we applied Lemma 8 and defined $\zeta_{X_{0}}^{2}=\|X_{0}\|_{\dot{H}^{1}}^{2}-\|X_{0*}\|_{\dot{H}^{1}}^{2}$. The above estimate is true as long as $X_{0}\in S_{\varepsilon_{0}}$ and $t\in[0,T_{0}]$.

In what follows, we shall prove the Theorem with

[TABLE]

We also take $\xi_{*}\leq T_{*}/2$ such that

[TABLE]

where $T_{*}=T_{*}(T_{0},C_{6},1/(2\pi))\in(0,T_{0}]$ given by Lemma 9, $C_{4}=C_{4}(C_{6},1/(2\pi))$ defined in (124) and $C_{7}$ defined in (131) are all universal constants. Hence, both $\varepsilon_{*}$ and $\xi_{*}$ are universal. We fix $T^{\prime}=2\xi_{*}$, which is also a universal constant. By Lemma 8 and the assumption that $\|X_{0}-X_{0*}\|_{\dot{H}^{1}}\leq\xi_{*}$,

[TABLE]

We are going to use mathematical induction to show existence of the global solution. First we focus on the local solution $X(s,t)$ for $t\in[0,T^{\prime}]$. By (131) and (134),

[TABLE]

We apply Lemma 9 to obtain the constant $c_{*}=c_{*}(C_{6},1/(2\pi),T^{\prime})$, and claim that the assumption (111) holds if $\|X_{0}-X_{0*}\|_{\dot{H}^{5/2}(\mathbb{T})}\geq\varepsilon_{*}/2$. In fact, by (125), (133) and (135),

[TABLE]

Therefore, if $\|X_{0}-X_{0*}\|_{\dot{H}^{5/2}(\mathbb{T})}\in[\varepsilon_{*}/2,\varepsilon_{*}]$, by (136) and Lemma 9, there exists $t_{1}\in[T^{\prime}/4,T^{\prime}]$, s.t.

[TABLE]

Otherwise, if $\|X_{0}-X_{0*}\|_{\dot{H}^{5/2}(\mathbb{T})}\leq\varepsilon_{*}/2$, by the fact that $T^{\prime}\leq T_{*}$ and Lemma 9, there exists $t_{1}\in[T^{\prime}/4,T^{\prime}]$, s.t.

[TABLE]

This implies that for all $X_{0}\in S_{\varepsilon_{*}}$, we can always find $t_{1}\in[T^{\prime}/4,T^{\prime}]$, such that the unique local solution of (7) in $C_{[0,t_{1}]}H^{5/2}\cap L^{2}_{t_{1}}H^{3}(\mathbb{T})$ satisfies that

[TABLE]

We note that $T^{\prime}$ is a universal constant.

Suppose we have found $t_{k}$’s for $k\leq n$, satisfying that for $\forall\,k=1,\cdots,n$,

$t_{k}\in[T^{\prime}/4,T^{\prime}]$. 2. 2.

There exists a unique solution $X$ of (7) in $C_{[0,T_{n}]}H^{5/2}\cap L^{2}_{T_{n}}H^{3}(\mathbb{T})$, where $T_{k}=\sum_{i=1}^{k}t_{i}$ for $i=1,\cdots,n$, such that

[TABLE]

Now let us restart the equation at $t=T_{n}$. To be more precise, we consider

[TABLE]

with $X(\cdot,T_{n})\in S_{\varepsilon_{*}}=S_{\varepsilon_{0}}$ given. As before, there exists a unique local solution $X(s,t)$ for $t\in[T_{n},T_{n}+T_{0}]$ satisfying the uniform estimates (128) and (129) for solutions starting in $S_{\varepsilon_{*}}$. Moreover, with $T_{*}$ and $T^{\prime}$ defined as before,

[TABLE]

By (131),

[TABLE]

where $X_{T_{n}}(s)\triangleq X(s,T_{n})$. Since the solution obtained in $[0,T_{n}]$ satisfies the assumption of Lemma 2, by (38),

[TABLE]

Note that $\|(X_{T_{n}})_{*}\|_{\dot{H}^{1}}=\|X_{0*}\|_{\dot{H}^{1}}$. Hence, $\|X-(X_{T_{n}})_{*}\|_{L^{\infty}_{[T_{n},T_{n}+T^{\prime}]}\dot{H}^{1}}\leq(C_{7}+2)T^{\prime}$. To this end, we simply argue to show $\exists\,t_{n+1}\in[T^{\prime}/4,T^{\prime}]$, s.t. there exists a unique local solution in $C_{[0,T_{n+1}]}H^{5/2}\cap L^{2}_{T_{n+1}}H^{3}(\mathbb{T})$ with $T_{n+1}=T_{n}+t_{n+1}$ and $X(T_{n+1})\in S_{\varepsilon_{*}}$. Estimates (137) and (138) in the new time interval $[0,T_{n+1}]$ follow as before. Since $T_{n}\geq nT^{\prime}/4$ with $T^{\prime}>0$ being a universal constant, $T_{n}\rightarrow+\infty$ as $n\rightarrow\infty$. The existence of global solution is thus established. The uniqueness follows from Theorem 2. That $X_{t}\in L^{2}_{[0,+\infty),loc}H^{2}(\mathbb{T})$ follows from Theorem 1. Estimates (22), (23) and (24) are established in the induction. ∎

Remark 10*.*

Instead of (132), we may take arbitrary $\varepsilon_{*}\in(0,\varepsilon_{0}]$, and the same proof still works. ∎

The main idea in the proof of Theorem 3 is that when the string configuration is close to an equilibrium, $\|X_{0}-X_{0*}\|_{\dot{H}^{1}}$ sets a bound for $\|X-X_{*}\|_{\dot{H}^{5/2}}$ in an indirect way (at least within a short time). In the same spirit, we prove the following corollary with refined estimates. It will be useful in the proof of Theorem 4.

Corollary 4**.**

Let $X_{0}\in H^{5/2}(\mathbb{T})$ satisfy all the assumptions of Theorem 3 and let $X$ be the unique global solution of (7) starting from $X_{0}$ obtained in Theorem 3. Then for any given $\xi\in(0,\xi_{*}]$, if in addition

[TABLE]

then the solution $X$ satisfies that for $\forall\,t\geq 0$,

[TABLE]

with

[TABLE]

where $C_{4}=C_{4}(C_{6},1/(2\pi))$ and $C_{7}$ are universal constants defined in (124) and (131) respectively.

Remark 11*.*

We only define $\varepsilon_{\xi}$ for $\xi>0$ in order to avoid abusing the notation $\varepsilon_{0}$ defined in the proof of Theorem 3.

Proof.

We follow exactly the proof of Theorem 3 until the definition of $T^{\prime}$. Now we define $T^{\prime}=2\xi$ instead. It is worthwhile to note that

[TABLE]

For the solution $X(s,t)$ in $t\in[0,T^{\prime}]$, (135) still holds. With $c_{*}=C_{4}(C_{6},1/(2\pi))(|\ln T^{\prime}|+1)^{2}$ as before, we have a similar estimate as (136)

[TABLE]

Therefore, if $\|X_{0}-X_{0*}\|_{\dot{H}^{5/2}(\mathbb{T})}\geq\varepsilon_{\xi}/2$, the assumption (111) holds. By Lemma 9, there exists $t_{1}\in[T^{\prime}/4,T^{\prime}]$, s.t.

[TABLE]

Otherwise, if $\|X_{0}-X_{0*}\|_{\dot{H}^{5/2}(\mathbb{T})}\leq\varepsilon_{\xi}/2$, by Lemma 9, there exists $t_{1}\in[T^{\prime}/4,T^{\prime}]$, s.t.

[TABLE]

This implies that there always exists $t_{1}\in[T^{\prime}/4,T^{\prime}]$, s.t.

[TABLE]

Now suppose that we have found $t_{k}$’s for $k\leq n$, satisfying that for $\forall\,k=1,\cdots,n$, $t_{k}\in[T^{\prime}/4,T^{\prime}]$, and

[TABLE]

where $T_{k}=\sum_{i=1}^{k}t_{i}$ for $i=1,\cdots,n$. Now we consider the equation in $t\in[T_{n},T_{n}+T^{\prime}]$. As in (140),

[TABLE]

Here we used the energy estimate (38) again. Hence, as in (144), we have $c_{*}\|X-X_{0*}\|_{L^{\infty}_{[T_{n},T_{n}+T^{\prime}]}\dot{H}^{1}}\leq\varepsilon_{\xi}/2$. We argue as before to find that there always exists $t_{n+1}\in[T^{\prime}/4,T^{\prime}]$, s.t.

[TABLE]

By induction, there exists a sequence $\{t_{k}\}_{k\in\mathbb{Z}_{+}}$, $t_{k}\in[T^{\prime}/4,T^{\prime}]$, such that for $\forall\,k\in\mathbb{Z}_{+}$,

[TABLE]

where $T_{k}=\sum_{i=1}^{k}t_{i}\rightarrow+\infty$. Since $t_{k}\leq T^{\prime}\leq T_{*}\leq 1$, by Lemma 9,

[TABLE]

Note that here we are abusing the notation $T_{0}$ by defining $T_{0}=0$; it does not refer to the $T_{0}$ in Theorem 1. Using the fact that $X\in C_{[0,+\infty)}H^{5/2}(\mathbb{T})$, for $\forall\,t\in[T_{k-1},T_{k}]$,

[TABLE]

This completes the proof. ∎

6 Exponential Convergence to Equilibrium Configurations

In this section, we shall prove that the global-in-time solution near equilibriums obtained in Theorem 3 converges exponentially in the $H^{s}$-sense to an equilibrium configuration as $t\rightarrow+\infty$. See the statement of Theorem 4. In the sequel, we shall always consider the contour dynamic formulation (7), with $X_{0}\in H^{5/2}(\mathbb{T})$ satisfying (20) and (21) with $\varepsilon_{*},\xi_{*}>0$ found in Theorem 3. Without loss of generality, we assume $R_{X_{0}}=1$.

6.1 A lower bound of the rate of energy dissipation

A key step to prove the exponential convergence of the global solution near equilibrium is to establish a lower bound of the rate of energy dissipation $\int_{\mathbb{R}^{2}}|\nabla u_{X}|^{2}\,dx$ (see Lemma 2) in terms of $\|X-X_{*}\|_{\dot{H}^{1}}$ provided that the latter is sufficiently small.

Let $S_{\varepsilon}$ be defined as in (126). Let $\varepsilon_{*}^{\prime}\in(0,\varepsilon_{*})$ to be determined. Let $\Omega_{X}\subset\mathbb{R}^{2}$ denote the bounded open domain enclosed by $X(\mathbb{T})$ where $X\in S_{\sqrt{2}\varepsilon^{\prime}_{*}}$. Here the constant $\sqrt{2}$ comes from the estimate (22) of the global solution. Define the collection of all such domains to be

[TABLE]

We assume that $\varepsilon_{*}^{\prime}$ is sufficiently small, such that domains in $\mathcal{M}_{\varepsilon_{*}^{\prime}}$ satisfy uniform $C^{1}$-regularity condition with uniform constants (see [25] in § 4.10 for the rigorous definition). Indeed, this is achievable due to the implicit function theorem and the Sobolev embedding $H^{5/2}(\mathbb{T})\hookrightarrow C^{1,\alpha}(\mathbb{T})$ for $\forall\,\alpha\in(0,1)$.

Let $u_{X}$ be the velocity field determined by a configuration $X\in S_{\sqrt{2}\varepsilon^{\prime}_{*}}$. Let

[TABLE]

where $l$ is the arc-length parameter of $\partial\Omega_{X}$. Then by the boundary trace theorem [25],

[TABLE]

Here we used the fact that $dl=|X^{\prime}(s)|\,ds$, since $s$ is a monotone parameterization of $\partial\Omega_{X}$. Thanks to the uniform $C^{1}$-regularity of $\Omega_{X}\in\mathcal{M}_{\varepsilon_{*}^{\prime}}$, the constant $C$ is uniform for $\forall\,X\in S_{\sqrt{2}\varepsilon^{\prime}_{*}}$. We derive that

[TABLE]

Here $\bar{u}_{X}=|\mathbb{T}|^{-1}\int_{\mathbb{T}}u_{X}(X(s))\,ds$. Again, the constant $C_{5}$, which first showed up in the proof of Theorem 3, comes from the Sobolev embedding $H^{5/2}(\mathbb{T})\hookrightarrow C^{1}(\mathbb{T})$ and is independent of $X\in S_{\sqrt{2}\varepsilon^{\prime}_{*}}$. Taking $\varepsilon_{*}^{\prime}\leq(2C_{5})^{-1}$, we obtain that

[TABLE]

for some universal constant $C$ independent of $X\in S_{\sqrt{2}\varepsilon^{\prime}_{*}}$.

Now we turn to derive a lower bound for $\int_{\mathbb{T}}|u_{X}(X(s))-\bar{u}_{X}|^{2}\,ds$. We are going to perform linearization of $u_{X}(X(s))$ around the equilibrium configuration $X_{*}$. Fix $X\in S_{\sqrt{2}\varepsilon^{\prime}_{*}}$, with $\varepsilon_{*}^{\prime}\leq\min\{1,\varepsilon_{*}\}$ satisfying all the assumptions above and to be determined. Let $D(s)=X(s)-X_{*}(s)$ and

[TABLE]

By definition, $\|D\|_{\dot{H}^{5/2}(\mathbb{T})}\leq\sqrt{2}\varepsilon_{*}^{\prime}$. It is easy to show that with $\varepsilon_{*}^{\prime}$ sufficiently small,

[TABLE]

where $C$ is a universal constant. Note that with $\varepsilon_{*}^{\prime}$ being sufficiently small, $X_{\eta}$ is also a non-self-intersecting string configuration, but it may not be in $S_{\sqrt{2}\varepsilon^{\prime}_{*}}$, as $R_{X_{\eta}}=1$ is not necessarily true.

The following lemma shows that $u_{X}(X(s))$, as a function of $X$, can be well approximated by linearization around $X_{*}$.

Lemma 10**.**

Assume $\varepsilon_{*}^{\prime}\leq\min\{1,\varepsilon_{*},(2C_{5})^{-1}\}$ is sufficiently small such that domains in $\mathcal{M}_{\varepsilon_{*}^{\prime}}$ satisfy uniform $C^{1}$-regularity condition with uniform constants, and (147) and (148) hold. Then

[TABLE]

where

[TABLE]

with $C$ being a universal constant.

Proof.

Recall that $u_{X}$ is given by (33). By (57),

[TABLE]

where $L_{X}=L_{X}(s,s^{\prime})$ and $M_{X}=M_{X}(s,s^{\prime})$ are defined in (39) and (40). The subscripts stress that they are determined by $X$. Since $u_{X_{*}}\equiv 0$, by mean value theorem with respect to $\eta$, there exists an $\eta_{*}\in[0,1]$ such that,

[TABLE]

In Lemma 17 in the Appendix A.3, we will show that the $\eta$-derivative and the integral in $u_{X_{\eta}}$ commute. Hence,

[TABLE]

We then replace all the $X_{\eta_{*}}$ in (151) by $X_{*}$, i.e. $\eta=0$, and introduce some error denoted by $\mathcal{R}_{X}(s)$. In this way, we obtain the representation (149). To show (150), for conciseness, we only look at one part of $\mathcal{R}_{X}(s)$, which is the error in approximating the first term on the right hand side of (151),

[TABLE]

Note that (54) and (148) imply that $|L_{X_{*}}|,|L_{X_{\eta_{*}}}|\geq C$ for some universal constant $C$. By Lemma 3, (147), and (148),

[TABLE]

In a similar manner, we can prove the same bound for the other terms in $\mathcal{R}_{X}$. Thus we proved (150). ∎

The following lemma calculates the leading term of $u_{X}(X(s))$ in (149).

Lemma 11**.**

Assume $X_{*}(s)=(\cos s,\sin s)^{T}$. Then

[TABLE]

Here $\mathcal{H}$ denotes the Hilbert transform on $\mathbb{T}$ [22].

The proof is simply a long calculation. We leave it to the Appendix A.3.

Lemma 12**.**

There is a universal $\varepsilon_{*}^{\prime}>0$ and a universal constant $C>0$, such that

[TABLE]

where $S_{\varepsilon}$ is defined in (126). In particular, this implies that

[TABLE]

with some universal constant $C>0$.

Proof.

We always assume that $\varepsilon_{*}^{\prime}$ satisfies the assumptions in Lemma 10. By Lemma 11,

[TABLE]

Obviously,

[TABLE]

By Parseval’s identity and the fact that $\hat{D}_{-k}=\overline{\hat{D}_{k}}$,

[TABLE]

Recall that $D(s)$ satisfies the constraints (99) and (100), with $Y_{*}$ replaced by $X_{*}$. (100) imples that $(\mathrm{Im}\,\hat{D}_{1,1}-\mathrm{Re}\,\hat{D}_{1,2})^{2}=2(\mathrm{Im}\,\hat{D}_{1,1})^{2}+2(\mathrm{Re}\,\hat{D}_{1,2})^{2}$; (99) together with (101) implies that

[TABLE]

Here we used the fact that $D$ has mean zero on $\mathbb{T}$, and thus $\|D\|_{L^{2}(\mathbb{T})}\leq C\|D\|_{\dot{H}^{5/2}(\mathbb{T})}\leq C\varepsilon_{*}^{\prime}$. Hence, we use (156) to derive that

[TABLE]

Here $C$ is a universal constant. To this end, we use Lemma 10 and (155) to derive that

[TABLE]

Again, $C$ is a universal constant. Taking $\varepsilon_{*}^{\prime}$ sufficiently small, but still universal, we proved the desired lower bound (153) with some universal constant $C$. (154) follows immediately from (145), (153) and Lemma 8. ∎

With Lemma 12, we conclude by (37) and Lemma 8 that

Corollary 5**.**

Let $X_{0}$ satisfy all the assumptions in Theorem 3 so that $X$ is the unique global-in-time solution of (7) starting from $X_{0}$. Then there exist universal constants $\varepsilon_{*}^{\prime},\alpha>0$, such that if in addition $X(\cdot,t)\in S_{\sqrt{2}\varepsilon_{*}^{\prime}}$,

[TABLE]

where $S_{\varepsilon}$ is defined in (126).

6.2 Proof of exponential convergence to equilibrium configurations

Before we prove Theorem 4, we first state the following simple lemma.

Lemma 13**.**

Let $\varepsilon_{\xi}$ be defined as in (143), i.e. $\varepsilon_{\xi}=2C_{4}(C_{6},1/(2\pi))(C_{7}+2)(|\ln(2\xi)|+1)^{2}(2\xi)$, where $C_{4}$, $C_{6}$ and $C_{7}$ are universal constant. Then $\varepsilon_{\xi}$ is increasing on $\xi\in(0,1/(2e)]$. Moreover, for $\forall\,\xi\in(0,1/(2e)]$ and $\forall\,c\geq e$,

[TABLE]

The first inequality is true even for $\forall\,c\geq 1$. $\beta_{c}$ is decreasing in $c\geq e$ and $\beta_{c}\leq\frac{9}{4e}<1$ for $\forall\,c\geq e$.

Its proof is a simple calculation, which we shall omit. Now we are able to prove Theorem 4.

Proof of Theorem 4.

As before, we assume $R_{X_{0}}=1$. For convenience, we denote

[TABLE]

Note that the case $\mathcal{F}(0)=0$ is trivial; we shall only discuss the case $\mathcal{F}(0)>0$ in the sequel.

Let $\varepsilon_{*}^{\prime}$ be defined as in Corollary 5. We may assume $\varepsilon_{*}^{\prime}\leq 1$. Take $\xi_{**}\leq 1/(16e)$ such that

[TABLE]

where $C_{4}$, $C_{6}$ and $C_{7}$ are universal constants defined in (124), (128) and (131) respectively. Such $\xi_{**}$ is uniquely achievable as a universal constant by the assumptions $C_{4}\geq 1$ and $\varepsilon_{*}^{\prime}\leq 1$, since it is required that

[TABLE]

while $x(|\ln x|+1)^{2}$ monotonically maps $(0,1/e]$ onto $(0,4/e]$. Hence, by Corollary 4, the solution $X$ satisfies that for $\forall\,t\geq 0$,

[TABLE]

Here we used Lemma 13 to find that $\varepsilon_{\mathcal{F}(0)}\leq\varepsilon_{8\xi_{**}}=\sqrt{2}\varepsilon_{*}^{\prime}$. If $2\mathcal{G}(0)\leq\varepsilon_{\mathcal{F}(0)}$, we take $t_{*}=0$; otherwise, take $t_{*}$ such that

[TABLE]

Hence, we have $X(t)\in S_{\varepsilon_{\mathcal{F}(0)}}\subset S_{\sqrt{2}\varepsilon_{*}^{\prime}}$ if $t\geq t_{*}$, which allows us to apply Corollary 5 for $t\geq t_{*}$. By Lemma 2 and Lemma 8, we derive that

[TABLE]

By Corollary 5, for some universal $\alpha>0$, and $\forall\,t>0$,

[TABLE]

Note that $8\mathcal{F}(0)\in(0,1/(2e)]$ by the assumption on $\xi_{**}$, in which interval $\varepsilon_{\xi}$ is increasing in $\xi$. Without loss of generality, we may assume $\alpha<\frac{1}{8}$. We additionally take $t_{**}>0$ to be a universal constant such that

[TABLE]

To this end, we shall use mathematical induction to show (25). Let us summarize what has been proved so far:

For $\forall\,t\geq 0$, $\mathcal{G}(t)\leq\max\{2e^{-t/8}\mathcal{G}(0),\varepsilon_{\mathcal{F}(0)}\}$. 2. 2.

Wit the choice of $t_{*}$ in (160), for $\forall\,t\in[t_{*},t_{*}+2t_{**}]$, $\mathcal{G}(t)\leq\varepsilon_{\mathcal{F}(0)}\leq\sqrt{2}\varepsilon_{*}^{\prime}$. 3. 3.

For $\forall\,t>0$, $\mathcal{F}(t_{*}+t)\leq 8\mathcal{F}(0)e^{-\alpha t}$. In particular, with the choice of $t_{**}$ in (161), for $\forall\,k\in\mathbb{Z}_{+}$, $k\geq 2$, $\mathcal{F}(t_{*}+kt_{**})\leq e^{-\alpha(k-1)t_{**}}\mathcal{F}(0)$.

Suppose that

[TABLE]

has been proved for some $k\geq 2$, $k\in\mathbb{Z}_{+}$ (indeed, the case $k=2$ has been established above). By Corollary 4, $\forall\,t\in[0,t_{**}]$,

[TABLE]

In particular,

[TABLE]

We claim that with the choice of $t_{**}$ in (161), the first term on the right hand side is always smaller. Indeed, by (161), and the lower bound in Lemma 13,

[TABLE]

Hence, we proved that

[TABLE]

Therefore, by induction, (162) is true for all $k\in\mathbb{Z}_{+}$, $k\geq 2$; so is (163). With the choice of $t_{**}$, we use (163) and the upper bound in Lemma 13 to derive that for $\forall\,t\in[0,t_{**}]$,

[TABLE]

Note that $t_{**}$ and $\beta_{8}<1$ are both universal constants. Hence, combining this with the fact that $\mathcal{G}(t_{*}+t)\leq\varepsilon_{\mathcal{F}(0)}$ for $\forall\,t\geq 0$, we find that there exist universal constants $\alpha_{*}\leq 1/8$ and $C>1$, such that for $\forall\,t\geq 0$,

[TABLE]

If $t_{*}=0$, we readily proved that for $\forall\,t\geq 0$,

[TABLE]

where $C$ and $\alpha_{*}$ are universal constants. If $t_{*}>0$, by (160) and the fact that $\alpha_{*}\leq 1/8$, $\varepsilon_{\mathcal{F}(0)}=2e^{-t_{*}/8}\mathcal{G}(0)\leq 2e^{-\alpha_{*}t_{*}}\mathcal{G}(0)$. Hence, by (164), for $\forall\,t\geq 0$,

[TABLE]

On the other hand, since $\alpha_{*}\leq 1/8$, we also know that for $t\in[0,t_{*}]$,

[TABLE]

Combining (166) and (167), we proved that

[TABLE]

with some universal constants $\alpha_{*}$ and $C$. Combining (165) and (168), we complete the proof of (25).

In order to prove (26), we use the fact $u_{X_{*}}(x)\equiv 0$ to derive that

[TABLE]

By Corollary 1 and Corollary 2,

[TABLE]

Here $C$ is a universal constant thanks to the uniform estimates of solutions obtained in Theorem 3. Hence, by (25),

[TABLE]

Here $B(X_{0})$ is defined in (25) and $C$ is a universal constant. This implies that $X(s,t)$ is a Cauchy sequence in $H^{1}(\mathbb{T})$, which converges to some $X_{\infty}(s)\in H^{1}(\mathbb{T})$. Take $t^{\prime}\rightarrow+\infty$ in (169) and we find

[TABLE]

On the other hand, by virtue of the bound (24) of $\|X\|_{\dot{H}^{5/2}(\mathbb{T})}(t)$, we may take $\tilde{X}_{w,\infty}\in H^{5/2}(\mathbb{T})$ as an arbitrary weak limit (up to a subsequence) of $\{\tilde{X}(t)\}_{t\geq 0}$ in $H^{5/2}(\mathbb{T})$. Note that we only have the bound on the $H^{5/2}$-seminorm of $X(t)$, so we can only extract weak limit for $\{\tilde{X}(t)\}_{t\geq 0}$ instead of $\{X(t)\}_{t\geq 0}$ at this moment. By compact Sobolev embedding, $\tilde{X}_{w,\infty}$ is a strong $H^{1}(\mathbb{T})$-limit of a subsequence of $\{\tilde{X}(t)\}_{t\geq 0}$. Since $\tilde{X}(t)\rightarrow\tilde{X}_{\infty}$ in $H^{1}(\mathbb{T})$, one must have $\tilde{X}_{w,\infty}=\tilde{X}_{\infty}$. And this is true for all weak limits of $\{\tilde{X}(t)\}_{t\geq 0}$. Hence, $X_{\infty}\in H^{5/2}(\mathbb{T})$ and satisfies $\|X_{\infty}\|_{\dot{H}^{5/2}(\mathbb{T})}\leq C$, with the same universal constant $C$ as in (24). By (25) and the convergence $X(t)\rightarrow X_{\infty}$ in $H^{1}(\mathbb{T})$, we know that $\|X_{\infty}-X_{\infty,*}\|_{\dot{H}^{1}(\mathbb{T})}=0$. Hence $X_{\infty}=X_{\infty,*}$ is an equilibrium configuration.

To this end, we derive (26) as follows

[TABLE]

Note that both $X_{*}$ and $X_{\infty}$ are equilibrium configurations. Since $X_{\infty}(s,t)-X_{*}(s,t)$ as a function of $s\in\mathbb{T}$ only contains Fourier modes with wave numbers [math] and $\pm 1$, we can replace the $H^{5/2}$-seminorm in the last term by $H^{1}$-seminorm without changing its value, i.e.

[TABLE]

In the last inequality, we used (25) and (170). This completes the proof of (26). ∎

7 Conclusion and Discussion

In this paper, we transform the Stokes immersed boundary problem (1)-(5) in two dimensions into a contour dynamic formulation (7) via the fundamental solution of the Stokes equation. We proved that there exists a unique local solution of the contour dynamic formulation (Theorem 1 and Theorem 2), provided that the initial data is an $H^{5/2}$-function in Lagrangian coordinate and satisfies the well-stretched condition (6). If in addition the initial configuration is sufficiently close to an equilibrium, the solution should exist globally in time (Theorem 3), and it converges exponentially to an equilibrium as $t\rightarrow+\infty$ (Theorem 4). Regularity of the ambient flow field can thus be recovered through the fundamental solution of the Stokes equation (Lemma 1 and Lemma 2).

In the contour dynamic formulation (7), the string motion is given by a singular integral, which depends nonlinearly functional on the string configuration. The starting point of the proofs in this paper is that the principal part of the singular integral in the contour dynamic formulation introduces dissipation, which essentially results from the dissipation in the Stokes flow. Then it suffices to show that the remainder term is regular in some sense and can be well-controlled by the dissipation. The same approach may also apply to the higher dimensional case, where a 2-D closed membrane is immersed and moving in a 3-D Stokes flow, although the description of the 2-D membrane needs some extra efforts. Note that the equilibrium shape of the membrane may not necessarily be a sphere. We shall address this problem in a forthcoming work.

In this paper, we only consider the simplest case where the 1-D string is modeled by a Hookean material with zero resting length in the force-free state. See the local elastic energy density (4). In particular, the material always tends to shorten its length in all time. Other types of elastic constitutive law can be also considered. In fact, most of the discussion in this paper may also apply to more general elastic energy of other forms. We do not dig deep into this topic here, but it would be interesting to find out what conditions are needed for the energy density so that the current approach still works.

Appendix A Appendix

A.1 Study of the Flow Field

In this section, we shall prove Lemma 1 and Lemma 2 that characterize properties of the flow field $u_{X}$. Roughly speaking, Lemma 1 claims that under certain assumptions on $X$, $u_{X}\in C(\mathbb{R}^{2})$ and $\nabla u_{X}\in L^{2}(\mathbb{R}^{2})$; while Lemma 2 proves an energy estimate of the whole system, which says that under certain assumptions on $X$, the decrease in the elastic energy of the string is fully accounted by the energy dissipation in the Stokes flow. See their precise statements in Section 2.2.

Proof of Lemma 1.

If $x\not\in\Gamma_{t}$, since $G(x-X(s^{\prime},t))$ in (28) is smooth at $x$, $u_{X}$ is also smooth at $x$. We then turn to show continuity of $u_{X}$ at $X(s,t)\in\Gamma_{t}$.

Take any arbitrary $x\in\mathbb{R}^{2}$, and let $s_{x}$ be defined as in (30). Note that $s_{x}$ may not be unique; take an arbitrary one if it is the case. We first show that

[TABLE]

Indeed, if $|s^{\prime}-s_{x}|\leq 2\lambda^{-1}\mathrm{dist}(x,X(\mathbb{T}))$, then by definition of $s_{x}$,

[TABLE]

If $|s^{\prime}-s_{x}|\geq 2\lambda^{-1}\mathrm{dist}(x,X(\mathbb{T}))$, by triangle inequality,

[TABLE]

This proves (171). If $s_{x}$ and $s_{x}^{\prime}$ both satisfy (30), (171) implies that $|s_{x}-s_{x}^{\prime}|\leq 2\lambda^{-1}\mathrm{dist}(x,X(\mathbb{T}))$.

Next, we shall take the limit $x\rightarrow X(s,t)$ and apply dominated convergence theorem to (31). Here we do not assume $x\not\in\Gamma_{t}$. For $s^{\prime}\not=s$, using the formula in (31) and (33), it is easy to find that

[TABLE]

Here we used the fact that $X^{\prime}(s_{x})\rightarrow X^{\prime}(s)$ as $x\rightarrow X(s,t)$. This is because (171) implies that $s_{x}\rightarrow s$ and $X\in H^{2}(\mathbb{T})$ implies that $X^{\prime}\in C^{1/2}(\mathbb{T})$.

On the other hand, by (171),

[TABLE]

Here $\mathcal{M}$ is the centered Hardy-Littlewood maximal operator on $\mathbb{T}$ [22]. Note that the bound is independent of $x$. Since $\mathcal{M}$ is bounded from $L^{2}(\mathbb{T})$ to $L^{2}(\mathbb{T})$, $C\lambda^{-1}|X^{\prime}(s^{\prime})||\mathcal{M}X^{\prime\prime}(s^{\prime})|\in L^{1}(\mathbb{T})$. Therefore, by dominated convergence theorem, $u_{X}(x)\rightarrow u_{X}(X(s))$ as $x\rightarrow X(s)$. This completes the proof of the continuity of $u_{X}$.

To show $\nabla u_{X}\in L^{2}(\mathbb{R}^{2})$, we first introduce a mollifier $\varphi(x)\in C^{\infty}_{0}(\mathbb{R}^{2})$, such that $\varphi\geq 0$, $\mathrm{supp}\,\varphi\subset B(0,1)$ and $\int_{\mathbb{R}^{2}}\varphi(x)\,dx=1$. Define $\varphi_{\varepsilon}(x)=\varepsilon^{-2}\varphi(x/\varepsilon)$. Let $f_{\varepsilon}=\varphi_{\varepsilon}*f$ and let $(u_{\varepsilon},p_{\varepsilon})$ solves the Stokes equation with $\varepsilon<1$,

[TABLE]

It is obvious that $u_{\varepsilon}=\varphi_{\varepsilon}*u_{X}$ and $p_{\varepsilon}=\varphi_{\varepsilon}*p_{X}$. On the other hand, since $f\in\mathscr{M}(\mathbb{R}^{2})$ under the assumption of the lemma, $f_{\varepsilon}$ is smooth and so are $u_{\varepsilon}$, and $p_{\varepsilon}$.

We also introduce a cut-off function $\phi\in C^{\infty}_{0}(\mathbb{R}^{2})$, such that $\phi\geq 0$; $\phi(y)=1$ for $|y|\leq 1$; $\phi(y)=0$ for $|y|\geq 2$; and $|\nabla\phi(y)|\leq C$ for some universal constant $C$. Define $\phi_{r}(x)\triangleq\phi(x/r)$. For given $t$, assume $\Gamma_{t}\subset B_{R}(0)=\{x\in\mathbb{R}^{2}:\;|x|<R\}$ with some $R>2$. We multiply the regularized Stokes equation (172) on both sides by $\phi_{r}u_{\varepsilon}$, with $r=2R+1$, and take integral on $\mathbb{R}^{2}$. By integration by parts, we obtain that

[TABLE]

Note that $\nabla\phi$ is supported on $B_{2r}(0)\backslash B_{r}(0)$, which is away from $X(\cdot,t)$. In the region $B_{6R}(0)\backslash B_{2R}(0)$, which contains an $\varepsilon$-neighborhood of $B_{2r}(0)\backslash B_{r}(0)$ since $R>2>2\varepsilon$, $u_{X}$, $\nabla u_{X}$ and $p_{X}$ have the following $L^{\infty}$-bound due to (31) and (32) with $C_{x}=0$.

[TABLE]

Therefore, the regularized solutions also enjoy similar bounds in $B_{2r}(0)\backslash B_{r}(0)$, namely

[TABLE]

Note that the constants $C$’s are uniform in $\varepsilon$. Applying these estimates in (173), we find that

[TABLE]

It is known that

[TABLE]

This gives

[TABLE]

Therefore, by (34),

[TABLE]

Here we used the fact that $\Gamma_{t}\subset B_{r}(0)$. This implies that for fixed $r$ and any sequence $\varepsilon_{k}\rightarrow 0^{+}$, there exists a subsequence $\varepsilon_{k_{l}}\rightarrow 0^{+}$ and $\tilde{u}\in H^{1}(B_{r}(0))$ such that $\nabla u_{\varepsilon_{k_{l}}}\rightharpoonup\nabla\tilde{u}$ in $L^{2}(B_{r}(0))$. Combining this with the fact $u_{\varepsilon}\rightarrow u_{X}$ in $C(B_{r}(0))$, we can even obtain $u_{\varepsilon_{k_{l}}}\rightarrow\tilde{u}$ in $L^{2}(B_{r}(0))$ and thus $\tilde{u}=u_{X}\in H^{1}(B_{r}(0))$. Since the sequence $\{\varepsilon_{k}\}$ is arbitrary and $r$ (or equivalently $R$) can be arbitrarily large, we conclude that $u_{X}\in H^{1}_{loc}(\mathbb{R}^{2})$ and $\nabla u_{\varepsilon}\rightarrow\nabla u_{X}$ in $L^{2}_{loc}(\mathbb{R}^{2})$. Here we obtain local strong convergence as a property of the mollification. This together with (178) implies that

[TABLE]

Take $r\rightarrow\infty$ and we find

[TABLE]

This completes the proof. ∎

As is mentioned before, Lemma 2 is devoted to an energy estimate of the system, which will be used in the proof of Theorem 3 and Theorem 4.

Proof of Lemma 2.

Since $X\in C_{T}H^{2}(\mathbb{T})$, $\Gamma_{t}$ stays in a bounded set in $t\in[0,T]$. We may assume $\Gamma_{t}\subset B_{R}(0)=\{x\in\mathbb{R}^{2}:\;|x|<R\}$ for $t\in[0,T]$ with some $R>2$. Again let $r=2R+1$.

We start from the local energy estimate of the regularized solution in the proof of Lemma 1. By (173) and the decay estimates (174)-(176), we find that

[TABLE]

By the convergence (177) and $\nabla u_{\varepsilon}\rightarrow\nabla u_{X}$ in $L^{2}_{loc}(\mathbb{R}^{2})$, it becomes

[TABLE]

Take $r\rightarrow\infty$ and we find

[TABLE]

The last equality, estabilished in [8] in Chapter III, § 1.4, holds in the scalar distribution sense. By some limiting argument and the assumption that $X\in C_{T}H^{2}(\mathbb{T})$, we may take integral on both sides of (180) in $t$ from [math] to $T$ to obtain (38). ∎

A.2 A Priori Estimates Involving $\mathcal{L}$

We state several a priori estimates involving the operator $\mathcal{L}$ without proofs.

Lemma 14**.**

For $\forall\,v_{0}\in H^{l}(\mathbb{T})$ with arbitrary $l\in\mathbb{R}_{+}$,

$\|\mathrm{e}^{t\mathcal{L}}v_{0}\|_{\dot{H}^{l}(\mathbb{T})}\leq\mathrm{e}^{-t/4}\|v_{0}\|_{\dot{H}^{l}(\mathbb{T})}$; 2. 2.

$\mathrm{e}^{t\mathcal{L}}v_{0}\in C([0,+\infty);H^{l}(\mathbb{T}))$; 3. 3.

$\mathrm{e}^{t\mathcal{L}}v_{0}\rightarrow v_{0}$* in $H^{l}(\mathbb{T})$ as $t\rightarrow 0^{+}$.*

Lemma 15**.**

Given $T>0$, let $h\in L^{2}_{T}H^{l}(\mathbb{T})$. The model equation

[TABLE]

has a unique solution $v\in L_{T}^{\infty}H^{l+1/2}\cap L_{T}^{2}H^{l+1}(\mathbb{T})$ with $v_{t}\in L_{T}^{2}H^{l}(\mathbb{T})$, satisfying the following a priori estimates: for $\forall\,t\in[0,T]$,

[TABLE] 2. 2.

[TABLE]

Hence,

[TABLE]

In particular,

[TABLE] 3. 3.

[TABLE]

A.3 Auxiliary Calculations

The following lemma is used to derive a simplification of $\Gamma_{1}(s,s^{\prime})$ used in Section 3.2.

Lemma 16**.**

Let $\Gamma_{1}(s,s^{\prime})$ be defined by (73), i.e.

[TABLE]

Then with the notations introduced in (39), for $\forall\,s,s^{\prime}\in\mathbb{T}$, $s^{\prime}\not=s$, we have (74).

Proof.

We shall simplify $\Gamma_{1}(s,s^{\prime})$ by exploring cancelations between its terms. Using the notations introduced in (39), we calculate that

[TABLE]

We simplify the first four terms by plugging in $X^{\prime}(s^{\prime})=X^{\prime}(s)+\tau M$,

[TABLE]

Hence,

[TABLE]

Using $X^{\prime}(s)=L-\tau N$, we calculate that

[TABLE]

Similarly,

[TABLE]

For $A_{3}$, we split the last term into four and look for cancellations with the other four terms. That is,

[TABLE]

Here we used $X^{\prime}(s)=L-\tau N$ and $X^{\prime}(s^{\prime})=L+\tau(M-N)$. Therefore,

[TABLE]

Combining (184), (185) and (186), we obtain the desired simplification (74). ∎

The following lemma states that the $\eta$-derivative of $u_{X_{\eta}}(X_{\eta}(s))$ can commute with the integral in the representation of $u_{X_{\eta}}(X_{\eta}(s))$. It will be used in the proofs of Lemma 10 and Lemma 11.

Lemma 17**.**

Let $\eta\in[0,1]$ and let $X_{\eta}$ be defined as in (146). Let

[TABLE]

Then under the same assumptions as in Lemma 10, for $\forall\,\eta\in[0,1]$,

[TABLE]

Proof.

By definition,

[TABLE]

We shall check the conditions of the dominated convergence theorem to show that the limit and the integral commute. In particular, we need to show that for $\forall\,s\in\mathbb{T}$, there exists an $s^{\prime}$-integrable function $h(s,s^{\prime})$, such that for $\forall\,\eta_{1},\eta_{2}\in[0,1]$, $\eta_{1}\not=\eta_{2}$,

[TABLE]

By definition, we calculate that

[TABLE]

For conciseness, we only show estimate of one of the terms above. Thanks to the uniform estimates (147) and (148),

[TABLE]

where $C$ is a universal constant. The other terms in (189) can be estimated in a similar fashion. Hence, we obtain that, for $s^{\prime}\not=s$,

[TABLE]

where $\mathcal{M}$ is again the centered Hardy-Littlewood maximal operator on $\mathbb{T}$. Hence (188) is proved with $h(s,s^{\prime})=C(|\mathcal{M}X^{\prime\prime}_{*}(s^{\prime})|+|\mathcal{M}D^{\prime\prime}(s^{\prime})|)\in L^{1}_{s^{\prime}}(\mathbb{T})$. Note that $h$ is independent of $\eta_{1}$, $\eta_{2}$ and $s$. By dominated convergence theorem, the limit and the integral in (187) commute, which proves the Lemma. ∎

Finally, we come to prove Lemma 11, which calculates the leading term of $u_{X}(X(s))$ in (149).

Proof of Lemma 11.

This time, we used (28) as the representation of $u_{X_{\eta}}(X_{\eta}(s))$,

[TABLE]

It has been showed before (see Section 2.1) that, with the well-stretched condition (6), the integral with logarithmic singularity is well-defined. Hence, by virtue of Lemma 17,

[TABLE]

We split $X_{*}^{\prime\prime}(s^{\prime})$ into two terms, namely, $X_{*}^{\prime\prime}(s^{\prime})=-X_{*}(s^{\prime})=-\frac{1}{2}(X_{*}(s^{\prime})-X_{*}(s))-\frac{1}{2}(X_{*}(s^{\prime})+X_{*}(s))$.

[TABLE]

Here we used the fact that $(X_{*}(s^{\prime})-X_{*}(s))\cdot(X_{*}(s^{\prime})+X_{*}(s))=0$ and $\int_{\mathbb{T}}D(s^{\prime})\,ds^{\prime}=0$. Since $X_{*}(s)=(\cos s,\sin s)^{T}$, we plug this into (190) and find that

[TABLE]

Here $\mathcal{H}$ is the Hilbert transform on $\mathbb{T}$ [22]. Moreover,

[TABLE]

and

[TABLE]

In the last line, we used the fact that only the Fourier modes of $D^{\prime\prime}(s^{\prime})$ with wave numbers $\pm 1$ contributes to the integral, and thus replacing $D^{\prime\prime}(s^{\prime})$ by $-D(s^{\prime})$ does not change the integral. Combining the above calculations with (190), we find the desired result (152). ∎

Acknowledgement

We want to thank Prof. Michael Shelley and Prof. Wei Wang for helpful discussions. We also would like to thank Prof. Jim Portegies for his early notes that contain some preliminary but inspiring calculations on this problem. We also thank anonymous referee for helpful comments and suggestions. This work was partially supported by National Science Foundation under Award Number DMS-1501000.