
==== Front
J Evol Equ
J Evol Equ
Journal of Evolution Equations
1424-3199
1424-3202
Springer International Publishing Cham

38966404
988
10.1007/s00028-024-00988-1
Article
Existence and convergence of the length-preserving elastic flow of clamped curves
http://orcid.org/0000-0002-3030-414X
Rupp Fabian fabian.rupp@univie.ac.at

1
Spener Adrian 2
1 https://ror.org/03prydq77 grid.10420.37 0000 0001 2286 1424 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
2 https://ror.org/032000t02 grid.6582.9 0000 0004 1936 9748 Institute of Applied Analysis, Ulm University, Helmholtzstraße 18, 89081 Ulm, Germany
2 7 2024
2 7 2024
2024
24 3 5928 5 2024
© The Author(s) 2024
2024
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We study the evolution of curves with fixed length and clamped boundary conditions moving by the negative L2-gradient flow of the elastic energy. For any initial curve lying merely in the energy space we show existence and parabolic smoothing of the solution. Applying previous results on long-time existence and proving a constrained Łojasiewicz–Simon gradient inequality we furthermore show convergence to a critical point as time tends to infinity.

Keywords

Nonlocal geometric evolution equation
Clamped boundary conditions
Elastic energy
Willmore functional
Łojasiewicz–Simon gradient inequality
Mathematics Subject Classification

Primary 53E40
Secondary 35K52
35K55
http://dx.doi.org/10.13039/501100001659 Deutsche Forschungsgemeinschaft 404870139 http://dx.doi.org/10.13039/501100002428 Austrian Science Fund 32788-N ESP 557 Rupp Fabian University of ViennaOpen access funding provided by University of Vienna.

issue-copyright-statement© Springer Nature Switzerland AG 2024
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pmcIntroduction and main results

For an immersed curve f:I:=[0,1]→Rd, d≥2, its Euler–Bernoulli energy or simply elastic energy is defined byE(f):=12∫I|κ→|2ds.

Here ds:=γdx, where γ:=|∂xf| denotes the arc-length element, and κ→:=∂s2f is the curvature vector field, where ∂s:=γ-1∂x is the arc-length derivative.

In this article, we deform an initial curve f0 in such a way that its elastic energy decreases as fast as possible, while keeping the (total) length L(f):=∫Ids fixed. This yields the geometric evolution equation1.1 ∂t⊥f=-∇s2κ→-12|κ→|2κ→+λκ→.

Here ∇s denotes the connection on the normal bundle along f, i.e. ∇s:=P⊥∂s, where P⊥X:=X⊥f:=X-⟨X,∂sf⟩∂sf denotes the orthogonal projection along f of any vector field X along f. The Lagrange multiplier λ depends on the solution f and is given by1.2 λ(f)=λ(f)(t)=∫I∇s2κ→+12|κ→|2κ→,κ→ds∫I|κ→|2ds.

Here ⟨·,·⟩ denotes the Euclidean inner product. Note that the evolution (1.1) is geometric, i.e. if a smooth f satisfies (1.1), then for any smooth family of reparametrizations Φ:[0,T)×I→I so does f^(t,x):=(f∘Φ)(t,x):=f(t,Φ(t,x)). In addition to the evolution (1.1), we prescribe clamped boundary conditions, fixing position and the unit tangent of the curve at the endpoints of I. For an immersed curve f0 we hence study the following initial boundary value problem.1.3 ∂tf=-∇s2κ→-12|κ→|2κ→+λκ→+θ∂sfon(0,T)×If(0,x)=f0(x)forx∈If(t,y)=pyfor0≤t<T,y∈∂I∂sf(t,y)=τyfor0≤t<T,y∈∂I,

where the unknown θ:[0,T)×I→R, θ=⟨∂tf,∂sf⟩ is the tangential velocity. By the integral representation of λ, (1.3) becomes a nonlocal quasilinear system which is also degenerate parabolic by its geometric nature. We assume that the boundary data py∈Rd,τy∈Sd-1⊂Rd satisfy the compatibility conditions1.4 f0(y)=pyand∂sf0(y)=τyfory∈∂I.

Note that (1.3) is preserved under a smooth family of reparametrizations Φ which keeps the boundary ∂I fixed, where the tangential velocity might change.

It is not difficult to see that λ is chosen exactly in such a way that the length remains fixed during the flow, since along any sufficiently smooth solution of (1.3) we have1.5 ddtL(f)=-∫I⟨κ→,∂tf⟩ds=∫I∇s2κ→+12|κ→|2κ→,κ→ds-λ∫I|κ→|2ds=0,

whereas the energy indeed decreases since by (1.5)1.6 ddtE(f)=∫I⟨∇E(f),∂tf⟩ds=∫I⟨∇E(f)-λκ→,∂t⊥f⟩ds=-∫I|∂t⊥f|2ds,

using that the L2(dsf)-gradient of E is given by ∇E(f)=∇s2κ→+12|κ→|2κ→. In the above calculations, we also used the fact that all boundary terms vanish due to the boundary conditions. In order for λ to be well-defined, we need to ensure that f(t):=f(t,·) is not a piece of a straight line. This can be guaranteed with no restrictions on τ0,τ1 by requiring1.7 |p0-p1|<ℓ:=L(f0),

so E(f0)>0, see Sect. 2.1 for a more detailed analysis of λ.

In [10], long-time existence for smooth solutions of (1.3) with tangential velocity θ≡0 under assumption (1.7) was shown with the help of interpolation inequalities. For the short-time existence the authors of [10] refer to the beginning of Section 3 in [17], where the short-time existence in the setting of Hölder spaces is only sketched for the case of closed curves. Moreover, the uniform bounds in [10, Theorem 1.1] imply subconvergence after reparametrization as t→∞. However, different sequences could still have different limits.

The contribution of this paper is twofold: First, we give a rigorous and fairly concise proof of short-time existence and parabolic smoothing for the elastic flow (1.3). Compared to the previous classical existence results for elastic flows, where the initial datum is assumed to be smooth [10, 17, 42] or at least with Hölder continuous second derivative [50], one major improvement is that we allow for rough initial values, lying merely in the natural energy space, see Remark 2.11 for a detailed discussion. In contrast to existence theorems relying on the minimizing movement scheme (cf. [5, 6, 33, 39, 41]), our methods rely on maximal regularity, yielding here smooth solutions, cf. Theorem 1.1 below, while still allowing for rough initial data. The price for this substantial improvement is that the necessary contraction estimates become quite technical and rely delicately on the precise structure of (1.3). This is the first existence result for an elastic flow with general initial data of such weak regularity.

Theorem 1.1

Let f0∈W2,2(I;Rd) be immersed, let p0,p1∈Rd and τ0,τ1∈Sd-1 satisfy (1.4) and (1.7). Then, there exist T>0 and a solution f∈W1,2(0,T;L2(I,Rd))∩L2(0,T;W4,2(I;Rd)) of (1.3).

Moreover, we show that under the assumptions (1.4) and (1.7), the solution in Theorem 1.1 instantaneously becomes smooth, both in space and time, cf. Theorem 3.1.

Secondly we prove and apply a constrained Łojasiewicz–Simon gradient inequality (cf. [45]) to deduce convergence of the flow, where a new estimate (see Lemma 4.10) substantially simplifies the argument for the convergence result compared to previous works, cf. [8, 13].

Theorem 1.2

Let f0∈W2,2(I;Rd) be an immersed curve and suppose p0,p1∈Rd and τ0,τ1∈Sd-1 satisfy (1.4) and (1.7). Then, there exists a smooth family of curves f:(0,∞)×I→Rd solving (1.3), such that (i) f(t)→f0 in W2,2(I;Rd) as t→0;

(ii) f(t)→f∞ smoothly after reparametrization as t→∞, where f∞ is a constrained clamped elastica, i.e. a solution of 1.8 -∇s2κ→-12|κ→|2κ→+λκ→=0onIf(y)=pyfory∈∂I∂sf(y)=τyfory∈∂I

for some λ∈R.

Together with the previously mentioned work [10] this paper completes the study of the existence and convergence of the elastic flow of clamped curves with fixed length. Unfortunately, due to the low regularity of the initial curves considered here, we are not able to show uniqueness for the solution of the geometric evolution equation (1.3).

In the smooth category, one can show uniqueness “up to reparametrization” by a PDE argument similar to [22]. However, due to our low regularity we were not able to prove sufficient contraction estimates. The reason for that is the rigid characterization of Lipschitz properties of Nemytskii operators, see for instance [4, Theorem 3.10, Theorem 7.9].

The elastic energy of curves has already been studied by Bernoulli. The analysis of the elastic flow, i.e. the one-dimensional analogue of the Willmore flow, started with [42] and [17]. The boundary value problem for the elastic flow was considered in [25] for clamped curves and in [9] for natural second-order boundary conditions, see also [26, 27, 52] for related second-order evolutions. For further related literature on elastic flows, we refer to [5, 6, 35, 39, 41, 50]. Recent research has also studied the geometric evolution of networks and previously achieved results were applied to the elastic flow of networks, see e.g. [11, 15, 20, 21, 37]. Moreover, the elastic flow with different ambient geometries has been considered in [14, 34, 43], especially, the case of hyperbolic space [34] is of interest, cf. [12, 24]. Additionally, we mention the elastic flow of closed curves under a length and area constraint [38].

The Łojasiewicz–Simon gradient inequality is a remarkable result on (real) analytic functions which was first proven in Rd [28] and later generalized to infinite dimensions [49], see also [7]. Nowadays, it is the fundamental tool for investigating the asymptotic properties of gradient flows with analytic energies, which has been used for many geometric evolution equations, see for instance [8, 13, 19, 30, 31, 40, 47, 48] and also [36] for a different approach. The fixed-length constraint in (1.3) and (1.5) obstructs the use of [7] to deduce the gradient inequality, which is why we apply a recent extension to constrained energies [45]. We emphasize that this article is the first application of the constrained Łojasiewicz–Simon gradient inequality for a constrained gradient flow.

This article is structured as follows. In Sect. 2, we pick a specific tangential velocity such that (1.3) becomes a parabolic system, which we reduce to a fixed point equation. The existence of a fixed point is then established on a small time interval, using the concept of maximal Lp-regularity together with appropriate contraction estimates. Section 3 is devoted to show instantaneous smoothing of our solution, both in space and in time. After that, we prove long-time existence and a refined Łojasiewicz–Simon gradient inequality to finally prove Theorem 1.2 in Sect. 4. For the sake of readability, some details on the contraction estimates and the parabolic smoothing have been moved to the appendix or can be found in the first author’s dissertation [46, Chapter 3].

Short-time existence

The goal of this section is to prove Theorem 1.1. As in [20], we prescribe an explicit tangential motion to transform (1.3) into a quasilinear parabolic system. We then perform a linearization and use the theory of maximal Lp-regularity and suitable contraction estimates to prove Theorem 1.1 using a fixed point argument. We consider an initial datum merely lying in WImm2,2(I;Rd), the space of W2,2-immersions. This is a natural space for the elastic energy, since it is the roughest Sobolev space where E remains finite.

On the Lagrange multiplier

To ensure that the Lagrange multiplier is well-defined, one needs to prevent the denominator from vanishing. Write λ(f)=:N(f)2E(f), where N(f) denotes the numerator in (1.2) and observe that for a solution of (1.3) we have|f(t,0)-f(t,1)|=|p0-p1|<ℓ=L(f(t))for allt∈[0,T),

using the boundary conditions, (1.7) and (1.5). In particular, f(t) cannot be part of a straight line, so E(f(t))>0 for all t∈[0,T). Moreover, we observe that after integration by parts we have2.1 N(f)=∫I⟨∇E(f),κ→⟩ds=⟨∇sκ→,κ→⟩|∂I-∫I|∇sκ→|2ds+12∫I|κ→|4ds.

Note that in (2.1), no derivatives of second order of the curvature appear, which means that the Lagrange multiplier is formally of lower order compared to ∇E(f). This is extremely useful later on, since we can rely on the well-studied property of maximal Lp-regularity for a local operator in the linearization and treat the Lagrange multiplier as a nonlinearity in the fixed point argument.

From the geometric problem to a quasilinear PDE

As a next step, we explicitly compute the right-hand side of (1.1). By Proposition A.1∇E(f)=∇s2κ→+12|κ→|2κ→=A(f)⊥,

where2.2 A(f):=∂x4fγ4-6⟨∂x2f,∂xf⟩γ6∂x3f-4⟨∂x3f,∂xf⟩γ6∂x2f-52|∂x2f|2γ6∂x2f+352⟨∂x2f,∂xf⟩2γ8∂x2f=:∂x4fγ4+F~(γ-1,∂xf,∂x2f,∂x3f).

In order to solve (1.3), we study the following evolution problem, prescribing an explicit tangential motion θ=μ to make the problem parabolic. We want to find a family of immersions f:[0,T)×I→Rd satisfying2.3 ∂tf=-∇s2κ→-12|κ→|2κ→+μ∂sf+λκ→on(0,T)×If(0,x)=f0(x)forx∈If(t,y)=pyfor0≤t<T,y∈∂I∂xf(t,y)=τyγ0(y)for0≤t<T,y∈∂I.

with λ as in (1.2) and μ=μ(f):[0,T)×I→R given by μ:=-⟨A(f),∂sf⟩. Note that the first-order boundary conditions are a linear version of the general boundary conditions in (1.3) and thus easier to handle. The system (2.3) is often referred to as the analytic problem.

For 1<p<∞ and T>0, we consider the space of solutionsXT,p:=W1,p0,T;Lp(I;Rd)∩Lp0,T;W4,p(I;Rd)

and the space of dataYT,p1:=Lp0,T;Lp(I;Rd).

The space of initial data is given by the Besov spaceYp2:={f(0)∣f∈XT,p}=Bp,p4(1-1p)(I;Rd),

see for instance [16, Section 2]. We also consider the solution space with vanishing trace at time t=0 given by0XT,p:={f∈XT,p∣f(0)=0}.

For convenience, we also set YT,p:=YT,p1×Yp2.

Linearization of the analytic problem

If we linearize (2.3) for λ≡0, we obtain a linear parabolic system. This system is a local PDE which we can apply maximal regularity theory to. First, assuming λ≡0 and using (2.2), the evolution in (2.3) has the form∂tf=-A(f)=:-∂x4fγ4-F~(γ-1,∂xf,∂x2f,∂x3f)

with A as in (2.2). If we freeze coefficients for the highest order term at the initial datum f0 we get2.4 ∂tf+∂x4fγ04=1γ04-1γ4∂x4f-F~(γ-1,∂xf,∂x2f,∂x3f)=:F(γ-1,∂xf,∂x2f,∂x3f,∂x4f),

where γ0:=γ(0,·)=|∂xf0| and F~ is as in (2.2). The linearized system we associate to (2.3) with λ≡0 is2.5 ∂tf+1γ04∂x4f=Fon(0,T)×If(0,x)=f0(x)forx∈If(t,y)=pyfor0≤t<T,y∈∂I∂xf(t,y)=τyγ0(y)for0≤t<T,y∈∂I.

We can now apply the general Lp-theory for parabolic systems to obtain the following classical maximal regularity result, whose proof can be found in [46, Chapter 3, Section 2.3]. For the definition of the spaces for the boundary data DT,pi with i=0,1, see (B.2).

Theorem 2.1

Let p∈(53,∞), 0<T≤T0. Suppose a∈C([0,T0]×I;R) such that a(t,x)≥α for some α>0 and all t∈[0,T0],x∈I. Let (ψ,f0)∈YT,p, b0∈DT,p0 and b1∈DT,p1 such that the following compatibility conditions are satisfied:2.6 b0(0,y)=f0(y)fory∈∂I,b1(0,y)=∂xf0(y)fory∈∂I.

Then, there exists a unique f∈XT,p such that2.7 ∂tf+a∂x4f=ψon(0,T)×If(0,x)=f0(x)forx∈If(t,y)=b0(t,y)for0≤t<T,y∈∂I∂xf(t,y)=b1(t,y)for0≤t<T,y∈∂I,

and there exists C=C(p,T,a)>0 such that2.8 ‖f‖XT,p≤C‖ψ‖YT,p1+‖f0‖Yp2+‖b0‖DT,p0+‖b1‖DT,p1.

Moreover, if b0=0 and b1=0, then we may choose C=C(p,T0,a) independent of T≤T0.

Now, we want to solve (2.3) for initial data f0∈W2,2(I;Rd) using a fixed point argument. Note that B2,22(I;Rd)=W2,2(I;Rd) by (B.1), so p=2 is a fine setup to deal with the desired initial data, see Remark 2.11 for a more detailed discussion. We observe that the linearized system (2.5) can be viewed as a special case of Theorem 2.1 with a=1γ04, b0=(p0,p1), b1=(τ0,τ1) and ψ=F.

Throughout the rest of this section, we exclusively work with p=2. To simplify notation the spaces XT,YT,D0,D1 denote the respective spaces with p=2.

Contraction estimates

The key ingredient in the proof of the short-time existence is a contraction estimate for the nonlinearity in (2.3). We fix an initial datum f0∈WImm2,2(I;Rd) and boundary conditions p0,p1∈Rd and τ0,τ1∈Sd-1 satisfying (1.4) and (1.7). For a reference flow f¯∈XT=1 with f¯(0)=f0, and some M and T∈(0,1] we define2.9 B¯T,M:=f∈XT∣f(0)=f0and‖f-f¯‖XT≤M.

We denote by T the existence time and by M the contraction radius. Since we take T,M>0 small later on, it is no restriction to only consider T,M≤1. Later, we choose a specific reference flow f¯, see Definition 2.5.

First, the following lemma yields uniform bounds from below on the arc-length element and the elastic energy for small times, ensuring that the system (2.3) does not immediately become singular. A detailed proof can be found in [46, Chapter 3, Section 2.4.1].

Lemma 2.2

For T=T(f¯)∈(0,1] small enough and M∈(0,1], any f∈B¯T,M satisfies γ(t,x)≥infIγ02 for all (t,x)∈[0,T)×I. In particular, all curves f(t,·) are immersed.

Lemma 2.3

For T=T(f¯)∈(0,1] small enough and M∈(0,1], any f∈B¯T,M satisfies E(f(t))≥E(f0)3>0 (cf. (1.7)) for all t∈[0,T),.

We now state the crucial contraction property of the nonlinearities. Since the space of initial data is the energy space, cf. Remark 2.11, the necessary estimates are quite involved and rely on the special structure of (1.1). For the sake of readability, some of the details and the proof of the following lemma are moved to Appendix C.

Lemma 2.4

Let q∈(0,1). Then the following mapsF:B¯T,M→YT1,F(f):=F(γ-1,∂xf,∂x2f,∂x3f,∂x4f)Λ:B¯T,M→YT1,Λ(f):=λ(f)κ→fN:B¯T,M→YT1,N(f):=F(f)+Λ(f),

are well-defined q-contractions (i.e. Lipschitz continuous with Lipschitz constant q) for T=T(q,f¯), M=M(q,f¯)∈(0,1] small enough, with F as in (2.4) and λ as in (1.2).

The fixed point argument

We now reduce the analytic problem (2.3) to a fixed point equation and show local existence and uniqueness via the contraction principle. To that end, we first choose a specific reference solution f¯ in (2.9) on the time interval [0,1]⊃[0,T] for 0<T≤1.

Definition 2.5

We define the reference solution f¯ to be the unique solution of the following initial boundary value problem.∂tf¯+∂x4f¯γ04=0on[0,1)×If¯(0,x)=f0(x)forx∈If¯(t,y)=pyfor0≤t<1,y∈∂I∂xf¯(t,y)=τyγ0(y)for0≤t<1,y∈∂I.

Existence and uniqueness in the classW1,20,1;L2(I;Rd)∩L20,1;W4,2(I;Rd)

follows from Theorem 2.1. Note that the restriction of the solution to any time interval [0, T] is the unique solution in the class XT for all 0<T≤1.

Fix q∈(0,1) and take T=T(q,f¯)∈(0,1],M=M(q,f¯)∈(0,1] small enough such that Lemmas 2.2 to 2.4 hold. Let f∈B¯T,M. Then, we have N(f)∈YT1, cf. Lemma 2.4. For ψ:=N(f),b0:=(p0,p1), b1:=(τ0,τ1), a:=γ0-4∈C([0,1]×I) the compatibility conditions (2.6) are satisfied, since by (1.4) we haveb0(0,y)=f0(y)fory∈∂I,b1(0,y)=τyγ0(y)=∂xf0(y)fory∈∂I.

Hence, by Theorem 2.1, there exists a unique solution g∈XT of the linear initial boundary value problem2.10 ∂tg+∂x4gγ04=N(f)on(0,T)×Ig(0,x)=f0(x)forx∈Ig(t,y)=pyfor0≤t<T,y∈∂I∂xg(t,y)=τyγ0(y)for0≤t<T,y∈∂I.

Definition 2.6

We define the map Φ:B¯T,M→XT,Φ(f):=g, where g∈XT is the unique solution to (2.10).

Remark 2.7

Finding a solution of (2.3) in the ball B¯T,M⊂XT is equivalent to finding a fixed point of the map Φ in Definition 2.6.

We now show that Φ is a contraction on B¯T,M for T,M>0 small enough.

Proposition 2.8

Let q∈(0,1). Then there exist M=M(q,f¯)∈(0,1], T=T(q,M,f¯)∈(0,1] such that Φ:B¯T,M→B¯T,M is well-defined and a q-contraction, i.e.2.11 ‖Φ(f)-Φ(f~)‖XT≤q‖f-f~‖XT

for all f,f~∈B¯T,M.

Proof

The contraction property: Let q∈(0,1) and f,f~∈B¯T,M and let g=Φ(f), g~=Φ(f~). We observe that g-g~ vanishes at t=0 and at the boundary ∂I up to first order. Hence, by Definition 2.6 and (2.8), for some C=C(f0)=C(f¯)>0, independent of T∈(0,1], we have2.12 ‖g-g~‖XT≤C‖N(f)-N(f~)‖L2(0,T;L2).

Taking T=T(q,f¯),M=M(q,f¯)∈(0,1] small enough so that Lemma 2.4 can be applied with q replaced by q/(2C), we have2.13 ‖N(f)-N(f~)‖L2(0,T;L2)≤q2C‖f-f~‖XT.

Equations (2.12) and (2.13) imply (2.11).

Well-definedness: Let f∈B¯T,M,g=Φ(f). Again by (2.8) we find2.14 ‖g-f¯‖XT≤C‖N(f)-0‖L2(0,T;L2)≤C‖N(f)-N(f¯)‖L2(0,T;L2)+‖N(f¯)‖L2(0,T;L2)≤q2‖f-f¯‖XT+C‖N(f¯)‖L2(0,T;L2)≤q2M+C‖N(f¯)‖L2(0,T;L2),

where we applied (2.13) with f~=f¯ in the third step. Now, by dominated convergence we have ‖N(f¯)‖L2(0,T;L2)≤M2C reducing T=T(q,M,f¯)∈(0,1] if necessary. Then, from (2.14) we conclude ‖Φ(f)-f¯‖XT≤M. □

Theorem 2.9

Let f0∈WImm2,2(I;Rd), p0,p1∈Rd, τ0,τ1∈Sd-1 satisfying (1.4) and (1.7). Then there exist M>0 and T>0 such that the system (2.3) has a unique solution f∈B¯T,M⊂W1,20,T;L2(I;Rd)∩L20,T;W4,2(I;Rd).

Proof

For T,M>0 as in Proposition 2.8 with q=12, the map Φ:B¯T,M→B¯T,M is a contraction in the complete metric space B¯T,M and hence has a unique fixed point f∈B¯T,M by the contraction principle. Since any fixed point of Φ is a solution of (2.3) in B¯T,M and vice versa, the claim follows. □

Remark 2.10

By the construction of our solution and Lemma 2.2 and Lemma 2.3 the arc-length element |∂xf| and the elastic energy of the solution f in Theorem 2.9 are bounded from below and above, uniformly in t∈[0,T).

This immediately implies Theorem 1.1.

Proof of Theorem 1.1

By Theorem 2.9, there exist T>0 and a solution f of (2.3) such that f∈W1,2(0,T;L2(I;Rd))∩L2(0,T;W4,2(I;Rd)). Consequently, f solves (1.3), since at the boundary we have∂sff(t,y)=∂xf(t,y)|∂xf(t,y)|=γ0(y)τy|γ0(y)τy|=τyfort∈[0,T),y∈∂I.

□

Remark 2.11

Our assumption on the regularity of the initial datum is very natural. On the one hand, the space WImm2,2(I;Rd) is the correct energy space associated to the elastic energy, so we would like to obtain short-time existence for an initial datum in WImm2,2(I;Rd). In view of the linear problem in Theorem 2.1, working in the Sobolev scale one would hence need to pick p∈(1,∞) such that W2,2(I;Rd)↪B4(1-1p)p,p(I;Rd)=Yp2.

However, in order to estimate the denominator of the Lagrange multiplier λ, we want continuity of our solution with values in W2,2(I;Rd). Using Proposition B.1 (i), this can be achieved if B4(1-1p)p,p(I;Rd)↪W2,2(I;Rd).

Clearly, this can only work for p=2. Moreover, for the same reason as above, even the introduction of time-weighted Sobolev spaces would not provide solutions with lower initial regularity.

Theorem 2.12

The solution f∈B¯T,M in Theorem 2.9 is the unique solution of (2.3) in the whole space W1,20,T;L2(I;Rd)∩L20,T;W4,2(I;Rd).

Proof

First we note that any restriction of the solution f∈B¯M to a smaller time interval [0,T~] is again the unique solution of (2.3) in B¯M on [0,T~] by Theorem 2.9. Now, we let T1,T2>0 and assume that fi∈W1,20,Ti;L2(I;Rd)∩L20,Ti;W4,2(I;Rd), i=1,2 are two families of immersions satisfying (2.3) with f0∈WImm2,2(I). Without loss of generality we may assume that T1≤T2. We claim that f2|[0,T1]=f1.

To show the claim we define t¯=sup{t∈[0,T1):f1(s)=f2(s)∀0≤s≤t}. Note that t¯ is well-defined by Proposition B.1 (i). We need to show that t¯=T1. To do so we first prove that t¯>0. Indeed, for T↘0, we have ‖fi|[0,T]‖XT→0 by the dominated convergence theorem, and the same holds for the reference flow f¯ from Definition 2.5. Thus, for T>0 small enough, fi|[0,T]∈B¯M for i=1,2. Further decreasing T>0 if necessary we obtain from Theorem 2.9 that f1|[0,T]=f2|[0,T] is the unique solution f∈B¯M. Thus, f1(s)=f(s)=f2(s) for all 0≤s≤T, showing that t¯≥T>0.

We now assume that t¯<T1. Since fi∈XT1↪BUC([0,T1],W2,2(I;Rd)) and both solutions are immersed for all times, we find that f0:=f1(t¯)∈WImm2,2(I;Rd). Whence, by Theorem 2.9, there exist M>0, T>0 such that (2.3) has a unique solution f∈B¯M. Observing that fi(t¯+t,·)|0≤t≤T1-t¯, i=1,2 are both solutions to (2.3) with the same initial value f0, we find by similar arguments as above that f1(t¯+·)=f=f2(t¯+·) on [0, T), contradicting the definition of t¯. □

Parabolic smoothing

The goal of this section is to show that our solution f from Theorem 2.9 instantaneously becomes smooth.

Theorem 3.1

Let f0∈WImm2,2(I;Rd) such that (1.4) and (1.7) are satisfied. Then, there exists 0<T1≤T such that the solution f in Theorem 1.1 is smooth on (0,T1), i.e. f∈C∞((0,T1)×I;Rd).

A close examination of the contraction estimates in Appendix C reveals that the critical embeddings are used, for instance in (C.12), (C.14) and (C.15). Thus, higher integrability of the nonlinearity cannot be obtained by standard estimates relying on Hölder’s inequality. Therefore, we cannot directly start the usual bootstrap argument to show smoothness. Instead, we use an instantaneous gain of regularity in the time variable, relying on Angenent’s parameter trick [2, 3], see also [18] and [44, Chapter 9]. Since we want to conclude smoothness in space up to the boundary, we first show increased regularity in time before deducing global smoothness up to the boundary by using parabolic Schauder theory. For the sake of readability, we omit the proof of the following proposition and refer to [46, Chapter 3, Section 3.1].

Proposition 3.2

Let f∈W1,2(0,T;L2(I;Rd))∩L2(0,T;W4,2(I;Rd)) be the unique solution of (2.3), given by Theorem 2.9. Then there exists 0<T1<T such that f∈Cω((0,T1);W2,2(I,Rd)).

Next, we use the higher time regularity to improve the integrability of λ, which then allows us to start a bootstrap argument. First, we recall the following modification of [10, Lemma 4.3].

Lemma 3.3

Let f∈XT,2 be a solution of (1.3). Then, we have|λ|(ℓ-|p1-p0|)≤2ℓ‖∂t⊥f‖L1(ds)+∫I|κ→|2ds+∫I|∇sκ→|ds.

Proof

We proceed as in [10, Lemma 4.3]. Let l:[0,T)×I→Rd be the parametrization of the line segment from p0 to p1 given byl(t,x):=p0+φ(t,x)ℓ(p1-p0),

with φ(t,ξ):=∫0ξ|∂xf|dx for (t,ξ)∈[0,T)×I. Then l(t,0)=p0, l(t,1)=p1 and ∂sl(t,·)=1ℓ(p1-p0). Therefore, using ∇s2κ→+12|κ→|2κ→=∇s∇sκ→+12|κ→|2∂sf (cf. [10, p. 1048]), we find after integrating by parts∫I⟨∂t⊥f,f-l⟩ds=λ-12|κ→|2∂sf-∇sκ→,f-l|∂I+12∫I|κ→|2ds-λ∫Ids-1ℓ∫I∇sκ→+12|κ→|2∂sf-λ∂sf,p1-p0ds.

Consequently, since f=l on the boundary, we have|λ|(ℓ-|p1-p0|)=1-|p1-p0|ℓ|λ|∫Ids≤∫I|∂t⊥f|ds‖f-l‖∞+12∫I|κ→|2ds+|p1-p0|2ℓ∫I|κ→|2ds+|p1-p0|ℓ∫I|∇sκ→|ds.

Using (1.7) and the simple estimate ‖f-l‖∞≤2ℓ yields the claim. □

Note that a priori, the Lagrange multiplier λ is only L2(0,T) for f∈XT,2. The next proposition improves this integrability, at least on a small timescale bounded away from zero.

Lemma 3.4

Let f be the solution of (2.3) from Theorem 2.9 and let T1>0 as in Proposition 3.2. Then, for any 0<ε<T1 we have λ(f)∈L4(ε,T1).

Proof

As a consequence of Proposition 3.2, we have ∂tf∈Cω((0,T1);W2,2(I;Rd)) and thus we get ∂tf∈C0([ε,T1]×I;Rd). Hence, Lemma 3.3 and (1.7) yield that λ has the same integrability on (ε,T1) as ∫I|∇sκ→|ds. By Proposition A.1 (ii) and the uniform bounds on the arc-length element, cf. Remark 2.10, it suffices to show ∂x3f∈L4(ε,T1;L1(I;Rd)), since ∂x2f∈C0([ε,T1];L2(I;Rd)). In fact using Proposition B.3 (i) as in (C.12), we even get ∂x3f∈L4(ε,T1;L2(I;Rd)). □

The improved integrability of λ in Lemma 3.4 enables us to start a bootstrap argument to increase the Sobolev regularity of our solution in Theorem 2.9. Note that by Sobolev embeddings, in order to prove smoothness of our solution it suffices to reach XT,p with p>5, see Lemma 3.6.

Lemma 3.5

Let f be as in Theorem 2.9, let T1>0 be as in Proposition 3.2 and let 0<ε<T1. Then f∈W1,20(ε,T1;L20(I;Rd))∩L20(ε,T1;W4,20(I;Rd)).

Proof

See [46, Chapter 3, Section 3.3]. □

Finally, Theorem 3.1 follows from parabolic Schauder theory and the following

Lemma 3.6

Let f be the solution of (2.3) constructed in Theorem 2.9. If there exist p>5 and ε>0 such that f∈W1,pε,T1;Lp(I;Rd)∩Lpε,T1;W4,p(I;Rd) then f∈C∞((ε,T1)×I;Rd).

Proof

See [46, Chapter 3, Section 3.4]. □

Now, Theorem 3.1 is immediate.

Proof of Theorem 3.1

The solution f in Theorem 1.1 is exactly the solution f in Theorem 2.9. By Lemma 3.5 we have f∈W1,20(ε,T1;L20(I;Rd))∩L20(ε,T1;L20(I;Rd)) for any 0<ε<T1. Hence, by Lemma 3.6, we find that f∈C∞((ε,T1)×I;Rd)). □

Long-time behaviour and the proof of Theorem 1.2

In this section, we use the long-time existence result in [10] to show the existence of a global solution of (1.3). Moreover, we prove and use a refined Łojasiewicz–Simon gradient inequality to conclude convergence after reparametrization.

Long-time existence after reparametrization

As a first step towards proving Theorem 1.2, we establish long-time existence and subconvergence after reparametrization for our solution. The key ingredient is the smoothness of our solution and [10, Theorem 1.1].

Theorem 4.1

Let f∈W1,2(0,T;L2(I;Rd))∩L2(0,T;W4,2(I;Rd)) be as in Theorem 2.9 and let 0<ε<T. Then, there exist ε¯∈(ε,T) and f^∈C∞((0,∞)×I;Rd) satisfying (1.3) such that (i) f^(t,x)=f(t,x) for all 0≤t≤ε,x∈I;

(ii) f^(t,·) has zero tangential velocity for all t≥ε¯;

(iii) f^ subconverges smoothly as t→∞, after reparametrization with constant speed, to a constrained elastica, i.e. a solution (1.8).

Proof

By Theorem 3.1, the solution f in Theorem 2.9 is instantaneously smooth. Thus, to simplify notation we may assume f∈C∞([ε,T]×I;Rd) for some ε>0 after possibly reducing T>0. Moreover, we may also assume a uniform bound from below on the arc-length element using Remark 2.10.

Let θ:=⟨∂tf,∂sff⟩ be the tangential velocity of f. By the smoothness of f and the bound on the arc-length element, the function (t,r)↦θ(t,r)|∂xf(t,r)| is globally Lipschitz continuous on [ε,T]×I. For each x∈I, we consider the initial value problem4.1 ∂tΦ(t,x)=-θ(t,Φ(t,x))|∂xf(t,Φ(t,x))|Φ(ε,x)=x.

By classical ODE theory, there exist ε<T^≤T and a smooth family of reparametrizations Φ:[ε,T^]×I→I satisfying (4.1) and4.2 Φ(t,y)=yfort∈[ε,T^],y∈∂I∂xΦ(t,x)>0for all(t,x)∈[ε,T^]×I.

Therefore, Φ(t,·) is strictly increasing and a diffeomorphism of I for each t∈[ε,T^]. A direct computation yields that the reparametrization f1(t,x):=f(t,Φ(t,x)) satisfies∂tf1(t,x)=∂tf(t,Φ(t,x))+∂xf(t,Φ(t,x))∂tΦ(t,x)=∂t⊥f(t,Φ(t,x))+θ(t,Φ(t,x))∂sff(t,Φ(t,x))+∂xf(t,Φ(t,x))∂tΦ(t,x)=∂t⊥f(t,Φ(t,x))=-∇sf12κ→f1(t,x)-12|κ→f1(t,x)|2κ→f1(t,x)+λ(f1)(t)κ→f1(t,x),

using that f solves (1.3) and the transformation of the geometric quantities. For the boundary conditions, let t∈[ε,T^], y∈∂I and note that f1(t,y)=f(t,y)=py and ∂sf1f1(t,y)=∂sff(t,y)=τy by (4.2). Consequently, f1 is a smooth solution of (1.3) on [ε,T^] with tangential velocity zero and smooth initial datum f(ε). By [10, Theorem 1.1], f1 can be extended to a global smooth solution f¯ on [ε,∞) which subconverges, after reparametrization with constant speed, to a constrained elastica as t→∞.

In particular, we have the identity4.3 f¯(t,x)=f(t,Φ(t,x))for allε≤t≤T^.

Now, let ε<ε¯<T^ and Ψ:[0,T^]×I→I be a smooth family of reparametrizations with4.4 Ψ(t,x)=xfor all0≤t≤ε;Ψ(t,x)=Φ(t,x)for allε¯≤t≤T^.

The existence of such a Ψ is proven in Lemma D.1. We now definef^(t,x):=f(t,Ψ(t,x))for0≤t≤T^,x∈If¯(t,x)fort≥ε¯,x∈I.

Note that f^ is clearly smooth in x for every t≥0 fixed. It is also smooth in t for fixed x∈I, by (4.3) and (4.4). Property (i) follows from (4.4). Furthermore, by definition of f^ on [ε¯,∞)×I we find that f^=f¯ has zero tangential velocity and hence (ii) is satisfied. The last property follows since the asymptotic behaviour of f^ is inherited from f¯. □

The length-preserving elastic flow as a gradient flow on a Hilbert manifold

In this section, we show that the flow (1.3) is in fact a gradient flow on a suitable submanifold of curves.

Proposition 4.2

Let p0,p1∈Rd,τ0,τ1∈Sd-1 and ℓ∈R such that (1.7) holds. ThenX:=f∈WImm4,2(I;Rd)∣f(y)=pyand∂sf(y)=τyfory∈∂I,L(f)=ℓ.

is a weak Riemannian splitting analytic submanifold of W4,2(I;Rd) with codimension 4d-1.

Proof

By the Sobolev embedding W4,2(I;Rd)↪C1(I;Rd), the set of W4,2-immersions denoted by WImm4,2(I;Rd) is open in W4,2(I;Rd). The functionG:WImm4,2(I;Rd)→R×(Rd)2×(Sd-1)2,G(f):=L(f)f(0)f(1)∂sff(0)∂sff(1)

is an analytic map. Moreover, its differential is given bydGf:W4,2(I;Rd)→R×(Rd)2×T∂sf(0)Sd-1×T∂sf(1)Sd-1,dGf(u)=-∫I⟨κ→f,u⟩dsfu(0)u(1)∂xu(0)|∂xf(0)|-⟨∂xu(0),∂xf(0)⟩∂xf(0)|∂xf(0)|3∂xu(1)|∂xf(1)|-⟨∂xu(1),∂xf(1)⟩∂xf(1)|∂xf(1)|3

for f∈WImm4,2(I;Rd) and u∈W4,2(I;Rd). It is not difficult to see that dGf is surjective if f∈X=G-1{(ℓ,p0,p1,τ0,τ1)T}. Indeed, let α∈R, qy∈Rd, zy∈T∂sf(y)Sd-1 for y=0,1. We have T∂sf(y)Sd-1={z∈Rd∣⟨z,∂xf(y)⟩=0}. Clearly, we can find an immersed curve u∈W4,2(I;Rd) with u(y)=qy and ∂xu(y)|∂xf(y)|=zy for y=0,1. Now, using the characterization of the tangent space, for v∈C0∞(I;Rd) we finddGf(u+v)=-∫I⟨κ→f,u+v⟩dsfq0q1z0z1,

since adding v does not change the boundary behaviour. Moreover, as κ→f≢0 using f∈X and (1.7), we can choose v such that ∫I⟨κ→f,v⟩dsf=ε≠0. Setting β:=∫I⟨κ→f,u⟩dsf and w:=u-α+βδv, we find ∫I⟨κ→f,w⟩dsf=β-(α+β)=-α; hence, we have shown dGf(w)=(α,q0,q1,z0,z1), so dGf is surjective.

Consequently, X⊂W4,2(I;Rd) is a splitting submanifold by [1, Theorem 3.5.4] with codimension 1+2d+2(d-1)=4d-1. Like in [45], the analytic form of the implicit function theorem can be used to show that X is in fact analytic. The tangent space is given by4.5 TfX=kerdGf=u∈W4,2(I;Rd)∣u=0on∂I,∂xfu=0on∂I,∫I⟨κ→f,u⟩dsf=0.

Since (1.3) is a L2(dsf) gradient flow, it is natural to endow X with the Riemannian metric ⟨u,v⟩L2(dsf)=∫I⟨u,v⟩dsf for u,v∈TfX. Note that since TfX is certainly not complete with respect to the induced norm, the metric is only weakly Riemannian (cf. [1, Definition 5.2.12]). □

It is not difficult to see that by (4.5) the right-hand side of the evolution (1.1) is the projection of the full L2(dsf)-gradient ∇E(f) onto the L2(dsf)-closure of the tangent space TfX. This implies that (1.1) is the gradient flow of E on the manifold X.

The constrained Łojasiewicz–Simon gradient inequality

In this subsection, we establish a Łojasiewicz–Simon inequality for E on X. To do so, we have to deal with the invariance of both energies E and G, which unfortunately creates large kernels for their first and second variations. Like in [8, 13], we work around this issue by restricting the energy to normal directions and using the implicit function theorem.

In the following, we always assume that the assumptions (1.4) and (1.7) are satisfied.

Definition 4.3

Fix f¯∈X and define Vc:=W4,2(I;Rd)∩W02,2(I;Rd). We define the space of normal vector fields along f¯ byW4,2,⊥(I;Rd):={f∈W4,2(I;Rd)∣⟨f,∂xf¯⟩=0onI}.

Moreover, we define H⊥:=L2,⊥(I;Rd):={u∈L2(I;Rd)∣⟨u,∂xf¯⟩=0a.e.} and Vc⊥:=Vc∩W4,2,⊥(I;Rd). Both are Hilbert spaces and the L2-orthogonal projection onto H⊥ is given by the pointwise projection P⊥(f):=f-⟨f,∂sf¯⟩∂sf¯.

Moreover, by the embedding W4,2(I;Rd)↪C1(I;Rd) there exists ε>0 small enough such that for all u∈W4,2,⊥(I;Rd) with ‖u‖W4,2<ε, the curve f=f¯+u is immersed. Defining Uε:={u∈Vc⊥∣‖u‖W4,2<ε} we consider the energiesL:Uε→R,L(u)=L(f¯+u)andE:Uε→R,E(u)=E(f¯+u).

We have the following result.

Proposition 4.4

(cf. [13, Proof of Theorem 3.1, Remark 3.3]) The energy E satisfies the following properties. E:Uε→R is analytic;

its gradient ∇E:Uε→H⊥ is analytic;

the derivative (∇E)′(0):Vc⊥→H⊥ is Fredholm with index zero.

It is well known that this is sufficient to prove a Łojasiewicz–Simon gradient inequality for E (cf. [7, Corollary 3.11]), [13, Theorem 3.1], [45, Theorem 1.2], [43, Corollary 2.6]). However, in order to conclude a constrained or refined Łojasiewicz–Simon gradient inequality, cf. (16) in [45], we also need to analyse the length functional.

Proposition 4.5

The energy L satisfies the following properties. L:Uε→R is analytic.

The gradient map ∇L:Uε→H⊥ is analytic.

The derivative (∇L)′(0):Vc⊥→H⊥ is compact.

L(0)=ℓ and ∇L(0)≠0.

Proof

  The map Uε→C(I;Rd),u↦|∂x(f¯+u)| is analytic by [13, Lemma 3.4, 1.], and hence so is L.

The H⊥-gradient of L is given by ∇L(u)=-P⊥κ→f¯+u|∂x(f¯+u)|. Note that the map Uε→L2(I;Rd),u↦κ→f¯+u is analytic by [13, Lemma 3.4, 3.]. Since the multiplication L2(I;Rd)×L∞(I;R)→L2(I;Rd),(f,ϕ)↦fϕ is analytic, so is the map Uε→L2(I;Rd),u↦κ→f¯+u|∂x(f¯+u)|. The continuity and linearity of P⊥:L2(I;Rd)→H⊥ yield the claim.

We compute the second derivative using standard formulas for the variation of geometric quantities (see for instance [17, Lemma 2.1]). We have (∇L)′(0)u=ddtt=0∇L(tu)=-ddtt=0P⊥κ→f¯+u|∂x(f¯+u)|=-P⊥ddtt=0κ→f¯+tu|∂xf¯|-P⊥κ→f¯ddtt=0|∂x(f¯+tu)|=-∇sf¯2u+⟨u,κ→f¯⟩κ→f¯|∂xf¯|+κ→f¯⟨u,κ→f¯⟩|∂xf¯|.

In particular, the operator (∇L)′(0):Vc⊥→H⊥ is only of second order in u, hence compact by the Rellich–Kondrachov Theorem [23, Theorem 7.26].

L(0)=L(f¯)=ℓ since f¯∈X. Since we have |f¯(1)-f¯(0)|=|p1-p0|<ℓ, f¯ cannot be part of a straight line; hence, κ→f¯≢0 and also |∂xf¯|≠0 since f¯ is immersed.

□

This enables us to conclude the inequality in normal directions.

Theorem 4.6

Suppose f¯∈X is a constrained elastica. Then, there exist C,σ>0 and θ∈(0,12] such that for all f=f¯+u∈X with u∈Vc⊥ and ‖u‖W4,2≤σ we have|E(f)-E(f¯)|1-θ≤C‖∇L2(dsf)E(f)+λ(f)∇L2(dsf)L(f)‖L2(dsf).

Proof

First, we verify the conditions of [45, Corollary 5.2] for the energy E and the constraint G(u)=L(u)-ℓ on the spaces V=Vc⊥,H=H⊥. Note that ∇G=∇L. Clearly, Vc⊥↪H⊥ densely. Assumptions (ii) and (iii) follow from Proposition 4.4, whereas assumptions (iv)–(-vi) are satisfied by Proposition 4.5. Note that u=0 is a constrained critical point of E on M=G-1({0}) since f¯ is a constrained elastica.

Then, by [45, Corollary 5.2] E|M satisfies a constrained Łojasiewicz–Simon gradient inequality, i.e. there exist C,σ>0 and θ∈(0,12] such that for all u∈M with ‖u‖W4,2≤σ we have|E(u)-E(0)|1-θ≤C‖P(u)∇E(u)‖L2,

where P(u):H⊥→H⊥ denotes the orthogonal projection onto the closure of the tangent space TuM¯={y∈H⊥∣⟨∇L(u),y⟩L2=0} (cf. [45, Proposition 3.3]). Therefore, forλ(f)=⟨κ→f,∇E(f)⟩L2(dsf)‖κ→f‖L2(dsf)2

as in (1.2) with f=f¯+u we have the estimate‖P(u)∇E(u)‖L2=‖P(u)∇E(u)+λ∇L(u)‖L2≤‖∇E(u)+λ∇L(u)‖L2.

Moreover, we have ∇E(u)=∇L2(dsf)E(f)|∂xf| and ∇L(u)=∇L2(dsf)L(f)|∂xf|. Consequently,‖P(u)∇E(u)‖L2≤‖P(u)∇E(u)+λ∇L(u)‖L2≤‖∇L2(dsf)E(f)|∂xf|+λ∇L2(dsf)L(f)|∂xf|‖L2≤‖∂xf‖L∞12‖∇L2(dsf)E(f)+λ∇L2(dsf)L(f)‖L2(dsf).

Reducing σ>0 if necessary, we may assume that ‖∂xf‖L∞ is uniformly bounded for ‖f-f¯‖W4,2≤σ by the Sobolev embedding theorem. This proves the claim. □

We use this to prove the full constrained Łojasiewicz–Simon gradient inequality for not necessarily normal variations via the following reparametrization argument.

Lemma 4.7

([13, Lemma 4.1]) Let f¯∈W5,2(I;Rd) be a regular curve. Then, there exists σ>0 such that for all ψ∈Vc with ‖ψ‖W4,2≤σ, there exists a W4,2-diffeomorphism Φ:I→I such that4.6 (f¯+ψ)∘Φ=f¯+η

for some η∈Vc⊥. Moreover, given σ>0 there exists σ~=σ~(f¯,σ)>0 such that for all ψ∈Vc with ‖ψ‖W4,2≤σ~ we have the above representation with ‖η‖W4,2≤σ.

Theorem 4.8

Let f¯∈X∩W5,2(I;Rd) be a constrained elastica. Then there exist C,σ>0 and θ∈(0,12] such that|E(f)-E(f¯)|≤C‖∇L2(dsf)E(f)+λ(f)∇L2(dsf)L(f)‖L2(dsf),

for all f∈X such that ‖f-f¯‖W4,2≤σ.

Proof

Let C,σ>0,θ∈(0,12] as in Theorem 4.6, f¯∈X be a constrained critical point of E on X. By the regularity assumption on f¯, we may use Lemma 4.7.

Thus, we find σ~>0 such that (4.6) holds for all ψ∈Vc with ‖ψ‖W4,2≤σ~ for some η∈Vc⊥ with ‖η‖W4,2≤σ. Let f∈X such that ‖f-f¯‖W4,2≤σ~. Then by Lemma 4.7, there exist a diffeomorphism Φ:I→I and η∈Vc⊥ such that f∘Φ=f¯+η.

Note that with f,f¯∈X we also get f∘Φ=f¯+η∈X, since L(f)=L(f∘Φ)=ℓ. Since the elastic energy is invariant under reparametrization, we hence get using Theorem 4.64.7 E(f)-E(f¯)1-θ=E(f¯+η)-E(f¯)1-θ≤C‖∇L2(dsf¯+η)E(f¯+η)+λ(f¯+η)∇L2(dsf¯+η)L(f¯+η)‖L2(dsf¯+η).

Since λ and the gradients are geometric, i.e. transform correctly under reparametrizations, we haveλ(f¯+η)=λ(f∘Φ)=λ(f),∇L2(dsf¯+η)E(f¯+η)=∇L2(dsf)E(f)∘Φ∇L2(dsf¯+η)L(f¯+η)=∇L2(dsf)L(f)∘Φ.

Consequently, we obtain‖∇L2(dsf¯+η)E(f¯+η)+λ(f¯+η)∇L2(dsf¯+η)L(f¯+η)‖L2(dsf¯+η)=‖∇L2(dsf)E(f)∘Φ+λ(f)∇L2(dsf)L(f)∘Φ‖L2(dsf∘Φ)=‖∇L2(dsf)E(f)+λ(f)∇L2(dsf)L(f)‖L2(dsf).

Together with (4.7), this implies the Łojasiewicz–Simon gradient inequality for the elastic energy on X. □

Convergence

In previous works (see e.g. [8, p. 358 – 359] and [13, p. 2188 – 2191]), a lot of PDE theory and a priori parabolic Schauder estimates are needed to apply the Łojasiewicz–Simon gradient inequality to conclude convergence for geometric problems. In this section, we introduce a novel inequality (see Lemma 4.10) which enables us to significantly shorten this lengthy argument in the proof of Theorem 1.2. We exploit the explicit structure of the constant speed reparametrization and the length bound to control the full velocity of the constant speed parametrization by the purely normal velocity of the original evolution.

Definition 4.9

Let T∈(0,∞] and let f:[0,T)×I→Rd be a family of immersed curves in Rd. The constant speed L(f(t)) reparametrization f~(t) of f(t) is given by f~(t,x):=f(t,ψ(t,x)) where ψ(t,·):I→I is the inverse of φ(t,·):I→I given byφ(t,x):=1L(f(t))∫0x|∂xf(t,z)|dz=1L(f(t))∫0xdsf(t).

Lemma 4.10

Suppose T∈(0,∞] and f:[0,T)×I→Rd is a family of curves in Rd, such that f(t,0)=p0,f(t,1)=p1 and L(f(t))>0 for all t∈(0,T]. Then, if f~(t) is the constant speed L(f(t)) reparametrization of f(t), for all t∈[0,T) we have‖∂tf~(t)‖L2(dx)≤2L(f(t))+16E(f(t))‖∂tf‖L2(dsf(t)).

In particular, if f evolves by the length-preserving elastic flow (1.3), we have‖∂tf~(t)‖L2(dx)≤C‖∂tf‖L2(dsf(t)),

for all t∈(0,T], where C=2ℓ+16E(f0).

Proof

Recall that by Definition 4.9 we haveψ(t,φ(t,x))=φ(t,ψ(t,x))=xfor allt∈[0,T),x∈I.

For the derivatives of φ and ψ we thus obtain (i) ∂tφ(t,x)=-∂tL(f(t))L(f(t))2∫0xdsf(t)-1L(f(t))∫0x⟨∂tf,κ→f(t)⟩dsf(t);

(ii) ∂xφ(t,x)=|∂xf(t,x)|L(f(t));

(iii) ∂xψ(t,φ(t,x))=∂xφ(t,x)-1=L(f(t))|∂xf(t,x)|;

(iv) ∂tψ(t,φ(t,x))=-∂xψ(t,φ(t,x))∂tφ(t,x)=L(f(t))|∂xf(t,x)|∂tL(f(t))L(f(t))2∫0xdsf(t)+1L(f(t))∫0x⟨∂tf,κ→f(t)⟩dsf(t).

Now, we estimate‖∂tf~(t)‖L2(dx)2≤2∫01|(∂tf)(t,ψ(t,x))|2dx+2∫01|(∂xf)(t,ψ(t,x))|2|∂tψ(t,x)|2dx.

Taking y=ψ(t,x) and using ψ(t,0)=0, ψ(t,1)=1, we find‖∂tf~(t)‖L2(dx)2≤2(∫01|∂tf(t,y)|21∂xψ(t,φ(t,y))dy+|∂xf(t,y)|2|∂tψ(t,φ(t,y))|21∂xψ(t,φ(t,y))dy)=:2(A+B).

For the first integral, we clearly haveA=∫01|∂tf(t,y)|2|∂xf(t,y)|L(f(t))dy=1L(f(t))‖∂tf‖L2(dsf(t))2.

For the second part, note that by (iv), we haveB=∫01|∂tL(f(t))L(f(t))∫0ydsf(t)+∫0y⟨∂tf,κ→f(t)⟩dsf(t)|2|∂xf(t,y)|L(f(t))dy.

Now, using the boundary conditions, we have ∂tL(f(t))=-∫I⟨κ→f(t),∂tf(t)⟩dsf(t) and the Cauchy–Schwarz inequality yieldsB≤2∫01(∂tL(f(t))L(f(t))∫0ydsf(t)2+∫0y⟨∂tf(t),κ→f(t)⟩dsf(t)2)|∂xf(t,y)|L(f(t))dy≤2∫01(∫01|⟨∂tf(t),κ→f(t)⟩|dsf(t)2+∫01|⟨∂tf(t),κ→f(t)⟩|dsf(t)2)|∂xf(t,y)|L(f(t))dy=4∫01|⟨∂tf(t),κ→f(t)⟩|dsf(t)2≤4‖∂tf(t)‖L2(dsf(t))2‖κ→f(t)‖L2(dsf(t))2=8E(f(t))‖∂tf(t)‖L2(dsf(t))2.

□

Remark 4.11

Note that in the proof of Lemma 4.10, we only used the boundary conditions to conclude that no boundary terms appear when integrating by parts. In particular, Lemma 4.10 also holds in the case of closed curves.

Finally, we can prove our main convergence result.

Proof of Theorem 1.2

Let ε>0 and let f^∈C∞((0,∞)×I;Rd), ε¯>ε be as in Theorem 4.1. The first statement of Theorem 1.2 follows from property (i) in Theorem 4.1, and the fact that the solution f in Theorem 2.9 lies in XT,2↪BUC([0,T];W2,2(I;Rd)) by Proposition B.1 and (B.1).

For the convergence statement, let f~ be the constant speed ℓ reparametrization of f^, cf. Definition 4.9, and note that f~∈C∞((0,∞)×I;Rd). By Theorem 4.1 (iii), there exist a sequence tn→∞ and a smooth regular curve f∞:I→Rn, such that f~(tn)→f∞ in Ck(I;Rn) for all k∈N0. Moreover, as a consequence of Theorem 4.1, f∞ is a smooth constrained elastica, i.e. a smooth solution of (1.8).

Recall from Theorem 4.1 (ii) that f^ has tangential velocity zero for t sufficiently large. Thus, we can without loss of generality assume E(f^(t))=E(f~(t))>E(f∞), since otherwise f^(t) would be eventually constant by (1.6), and hence convergent. Moreover, since E(f^(t)) is nonincreasing, we have that limt→∞E(f^(t))=limn→∞ E(f~(tn))=E(f∞).

Since f∞ is smooth, by Theorem 4.8, there exist σ,CLS>0 and θ∈(0,12] such that we have a refined Łojasiewicz–Simon inequality, i.e. for all g∈X satisfying ‖g-f∞‖W4,2≤σ we have4.8 |E(g)-E(f∞)|1-θ≤CLS‖∇L2(dsg)E(g)+λ(g)∇L2(dsg)L(g)‖L2(dsg).

Passing to a subsequence, we can assume ‖f~(tn,·)-f∞‖W4,2<σ for all n. Definesn:=sups≥tn∣‖f~(t,·)-f∞‖W4,2<σfor allt∈[tn,s]

and note that sn>tn since f~ is smooth. Define G(t):=E(f~(t))-E(f∞)θ. By our assumption E(f~(t))>E(f∞), so we can compute on [tn,sn) using that f^ solves (1.3) with θ≡0, so ∂tf^=-∇E(f^)-λ∇L(f^) and the fact that E is geometric, i.e. invariant under reparametrization-ddtG=θE(f~)-E(f∞)θ-1-ddtE(f^)=θE(f~)-E(f∞)θ-1-∇L2(dsf^)E(f^),∂tf^L2(dsf^)=θE(f~)-E(f∞)θ-1‖∇L2(dsf^)E(f^)+λ(f^)∇L2(dsf^)L(f^)‖L2(dsf^)‖∂tf^‖L2(dsf^).

However, the quantity ‖∇L2(dsf^)E(f^)+λ(f^)∇L2(dsf^)L(f^)‖L2(dsf^) is geometric, too. Thus-ddtG=θE(f~)-E(f∞)θ-1‖∇L2(dsf~)E(f~)+λ(f~)∇L2(dsf~)L(f~)‖L2(dsf~)‖∂tf^‖L2(dsf^)≥θCLS‖∂tf^‖L2(dsf^).

on [tn,sn) by (4.8) and our choice of sn. Therefore, by Lemma 4.10 we have4.9 -ddtG(t)≥C‖∂tf~‖L2(dx),

for all t∈[tn,sn), where C=C(ℓ,E(f0),θ,CLS)>0. Let t∈[tn,sn). Then4.10 ‖f~(t)-f~(tn)‖L2(dx)≤∫tnt‖∂tf~(τ)‖L2(dx)dτ≤1CG(tn)→0

using (4.9) and E(f~(tn))→E(f∞) as n→∞. We now assume that all of the sn are finite. Then, by continuity (4.10) also holds for t=sn. By the subconvergence result in Theorem 4.1, passing to a subsequence we have f~(sn)→ψ smoothly as n→∞. Moreover, by continuity and the definition of sn, we have that ‖ψ-f∞‖W4,2=σ, whereas ‖ψ-f∞‖L2(dx)=limn→∞‖f~(sn)-f~(tn)‖L2(dx)=0 by (4.10), a contradiction.

Consequently, there has to exist some n0∈N such that sn0=∞, and this yields ‖f~(t)-f∞‖W4,2<σ for all t≥tn0. This means that (4.9) holds for any t≥tn0; thus, t↦‖∂tf~(t)‖L2(dx)∈L1([0,∞);R). Hence, for all tn0≤t≤t′ we have‖f~(t)-f~(t′)‖L2(dx)≤∫t′‖∂tf~(τ)‖L2(dx)dτ→0,

as t,t′→∞ by the dominated convergence theorem. Therefore, limt→∞f~(t) exists in L2(dx) and thus equals f∞. A subsequence argument shows that for any k∈N0 we have ‖f~(t)-f∞‖Ck(I;Rd)→0 as t→∞, i.e. the convergence is smooth. □

Appendix A: Explicit formulas in coordinates

In this section, we present the explicit representation of the geometric quantities appearing in this article. They can be obtained by a straight forward calculation, see e.g. [20, (2.3)].

Proposition A.1

Suppose f:I→Rd is a smooth immersion. With the arc-length element γ=|∂xf| we have (i) κ→f=∂s2f=∂x2fγ2-⟨∂x2f,∂xf⟩γ4∂xf=(∂x2f)⊥γ4;

(ii) ∇sfκ→f=∂x3fγ3-⟨∂x3f,∂xf⟩γ5∂xf-3⟨∂x2f,∂xf⟩γ5∂x2f+3⟨∂x2f,∂xf⟩2γ7∂xf;

(iii) ∇sf2κ→f=[∂x4fγ4-6⟨∂x2f,∂xf⟩γ6∂x3f-4⟨∂x3f,∂xf⟩γ6∂x2f-3|∂x2f|2γ6∂x2f+18⟨∂x2f,∂xf⟩2γ8∂x2f]⊥f;

(iv) ∇E(f)=[∂x4fγ4-6⟨∂x2f,∂xf⟩γ6∂x3f-4⟨∂x3f,∂xf⟩γ6∂x2f-52|∂x2f|2γ6∂x2f+352⟨∂x2f,∂xf⟩2γ8∂x2f]⊥f.

Here ∇E(f) denotes the L2(dsf)-gradient of E at f.

Lemma A.2

Let f be a smooth immersed curve with arc-length element γ. Then (i) |κ→f|4=γ-8|∂x2f|4-2γ-10|∂x2f|2⟨∂x2f,∂xf⟩2+γ-12⟨∂x2f,∂xf⟩4;

(ii) |∇sfκ→f|2=γ-6|∂x3f|2-γ-8⟨∂x3f,∂xf⟩2-6γ-8⟨∂x3f,∂x2f⟩⟨∂x2f,∂xf⟩+6γ-10⟨∂x3f,∂xf⟩⟨∂x2f,∂xf⟩2+9γ-10⟨∂x2f,∂xf⟩2|∂x2f|2-9γ-12⟨∂x2f,∂xf⟩4;

(iii) ⟨∇sfκ→f,κ→f⟩=γ-5⟨∂x3f,∂x2f⟩-γ-7⟨∂x3f,∂xf⟩⟨∂x2f,∂xf⟩-3γ-7⟨∂x2f,∂xf⟩|∂x2f|2+3γ-9⟨∂x2f,∂xf⟩3.

Appendix B: Function spaces

In this section, we collect all relevant information on the function spaces for maximal Lp-regularity. Most of the embedding results are collected in [16], see also [32], where even a polynomial weight t1-μ in time is allowed. As discussed in Remark 2.11, time weights do not allow to prove short-time existence with weaker initial data, and hence we restrict ourselves to the case μ=1.

Let J⊂R be an interval, 1≤p<∞. For any s∈(0,∞)\N and a Banach space E, the (E-valued) Sobolev–Slobodetskii space and the Bessel potential space, respectively, are given by real and complex interpolationWs,p(J;E):=W[s],p(J;E),W[s]+1,p(J;E)s-[s],p;Hs,p(J;E):=W[s],p(J;E),W[s]+1,p(J;E)s-[s];

where Wk,p(J;E) denotes the usual Bochner–Sobolev space for k∈N0. Recall from [51, Theorem 2.4.1 (a), Definition 4.2.1] that the Besov spaces are given byBp,qs(I;Rd):=Wm1,p(I;Rd),Wm2,p(I;Rd)θ,q,

where s=(1-θ)m1+θm2, θ∈(0,1),m1,m2∈N0,m1<m2,p,q∈[1,∞). By [51, Definition 2.3.1 (d) and Theorem 2.3.2 (d)] we have the relationB.1 Bp,ps(I;Rd)=Ws,p(I;Rd)fors∉N;B2,2s(I;Rd)=Ws,2(I;Rd)fors>0.

Moreover, recall from [16, Section 2], that in the setting of the maximal regularity spaces XT,p, the spaces of zeroth- and first-order boundary data are given byB.2 DT,p0:=W1-14p,p(0,T;Lp(∂I;Rd))≅W1-14p,p(0,T;(Rd)2);DT,p1:=W34-14p,p(0,T;Lp(∂I;Rd))≅W34-14p,p(0,T;(Rd)2).

We now recall the Sobolev embeddings for the maximal regularity space XT,p. We emphasize that the operator norms of the embeddings below might blow up as T→0+. However, in the solution space with vanishing trace, i.e. the space 0XT,p, they are bounded independently of T.

Proposition B.1

Let 0<T≤∞, let p≥2 and let k∈{0,⋯,4}. (i) XT,p↪BUC([0,T],Bp,p4(1-1p))(I;Rd)) with the estimate ‖f‖BUC([0,T];Bp,p4(1-1p)(I;Rd))≤C(p)‖f‖XT,pforf∈0XT,p.

(ii) XT,p↪Cα([0,T];C1,α(I;Rd)) for some α∈(0,1) with the estimate ‖f‖Cα([0,T];C1,α(I;Rd))≤C(p,α)‖f‖XT,pforf∈0XT,p.

(iii) The k-th spatial derivative is continuous as a map ∂xk:XT,p→H4-k4,p(0,T;Lp(I;Rd))∩Lp(0,T;H4-k,p(I;Rd))↪W(4-k)θ4,p(0,T;W(4-k)(1-θ),p(I;Rd))for allθ∈(0,1),

with the estimate ‖∂xkf‖H4-k4,p(0,T;Lp(I;Rd))∩Lp(0,T;H4-k,p(I;Rd))≤C(k,p)‖f‖XT,pforf∈0XT,p.

(iv) The spatial trace of the k-th spatial derivative is continuous as a map tr∂I∂xk:XT→W4-k4-18,p(0,T;Lp(∂I;Rd))≅W4-k4-18,p(0,T;(Rd)2),

with the estimate ‖tr∂I∂xkf‖W4-k4-18,p(0,T;(Rd)2)≤C(k,p)‖f‖XT,pforf∈0XT,p.

Proof

For T=∞ and I replaced by R, the statements follow from the corresponding results in [16, Section 3]. The statements in our case can then be obtained by considering appropriate temporal and spatial extension operators, see for instance [32, Lemma 2.5 and (3.2)]. When restricted to 0XT,p, the operator norm of the temporal extension does not depend on T by [32, Lemma 2.5], which implies the above T-independent estimates. □

Remark B.2

For a Hilbert space E and s∈(0,∞),p=2, the Bessel potential spaces coincide with the Slobodetskii spaces, i.e. Hs,2(0,T;E)=Ws,2(0,T;E) with equivalence of norms, cf. [29, Corollary 4.37]. A particular consequence of this is that in the case p=2 we get from Proposition B.1 (iii) that for k∈N, k≤4, and T∈(0,1] we have∂xk:XT,2→W4-k4,2(0,T;L2(I;Rd))∩L2(0,T;W4-k,2(I;Rd))

is continuous, with the estimate‖∂xkf‖W4-k4,2(0,T;L2(I;Rd))∩L2(0,T;W4-k,2(I;Rd))≤C(k)‖f‖XT,2forf∈0XT,2.

A crucial tool in proving the contraction estimates in Sect. 2.4 is the precise control of the integrability of the spatial derivatives and their spatial trace, with operator norm bounded independent of T. As in Sect. 2.4, we restrict to the case p=2 here.

Proposition B.3

Let T∈(0,1], k∈N, k≤4, ρ1,ρ2∈[1,∞). (i) If there exists θ∈[0,1] such that 4-k4θ-12≥-1ρ1 and (4-k)(1-θ)-12≥-1ρ2 then ∂xk:XT,2→Lρ1(0,T;Lρ2(I;Rd)) with the estimate ‖∂xkf‖Lρ1(0,T;Lρ2(I;Rd))≤C(k,θ,ρ1,ρ2)‖f‖XT,2for allf∈0XT,2.

(ii) If 4-k4-58≥-1ρ1, then tr∂I∂xk:XT,2→Lρ1(0,T;(Rd)2) with the estimate ‖tr∂I∂xkf‖Lρ1(0,T;(Rd)2≤C(k,ρ1)‖f‖XT,2for allf∈0XT,2.

Proof

We first prove the estimates. (i) Using first Proposition B.1 (iii) and Remark B.2, then interpolation, and in the last line the usual Sobolev embedding both in the temporal and spatial variable, we find B.3 ∂xk:0X1,2→W4-k4,2(0,1;L2(I;Rd))∩L2(0,1,W4-k,2(I;Rd))↪W4-k4θ,2(0,1;W(4-k)(1-θ),2(I;Rd))↪Lρ1(0,1;Lρ2(I;Rd)).

Now, by [32, Lemma 2.5], there exists an extension operator ET from (0, T) to (0,1) such that ET:0XT,2→0X1,2 has operator norm independent of T. Then, for any f∈0XT,2 we have using (B.3) ‖∂xkf‖Lρ1(0,T;Lρ2(I;Rd))≤‖∂xk(ETf)‖Lρ1(0,1;Lρ2(I;Rd))≤C(k,θ,ρ1,ρ2)‖ETf‖X1,2≤C(k,θ,ρ1,ρ2)‖f‖XT,2.

(ii) Since tr∂I∂kf only depends on the temporal variable, we first use Proposition B.1 (iv) and Remark B.2 and then the Sobolev embedding to find B.4 tr∂I∂xk:0X1,2→W4-k4-18,2(0,1;(Rd)2)↪Lρ1(0,1;(Rd)2).

Again, using the extension operator, we find for any f∈0XT,2‖tr∂I∂xkf‖Lρ1(0,T;(Rd)2)≤‖tr∂I∂xk(ETf)‖Lρ1(0,1;(Rd)2)≤C(k,ρ1)‖ETf‖X1,2≤C(k,ρ1)‖f‖XT,2.

The mapping properties follow from (B.3) and (B.4). □

Appendix C: Details of the contraction estimates

First, the following definition describes the structure of the nonlinearities in (2.3) which guarantees the desired contraction properties.

Definition C.1

Let (a,b)∈N02. We denote by A(a,b) the set of bounded multilinear mapsC.1 φ:(Rd)m×(Rd)a×(Rd)b→Rw

for some w∈N, m∈N0. Then, we define the set A(a,b) of multilinear maps of type (a, b) as the set of all maps f↦Φ(f) acting viaΦ(f)(t,x)=φ(∂xf(t,x),⋯,∂xf(t,x)⏟m-times,∂x2f(t,x),⋯,∂x2f(t,x)⏟a-times,∂x3f(t,x),⋯,∂x3f(t,x)⏟b-times),

for almost every (t,x)∈(0,T)×I where φ∈A(a,b).

Remark C.2

Note that we do not keep track of m, the number of first-order derivatives appearing in Φ∈A(a,b). This is justified since by Proposition B.1 (ii), the derivatives of first order of f∈XT are in C([0,T]×I;Rd) and hence do not affect the integrability of Φ(f).

Example C.3

The map f↦Φ(f)=⟨∂x2f,∂xf⟩∂x3f is in A(1,1), since the derivatives of second and third order only appear linearly.

The following proposition yields for which parameters (a, b) we get a contraction. Note that nonlinearities with this structure appear in F~ in (2.2) and λ in (2.1). As in Sect. 2.4, we assume T,M≤1 and set XT=XT,2 to simplify notation.

Proposition C.4

Let q∈(0,1) and let Φ∈A(a,b). Then, for T=T(q,f¯),M=M(q,f¯)∈(0,1] small enough, each of the following nonlinear maps is a well-defined q-contraction, i.e. Lipschitz continuous with Lipschitz constant q. (i) B¯T,M→L2(0,T;L2),f↦Φ(f), if (a,b)=(1,1) or (a,b)=(3,0).

(ii) B¯T,M→L2(0,T),f↦∫IΦ(f)dx, if (a,b)=(0,2),(a,b)=(2,1) or (a,b)=(4,0).

(iii) B¯T,M→L2(0,T;(Rd)2),f↦tr∂IΦ(f), if (a,b)=(1,1) or (a,b)=(3,0).

(iv) B¯T,M→L2(0,T;L2),f↦γ0-4-γ-4∂x4f.

The following general functional analytic result gives sufficient conditions for a multilinear map to be a q-contraction for T>0 small. It is the key ingredient in the proof of Proposition C.4.

Lemma C.5

Let 1≤q1≤∞ and suppose (f1,⋯,fr)↦μ(f1,⋯,fr) is a multilinear map such that for all f1,⋯,fr∈XT,2 we haveC.2 ‖μ(f1,⋯,fr)‖Lq1(0,T;Z)≤C∏j=1r‖S∂xjfj‖Xj.

Here, we have d1,⋯,dr∈{0,⋯,3}, S∈{Id,tr∂I} and Z,X1,⋯,Xr are Banach spaces such that there exists C∈(0,∞) independent of T with (i) ∂xi:XT→Xi and for f∈0XT we have ‖S∂xif‖Xi≤C‖f‖XT for all i=1,⋯,r.

(ii) for all j=1,⋯r one of the following conditions is satisfied. There exists α>0 with ‖S∂xjf‖Xj≤CTα‖f‖XT for all f∈0XT.

There exists k≠j with ‖S∂xkf‖Xk→0 as T→0 for all f∈XT.

Then, setting μ(f)=μ(f,⋯,f), we have μ(f)∈Lq1(0,T;Z) for all f∈XT and for any q∈(0,1), there exist M=M(q,r,f¯),T=T(q,r,f¯)∈(0,1] small enough, such that for all f,f~∈B¯T,M we have‖μ(f)-μ(f~)‖Lq1(0,T;Z)≤q‖f-f~‖XT.

Remark C.6

When applying Lemma C.5, we always work with Banach spaces of the type Xj=Lpj(0,T;Lqj) and Z=Lp0, for some p0,pj,qj∈[1,∞]. Note that (ii) b) is always satisfied if there exists k≠j with pk<∞, since then limT→0‖f‖Lpk(0,T;Lqk)→0 by dominated convergence.

Proof of Lemma C.5

Let f,f~∈B¯T,M. Adding and subtracting zeroes and using the multilinearity, we getμ(f)-μ(f~)=μ(f-f~,f,⋯,f)+μ(f~,f-f~,f,⋯,f)+⋯+μ(f~,⋯,f-f~,f)+μ(f~,⋯,f~,f-f~).

Thus, using (C.2), we getC.3 ‖μ(f)-μ(f~)‖Lq1(0,T;Z)≤C∑j=1r‖S∂x1f~‖X1⋯‖S∂xj(f-f~)‖Xj⋯‖S∂xrf‖Xr.

We now show that the contraction property is valid for each summand in (C.3). Note that for all k∈{1,⋯,r} by (i) we haveC.4 ‖S∂xkf‖Xk≤‖S∂xk(f-f¯)‖Xk+‖S∂xkf¯‖Xk≤C‖f-f¯‖XT+‖S∂xkf¯‖Xk≤CM+‖S∂xkf¯‖Xk.

In particular, for T≤1, M≤1 we findC.5 ‖S∂xkf‖Xk,‖S∂xkf~‖Xk≤C(f¯).

Now, let j∈{1,⋯,r}. If (ii) a) is satisfied, using f(0)=f~(0)=f0, we find‖S∂x1f~‖X1⋯‖S∂xdj-1f~‖Xj-1‖S∂xj(f-f~)‖Xj‖S∂xdj+1f‖Xj+1⋯‖S∂xrf‖Xr≤C(f¯)Tα‖f-f~‖XT≤qCr‖f-f~‖XT,

for T=T(α,f¯)>0 small enough. Otherwise, if (ii) b) is satisfied, we estimate using (i) for the j-th factor, (C.4) for the k-th factor and (C.5) for the remaining factors, to get‖S∂x1f~‖X1⋯‖S∂xdj-1f~‖Xj-1‖S∂xj(f-f~)‖Xj‖S∂xdj+1f‖Xj+1⋯‖S∂xrf‖Xr≤C(f¯)M+‖S∂xkf¯‖Xk‖f-f~‖XT.

By (ii) b), limT→0‖S∂xkf¯‖Xk=0. Consequently, for T=T(q,r,f¯),M= M(q,r,f¯)∈(0,1] small enough we find‖S∂x1f~‖X1⋯‖S∂xdj-1f~‖Xj-1‖S∂xj(f-f~)‖Xj‖S∂xdj+1f‖Xj+1⋯‖S∂xrf‖Xr≤qCr‖f-f~‖XT.

All in all, we have proven‖μ(f)-μ(f~)‖Lq1(0,T;Z)≤q‖f-f~‖XTforf,f~∈B¯T,M.

□

Together with the embedding results in Proposition B.1 and Proposition B.3, we can now prove Proposition C.4.

Proof of Proposition C.4

Let f,f~∈B¯T,M⊂XT with T=T(q,f¯),M=M(q,f¯) ∈(0,1] small enough such that Lemma 2.2 is satisfied. The strategy for the proof of cases (i)–(iii) is to apply Lemma C.5. To that end, we use Hölder’s inequality in time and space and then verify the assumptions of Lemma C.5 using Proposition B.3. In the following, we denote by f1,⋯,fm, g,g1,g2,g3,g4,h,h1,h2 general functions in B¯T,M.

Case (i): If (a,b)=(1,1), by Hölder’s inequality we haveC.6 ‖φ(∂xf1,⋯,∂xfm,∂x2g,∂x3h)‖L2(0,T;L2)≤C(φ)∏j=1m‖∂xfj‖∞‖∂x2g‖L4(0,T;L8)‖∂x3h‖L4(0,T;L83).

Using Proposition B.1 (ii) we have∂x:XT→Cα([0,T];C(I;Rd)),with the estimate‖∂xf‖Cα([0,T];C(I;Rd))≤C‖f‖XTforf∈0XT.

Therefore, we findC.7 ∂x:XT→C([0,T]×I;Rd),with the estimate‖∂xf‖∞≤CTα‖f‖XTforf∈0XT,

such that (ii) a) in Lemma C.5 is satisfied. Next, using Proposition B.3 (i) with k=2 and θ=34 yieldsC.8 ∂x2:XT→L8(0,T;L4),with the estimate‖∂x2f‖L8(0,T;L4)≤C‖f‖XTfor allf∈0XT,

since 4-24·34-12≥-18 and (4-2)(1-34)-12≥-14. Similarly for the third derivative with θ=12 we getC.9 ∂x3:XT→L83(0,T;L4),with the estimate‖∂x3f‖L83(0,T;L4)≤C‖f‖XTfor allf∈0XT.

Thus, condition (ii) a) in Lemma C.5 is satisfied for the fist m factors in (C.6) by (C.7), whereas for the remaining factors condition (ii) b) holds. More precisely, for j=m+1 choosing k=m+2 works and conversely j=m+2,k=m+1, using Remark C.6.

The case (a,b)=(3,0) can be treated similarly, using Hölder to obtain‖φ(∂xf1,…,∂xfm,∂x2g1,∂x2g2,∂x2g3)‖L2(0,T;L2)≤C∏j=1m‖∂xfj‖∞∏j=13‖∂x2gj‖L6(0,T;L6),

and then Proposition B.3 (i) with k=2, θ=23 to getC.10 ∂x2:XT→L6(0,T;L6),with the estimate‖∂x2f‖L6(0,T;L6)≤C‖f‖XTfor allf∈0XT.

Case (ii): First, we have the following basic estimate∫IΦ(f)dx-∫IΦ(f~)dxL2(0,T)≤‖Φ(f)-Φ(f~)‖L2(0,T;L1).

It hence suffices to show that XT→L2(0,T;L1),f↦Φ(f) is a q-contraction. To that end, we use Lemma C.5 with Z=L1,S=Id.

If (a,b)=(0,2), we have by Hölder’s inequalityC.11 ‖φ(∂xf1,⋯∂xfm,∂x3h1,∂x3h2)‖L2(0,T;L1)≤C∏j=1m‖∂xfj‖∞∏j=12‖∂x3hj‖L4(0,T;L2).

Now, using Proposition B.3 (i) with k=3 and θ=1, we haveC.12 ∂x3:XT→L4(0,T;L2(I;Rd))with the estimate‖∂x3f‖L4(0,T;L2)≤C‖f‖XTfor allf∈0XT.

Consequently, the last two factors in (C.11) satisfy condition (i) and (ii) b) in Lemma C.5, cf. Remark C.6. For the first m factors, we may once again use (C.7) to deduce that conditions (i) and (ii) a) in Lemma C.5 are satisfied.

If (a,b)=(2,1), we proceed similarly, first using Hölder to get‖φ(∂xf1,⋯∂xfm,∂x2g1,∂x2g2,∂x3h)‖L2(0,T;L1)≤C∏j=1m‖∂xfj‖∞∏j=12‖∂x2gj‖L8(0,T;L4)‖∂x3h‖L4(0,T;L2),

and then applying (C.7), (C.8) and (C.12). For (a,b)=(4,0), we may apply Hölder’s inequality to obtain‖φ(∂xf1,⋯∂xfm,∂x2g1,∂x2g2,∂x2g3,∂x2g4)‖L2(0,T;L1)≤C∏j=1m‖∂xfj‖∞∏j=14‖∂x2gj‖L8(0,T;L4),

and then use (C.7) and (C.8).

Case (iii): Again, we use Lemma C.5, now with Z=(Rd)2 and S=tr∂I. If (a,b)=(1,1) we obtain by Hölder’s inequalityC.13 ‖tr∂Iφ(∂xf1,⋯,∂xfm,∂x2g,∂x3h)‖L2(0,T;(Rd)2)≤C∏j=1m‖tr∂I∂xfj‖∞‖tr∂I∂x2g‖L8(0,T;(Rd)2)‖tr∂I∂x3h‖L83(0,T;(Rd)2).

Note that by Proposition B.3 (ii), we haveC.14 tr∂I∂x2:XT→L8(0,T;(Rd)2),with the estimate‖tr∂I∂x2f‖L8(0,T;(Rd)2)≤C‖f‖XTfor allf∈0XT,

whereas for the third derivative, we obtainC.15 tr∂I∂x3:XT→L83(0,T;(Rd)2),with the estimate‖tr∂I∂x3f‖L83(0,T;(Rd)2)≤C‖f‖XTfor allf∈0XT.

As in cases (i) and (ii), we then use the mapping properties and the estimates in (C.7), (C.14) and (C.15) together with Remark C.6 to verify that the assumptions of Lemma C.5 are satisfied.

If (a,b)=(3,0), we proceed similarly, first using Hölder to obtain‖tr∂Iφ(∂xf1,⋯,∂xfm,∂x2g1,∂x2g2,∂x2g3)‖L2(0,T;(Rd)2)≤C∏j=1m‖∂xfj‖∞∏j=13‖∂x2gj‖L6(0,T;(Rd)2),

and then (C.7) for the first-order terms and Proposition B.3 (ii) with k=2, yieldingtr∂I∂x2:XT→L6(0,T;(Rd)2),with the estimate‖tr∂I∂x2f‖L6(0,T;(Rd)2)≤C‖f‖XTfor allf∈0XT.

Case (iv): Let q∈(0,1). For f,f~∈B¯T,M, we have|(γ0-4-γ-4)∂x4f-(γ0-4-γ~-4)∂x4f~|≤|γ0-4-γ-4||∂x4f-∂x4f~|+|γ~-4-γ-4||∂x4f~|.

Thus, we may estimateC.16 ‖(γ0-4-γ-4)∂x4f-(γ0-4-γ~-4)∂x4f~‖L2(0,T;L2)≤‖γ0-4-γ-4‖∞‖∂x4f-∂x4f~‖L2(0,T;L2)+‖γ~-4-γ-4‖∞‖∂x4f~‖L2(0,T;L2).

For the first term, we use the mean value theorem and Lemma 2.2 to concludeC.17 |γ0(x)-4-γ(t,x)-4|≤4infIγ02-5|∂xf0(x)-∂xf(t,x)|.

Consequently, using Proposition B.1 (ii) we may estimate‖γ0-4-γ-4‖∞≤C(γ0)sup(t,x)∈[0,T]×I|∂xf(0,x)-∂xf(t,x)|≤C(γ0)sup(t,x)∈[0,T]×I‖f(0)-f(t)‖C1+α(I;Rd)≤C(γ0)sup(t,x)∈[0,T]×Itα‖f‖Cα([0,T];C1+α(I;Rd))≤C(γ0)Tα‖f-f¯‖Cα([0,T];C1+α(I;Rd))+‖f¯‖Cα([0,T];C1+α(I;Rd))≤C(γ0)Tα‖f-f¯‖XT+‖f¯‖Cα([0,T];C1+α(I;Rd))≤C(f¯)Tα.

Combined with the simple estimate ‖∂x4f-∂x4f~‖L2(0,T;L2)≤‖f-f~‖XT this yields a q2-contraction estimate for the first part of (C.16), taking T=T(q,f¯)∈(0,1] small enough. For the remaining part, we use (C.17) with γ0 replaced by γ~ to conclude‖γ~-4-γ-4‖∞≤4infIγ02-5‖∂xf-∂xf~‖∞≤C(f¯)Tα‖f-f~‖XT,

and ‖∂x4f‖L2(0,T;L2)≤‖f-f¯‖XT+‖∂x4f¯‖L2(0,T;L2)≤C(f¯). Consequently, if T=T(q,f¯)∈(0,1] is small enough, the second part of (C.16) is a q2-contraction. □

It is not difficult to see that the statement of Proposition C.4 remains true if one allows multiplication by powers of the arc-length element.

Corollary C.7

Let q∈(0,1),ℓ∈N, Φ∈A(a,b). For T=T(q,ℓ)∈(0,1], M= M(q,ℓ)∈(0,1] small enough, each of the following maps is a well-defined q-contraction. (i) B¯T,M→L2(0,T;L2),f↦γ-ℓΦ(f), if (a,b)=(1,1) or (a,b)=(3,0).

(ii) B¯T,M→L2(0,T),f↦∫Iγ-ℓΦ(f)dx, if (a,b)=(0,2),(a,b)=(2,1) or (a,b)=(4,0).

(iii) B¯T,M→L2(0,T(Rd)2),f↦tr∂Iγ-ℓΦ(f), if (a,b)=(1,1) or (a,b)=(3,0).

Proof

Well-definedness: By Lemma 2.2 we can estimate |γ-ℓΦ(f)|≤infγ02|Φ(f)| for all T,M>0 small enough. Thus f↦γ-ℓΦ(f) maps into the correct space by Lemma C.5.

Contraction: Let q∈(0,1) and let f,f~∈B¯T,M. For the first case, taking T,M>0 small enough, we have‖γ-ℓΦ(f)-γ~-ℓΦ(f~)‖L2(0,T;L2)≤‖γ-ℓ-γ~-ℓ‖∞‖Φ(f)‖L2(0,T;L2)+‖γ~-ℓ‖∞‖Φ(f)-Φ(f~)‖L2(0,T;L2)≤‖γ-ℓ-γ~-ℓ‖∞‖Φ(f)-Φ(f¯)‖L2(0,T;L2)+‖Φ(f¯)‖L2(0,T;L2)+‖γ~-ℓ-γ¯-ℓ‖∞+‖γ¯-ℓ‖∞‖Φ(f)-Φ(f~)‖L2(0,T;L2)≤C(f¯)‖γ-ℓ-γ~-ℓ‖∞+‖γ~-ℓ-γ¯-ℓ‖∞+‖γ¯-ℓ‖∞q2‖f-f~‖XT

using Proposition C.4 for q2∈(0,1) to be chosen. With similar estimates one finds∫Iγ-ℓΦ(f)dx-∫Iγ~-ℓΦ(f~)dxL2(0,T)≤C(f¯)‖γ-ℓ-γ~-ℓ‖∞+‖γ~-ℓ-γ¯-ℓ‖∞+‖γ¯-ℓ‖∞q2‖f-f~‖XT

and‖tr∂Iγ-ℓΦ(f)-tr∂Iγ~-ℓΦ(f~)‖L2(0,T;(Rd)2)≤C(f¯)‖γ-ℓ-γ~-ℓ‖∞+‖γ~-ℓ-γ¯-ℓ‖∞+‖γ¯-ℓ‖∞q2‖f-f~‖XT.

We now prove that for any q∈(0,1) the map B¯T,M→L∞((0,T)×I), f↦γ-ℓ is a q-contraction for T,M>0 small enough. We find as in (C.17)‖γ-ℓ-γ~-ℓ‖∞≤C(ℓ,f¯)‖f-f~‖C0([0,T];C1(I;Rd))≤C(ℓ,f¯)Tα‖f-f~‖XT≤q2‖f-f~‖XT,

for T=T(q,ℓ,f¯)∈(0,1] small enough using Proposition B.1 (ii) and the fact that f(0)=f~(0)=f0. Thus, we findC(f¯)‖γ-ℓ-γ~-ℓ‖∞+‖γ~-ℓ-γ¯-ℓ‖∞+‖γ¯-ℓ‖∞q2‖f-f~‖XT≤q2‖f-f~‖XT+M+‖γ¯-ℓ‖∞q2‖f-f~‖XT≤q‖f-f~‖XT,

choosing first q2=q2(q,ℓ,f¯)∈(0,1) sufficiently small and passing to a smaller T=T(q,ℓ,f¯) and M=M(q,ℓ,f¯)∈(0,1] if necessary. □

Proof of Lemma 2.4

First, taking T=T(f¯),M∈(0,1] small enough such that Lemmas 2.2 and 2.3 hold, all terms are defined almost everywhere. We observe that F(f) is a sum of terms as in Corollary C.7 (i) and Proposition C.4 (iv) by (2.2), hence well-defined and a q-contraction for all q∈(0,1), if T=T(q,f¯), M=M(q,f¯)∈(0,1] are small enough.

For Λ we need to do one additional estimate. For f∈B¯T,M and T,M>0 the scalar-valued function λ is in L2(0,T), since by Lemma 2.3 the energy E(f) in the denominator of λ (cf. Sect. 2.1) is bounded from below uniformly in t, whereas the nominator N(f) is in L2(0,T) by Corollary C.7 (ii) and (iii) and by the explicit formulas in Lemma A.2 and (2.1). The term κ→f is in L∞(0,T) by the embedding XT↪BUC([0,T];W2,2(I;Rd)), cf. Proposition B.1 (i),(B.1) and Proposition A.1.

Now, the crucial step is the proof of the contraction estimate for Λ. To that end, let f,f~∈B¯T,M. Then, writing λ(f)=N(f)2E(f) as in Sect. 2.1, we find for almost every (t, x)C.18 |λ(f)(t)κ→f(t,x)-λ(f~)(t)κ→f~(t,x)|≤|λ(f)(t)-λ(f~)(t)||κ→f(t,x)|+|λ(f~)(t)||κ→f(t,x)-κ→f~(t,x)|≤12E(f(t))E(f~(t))|N(f)(t)||E(f(t))-E(f~(t))||κ→f(t,x)|+12E(f(t))|N(f)(t)-N(f~)(t)||κ→f(t,x)|+|λ(f)(t)||κ→f(t,x)-κ→f~(t,x)|≤C(f0)|N(f)(t)||E(f(t))-E(f~(t))||κ→f(t,x)|+C(f0)|N(f)(t)-N(f~)(t)||κ→f(t,x)|+|λ(f)(t)||κ→f(t,x)-κ→f~(t,x)|,

using that by Lemma 2.3 the elastic energy is bounded from below. Taking the L2L2-norm in (C.18), we are left with three terms. The first one isC.19 ‖|N(f)||E(f)-E(f~)||κ→f|‖L2(0,T;L2)≤‖N(f)‖L2(0,T)‖E(f)-E(f~)‖L∞(0,T)‖κ→f‖L∞(0,T;L2).

Now, note that N(f) is a sum of terms as in Corollary C.7 (ii) and (iii) by (2.1) and the explicit formulas in Lemma A.2. Therefore, for any q∈(0,1), we haveC.20 ‖N(f)-N(f~)‖L2(0,T)≤q‖f-f~‖XT,

if we take T=T(q,f¯),M=M(q,f¯)∈(0,1] small enough. In particular, we can assume that f↦N(f) is 1-Lipschitz.

For the elastic energy term, note that E is analytic, hence C1 on the space of W2,2-immersions, cf. Proposition 4.4, in particular it is locally Lipschitz continuous in a neighbourhood of f0∈WImm2,2(I;Rd). Hence, there exists C(f0)>0 such that |E(h)-E(h~)|≤C(f0)‖h-h~‖W2,2(I;Rd) for all h and h~ satisfying ‖h-f0‖W2,2<δ and ‖h~-f0‖W2,2≤δ.

By Proposition B.1(i), we have 0XT↪BUC(0,T;W2,2) with operator norm independent of T∈(0,1]. Consequently, we have‖f(t)-f0‖W2,2≤‖f(t)-f¯(t)‖W2,2+‖f¯(t)-f0‖W2,2≤CM+‖f¯(t)-f¯(0)‖W2,2≤δ

for T=T(f¯),M∈(0,1] small enough, and similarly ‖f~(t)-f0‖W2,2≤δ. But then, using Proposition B.1(i), we have the estimateC.21 ‖E(f)-E(f~)‖L∞(0,T)≤C(f0)‖f-f~‖L∞(0,T;W2,2)≤C(f¯)‖f-f~‖XT.

For the curvature term κ→f, note that WImm2,2(I;Rd)→L2(I;Rd),f↦κ→f=∂sf2f is analytic (cf. Proposition 4.4), in particular Lipschitz continuous near f0. The same argument as above yieldsC.22 ‖κ→f-κ→f~‖L∞(0,T;L2)≤C(f0)‖f-f~‖L∞(0,T;W2,2)≤C(f¯)‖f-f~‖XT.

Now, we estimateC.23 ‖κ→f‖L∞(0,T;L2)≤‖κ→f-κ→f¯‖L∞(0,T;L2)+‖κ→f¯‖L∞(0;T;L2)≤C(f¯)‖f-f¯‖XT+‖κ→f¯‖L∞(0;T;L2)≤C(f¯)

and using (C.20), we obtain the boundC.24 ‖N(f)‖L2(0,T)≤‖N(f)-N(f¯)‖L2(0,T)+‖N(f¯)‖L2(0,T)≤‖f-f¯‖XT+‖N(f¯)‖L2(0,T)≤M+‖N(f¯)‖L2(0,T).

If we now combine (C.21), (C.23) and (C.24), we obtain from (C.19)C(f0)‖|N(f)||E(f)-E(f~)||κ→f|‖L2(0,T;L2)≤M+‖N(f¯)‖L2(0,T)C(f¯)‖f-f~‖XT≤q4‖f-f~‖XT

if we take T=T(q,f¯),M=M(q,f¯)∈(0,1] small enough.

For the second term in (C.18), using (C.20) and (C.23) we haveC(f0)‖N(f)(t)-N(f~)(t)‖L2(0,T)‖κ→f(t,x)‖L∞(0,T;L2)≤C(f¯)q2‖f-f~‖XT≤q4‖f-f~‖XT,

taking q2=q2(q,f¯)∈(0,1) small enough and possibly reducing T=T(q,f¯), M= M(q,f¯)∈(0,1] if necessary.

For the last term in (C.18), using Lemma 2.3, (C.24) and (C.22), we haveN(f)E(f)L2(0,T)‖κ→f-κ→f~‖L∞(0,T;L2)≤3E(f0)M+‖N(f¯)‖L2(0,T)C(f¯)‖f-f~‖XT≤q4‖f-f~‖XT.

taking M=M(q,f¯),T=T(q,f¯)∈(0,1] small enough. All in all, we have now shown that taking T=T(q,f¯),M=M(q,f¯)∈(0,1] small enough, we have‖Λ(f)-Λ(f~)‖L2(0,T;L2)=‖λ(f)κ→f-λ(f~)κ→f~‖L2(0,T;L2)≤3q4‖f-f~‖XT,

which proves that Λ:B¯T,M→L2(0,T;L2) is a 3q4-contraction. Reducing T= T(q,f¯)∈(0,1], M=M(q,f¯)∈(0,1] if necessary, we may assume that F is a q4-contraction; hence, N:B¯T,M→YT1 is a q-contraction for T=T(q,f¯),M=M(q,f¯)∈(0,1] small enough. □

Appendix D: A gluing lemma for reparametrizations

In Theorem 4.1, we used the fact that two smooth reparametrizations can be interpolated by another smooth reparametrization. We state this gluing result here in a slightly more general form for possible future reference.

Lemma D.1

Let 0<t1<t2<T and Φ1:[0,t2]×I→I, Φ2:[t1,T]×I→I be smooth families of reparametrizations, such that Φi(t,·) is strictly increasing for all suitable t and i=1,2. Then, there exists a smooth family of strictly increasing reparametrizations Ψ:[0,T]×I→I satisfyingΨ(t,x)=Φ1(t,x),for all0≤t≤t1,x∈IΨ(t,x)=Φ2(t,x),for allt2≤t≤T,x∈I.

Proof

Let δ>0 be sufficiently small and η:[0,T]→R,0≤η≤1 be a smooth cut-off function, satisfyingη(t)=1,for0≤t≤t1+δ0,fort≥t2-δ.

Then it is not difficult to check that the function Ψ:[0,T]×I→R given byΨ(t,x):=Φ1(t,x)for0≤t≤t1,x∈IΦ1(t,x)η(t)+Φ2(t,x)(1-η(t)),fort∈[t1,t2],x∈IΦ2(t,x)fort2≤t≤T,x∈I

is smooth and satisfies all the desired properties. □

Acknowledgements

Fabian Rupp has been supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)-Projektnummer: 404870139 and by the Austrian Science Fund (FWF), grant numbers 10.55776/P32788 and 10.55776/ESP557. The authors would like to thank Anna Dall’Acqua, Marius Müller and Rico Zacher for helpful discussions and comments. Moreover, the authors are grateful to the referees for their valuable feedback on the original manuscript.

Funding

Open access funding provided by University of Vienna.

Data availability

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Declaration

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

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