Elastic flow of networks: long-time existence result

We study and give a long time existence result for elastic motion with a penalization term on the length and some extra topological conditions that prevents the appearance of some pathological cases: for instance, the triple junction should not be allowed to collapse on one of the boundary points Pi. As far as we know, research on elastic ow of networks is still at its beginning stage inmathematics. A lot of literature is focused on the case where the motion occurs by mean curvature ow (also called curve shortening ow, a second order ow that decreases length of the curves), see for instance the survey paper [20]. A


Introduction
We consider the long-time evolution for the elastic ow of a three-network in R n (n ≥ ) as depicted in Figure 1, that is with three xed boundary points P , P , P and one moving triple junction. That is, a threepointed curved star with a star-center that may move in time. We study and give a long time existence result for elastic motion with a penalization term on the length and some extra topological conditions that prevents the appearance of some pathological cases: for instance, the triple junction should not be allowed to collapse on one of the boundary points P i .
As far as we know, research on elastic ow of networks is still at its beginning stage in mathematics. A lot of literature is focused on the case where the motion occurs by mean curvature ow (also called curve shortening ow, a second order ow that decreases length of the curves), see for instance the survey paper [20]. A short time existence result for the elastic ow of a network of planar curves has been given recently in [13] while the stationary case (in the special case of so-called 'Theta' networks) has been considered in [7]. Elastic ow with junctions is considered numerically in [1]: in particular an appropriate variational formulation and two types of junction conditions, the so called C respectively C boundary conditions, are discussed. In that work the authors concentrated in the derivation of the Euler-Lagrange equations and did many numerical simulations.
Networks and ow of networks arise naturally in the study of multiphase systems and of the dynamics of their interfaces, see for instance [14,18]. Elastic networks appear in some investigation in mechanical engineering or material sciences related to polymer gels, ber or protein networks, e.g., [3], [15]. In these physical systems, junctions between elastic beams play an important role in determining mechanical properties, e.g., rigidity or deformability.
Before stating our main result, we introduce brie y the set up of our work and recall some well known facts. The elastic energy of a smooth regular curve (an immersion) f :Ī → R n , n ≥ , I = ( , ) is given by where ds = |∂x f |dx is the arc-length element and κ is the curvature vector of the curve. De ning ∂s = |∂x f | − ∂x f , then κ = ∂ s f . The length of a curve f is given by For λ ≥ let This is the energy that we consider: the length of the curve is allowed to change in time but its growth is penalized according to the weight λ. In the following we call Γ = {f , f , f } a three-pointed star network or simply network. For λ = (λ , λ , λ ) ∈ R + , the energy of the network Γ = {f , f , f } is given by ( 1.5) Here and in the following we agree that E(Γ) := E (Γ). We let the network Γ evolve in time according to an L -gradient ow for the energy E λ . Natural boundary conditions are imposed on the three curves (more details are given in Section 2). Our main result is the following: (1.7) Moreover let Γ satisfy appropriate compatibility conditions (speci ed in Remark C. 3), and be such that at the triple junction at least two curves form a strictly positive angle. Then the following holds: (i) Long-time existence result: the equations with boundary conditions Smooth solution means that the three parametrization of the three curves are smooth functions in the time and space variable. The extension of the long-time existence result to the case λ i ≥ is discussed in Remark 6.3 below.
Note that the above theorem must be understood in a geometrical sense: that is the existence of a global parametrization of the ow is meant up to reparametrization. So our result states that we are able to nd a global in time smooth motion of the network, provided two topological constraints are ful lled during the ow: namely that the lengths of the curves are uniformly bounded from below and that the the curves never entirely "collapse" to a con guration where all tangents vectors are parallel at the triple junctions. In our proof the topological constraints occurs naturally as follows: the bound from below on the lengths of the curves is needed to apply interpolation inequalities (cf. for instance Lemma 3.10 and Lemma 3.11 below); that the dimension of the space spanned by the unit tangents at the triple junction should always be bigger or equal to two arise when we express the tangential components at the boundary in terms of geometric quantities (cf. Remark 5.1 and Remark 5.2 below.) Not surprisingly it arises also in the proof of short-time existence of the ow provided in [6] and [13]. At the moment we have no means to control these topological constraints: whether and how this could be done is subject to future studies (see also Remark 6.5 below).
To achieve our goal, we will consider in place of (1.8) equations of type ∂ t f i = −∇ s κ i − | κ i | κ i + λ i κ i + φ i ∂s f i on ( , T) × I for i = , , , (1.10) where φ i are smooth functions. Note that the presence of the tangential components is necessary in order for the ow to ful ll the topological constraint that the curves stay "glued" at the triple junction (concurrency condition), with the latter being able to move freely in time. A proper choice of tangential component is necessary and is discussed in details in Section 2.3 and in Section C. Our strategy can be summarized as follows: starting from a short-time existence result (see Section 2.3) we reparametrize the ow in such a way that for each curve the maps φ i linearly interpolate their values between the boundary points. For this choice of parametrizations we consider the long-time behavior of the network, and show that if the ow does not exist globally then we obtain a contradiction. This is achieved by obtaining uniform bounds for the curvature and its derivatives, together with a control on the arc-length, up to the maximal time of existence < T < ∞. With these estimates we are able to extend the ow smoothly up to T and then restart the ow, contradicting the maximality of T.
In its essence the strategy of our proof is not di erent from our previous works on long-time existence for open elastic curves in R n (see [4], [5], [8], [9], which in turn rely on ideas of [10] and [22]): we use inequalities of Gagliardo-Nirenberg type, exploit the boundary conditions to reduce the order of some boundary terms, and rely heavily on interpolation estimates presented in [5]. However, the treatment of the tangential components is completely new and far from trivial. In particular the "algebra" for the maps φ i (that is how their derivatives in time and space behave with respect to the order of the studied PDEs, see Remark 4.3 for more details) must be thoroughly understood. Furthermore, an accurate choice of the "right" vector eld (speci cally ϕ in Lemma 3.2) for which uniform bounds are derived is absolutely crucial for any of the presented arguments to work. Finally, because of the interaction of the three curves proofs become increasingly technical and lengthy, and several new lemmas are derived in order to make our arguments more concise and more transparent.
The paper is organized as follows: after introducing the notation and motivating the de nition of the ow in Section 2, we collect several preliminary estimates and interpolation estimates in Section 3. The treatment of the boundary term is given in Section 4, whereas the in uence of the tangential components at the boundary is discussed in Section 5. Finally, in Section 6 we prove the main Theorem 1.1. The proof of the latter is divided in several steps: we have an initial step, where rst bounds on the curvature vectors are derived. In the second step, we show the starting procedure of an induction argument: it is at this point that a proper choice of φ i starts playing an important role. After the somewhat cumbersome induction step, where uniform estimates for the derivative of the curvature vectors are derived, we are nally able to conclude long-time existence by the contradiction procedure mentioned above. To ease the presentation many technical proofs are collected in the Appendix where we also cite the results on short-time existence that are needed for our present work.
During the time between the submission of this work and its revision an interesting preprint [12] appeared where long-time existence for planar networks moving by elastic energy (as in this work) is studied but with the di erence that weaker function spaces are used. Moreover, in [21] the authors provide a long-time existence result for a di erent ow of the elastic energy in the case of planar networks with curves of xed lengths. The relation between the setting and the results presented here and those depicted in [21] are discussed there.

Set up and notation . First variation and natural boundary conditions
First of all recall, that for su ciently smooth f , ϕ :Ī → R n , with f a regular curve, the rst variation of the length is given by Here ∇s is an operator that on a smooth vector eld ϕ acts as follows i.e. it is the normal projection of ∂s ϕ. Let consider a variation Γ tη of the network Γ given by t ∈ (−δ, δ) and su ciently smooth vector elds {η , η , η }, η i : I → R n , such that η i ( ) = for i = , , and η i ( ) = η j ( ) for i, j = , , , so that Γ tη = {f + tη , f + tη , f + tη } is still a three-pointed star network.
Then by (2.1) and (2.2) we nd Notice that here and in the rest of the work for simplicity of notation we simply write ds instead of ds i and also in the derivatives we simply write ∇s instead of the correct ∇ s i . Choosing rst test functions with compact support we see that each critical point has to satisfy for any test function η i and hence that κ i ( ) = for i = , , .

The flow
As mentioned in the introduction we take an L -ow for the energy E λ with the condition that the network keeps its topological properties along the ow. For this one needs a tangential component. The problem we consider is then (1.10) with φ i , i = , , , smooth functions (whose role and de nition is discussed below), with boundary conditions given in (1.9) and initial value For the latter we ask that Γ = {f , , f , , f , } be a network of regular smooth curves parametrized by constant speed, satisfying (1.6), (1.7) and being such that at the triple junction at least two curves form a strictly positive angle (the latter condition is necessary for our short-time existence result given in Section C). Moreover the initial network needs to satisfy a set of compatibility conditions. These are required to ensure that the solution of the parabolic problem is smooth up to the initial time t = . We ask therefore that f i, , i = , , satisfy compatibility condition of any order according to Remark C.3: indeed, with this choice and together with a very speci c choice of tangential components φ i := φ * i as de ned in (C1) we can start the ow smoothly (see Appendix C for the short-time existence result).

. Short time existence and reparametrization
Our starting point is the short time existence result Theorem C.1, stating that given an initial network Γ of smooth curves satisfying (1.6), (1.7) and suitable compatibility conditions (cf. Appendix C), then there exists an interval of time where our problem (1.10) (with φ i := φ * i as de ned in (C1)) admits a smooth regular solution, meaning that f i ∈ C ∞ ([ , T) × [ , ]), i = , , , are regular parametrizations. A proof of this result is provided in [6]. A short-time existence for planar curves, in appropriate Hölder spaces can be found in [13].
Before we proceed some comments are in order, in particular more information must be given on the role/choice of the tangential components φ i in (1.10). First of all, notice that to construct a short-time existence solution for (1.8) (together with the chosen initial and boundary conditions), one typically proceeds by 1) introducing a suitable choice of tangential components φ i (in order to factor out the degeneracies due to the geometric invariances; in our case we take φ i := φ * i as de ned in (C1)), 2) applying a linearization procedure and Solonnikov-theory (see [23]), 3) employing a xed point argument to show short-time existence for the systems of non-linear equations under consideration (see [2]). In particular we see that at a rst sight any tangential component φ i "destroys the geometric nature" of the equations (1.8). This is, however, not entirely true. It is well known, that tangential components can be modi ed by a reparametrization and that indeed all geometrical quantities (tangents, curvature vectors, length, etc.) are invariant under reparametrization. That, for the geometric motion, φ i plays no role in the interior of each curves becomes evident also during the computations performed in this paper. The role of the tangential components φ i becomes tangible only at the boundary of each curve, when enforcing the concurrency condition at the triple junction (see (1.4)) and in uencing variation of the length of each curve (cf. also Remark 3.3 below). On the other hand even at the boundary the tangential components are determined by geometric quantities (see Remark 5.1, in particular (5.3) below). So the "freedom of choice" in the tangential components is in principle only allowed in the interior of the curve, where, as we have already stated, the geometric quantities do not "register" it. Needles to say, we want to avoid tangential components, hence parametrizations, that destroy the regularity properties of the ow.
For the long-time existence proof it is important to have a good control of the tangential components also in the interior of the curves. To that end we reparametrize our short-time solution f = (f , f , f ) from Theorem C.1 in such a way that the new tangential component φ G i interpolates linearly between the boundary data of φ * i at x = and x = (where φ * i (t, ) = , since at x = the velocities vanish due to the boundary conditions). This transformation employes the fact that the tangential components are geometrical at the boundary points and hence independent of the parametrization (see Remark 5.1). To this end, we choose a family of smooth di eomorphisms ϕ i (t, ·) : I → I, i = , , , such that ϕ i (t, x) = x for (t, x) ∈ [ , T ) × ∂I, for some < T ≤ T, and such that forf Existence and uniqueness follow from Lemma C.4. A straightforward computation gives then that In other words we can reparametrize the ow in such a way that the tangential component interpolates linearly its boundary values. At the same time problem (1.8) is satis ed on [ , T ). As we will see in the proof of the long time existence, this will be of great help in many estimates. Also note that as well as so thatf i ful lls all the same boundary conditions as f i , i = , , . Again, let us stress the fact that by Remark 5.1, in particular (5.3) below, the tangential components φ G i = φ * i at the boundary are terms that can be given in terms of geometric quantities (curvature vectors, tangents, etc.), which of course do not change under reparametrizations of the ow. In particular φ G i can be written in terms of geometric quantities for the networkf (cf. (5.3) and Remark 5.1) and we can write This is to say that we have achieved an expression that is independent of the di eomorphisms ϕ i used above in the derivation. We refer to (2.5), (2.6), with boundary conditions (1.9) (with f replaced byf ) as the geometric problem. The observations made so far yield that given an initial data satisfying the assumptions of Theorem 1.1, then there exists a short-time solution to the geometric problem. For the sake of simplicity we drop the notationf .
Summarizing, from now on we consider the geometric problem (1.10), (1.9), whereby φ i (:= φ G i ) linearly interpolates between its boundary values, i.e it satis es (2.6) (see also (6.6)). It is for this problem that we want to obtain a long-time existence result.

Preliminaries
First of all we state a simple fact that will be used repeatedly in the computations that follow: for ϕ any smooth normal eld along f and h a scalar map we have that for any m ∈ N (using (2.3)) where ∇ t ϕ = ∂ t ϕ − ∂ t ϕ, ∂s f ∂s f .
. . . Decrease of the energy along the flow.
As a rst application of the above lemma we show that the energy decreases along the ow. Let Γ(t) = {f (t, ·), f (t, ·), f (t, ·)} be a three-pointed star network moving according the elastic ow as considered in Section 2.2. Then by (1.2), (3.4) and (3.9) we nd Integrating by parts and using that the curvature is zero at both boundary points As f (t, x = ) is xed in time there is no contribution by the boundary terms at x = . At x = one uses that ∂ t f i = ∂ t f j . Under natural boundary conditions at zero the boundary term also vanishes and we nd that the energy is indeed decreasing.  In other words at the moving point (think here at the triple junction) we infer It is not surprising that the length plays a role, since φ(t, ) determines how the curve grows or shrinks.
As in [10,Lem.2.3] and [8,Sec.3] we denote by the product ϕ * ϕ * · · · * ϕ k the product of k normal vector elds ϕ i (i = , .., k) de ned as ϕ , ϕ · .. · ϕ k− , ϕ k− ϕ k if k is odd and as ϕ , ϕ · .. · ϕ k− , ϕ k , if k is even. The expression P a,c b ( κ) stands for any linear combination of terms of the type (∇s) i κ * · · · * (∇s) i b κ with i + . . . + i b = a and max i j ≤ c with universal, constant coe cients. Notice that a gives the total number of derivatives, b denotes the number of factors and c gives a bound on the highest number of derivatives falling on one factor. With a slight abuse of notation, |P a,c b ( ϕ)| denotes any linear combination with non-negative coe cients of terms of type |∇ i s ϕ| · |∇ i s ϕ| · ... · |∇ i b s ϕ| with i + · · · + i b = a and max i j ≤ c .
The second claim follows by induction.
The previous Lemma will help us to transform estimates obtained for ∇ m s κ into estimates for the full derivative ∂ m s κ. The reason why we work with the operator ∇s (instead of the full derivative ∂s) is that it naturally appears in the evolution equation and it is therefore the most natural choice to work with.
If λ is a given xed constant then we also write Notice that the rst part of the statement is more precise. In the second expression, we allow for more terms than the ones that are actually there. In some computations we need the precise expression and hence the rst. When using interpolation estimates we use the second expression since it is much shorter and the extra terms do not create any problem in the analysis since they have (at most) the same order of the ones we already have.
Proof. See [10, Lemma 2.3] and [4,Lemma 2.3] in the case that there is no tangential component. For m = the claim follows directly from (3.9). For m = we nd using (3.10) and (3.9) (and writing as usual V for the normal component of the velocity) we nd using that P , ( κ) allows for more terms than P , ( κ). The general statement follows with an induction argument.
Notice that no derivatives of the tangential component φ appear. In the following lemma we collect further important formulae.
whereas for ν even we have Proof. The proof is obtained by a straight forward generalisation of [8, Lemma 3.1]. We report all details for the sake of the reader in Appendix B.1.
Finally we give some estimates that will be used repeatedly for boundary terms.
Lemma 3.7. We have that for any x ∈ [ , ] there holds The above lemma will be used in conjunction with interpolation estimates shown below: in particular (3.21) will be used when b = .
Proof. Let x ∈ [ , ]. We rst start by showing our statement when b is odd. To motivate our proof's strategy, observe that by embedding theory we know that for any normal vector eld ϕ : I → R n along f we have On the other hand, to apply interpolation inequalities later on, where the number of factors b must be b ≥ , it is better (instead of applying Cauchy-Schwarz to the L -norms) to consider using that ∂s|ϕ| = ϕ, ∇s ϕ . Then this gives (3.21). When b is even we use the inequality ϕ L ∞ (I) ≤ C ∂s ϕ L (I) + C L(f ) ϕ L (I) , which holds for any scalar map ϕ. Choosing ϕ = P a,c b ( κ)(x) we obtain (3.22).

. Interpolation inequalities
Interpolation inequalities are crucial in the proof of long-time existence. Consider the scale invariant norms for k ∈ N and p ∈ [ , ∞) , (as in [10]) and the usual L p -norm ∇ i s κ p L p := I |∇ i s κ| p ds. Most of the following results (that we brie y state without proof) can be found in several papers (e.g. [8,10,17]). We give the precise reference to the paper where a complete proof can be found. These inequalities are satis ed by closed and open curves and allow also for the boundary points of the curve to move in time. One needs only a control from below on the length of the curve. Lemma 3.8 (Lemma 4.1 [8]). Let f : I → R n be a smooth regular curve. Then for all k ∈ N, p ≥ and ≤ i < k we have with α = (i + − p )/k and C = C(n, k, p). Corollary 3.9 (Corollary 4.2 [8]). Let f : I → R n be a smooth regular curve. Then for all k ∈ N we have Lemma 3.10 (Lemma 3.4 [5]). Let f : I → R n be a smooth regular curve. For any a, c, Lemma 3.11 (Lemma 3.5 [5]). Let f : I → R n be a smooth regular curve and ∈ N . If A, B ∈ N with B ≥ , and A + B < + then we have and for any ε ∈ ( , )

Treatment of the boundary terms
Similar to [4,Lemma 2.4] we see that at the xed boundary points the derivatives of the curvature of any even order vanish.

Lemma 4.1. Let f be a smooth solution of
to the boundary conditions f (t, ) = P (for some P ∈ R n ) and κ(t, ) = for all t ≥ . Then φ(t, ) = and ∇ l s κ(t, ) = for all l ∈ N and for all times t ∈ ( , T).
Proof. For l = , the claim follows immediately from the boundary conditions and the fact that f is a smooth solution. Notice that because of the boundary conditions φ(t, ) = for all t. The case l = is a consequence of (3.9): indeed at x = we have The general statement follows from an induction argument. Indeed, assume that the claim is true up to n, n ∈ N. By the induction assumption it follows that Since φ(t, ) = for all t and in all the other terms on the right-hand side there is an odd number of factors and an even number of derivatives, we see that ∇ n+ s κ(t, ) = for all t ∈ ( , T).
We consider now the triple junction where the tangential component plays an important role.

Remark 4.3.
In this lemma we see the 'algebra' of the tangential component. More precisely, looking at the sets S l i as de ned in (4.4) and, in particular, at the special de nition of the length of the multiindex, one sees that a factor φ i takes the place of three derivatives of the curvature, while a factor ∂ t φ i = φ ( ) i takes the place as + l derivatives of the curvature. The order of these terms is given in Lemma 5.3 below.

Treatment of the tangential component
Here we study the tangential component at the junction point, the only point where the problem gives us information on the φ i 's (see Section 2.3). It is exactly here that we need the topological condition that the dimension of the space spanned by the unit tangents at the junction is at least two, see (5.1) below.

Remark 5.1. In order that the curves remain attached, it is necessary that
) for all t ∈ ( , T) and i, j = , , .
This follows from di erentiating with respect to t the equality f i (t, ) = f j (t, ).
The condition is also su cient. Indeed, since the initial datum is attached we have for any i, j = , , This gives in particular a condition on the tangential part. Indeed, since the curvature is zero at it is necessary that at for i, j = , , That is, at zero (i.e. at the triple junction) Let us elaborate on this a bit further. For the sake of notation denote Notice that here (and only in this and the following remark) we denote the tangential vectors at the junction point by T i . Also we write φ i meaning φ i (t, ). Using the above identity yields that and after addition where the subindex have to be understood modulo 3. This yields the system

The above real and symmetric matrix is positive de nite (by Sylvester's criterion) if and only if its determinant is strictly positive. A straight forward calculation gives that
with equality if and only if T = T = T or (T i = T i+ and T i+ = −T i ) for some i = , , . These degenerate situations are always excluded if we assume that Since the inverse of the matrix is given by
Now using (3.7), (3.6) and the fact that κ i = at the junction we infer that with normal component (cf. Lemma 3.5 and use κ i = ) For the sake of notation we write again Using (5.4) and (5.5) yields that and after addition for any i = , , , where the subindex have to be understood modulo 3. This yields the system which we have already solved in Remark 5.1. Therefore we nd again that the matrix is invertible if we assume (5.1). By the expression for the inverse of the matrix given in (5.2), we see for instance from (5.6) that and similar formulas hold for ∂ t φ ( ), ∂ t φ ( ).
More generally an induction argument, that uses (3.6), (3.7), κ i = at the boundary, Lemma 3.5, (3.19), and (3.20), gives that for any m ∈ N, m ≥ , at the boundary points we have Using for all t ∈ ( , T) and i, j = , , , (5.10) and (5.8) the same arguments as above give then for m ∈ N, m ≥ Next we give estimates for the tangential components φ i ( and their time derivatives) at the triple junction.
To that end we will use repeatedly Lemma 3.7.
. Furthermore let Assumption 6.1 (see below) hold and assume that there exists a constant C > such that

12)
and any t ∈ ( , T). Then we have for any ∈ N and i = , , that More generally, we have for any ∈ N and i = , , that Proof. An expression for φ i (t, ) is given in Remark 5.1 (see (5.3) for φ ). Again we write here φ i ( ) meaning φ i (t, ) for t ∈ ( , T). Using Assumption 6.1 we nd with Lemma 4.1 and Lemma 3.10 (for any ≥ ) An expression for ∂ t φ i (t, ) is given in Remark 5.2 (see for instance (5.7) for ∂ t φ ). Again using Assumption 6.1 we nd By (3.21), Lemma 3.11 and the uniform bound on length and the L -norm of the curvature we infer for any ≥ Also, using once again that ∇ s κ i ( ) = , by Lemma 3.11 Therefore we get and similarly together with the induction assumptions where we have used Lemma A.2 with γ = + m ( + ) (on recalling (4.4) note that m− l= β l ( + ) < m− l= β l ( + ) = |β| .) For the remaining terms in (5.11) (with m replaced by m + ) we observe that where we have used Lemma 3.7, (5.12), Lemma 3.11, and Lemma A.2. The claim now follows from (5.11) putting all estimates together.  3) the tangential components φ i grow linearly in the interior of each curves. We will give notice when this fact plays a role in the proof.

Long-time existence result
We assume by contradiction that the solution does not exist globally in time, that is there exists < T < ∞ where T denotes the maximal time of existence. In view of our Assumption 6.1 this implies that at least one curve ceases to be smooth or regular at t = T.
Since (1.10) is a gradient ow, the energy is decreasing in time and in particular the L -norm of the curvature is uniformly bounded in ( , T). Indeed, max i= , , Similarly, since λ i > , i = , , , the length of the curves remains uniformly bounded from above.

First
Step Uniform estimate of ∇ s κ i L on ( , T), i = , , .
We wish now to estimate the derivatives of the curvature. We use here that the normal velocity in (1.10) can be written as −∇ s κ i + P , ( κ i ) + λ i κ i i = , , , and we simplify as much as possible the notation using the P a,c b ( κ i ). Using Lemma 3.2 (taking ϕ = ∇ s κ i ) and Lemma 3.5, and summing over i we nd Using Lemma 3.11 (with A = , B = , C ≤ , = ) one sees that the integrals on the right hand side can be controlled as follows with ε ∈ ( , ). Here we have used that fact that the length of the curves remains bounded away from zero. It remains to consider the boundary terms. At the xed boundary points (i.e. at x = ) all even derivatives of the curvature are zero by Lemma 4.1 and hence in reality we have a contribution from the boundary terms only from the junction point. As one can immediately see we have high derivatives on the boundary terms but using the boundary condition as done in Lemma 4.2 we can lower the order of the boundary terms. Let consider the three terms separately. Since κ = at the boundary and at the junction point Then by Lemma 4.2 at x = (recall that κ = at x = , therefore where we have used the fact that λ i are constant. By (3.22) we get Thanks to the interpolation inequality (3.27) (together with the uniform control on the lengths from below, see Assumption 6.1) we nd that the above terms can be controlled via absorbtion into the terms ∇ s κ i L . Next we study the term that depends on the tangential component of the ow equation and hence is new compared to our previous studies ( [4], [5], [8], [17], [9]). The term we need to control is The factor φ i ( ) can be expressed using the formulas for the tangential component of the velocity at x = (see (5.3) for φ ). Then by Assumption 6.1 and (5.13) (with = ) we nd Combining the estimates above and using the uniform bounds on the lengths we infer using Young's inequality |ab| ≤ C(|a| p + |b| q ) with p = and q = on product terms of type b = max{ , κ i , } , a = max{ , κ j , } . Since the exponent of the factor κ i , is smaller than we can control this term.
Proceeding similarly for the second boundary term in (6.1) we nd with Lemma 4.2 for i = , , As before with (3.22) and Lemma 3.11 the rst term on the right hand side can be controlled by ∇ s κ since λ i are constant and by Assumption 6.1. The second term coincides with one of the term coming from the tangential component treated above and hence it is also controlled. The last boundary term in (6.1) coincides with terms that we have already treated and hence this also behaves as κ i , . By Corollary 3.9 and Young's inequality we nally achieve that for ε ∈ ( , ) From (6.1), (6.2) and the estimate above choosing ε appropriately we nd that and hence, using the smoothness of the ow and of the initial data, we infer that ∇ s κ i L are uniformly bounded on [ , T) for i = , , .

Second
Step Uniform estimate of ∇ s κ i L on ( , T), i = , , for special choice of φ i Since equations (1.8) are of fourth order it is natural, after having controlled the second order derivative of the curvature, to control now the sixth order derivative of the curvature and then (later ) in the general step the derivative of order + m for m ∈ N. With interpolation inequalities we then get also estimates for the intermediate terms.
In this step we proceed as in the previous one applying Lemma 3.2. The crucial point is the choice of the normal vector eld ϕ. The main di culties in the case of networks are the boundary terms and the tangential components. Concerning the boundary terms it is necessary to lower their order (using the boundary conditions). Recall that since the last boundary condition (see (1.9)) involves a sum it was fundamental in (6.3) that −∇ s κ i is equal to the normal component of ∂ t f i at the triple junction. In order to use the same idea at this step instead of working with ∇ s κ i directly we are going to work with the vector eld ∇ t S ,i where S ,i = −∇ s κ i + φ i ∂s f i that we call speed (motivated by the fact that indeed at the triple junction S ,i = ∂ t f i ). This choice is due to the fact that at the triple junction ∂ t f i = ∂ t f j . Instead of working with ∂ t f i we can work with shorter expressions by the following considerations. Since in the boundary terms in (3.11) scalar products with normal vector elds of the same curve appear, only the normal component of the vector eld is relevant and hence we could work with ∇ t ∂ t f i . In order to simplify further the computations we use that at the junction point ∂ t f i = −∇ s κ i + φ i ∂s f i and hence work with ∇ t S ,i . As we will see, ∇ t S ,i goes like −∇ s κ i plus lower order terms.
The other di culty is due to the tangential component characterised by the functions φ i . These functions are needed to keep the network connected and hence play a role only at the triple junction. Because of this we have informations only at the triple junction (see Remark 5.1 and 5.2), more precisely on φ i ( ) and ∂ m t φ i ( ) for m ∈ N. On the other hand, when computing the evolution of several quantities, derivatives with respect to s of φ i appear. In Section 2.3, in order to simplify the computations, we made a special choice of φ i taking the linear interpolation between the value at the junction point, i.e. φ i ( ), and the other boundary point where φ i ( ) = (since the point is kept xed in time). More precisely, we have choosen to work with As a consequence, for i = , , and for any t ∈ ( , T) Moreover using (3.12) we can write From here on we will use these special choices of φ i 's without further notice. Observe that in the rst step φ i were arbitrary. We can now start with the computations. By Lemma 3.2 with ϕ = ∇ t S ,i and summing over i we nd We need to compute these terms. Since Due to our choice of φ i and since the length is uniformly bounded from below, we have (up to a constant) the same estimate from above for |φ i | and |∂s φ i |, namely Also from (6.8), the uniform bounds for the curvature derived in the rst step, and interpolation inequalities we infer As a consequence we nd on [ , ] Next, using the simple inequalities and (6.11) we infer that (for instance use the expressions derived above in (6.10) to write ∇ s ∇ t S ,i = ∇ s κ i +rest and take a = ∇ s κ i , b = rest and ϵ = , then use that rest = . . . to evaluate the b-term) We rst estimate the integral terms. By Lemma 3.11 (more precisely, (3.27 )). With the same arguments and now using also (5.14) we estimate i= j= using again the general Young inequality of Lemma A.1. It remains to treat the boundary terms. We may write Since S ,i = ∂ t f i at the boundary points, the velocities and their time derivatives coincide at the boundary points (and vanish at x = where the points are xed in time) we nd by (6.10), we get using (4.6) the following order reduction Note that we can write the rst term as By Lemma 3.11 and (5.13), again with = , with the same arguments as above we see that the integrals on the right hand side can be bounded by For the last term we observe that since κ i ( ) = and ∂ t f ( ) = it follows ∇ s κ i ( ) = and hence with Lemma 3.11 By (5.14) we nally get On the other hand with (3.7), Lemma 3.5, (3.6), and using that κ i = at the boundary we nd so that with Lemma 3.7, Lemma 3.11, (5.13) and (5.14) we obtain Since the sum of the exponents of the several terms is smaller than we can use Lemma A.1 and we obtain with ε ∈ ( , ). It remains to evaluate the last boundary term i= [ ∇s∇ t S ,i , ∇ s ∇ t S ,i ] in (6.9). First of all note that by (6.10), the fact that κ i = , (4.2) (at x = ) respectively Lemma 4.1 ( at x = ), we infer that at the boundary points we have (since there are some cancellations and κ i = ) We estimate each term separately at the boundary x = and x = .
(At x = many terms do actually vanish since here ∂ t φ i = = φ i , however we do not distinguish the treatment of the terms.) We use Lemma 3.7, (6.7), Lemma 3.11, (5.13), (5.14) to obtain at and similarly Finally using Lemma A.1, Corollary 3.9, and the uniform bounds on the length we obtain Putting all estimates together we nally obtain Choosing ε appropriately and a Gronwall Lemma give sup ( ,T) i= I |∇ t S ,i | ds ≤ C, with C = C(δ, λ j , f j, , L(f j ), E λ j (f j, )), j = , , . Going back to (6.12), integrating and by the bound on I |∇ t S ,i | ds we nd for all t ∈ ( , T) All this together with the uniform bound on the lengths, Lemma A.1 and Corollary 3. . Again here C depends on δ,λ j , f j, , L(f j ) and E λ j (f j, ) for j = , , .

Induction
Step: In the following we denote by The induction hypothesis reads (for some m ∈ N): (6.14) where C = C(δ, λ j , f j, , L(f j ), E λ j (f j, )) for j = , , . Thus let us consider S m+ ,i and derive the corresponding estimates. By Lemma 3.2 with ϕ = ∇ t S m+ ,i and summing over i we nd We need to compute these terms. For m ≥ we have that by calculations similar the ones performed in Remark 5.2 (that is inductively using (3.6), (3.7), Lemma 3.5, Lemma 3.6) Using Lemma A.3, where the behaviour of the derivatives with respect to s of time derivatives of φ i is investigated, (and the inequality |a + b| ≥ |a| − C|b| ) we obtain where we have used the induction hypothethis (6.14) in the last step. Similarly and again with Lemma A.3 where for the second inequality we have used the induction hypothethis (6.14) and where the term mutiplying |∂ m t φ i (t, )| actually appears only when m = .
From (6.15) we obtain adding the term I |∇ t S m+ ,i | ds and using the expressions derived above (and recalling (6.11) and the fact that m ≥ ) where the last two integrals appear only if m = . We rst estimate the integral terms. By Lemma 3.11 with = m + we obtain with ε ∈ ( , Since from (6.19) we get using (4.7) Therefore we infer using (6.14) and Lemma A.
We write [[a,b] Next we observe that by (6.16) we have Putting all estimates together, using Lemma A.1 and Corollary 3.9 we obtain It remains to evaluate the last boundary term i= [ ∇s∇ t S m+ ,i , ∇ s ∇ t S m+ ,i ] . First of all note that by (6.17), Lemma A.3, Lemma 3.7, Lemma 3.11 with = m + , and (5.15), we have for x ∈ I Putting the estimates together using Lemma A.1 and Corollary 3.9 we obtain . Putting the estimates together, obtained for the boundary terms and the integral terms in (6.21), we can nally state Choosing ε appropriately and applying Gronwall Lemma give that with P m− a polynomial of degree at most m − . A bound on ∂ x κ i C (Ī) follows from (6.23) taking h = κ i and from bounds on ∂ s κ i C (Ī) (see (6.22)) and on ∂ x γ i C (Ī) . Thus it remains to estimate ∂ x γ i C (Ī) for ∈ N . We start by showing that γ i = |∂x f i | is uniformly bounded from above and below. Upon recalling (3.4) we see that each function γ i , i = , , , satis es the following parabolic equation By regularity of the initial datum we have that /c ≤ γ i ( ) ≤ c for some positive c . From the estimates given in (6.22), (6.7), and the uniform estimates for the tangential components and for the lengths of the curves, it follows that the coe cients ∂s φ i C (Ī) + κ i , V i C (Ī) in (6.24) are uniformly bounded and hence we infer that /C ≤ γ i ≤ C, with C having the same dependencies as the constant in (6.22) as well as T. In order to prove bounds on ∂ m x γ i we proceed by induction. Let us assume that we have shown for all ≤ i ≤ m + , q = , , . Di erentiating (6.24) (m + )-times with respect to x, and recalling that ∂x(∂s φ i ) = by (6.7), we nd for some coe cients c(i, j, m, q) and q = , , . Together with (6.25), (6.26) we derive for i = , , . Next note that (6.22) implies which in turns gives uniform estimates for ∂ m x V i C (Ī) , i = , , , in view of (6.23) and the bounds for the length elements and its derivatives.
Finally, since the points P i , i = , , are xed by the uniform bounds on the length, we derive that f i C is uniformly bounded. This together with the uniform C -bounds on the curvature κ i , the velocity V i , γ i , φ i (recall Lemma A.3, (6.7)) and all their derivatives, allow for a smooth extension of f i up to t = T.

Fifth
Step: Long-time existence We want now to restart the ow at t = T and by the short-time existence result get a contradiction to the maximality of T. To do that we have to verify that the network f (T) satis es the compatibility conditions required by Theorem C.1 and that the glueing of the curves across t = T occurs in a smooth way. This requires some care. For clarity let us write down the strategy recalling also the major steps performed so far. Also, since several reparametrizations are necessary, let us distinguish again carefully between tangential components. Due to our assumptions on the initial datum, ϕ i ( , x) = x. Note that ϕ i is smooth and a straightforward computation that uses integration by parts yields Due to the regularity of f i we see that and where κ i denotes the curvature vector of the curvef i ) with boundary conditions This holds also for their time-derivatives. Moreover, at the boundary the tangential components φ ** i = φ G i = φ * i can be expressed in geometric terms. 6. Observation: On the equivalence of the compatibility conditions for and ∂ x f = for curves parametrized by constant speed. At the boundary we observe: Step 1: Step 2: Knowing that One proceeds analogously for higher order derivatives. 7. Hence, at t = T the compatibility conditions for the analytical problem (see Remark C.3) are satis ed. 8. We restart the ow at f (T) via Theorem C.1 (hence solving the analytical problem) and get a smooth solution in some time interval [T, T + ϵ]. 9. Reparametrize the solution in such a way that the tangential component linearly interpolates (using Lemma C. 4 and choosing the initial di eomorphism ψ i (x) = ϕ i (T, x) (see (6.27)), so that the original parametrization f (T) is attained again) 10. We obtain a smooth solution of the geometric probelm (1.10), (1.9), with φ i := φ G i satisfying (6.6), together with f (T) as initial data on some interval [T, T +ε], for <ε < ϵ. The latter is by construction a smooth extension of the geometric problem. This yields a contradiction, that is T can not be nite.

Sixth
Step: Sub-convergence The statement follows from an adaptation of the arguments depicted in [8, § 5 (Step nine)] to the present case. Remark 6.3. The long-time existence result can be extended to the case λ i ≥ with just few modi cations. Indeed, in order to derive the bounds on the curvatures and on the tangential components one needs only bounds on the lengths from below. The fact that λ i > has been used in the proof above to derive a bound from above on the length. In the case λ i ≥ since by (3.4) (see also Remark 3.3) the lengths grow at most linearly one has also a bound from above on the length in nite time. This is su cient to conclude the argument by contradiction. See [8, (5.14)] for a similar argument. The presence of the tangential component does not create any di culty. , ε] of (C2), (C4) (and also of (1.8), (1.9)) that is instantaneously smooth. Moreover, at any t ∈ ( , ε) compatibility conditions of any order are satis ed at the boundary. Hence this solution at some time t ∈ ( , ε) is an admissible initial value to apply the same construction presented in Section 2.3 and for our main result. Remark 6.5. We think that Assumption 6.1 is not just a technical assumption. In fact it is possible to picture situations where at least one of the requirements might stop holding true. For instance let us look at the condition imposed on the lengths of the curves. If we set λ = λ = λ = a network made up of three straight segments that meet at angles of π is a critical point for the energy functional. One thinks immediately at the "Mercedez-Benz symbol" but depending on the relative position of the points P , P and P it might be energetically better to have the junction point at one of the prescribed points. In this case, we will expect that along the elastic ow starting with an initial datum near to this con guration one of the curves collapses to a point. Remark 6.6. For the sake of simplicity, we have considered so far only the case of a single triod. This gave us the opportunity to explore how to treat equations and boundary conditions, when either one endpoint of a curve is xed or when it is connected to a triple junction. It is now possible to consider more complicated networks, composed of m curves (m ≥ ) meeting only in triple junctions and having at least one xed point P i in space (which guarantees that the network doe not translate in space). For this type of con gurations one can easily generalize our main result.
Proof. The proof goes by induction. The case n = is simply the standard Young inequality. Now suppose the claim holds for n − . Then since by hypothesis we have (i + . . . + in)/( − i ) < we infer applying Young and the induction hypothesis that a i · a i · . . . · a in n ≤ ϵa + Cϵ(a i · . . . · a in n ) /( −i ) ≤ ϵa + Cϵ δ(a + . . . + a n n ) + Cϵ C δ for any δ > . Choosing δ < ϵ/Cϵ the claim follows.
where r x m (t) := where The rst three claims follows now by an induction argument. More precisely: for m = starting from and using the expression for the derivatives of L(f i ), B, and φ i (x, t) we rst derive where we have used Lemma 3.7, Lemma 3.11, and (6.13). Next we infer using (A4) and (6.14) that Repeating the same arguments inductively we obtain the fourth claim in the lemma. Next, we observe from (A2), (A3), Lemma 3.7, Lemma 3.11, and (6.13) that The nal three claims of the lemma are again proved using an induction arguments and employing all estimates achieved so far: it is important that one proves the claim rst for k = , then k = and nally k = .
Lemma A.4. Let φ i be the tangential component in (1.10), and j, p ∈ N . Let S j p be de ned as in (4.4). Then Since |β | = |β| + , the proof of (i) is obtained from replacing β by β in the term [ proofs of (ii) is straightforward by using the de tion of the notation S j p in (4.4). The proofs of (iii) and (iv) are also straightforward, by applying (i), (ii) above, and (3.20) and (3.19).

B. Proof of parts of Lemma 4.2
Proof of (4.3) in Lemma 4.2. Based on (4.2), the proof follows from an induction argument. Suppose that (4.3) holds for some m ∈ N bigger than or equal to . Then we take the covariant derivatives, ∇ t , of (4.3). The lefthand side is simply obtained from using Lemma 3.5, that is (4.7) for m = . In these computations we use that the term multiplying φ i in the tangential component of (3.19) vanishes at the boundary since there κ i = .
The proof of (4.7) follows then from an induction argument. Since (4.5), (4.6), and (B5) are the cases of m = , , in (4.7), we suppose that (4.7) holds for some m ∈ N . Then we take the partial di erentiation, ∂ t , of (4.7). By using Lemma 3.5, (3.7), and the fact that κ i = we obtain for the left-hand side, The proof follows from using (B6) and (B7).
then the tangent vectors of f i andf i coincide, that is ∂sf i (t, x) = ∂s f i (t, ϕ i (t, x)). The same holds for all geometric quantities since they are invariant under reparametrization, that is κ i (t, x) = κ i (t, ϕ(t, x)) and so on. Next observe that ∂ xfi (t, x) = ∂ x f i (t, ϕ i (t, x))(∂x ϕ i (t, x)) + ∂x f i (t, ϕ i (t, x))∂ x ϕ i (t, x), In particular by (C9) and (C6) it follows that at the boundary x ∈ { , } we have and hence κ = at the boundary. In fact if ψ i (x) = x the above computation even yields that ∂ xfi (t = , x) = for x ∈ { , }. The other boundary conditions are satis ed since they involve pure geometric quantities. Finally notice that using the ow equation x) ∂s f i (t, ϕ i (t, x)) and hence x) ∂s f i (t, ϕ i (t, x)) .
It follows that the di eomorphism ϕ i has to solve the rst order ODE with initial datum ϕ i ( , x) = ψ(x) for each x ∈ [ , ]. The right hand side in (C12) can be written as G(t, x, ϕ i (t, x)), with G(t, x, y) a smooth functions in its variables. Here x plays the role of a parameter, x ∈ [ , ]. Since φ I i (t, x) = φ II i (t, x) for x ∈ { , } and all t, we see that the solution at x = , is ϕ i (t, x) = x for all t, i.e. (C9) is satis ed. The existence, uniqueness and regularity of the solution follow from [19,Sec.1.3] and [16,Chap.9 and App.D]. The smoothness of the solution together with the assumption on the initial datum ∂x ψ i > imply that also (C10) is satis ed.