A regularized gradient flow for the p-elastic energy

Simon Blatt; Christopher Hopper; Nicole Vorderobermeier

doi:10.1515/anona-2022-0244

Open Access Published by De Gruyter May 5, 2022

A regularized gradient flow for the p-elastic energy

Simon Blatt , Christopher Hopper and Nicole Vorderobermeier

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2022-0244

Abstract

We prove long-time existence for the negative L 2 -gradient flow of the p-elastic energy, p ≥ 2 , with an additive positive multiple of the length of the curve. To achieve this result, we regularize the energy by cutting off the degeneracy at points with vanishing curvature and add a small multiple of a higher order energy, namely, the square of the L 2 -norm of the normal gradient of the curvature κ . Long-time existence is proved for the gradient flow of these new energies together with the smooth subconvergence of the evolution equation’s solutions to critical points of the regularized energy in W 2 , p . We then show that the solutions to the regularized evolution equations converge to a weak solution of the negative gradient flow of the p-elastic energies. These latter weak solutions also subconverge to critical points of the p-elastic energy.

Keywords: geometric evolution equation; degenerate evolution equation; fourth order; p-elastic curves

MSC 2010: 53C44; 53A04

1 Introduction

We continue our investigation that was started in [1] of the negative gradient flow of the sum of p-elastic energy and a positive multiple λ > 0 of the length of a regular curve f : R / Z → R n . More precisely, we consider the energy

ℰ ( f ) = ℰ ( p ) ( f ) + λ ℒ ( f ) ,

where

ℰ ( p ) ( f ) = 1 p ∫ R / Z ∣ κ f ∣ p d s

denotes the p-elastic energy and ℒ ( f ) = ∫ R / Z ∣ f ′ ∣ d x denotes the length of the curve. We use κ = κ f = ∂ s 2 f for the curvature vector of the curve f , where ∂ s = ∂ x ∣ f ′ ∣ stands for the derivative with respect to arc-length, and d s = ∣ f ′ ∣ d x .

In [1], we used de Giorgi’s minimizing movement scheme together with approximate normal graphs to prove short-time existence for the weak negative gradient flow of the energy ℰ . We could start the gradient flow for any initial regular curve of class W 2 , p and give a lower bound on the time of existence that only depended on p , λ , and the energy of the initial curve.

Let us introduce some notations to specify the evolution equation we want to study. For an energy G : X → R on a vector space X , we denote by

δ v G ( x ) = d d τ G ( x + τ v ) τ = 0

the first variation of G at x in direction v . If X is a subset of all regular curves in C 0 , 1 ( R / Z , R n ) and f = x ∈ X , we say that ∇ L 2 G ( f ) ∈ L 2 ( d s ) is the L 2 -gradient of G at f if

δ v G ( f ) = ∫ R / Z ⟨ ∇ L 2 G ( f ) , v ⟩ d s

for all v ∈ X . We can now write down the evolution equation we are interested in. Given an initial curve f 0 ∈ W 2 , p parametrized with the constant speed, we seek weak solutions to the initial value problem

(1.1) ∂ t ⊥ f = − ∇ L 2 ℰ ( f ) on [ 0 , T ) × R / Z f ( 0 , ⋅ ) = f 0 ,

where ∂ t ⊥ f = ( ∂ t f ) ⊥ = ∂ t f − ⟨ ∂ t f , ∂ s f ⟩ ∂ s f denotes the normal velocity of the family of curves. In the case p = 2 , one can use the smoothing effects of the equation to find a time-dependent reparametrization of this family of curves that solves (1.1) with the full velocity ∂ t f in place of ∂ t ⊥ f . We do not expect this to be possible in the case p ≠ 2 .

We will only consider the nonsingular case p ∈ [ 2 , ∞ ) of the energy ℰ ( p ) and leave the singular case p < 2 for a later study. On a heuristic level in the degenerate case, the diffusion is weakened whenever the curvature is small, while the diffusion effects then obtain strong in the singular case. It is an interesting open question, whether the techniques in this article can be adapted to also treat the singular case.

In this article, we prove the following fundamental result.

Theorem 1.1^[1]

Given any regular closed curve f 0 ∈ W 2 , p ( R / Z , R n ) parametrized with constant speed, there is a family of regular curves f : [ 0 , ∞ ) × R / Z → R n , f ∈ H 1 ( [ 0 , ∞ ) , L 2 ( R / Z , R n ) ) ∩ L ∞ ( [ 0 , ∞ ) , W 2 , p ( R / Z , R n ) ) ∩ C 1 2 ( [ 0 , ∞ ) , L 2 ( R / Z , R n ) ) solving the initial value problem (1.1) in the weak sense, i.e., for all V ∈ C c ∞ ( [ 0 , ∞ ) × R / Z , R n ) , we have

∫ 0 ∞ ∫ R / Z ⟨ ∂ t ⊥ f , V ⟩ d s d t = − ∫ 0 ∞ ( δ V ℰ ( f ) ) d t .

For all times t ∈ [ 0 , ∞ ) , the curves f t = f ( t , ⋅ ) are parametrized with constant speed and there is a subsequence t n → ∞ such that the translated curves f t n ( ⋅ ) − f t n ( 0 ) converge in W 2 , p to a critical point of ℰ . The solution furthermore satisfies for almost all times 0 ≤ t 0 < t 1 < ∞ the energy identity

∫ t 0 t 1 ∫ R / Z ∣ ∂ t ⊥ f ∣ 2 d s d t ≤ ℰ ( f t 0 ) − ℰ ( f t 1 ) .

This result dramatically improves our previous findings in [1] and is based on a completely different approach. Instead of using de Giorgi’s minimizing movement scheme, we approximate the solution by solutions to the negative gradient flow of regularized energies. We cut off the degeneracy of the energy ℰ ( p ) near κ = 0 and add a small multiple of a higher order term to obtain energy for which one can prove long-time existence of the related gradient flow adapting the techniques in [7]. In Section 3, we prove uniform estimates of the solutions in some higher order Besov space. We use these in Section 4 to show that the solutions of the gradient flow of the regularized energies converge locally in the Bochner space L p ( ( 0 , ∞ ) , W 2 , p ( R / Z , R n ) ) to a solution of our initial problem after reparametrizing the curves with constant speed. A further essential ingredient is the fact that the added higher order part converges to zero (c.f. Corollary 3.8).

Elastica have led to major breakthroughs ever since James Bernoulli had challenged the mathematical world to invent a mathematical model for the elastic beam and solve the resulting equations in 1691. Apart from the invention of the curvature of a curve by James Bernoulli, this led to the remarkable classification of planar elastica by Euler [8,21] in Euclidean space and by Langer and Singer on the sphere and in hyperbolic space [11]. For a very nice introduction to the history of this problem, we recommend the article [24].

In the last decades, one has successfully investigated the related gradient flows in the quadratic case p = 2 . Both, for the flow in Euclidean space [7] and for the flow of curves on manifolds [4,6,16,17,22], one has now quite a complete picture of the behavior of the evolution equation in this special case. In [9,14], the Willmore flow of planar networks is considered.

While the analysis of the Euler elastica goes back more than three centuries, the nonquadratic case p ≠ 2 was treated only quite recently. Watanabe [25] found that critical points of the p-elastic energy for p ≠ 2 can have a very different behavior than in the quadratic case. He was able to classify all p-elastic curves using new variants of elliptic integrals.

In [18,19], a second-order version of the flow was considered for both curves and networks.

2 The regularized equations

To construct solutions to (1.1), we approximate the energy ℰ by regularized energies. We get rid of the degeneracy of ℰ by introducing the regularization

ℰ δ ( p ) ( f ) = 1 p ∫ R / Z ( ∣ κ ∣ 2 + δ 2 ) p 2 d s

for any δ > 0 . We will furthermore add a small positive multiple of the energy

ℱ ( f ) = 1 2 ∫ R / Z ∣ ∇ s κ ∣ 2 d s ,

where ∇ s ϕ = ∂ s ϕ − ⟨ ∂ s ϕ , τ ⟩ τ indicates the normal part of ∂ s ϕ and τ = ∂ s f denotes the unit tangent along f . For any p ≥ 2 , ε > 0 , δ > 0 , and λ > 0 , let

(2.1) ℰ ε , δ , λ ( p ) ( f ) = ε ℱ ( f ) + ℰ δ ( p ) ( f ) + λ ℒ ( f ) .

The negative L 2 -gradient flow for the regularized energy ℰ ε , δ , λ ( p ) is given by

(2.2) ∂ t f = − ∇ L 2 ℰ ε , δ , λ ( p ) ( f ) ,

where ∇ L 2 ℰ ε , δ , λ ( p ) ( f ) = ε ∇ L 2 ℱ ( f ) + ∇ L 2 ℰ δ ( p ) ( f ) + λ ∇ L 2 ℒ ( f ) . A straightforward calculation following [7] yields for smooth f

(2.3) ∇ L 2 ℱ ( f ) = − ∇ s 4 κ + ∣ κ ∣ 2 ∇ s 2 κ + ⟨ ∇ s κ , κ ⟩ ∇ s κ − 3 2 ∣ ∇ s κ ∣ 2 κ ,

(2.4) ∇ L 2 ℰ δ ( p ) ( f ) = ( κ 2 + δ 2 ) p − 2 2 ( ∇ s 2 κ + ∣ κ ∣ 2 κ ) + ( p − 2 ) ( κ 2 + δ 2 ) p − 4 2 ( ⟨ κ , ∇ s 2 κ ⟩ κ + ∣ ∇ s κ ∣ 2 κ + 2 ⟨ κ , ∇ s κ ⟩ ∇ s κ ) + ( p − 4 ) ( p − 2 ) ( κ 2 + δ 2 ) p − 6 2 ⟨ κ , ∇ s κ ⟩ 2 κ − 1 p ( κ 2 + δ 2 ) p 2 κ ,

and ∇ L 2 ℒ ( f ) = − κ . For the convenience of the reader, we will give the details of these calculations in the next section.

This section is devoted to the proof of the following theorem. The proof follows the path paved in [7].

Theorem 2.1

For any fixed numbers p > 2 , δ > 0 , ε > 0 , λ > 0 , and any smooth initial closed curve f 0 ∈ C ∞ ( R / Z , R n ) , there exists a smooth solution f t : R / Z → R n , 0 ≤ t < ∞ , to the regularized L 2 -gradient flow (2.2) of ℰ ε , δ , λ ( p ) . Furthermore, after reparametrization by arc-length and suitable translation, for t i → ∞ , a subsequence of the curves f t i smoothly converges to a critical point of ℰ ε , δ , λ ( p ) .

Remark 2.2

Note that the asymptotic behavior stated in this theorem will not be used in the rest of the article. Indeed, our proof of the asymptotic behavior as stated in Theorem 1.1 does not use the approximations at all.
One can prove full convergence of the flow as t → ∞ using Łojasiewicz-Simon gradient estimates as in [3,5,15,16,22]. In contrast to that, full convergence for solutions of the negative gradient flow of the energy ℰ for t → ∞ is an interesting open problem.

2.1 Equations of evolution and inequalities

We will use the following toolbox from [7] to derive the evolution equations of geometric quantities. Note that ∇ s stands for the normal derivative as mentioned above.

Lemma 2.3

([7, Lemma 2.1]) Let f : [ 0 , T ) × R / Z → R n be a time-dependent curve and ϕ any normal field along f . If f satisfies ∂ t f = V + φ τ , where V is the normal velocity and φ = ⟨ ∂ t f , τ ⟩ , we have

(2.5) ∇ s ϕ = ∂ s ϕ + ⟨ ϕ , κ ⟩ τ ,

(2.6) ∂ t ( d s ) = ( ∂ s φ − ⟨ κ , V ⟩ ) d s ,

(2.7) ∂ t ∂ s − ∂ s ∂ t = ( ⟨ κ , V ⟩ − ∂ s φ ) ∂ s ,

(2.8) ∂ t τ = ∇ s V + φ ⋅ κ ,

(2.9) ∂ t ϕ = ∇ t ϕ − ⟨ ∇ s V + φ κ , ϕ ⟩ τ ,

(2.10) ∇ t κ = ∇ s 2 V + ⟨ κ , V ⟩ κ + φ ⋅ ∇ s κ ,

(2.11) ( ∇ t ∇ s − ∇ s ∇ t ) ϕ = ( ⟨ κ , V ⟩ − ∂ s φ ) ∇ s ϕ + ⟨ κ , ϕ ⟩ ∇ s V − ⟨ ∇ s V , ϕ ⟩ ⋅ κ .

Using the aforementioned formulas, a straightforward calculation yields the formulas for the L 2 -gradients of ℱ and ℰ δ ( p ) given in (2.3) and (2.4). Equations (2.6) and (2.10) tell us that under the condition that ∂ t f is normal to f , we have

∂ t ℰ δ ( p ) ( f t ) = ∫ R / Z ( κ 2 + δ 2 ) p − 2 2 ⟨ κ , ∂ t κ ⟩ d s + 1 p ∫ R / Z ( κ 2 + δ 2 ) p 2 ∂ t ( d s ) = ∫ R / Z ( κ 2 + δ 2 ) p − 2 2 ⟨ κ , ∇ s 2 V ⟩ d s + ∫ R / Z ( κ 2 + δ 2 ) p − 2 2 ∣ κ ∣ 2 ⟨ κ , V ⟩ d s − 1 p ∫ R / Z ( κ 2 + δ 2 ) p 2 ⟨ κ , V ⟩ d s = ∫ R / Z ⟨ ∇ s 2 ( ( κ 2 + δ 2 ) p − 2 2 κ ) , V ⟩ d s + ∫ R / Z ( κ 2 + δ 2 ) p − 2 2 ∣ κ ∣ 2 ⟨ κ , V ⟩ d s − 1 p ∫ R / Z ( κ 2 + δ 2 ) p 2 ⟨ κ , V ⟩ d s ,

where we used integration by parts in the last step. So we obtain

∇ L 2 ℰ δ ( p ) ( f ) = ∇ s 2 ( ( κ 2 + δ 2 ) p − 2 2 κ ) + ( κ 2 + δ 2 ) p − 2 2 ∣ κ ∣ 2 κ − 1 p ( κ 2 + δ 2 ) p 2 κ .

Together with

∇ s 2 ( ( κ 2 + δ 2 ) p − 2 2 κ ) = ( κ 2 + δ 2 ) p − 2 2 ∇ s 2 κ + 2 ( p − 2 ) ( κ 2 + δ 2 ) p − 4 2 ⟨ κ , ∇ s κ ⟩ ∇ s κ + ( p − 2 ) ( κ 2 + δ 2 ) p − 4 2 ⟨ κ , ∇ s 2 κ ⟩ + ( κ 2 + δ 2 ) p − 4 2 ∣ ∇ s κ ∣ 2 + ( p − 4 ) ( κ 2 + δ 2 ) p − 6 2 ⟨ κ , ∇ s κ ⟩ 2 κ ,

this proves (2.4).

To calculate the L 2 -gradient of ℱ , we first observe using (2.6)

∂ t ℱ ( f ) = ∫ R / Z ⟨ ∇ s κ , ∇ t ∇ s κ ⟩ d s + 1 2 ∫ R / Z ∣ ∇ s κ ∣ 2 ∂ t ( d s ) = ∫ R / Z ⟨ ∇ s κ , ∇ t ∇ s κ ⟩ d s − 1 2 ∫ R / Z ∣ ∇ s κ ∣ 2 ⟨ κ , V ⟩ d s .

From (2.11) and (2.10), we obtain

∇ t ∇ s κ = ∇ s ( ∇ s 2 V + ⟨ κ , V ⟩ κ ) + ⟨ κ , V ⟩ ∇ s κ + ∣ κ ∣ 2 ∇ s V − ⟨ ∇ s V , κ ⟩ κ = ∇ s 3 V + ⟨ ∇ s κ , V ⟩ κ + 2 ⟨ κ , V ⟩ ∇ s κ + ∣ κ ∣ 2 ∇ s V

and hence,

∂ t ℱ ( f ) = ∫ R / Z ⟨ ∇ s κ , ∇ s 3 V ⟩ d s + ∫ R / Z ∣ κ ∣ 2 ⟨ ∇ s κ , ∇ s V ⟩ d s + ∫ R / Z ⟨ ( ⟨ ∇ s κ , κ ⟩ ∇ s κ + 2 ∣ ∇ s κ ∣ 2 κ ) , V ⟩ d s − 1 2 ∫ R / Z ∣ ∇ s κ ∣ 2 ⟨ κ , V ⟩ d s = − ∫ R / Z ⟨ ∇ s 4 κ , V ⟩ d s − ∫ R / Z ⟨ ∇ s ( ∣ κ ∣ 2 ∇ s κ ) , V ⟩ d s + ∫ R / Z ⟨ ( ⟨ ∇ s κ , κ ⟩ ∇ s κ + 3 2 ∣ ∇ s κ ∣ 2 κ ) , V ⟩ d s ,

where we used integration by parts in the last step. From this, we can read off that

∇ L 2 ℱ ( f ) = − ∇ s 4 κ − ∇ s ( ∣ κ ∣ 2 ∇ s κ ) + ⟨ ∇ s κ , κ ⟩ ∇ s κ + 3 2 ∣ ∇ s κ ∣ 2 κ = − ∇ s 4 κ − ∣ κ ∣ 2 ∇ s 2 κ − ⟨ κ , ∇ s κ ⟩ ∇ s κ + 3 2 ∣ ∇ s κ ∣ 2 κ .

Thus, equation (2.3) is proven.

As often the precise algebraic form of the terms does not matter, we will use the ∗ notation introduced in [7] to shorten the notation. For vectors ϕ 1 , … , ϕ k along f , we denote by ϕ 1 ∗ ⋯ ∗ ϕ k any multilinear combination of these vectors. For p > 2 and δ > 0 , we let ψ : [ 0 , ∞ ) → R be given by ψ ( x ) = ( x + δ 2 ) p 2 . For a vector field ϕ , we denote by P ν μ ( ϕ ) any linear combination of terms of the type

( ψ ( i 0 ) ( ∣ κ ∣ 2 ) ) j ∇ s i 1 ϕ ∗ ⋯ ∗ ∇ s i ν ϕ

with universal constant coefficients, where μ is greater than or equal to the total number of derivatives i 1 + ⋯ + i ν , i l ∈ N 0 , and j ∈ { 0 , 1 } . If P ν μ ( ϕ ) only contains terms with derivatives of ϕ up to order ω ∈ N , we indicate that by writing P ν μ , ω ( ϕ ) if needed. We will write P μ ( ϕ ) for any linear combination of terms P ν μ , ν ∈ N , and P μ , ω ( ϕ ) for any linear combination of terms P ν μ , ω , ν ∈ N .

Note that the factor ( ψ ( i 0 ) ( ∣ κ ∣ 2 ) ) j hardly plays a role in the upcoming computations as along the flow ℱ is bounded by the initial energy, and hence, by Sobolev’s embeddings, ‖ κ ‖ L ∞ is bounded as well. Hence, the factor is controlled by a constant dependent on δ . Notice also that the formula ∇ s P μ ( κ ) = P μ + 1 ( κ ) holds and that P μ ( κ ) can contain summands P μ ˜ ( κ ) of lower order 0 ≤ μ ˜ ≤ μ − 1 .

Using this notation, the evolution equation (2.2) reduces to

∂ t f = ε ( ∇ s 4 κ + P 2 ( κ ) ) + P 2 ( κ ) + λ κ .

We will use the following lemma to capture the main structure of the evolution equation of the L 2 -norm of higher derivatives of κ .

Lemma 2.4

Suppose f ∈ C ∞ ( [ 0 , T ) × R / Z , R n ) moves in a normal direction with velocity ∂ t f = V , ϕ is a normal vector field along f , δ > 0 , and ∇ t ϕ − ε ∇ s 6 ϕ = Y . Then,

(2.12) d d t 1 2 ∫ R / Z ∣ ϕ ∣ 2 d s + ε ∫ R / Z ∣ ∇ s 3 ϕ ∣ 2 d s = ∫ R / Z ⟨ Y , ϕ ⟩ d s − 1 2 ∫ R / Z ∣ ϕ ∣ 2 ⟨ κ , V ⟩ d s .

Furthermore, χ = ∇ s ϕ satisfies the equation:

(2.13) ∇ t χ − ε ∇ s 6 χ = ∇ s Y + ⟨ ϕ , κ ⟩ ∇ s V − ⟨ ϕ , ∇ s V ⟩ κ + ⟨ κ , V ⟩ χ .

Proof

Analogous to [7, Lemma 2.2], (2.12) follows from the evolution equations (2.6) and (2.9) as well as (2.13) from (2.11).□

We observe as in [7, Lemma 2.3]:

Lemma 2.5

Suppose ∂ t f = ε ( ∇ s 4 κ + P 2 ( κ ) ) + λ κ + P 2 ( κ ) , where ε , λ > 0 . Then, for any m ≥ 0 , the derivatives of the curvature ϕ m = ∇ s m κ satisfy

∇ t ϕ m − ε ∇ s 6 ϕ m = ε P m + 4 ( κ ) + λ ( ∇ s m + 2 κ + P m ( κ ) ) + P m + 4 ( κ ) .

Proof

The case of m = 0 is a consequence of (2.10). By applying (2.13), we obtain inductively

∇ t ϕ m + 1 − ε ∇ s 6 ϕ m + 1 = ∇ s ( ∇ t ϕ m − ε ∇ s 6 ϕ m ) + ( ⟨ ϕ m , κ ⟩ ∇ s ∂ t f − ⟨ ϕ m , ∇ s ∂ t f ⟩ κ + ⟨ κ , ∂ t f ⟩ ϕ m + 1 ) = ε P m + 5 ( κ ) + λ ( ∇ s m + 3 κ + P m + 1 ( κ ) ) + P m + 5 ( κ ) + ( ε P m + 5 ( κ ) + λ P m + 1 ( κ ) + P m + 5 ( κ ) ) = ε P m + 5 ( κ ) + λ ( ∇ s m + 3 κ + P m + 1 ( κ ) ) + P m + 5 ( κ ) ,

which proves the lemma.□

Counting the number of factors containing ∇ s κ or higher derivatives thereof, we obtain the following fact about terms of type P μ , ω ( κ ) .

Lemma 2.6

Let f : R / Z → R n be a smooth closed curve and μ ∈ N . Then, we have

(2.14) P μ , ω ( κ ) = ∑ k = 1 μ P k μ − k , ω − 1 ( ∇ s κ ) ∗ P 0 ( κ ) + P 0 ( κ ) ,

for any ω ∈ N , 1 ≤ ω ≤ μ .

Proof

A term of type P μ , ω ( κ ) consists of a linear combination of terms

( ψ ( i 0 ) ( ∣ κ ∣ 2 ) ) j ∇ s i 1 κ ∗ ⋯ ∗ ∇ s i ν κ

with universal constant coefficients, where μ is greater than or equal to the total number of derivatives i 1 + ⋯ + i ν , i l ∈ { 0 , … , ω } , and j ∈ { 0 , 1 } . Changing the order of indices, we can achieve that i 1 , … , i k ≥ 1 and i k + 1 , … , i ν = 0 . Since i 1 + ⋯ + i k ≤ μ , we obtain k ≤ μ . Hence,

( ψ ( i 0 ) ( ∣ κ ∣ 2 ) ) j ∇ s i 1 κ ∗ ⋯ ∗ ∇ s i ν κ = ( ( ψ ( i 0 ) ( ∣ κ ∣ 2 ) ) j ∇ s i 1 κ ∗ ⋯ ∗ ∇ s i k κ ) ( ∇ s i k + 1 κ ∗ ⋯ ∗ ∇ s i ν κ ) = P k μ − k , ω − 1 ( ∇ s κ ) ∗ P 0 ( κ ) . □

In a next step, we state a variant of the Gagliardo-Nirenberg interpolation inequality for higher order curvature functionals for curves in R n . For that, we define scale invariant norms ‖ κ ‖ k , q = ∑ i = 0 k ‖ ∇ s i κ ‖ q , where

(2.15) ‖ ∇ s i κ ‖ q = ℒ ( f ) i + 1 − 1 q ∫ R / Z ∣ ∇ s i κ ∣ q d s 1 q .

Lemma 2.7

([7, Lemma 2.4]) For any smooth closed curve f : R / Z → R n and any k ∈ N , q ≥ 2 , and 0 ≤ i < k , we have

‖ ∇ s i κ ‖ q ≤ c ‖ κ ‖ 2 1 − α ‖ κ ‖ k , 2 α ,

where α = i + 1 2 − 1 q / k and c = c ( n , k , q ) .

An immediate consequence of this lemma for terms of type P ν μ , k − 1 is the following:

Lemma 2.8

([7, Proposition 2.5]) Let k ∈ N and f as in the previous lemma. For any term P ν μ , k − 1 ( κ ) with ν ≥ 2 , we have

(2.16) ∫ R / Z ∣ P ν μ , k − 1 ( κ ) ∣ d s ≤ c ℒ ( f ) 1 − μ − ν ‖ κ ‖ L ∞ ‖ κ ‖ 2 ν − γ ‖ κ ‖ k , 2 γ ,

where γ = μ + 1 2 ν − 1 / k and c = c ( n , k , μ , ν , δ ) > 0 . Moreover, if μ + 1 2 ν < 2 k + 1 , then γ < 2 , and we have for any η > 0

(2.17) ∫ R / Z ∣ P ν μ , k − 1 ( κ ) ∣ d s ≤ c ‖ κ ‖ L ∞ η ∫ R / Z ∣ ∇ s k κ ∣ 2 d s + η − γ 2 − γ ∫ R / Z ∣ κ ∣ 2 d s ν − γ 2 − γ + ℒ ( f ) 1 − μ − ν 2 ∫ R / Z ∣ κ ∣ 2 d s ν 2

for another constant c = c ( n , k , μ , ν , δ ) > 0 .

Proof

We prove this statement along the lines of [7, Proposition 2.5]. First note that κ is uniformly bounded on R / Z , and it is therefore sufficient to focus on ∇ s i 1 κ ∗ ⋯ ∗ ∇ s i ν κ with i 1 + ⋯ + i ν = μ and i l ≤ k − 1 instead of P ν μ , k − 1 ( k ) subsequently. We then observe by Hölder’s inequality, the notion of scale invariant norms in (2.15), and Lemma 2.7 that

∫ R / Z ∣ ∇ s i 1 κ ∗ ⋯ ∗ ∇ s i ν κ ∣ d s ≤ ∏ l = 1 ν ‖ ∇ s i l κ ‖ L ν = ℒ 1 − μ − ν ( f ) ∏ l = 1 ν ‖ ∇ s i l κ ‖ ν ≤ c ℒ 1 − μ − ν ( f ) ∏ l = 1 ν ‖ κ ‖ 2 1 − α l ‖ κ ‖ k , 2 α l

for α l = i l + 1 2 − 1 ν / k and some positive constant c = c ( n , k , ν ) . As ∑ l = 1 ν α l = γ , the first inequality (2.16) follows. For the second claim (2.17), we recall the following standard interpolation result

(2.18) ‖ κ ‖ k , 2 2 ≤ c ( k ) ( ‖ ∇ s k κ ‖ 2 2 + ‖ κ ‖ 2 2 ) ,

from which we deduce together with γ < 2 and the equivalence of p-norms on R 2

‖ κ ‖ 2 ν − γ ‖ κ ‖ k , 2 γ ≤ c ( k ) ( ‖ κ ‖ 2 ν − γ ‖ ∇ s k κ ‖ 2 γ + ‖ κ ‖ 2 ν ) .

Now taking account of the scaling and applying Young’s inequality for η > 0 , we observe

(2.19) ℒ ( f ) 1 − μ − ν ‖ κ ‖ 2 ν − γ ‖ κ ‖ k , 2 γ ≤ c ℒ ( f ) 1 − μ − ν ( ‖ κ ‖ 2 ν − γ ‖ ∇ s k κ ‖ 2 γ + ‖ κ ‖ 2 ν ) ≤ c ‖ κ ‖ L 2 ν − γ ‖ ∇ s k κ ‖ L 2 γ + c ℒ ( f ) 1 − μ − ν 2 ‖ κ ‖ L 2 ν ≤ η ‖ ∇ s k κ ‖ L 2 γ + c η − γ 2 − γ ‖ κ ‖ L 2 2 ν − γ 2 − γ + c ℒ ( f ) 1 − μ − ν 2 ‖ κ ‖ L 2 ν

for some constant c = c ( k ) > 0 .□

The interpolation inequality (2.17) can be transferred to any term P ν μ ( ∇ s κ ) that only involves derivatives of ∇ s κ . The aim is to interpolate the L 1 -norm of P ν μ ( ∇ s κ ) between the L 2 -norms of ∇ s κ and ∇ s k + 1 κ .

Lemma 2.9

Let k ∈ N and f as in Lemma 2.7. For any term P ν μ , k − 1 ( ∇ s κ ) with ν ≥ 2 such that μ + 1 2 ν < 2 k + 1 and η > 0 , we obtain

(2.20) ∫ R / Z ∣ P ν μ , k − 1 ( ∇ s κ ) ∣ d s ≤ c ‖ κ ‖ L ∞ η ∫ R / Z ∣ ∇ s k + 1 κ ∣ 2 d s + η − γ 2 − γ ∫ R / Z ∣ ∇ s κ ∣ 2 d s ν − γ 2 − γ

(2.21) + ℒ ( f ) 1 − μ − ν 2 ∫ R / Z ∣ ∇ s κ ∣ 2 d s ν 2 ,

where γ = μ + 1 2 ν − 1 / k and c = c ( n , k , μ , ν , δ ) > 0 .

Proof

Lemma 2.7 also applies to derivatives of ∇ s κ , more precisely we have

‖ ∇ s i ( ∇ s κ ) ‖ q ≤ c ‖ ∇ s κ ‖ 2 1 − α ‖ ∇ s κ ‖ k , 2 α ,

where α = i + 1 2 − 1 q / k and c = c ( n , k , q ) . Therefore, the argument in the proof of Lemma 2.8 yields the statement.□

The next two lemmas illustrate some relations between the full derivatives and normal derivatives of κ . Here, Q ν μ ( κ ) denotes any linear combination of terms of the type ∇ s i 1 κ ∗ ⋯ ∗ ∇ s i ν κ , where i 1 + ⋯ + i ν = μ .

Lemma 2.10

([7, Lemma 2.6]) We have the identities

∇ s κ − ∂ s κ = ∣ κ ∣ 2 τ , ∇ s m κ − ∂ s m κ = ∑ i = 1 m 2 Q 2 i + 1 m − 2 i ( κ ) + ∑ i = 1 m + 1 2 Q 2 i m + 1 − 2 i ( κ ) τ .

One can inductively derive the following estimates using these equalities. They allow us to derive estimates for the full derivatives from estimates of the normal derivatives.

Lemma 2.11

([7, Lemma 2.7]) Assume the bounds ‖ κ ‖ L 2 ≤ Λ 0 and ‖ ∇ s m κ ‖ L 1 ≤ Λ m for m ≥ 1 . Then, for any m ≥ 1 , one has

‖ ∂ s m − 1 κ ‖ L ∞ + ‖ ∂ s m κ ‖ L 1 ≤ c m ( Λ 0 , … , Λ m ) .

2.2 Short-time existence

For the flow considered in this paper, short-time existence is a standard matter. An approach for more general parabolic geometric evolution equations for hyper-surfaces can, for example, be found in [10,13]. We only shortly sketch a proof for smooth initial curves here along the lines in [7].

By using ∇ s ϕ = ∂ s ϕ + ⟨ κ , ϕ ⟩ τ , we obtain

∇ s 4 κ = ∣ f ′ ∣ − 6 ( ∂ x 6 f − ⟨ ∂ x 6 f , τ ⟩ τ ) + lower order terms .

So by using the standard theory, we obtain a smooth solution f ˜ : R / Z × [ 0 , T ) → R n to the quasilinear parabolic equation:

∂ t f ˜ = ∇ L 2 ℰ ε , δ , λ ( p ) ( f ˜ ) + ∣ f ˜ ′ ∣ − 6 ⟨ ∂ x 6 f ˜ , τ ˜ ⟩ τ ˜

with initial data f 0 , i.e., a smooth family of smooth curves with the right normal velocity. We set σ ( t , ⋅ ) = ∣ f ˜ ′ ∣ − 7 ⟨ ∂ x 6 f ˜ t , τ ˜ t ⟩ and solve the initial value problem ∂ t ϕ = σ ( t , ϕ ) with ϕ ( 0 , ⋅ ) = i d . Then, one easily calculates that f ( t , x ) = f ( t , ϕ ( t , x ) ) solves the initial value problem (2.2).

2.3 Long-time existence and asymptotic behavior

As mentioned earlier, we can use the ∗ -factor notation and its linear combinations to write the evolution equation (2.2) in the form

(2.22) ∂ t f = ε ( ∇ s 4 κ + P 2 ( κ ) ) + P 2 ( κ ) + λ κ .

Let us first deduce some obvious uniform bounds on ℒ ( f t ) , ‖ κ ‖ L ∞ , and ‖ ∇ s κ ‖ L 2 for solutions of (2.2). Note that ℰ ε , δ , λ ( p ) ( f t ) is monotonically decreasing in time t as follows:

(2.23) d d t ℰ ε , δ , λ ( p ) ( f t ) = − ∫ R / Z ∣ ∇ L 2 ℰ ε , δ , λ ( p ) ( f t ) ∣ 2 d s = − ∫ R / Z ∣ ∂ t f t ∣ 2 d s .

We therefore obtain from the nonnegativity of each term in the energy (2.1)

(2.24) λ ℒ ( f t ) ≤ ℰ ε , δ , λ ( p ) ( f t ) ≤ ℰ ε , δ , λ ( p ) ( f 0 ) , ∫ R / Z ∣ ∇ s κ ∣ 2 d s ≤ 2 ε ℰ ε , δ , λ ( p ) ( f 0 ) , and ∫ R / Z ∣ κ ∣ p d s ≤ ∫ R / Z ( κ 2 + δ 2 ) p 2 d s ≤ p ℰ ε , δ , λ ( p ) ( f 0 )

for all T > t > 0 . Fenchel’s and Hölder’s inequality imply that

2 π ≤ ∫ R / Z ∣ κ ∣ d s ≤ ℒ ( f ) 1 − 1 p ∫ R / Z ∣ κ ∣ p d s 1 p ,

and hence

(2.25) ℒ ( f t ) ≥ ( 2 π ) p p − 1 ∫ R / Z ∣ κ ∣ p d s 1 p − 1 ≥ ( 2 π ) p p − 1 ( p ℰ ε , δ , λ ( p ) ( f 0 ) ) 1 p − 1 .

From the inequalities in (2.24) and (2.25), we thereby gain the following bounds for both the length and the L 2 -norm of ∇ s κ :

(2.26) ( 2 π ) p p − 1 ( p ℰ ε , δ , λ ( p ) ( f 0 ) ) 1 p − 1 ≤ ℒ ( f t ) ≤ ℰ ε , δ , λ ( p ) ( f 0 ) λ ,

(2.27) ‖ ∇ s κ ‖ L 2 2 ≤ 2 ε ℰ ε , δ , λ ( p ) .

Also the L ∞ -norm of κ is bounded, which can be seen as follows. Hölder’s inequality gives

‖ κ ‖ L 2 ≤ ℒ ( f t ) 1 2 − 1 p ‖ κ ‖ L p ≤ ℒ ( f t ) 1 2 − 1 p ∫ R / Z ∣ κ 2 + δ 2 ∣ p 2 d s 1 p ≤ ( p λ ) 1 p λ − 1 2 ℰ ε , δ , λ ( p ) ( f 0 ) 1 2

and

‖ ∇ s κ ‖ L 1 ≤ ℒ ( f t ) 1 2 ‖ ∇ s κ ‖ L 2 ≤ ( λ ε 2 ) − 1 2 ℰ ε , δ , λ ( p ) ( f 0 ) .

Hence, Lemma 2.11 tells us that

(2.28) ‖ κ ‖ L ∞ < c ( p , ε , λ , ℰ ε , δ , λ ( p ) ( f 0 ) ) .

To establish the long-time existence of a solution, let us assume that the maximal time T of smooth existence for the flow (2.2) satisfies T < ∞ , i.e., let us assume that the flow does not smoothly exist for all times. Considering Lemmas 2.4 and 2.5 and the flow equation (2.22), we obtain

(2.29) d d t 1 2 ∫ R / Z ∣ ∇ s m κ ∣ 2 d s + ε ∫ R / Z ∣ ∇ s m + 3 κ ∣ 2 d s = ε ∫ R / Z ⟨ P m + 4 ( κ ) − 1 2 ⟨ ∇ s 4 κ + P 2 ( κ ) , κ ⟩ ∇ s m κ , ∇ s m κ ⟩ d s + ∫ R / Z ⟨ P m + 4 ( κ ) − 1 2 ⟨ P 2 ( κ ) , κ ⟩ ∇ s m κ , ∇ s m κ ⟩ d s + λ ∫ R / Z ⟨ ∇ s m + 2 κ + P m ( κ ) − 1 2 ∣ κ ∣ 2 ∇ s m κ , ∇ s m κ ⟩ d s = ε ∫ R / Z ⟨ P m + 4 ( κ ) , ∇ s m κ ⟩ d s + ∫ R / Z ⟨ P m + 4 ( κ ) , ∇ s m κ ⟩ d s + λ ∫ R / Z ⟨ P m + 2 ( κ ) , ∇ s m κ ⟩ d s .

In order to apply interpolation estimates on the right-hand side, we need to integrate by parts once or twice, in case that terms like ∇ s m + 3 κ or ∇ s m + 4 κ appear in P m + 4 ( κ ) , to achieve from (2.29)

(2.30) d d t 1 2 ∫ R / Z ∣ ∇ s m κ ∣ 2 d s + ε ∫ R / Z ∣ ∇ s m + 3 κ ∣ 2 d s = ε ∫ R / Z P 2 m + 4 , m + 2 ( κ ) d s + ∫ R / Z P 2 m + 4 , m + 2 ( κ ) d s + λ ∫ R / Z P 2 m + 2 , m + 2 ( κ ) d s .

It suffices to estimate, as P 2 m + 2 , m + 2 ( κ ) is included in the notation of P 2 m + 4 , m + 2 ( κ ) , by using Lemma 2.6 and Hölder’s inequality:

∫ R / Z P 2 m + 4 , m + 2 ( κ ) d s ≤ ∫ R / Z ∣ P 2 m + 4 , m + 2 ( κ ) ∣ d s ≤ ∑ i = 1 2 m + 4 ∫ R / Z ∣ P i 2 m + 4 − i , m + 1 ( ∇ s κ ) ∣ ⋅ ∣ P 0 ( κ ) ∣ d s + ∫ R / Z ∣ P 0 ( κ ) ∣ d s ≤ ∑ i = 1 2 m + 4 c 0 ‖ κ ‖ L ∞ ∫ R / Z ∣ P i 2 m + 4 − i , m + 1 ( ∇ s κ ) ∣ d s + c 1 ‖ κ ‖ L ∞

for some constants c 0 , c 1 > 0 depending on m and δ . Note that the case i = 1 can be ruled out thanks to the underlying precise gradient flow equation, including the gradients (2.3) and (2.4), and integration by parts if needed.

Now let us assume the initial bound ℰ ε , δ , λ ( p ) ( f t ) ≤ Λ for 0 ≤ t < T . Then, the previous inequality reduces by the uniform bound on κ in (2.28) to

(2.31) ∫ R / Z P 2 m + 4 , m + 2 ( κ ) d s ≤ ∑ i = 2 2 m + 4 c 0 ( m , p , ε , δ , λ , Λ ) ∫ R / Z ∣ P i 2 m + 4 − i , m + 1 ( ∇ s κ ) ∣ d s + c 1 ( m , p , ε , δ , λ , Λ ) .

Finally, we interpolate the remaining integral between ‖ ∇ s m + 3 κ ‖ L 2 and ‖ ∇ s κ ‖ L 2 by Lemma 2.9 with k = m + 2 , μ = 2 m + 4 − i , ν = i , and γ = 2 m + 4 − i 2 − 1 m + 2 . Note that Lemma 2.9 is applicable since κ is uniformly bounded by (2.28), i = ν ≥ 2 , and μ + 1 2 ν = 2 m + 4 − i + 1 2 i < 2 m + 4 + 1 = 2 k + 1 . So we have for any small η > 0 that

(2.32) ∫ R / Z ∣ P i 2 m + 4 − i , m − 1 ( ∇ s κ ) ∣ d s ≤ c 2 η ∫ R / Z ∣ ∇ s m + 3 κ ∣ 2 d s + η − γ 2 − γ ∫ R / Z ∣ ∇ s κ ∣ 2 d s i − γ 2 − γ + ℒ ( f ) − 2 m − 3 + i 2 ∫ R / Z ∣ ∇ s κ ∣ 2 d s i 2

for a constant c 2 = c 2 ( i , m , n , p , ε , δ , λ , Λ ) > 0 . By using 2 ≤ i ≤ 2 m + 4 , the bound on ‖ ∇ s κ ‖ L 2 in (2.27), and the boundedness of the length term in (2.26), the interpolation estimate (2.32) reduces to

(2.33) ∫ R / Z ∣ P i 2 m + 4 − i , m + 1 ( ∇ s κ ) ∣ d s ≤ η c 2 ∫ R / Z ∣ ∇ s m + 3 κ ∣ 2 d s + c ( m , n , p , ε , δ , λ , Λ , η ) .

Hence, the estimates (2.30), (2.31), and (2.33) lead to

(2.34) d d t ∫ R / Z ∣ ∇ s m κ ∣ 2 d s + ε ∫ R / Z ∣ ∇ s m + 3 κ ∣ 2 d s ≤ c ( m , n , p , ε , δ , λ , Λ ) ,

where we absorbed ‖ ∇ s m + 3 κ ‖ L 2 2 , that appears on the right-hand side in (2.33), by choosing η = ε 2 c 0 c 2 ( 2 m + 4 ) > 0 small enough. So we obtain by estimate (2.34)

‖ ∇ s m κ ‖ L 2 2 ( t ) ≤ ‖ ∇ s m κ ‖ L 2 2 ( 0 ) + c ( m , n , p , ε , δ , λ , Λ ) T

for all m ∈ N 0 .

From this, we conclude using the upper bound on the length of the curve, ‖ ∇ s m κ ‖ L 1 ≤ ℒ ( f t ) 1 2 ‖ ∇ s m κ ‖ L 2 2 , and Lemma 2.11 that

‖ ∂ s m κ ‖ L ∞ ≤ c ( m , n , p , ε , δ , λ , Λ , f 0 , T )

for all m ∈ N 0 . We now argue exactly as in the proof of [7, Theorem 3.1] to first estimate

1 c ( n , p , ε , δ , λ , Λ , f 0 , T ) ≤ ∣ f ′ ∣ ≤ c ( n , p , ε , δ , λ , Λ , f 0 , T )

and then obtain by an inductive argument

‖ ∂ x m κ ‖ L ∞ ≤ c ( m , n , p , ε , δ , λ , Λ , f 0 , T )

holds for all m ∈ N 0 . These bounds imply that f smoothly extends to [ 0 , T ] × R / Z and together with the short-time existence result even beyond T , from which we iteratively obtain long-time existence.

Now we draw our attention to the asymptotic behavior of long-time solutions to the regularized gradient flow 2.2. For any such solution f ( t , ⋅ ) , we can choose, after reparametrization by arc-length and an appropriate choice of translations, a subsequence f ˜ ( t i , ⋅ ) − p i that smoothly converges to a limit curve f ∞ as t i → ∞ .

By the standard interpolation inequality as stated in (2.18) for k = m + 3 , the bound on the length of the curve (2.26), and estimate (2.34), we infer

d d t ∫ R / Z ∣ ∇ s m κ ∣ 2 d s + c 0 ∫ R / Z ∣ ∇ s m κ ∣ 2 d s ≤ c ( m , n , p , ε , δ , λ , f 0 ) ,

where c 0 = c 0 ( p , ε , δ , λ , f 0 ) > 0 , which gives us the bound on ‖ ∇ s m κ ‖ L 2 2 ( t ) ≤ ‖ ∇ s m κ ‖ L 2 2 ( 0 ) + c ( m , n , p , ε , δ , λ , f 0 ) . Together with the bound on the length (2.26) and Lemmas 2.11 and 2.10, we obtain as a consequence

(2.35) ‖ ∂ s m κ ‖ L ∞ + ‖ ∇ s m κ ‖ L ∞ ≤ c ( m , n , p , ε , δ , λ , f 0 ) ,

and hence also

(2.36) ‖ ∇ t ( ∇ s m κ ) ‖ L ∞ ≤ c ( m , n , p , ε , δ , λ , f 0 )

by Lemma 2.5.

Next we define u ( t ) = ‖ ∂ t f ‖ L 2 2 ( t ) . Then, we have u ( t ) ∈ L 1 ( ( 0 , ∞ ) ) by the energy identity (2.23) and ∣ u ′ ( t ) ∣ ≤ c ( m , n , p , ε , δ , λ , f 0 ) by (2.26), (2.35), and (2.36). Thus, we obtain that u ( t ) → 0 as t → ∞ , from which we conclude that f ∞ is a critical point of ℰ ε , δ , λ ( p ) .

This completes the proof of Theorem 2.1.

3 Higher order estimates for the regularized energies

In this section, we prove the essential a priori estimates that will allow us to obtain a solution to (2.2) sending ε , δ ↓ 0 in Section 4. To this end, we first have to bring the evolution equations in a suitable form.

3.1 The first variation of the energy

We will once more calculate the first variation of the energy ℰ ε , δ , λ ( p ) . This time we will not decompose the direction into a normal and tangential part as later on we test the equation on a finite difference f ( ⋅ + h ) − f , which need not be pointing in the normal direction. We consider a smooth time-dependent family of curves f ( t , x ) = f : [ 0 , T ) × R / Z → R n and set V = ∂ t f . Note that in abuse of notation, V no longer stands for the normal part of the variation but for the complete variational direction.

We obtain the following variant of [7, Lemma 2.1].

Lemma 3.1

(First variation of geometric quantities) We have

∂ t ( d s ) = ⟨ τ , ∂ s V ⟩ d s ,
∂ t ∂ s − ∂ s ∂ t = − ⟨ ∂ s V , τ ⟩ ∂ s ,
∂ t τ = ∇ s V ,
∂ t κ = ( ∂ s 2 V ) ⊥ − 2 ⟨ ∂ s V , τ ⟩ κ − ⟨ ∂ s V , κ ⟩ τ ,
∇ t ∇ s κ = ( ∂ s 3 V ) ⊥ − 3 ⟨ ∂ s 2 V , τ ⟩ κ − 3 ⟨ ∂ s V , τ ⟩ ( ∂ s κ ) ⊥ − 3 ⟨ ∇ s V , κ ⟩ κ + ∣ κ ∣ 2 ∇ s V .

Proof

First, by using f ′ = ∂ x f , we obtain

∂ t ∣ f ′ ∣ = f ′ ∣ f ′ ∣ , ∂ x V = ⟨ τ , ∂ s V ⟩ ∣ f ′ ∣ .

Hence, from d s = ∣ f ′ ∣ d x , we obtain

∂ t ( d s ) = ( ∂ t ∣ f ′ ∣ ) d x = ⟨ τ , ∂ s V ⟩ d s

and

∂ t τ = ∂ t f ′ ∣ f ′ ∣ − f ′ ∣ f ′ ∣ ⟨ τ , ∂ s V ⟩ = ∂ s V − ⟨ τ , ∂ s V ⟩ τ = ∇ s V .

This proves parts (i) and (iii). Similarly, differentiating ∂ s = ∂ x ∣ f ′ ∣ yields

∂ t ∂ s = 1 ∣ f ′ ∣ ∂ t ∂ x − ⟨ τ , ∂ s V ⟩ 1 ∣ f ′ ∣ ∂ x = ∂ s ∂ t − ⟨ τ , ∂ s V ⟩ ∂ s .

To obtain (iv), we calculate using the formulas (ii) and (iii)

∂ t κ = ∂ t ∂ s τ = ∂ s ∂ t τ − ⟨ τ , ∂ s V ⟩ ∂ s τ = ∂ s ( ∂ s V − ⟨ ∂ s V , τ ⟩ τ ) − ⟨ τ , ∂ s V ⟩ κ = ( ∂ s 2 V ) ⊥ − ⟨ ∂ s V , κ ⟩ τ − ⟨ ∂ s V , τ ⟩ κ − ⟨ τ , ∂ s V ⟩ κ = ( ∂ s 2 V ) ⊥ − 2 ⟨ ∂ s V , τ ⟩ κ − ⟨ ∂ s V , κ ⟩ τ .

Finally, from

∇ t ∂ s κ = ( ∂ s ∂ t κ − ⟨ ∂ s V , τ ⟩ ∂ s κ ) ⊥ = ( ∂ s ( ∂ s 2 V − ⟨ ∂ s 2 V , τ ⟩ τ − 2 ⟨ ∂ s V , τ ⟩ κ − ⟨ ∂ s V , κ ⟩ τ ) − ⟨ ∂ s V , τ ⟩ ∂ s κ ) ⊥ = ( ( ∂ s 3 V ) ⊥ − ⟨ ∂ s 2 V , τ ⟩ κ − 2 ⟨ ∂ s 2 V , τ ⟩ κ − 2 ⟨ ∂ s V , κ ⟩ κ − 2 ⟨ ∂ s V , τ ⟩ ∂ s κ − ⟨ ∂ s V , κ ⟩ κ − ⟨ ∂ s V , τ ⟩ ∂ s κ ) ⊥ = ( ∂ s 3 V ) ⊥ − 3 ⟨ ∂ s 2 V , τ ⟩ κ − 3 ⟨ ∂ s V , τ ⟩ ∇ s κ − 3 ⟨ ∂ s V , κ ⟩ κ

together with

∇ t ( ⟨ ∂ s κ , τ ⟩ τ ) = ⟨ ∂ s κ , τ ⟩ ∇ t τ = ⟨ ∂ s κ , τ ⟩ ∇ s V = − ∣ κ ∣ 2 ∇ s V ,

we obtain (v).□

By using this lemma with f t = f + t V , we obtain for the first variation of ℱ :

(3.1) δ V ℱ ( f ) ≔ d d t ℱ ( f + t V ) t = 0 = ∫ R / Z ⟨ ∇ s κ , δ V ( ∇ s κ ) ⟩ d s + 1 2 ∫ R / Z ∣ ∇ s κ ∣ 2 δ V ( d s ) = ∫ R / Z ⟨ ∇ s κ , ( δ V ( ∇ s κ ) ) ⊥ ⟩ d s + 1 2 ∫ R / Z ∣ ∇ s κ ∣ 2 ⟨ τ , ∂ s V ⟩ d s ,

where

( δ V ( ∇ s κ ) ) ⊥ = ( ∂ s 3 V ) ⊥ − 3 ⟨ ∂ s 2 V , τ ⟩ κ − 3 ⟨ ∂ s V , τ ⟩ ∇ s κ − ⟨ ∇ s V , κ ⟩ κ + ∣ κ ∣ 2 ∇ s V .

Similarly,

(3.2) δ V ℰ δ ( p ) ( f ) = ∫ R / Z ( ∣ κ ∣ 2 + δ 2 ) p − 2 2 ⟨ κ , δ V κ ⟩ d s + 1 p ∫ R / Z ( ∣ κ ∣ 2 + δ 2 ) p 2 ⟨ τ , ∂ s V ⟩ d s

and

δ V κ = ( ∂ s 2 V ) ⊥ − 2 ⟨ ∂ s V , τ ⟩ κ − ⟨ ∂ s V , κ ⟩ τ .

The first variation of the length of course is

(3.3) δ V ℒ ( f ) = ∫ R / Z ⟨ τ , ∂ s V ⟩ d s .

Let us bring the first variation of ∇ s κ in a more suitable form for our task, using again a slight variant of the ∗ -notation introduced in [7]. We obtain

(3.4) ( δ V ( ∇ s κ ) ) ⊥ = ( ∂ s 3 V + ∂ s 2 V ∗ τ ∗ κ + ∂ s V ∗ ( τ ∗ ∂ s κ + κ ∗ κ ) ) ⊥ .

3.2 An a priori estimate

We will now prove that for solutions of the negative gradient flow for the regularized energies ℰ ε , δ , λ ( p ) , we can bound a better norm than the W 2 , p -norm of our curves, using a finite difference method as in [2]. To this end, we consider a regular curve f ∈ C 0 , 1 ( R / L Z , R n ) of length L parametrized by arc-length. We measure the additional regularity using the Besov-Nikolski space B p , ∞ s ( R / L Z ) for s ∈ ( 0 , 1 ) and p ∈ [ 1 , ∞ ) , which consists of all functions u ∈ L p ( R / L Z , d x ) for which the semi-norm

∣ u ∣ B p , ∞ s ≔ sup h ‖ u h − u ‖ L p ( d x ) ∣ h ∣ s

is finite. Here, u h ( x ) ≔ u ( x + h ) denotes the function shifted by the value h . This space is equipped with the norm

‖ u ‖ B p , ∞ s = ‖ u ‖ L p + ∣ u ∣ B p , ∞ s .

Later on we will also use the Besov-space B p , ∞ 1 + s ( R / L Z ) whose norm can be given by

‖ u ‖ B p , ∞ 1 + s = ‖ u ‖ W 1 , p + ‖ u ′ ‖ B p , ∞ s .

Theorem 3.2

(A priori estimate) Let f : R / L Z → R n be a smooth curve parametrized by arc-length, which satisfies the equation

(3.5) ∇ L 2 ℰ ε , δ , λ ( p ) ( f ) = g

for a g ∈ L 2 ( d s ) and an 0 < ε < 1 . Then,

(3.6) ε ‖ ∂ s 3 f ‖ B 2 , ∞ 1 4 2 + ‖ ∂ s 2 f ‖ B p , ∞ 1 2 p p ≤ C ( 1 + ‖ g ‖ L 2 )

for some constant C = C ( p , λ , ℰ ε , δ , λ ( p ) ( f ) ) .

Till the end of the proof of Theorem 3.2 let us assume that the curve f : R / L Z → R n is parametrized by arc-length so that x is equal to the arc-length parameter s .

We note that Fenchel’s together with Hölder’s inequality give

2 π ≤ ∫ R / L Z ∣ κ ∣ d s ≤ ℒ ( f ) 1 − 1 p ∫ R / L Z ∣ κ ∣ p d s 1 p ≤ ℒ ( f ) 1 − 1 p ( p ℰ ε , δ , λ ( p ) ( f ) ) 1 p ,

which implies

ℒ ( f ) ≥ ( 2 π ) p p − 1 ( p ℰ ε , δ , λ ( p ) ( f ) ) 1 p − 1 .

Since trivially

ℒ ( f ) ≤ ℰ ε , δ , λ ( p ) ( f ) λ ,

we see that the length of the curve is bounded from above and from below away from zero if the regularized energy is controlled.

We note that by using Hölder’s inequality, we obtain

(3.7) ε 2 ∫ R / L Z ∣ ∇ s κ ∣ 2 d s + 1 p ℒ ( f ) p 2 − 1 ∫ R / L Z ∣ κ ∣ 2 d s p 2 ≤ ε ℱ ( f ) + ℰ δ ( p ) ( f ) ≤ ℰ ε , δ , λ ( p ) ( f )

as p ≥ 2 . From ∇ s κ = ∂ s κ + ∣ κ ∣ 2 τ together with Gagliardo-Nirenberg’s interpolation inequality (cf. Lemma 2.7) and Young’s inequality, we obtain

‖ ∂ s κ ‖ L 2 ≤ C ( ‖ ∇ s κ ‖ L 2 + ‖ κ ‖ L 4 2 ) ≤ C ( ‖ ∇ s κ ‖ L 2 + ‖ ∇ s κ ‖ L 2 1 2 ‖ κ ‖ L 2 3 2 ) ≤ C ( ‖ ∇ s κ ‖ L 2 + ‖ κ ‖ L 2 3 ) ≤ C ( ‖ ∇ s κ ‖ L 2 + 1 ) .

Hence, (3.7) implies ε ‖ ∂ s κ ‖ L 2 2 ≤ C , where the constant C > 0 only depends on the energy ℰ ε , δ , λ ( p ) ( f ) . From equation (3.5), we obtain

∇ L 2 ℰ ε , δ , λ ( p ) ( f h ) = g h and ∇ L 2 ℰ ε , δ , λ ( p ) ( f ) = g ,

where for a function u we set u h ( x ) ≔ u ( x + h ) . We test these equations with V = f h − f , subtract the results, and integrate over R / L Z to obtain

(3.8) ε δ V ( ℱ ( f h ) − ℱ ( f ) ) + δ V ( ℰ δ ( p ) ( f h ) − ℰ δ ( p ) ( f ) ) + λ ( δ V ( ℒ ( f h ) − ℒ ( f ) ) = ∫ R / L Z ( g h − g ) ( f h − f ) d s .

We will estimate this further using that

(3.9) höl 1 2 τ = höl x ≠ y ∈ [ − L , 2 L ] ∣ τ ( x ) − τ ( y ) ∣ ∣ x − y ∣ 1 2 ≤ C ‖ κ ‖ L 2 ( R / L Z ) , höl 1 2 κ ≤ C ‖ ∂ s κ ‖ L 2 ( R / L Z ) ,

which can easily seen to be true observing that the fundamental theorem of calculus together with Hölder’s inequality gives for a differentiable function u : R / L Z → R n and x , h ∈ R with ∣ h ∣ < 3 L ,

∣ u ( x + h ) − u ( x ) ∣ = ∫ 0 h u ′ ( x + t ) d t ≤ ∫ 0 h ∣ u ′ ( x + t ) ∣ d t ≤ 3 ∣ h ∣ 1 2 ‖ u ′ ‖ L 2 ( R / L Z ) .

Hence, we also have

(3.10) ‖ ∂ s V ‖ L ∞ ≤ C ∣ h ∣ 1 2 ‖ κ ‖ L 2 ‖ ∂ s 2 V ‖ L ∞ ≤ C ∣ h ∣ 1 2 ‖ ∂ s κ ‖ L 2 .

We start discussing the terms coming from the first variation of ℱ in (3.8).

Lemma 3.3

Under the assumptions of Theorem 3.2, we have

ε δ V ( ℱ ( f h ) − ℱ ( f ) ) = ε ∫ R / L Z ∣ ∂ s 3 f h − ∂ s 3 f ∣ 2 d s + R ℱ

with ∣ R ℱ ∣ ≤ C ∣ h ∣ 1 2 .

Proof

To prove this claim, we use ∇ s κ = ∂ s κ − ⟨ ∂ s κ , τ ⟩ τ = ∂ s κ + ∣ κ ∣ 2 τ to decompose

ε ∫ R / L Z ⟨ ∇ s κ h , δ V ( ∇ s κ h ) ⟩ d s − ∫ R / L Z ⟨ ∇ s κ , δ V ( ∇ s κ ) ⟩ d s = ε ∫ R / L Z ⟨ ∂ s 3 f h + ∣ κ h ∣ 2 τ h , ∂ s 3 V + ∂ s 2 V ∗ τ h ∗ κ h + ∂ s V ∗ ( τ h ∗ ∂ s κ h + κ h ∗ κ h ) ⟩ d s − ε ∫ R / L Z ⟨ ∂ s 3 f + ∣ κ ∣ 2 τ , ∂ s 3 V + ∂ s 2 V ∗ τ ∗ κ + ∂ s V ∗ ( τ ∗ ∂ s κ + κ ∗ κ ) ⟩ d s = ε ∫ R / L Z ∣ ∂ s 3 f h − ∂ s 3 f ∣ 2 d s + ε ∣ I + I I + I I I ∣ ,

where

I = ∫ R / L Z ⟨ ∣ κ h ∣ 2 τ h − ∣ κ ∣ 2 τ , ∂ s 3 V ⟩ d s , I I = ∫ R / L Z ⟨ ∂ s 3 f h , ∂ s 2 V ∗ κ h ∗ τ h + ∂ s V ∗ ( τ h ∗ ∂ s κ h + κ h ∗ κ h ) ⟩ d s − ∫ R / L Z ⟨ ∂ s 3 f , ∂ s 2 V ∗ κ ∗ τ + ∂ s V ∗ ( τ ∗ ∂ s κ + κ ∗ κ ) ⟩ d s , I I I = ∫ R / L Z ⟨ ∣ κ h ∣ 2 τ h , ∂ s 2 V ∗ κ h ∗ τ h + ∂ s V ∗ ( τ h ∗ ∂ s κ h + κ h ∗ κ h ) ⟩ d s − ∫ R / L Z ⟨ ∣ κ ∣ 2 τ , ∂ s 2 V ∗ κ ∗ τ + ∂ s V ∗ ( τ ∗ ∂ s κ + κ ∗ κ ) ⟩ d s .

For the term I, we see

∣ I ∣ ≤ ‖ ∂ s 3 V ‖ L 2 ‖ ∣ κ h ∣ 2 τ h − ∣ κ ∣ 2 τ ‖ L 2 ≤ 2 ‖ ∂ s κ ‖ L 2 ‖ ∣ κ h ∣ 2 τ h − ∣ κ ∣ 2 τ ‖ L 2 .

The equality

∣ κ h ∣ 2 τ h − ∣ κ ∣ 2 τ = ( ∣ κ h ∣ − ∣ κ ∣ ) ∣ κ h ∣ τ h + ∣ κ ∣ ( ∣ κ h ∣ − ∣ κ ∣ ) τ h + ∣ κ ∣ 2 ( τ h − τ )

together with (3.9) and Gagliardo-Nirenberg interpolation estimates (cf. Lemma 2.7) implies that

‖ ∣ κ h ∣ 2 τ h − ∣ κ ∣ 2 τ ‖ L 2 ≤ ∣ h ∣ 1 2 2 höl 1 2 κ ‖ κ ‖ L 2 + höl 1 2 τ ‖ κ ‖ L 4 2 ≤ C ∣ h ∣ 1 2 ( ‖ ∂ s κ ‖ L 2 ‖ κ ‖ L 2 + ‖ κ ‖ L 2 ‖ κ ‖ L 4 2 ) ≤ C ∣ h ∣ 1 2 ‖ ∂ s κ ‖ L 2 ‖ κ ‖ L 2 + ‖ ∂ s κ ‖ L 2 1 2 ‖ κ ‖ L 2 1 + 3 2 ≤ C ∣ h ∣ 1 2 ( ‖ ∂ s κ ‖ L 2 + 1 )

as ‖ κ ‖ L 2 is bounded. Hence,

ε ∣ I ∣ ≤ C ∣ h ∣ 1 2 ε ( ‖ ∂ s κ ‖ L 2 2 + ‖ ∂ s κ ‖ L 2 ) ≤ C ∣ h ∣ 1 2 ,

where we applied (3.7). To see that the same estimate holds for the terms in II, we observe by (3.10)

∫ R / L Z ⟨ ∂ s 3 f , ∂ 2 V ∗ κ ∗ τ ⟩ d s ≤ C ‖ ∂ s κ ‖ L 2 ‖ κ ‖ L 2 ‖ ∂ 2 V ‖ L ∞ ≤ C ∣ h ∣ 1 2 ‖ ∂ s κ ‖ L 2 2 ≤ C h 1 2

and

∫ R / L Z ⟨ ∂ s 3 f , ∂ s V ∗ ( τ ∗ ∂ s κ + κ ∗ κ ) ⟩ d s ≤ C ‖ ∂ s κ ‖ L 2 ‖ ∂ s V ‖ L ∞ ( ‖ ∂ s κ ‖ L 2 + ‖ κ ‖ L 4 2 ) ≤ C ∣ h ∣ 1 2 ‖ ∂ s κ ‖ L 2 2 ‖ κ ‖ L 2 + ‖ ∂ s κ ‖ L 2 1 + 1 2 ‖ κ ‖ L 2 1 + 3 2 ≤ C ∣ h ∣ 1 2 ( ‖ ∂ s κ ‖ L 2 2 + 1 ) ≤ C ∣ h ∣ 1 2 ,

where again we used (3.7) and the interpolation estimates stated in Lemma 2.7. As the other terms in I I involving f h instead of f can be estimated in precisely the same way, we obtain

ε ∣ I I ∣ ≤ C ∣ h ∣ 1 2 ε ( ‖ ∂ s κ ‖ L 2 2 + 1 ) ≤ C ∣ h ∣ 1 2 .

Similarly,

∣ I I I ∣ ≤ C ( ‖ κ ‖ L 3 3 ‖ ∂ s 2 V ‖ L ∞ + ‖ ∂ s κ ‖ L 2 ‖ κ ‖ L 4 2 ‖ ∂ s V ‖ L ∞ + ‖ κ ‖ L 4 4 ‖ ∂ s V ‖ L ∞ ) ≤ C ∣ h ∣ 1 2 ‖ ∂ s κ ‖ L 2 1 + 1 2 ‖ κ ‖ L 2 2 + 1 2 + ‖ ∂ s κ ‖ L 2 1 + 1 2 ‖ κ ‖ L 2 1 + 3 2 + ‖ ∂ s κ ‖ L 2 ‖ κ ‖ L 2 4 ≤ C ∣ h ∣ 1 2 ( ‖ ∂ s κ ‖ L 2 2 + 1 ) .

Hence, also

ε ∣ I I I ∣ ≤ C ∣ h ∣ 1 2 .

Finally,

∫ R / L Z ∣ ∇ s κ ∣ 2 ⟨ τ , ∂ s V ⟩ d s ≤ C ∣ h ∣ 1 2 ‖ ∇ s κ ‖ L 2 2 ‖ κ ‖ L 2 ≤ C ∣ h ∣ 1 2 ‖ ∇ s κ ‖ L 2 2 ≤ C ∣ h ∣ 1 2

and also

∫ R / L Z ∣ ∇ s κ h ∣ 2 ⟨ τ h , ∂ s V ⟩ d s ≤ C ∣ h ∣ 1 2 ,

this gives using the formula (3.1) for the first variation of ℱ that

ε δ V ( ℱ ( f h ) − ℱ ( g ) ) = ε ∫ R / L Z ∣ ∂ s 3 f h − ∂ s 3 f ∣ 2 d s + R ℱ

with

∣ R ℱ ∣ ≤ C ∣ h ∣ 1 2 . □

The terms in (3.8) containing the first variation of ℰ δ ( p ) can be estimated as follows.

Lemma 3.4

Under the assumptions of Theorem 3.2, we have

δ V ℰ δ ( p ) ( f h ) − δ V ℰ δ ( p ) ( f ) = ∫ R / L Z ⟨ ( ∣ κ h ∣ 2 + δ 2 ) p − 2 2 κ h − ( ∣ κ ∣ 2 + δ 2 ) p − 2 2 κ , ∂ s 2 V ⟩ d s + ℛ ℰ δ ( p )

with ∣ ℛ ℰ δ ( p ) ∣ ≤ C ∣ h ∣ 1 2 .

Proof

We decompose

∫ R / L Z ( ∣ κ h ∣ 2 + δ 2 ) p − 2 2 ⟨ κ h , δ V ( κ h ) ⟩ − ( ∣ κ ∣ 2 + δ 2 ) p − 2 2 ⟨ κ , δ V ( κ ) ⟩ d s = ∫ R / L Z ( ∣ κ h ∣ 2 + δ 2 ) p − 2 2 ⟨ κ h , ∂ s 2 V + ∂ s V ∗ κ h ∗ τ h ⟩ − ( ∣ κ ∣ 2 + δ 2 ) p − 2 2 ⟨ κ , ∂ s 2 V + ∂ s V ∗ κ ∗ τ ⟩ d s = ∫ R / L Z ⟨ ( ∣ κ h ∣ 2 + δ 2 ) p − 2 2 κ h − ( ∣ κ ∣ 2 + δ 2 ) p − 2 2 κ , ∂ s 2 V ⟩ d s + I ,

where this time

I = ∫ R / L Z ( ∣ κ h ∣ 2 + δ 2 ) p − 2 2 ⟨ κ h , ∂ s V ∗ κ h ∗ τ h ⟩ d s − ∫ R / L Z ( ∣ κ ∣ 2 + δ 2 ) p − 2 2 ⟨ κ , ∂ s V ∗ κ ∗ τ ⟩ d s .

Observe that

∣ I ∣ ≤ C ‖ ∂ s V ‖ L ∞ ∫ R / L Z ( ∣ κ ∣ 2 + δ 2 ) p − 2 2 ∣ κ ∣ 2 d s ≤ C ‖ κ ‖ L 2 ℰ δ ( p ) ( f ) ∣ h ∣ 1 2 ≤ C ∣ h ∣ 1 2 .

Furthermore, we have

∫ R / L Z ( ∣ κ ∣ 2 + δ 2 ) p 2 ⟨ ∂ s V , τ ⟩ d s ≤ p ℰ δ ( p ) ( f ) ‖ ∂ s V ‖ L ∞ ≤ C ∣ h ∣ 1 2

and by symmetry,

∫ R / L Z ( ∣ κ h ∣ 2 + δ 2 ) p 2 ⟨ ∂ s V , τ h ⟩ d s ≤ C ∣ h ∣ 1 2 .

Together with formula (3.2), this proves the claim.□

Lemma 3.5

Under the assumptions of Theorem 3.2, we have

δ V ℒ ( f h ) − δ V ℒ ( f ) = ℛ ℒ

with ∣ ℛ ℒ ∣ ≤ C ∣ h ∣ 1 2 .

Proof

The first variation of length (3.3) together with (3.10) leads to

δ V ℒ ( f h ) − δ V ℒ ( f ) ≤ C ‖ ∂ s V ‖ L ∞ ≤ C ∣ h ∣ 1 2 ,

which proves the statement.□

The following lemma tells us that the main part in Lemma 3.4 has similar monotonicity properties as the p-Laplace operator.

Lemma 3.6

Under the assumptions of Theorem 3.2, we have

∫ R / L Z ( ∣ κ h ∣ 2 + δ 2 ) p − 2 2 κ h − ( ∣ κ ∣ 2 + δ 2 ) p − 2 2 κ , ∂ s 2 V d s ≥ 1 4 p − 1 ∫ R / L Z ∣ κ h − κ ∣ p d s .

Proof

For vectors w , v , we set w t = v + ( w − v ) t . By using the fundamental theorem of calculus, we obtain

( ∣ w ∣ 2 + δ 2 ) p − 2 2 w − ( ∣ v ∣ 2 + δ 2 ) p − 2 2 v , w − v = ∫ 0 1 ( p − 2 ) ( ∣ w t ∣ 2 + δ 2 ) p − 4 2 ⟨ w t , ( w − v ) ⟩ 2 d t + ∫ 0 1 ( ∣ w t ∣ 2 + δ 2 ) p − 2 2 ∣ w − v ∣ 2 d t ≥ ∫ 0 1 ( ∣ w t ∣ 2 + δ 2 ) p − 2 2 ∣ w − v ∣ 2 d t .

If we now assume that ∣ v ∣ ≥ ∣ w ∣ , we obtain for t ∈ 0 , 1 4

∣ w t ∣ ≥ ∣ v ∣ − t ( ∣ w ∣ + ∣ v ∣ ) ≥ ∣ v ∣ − 1 2 ∣ v ∣ = ∣ v ∣ 2 ≥ 1 4 ∣ w − v ∣ .

So,

( ∣ w ∣ 2 + δ 2 ) p − 2 2 w − ( ∣ v ∣ 2 + δ 2 ) p − 2 2 v , w − v ≥ ∫ 0 1 4 ( ∣ w t ∣ 2 + δ 2 ) p − 2 2 ∣ w − v ∣ 2 d t ≥ 1 4 p − 1 ∣ w − v ∣ p ,

and hence,

∫ R / L Z ( ∣ κ h ∣ 2 + δ 2 ) p − 2 2 κ h − ( ∣ κ ∣ 2 + δ 2 ) p − 2 2 κ , ∂ s 2 V d s ≥ 1 4 p − 1 ∫ R / L Z ∣ κ h − κ ∣ p d s . □

Now we can prove Theorem 3.2.

Proof of Theorem 3.2

Note that

∫ R / L Z ( g h − g ) V d s ≤ C ‖ g ‖ L 2 ∣ h ∣ 1 2 .

By using equation (3.5) together with Lemmas 3.3 and 3.4 and the aforementioned estimate, we obtain

ε ∫ R / L Z ∣ ∂ s 3 f h − ∂ s 3 f ∣ 2 d s + ∫ R / L Z ( ∣ κ h ∣ 2 + δ 2 ) p − 2 2 κ h − ( ∣ κ ∣ 2 + δ 2 ) p − 2 2 κ , ∂ s 2 V d s ≤ C ∣ h ∣ 1 2 ( ‖ g ‖ L 2 + 1 ) .

Together with Lemma 3.6 and the definition of the Besov spaces, we hence obtain

ε ‖ ∂ s 3 f ‖ B 2 , ∞ 1 4 2 + ‖ κ ‖ B p , ∞ 1 2 p p ≤ C ( ‖ g ‖ L 2 + 1 ) . □

We can extend Theorem 3.2 to arbitrary curves f : R / Z → R n re-parametrizing the curve by arc-length. Let L = ℒ ( f ) be the length of f and σ f : R / L Z → R / Z be a re-parametrization such that f ∘ σ f is parametrized by arc-length. For u : R / Z → R n , we set

‖ u ‖ L p ( d s ) = ‖ u ∘ σ f ‖ L p ( R / L Z ) = ∫ R / L Z ∣ u ∣ p d s 1 p ,

‖ u ‖ W k , p ( d s ) = ‖ u ∘ σ f ‖ W k , p ,

and

‖ u ‖ B p , ∞ s ( d s ) = ‖ u ∘ σ f ‖ B p , ∞ s .

Applying Theorem 3.2 to the re-parametrization of f by arc-length, we immediately obtain the following corollary for regular curves.

Corollary 3.7

(A priori estimate) Let f : R / Z → R n be a smooth regular curve, which satisfies the equation:

(3.11) ∇ L 2 ℰ ε , δ , λ ( p ) ( f ) = g

for a g ∈ L 2 ( d s ) and an 0 < ε < 1 . Then,

(3.12) ε ‖ ∂ s 3 f ‖ B 2 , ∞ 1 4 ( d s ) 2 + ‖ ∂ s 2 f ‖ B p , ∞ 1 2 p ( d s ) p ≤ C ( 1 + ‖ g ‖ L 2 ( d s ) )

for some constant C = C ( p , λ , ℰ ε , δ , λ ( p ) ( f ) ) .

Let us list some immediate consequences for solutions to the gradient flow for the regularized energies. First, we observe that the highest order part grows slower as ε goes to 0 than the trivial bound ε ℱ ( f ) < C might suggest:

Corollary 3.8

For any smooth solution f ε , δ : [ 0 , T ) × R / Z → R n of equation (2.2) we obtain

ε ∫ t t + T ∫ R / Z ∣ ∂ s 3 f ε , δ ∣ 2 d s d t ≤ ε 1 5 C ( T + 1 )

where the constant C = C ( p , λ , ℰ ε , δ , λ ( p ) ( f 0 ε , δ ) ) > 0 depends only on p , λ , and ℰ ε , δ , λ ( p ) ( f 0 ε , δ ) .

Proof

Shifting time, we can assume without loss of generality that t = 0 . As the solution satisfies

∂ t f ε , δ = − ∇ L 2 ℰ ε , δ , λ ( p ) ( f ε , δ ) ,

we get integrating inequality (3.12) over time, and using Hölder’s inequality

(3.13) ε ∫ 0 T ‖ ∂ s 3 f ε , δ ‖ B 2 , ∞ 1 4 ( d s ) 2 d t ≤ C ∫ 0 T ( ‖ ∂ t f ε , δ ‖ L 2 ( d s ) + 1 ) d t ≤ C T 1 2 ∫ 0 T ∫ R / Z ∣ ∂ t f ε , δ ∣ 2 d s d t 1 2 + C T ≤ C ( T 1 2 + T ) .

Together with the well-known interpolation estimate for Besov spaces^[2]

‖ ∂ s 3 f ε , δ ‖ L 2 ( d s ) ≤ C ( ℒ ( f ε , δ ) ) ‖ ∂ s 2 f ‖ B 2 , ∞ 1 + 1 4 ( d s ) 4 5 ‖ κ ‖ L 2 ( d s ) 1 5 ,

and the fact that the length of the curves is bounded uniformly from below and above, this implies using Young’s inequality

ε ∫ 0 T ∫ R / Z ∣ ∂ s 3 f ε , δ ∣ 2 d s d t ≤ ε 1 5 C ∫ 0 T ε 4 5 ‖ ∂ s 3 f ε , δ ‖ B 2 , ∞ 1 4 ( d s ) 8 5 + 1 d t ≤ ε 1 5 C ∫ 0 T ε ‖ ∂ s 3 f ε , δ ‖ B 2 , ∞ 1 4 ( d s ) 2 + 1 d t ≤ C ε 1 5 ( T + 1 ) . □

Furthermore, integrating again over the estimate in Corollary 3.7 and using Hölder’s inequality together with the uniform bound on the L p -norm of κ , we obtain the following corollary.

Corollary 3.9

For any solution f ε , δ : [ 0 , T ) × R / Z → R n of equation (2.2), we obtain

∫ t t + T ‖ κ ‖ B p , ∞ 1 2 p ( d s ) p d t ≤ C ( T + 1 )

for any t ≥ 0 , where the constant C = C ( p , λ , ℰ ε , δ , λ ( p ) ( f 0 ε , δ ) ) > 0 depends only on p , λ , and ℰ ε , δ , λ ( p ) ( f 0 ε , δ ) .

4 Convergence to solutions

4.1 The case of smooth initial data

We now show the following version of Theorem 1.1 for smooth initial data. Note that we deal with curves of constant speed in order to go to the limit in the equations.

Theorem 4.1

Given any regular closed curve f 0 : R / Z → R n of class C ∞ parametrized with constant speed, there is a family of regular curves f : [ 0 , ∞ ) × R / Z → R n , f ∈ H 1 ( [ 0 , ∞ ) , L 2 ( R / Z , R n ) ) ∩ L ∞ ( [ 0 , ∞ ) , W 2 , p ( R / Z , R n ) ) ∩ C 1 2 ( [ 0 , ∞ ) , L 2 ( R / Z , R n ) ) solving the initial value problem

∂ t ⊥ f = − ∇ L 2 ℰ ( f ) f ( 0 , ⋅ ) = f 0

in the weak sense, i.e., for all V ∈ C c ∞ ( [ 0 , ∞ ) × R / Z , R n ) , we have

∫ 0 ∞ ∫ R / Z ⟨ ∂ t ⊥ f , V ⟩ d s d t = − ∫ 0 ∞ ( δ V ℰ ( f ) ) d t .

Furthermore, there is a set N ⊂ ( 0 , ∞ ) of Lebesque measure 0 such that for all t 0 , t 1 ∈ [ 0 , 1 ) ⧹ N , t 0 < t 1 , the energy identity

∫ t 0 t 1 ∫ R / Z ∣ ∂ t ⊥ f ∣ 2 d s d t ≤ ℰ ( t 0 ) − ℰ ( t 1 )

holds. The solution satisfies the estimates

‖ ∂ t f ‖ L 2 ( ( 0 , ∞ ) × R / Z , R n ) ≤ C , ‖ κ ‖ L p ( ( t , t + T ) , B p , ∞ 1 2 p ) ≤ C ( T + 1 ) ,

for all t , T > 0 and

‖ f t 1 − f t 0 ‖ L 2 ≤ C ∣ t 1 − t 0 ∣ 1 2

for all t 0 , t 1 ∈ [ 0 , ∞ ) , where C > 0 only depends on ℰ ( f 0 ) , p, and λ .

To construct this solution, we take solutions f ε , δ to the gradient flow of the regularized energies (2.2) with 0 < ε , δ ≤ 1 . We will now carefully reparametrize this family such that each curve is parametrized by constant speed. Since f ε , δ is a solution of the gradient flow of ℰ ε , δ , λ ( p ) , we obtain

∫ 0 ∞ ∫ R / Z ∣ ∂ t f ε , δ ∣ 2 d s d t ≤ ℰ ε , δ , λ ( p ) ( f 0 ε , δ ) .

We now reparametrize the solution by constant speed setting

f ˜ ε , δ ( t , x ) = f ε , δ ( t , σ t ε , δ ( x ) )

where σ t ε , δ is the inverse of the function ϕ t ε , δ ( x ) = 1 ℒ ( f ε , δ ) ∫ 0 x ∣ ( f t ε , δ ( y ) ) ′ ∣ d y . Of course, these reparametrized solutions solve the equation

∂ t ⊥ f ˜ ε , δ = − ∇ L 2 ℰ ε , δ , λ ( p ) ( f ˜ ε , δ ) ,

but they also satisfy

(4.1) ∣ ( f ˜ ε , δ ) ′ ∣ = ℒ ( f ˜ ε , δ ) ,

(4.2) ∂ t ⊥ f ˜ ε , δ ( t , 0 ) = ∂ t f ˜ ε , δ ( t , 0 ) ,

and

(4.3) ∫ 0 ∞ ∣ ∂ t f ˜ ε , δ ( t , 0 ) ∣ 2 d t ≤ C .

We will now show that they furthermore have the following properties.

Lemma 4.2

The reparametrized solutions f ˜ ε , δ are parametrized with constant speed, solve the equation ∂ t ⊥ f ˜ ε , δ = − ∇ L 2 ℰ ε , δ , λ ( p ) ( f ˜ ε , δ ) , and satisfy the following estimates:

ε ∫ 0 T ∫ R / Z ∣ ∇ s κ ∣ 2 d s d t ≤ C ε 1 5 ( T + 1 ) ,
∫ 0 T ‖ κ ‖ B p , ∞ 1 2 p p d t ≤ C ( T + 1 ) ,
∫ 0 ∞ ∫ R / Z ∣ ∂ t f ˜ ε , δ ∣ 2 d x d t ≤ C ,
‖ f ˜ t 1 ε , δ − f ˜ t 0 ε , δ ‖ L 2 ≤ C ∣ t 1 − t 0 ∣ 1 2 for all t 1 , t 0 ∈ [ 0 , ∞ ) with ∣ t 1 − t 0 ∣ ≤ 1 ,

where C > 0 is a constant only depending on the initial energy ℰ ε , δ , λ ( p ) ( f 0 ) , p, λ , and n .

Proof

The estimate (1) and (2) follow directly from Corollaries 3.8 and 3.9, the fact that f ˜ ε , δ have constant speed equal to its length ℒ ( f ˜ ε , δ ) and that this length is bounded from above and below. Differentiating equation (4.1), we obtain

ℒ ( f ˜ ε , δ ) ∂ t ℒ ( f ˜ ε , δ ) = ⟨ ( f ˜ ε , δ ) ′ , ∂ x ∂ t f ˜ ε , δ ⟩ = − ℒ ( f ˜ ε , δ ) 2 ⟨ κ , ∂ t f ˜ ε , δ ⟩ + ∂ x ⟨ ( f ˜ ε , δ ) ′ , ∂ t f ˜ ε , δ ⟩ ,

and hence, using that the length of the curves is bounded from below and above,

∣ ∂ x ⟨ ( f ˜ ε , δ ) ′ , ∂ t f ˜ ε , δ ⟩ ∣ ≤ C ( ∣ ∂ t ℒ ( f ˜ ε , δ ) ∣ + ∣ ⟨ κ , ∂ t ⊥ f ˜ ε , δ ⟩ ∣ ) .

Integrating over this estimate, we obtain by applying the fundamental theorem of calculus, Hölder’s inequality, and (4.2)

∣ ⟨ ( f ˜ ε , δ ) ′ , ∂ t f ˜ ε , δ ⟩ ∣ ≤ C ( ∣ ∂ t ℒ ( f ˜ ε , δ ) ∣ + ‖ κ ‖ L 2 ‖ ∂ t ⊥ f ˜ ε , δ ‖ L 2 ) + ∣ ⟨ ( f ˜ ε , δ ) ′ ( t , 0 ) , ∂ t f ˜ ε , δ ( t , 0 ) ⟩ ∣ ≤ C ( ∣ ∂ t ℒ ( f ˜ ε , δ ) ∣ + ‖ κ ‖ L 2 ‖ ∂ t ⊥ f ˜ ε , δ ‖ L 2 ) .

Combined with

∣ ∂ t ℒ ( f ˜ ε , δ ) ∣ = ∫ R / Z ⟨ κ , ∂ t ⊥ f ˜ ε , δ ⟩ d s ≤ ‖ κ ‖ L 2 ‖ ∂ t ⊥ f ˜ ε , δ ‖ L 2 ≤ C ‖ ∂ t ⊥ f ˜ ε , δ ‖ L 2 ,

this gives

∣ ∂ t T f ˜ ε , δ ∣ ≤ C ( ‖ ∂ t ⊥ f ˜ ε , δ ‖ L 2 + ‖ κ ‖ L 2 ‖ ∂ t ⊥ f ˜ ε , δ ‖ L 2 ) .

Integrating over space and time and using Hölder’s inequality, we hence obtain

∫ 0 ∞ ∫ R / Z ∣ ∂ t T f ˜ ε , δ ∣ 2 d s d t ≤ C ∫ 0 ∞ ∫ R / Z ∣ ∂ t ⊥ f ˜ ε , δ ∣ 2 d s d t ≤ C .

As d s = ℒ d x and the length is bounded from below, this proves property (3).

We can derive the Hölder estimate (4) using a standard estimate for L 2 -gradient flows. Differentiating the quantity ∫ R / Z ∣ f ˜ t ε , δ − f ˜ t 0 ε , δ ∣ 2 d x for a fixed time t 0 ∈ ( 0 , ∞ ) , we obtain that

d d t ‖ f ˜ t ε , δ − f ˜ t 0 ε , δ ‖ L 2 2 = 2 ∫ R / Z ⟨ f ˜ t ε , δ − f ˜ t 0 ε , δ , ∂ t f ˜ ε , δ ⟩ d x ≤ 2 ‖ f ˜ t ε , δ − f ˜ 0 ε , δ ‖ L 2 ‖ ∂ t f ˜ ε , δ ‖ L 2 .

Hence, by the fundamental theorem of calculus,

‖ f ˜ t ε , δ − f ˜ t 0 ε , δ ‖ L 2 ≤ C ∣ t − t 0 ∣ 1 2 .

So also (4) is proven.□

It is now straightforward to prove convergence of the solutions f ˜ ε , δ to a weak solution that has all the properties mentioned in Theorem 4.1. The essential ingredient is that we can choose a subsequence converging in the Bochner space L p ( ( 0 , T ) , W 2 , p ( R / Z , R n ) ) for all T > 0 .

Proof of Theorem 4.1

Let ε n , δ n → 0 and let us set f ( n ) = f ˜ ε n , δ n . By Lemma 4.2, (3), the time derivatives are uniformly bounded in L 2 ( ( 0 , ∞ ) × R / Z , R n ) . Choosing an appropriate subsequence, we can assume that ∂ t f ( n ) converges to ∂ t f weakly in L 2 ( ( 0 , ∞ ) × R / Z , R n ) . Using that the solutions f ( n ) are uniformly bounded in L ∞ W 2 , p , the compact embedding W 2 , p ( R / Z , R n ) ↪ L 2 ( R / Z , R n ) , and a standard diagonal sequence argument, we can furthermore assume after going to a subsequence that f ( n ) converges in L 2 for all times t ∈ Q ∩ ( 0 , ∞ ) . Due to the uniform control of the Hölder constant (4) in Lemma 4.2, this subsequence then also converges locally in L ∞ L 2 to f : [ 0 , ∞ ) × R / Z → R n . Interpolating once more, using that the W 2 , p -norm of the curves is uniformly bounded, we also obtain convergence in L ∞ W 1 , ∞ locally in time.

Applying Ehrling’s lemma to the compact embeddings

B p , ∞ 2 + 1 2 p ( R / Z ) → W 2 , p ( R / Z ) → L 2 ( R / Z )

for every ε > 0 , we obtain an C ε such that

‖ f ( n ) − f ( m ) ‖ W 2 , p ≤ ε ‖ f ( n ) − f ( m ) ‖ B p , ∞ 2 + 1 2 p + C ε ‖ f ( n ) − f ( m ) ‖ L 2 ≤ ε ‖ ∂ x 2 ( f ( n ) − f ( m ) ) ‖ B p , ∞ 1 2 p + C ε ‖ f ( n ) − f ( m ) ‖ L 2 .

If we integrate over this interpolation estimate and use property (2) of Lemma 4.2 and that the f ( n ) converge to f in L ∞ L 2 , we see that

∫ 0 T ∫ R / Z ‖ f ( n ) − f ( m ) ‖ W 2 , p p d s d t ≤ ε C ( T + 1 ) + C ε T ‖ f ( n ) − f ( m ) ‖ L ∞ ( ( 0 , T ) , L p ) p → ε C ( T + 1 )

as n and m go to ∞ . As this hold for all ε > 0 , this implies that the f ( n ) even converge in L p W 2 , p ( ( 0 , T ) × R / Z ) to f for all T > 0 . Furthermore, we know from property (1) of Lemma 4.2 that

(4.4) ε n ∫ 0 T ∫ R / Z ∣ ∇ s κ ∣ 2 d s d t ≤ C ε n 1 5 → 0 .

From the evolution equation, we see that for all test functions V ∈ C c ∞ ( ( 0 , ∞ ) × R / Z , R n ) , we have

(4.5) ∫ 0 ∞ ∫ R / Z ⟨ ∂ t ⊥ f ( n ) , V ⟩ d s d t = ∫ 0 ∞ ( ε n δ V ℱ ( f ( n ) ) + δ V ℰ δ n ( p ) ( f ( n ) ) + λ δ V ℒ ( f ( n ) ) ) d t .

As ∂ t f ( n ) converges weakly to ∂ t f in L 2 , τ n converges strongly to τ in L ∞ , and ∂ t ⊥ f ( n ) = ∂ t f ( n ) − ⟨ ∂ t f ( n ) , τ n ⟩ τ n , we obtain

∫ 0 ∞ ∫ R / Z ⟨ ∂ t ⊥ f ( n ) , V ⟩ d s d t → ∫ 0 ∞ ∫ R / Z ⟨ ∂ t ⊥ f , V ⟩ d s d t

as n goes to ∞ . The first variation of ℱ is given by

δ V ℱ ( f ) = ∫ R / Z ⟨ ∇ s κ , δ V ( ∇ s κ ) ⟩ d s + 1 2 ∫ R / Z ∣ ∇ s κ ∣ 2 ⟨ τ , ∂ s V ⟩ d s ,

where

δ V ( ∇ s κ ) = ( δ V ( ∇ s κ ) ) ⊥ = ( ∂ s 3 V + ∂ s 2 V ∗ τ ∗ κ + ∂ s V ∗ ( τ ∗ ∂ s κ + κ ∗ κ ) ) ⊥

(cf. equations (3.1) and (3.4)). Hence, by using Hölder’s inequality together with the multiplicative interpolation estimates in Lemma 2.7, we obtain

ε n ∣ δ V ℱ ( f ( n ) ) ∣ ≤ ε n C ( ‖ ∂ s 3 f ( n ) ‖ L 2 + ‖ ∂ s 3 f ( n ) ‖ L 2 ( ‖ κ n ‖ L 4 2 + ‖ κ n ‖ L 2 ) ) ≤ ε n C ( ‖ ∂ s 3 f ( n ) ‖ L 2 + ‖ ∂ s 3 f ( n ) ‖ L 2 3 2 + 1 ) ≤ ε n C ( ‖ ∂ s 3 f ( n ) ‖ L 2 2 + 1 ) .

Combining this with (4.4), we obtain

(4.6) ε n ∫ 0 ∞ δ V ℱ ( f ( n ) ) d t → 0

as n goes to ∞ .

To obtain control of the first variation of ℰ δ n ( p ) , we first observe that after taking a subsequence, we can furthermore assume that κ n converges to κ almost everywhere in space and time. By using the convexity of x → x p 2 for p > 2 , we see that

( ∣ κ n ∣ 2 + δ n 2 ) p 2 ≤ 2 p 2 − 1 ( ∣ κ n ∣ p + δ n p ) .

Since ∣ κ n ∣ p is uniformly integrable on [ 0 , T ) × R / Z for all T > 0 as it converges to ∣ κ ∣ p in L 1 ( [ 0 , T ) × R / Z ) , also

( ∣ κ n ∣ 2 + δ n 2 ) p 2

is uniformly integrable on this domain. As this expression converges pointwise almost everywhere to ∣ κ ∣ p as n → ∞ , we obtain again applying Vitali’s theorem that

( ∣ κ n ∣ 2 + δ n 2 ) p 2 → ∣ κ ∣ p

in L 1 ( ( 0 , T ) × R / Z ) as n → ∞ for all T > 0 . Similarly,

( ∣ κ n ∣ 2 + δ n 2 ) p 2 − 1 κ n → ∣ κ ∣ p − 2 κ

in L 1 ( ( 0 , T ) × R / Z ) as n goes to ∞ for all T > 0 . From

δ V ℰ δ ( p ) ( f ) = ∫ R / Z ( ∣ κ ∣ 2 + δ 2 ) p − 2 2 ⟨ κ , δ V κ ⟩ d s + 1 p ∫ R / Z ( ∣ κ ∣ 2 + δ 2 ) p 2 ⟨ τ , ∂ s V ⟩ d s

and

δ V κ = ( ∂ s 2 V ) ⊥ − 2 ⟨ ∂ s V , τ ⟩ κ − ⟨ ∂ s V , κ ⟩ τ

(cf. equation (3.2)) and since τ n converges to τ in L ∞ ( ( 0 , T ) × R / Z ) , we thus deduce that

(4.7) δ V ℰ δ n ( p ) ( f ( n ) ) → δ V ℰ ( p ) ( f )

as n → ∞ in L 1 ( ( 0 , ∞ ) ) . From

δ V ℒ ( f ) = ∫ R / Z ⟨ τ , ∂ s V ⟩ d s

and the fact that τ n converges to τ in L ∞ ( [ 0 , T ) × R / Z ) for all T > 0 , we see that

(4.8) δ V ℒ ( f ( n ) ) → δ V ℒ ( f )

in L 1 ( 0 , T ) for all T > 0 . By using (4.6), (4.7), and (4.8), we can let n go to infinity in (4.5) to obtain

∫ 0 ∞ ∫ R / Z ⟨ ∂ t ⊥ f , V ⟩ d s d t = ∫ 0 ∞ δ V ( ℰ ( p ) ( f ) + λ ℒ ( f ) ) d t = ∫ 0 ∞ δ V ℰ ( f ) d t

for all test functions V ∈ C c ∞ ( ( 0 , ∞ ) × R / Z , R n ) . Furthermore, the estimates (1), (2), (3), and (4) of Lemma 4.2 together with the fact that ℰ ε , δ , λ ( p ) ( f 0 ) converges to ℰ ( f 0 ) as ε , δ → 0 immediately give the estimates mentioned in the theorem.

Let us finally prove the energy inequality. Testing the evolution equation for the f ( n ) with ∂ t f ( n ) , one obtains that

(4.9) ∫ t 0 t 1 ∫ R / Z ∣ ∂ t ⊥ f ( n ) ∣ 2 d s d t ≤ ℰ ε , δ , λ ( p ) ( ( f ( n ) ) t 0 ) − ℰ ε , δ , λ ( p ) ( ( f ( n ) ) t 1 ) ,

since the solution is smooth in space and time.

We first show that the right-hand side of this inequality converges to the right-hand side of the energy inequality except for a null-set of times. By using Fischer-Riesz’s theorem, we can assume after going to a subsequence that f t ( n ) converges to f t in W 2 , p on all t ∈ [ 0 , ∞ ) except for a null-set N ⊂ [ 0 , ∞ ) . Arguing as above, we can deduce that

ℰ δ n ( p ) ( f t ( n ) ) → ℰ ( p ) ( f t )

for all such t . As ∫ 0 T ε n ℱ ( f t ( n ) ) d t → 0 , we can furthermore assume – after thinning out the sequence once more if necessary – that ε n ℱ ( f t ( n ) ) → 0 for almost all times t > 0 , and hence,

ℰ ε n , δ n , λ ( p ) ( f t ( n ) ) → ℰ ( f t )

for all t ∈ [ 0 , ∞ ) ⧹ N . Thus, the right-hand side of (4.9) converges to the right-hand side of the energy inequality we want to prove except for a null set of times N ⊂ ( 0 , ∞ ) .

We conclude the proof showing that if we let n tend to ∞ , the left-hand side can in (4.9) only drop. From

∂ t ⊥ f ( n ) = ∂ t f ( n ) + ⟨ ∂ t f ( n ) , τ n ⟩ τ n ,

we deduce that ∂ t ⊥ f ( n ) converges to ∂ t ⊥ f weakly in L 2 ( ( t 0 , t 1 ) × R / Z ) since we know that ∂ t f ( n ) converges weakly in L 2 ( ( t 0 , t 1 ) × R / Z ) to ∂ t f and that f ( n ) converges in L ∞ ( ( t 0 , t 1 ) × W 1 , ∞ ( R / Z ) ) to f . Hence, we obtain

∫ t 0 t 1 ∫ R / Z ⟨ ∂ ⊥ f , ∂ ⊥ f ( n ) ⟩ d s d t → ∫ t 0 t 1 ∫ R / Z ∣ ∂ ⊥ f ∣ 2 d s d t

as n → ∞ . Combining this with Cauchy’s inequality, we obtain

liminf n → ∞ ‖ ∂ t ⊥ f ( n ) ‖ L 2 ( ( t 0 , t 1 ) × L 2 ( d s ) ) ‖ ∂ t ⊥ f ‖ L 2 ( ( t 0 , t 1 ) × L 2 ( d s ) ) ≥ ‖ ∂ t ⊥ f ‖ L 2 ( ( t 0 , t 1 ) × R / Z ) 2

and hence,

∫ t 0 t 1 ∫ R / Z ∣ ∂ t ⊥ f ∣ 2 d s d t ≤ liminf n → ∞ ∫ t 0 t 1 ∫ R / Z ∣ ∂ t ⊥ f ( n ) ∣ 2 d s d t .

Going to the limit n → ∞ in equation (4.9), we hence obtain the desired energy inequality for all t 0 , t 1 ∈ [ 0 , ∞ ) ⧹ N . □

4.2 Arbitrary initial data

For an arbitrary initial regular curve f 0 of class W 2 , p , we pick a sequence of smooth regular curves f 0 ( n ) ∈ C ∞ converging to f 0 in W 2 , p with sup n ℰ ( f 0 ( n ) ) ≤ 2 ℰ ( f 0 ) . By using Theorem 4.1, we obtain a weak solution f ( n ) of the initial value problem

∂ t ⊥ f ( n ) = − ∇ L 2 ℰ ( f ( n ) ) f ( n ) ( 0 , ⋅ ) = f 0 ( n )

such that f ( n ) ( t , ⋅ ) is parametrized with constant speed. Furthermore, these solutions satisfy the estimates

‖ ∂ t f ( n ) ‖ L 2 ( ( 0 , ∞ ) × R / Z ) ≤ C , ‖ κ n ‖ L p ( ( t , t + T ) , B p , ∞ 1 2 p ) ≤ C ( T + 1 )

and

‖ f t 1 ( n ) − f t 0 ( n ) ‖ L 2 ≤ C ∣ t 1 − t 0 ∣ 1 2 for all t 1 , t 0 ∈ [ 0 , ∞ ) with ∣ t 1 − t 0 ∣ ≤ 1 ,

where C > 0 only depends on ℰ ( f 0 ) as sup n ℰ ( f 0 ( n ) ) ≤ 2 ℰ ( f 0 ) .

Following the line of arguments in the proof of Theorem 4.1, we will now pick a suitable subsequence and then let n go to infinity in the evolution equations for f ( n ) .

We can again assume that f ( n ) → f strongly in L loc p ( [ 0 , ∞ ) , W 2 , p ( R / Z ) ) to f and ∂ t f ( n ) converges weakly locally in L 2 ( [ 0 , ∞ ) , L 2 ( R / Z ) ) to ∂ t f . From the evolution equation, we see that for all test functions V ∈ C c ∞ ( ( 0 , ∞ ) × R / Z , R n ) , we have

(4.10) ∫ 0 ∞ ∫ R / Z ⟨ ∂ t ⊥ f ( n ) , V ⟩ d s d t = ∫ 0 ∞ ( δ V ℰ ( p ) ( f ( n ) ) + λ δ V ℒ ( f ( n ) ) ) d t .

As ∂ t f ( n ) converges weakly to ∂ t f in L 2 ( ( 0 , ∞ ) × R / Z ) , τ n converges strongly to τ in L ∞ , and ∂ t ⊥ f ( n ) = ∂ t f ( n ) − ⟨ ∂ t f ( n ) , τ n ⟩ τ n , we obtain

∫ 0 ∞ ∫ R / Z ⟨ ∂ t ⊥ f ( n ) , V ⟩ d s d t → ∫ 0 ∞ ∫ R / Z ⟨ ∂ t ⊥ f , V ⟩ d s d t

as n goes to ∞ . From

δ V ℰ ( p ) ( f ) = ∫ R / Z ∣ κ ∣ p − 2 ⟨ κ , δ V κ ⟩ d s + 1 p ∫ R / Z ∣ κ ∣ p ⟨ τ , ∂ s V ⟩ d s

and

δ V κ = ( ∂ s 2 V ) ⊥ − 2 ⟨ ∂ s V , τ ⟩ κ − ⟨ ∂ s V , κ ⟩ τ ,

we thus deduce that

δ V ℰ ( p ) ( f ( n ) ) → δ V ℰ ( p ) ( f )

as n → ∞ in L loc 1 ( ( 0 , ∞ ) ) . From

δ V ℒ ( f ) = ∫ R / Z ⟨ τ , ∂ s V ⟩ d s ,

we obtain

δ V ℒ ( f ( n ) ) → δ V ℒ ( f ) .

Hence, we can let in equation (4.10) n go to infinity to obtain

∫ 0 ∞ ∫ R / Z ⟨ ∂ t ⊥ f , V ⟩ d s d t = ∫ 0 ∞ ( δ V ℰ ( p ) ( f ) + λ δ V ℒ ( f ) ) d t

for all test functions V ∈ C c ∞ ( ( 0 , ∞ ) × R / Z , R n ) . Furthermore, we have

‖ ∂ t f ‖ L 2 ( ( 0 , ∞ ) × R / Z ) ≤ C and ‖ κ ‖ L p ( ( t , t + T ) , B p , ∞ 1 2 p ) ≤ C ( T + 1 ) .

Also the energy inequality can be shown along the lines of the proof of Theorem 4.1 going to the limit in the energy inequality for the f ( n ) using the convergence in L p ( ( t 0 , t 1 ) , W 2 , p ( R / Z , R n ) ) to deal with the right-hand side and the weak convergence of ∂ t f ( n ) in L 2 ( ( t 0 , t 1 ) × R / Z ) to see that the left-hand side can only drop.

5 Asymptotics of the solution

We use the last estimates to choose t n → ∞ such that both ‖ ∂ t f ‖ L 2 ( [ t n , t n + 1 ] × R / Z ) → 0 and ‖ κ ‖ L p ( [ t n , t n + 1 ] , B p , ∞ 1 2 p ) is uniformly bounded. We now set

f ( n ) : [ 0 , 1 ] × R / Z → R n , f ( n ) ( t , x ) = f ( t n + t , x ) − f ( t n , 0 ) .

Again we can assume after going to a subsequence that f ( n ) → f strongly in L p ( [ 0 , 1 ] , W 2 , p ( R / Z ) ) . From the evolution equation, we see that for all test functions V ∈ C c ∞ ( ( 0 , 1 ) × R / Z , R n ) , we have

(5.1) ∫ 0 1 ∫ R / Z ⟨ ∂ t ⊥ f ( n ) , V ⟩ d s d t = ∫ 0 1 ( δ V ℰ ( p ) ( f ( n ) ) + λ δ V ℒ ( f ( n ) ) ) d t .

We will now for a third time show that we can go to the n → ∞ in this equation. As ∂ t f ( n ) converges to 0 in L 2 , τ n is bounded, and ∂ t ⊥ f ( n ) = ∂ t f ( n ) − ⟨ ∂ t f ( n ) , τ n ⟩ τ n , we obtain

∫ 0 1 ∫ R / Z ⟨ ∂ t ⊥ f ( n ) , V ⟩ d s d t → 0

as n goes to ∞ . From

δ V ℰ ( p ) ( f ) = ∫ R / Z ∣ κ ∣ p − 2 ⟨ κ , δ V κ ⟩ d s + 1 p ∫ R / Z ∣ κ ∣ p ⟨ τ , ∂ s V ⟩ d s

and

δ V κ = ( ∂ s 2 V ) ⊥ − 2 ⟨ ∂ s V , τ ⟩ κ − ⟨ ∂ s V , κ ⟩ τ

together with the convergence of κ n to κ in L p ( [ 0 , T ) × R / Z ) and of τ n to τ in L p ( [ 0 , T ) × R / Z ) for all T > 0 , we deduce that

δ V ℰ ( p ) ( f ( n ) ) → δ V ℰ ( p ) ( f )

as n → ∞ in L 1 ( ( 0 , ∞ ) ) . From

δ V ℒ ( f ) = ∫ R / Z ⟨ τ , ∂ s V ⟩ d s ,

we derive

δ V ℒ ( f ( n ) ) → δ V ℒ ( f ) .

Hence, we can let n go to infinity in equation (5.1) to obtain

(5.2) 0 = ∫ 0 1 ( δ V ℰ ( p ) ( f ) + λ δ V ℒ ( f ) ) d t

for all test functions V ∈ C c ∞ ( ( 0 , 1 ) × R / Z , R n ) .

For V ˜ ∈ C ∞ ( R / Z , R n ) and a nonvanishing ϕ ∈ C c ∞ ( ( 0 , 1 ) , [ 0 , 1 ] ) , we use equation (5.2) with V ( t , x ) = ϕ ( t ) V ( x ) ˜ to obtain

0 = ∫ 0 1 ϕ d t ( δ V ˜ ℰ ( p ) ( f ) + λ δ V ˜ ℒ ( f ) ) .

Hence, we finally obtain

δ V ˜ ℰ ( p ) ( f ) + λ δ V ˜ ℒ ( f ) = 0

for all V ˜ ∈ C ∞ ( R / Z , R n ) , as ∫ 0 1 ϕ d t > 0 . So f is a critical point of ℰ . This finishes the proof of Theorem 1.1.

Acknowledgments

Special thanks to Armin Schikorra for pointing the authors to the approach to higher differentiability in [2].

Funding information: The authors acknowledges support by the Austrian Science Fund (FWF), Grant P 29487.
Conflict of interest: The authors state no conflict of interest.

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Received: 2021-04-28

Revised: 2021-12-16

Accepted: 2022-02-04

Published Online: 2022-05-05

This work is licensed under the Creative Commons Attribution 4.0 International License.

A regularized gradient flow for the p-elastic energy

Abstract

1 Introduction

Theorem 1.1[1]

2 The regularized equations

Theorem 2.1

Remark 2.2

2.1 Equations of evolution and inequalities

Lemma 2.3

Lemma 2.4

Proof

Lemma 2.5

Proof

Lemma 2.6

Proof

Lemma 2.7

Lemma 2.8

Proof

Lemma 2.9

Proof

Lemma 2.10

Lemma 2.11

2.2 Short-time existence

2.3 Long-time existence and asymptotic behavior

3 Higher order estimates for the regularized energies

3.1 The first variation of the energy

Lemma 3.1

Proof

3.2 An a priori estimate

Theorem 3.2

Lemma 3.3

Proof

Lemma 3.4

Proof

Lemma 3.5

Proof

Lemma 3.6

Proof

Proof of Theorem 3.2

Corollary 3.7

Corollary 3.8

Proof

Corollary 3.9

4 Convergence to solutions

4.1 The case of smooth initial data

Theorem 4.1

Lemma 4.2

Proof

Proof of Theorem 4.1

4.2 Arbitrary initial data

5 Asymptotics of the solution

Acknowledgments

References

Journal and Issue

Articles in the same Issue

Theorem 1.1^[1]