Banach gradient flows for various families of knot energies

We establish long-time existence of Banach gradient flows for generalised integral Menger curvatures and tangent-point energies, and for O’Hara’s self-repulsive potentials Eα,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E^{\alpha ,p}$$\end{document}. In order to do so, we employ the theory of curves of maximal slope in slightly smaller spaces compactly embedding into the respective energy spaces associated to these functionals and add a term involving the logarithmic strain, which controls the parametrisations of the flowing (knotted) loops. As a prerequisite, we prove in addition that O’Hara’s knot energies Eα,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E^{\alpha ,p}$$\end{document} are continuously differentiable.


Introduction
It is an interesting and analytically challenging problem in geometric knot theory to evolve knots according to the gradient flow of a self-repelling interaction energy.Such energies are called knot energies, and one may categorise them into two types.for α > 0 and p > 1 satisfying 2 ≤ αp < 2p + 1.Here, d γ (x, y) denotes the intrinsic distance between the points γ(x) and γ(y) along the curve.Zh.-X.He [He00] had shown short-time existence for the L2 -gradient flow of the Möbius1 energy E 2,1 for smooth initial data, before S. Blatt investigated this flow systematically for the subfamily E α,1 .He established long-time existence results and convergence to a critical point: for α ∈ (2, 3) in [Bla18], and for α = 2 and initial data sufficiently close to a local minimiser in [Bla12c].Moreover, again for E 2,1 he proved an ε-regularity result together with a blow-up analysis in [Bla20], resulting in the convergence to the round circle if one restricts the L 2 -flow to planar loops.According to [BR13,p. 31] such strong results on the L 2 -flow for the general energy family E α,p may be out of reach because of a degenerate elliptic operator in the first variation formula for E α,p .As shown in [Bla12a] the underlying energy space of the Möbius energy E2,1 is a fractional Sobolev space, the Sobolev-Slobodeckiǐ 2 space W 3 2 ,2 (R/Z, R n ), which is a Hilbert space.This fact was recently used by Ph.Reiter and H. Schumacher to establish short-time existence for a Hilbert gradient flow in that space for E 2,1 with a method, however, that seems to be restricted to E 2,1 ; see [RS21, Remark after Theorem 1.2].
It is the purpose of the present paper to prove long-time existence results for the gradient flows of the three two-parameter energy families E α,p , intM (p,q) , and TP (p,q)  for the respective complete range of parameters given in (1.1), (1.4), and (1.5).Since the underlying energy spaces are Sobolev-Slobodeckiǐ spaces that are in general only Banach spaces, we employ the general metric gradient approach of Ambrosio et al. [AGS08] to construct curves of maximal slope, which turn out to be solutions of the doubly-nonlinear gradient flow equation.In order to do this, we need to control the curves' parametrisations along the flow, and for that we use the same constraint as in [KSSvdM22] but in a different manner.We add to the knot energy a suitable norm of the logarithmic strain Σ(γ) := log |γ | (1.6) as a lower order penalty term, instead of projecting onto the null space of its differential DΣ [γ] as in [KSSvdM22].As a second ingredient we restrict the total energy to a reflexive and uniformly convex Banach space C compactly embedded in the respective energy space B of closed curves, to account for the quite restrictive assumptions in the general existence theory for curves of maximal slope.
To summarise these ideas, given any knot energy E : B → (−∞, ∞], and some number κ > 1, we consider the total energy φ : C → (−∞, ∞] defined as φ(γ) := E(γ) + Σ(γ) κ A , if γ ∈ C is regular and injective +∞, else, (1.7) where the Banach space A consists of real-valued functions with exactly one order lower in differentiability than the curves γ ∈ B, since the scalar-valued logarithmic strain consumes one derivative.By means of the θ-duality mapping J C,θ : C → 2 C * with θ ∈ (1, ∞) defined by where C * denotes the dual space of C, we state our main result.
Remark 1.3.1.Notice that in general the duality mapping J C,θ defined in (1.8) is setvalued, so that one would expect to at most solve the differential inclusion −Dφ[u(•)] ∈ J C,θ (u (•)) on [0, ∞) instead of the gradient flow equation (1.9).The specific properties of the Banach space C := C ε in Theorem 1.1, however, imply that J Cε,θ is not only single-valued, but also a homeomorphism between C ε and C * ε .2. The choice of the Banach space B in the alternatives (i), (ii), and (iii) of Theorem 1.1 is maximal in the following sense: A regular injective curve γ ∈ B has finite energy E, and, on the other hand, if a regular injective C 1 -curve γ satisfies E(γ) < ∞ then its suitably rescaled arc length parametrisation is contained in B; see [ 3.There is complete freedom in the choice of the parameters κ, θ ∈ (1, ∞), and ε > 0 in Theorem 1.1.For κ = 1 we still obtain a curve of maximal slope for the total energy φ with respect to a strong upper gradient which, however, does not coincide with Dφ C * any more; see Proposition 3.3.But it is presently unclear if that curve of maximal slope also solves a differential inclusion.
The limiting process, ε 0, on the other hand, approximating the correct energy space B for the respective knot energy E in cases (i)-(iii) of Theorem 1.1, yields a subsequence of solutions of (1.9) that converges pointwise weakly to a limit mapping u * ∈ AC θ ([0, ∞), B), provided that the initial curves γ 0,ε are well-prepared; see Corollary 4.3 and the more general result Proposition 2.2 for metric spaces, which is similar in spirit as [Ser11, Theorem 2], where a more general limiting process is investigated.There, however, the existence of the limiting curve is assumed, see also [Ser11,Remark 1].While we know that the energy does not exceed its initial value φ(u * (0)), it is unclear whether u * is a curve of maximal slope in the limiting energy space B. It would be if we had a weakly lower semicontinuous strong upper gradient for φ as shown in Lemma 2.3, a condition that is also used in [BCGS16, Theorem 2.5] and in [Ser11,Theorem 2].
In addition, multiplying the term Σ(γ) κ B in (1.7) with a small prefactor ϑ > 0 and sending ϑ to zero gives rise to yet another interesting limiting process, similarly as in [GRvdM17], where such a procedure was used to study elastic knots.
4. We do not know at this point if the solution u(•) of (1.9) is unique.Moreover, it is open whether u(t) subconverges or even converges to a critical point of the total energy φ as t → ∞, not to speak of any information about convergence rates.These convergence issues would possibly require to study the second variation of the total energy φ and a suitable Łojasiewicz-Simon inequality as carried out for the L 2 -flow of the Möbius energy E 2,1 in [Bla12c, Sections 4 & 5]; see also the initial analysis of the kernel of the second variation of E 2,1 in [Bie21, Section 3].
5. Let us finally mention why we preferred the metric gradient approach to the study of ordinary differential equations in Banach spaces.First of all, the Picard-Lindelöf theory requires Lipschitz continuity of the right-hand side of the equation.This seems out of reach in the present context, where the duality mapping J C,θ fails to be Lipschitz unless the underlying Banach space is Lindenstrauß convex [Zem91].Indeed, here we deal with Sobolev-Slobodeckiǐ spaces, and the Lindenstrauß convexity requires an integrability that is at most quadratic; see [Byn76,Theorem 7] in combination with [CR15, Proposition 3.6].However, more general existence results with a compact operator on the right-hand side such as [Mar76, Ch.VI, Thm.3.1] could probably be used to obtain at least short-time existence.To extend this to long-time existence would then require further estimates which do not seem to provide a short cut.An interesting alternative approach could be a vanishing viscosity method as performed recently for the p-curvature integral by Blatt and N. Vorderobermeier, and this might lead to stronger results; see [BHV21] in comparison to [BVH21].
The paper is structured as follows.In Section 2 we recall the basic notions of the metric gradient flow approach following [AGS08] but slightly adapted to our context.There, we also also revisit how these notions manifest in Banach spaces.In Section 3 we prove an abstract existence theorem for curves of maximal slope for the total energy φ in (1.7) under certain assumptions on an otherwise arbitrary knot energy E; see Theorem 3.2.We also treat in Proposition 3.3 the limiting case κ = 1.In Section 4 we verify in detail the assumptions of Theorem 3.2 for the three energy families E α,p , intM (p,q) , and TP (p,q) , thus proving Theorem 1.1.One of these assumptions is the continuous differentiability of the knot energy, which is known for the generalised integral Menger curvature and tangent-point energies, but -to the best of our knowledge -for O'Hara's energies E α,p so far only for p = 1; see [BR13, Theorem 1.1].Kawakami and Nagasawa [KN20] established L 1 -bounds for the integrands of the first and second variation of a variant of O'Hara's energy that coincides with E α,p only on curves parametrised by arc length.Since we need bounds for more general parametrisations and their ansatz seems non-transferable, we have included a full proof of continuous differentiability of E α,p in Section 5; see Theorem 5.1, which may be of independent interest.Let us add that the limiting process ε → 0 for solutions of (1.9) is treated in Proposition 2.2 in the general metric setting, and specified in Corollary 4.3 to the situation described in Theorem 1.1.The appendix contains some technical material on the geometry of Sobolev-Slobodeckiǐ spaces and some differentiation rules.

Curves of Maximal Slope in Metric Spaces
Given a complete metric space (S , d), an interval I ⊂ R, and a number θ ≥ 1, a θabsolutely continuous curve is a curve u : I → S such that there exists a map m ∈ L θ (I) with the property d u(s), u(t) ≤ t s m(r) dr for all s, t ∈ I with s < t.
Notice that this notion of a strong upper gradient modifies slightly that of [AGS08, Definition 1.2.1] in that inequality (2.1) is only required for absolutely continuous curves whose image is contained in the effective domain D(φ).This restriction does not affect Theorem 2.1 below upon which our existence results build.In the situations encountered in this work, the local slope turns out to be a strong upper gradient.
With these notions at hand, one can define a θ-curve of maximal slope for φ with respect to the strong upper gradient g starting at u 0 ∈ D(φ).It is a curve u ∈ AC θ loc [0, ∞), S that satisfies u(0) = u 0 and the energy dissipation equality implies that (2.2) holds.In that case, we also have the identities In fact, one can also require (2.2) only to hold with "≤" instead of equality, since the converse inequality is always satisfied as a consequence of Young's inequality and (2.1).
A common way to obtain a curve of maximal slope is by carrying out three steps.Firstly, for a given step size, a discretised version of the energy dissipation equation is solved by iteratively solving minimisation problems associated with the step size.Secondly, using an interpolation method, each of the thus obtained piecewise constant solutions is transformed into a continuous curve.These curves form a relatively compact set and a limit curve is extracted.This limit curve is commonly referred to as a (generalised) minimising movement and satisfies a weaker form of the energy dissipation equality.Thirdly, it remains to check that the minimising movement in fact also satisfies the energy dissipation equality.General assumptions on the functional have been formulated that ensure that each of the previous steps can be executed.We state these assumptions collected from [AGS08, Section 2.1 & Remark 2.3.4] and the corresponding existence theorem.
Assumption (M) on the metric space.The complete metric space (S , d) is endowed with an additional weak topology σ which is Hausdorff, weaker than the topology induced by the metric d, and such that d is sequentially weakly lower semi-continuous5 , i.e., for all sequences (Φ2) The functional φ is coercive in the sense that there exists u * ∈ S , B ∈ R, and C > 0 such that φ(u) ≥ B − Cd(u, u * ) θ for all u ∈ S .

Assumptions on the functional
(Φ3) Every d-bounded subset of a sublevel set of φ is relatively weakly sequentially compact, i.e., for a sequence (Φ4) The local slope |∂φ| of φ is weakly sequentially lower semi-continuous on d-bounded subsets of sublevel sets of φ, that is, for every sequence (Φ5) The local slope of φ is a strong upper gradient on6 D(φ).
Proof.Let us first fix some notation.For a given metric space S ∈ {S 0 , S ε }, a locally absolutely continuous curve v ∈ AC loc ([0, ∞), S ) and a point v ∈ S , we denote by |v | S (r) the metric derivative of v at r ≥ 0 taken with respect to the metric on S .Moreover, since we are only interested in small ε, we may assume that sup ε>0 φ(u 0,ε ) < ∞.
The arguments presented here are continuous analogues of those used in [AGS08, Lemma 3.2.2]and [AGS08, Corollary 3.3.4].The existence of a converging subsequence relies on the theorem of Arzelà-Ascoli [AGS08, Proposition 3.3.1].We will check its prerequisites, which are consequences of the energy dissipation equalities From the assumptions on the metrics d 0 , d ε , it follows that for almost every r ≥ 0. Notice that Assumption (Φ2) also holds for u * = u 0 , possibly with different constants.We deduce from Assumption (Φ2) and (2.5) with s = 0 that On the other hand, since u ε and d 0 (u ε (•), u 0 ) are locally absolutely continuous, we have , and recalling that β = θ θ−1 , we obtain for almost every r ≥ 0 Therefore, Choosing, e.g., δ = (2Cc θ 0 ) −1 , and using (2.4), we deduce for some constants c 1 , c 2 > 0 independent of ε and t the inequality It follows from Gronwall's inequality [Hal80, Corollary 6.6] that (2.8) Now fix T > 0. It follows immediately from (2.5) and from (2.8) that (2.9) Therefore, the maps u ε [0,T ] take their values in a d 0 -bounded sublevel set of φ, which, by Assumption (Φ3) is relatively weakly sequentially compact.Moreover, a consequence of (2.6) and (2.8) with t = T is that dr|, we deduce from (2.11) and (2.9) that there exists a map u * T : [0, T ] → S 0 , such that, up to a subsequence, (2.12) By a diagonal argument, we find u * : [0, ∞) → S 0 and A ∈ L θ loc ([0, ∞)) such that, up to another subsequence, (2.12) holds with u * for every t ≥ 0 and such that for every T > 0 there holds u * = u * T and A = A T almost everywhere on [0, T ].Since the metric d 0 is sequentially weakly lower semicontinuous and because of (2.11) and (2.12), we deduce that for all 0 ≤ s < t < ∞ for almost every r ≥ 0. Because of (2.12), Assumption (Φ1), (2.5) and (2.4), the energy remains bounded by its initial value: The following lemma shows that under additional assumptions the limiting curve is actually a curve of maximal slope.We will not need this result in the remainder of this paper.
Lemma 2.3.If in addition to the assumptions of Proposition 2.2 with c 0 = 1, we also have for a strong upper gradient g of φ with respect to S 0 and for every u , then the limiting curve u * is a θ-curve of maximal slope for φ starting at u 0 .
Proof.Let u * and (u k ) k be as in the proof of Proposition 2.2.Since c 0 = 1, we have |u ε | S 0 (r) ≤ |u ε | Sε (r) for a.e.r ≥ 0. Using this inequality, (Φ1), the weak convergence of |u k | S 0 to A together with the inequality (2.13), the newly introduced assumption on the upper gradients, and (2.4), we may pass to the limit inferior in the energy dissipation equality (2.5) with s = 0 to deduce that the limit curve u * satisfies

Curves of Maximal Slope in Banach Spaces
If the metric space is actually a real Banach space (S , d) = (B, • B ) with dual space (B * , • B * ), then the objects of the previous section can be characterised in terms of classical derivatives; see, e.g.[Mat22, Section 2.3] for a proof of the following result regarding the local slope and the metric derivative.
Proposition 2.4.Let (B, • B ) be a Banach space, φ : B → (−∞, ∞] a mapping, and suppose u : If one has a more specific Banach space C with additional properties, then θ-curves of maximal slope satisfy differential inclusions.The following result is a simplified and slightly modified version of [AGS08, Proposition 1.4.1] and is stated in terms of the duality mapping , Ω be a θ-curve of maximal slope for φ with respect to its local slope |∂φ|.
holds for all 0 ≤ s < t < ∞.By the chain rule we obtain

Curves of maximal slope for abstract knot energies regularised by the logarithmic strain
For most of the known knot energies E the underlying energy space is a Sobolev-Slobodeckiǐ space W 1+s, (R/Z, R n ) for some s ∈ (0, 1), > 1/s.In the still abstract setting of the present section we regularise any such E by adding a power of the norm of the logarithmic strain Σ defined in (1.6) to form a total energy φ as in (1.7).It was shown in [KSSvdM22] that Σ is continuously differentiable on regular injective curves of that class.

Lemma 3.1 ([KSSvdM22, Proposition 5.1]).
For any s ∈ (0, 1) and Here, and from now on we use the following abbreviated notation: For a given space F of closed curves we denote from now on by F i the subset of all injective curves, by F r the regular curves, and we write F a for the arc length-parametrised curves contained in F, and we combine these indices i, r and a as needed.
To quantify injectivity of closed curves γ we define the bi-Lipschitz constant BiLip(γ) by means of To apply the metric existence result, Theorem 2.1 of Section 2, we formulate in the present section a few general assumptions on the knot energy E so that the total energy φ in (1.7) satisfies Assumptions (Φ1)-(Φ5) of Section 2. In order to verify (Φ4), however, we will need to consider a slightly smaller reflexive Banach space C that is compactly embedded in W 1+s, (R/Z, R n ) to finally prove the abstract existence result of this section; see Theorem 3.2 below.
Assumptions on the knot energy E. Let s ∈ (0, 1) and Now we can state our abstract existence result for curves of maximal slope and solutions of the gradient flow equation.where J C,θ : C → 2 C * is the θ-duality mapping defined in (1.8).If, in addition, C is strictly convex and has a Gâteaux-differentiable norm on C \ {0}, then Proof.We proceed as follows.In the first four steps we verify Assumptions (M), and (Φ1)-(Φ5) of Section 2.1 to obtain by virtue of the metric existence result, Theorem 2.1, a curve of maximal slope for the total energy φ with respect to its local slope |∂φ|.Then in a final step we apply the more specific results for Banach spaces presented in Section 2.2 to obtain the differential inclusion (3.1) and finally the gradient flow equation (3.2).
Step 1: Assumptions (M), (Φ2), (Φ3) are satisfied.The Sobolev-Slobodeckiǐ space W 1+s, (R/Z, R n ) as well as the Banach space C are complete metric spaces with the metric induced by their respective norms.It follows from general Banach space theory (see, e.g., [Bre11, Propositions 3.3 & 3.5]) that these norms are weakly lower semi-continuous, and that the weak topology is Hausdorff, so that Assumption (M) for the complete metric space S := C is satisfied (as well as for the larger space W 1+s, (R/Z, R n )).The total energy φ is coercive since E and the norm of Σ are nonnegative, which verifies Assumption (Φ2).Moreover, W 1+s, for > 1 is reflexive by Lemma A.1, and C is reflexive by assumption.So, the sequential weak compactness of bounded sets follows from general theory again; see, e.g., [Bre11, Theorem 3.17].In particular, Assumption (Φ3) holds here.
Step 2: Assumption (Φ1) holds true.To prove this claim we first show that a sequence of bounded total energy cannot have weak cluster points outside of the set of injective and regular curves.This way, the energy may not jump to infinity.Afterwards, it is straight-forward to prove weak lower semi-continuity on W 1+s, ir and up to a further subsequence we may also assume that lim inf where we used Arzela-Ascoli's theorem in combination with the embedding result, Proposition A.3 (ii) in the appendix for k 1 := 1, k 2 := 1, 1 := , and any µ ∈ (0, s − 1 ), to by the very definition (1.7) of the total energy φ, and further that By means of (3.4) and the convergence in (3.3) we obtain that γ is regular, and for distinct parameters so that γ is injective, and therefore γ ∈ W 1+s, ir (R/Z, R n ).By Assumption (E2) the energy E is sequentially weakly lower semi-continuous on W 1+s, ir (R/Z, R n ).It remains to prove the same for the regularising logarithmic strain term, which also appears in the total energy φ; cf.(1.7).Indeed, writing it out for γ k , Step 3: Assumption (Φ4) is satisfied.By definition of the total energy φ in (1.7) every sublevel set of φ is contained in C ir , which according to Lemma A.2 in the appendix is an open subset of the full space C. By Lemma 3.1 and Assumption (E1) the logarithmic strain Σ and the energy E are continuously Fréchet differentiable on W 1+s, ir (R/Z, R n ).Furthermore, the choice κ > 1 and Lemma A.1 in the appendix for k := 0 imply that • κ W s, is also continuously differentiable.Hence, the total energy φ is of class is continuous with respect to the norm-topology on W 1+s, (R/Z, R n ).Since C embeds compactly, i.e. since ι is a compact operator, effectively converting weakly convergent sequences in C into strongly convergent ones in W 1+s, (R/Z, R n ), we obtain the weak sequential continuity of |∂φ|(•) on C ir .In particular, it is also weakly sequentially lower semi-continuous, so that Assumption (Φ4) is satisfied.
Step 4: Assumption (Φ5) holds.Let u ∈ AC(I, C) such that φ(u(t)) < ∞ for all t ∈ I.By the previous step, the local slope |∂φ| is weakly sequentially lower semi-continuous and therefore it is lower semicontinuous with respect to the strong topology.It follows that |∂φ|•u is lower semicontinuous and thus measurable.Since φ is continuously Fréchet-differentiable on C ir , it is locally Lipschitz-continuous, whence φ • u is locally absolutely continuous, and therefore differentiable a.e. on I. Furthermore, since the reflexive space C satisfies the Radon-Nikodym property; see [ABHN11, Corollary 1.2.7] u is also differentiable a.e. on I. Therefore, by the chain rule and Proposition 2. for all s, t ∈ I with s < t.This is the defining inequality (2.1) for the strong upper gradient g := |∂φ|.Thus, Assumption (Φ5) is verified.
Step 5: Proof of (3.1) and (3.2).The previous steps have established the validity of Assumptions (M), and (Φ1)-(Φ5) so that Theorem 2.1 for the choice S := C yields a curve of maximal slope for the total energy φ with respect to its local slope |∂φ| starting at γ 0 ∈ D(φ).Since C is reflexive it has the Radon-Nikodym property [ABHN11, Corollary 1.2.7], part (i) of Proposition 2.5 implies the validity of the differential inclusion (3.1).Under the stronger assumption that C is strictly convex and has a Gâteauxdifferentiable norm on C \ {0}, part (ii) of the same proposition yields the gradient flow equation (3.2).
, and κ = 1.Suppose that C ⊂ W 1+s, (R/Z, R n ) is a compactly embedded reflexive Banach space, that the energy E satisfies Assumptions (E1)-(E3), and γ 0 ∈ D(φ), where the total energy φ is defined as in (1.7).Then there exists a θ-curve of maximal slope u ∈ AC θ ([0, ∞), (C, • C )) for φ with respect to its strong upper gradient |∂φ| starting at γ 0 .Furthermore, the local slope at γ ∈ C takes the form where A := W s, (R/Z) Proof.The assumptions (M) and (Φ1)-(Φ3) follow exactly as in in the first two steps of Theorem 3.2.We now show that the local slope takes the form (3.5).If Σ(γ) = 0, then φ = E + Σ A is differentiable at γ and we can apply Proposition 2.4.Notice that D Let us now assume that Σ(γ) = 0.By differentiability, we have as γ − η C → 0, and therefore, as γ − η C → 0. Choosing a maximising sequence (v n ) n∈N ⊂ C for the supremum and setting η n := γ + vn n , we infer that lim sup Changing v to −v completes the representation formula (3.5) for the local slope.
Next, we prove that the local slope is sequentially weakly lower semicontinuous and thus satisfies Assumption (Φ4).Let (γ n ) n∈N ⊂ C that converges weakly to γ ∈ C.Then, by compact embedding, γ n converges strongly to γ in W := W 1+s, (R/Z, R n ).By Assumption (E1) and Lemma 3.1 we have We begin with the case where Σ(γ) = 0. Let v ∈ C with v C = 1.We have by (3.6) (3.9) Combining (3.7), (3.8) and (3.9), we obtain the inequality ( Taking the supremum over all v ∈ C with v C = 1 on the lefthand side, we infer the lower semicontinuity.The case where Σ(γ) = 0 follows directly from the continuity properties (3.6), since Σ(γ n ) = 0 for all sufficiently large n ∈ N.This proves that Assumption (Φ4) is satisfied.Finally, we show that the local slope is a strong upper gradient, i.e., we verify Assumption (Φ5).Since E and Σ are continuously Fréchet-differentiable, they are locally Lipschitz-continuous and so is φ.In particular, φ • u is absolutely continuous for every and [AGS08, Definition 1.2.1] that |∂φ| is a strong upper gradient for φ.Therefore, Assumption (Φ5) is satisfied.The existence of a curve of maximal slope now follows as in step 5 of Theorem 3.2.

Gradient flows for various knot energies
To verify Assumptions (E1)-(E3) of Theorem 3.2 for the three energy families E α,p , intM (p,q) , and TP (p,q) , the following two general results turn out to be useful.The first one yields sequential lower semi-continuity for a multiple integral functional, whereas the second guarantees uniform control over the bi-Lipschitz constants.
Proof of Theorem 1.1.In all three scenarios (i)-(iii) the Banach space B is chosen as the energy space of the respective knot energy E, as pointed out in Remark 1.3 of the introduction.These Sobolev-Slobodeckiǐ spaces are all of the type W 1+s, (R/Z, R n ) for some s ∈ (0, 1) and ∈ ( 1 s , ∞) as considered in Theorem 3.2.Moreover, the slightly smaller Sobolev-Slobodeckiǐ space C is in each of the cases (i)-(iii) a reflexive Banach space compactly embedded in B according to Lemma A.1 and Proposition A.3.In addition, Lemma A.1 for k := 1 guarantees that the respective norm • C on the smaller Banach space C ⊂ B is continuously Fréchet differentiable away from 0, which according to Theorem 3.2 yields that the differential inclusion (3.1) (if it holds at all) reduces to the gradient flow equation (3.2), which corresponds to (1.9) for each choice (i), (ii), and (iii).The C 1 -regularity in time follows from the continuity in time of the right-hand side of (1.9):The curve u is absolutely continuous in time with values in C ir , and the differential Dφ is continuous on C ir , once Assumption (E1) is verified.In addition, Lemma A.1 implies that the duality mapping J C,θ is a homeomorphism from C onto the dual space C * .To summarise, the right-hand side of (1.9) is the composition of continuous mappings and therefore continuous in time.
So, it remains to establish (3.1), and for that it suffices to check that the respective knot energy E satisfies Assumptions (E1)-(E3) in each case (i)-(iii).
To verify Assumption (E3) for E α,p assume first that γ ∈ C 1 ia (R/Z, R n ) with E α,p (γ) ≤ M , which by means of [Bla12b, Theorem 1.1]9 implies that γ is of class W Then, we may approximate γ with respect to the W ), and use the continuity of E α,p implied by the upcoming Theorem 5.1 to assume without loss of generality that E α,p (γ k ) ≤ E α,p (γ) + 1 ≤ M + 1 for all k ∈ N. Now, to these smooth arc length-parametrised approximants we can apply [O'H92, Theorem 2.3] to obtain a uniform bi-Lipschitz constant, at least for the restricted parameter range α ∈ ( α p , 2].But even if α > 2, we may use O'Hara's work.For curves η ∈ C 1 (R/Z, R n ) parametrised by arc length, one has |∆η| ≤ d η ≤ 1 2 .Consequently, both, the integrand of E α,p and the energy E α,p itself are non-decreasing in α.This means that if E α,p (η) ≤ M + 1 for α ≥ 2, so is E 2,p and we may use the bi-Lipschitz constant obtained for that energy.O'Hara's bi-Lipschitz estimate yields a constant K = K(α, p, M + 1) independent of γ such that This, together with the energy's invariance under reparametrisation and its positive (2 − αp)-homogeneity implies that E α,p satisfies the suppositions of Lemma 4.2 and thus also Assumption (E3).
Proof of Corollary 1.2.For simplicity we set C := C ε .Note that for our choice of C, (1.9) is in fact well-defined, as by Lemma A.1 the duality mapping J C,θ is a homeomorphism between C and C * (and in particular single-valued).Combining the continuity of J −1 C,θ with the continuity of u and Dφ (see the proof of Theorem 1.1 for the latter) we obtain that u is almost everywhere equal to the continuous function ).It remains to establish that u (t) exists for all t and is equal to v(t).Since u is absolutely continuous on compact intervals and C is reflexive, we have the fundamental theorem of calculus [ABHN11, Corollary 1.2.7,Definition 1.2.5, Proposition 1.2.3], so that 1 As v is continuous, we can immediately infer that the right-hand side converges to v(t) as h → 0 and hence u is of class C 1 ([0, ∞), C).
We continue by proving that the energy is non-increasing along the flow.According to Lemma A.1 the Banach space C is reflexive in each of the three cases (i)-(iii) of Theorem 1.1.The θ-duality mapping J C,θ : C → 2 C * is a duality map with weight ϕ(s) := s θ β in the language of [Cio90, Definition I.4.1].Therefore, we may apply [Cio90, Corollary II.3.5] to identify the inverse J −1 C,θ with the duality mapping on C * with weight ϕ −1 (s) = s β θ , which is just J C * ,β .Thus, by the chain rule and (1.9), for all t ≥ 0. Hence φ•u is non-increasing, so that [AGS08, Proposition 1.4.1]implies that u is a θ-curve of maximal slope with respect to the weak upper gradient10 Dφ[u(t)] C * .In fact, by Step 4 of Theorem 3.2 whose prerequisites were verified in the proof of Theorem 1.1, it is even a strong upper gradient.Finally, C continuously embeds into C 1 (R/Z, R n ) in all three cases of Theorem 1.1, and thus u is a C 1 -isotopy.All curves u(t) are embedded because φ(u(t)) < ∞ for all t ≥ 0 and so, by [Rei05,Bla09], [u(t)] = [γ 0 ] for all t ≥ 0, i.e. the knot class is preserved along the flow.
The following corollary shows that the gradient flows obtained in Theorem 1.1 for each ε > 0 admit a converging subsequence as ε → 0.
Proof of Corollary 4.3.The claim is a consequence of Proposition 2.2, whose prerequisites we now check.As was shown in step 1 of the proof of Theorem 3. 2, (B, • B ) with its weak topology satisfies Assumption (M).Moreover, for all γ In addition, as was shown in the proof of Theorem 1.1, φ satisfies the Assumptions (E1)-(E3) und thus, by steps 1 and 2 of the proof of Theorem 3.2, it also satisfies the Assumptions (Φ1)-(Φ3) with (B, • B ) as the underlying metric space.Furthermore, since we are only interested in small ε > 0, in view of (4.2) we may assume that sup ε>0 φ(γ 0,ε ) < ∞ and sup ε>0 γ 0,ε B < ∞.Moreover, by Corollary 1.2, u ε is a θ-curve of maximal slope for φ with respect to the strong upper gradient |∂φ| Cε = Dφ C * ε starting at γ 0,ε .Identifying (S 0 , d 0 ) with (B, • B ) let σ be the weak topology on B. Furthermore, for ε > 0 let (S ε , d ε ) := (C ε , • Cε ), g ε := |∂φ| Cε , and u 0,ε := γ 0,ε .Then these satisfy all the assumptions of Proposition 2.2 with c 0 = 1 which concludes the proof.Notice that since φ(u * (t)) ≥ 0 and φ(γ 0 ) < ∞, it follows that u * is actually absolutely continuous and not only locally absolutely continuous.Our method of proof for continuous differentiability of E α,p is inspired by [RS21, Section 3].There are several arguments which carry over completely; we include these in our proof for the reader's convenience.Several technical results needed in the proof of Theorem 5.1 below, namely Lemmata 5.2-5.6 are deferred to the end of this section.

O'Hara's knot energies are continuously differentiable
In the following, we will use the derivative with respect to arc length D γ η(u) := η (u) and the interval I γ (x, y) parametrising the arc of γ where d γ (x, y) is attained.To be more precise, I γ (x, y) is the interval containing x and one of y − 1, y, y + 1 such that d γ (x, y) = Iγ (x,y) |γ (t)| dt.This is well-defined whenever d γ (x, y) < L 2 , i.e. for almost all x, y ∈ R. When there is no risk of confusion, we omit the arguments x and y.Lastly, we need the minimal velocity Proof of Theorem 5.1.For p = 1, this was already proved in [BR13, Proposition 2.1], so we may assume p > 1.
We use the chain rule (see e.g.[Zei93, Proposition 4.10] for the general Banach space version) to prove our statement.As outer function, we choose a geometric L p -norm additionally depending on γ: We consider the integrand e α in (5.1) as a mapping from W • e α (γ) and it suffices to show that both functions are C 1 .Lemma 5.2 together with the embedding result, part (ii) of Proposition A.3, take care of the outer function, so we only need to look at e α .In order to do this, define Here, δ k denotes the k-th variation.As for all x ∈ R/Z there exists exactly one y = x ∈ R/Z such that I γ (x, y) is not well-defined, we work on ) and is the same as (R/Z) 2 up to a set of measure 0. For all γ ∈ W , y).This means that the G k exist for all such (x, y) and a simple calculation shows that To shorten notation, we left out the t-dependencies in these terms and will also do this in the following when there is no risk of confusion.Recall that d γ (x, y) = Iγ |γ (t)| dt.For fixed η and |τ | < 1 sufficiently small, the estimate v γ+τ η ≥ 1 2 v γ holds and so This, together with Lemma A.4, enables us to find integrable majorants for the τderivatives of |γ τ | := |γ (t)+τ η (t)|: They only consist of sums of inner products of D γτ γ τ , D γτ η and |γ τ | (whose derivatives again fit the pattern).Consequently, and we may calculate for (x, y) ∈ Σ, that and Up until now, these are only pointwise limits and we still need to show that F 1 is indeed the Fréchet-derivative of e α and also continuous with respect to γ.Let us first show that F 1 is a valid candidate for a derivative.
Claim 1.There is In order to show this, decompose F 1 as Lemma 5.5 gives us a fitting upper bound for the first term, Lemma 5.3 gives one for the second term when we choose ϕ = 0, ψ = α, η1 := γ and η2 := η 1 (note that L 1 = L 2 = ∆ dγ ).We will later need a similar bound for F 2 .
Claim 2. There is Ξ = Ξ(γ) > 0 depending continuously on γ such that F 2 (γ; The central ingredient is once again the right decomposition.Rewrite F 2 as (5.5) We can use Lemma 5.3 with ψ = α + 2, ϕ = 0, η1 = η3 = γ, η2 = η 1 and η4 = η 2 to deal with (5.2) and the same Lemma with ψ = α, ϕ = 0, η1 = η 1 and η2 = η 2 to find an upper bound for (5.3).In order to take care of (5.4), let us define . The first term is again bounded above via Lemma 5.5, so let us take a look at the second one.Define If we can estimate this by terms controlled via Lemma 5.6, we have an upper bound for the L p -norm of (5.4).To achieve this, first apply the Hölder inequality (with p = q = 2) and our usual upper bound for the line elements to reduce the problem to bounding Each summand in the last line fits the pattern of Lemma 5.6, so we are finished with bounding (5.4).
In dealing with (5.5), again the important technique is creative rewriting.In this case, we sum up two terms which differ only by exchanging η 1 and η 2 : The first factor of each summand on the left-hand side is bounded above by 2 , because of the upcoming estimate (5.21) and the fact that |G 1 (γ; The second factor of each summand on the left-hand side is L p -bounded because of Lemma 5.5 and thus, also (5.5) is L p -bounded.Now let us really prove that F 1 is the Fréchet-derivative.
From now on, fix γ ∈ W < ε one has not only that γ + η ∈ W (5.7) Furthermore, let γ t := γ + tη 1 and decompose Σ = U (η 1 ) ∪ V (η 1 ) with On U (η 1 ), e α (γ t ) is differentiable with respect to t because d γt is differentiable as long as I γt is fixed.By Taylor's theorem, we have for (x, y) ∈ U (η 1 ) ) dt , so taking the L p -norm on U (η 1 ) yields, with Jensen's inequality, Tonelli's variant of Fubini's theorem, Claim 2 and (5.6): Instead of trying to show that the same holds true on the "bad" set V (η 1 ), we will show in Claim 4 that there, e α (γ)(x, y) is locally Lipschitz continuous with respect to γ and has a Lipschitz constant that is uniform in (x, y).If that is the case, and consequently and can use this together with (5.8) to show that e α (γ , which is enough for Fréchet-differentiability.
Claim 4.There is a constant for all η (5.9) The key ingredient is the local Lipschitz continuity of the intrinsic distance d γ as a mapping from W ).Since Σ has full measure, it suffices to look at (x, y) ∈ Σ and differentiate between two cases: The first case is Then we may simply compute The other case is when the shortest connections do not match, i.e.I γ = I γ .In this case, we need more precise control of the integrands, so let us first find bounds for them.We know that ||γ (t)| − |γ (t)|| ≤ η L ∞ and so Performing the same estimates for an upper bound, we arrive at the fact that Then, we use the fact that the connection between γ(x) and γ(y) via I γ is longer than the one via I γ to estimate (5.10) which implies (5.11) Consequently, this time using the fact that I γ parametrises the longer connection between γ(x) and γ(y), we have Furthermore, we can use the definition of γ and the fact that 1 ≤ |γ (t)| vγ to obtain Combining the last two estimates yields (5.12) so we have proven the local Lipschitz property of d γ .
To apply this result to e α , we first provide a simple local Lipschitz estimate for x → x −α .Assume x, x + h > 0, then (5.13) Rewriting e α a bit, we obtain In order to use (5.13), we need to make sure that the γ-terms do not veer too far from their γ-counterparts.For the ∆-terms, consider that for all k ∈ Z, we have Taking the minimum over all k, the fraction on the right-hand side becomes 1 and so we obtain η L ∞ as an upper bound.By (5.7), this means that 2 BiLip(γ), so the first part of (5.14) is bounded above by (5.15) In order to apply (5.13) to the d γ -terms of (5.14), we need a lower bound for d γ (x, y).
To achieve this, assume for the moment that d γ (x, y) ≥ L(γ) 3 , which we will prove in (5.17).Then, by (5.12) and (5.7), Applying (5.13) and (5.12), we obtain that the second part of (5.14) is bounded above by (5.16) The last thing we need for Lipschitz continuity of e α on V (η 1 ) is that x and y cannot get too close.The tuple (x, y) is in V (η 1 ) if and only if I γ (x, y) = I γt (x, y), so it suffices to establish that this cannot happen when |x − y| R/Z is small.Assume d γ (x, y) ≤ L(γ) 3 .We will show that this is impossible thus establishing (5.17) below, since Under our assumption and by means of (5.10) (recall that γ = γ + η) and (5.7), 2 .
To wrap up the proof of Claim 3, we need that V (η 1 ) is small.
For (x, y) ∈ V (η 1 ), there is t ∈ [0, 1] such that the intrinsic distance is parametrised over I γ instead of I γ , so let us take a closer look at the corresponding integral.The fundamental theorem of calculus yields I γ |γ t (s)| ds = I γ |γ (s)| + t 1 0 D γτt γ τ t (s), η 1 (s) dτ ds and so, for the fitting t, Note that for each (x, y) ∈ V (η 1 ), there is exactly one x = x(x, t) such that d γt (x, x) = 1 2 L(γ t ) and because it does not matter which arc of γ t we travel through to get from γ t (x) to γ t (x), d γt (x, x) = d γt (x, y) + d γt (x, y).Thus, y) and so 2 )| = 2r.We may use this in combination with (5.18) to estimate Next, we prove that F 1 is continuous in γ.
Claim 6.The mapping as η 1 → 0, the Landau symbol being uniform in η 2 .Let us once again split the domain of integration into U (η 1 ) and V (η 1 ).On the former, we can use that t → F 1 (γ t ; η 2 ) is differentiable, as well as Jensen's inequality, Tonelli's variant of Fubini's theorem, Claim 2 and (5.6): On V (η 1 ) we may use Claim 4 and Claim 5. Note that because of (5.7) and (5.9), L eα (γ + η) ≤ CL eα (γ) for all η W 1+ αp−1 2p ,2p < ε and we can thus estimate Lemma 5.2.Let p > 1.Then, the map Proof.We prove continuous Gateaux-differentiability, which implies Fréchet-differentiability (see e.g.[Zei93, Proposition 4 Calculating and then bounding the derivative in the integrand, we obtain for |h| < 1: Estimating |g + thg| p ≤ C(p)(|g| p + |g| p) and using Young's inequality with exponents p p−1 and p on the first term, we find an integrable upper bound By Lebesgue's theorem, we may interchange integration and taking the limit h → 0 and thus arrive at Hölder's inequality implies that this is a linear and bounded operator with respect to (γ, g) and thus, we have found a Gateaux-derivative.To obtain C 1 -regularity, consider the difference )(γ, g)| and look at each summand on its own.The difference of the first summands is (5.20) The first term may be bounded above by using Hölder's inequality, yielding As we have an integrable majorant and can thus apply Lebesgue's theorem to show that (5.19) goes to 0 as g 2 → g 1 .The convergence of (5.20) to 0 follows from Hölder's inequality and the W 1,∞convergence of γ 2 → γ 1 .
Here, Ξ(γ) > 0 continuously depends on γ with respect to the W Remark 5.4.This as well as Lemma 5.5 and Lemma 5.6 are also valid for the case αp = 2 if one considers the intersection W We do not make the effort to include this case as we do not want to deal with the more complicated space in Theorem 5.1.

Proof. Let us begin by bounding the b-term
, which is clear for the second and third option for L i .For the first one, we calculate (5.21) We continue by defining a function ζ which helps reduce the integrand to its most relevant content: ζ : (0, ∞) → R, r → r 2+ϕ r 2+ψ −1 r 2 −1 .With this, we may rewrite the second part of the integrand: The last difference can be written in terms of the unit tangents according to Lemma 5.6.Combining our estimates, we arrive at .Then, the map With these identities, we may rewrite the integrand and then bound it via the Cauchy-Schwarz-inequality as follows: Integrating over (R/Z) 2 and using Cauchy-Schwarz again, we arrive at a new bound to which we can apply Lemma 5.6: for some constant C = C(α, β, n, p) > 0.
Proof.We prove this statement via substitutions with the inverse arc length function For the moment dropping the factor v −2−2p γ , employing Jensen's inequality, Tonelli's variant of Fubini's theorem, the substitution u = x + θ 2 y as well as the L-periodicity of the integrand with respect to u, we obtain the new estimate Two further substitutions, θ 2 by ϑ = θ 1 − θ 2 and y by w = ϑy yield a new upper bound.Note that we dropped the θ 1 -integral as its domain has measure 1 and nothing depends on θ 1 after enlarging the domain of integration for ϑ.We deduce from the following two results that Z is uniformly convex.Firstly, in [Day41, Theorem 2 & p. 507] it is shown for non-negative measures μ, numbers 1 < ˜ < ∞, and Banach spaces B that L ˜ (μ; B) is uniformly convex if B is.The space B := R n endowed with the Euclidean 2-norm satisfies this condition.Secondly, an immediate consequence of [Day41, Theorem 3 & p. 504] is that l ˜ -direct sums of finitely many uniformly convex Banach spaces are uniformly convex.It is straightforward to check that uniform convexity is inherited by subspaces and preserved by linear isometries.Therefore, Ψ(W) and W are uniformly convex.
The remaining claims follow from general results on Banach spaces.Since W is uniformly convex, it is reflexive by the Milman-Pettis Theorem; see e.g., [Cio90, Theorem II.2.9].Another consequence of the uniform convexity of W is that the norm of its dual space is Fréchet-differentiable away from 0, see [Cio90, Theorem II.Proof.The statement for C 1 -curves was shown in [KSSvdM22,Lemma B.3].For the Sobolev-Slobodeckiǐ spaces and therefore for Banach spaces continuously embedded in those, this claim follows from the following standard embedding result, Proposition A.3.
The following proposition gathers well-known embedding results for Sobolev-Slobodeckiǐ spaces.See, e.g., the appendix of [Mat22] for a proof based on the Besov space theory presented in [Tri06].
1 − 1 2 , 0 , then the identity operator id : Taking the minimum over all k ∈ Z yields the estimate.
With this and another change of variables, we may calculate Note that our assumptions on g imply its surjectivity.Finally, we may use that g −1 (L) = g −1 (0) + l and periodicity to see that the double integral on the right-hand side is indeed the desired seminorm.
The set of all such curves is denoted by AC θ (I, S ) (writing AC θ loc (I, S ) for the local variant of this space if m is only in L θ loc (I), and abbreviating AC(I, S ) := AC 1 (I, S )).According to [AGS08, Theorem 1.1.2]every u ∈ AC θ loc (I, S ) is metrically differentiable almost everywhere in the following sense: its metric derivative |u |(t) := lim s→t Any functional φ : S → (−∞, ∞] with non-empty effective domain D(φ) The differentiability of the norm • C on C \ {0} ensures that the duality mapping J C,θ is single-valued [Cio90, Corollary I.4.5].The reflexivity of C guarantees that J C,θ is surjective [Cio90, Theorem II.3.4], and the strict convexity of C implies that it is injective [Cio90, Theorem II.1.8] 8.Thus, the inclusion (2.15) is actually an identity and we deduce (2.16).