Averaging of time-periodic dissipation potentials in rate-independent processes

We study the existence and well-posedness of rate-independent systems (or hysteresis operators) with a dissipation potential that oscillates in time with period \begin{document}$ \varepsilon$\end{document} . In particular, for the case of quadratic energies in a Hilbert space, we study the averaging limit \begin{document}$ \varepsilon \to 0$\end{document} and show that the effective dissipation potential is given by the minimum of all friction thresholds in one period, more precisely as the intersection of all the characteristic domains. We show that the rates of the process do not converge weakly, hence our analysis uses the notion of energetic solutions and relies on a detailed estimates to obtain a suitable equi-continuity of the solutions in the limit \begin{document}$ \varepsilon \to 0$\end{document} .

1. Introduction. In most applications of hysteresis or rate-independent systems the hysteresis operator or the dissipation potential is time-independent while the system is driven by a time-dependent external loading, see [24,4,11,17]. However, there are also systems where the internal dissipative mechanism depends on time in a prescribed manner, see [19,12,2] and the references below for mathematical treatments of this case. Moreover, there are mechanical devices where friction is modulated time-periodically by using a rotating unbalance, as in a vibratory plate compactor used in construction areas, see Figure 1.1.
In this paper we are interested in cases where the dissipation process is oscillating periodically on a much faster time scale than the driving of the system by an external loading. Similar, time-dependent friction mechanisms occur during walking or crawling of animals or mechanical devices. Typically, there is a periodic gait, where the contact pressure of the dierent extremities oscillates periodically, and only those legs are moved for which the normal pressure is minimal. Simple mechanical toys, where this interplay can easily be studied, are the so-called descending woodpecker, the toy ramp walkers, and the rocking toy animals, see Figure 1.2. We refer to [9,6,8,7] for models on locomotion for micro-machines or animals and to [21] for the slip-stick dynamics of polymers on inhomogeneous surfaces. Application to time-dependent hysteresis in piezo-ceramic actuators are given in [2,1].
Another application arises by moving an elastic body like a rubber over a at surface, where the surface is prepared such that the friction coecient changes periodically. Then, the system under consideration might serve to model how microstructures on surfaces give rise to kinetic friction which is smaller than the static friction (also called stiction). Of course, this model does not account for the true microstructure of the surface, being in general of a stochastic nature.
In the present work, we will not investigate how the periodic oscillation is generated by the system itself. Instead, we will assume that the friction is induced by a given time-periodic dissipation mechanism. More precisely, we consider a rate-independent system (Y, E, R ε ), where Y is a reexive Banach space. The energy E : [0, T ]×Y → R is the energy functional, where the Gateaux dierential DE(t, y) ∈ Y * is the static restoring force and ∂ t E(t, y) ∈ R is the power of the external loadings. In this introduction and in most parts of this article we restrict ourself to the quadratic case, where Y is a Hilbert space and E(t, y) = 1 2 Ay, y − (t), y , ∀ γ > 0, v ∈ Y : R ε (t, γv) = γR ε (t, v).
The dissipation forces are given by the set-valued subdierential ∂R ε (t, v) ⊂ Y * of the convex function R ε (t, ·). We are interested in the case where the temporal behavior is characterized by the microscopic period 2πε > 0 through a functional Φ : [0, T ]×R×Y → [0, ∞] via R ε (t, v) = Φ( t, t/ε, v), (1.2) where now Φ(t, s, ·) is a 1-homogeneous dissipation potential, and Φ(t, ·, v) is periodic with period 2π on the real line. We assume that Φ has moduli of continuity ω 1 and ω 2 in the rst and second variable respectively and is lower semi-continuous in the third variable. In Section 4, Theorem 4.1, we provide a general existence result for the Cauchy- Indeed, we establish the existence of energetic solutions for general rate-independent systems with non-convex energies in Section 4. This part is a suitable generalization of the general existence theory based on incremental minimization developed in [15,17]. For this we develop a suitable calculus to generalize the denition , Y) to all functions of bounded variation, see Section 3.
Section 5 then provides the main result concerning the limit of fast oscillatory dissipation structures, i.e. ε 0 where R ε is given in the form (1.2). This result states that the solutions y ε : [0, T ] → Y of (1.3) converge uniformly to a function y 0 : [0, T ] → Y, which is again a solution to a rate-independent system (Y, E, R eff ), where the eective dissipation potential is given by an innitedimensional inf-convolution, namely In fact, using the 1-homogeneity of the dissipation potential Φ(t, s, ·) we see that the dual dissipation potential has the form Φ * (t, s, ξ) = χ K(t,s) (ξ), where K(t, s) = ∂Φ(t, s, 0) ⊂ Y * is a convex set containing 0 ∈ Y * . Here χ A (ξ) = 0 for ξ ∈ A and χ A (ξ) = ∞ for ξ ∈ A. Under suitable assumptions we show (see Proposition 3.6) K(t, s) . (1.5) This can be understood in the sense that the eective dissipation potential is given in terms of the minimum of all the possible friction thresholds. Because of the rateindependent nature, the system can take immediate advantage of a low threshold and move as far as necessary, see e.g. the solution y ε in Figure 2.1, which moves with fast velocity O(1/ε) on tiny intervals of length O(ε 2 ), or the zigzag pattern of the solution t → (y 1 (t), y 2 (t)) in Figure 2.2, whereẏ 1 (t)ẏ 2 (t) ≡ 0.
To be more precise, we assume that Y = F×Z for Hilbert spaces F and Z, where the component φ ∈ F acts as a purely elastic part of the state y = (φ, z), while z is the dissipative part. Consequently, we will assume that the dissipation potential Φ(t, s, (φ,ż)) is independent ofφ. Moreover, we assume that there exists a convex, lower semi-continuous positive 1-homogeneous functional ψ 0 on Z such that the following holds where ω 1 and ω 2 are moduli of continuity, i.e. continuous, nondecreasing functions with ω j (0) = 0.
Moreover, for ε 0 we have y ε (t) y(t) for all t ∈ [0, T ], where the limit function y ∈ C([0, T ]; Y) is the unique energetic solution of the eective rateindependent system (F×Y, E, R eff ) with y(0) = y 0 . In particular, if Φ is Lipschitz in the rst variable, we have where R eff is dened in (1.4) and characterized in (1.5). Remark 1.2. Parts of the theory developed below also works for reexive Banach spaces. In particular, the existence result in Theorem 4.1 can be proved in a general metric setting. However, the uniqueness result in Theorem 4.7 and the proof of the averaging result of Theorem 1.1 heavily rely on Proposition 4.6, which is proved using the quadratic structure of the energy and the coercivity of the linear operator A. Thus, the quadratic form generated by A induces a Hilbertian structure.
The proof of this result strongly uses the theory of energetic rate-independent systems as developed in [15,17]. The main reason for which we cannot pass to the limit ε 0 in the equation (1.3) is that the derivativesẏ ε do not exist in the sense of L p (0, T ; Y) and even if they exist, they do not converge weakly in any L p space.
The problem of the very weak convergence is already seen in the simple scalar model which is studied in Section 2.1 for illustrative purposes, see Figure 2.1. In Section 2.2 we study a two-dimensional case (i.e. Y = R 2 ) which can be seen as a strongly simplied model for a two-leg walker, where the weight of the body is periodically relocated from one leg to the other such that their motion occurs alternatingly.
Instead of weak convergence of the solutions ofẏ ε we will rather rely on equicontinuity properties of the family (y ε ) ε∈(0,1) , see Proposition 5.2. Thus, the derivativefree notion of energetic solutions is ideally suited for the limit passage in the proof of Theorem 1.1.
Despite the fact that hysteresis operators and rate-independent systems have been studied by many works (e.g. [24,4,11,17]), there seems to be only few work on time-dependent dissipation potentials, even though a rst result for time dependent R was already obtained by Moreau in 1977, see [19] and the follow up paper [14].
The conceptually closest existence result to our work has been obtained by Kre-j£í and Liero in [12], who combined the framework of Kurzweil-integrals with the concept of energetic solutions. Instead of assuming continuity of R(t, ·) with respect to t, they consider R * (t, ·) = χ K(v(t)) and assume Lipschitz continuity of v → K(v) ⊂ Y * in the Hausdor distance, where v lies in a Banach space V. They then obtain existence and uniqueness of solutions for data ∈ BV([0, T ], Y * ) and v ∈ BV([0, T ], V). Note that in that case the mapping t → R * (t, ·) = χ K(v(t)) (·) may even have jumps. More recent generalizations are given in [22,13,23,10].
Our work uses similar ideas for establishing continuous dependence on the data, but restricts to the continuous case. Note that the time-periodic setting with ε 0 does not allow for uniform bounds in BV, since even Lipschitz continuity of (t, s) → Φ(t, s, ·) will give a BV bound for t → R ε (t, ·) = Φ(t, t/ε, ·) of order O(1/ε). So, we will explicitly exploit the periodicity of s → Φ(t, s, ·) to prove the equicontinuity in where we choose for deniteness (t) = 5t − t 2 and ρ(s) = 2 + cos(s).
Moreover, we see that the derivativeẏ ε : [0, 4] → R is either 0 (namely on at parts) or˙ (t) + 1 ε sin(t/ε). On each interval [kπ/ε, (k+2)π/ε] the solution has one increasing region whose length is O(ε 2 ), whileẏ ε is of order O(1/ε). We observe the basic principle that y ε waits until ρ(t/ε) = 2 + cos(t/ε) gets very close to ρ min = 1 and then moves very quickly. Thus, the at regions dominate andẏ ε is not bounded in L p ([0, T ]) for any p > 1. We only haveẏ ε M ẏ 0 (convergence in measure when testing with continuous test functions). More precisely, the sequenceẏ ε is bounded Nevertheless, one can establish an asymptotic equicontinuity estimate in the form Below, we will derive similar estimates in the general setting, see Proposition 5.2.

2.2.
A two-dimensional model for walking. We now consider Y = Z = R 2 , where y = (y 1 , y 2 ) contains the coordinates of the two legs walking on a onedimensional line. This may serve as a model for a toy ramp walker as well as for a rocking animal, if one restricts to the only relevant case where the two left and the two right legs always move together. We take For a walker with symmetric legs one would assume ρ 1 (s+ 1 2 ) = ρ 2 (s) and ρ 1 (s) + ρ 2 (s) = const. = ρ * , where ρ * is the constant normal pressure induced by the total weight. However, this is not important for our purpose, we only need that ρ min j := min{ρ j (s) | s ∈ R} ≥ α > 0. (Note that for a walker, when moving the free leg, there is always some small friction in the joints.) Moreover, we want to impose that the two minima are not attained for the same phase s ∈ [0, 2π].
The associated dierential inclusion takes the form  (C) The path t → y(t) = (y 1 (t), y 2 (t)) ∈ R 2 shows a microscopic zigzag pattern.
Here the eective equation for ε 0 is obtained with This is most easily seen by using the dual characterization via Then, our formula (1.5) and the assumption ρ min We remind at this point that convex 1-homogeneous functionals automatically fulll a triangle inequality. We denote by BV ψ (0, T ; Z) the set of all functions u with nite variation var(u; ψ; 0, T ). In case ψ(·) = · Z , we simply write BV(0, T ; Z) := BV · Z (0, T ; Z).
In what follows, we say that a sequence of functions (u n ) n∈N ⊂ BV(0, T ; Z) converges weakly to u ∈ BV(0, T ; Z) if sup n var(u n ; ψ; 0, T ) + u n L 1 < ∞ and u n (t) Denition 3.1 (ψ 0 -regular dissipation potentials). Let ψ 0 : Z → R be a convex, lower-semicontinuous and positive 1-homogeneous functional. A map Ψ : The proof of the following results is postponed to Appendix A.
Lemma 3.2 (Denition of total dissipation). Let ψ 0 : Z → R be a convex, positive, lower semi-continuous 1-homogeneous functional and let Ψ : is independent from the choice of T K and The quantity Diss Ψ (u; s, t) is the total dissipation of the function u with respect to Ψ over the time-interval [s, t]. Lemma 3.3 (Properties of total dissipation). Let ψ 0 : Z → R be a convex, positive, lower semi-continuous 1-homogeneous functional and let Ψ 1 , Proof. We have which gives the desired result. Lemma 3.5 (Lower semi-continuity of total dissipation). Let ψ 0 : Z → R be a convex, positive, lower semi-continuous 1-homogeneous functional and let Ψ : We now consider R ε (t,ż) = Φ(t, t ε ,ż) and recall the denition of R eff in (1.4) and provide the useful characterization (1.5), which will be proved in Appendix A.4. Proposition 3.6 (Characterization of eective dissipation). Let ψ 0 and Φ satisfy 4. Existence of energetic solutions. In this section, we will provide two existence results. Theorem 4.1 is more general than needed for the proof of Theorem 1.1, but it could also be useful in other contexts. It can also be proved in a metric setting. This can be achieved by replacing z 1 − z 2 Z by d(z 1 , z 2 ), the weak convergence with a topology T Z that is weaker than d(·, ·) and by requiring that Ψ(t, z 1 , z 2 ) is lower semi-continuous with respect to T Z in the proof below. Theorem 4.7 deals with the special case of quadratic energies in a Hilbert space, which is the basis of the proof of Theorem 1.1.
4.1. The general case. We assume that F and Z are separable and reexive Banach spaces and consider Y = F × Z equipped with the product norm. We write y ∈ Y as y = (φ, z). Assume we are given a functional E : [0, T ]×Y → R continuous in the rst and lower semicontinuous in the second variable. We furthermore assume that There exist c E > 0 such that for all y * ∈ Y : Gronwall's inequality applied to Assumption (4.1) yields for all 0 ≤ s < t ≤ T Clearly, the sum of a λ-convex functional and a convex functional is λ-convex.
Finally, in order to prove continuity of solutions, we will also need Lipschitz continuity of the power ∂ t E(t, ·), namely There exists C for all t ∈ [0, T ] and all y 0 , y 1 ∈ Y . (4.5) The functional ψ 0 : Z → R is a convex, non-negative, lower semi-continuous 1homogeneous functional and Ψ : [0, T ]×Z → R is a ψ 0 -regular dissipation potential.
Similar to [17,16] it can be shown that a suitable weak formulation of the inclusion is given by the notion of energetic solutions dened via ∀ŷ ∈ Y : E(t, y(t)) ≤ E(t,ŷ) + Ψ(t,ŷ − y(t)) ,     t n → t and y n y ⇒ DE(t n , y n ) DE(t, y) and 9) and if (4.5) holds, then for all t, τ ∈ [0, T ] it holds with α and α from (3.1). In particular, y ∈ C([0, T ]; Y) with modulus of continuity σ → C σ+ω(σ) for some positive constant C.
In case that Ψ does not depend on time, we obtain the well-known result that the solution is Lipschitz continuous in time. Thus, we suppose that our result is optimal with regard to the regularity of the solution.
Applying (4.5) gives the assertion of the lemma.
Discretization scheme. We rst consider the case that (4.8a) holds and afterwards discuss how the proof has to be modied if, instead, (4.8b) or (4.8c) hold. Remark, that continuity of ψ 0 implies boundedness of ψ 0 on bounded subsets of Z since ψ 0 is 1-homogeneous. This in turn implies continuity of z → Ψ(t, z) for all t ∈ [0, T ]. We consider the following sequence of partitions of the interval [0, T ]. For every K ∈ N, we set t K k := k 2 K T , 0 ≤ k ≤ 2 K and dene for k ≥ 1 Clearly, D K k (z,ẑ) is weakly lower semicontinuous (respectively continuous if ψ 0 is continuous). We will use D K k in the discretization scheme (4.13) in order to be able to apply Lemma 3.5 in the limit K → ∞ in (4.20). Lemma 4.4. Let ψ 0 be weakly continuous. Let t ∈ [0, T ], z ∈ Z and z K z weakly Proof. Due to (3.2) there holds for all s ∈ [0, T ] and allẑ ∈ Z that Therefore, we nd Similarly, we obtain lim K→∞ D K k(K,t) (z K ,ẑ) ≥ Ψ(t,ẑ − z(t)). This concludes the proof.
Given the initial value y 0 ∈ Y and K ∈ N, we look for y K 1 , . . . (4.13) The existence of the minimizers y K k follows from (4.2) and the lower semicontinuity of E and Ψ.
Step 1. A priori estimates. To simplify the notation in this step, we x K and write t k = t K k , (φ k , z k ) = y k = y K k and D k (·, ·) = D K k (·, ·). We use (4.13) and the triangle inequality to nd in a rst step: ≥ E(t k , y k ) . (4.14) Using again the minimization property (4.13) of y k , we obtain the upper energy inequality Like on page 5253 in [17] or page 489 in [15], we nd (4.17) We nally observe that a combination of (4.1) and (4.3) yields In what follows, let denote the right-continuous piecewise constant interpolation of y K k = (φ K k , z K k ) dened by (4.13) and let Z K denote the piecewise linear interpolation of z K k . Furthermore, we dene Due to (4.16) and (4.18), we nd that Θ K is uniformly bounded in L ∞ (0, T ). Since Ψ is a ψ 0 -regular dissipation potential and since Z K are piecewise constant in time, we obtain from estimate (4.17) and property (3.1) that var(Z K ; ψ 0 ; 0, T ) = var(Ẑ K ; ψ 0 ; 0, T ) = Furthermore, from (4.16) and (4.3) we obtain that and hence Y K L ∞ (0,T ;Y) is bounded by (4.2). Summing up (4.15) over k, we obtain E(t k , y k ) + Diss Ψ ( Z K ; 0, t k ) ≤ E(0, y 0 ) +ˆt k 0 ∂ s E(s, Y K (s)) ds (4.19) Step 2. Selection of subsequence and passing to the limit. From the generalized Helly selection principle in [15, theorem. 5.1] and Step 1 we infer that there exists z ∈ BV ψ0 (0, T ; Z) and a subsequence of Z K (still indexed by K) such that Z K (t) z(t) and Z K (t) z(t) pointwise for all t ∈ [0, T ]. From Lemma 3.5 we obtain that (4.20) Furthermore, there exists θ ∈ L ∞ (0, T ) such that for a further subsequence (still indexed by K) it holds We furthermore dene the function θ sup : t → lim sup K→∞ Θ K (t), for which we nd Hence, z(t) and φ(t) are dened for all t ∈ [0, T ].
Step 3. Stability of the limit function. Given t ∈ [0, T ] and y(t) = (z(t), φ(t)), let k n t := max k ∈ N : t K k ≤ t . Since E is continuous in the rst and lower semicontinuous in the second variable, from (4.14) and Lemma 4.4 we obtain . Therefore, we can apply [15] Proposition 5.6 and obtain that θ sup (t) = lim n→∞ Θ K n t (t) = ∂ t E(t, y(t)) . Step 5. Lower energy estimate. Let t ∈ [0, T ]. We follow the proof of Proposition 2.1.23 in [17]. Since θ : t → ∂ t E(t, y(t)) is integrable, we can apply the methods from [5,Sec. 4.4] and obtain a sequence of partitions 0 ≤ t K We use (4.21) and observe that for all K ∈ N and j = 1, . . . , K we have Summing j over 1, . . . , K, letting K → ∞ and using (3.3) we obtain This nishes the proof of existence of energetic solutions.
The case of (4.8b). Steps 1,2 and 56 work the same way as above. In Step 4, the identity θ sup (t) = ∂ t E(t, y(t)) can be obtained directly from the continuity of ∂ t E assumed in (4.8b).
Step 3 is a consequence of the rst assumption in (4.8b).
4.2. The case of a quadratic energy. In this section, we make the following assumptions.
Assumption 4.5. The spaces F and Z are Hilbert spaces and Y = F×Z is equipped with the product norm. The energy has the form E(t, y) = 1 2 Ay, y Y − l(t), y Y , (4.25) where A : Y → Y * is positive denite, symmetric and bounded, and where l ∈ W 1,2 (0, T ; Y). The positive deniteness of A implies that E(t, ·) is λ-convex with λ independent from t. We write y 2 A := Ay, y . Concerning ψ 0 and Ψ, we assume ψ 0 : Z → R is a lower semi-continuous, convex, and positively 1-homogeneous functional satisfying (4.9) and Ψ : [0, T ] × Z → R is a ψ 0 -regular dissipation potential.
We start with a result on the dependence of the solutions on the right-hand side l and the dissipation potential Ψ. This result is close to the continuous dependence result in [12, theorem. 2.3], which is more general as it allows for more general uniformly convex energies as well as for temporal jumps which are treated by the Kurzweil integral. Our result is slightly more general in a dierent direction, because we do not need any a priori bounds on the temporal BV norm of t → Ψ(t, ·). This generalization is crucial for our application to dissipations R ε (t, ·) = Ψ(t, t/ε, ·) where no uniform bound is available.
Under the additional assumption that the solutions are dierentiable almost everywhere the following result would be easily derived, however it still holds in the general case, see Appendix B for a discussion and the full proof, which is done within the concept of energetic solutions.  Diss Ψ3−j (y j ; 0, t) − Diss Ψj (y j ; 0, t) . As an application of this proposition, we rst obtain the following well-posedness and Lipschitz continuity result.
Proof. Existence and continuity properties of solutions follow from Theorem 4.1 observing that (4.8c) is satised. The uniqueness of solutions and the contraction property are a direct consequence of (4.26) with l = l 1 = l 2 and Ψ = Ψ 1 = Ψ 2 .
A second result is obtained if we give a specic estimate between the two dissipation potentials, namely For the dierence of the loadings l 1 −l 2 we use the adapted norm l 1 (t)−l 2 (t) * := A −1 l 1 (t)−l 2 (t) A , and similarly for the derivative. We obtain the following explicit estimate. Corollary 4.8. Consider the situation of Proposition 4.6 and assume additionally (4.27) and y 1 (0) = y 2 (0), then for all t > 0 we obtain the estimate where ∆ = δ Diss ψ0 (y 1 ; 0, t) + Diss ψ0 (y 2 ; 0, t) .
5. Proof of Theorem 1.1. We now return back to the case that R ε is given in the oscillatory form R ε (t, v) = Φ(t, t/ε, v). We rst show that it is easy to pass to the limit in the energetic formulation if we are able to extract a weakly convergent subsequence. While in previous evolutionary Γ-convergence results for rate-independent systems (cf. [18] or [17,Sec 2.4]) it was sucient to use a uniform a priori bound for the dissipation and apply Helly's selection principle, this is not enough in the present case, since the oscillatory behavior of the dissipation potential destroys the usual arguments. ∀ŷ ∈ Y : E(t, y ε (t)) ≤ E(t,ŷ) + R ε (t,ŷ − y ε (t)) , E(t, y ε (t)) + Diss Rε (y ε , 0, t) = E(0, y 0 ) +ˆt 0 ∂ s E(s, y ε (s)) ds .

(5.2)
We now postulate the asymptotic equicontinuity which will be established in the next section in Proposition 5.2: Recall that a modulus of continuity is a continuous, nondecreasing function ω : Using y ε (0) = y 0 which is independent of ε, we also have a uniform bound and may apply the Arzela-Ascoli theorem (see e.g. [3,Prop. 3.3.1]) in the weak topology of Y restricted to a large ball. Thus, we nd a sequence ε k → 0 such that for a limit function y : [0, T ] → Y. The aim is now to show that this limit y is indeed a solution of the eective rate-independent system (Y, E, R eff ) with y(0) = y 0 . Since this solution is unique, we know that the whole family y ε converges (without selecting a subsequence).
To show that the limit y ∈ C([0, T ], Y) is an energetic solution, we have to establish the stability (4.6) and the energy balance (4.7), but now with R eff instead of Ψ.
Upper energy inequality. By the rst relation in (3.6) we have the lower estimate Diss Rε (y ε ; 0, T ) ≥ Diss R eff (y ε ; 0, T ) for all ε > 0 and all t ∈ (0, T ]. Using the lower semicontinuity of the total dissipation as stated in Lemma 3.5 and the weak lower semi-continuity of the energy E(t, ·), we can pass to the limit ε k → 0 in (5.2) to obtain E(t, y(t)) + Diss R eff (y, 0, t) ≤ E(0, y 0 ) +ˆt 0 ∂ s E(s, y(s)) ds .
Note that for the power integral on the right-hand side we can use the linearity of −´t 0 l (s), y ε k (s) ds, the weak convergence and Lebesgue's dominated convergence theorem.
Lower energy inequality. This can be obtained from the stability like in Step 5 of the proof of Theorem 4.1.
Taking the above three points together we have shown that the limit y : [0, T ] → Y is an energetic solution for the eective rate-independent system. Uniqueness. Since the uniqueness of the solution with the initial value y(0) = y 0 follows from Theorem 4.7, we see that this solution is the only possible accumulation point of the family (y ε ) ε∈(0,1) . Hence we conclude the convergence as stated in Theorem 1.1. In particular, the only missing point in the proof of the theorem is the equicontinuity stated in (5.3).

Uniform equicontinuity of solutions.
In what follows, we write y 2 A := y, Ay and y ∞ := y L ∞ (0,T ;Y) . The rst result is a basic lemma showing that we have a uniform L ∞ bound for all ε ∈ (0, 1).
The next result is the fundamental equicontinuity result, the proof of which is delicate, since there cannot be any uniform a priori bounds for the derivativesẏ ε Hence, E(t, y) is now dened for t ∈ [−T, 2T ] as well. Extending each y ε by y ε (t) = y 0 for t ∈ [−T, 0] and using (1.7), we see that y ε is the unique solution to (4.6)(4.7) on [−T, T ] with initial value y ε (−T ) = y 0 . We now consider τ and t with 0 ≤ τ < t ≤ T and want to estimate y ε (t)−y ε (τ ) A uniformly in ε ∈ (0, 1). There are unique k ∈ N 0 and θ ∈ [0, 2πε) with t = τ + 2πεk + θ.
Since y ε and y ε k both are energetic solutions with the same initial datum y 0 at t = −T , we can compare them with our estimate from Corollary 4.8. Note that y ε k is obtained with the shifted loading l k (t) = l(t−2πkε) and the shifted dissipation potential Φ(t−2πkε, s, ·). Using the a priori bound for the dissipation from Lemma 5.1 we obtain where ω 1 is the modulus of continuity of Φ(·, s, v). Clearly, we have l(s) − l k (s) ≤ 2πkε l L ∞ (0,T ;Y * ) . Moreover, for ρ > 0 we set and see that ω l is continuous with ω l (0) = 0. Hence, ω(s) = sup ρ∈[0,s] ω l (ρ) is a modulus of continuity and we nd where ω * is still a modulus of continuity.
Now we want to estimate the second term on the right-hand side in (5.6), namely y ε (τ +θ)−y ε (τ ) A , where θ ∈ [0, 2πε]. We will not be able to show equicontinuity, but we will obtain a uniform bound that vanishes for ε 0. To achieve this we rst show that the dissipation in intervals of the length 2πε is uniformly bounded.
A.2. Proof of Lemma 3.3. The rst statement is obviously true. The inequality var(u; ψ; s, t) ≥ lim sup is an immediate consequence of the denition of var(·). On the other hand, for n ∈ N we can chose a partition T n such that ψ u(t K k+1 ) − u(t K k ) ≥ var(u; ψ; s, t)− 1 n . Due to the triangle inequality, we can assume T n ⊃ T n−1 .
Integrating over time and some integration by parts lead us then to the estimate +ˆt 0 Ψ 1 (s,ẏ 1 (s))+Ψ 2 (s,ẏ 1 (s))−Ψ 1 (s,ẏ 1 (s))−Ψ 2 (s,ẏ 2 (s)) ds, which is essentially the desired estimate (4.26) as stated in Proposition 4.6. However, due to the temporal uctuations of Ψ i and the low temporal regularity of y i , we have to carry out all of these calculations in a time-discrete setting.
Proof of Proposition 4.6. As above we write y 2 A := Ay, y , and for xed t ∈ (0, T ] and N ∈ N we dene the partition t k = kt N . For all continuous functions a : [0, T ] → Y we let a k := a(t k ) and a k−1/2 := 1 2 (a k +a k−1 ). In addition to y j and l j we will also use E j (t) = E j (t, y j (t)) and σ j (t) = DE j (t, y j (t)) = Ay j (t) − l j (t), Subsequently using the quadratic structure of E(t, y) and (4.7) we obtain the l j , y j ds − Diss Ψj (y j ; t k−1 , t k ).