Modeling solutions with jumps for rate-independent systems on metric spaces

Rate-independent systems allow for solutions with jumps that need additional modeling. Here we suggest a formulation that arises as limit of viscous regularization of the solutions in the extended state space. Hence, our parametrized metric solutions of a rate-independent system are absolutely continuous mappings from a parameter interval into the extended state space. Jumps appear as generalized gradient flows during which the time is constant. The closely related notion of BV solutions is developed afterwards. Our approach is based on the abstract theory of generalized gradient flows in metric spaces, and comparison with other notions of solutions is given.


1.
Introduction. Rate-independent evolutionary systems arise in a very broad class of mechanical problems, usually in connection with hysteretic behavior: with no claim at completeness, we may mention for instance elastoplasticity, damage, the quasistatic evolution of fractures, shape memory alloys, delamination and ferromagnetism, referring to [20] for a survey of the modeling of rate-independent phenomena.
Because of their relevance in applications, the analysis of these systems has attracted some attention over the last decade, also in connection with the issue of the proper formulation of their time dependence, which raises interesting and challenging questions even for systems depending on a finite number of degrees of freedom.
This paper is concerned with the metric-variational formulation of such kind of rate-independent evolutions, generated from a vanishing-viscosity approximation. An important part is also devoted to the analysis of several solution notions, in order to compare and clarify the differences between the various approaches. Since we want to capture and enucleate the essential features of the variational mechanism governing the evolution process, we develop our theory in a finite-dimensional setting, thus avoiding the further complications induced by the technical tools of non-smooth analysis in metric spaces. Nonetheless, we try to present the theory in an intrinsic and compact form, which can be adapted to the more general frameworks needed for real applications to PDE's; such extensions will be discussed in a forthcoming paper.
In several situations rate-independent problems may be recast in the form of a doubly nonlinear evolution equation involving two energy functionals, namely ∂qR(q(t),q(t)) + ∂ q E(t, q(t)) 0 in Q for a.a. t ∈ (0, T ) , where Q is a separable Banach space, R : Q × Q → [0, ∞] a dissipation functional and E : [0, T ] × Q → (−∞, ∞] an energy potential, ∂q and ∂ q denoting their subdifferential with respect to the second variable. Rate-independence is rendered through 1-homogeneity of the functional R with respect to its second variable. Indeed, assuming that R(q, γv) = γR(q, v) for all γ ≥ 0 and (q, v) ∈ Q × Q, one has that equation (1) is invariant for time-rescalings. This captures the main feature of this kind of processes, which are driven by an external loading set on a time scale much slower than the time scale intrinsic to the system, but still fast enough to prevent equilibrium. Typically, this quasistatic behavior originates in the limit of systems with a viscous, rate-dependent dissipation. The formulation of rate-independent problems in terms of the subdifferential inclusion (1) has been thoroughly analyzed in [22], in the case of a reflexive Banach space. Existence of solutions to the Cauchy problem for (1) is proved through approximation by time discretization and solution of incremental minimization problems. However, in many applications the energy E(t, ·) is neither smooth nor convex, and the state space Q is often neither reflexive nor the dual of a separable Banach space (see e.g. [16] for energies having linear growth at infinity). Furthermore, Q may even lack a linear structure (in these cases we will denote it by the calligraphic letter Q), like for finite-strain elastoplasticity [19,18] or for quasistatic evolution of fractures [7].
In such situations, the differential formulation (1) cannot be used. In [21,24], the concept of energetic solution for general rate-independent energetic systems (Q, E, D) has been introduced, by replacing the infinitesimal metric R of the subdifferential formulation (1) by a global dissipation distance D : Q × Q → [0, ∞]. This formulation, see Section 5.1, is derivative-free, and thus applies to solutions with jumps and can be used in very general frameworks, like, for example, in a topological space Q, with E and D lower semicontinuous only, see [17,20,12]. Energetic solutions are very flexible and allow for a quite general existence theory; however, the global stability condition, asking that q(t) globally minimizes the map q → E(t, q) + D(q(t), q), implies that solutions jump earlier than physically expected, since they are forced to leave a locally stable state, see e.g. [19,Ex. 6.1] or [14,Ex. 6.3], and Example 1 below. Moreover, existence of energetic solutions is proved via time discretization and incremental global minimization, but, as discussed in [19,Sec. 6], local minimization would be more appropriate both from the perspective of modeling and of numerical algorithms.
In response to these issues, in [10] a vanishing viscosity approach was proposed to derive new solution types for rate-independent systems (Q, R, E). There, Q is assumed to be a finite-dimensional Hilbert space Q and E ∈ C 1 ([0, T ] × Q). The natural viscous approximation of (1) is obtained by adding a quadratic term to the dissipation potential, viz. R ε (q, v) = R(q, v) + ε 2 v 2 , and leads to the doubly nonlinear equation εq(t) + ∂qR(q(t),q(t)) + ∂ q E(t, q(t)) 0 for a.a. t ∈ (0, T ) .
Using dim Q < ∞, the existence of solutions q ε ∈ H 1 ([0, T ]; Q) is obvious and passing to the limit ε 0 in (2) leads to new solutions and to a finer description of the jumps, which occur later than for energetic solutions. The key idea (see Section 2) is that the limiting solution at jumps shall follow a path which somehow keeps track of the viscous approximation. To exploit this additional information, one has to go over to an extended state space: reparametrizing the approximating viscous solutions q ε of (2) by their arclength τ ε , and introducing the rescalings t ε = τ −1 ε and q ε = q ε • t ε , one studies the limiting behavior of the sequence {( t ε , q ε )} ε as ε ↓ 0. Hence, in [10] it was proved that (up to a subsequence), {( t ε , q ε )} ε converges to a pair ( t, q), whose evolution encompasses both dry friction effects and, when the system jumps, the influence of rate-dependent dissipation. In fact, the jump path may be completely described by a gradient flow equation, which leads to this interpretation: jumps are fast (with respect to the slow external time scale) transitions between two metastable states, during which the system switches to a viscous regime. Furthermore, solutions of the limiting rate-independent problem can be constructed by means of a time-discretization scheme featuring local, rather than global, minimization.
This paper provides the first step of the generalization of these ideas to the much more general metric framework using the concept of curves of maximal slope, which dates back to the pioneering paper [8]. We also refer to the recent monograph [1], the references therein, and to [26]. The general setup starts with a complete metric space (X, d) and introduces the metric velocity which is defined a.e. along an absolutely continuous curve q : [0, T ] → X. This replaces the norm of the derivative q in the smooth setting, and, in the same way, the norm of the (Gâteaux)-derivative or the subdifferential of a functional F : X → (−∞, ∞] is replaced by the local slope of F in q ∈ dom(Ψ), which is defined by where (·) + denotes the positive part. With these concepts, the viscous problem (2) has the equivalent metric formulation d dt for a.a. t ∈ (0, T ), see Section 3.1 for further details. It was proved in [26,Thm. 3.5] that, under suitable assumptions on E, for every ε > 0 the related Cauchy problem has at least one solution q ε ∈ AC([0, T ]; X).
Following the approach of [10], for every ε > 0 we now consider the (arclength) rescalings ( t ε , q ε ) associated with q ε , which in turn fulfill a rescaled version of (5), cf. (30). Under suitable assumptions, in Theorem 3.3 we shall show that, up to a subsequence, {( t ε , q ε )} converges to a limit curve ( t, q) ∈ AC([0, T ]; [0, T ] × X) such that and the energy identity Indeed, the very focus of this paper is on getting insight into the properties of parametrized metric solutions and comparing them with the other solution notions for rate-independent evolutions. That is why, in order to avoid technicalities and to highlight, rather, the features of our approach, throughout the next sections we shall work in a technically simpler setup, in which the state space X is a finitedimensional manifold, endowed with a (Finsler) distance d associated with a 1homogeneous dissipation functional R : TQ → [0, ∞), and an energy E ∈ C 1 ([0, T ]× Q). The investigation of a general infinite-dimensional or fully metric framework with applications to PDE's is postponed to a forthcoming paper.
The notion of parametrized metric solution ( t, q) generalizes the outcome of the finite-dimensional vanishing viscosity analysis of [10] and hence allows for the same mechanical interpretation, see Remark 3. Namely, according to whether either of the derivatives t or | q | is null or strictly positive, one distinguishes in (6b) three regimes: sticking, rate-independent evolution, and switching to a viscous regime (in correspondence to jumps of the system from one metastable state to another). In this metric setup as well, we show that the behavior of the system along a jump path is described by a generalized gradient flow. This can be seen more clearly when considering the non-parametrized solution q corresponding to the pair ( t, q). The latter functions are called BV solutions of (Q, d, E) and are pointwise limits of the un-rescaled vanishing viscosity approximations q ε , see Definition 4.2 and Section 4 for an analysis of their properties. In particular, we shall show how to pass, by means of a suitable transformation, from a (truly jumping) BV solution q to a ("virtually" jumping) parametrized solution ( t, q), and conversely.
In Section 5, we compare the notion of BV solutions with other solutions concepts, namely with the energetic solutions of [21,24], and with the approximable and local solutions of [14,27,5] (suitably rephrased in the metric setting, see Definitions 5.1, 5.2, and 5.3). In Section 5.3 we review the notion of Φ-minimal solutions of a rate-independent evolutionary system, proposed in [28] using a global variational principle in terms of a suitably defined partial order relation between trajectories. First, we conclude that the notion of local solution is the most general concept, including energetic and BV solutions, whereas BV solutions encompass approximable and Φ-minimal solutions. Moreover, our notion of BV solutions has "more structure", which makes it robust with respect to data perturbations (cf. Remark 6), whereas neither approximable nor Φ-minimal solutions are upper semicontinuous with respect to data perturbations.
Further insight into the comparison between the various solution notions is provided by the examples presented in Section 6, which are one or two-dimensional, such that the set of all solutions can be discussed easily. The one-dimensional case in fact relates to crack growth (under the assumption of a prescribed crack path), which was treated in [27,5,25,14]. The solution concepts developed there are also based on the vanishing viscosity method. In the latter case, the solution type in fact coincides with our notion of BV solution. Related ideas using the vanishing viscosity method for PDEs are found in [6], for a model for elastoplasticity problems with softening, in [15], for rate-independent problems with convex energies and jumps in the loading, and in [23], for general parabolic PDEs with rate-independent dissipation terms.
2. Setup and mechanical motivation. We consider a manifold Q that contains the states of our system. The energy E of the system depends on the time t ∈ [0, T ] and the state q ∈ Q. Throughout the paper, we shall assume that E ∈ C 1 (Q T ), where Q T = [0, T ] × Q denotes the extended state space. The evolution of the system is governed by a balance between the potential restoring force −D q E(t, q) and a frictional force f . The latter is given by a continuous dissipation potential R : TQ → [0, ∞), in the form f ∈ ∂qR(q,q). We generally assume that R(q, ·) : T q Q → [0, ∞) is convex and ∂qR(q,q) ⊂ T * q Q is the set-valued subdifferential. Hence, the system is governed by the differential inclusion in which we have introduced a small parameter ε > 0 to indicate that we are on a very slow time scale. Further, we suppose that R = R 1 + R 2 , where R 1 : TQ → [0, ∞) and R 2 : TQ → [0, ∞) are such that for every q ∈ Q R 1 (q, ·) is convex and homogeneous of degree 1, R 2 (q, ·) is convex and homogeneous of degree 2.
We call R 1 the potential of rate-independent friction and R 2 the potential of viscous friction.

Remark 1.
A prototype of the mechanical situation we aim to model arises in connection with a system of k particles (k ≥ 1) moving in R d , hence with state-space Q = q = (q 1 , . . . , q k ) : q i ∈ R d = R kd . We impose rate-independent friction R 1 and viscous friction R 2 via where |q j | is the Euclidean norm of the j-th particle velocity and µ, ν : R d → [0, ∞) are given continuous functions. For k = 1 the potentials R j are related by R 2 (q,q) = ν(q) 2µ 2 (q) R 2 1 (q,q), while for k = 2 their interplay is more complex.
Our aim is to understand the limiting behavior of the solutions to (8) for ε → 0. In fact, we expect that for ε → 0 the rate-independent friction dominates, but the solution q ε : [0, T ] → Q may develop sharp transition layers, withq of order 1/ε. In the limit we obtain a jump, but in order to characterize the jump path the viscous potential is crucial.
The key idea is to study the trajectories T ε = { (t, q ε (t)) | t ∈ [0, T ] } in the extended state space Q T . The point is that the limit of trajectories T ε may no longer be the graph of a function. To study the limits via differential inclusions, we may reparametrize the trajectories T ε in the form where t ε is supposed to be nondecreasing and absolutely continuous.
For passing to the limit it is now helpful to select a family of parametrizations via m ε ∈ L 1 loc ((0, ∞)) converging to m in L 1 loc ((0, ∞)), with m(s), m ε (s) > 0 for a.a. s ∈ (0, ∞), and to assume Note that this can always be achieved. In particular, when relation (9) leads to the arclength parametrization of T ε , which was considered in [10]. The total length is S ε := T + T 0 2R 2 (q ε (t), q ε (t)) dt. Since S ε → T up to a subsequence (thanks to standard energy estimates), it is not restrictive to assume that S ε is independent of ε by the simple linear rescalingm ε (s) = m ε (sS ε /S).
By the chain rule and the j-homogeneity we have for a.a. s ∈ (0, S ε ) . Now, using (9) and defining easy computations (cf. [10,Thm. 3.1] in the particular case of (10)) show that (8) is equivalent to In this formulation we may pass to the limit, and we expect to obtain the limit problem where Formulation (12) is in fact a generalization of the one in [10], where rigorous convergence proofs of problem (11) to (12) are derived (in the case Q is finite-dimensional). Although R is no longer 1-homogeneous, the limit problem is still rate-independent: upon adjusting the free function m, one sees immediately that system (12) is invariant under time reparametrizations.
In the present paper we concentrate on the case R 2 (q, v) = 1 2 R 1 (q, v) 2 , since it is this case which can be generalized to abstract metric spaces and hence to the infinite-dimensional setting. By introducing the dual norm of a co-vector the operators ∂R 1 (q, ·) and ∂R 2 (q, ·) can be characterized by and they satisfy ∂R 2 (q, v) = R 1 (q, v) ∂R 1 (q, v) and Proof. First, we note that, using the 1-homogeneity of R 1 , the second of (12) may be rewritten as t m + R 1 q, q m = 1 a.e. in (0, T ) .
Now, it is not difficult to see that where the equivalences in parentheses follow from (17). Combining (18) with the first of (12) and using that ∂R 1 is 0-homogeneous, we deduce (16). Conversely, starting from (16), we put m(s) := t (s) + R 1 ( q(s), q (s)) for a.a. s ∈ (0, T ) and note that, by the third of (16), m > 0 a.e. in (0, T ) and m ∈ L 1 (0, T ), since R 1 is continuous. Using (17) and arguing as in the above lines, one sees that, if the pair ( t , λ) satisfies the second of (16), then λ∂qR 1 ( q, q ) = ∂q R( q, q m ) , which allows us to deduce the differential inclusion in (12) from the one in (16).

Analysis with metric space techniques.
3.1. Problem reformulation in a metric setting. First of all, we complement the setup of Section 2 by specifying our assumptions on the rate-independent system (Q, R 1 , E), where Q is the ambient space, R 1 the dissipation functional, and E the energy functional. Namely, we require that Q is a finite-dimensional and smooth manifold, (3.Q) and the energy functional satisfies called Minkowski norm in the Finsler setting. Then, Hence, (Q, d) is a metric space, which we assume to be complete. Like in the previous section, we let R 2 ≡ 1 2 R 2 1 . For a curve q ∈ AC([0, T ]; Q), the Finsler length of its velocity q (t) is given by and satisfies Using R 1 , we define the associated local slope |∂ q E| : which is the conjugate norm with respect to the Minkowski norm R 1 (q, ·) of the differential of the energy in the cotangent space T * q Q. Using the smoothness of E we have that for every curve (t, q) ∈ AC([s 0 , s 1 ]; Q T ) the map s → E(t(s), q(s)) is absolutely continuous and the chain rule for E gives for a.a. s ∈ (s 0 , s 1 ). On the other hand, formulae (20) and (22) yield Therefore, every (t, q) ∈ AC([s 0 , s 1 ]; Q T ) fulfills a.e. in (s 0 , s 1 ) the chain rule inequality

SOLUTIONS WITH JUMPS FOR RATE-INDEPENDENT SYSTEMS 9
The metric formulation of doubly nonlinear equations. We now see how notions (20) and (22) so far introduced come into play in the reformulation of a class of doubly nonlinear evolution equations in the metric setting (Q, d).
3.2. Rate-independent limit of viscous metric flows. We now consider the case of small viscosity added to the rate-independent dissipation, namely We obtain ψ * for a.a. s ∈ (s 0 , s 1 ), with Clearly, for fixed α > 0 the limit as ε 0 of M ε (α, ν, ξ) gives ψ 0 (ν) + ψ * 0 (ξ). However, our purpose is to blow-up the time parametrization whenever jumps occur. Indeed, the finite-dimensional case (see [10]) suggests that jumps in the rate-independent evolution will occur at fixed rescaled time (i.e., when t = 0). Hence, we also have to consider the case α → 0 as ε → 0. For this, note that when ξ > 1 M ε (·, ν, ξ) assumes its minimum on [0, ∞) and obtain the following result.
Then, we have the following results: Γ-limsup estimate: Proof. Estimate (34a) is trivial for α > 0, as we have pointwise convergence then. If α = 0, we employ (32) and use that M inf is continuous.
The notion of parametrized metric solutions. The following notion of parametrized metric solution of the rate-independent system (Q, d, E) is proposed in a general form, replacing the function M 0 obtained in the vanishing viscosity limit with a generic function M satisfying (35). The proposed notion is fitted to the metric framework and does not need a differentiable structure. However, it strongly relies on the fact that the small viscous friction ε|q |(t) 2 is given in terms of the same metric velocity as the rate-independent friction, see also the assumption R 2 = 1 2 R 2 1 . We refer to [10,23] for settings avoiding this assumption.
Since the set Ξ can be easily characterized via we ultimately find that (42b) can be replaced by the simple relations (42c).
Passing to the limit in (52) we easily get Now, the smoothness of E and assumption (3.Q) yield that E, ∂ t E, and |∂ q E| are continuous with respect to both arguments, thus we readily infer from (50) that To proceed further, we integrate (30) over [s 0 , s 1 ] and obtain On the left-hand side we can pass to the limit ε → 0 using (54) and (56), while for the right-hand side we use Part (B) of Lemma 3.1: where we have used (55) and the monotonicity of M 0 (α, ·, ξ) for the last estimate. We see that ( t, q) fulfills (40) a.e. in (s 0 , s 1 ) , and it is therefore a (possibly) degenerate parametrized metric solution of (Q, d, E). Moreover, comparing the last inequalities with the integrated form of (43), we get M 0 t (r), η(r), |∂ q E| t(r), q(r) = M 0 t (r), | q |(r), |∂ q E| t(r), q(r) < ∞ for a.a. r ∈ (s 0 , s 1 ). Since M 0 (α, ·, ξ) is strictly monotone in its domain of finiteness, we get η(r) = | q |(r) for a.a. r ∈ (s 0 , s 1 ), thus obtaining (51). Using the first convergence in (54) we also find t + | q | = m and the last assertion follows.
Remark 6. Mimicking the argument of the proof of Theorem 3.3, under the same assumptions it is also possible to prove a result of stability with respect to initial data for parametrized metric solutions. Namely, let {( t n , q n )} be a sequence of parametrized metric solutions on a time interval [s 0 , s 1 ], such that ( t n (s 0 ), q n (s 0 )) = (t n 0 , q n 0 ) for every n ∈ N, with (t n 0 , q n 0 ) → (t 0 , q 0 ) and m n := t n + | q n | m in L 1 (s 0 , s 1 ). Then, there exists a parametrized metric solution ( t ∞ , q ∞ ), starting from (t 0 , q 0 ), such that, up to the extraction of a subsequence, ( t n , q n ) → ( t ∞ , q ∞ ) uniformly in [s 0 , s 1 ], with ( t n , | q n |) ( t ∞ , | q ∞ |) in L 1 (s 0 , s 1 ).
where sup is taken over all n ∈ N and all partitions τ 0 < τ 1 · · · < τ n−1 < τ n with τ 0 , τ n ∈ I. We set where we emphasize that functions are defined everywhere, as is common for rateindependent processes. For q ∈ BV([0, T ]; Q) and t ∈ [0, T ] the left and right limits exist: where we put q(0 − ) = q(0) and q(T + ) = q(T ). In general, the three values q(t − ), q(t), and q(t + ) may differ. We define the continuity set C q and the jump set J q by Indeed, our definition of "Var" is such that we have for all 0 ≤ r ≤ s ≤ t ≤ T Var(q, [r, s]) = d(q(r), q(r+)) + Var(q, (r, s)) + d(q(s−), q(s)) , The distributional derivative of V q defines a nonnegative Radon measure µ q such that and, more generally (see [11, 2.5.17 where T 0 ζ dV q denotes the Riemann-Stieltjes integral. As usual, µ q can be decomposed into a continuous (also called diffuse) part µ co q and a discrete part µ J q , where for a Borel set A ⊂ [0, T ] we have in accordance with formula (59) above. In the following technical lemma (whose proof is postponed to the end of this section), we will discuss the link between a BV map q : [0, T ] → Q and its graph q(t) = (t, q(t)) in the extended state space Q T , endowed with the distance d Q T ((t 0 , q 0 ), (t 1 , q 1 )) := |t 0 −t 1 | + d(q 0 , q 1 ). We denote by L [a,b] the Lebesgue measure on the interval [a, b], whereas L denotes a general one-dimensional Lebesgue measure.
Then, the following statements hold: The map t is uniquely determined, it is the right-continuous inverse of σ, and it is injective on C q := σ(C q ) = t −1 (C q ). The maps q and ρ are uniquely determined on the set C q and satisfy q(s) = q( t(s)) and τ ( ρ(s)) = t(s) for all s ∈ C q .
A straightforward consequence of the symmetry R 1 (q, −v) = R 1 (q, v) (see (3.R)) is that S α (t; q 0 , q 1 ) = S α (t; q 1 , q 0 ). Using the chain rule inequality (25) we find for all (t, q 0 , The notion of BV solution to the rate-independent system (Q, d, E), which we are going to introduce, relies on a version of the chain rule for BV functions with values in a metric space. In order to state it, for a general q ∈ BV([0, T ]; Q) and |∂ q E| (r, q(r)) µ co q (dr) Based on (70), we define a second functional Γ via Obviously, we have Σ 0 (q, [s, t]) ≥ Γ(q, [s, t]) ≥ 0 and both functionals Σ 0 (q, ·) and Γ(q, ·) fulfill the additivity property (60), when considered as functions on intervals.
Proof. The function t → E(t) := E(t, q(t)) is of bounded variation on [0, T ] and its jump set is contained in J q . We denote by η = d dt E its distributional derivative (a bounded Radon measure on (0, T )) and by η co its diffuse part, defined as η co (A) := η(A ∩ C q ) for all Borel sets A ⊂ [0, T ]. Thanks to (70), we have (72) if we show that We introduce the maps σ and q = ( t, q) as in Lemma 4.1 and we set E(s) := E( t(s), q(s)) for all s ∈ [0, S], so that E(t) = E(σ(t)) for all t ∈ [0, T ]. Since t, q are Lipschitz continuous and E is of class C 1 , the classical chain rule (23)-(25) yields On the other hand, since E is a Lipschitz map and since dσ dt = µ q , the general chain rule of [2] yields Taking into account (66), we conclude (notice that E • σ is well defined µ co q -a.e., since, for every Lebesgue negligible set N ⊂ C q = σ(C q ), formula (65) yields µ co q (σ −1 (N )) = 0). Now we are able to introduce the notion of BV solution. The formulation is more complicated than the one defining parametrized metric solutions, but it nicely reflects the different flow regimes of rate-independent flow, and the jumps. A shorter but much more implicit formulation will be given in Remark 5.
The above formulation of BV solutions looks quite lengthy compared to the more elegant forms of gradient-like flows, which can be characterized by one inequality, cf. e.g., (26) or (38c). However, this formulation reflects the mechanical interpretation of the three different flow types (namely sticking, slipping, and jumping) quite well. The following result presents a more compact form, which is however less tractable for further analysis.
Recalling (68) and the chain rule, for every t ∈ J q we find a curve y t satisfying condition (76d).
BV and parametrized metric solutions. We claim that the notion of BV solution is essentially the same as that of parametrized metric solution. Intuitively, (76a) corresponds to (41). Further, (76b) and (76c) are the analog in the BV setting of the first of (42c), which encompasses both sticking and rate-independent evolution (recall Remark 3). The jumping regime is accounted for by condition (76d): at jump times, the system switches to a viscous, rate-dependent behavior, following a path described by a generalized gradient flow, see (γ) in (76d).
In order to formalize these considerations, we return to the trajectories in Q T . Indeed, we may associate with each BV solution q BV a trajectory, by filling the jumps of the graph { (t, q BV (t)) | t ∈ [0, T ]) } with the curves y t ∈ AC([0, 1], Q), for t ∈ J qBV . Thus, we obtain By construction, T is a connected curve that has exactly the length Var(q, [0, T ])+T if we use the extended metric d Q T ((t 0 , q 0 ), (t 1 , q 1 )) := |t 0 −t 1 | + d(q 0 , q 1 ) on Q T . Hence, there exists an absolutely continuous parametrization of T, and it can be shown that this parametrized curve is a parametrized metric solution. Indeed, in Example 4 we shall show, that to a given BV solution, there may correspond infinitely many distinct parametrized metric solutions.
On the other hand, we can pass from parametrized metric solutions ( t, q) defined in [s 0 , s 1 ] to BV solutions by choosing At the jump times t we can in fact choose any point defines a connecting jump path as desired in (76d). We collect these remarks in the next proposition, whose proof easily follows from Lemma 4.1 (see also Remark 2).  The next result shows that BV solutions can be directly obtained as a vanishing viscosity limit, as in Theorem 3.3, but now rescaling is not needed. The imposed a priori bound on the total variation for the viscosity solutions q ε can be easily obtained from the energy inequality (26) under general assumptions on E. Corollary 1 (Vanishing viscosity limit (II)). Let (Q, d, E) satisfy (3.Q), (3.R), and (3.E). For every ε > 0 let q ε ∈ AC([0, T ]; Q) be a solution to (26) for ψ = ψ ε . Assume that q ε (0) → q 0 as ε 0 and Var(q ε , [0, T ]) ≤ C for all ε > 0 with a constant C independent of ε. Then, there exist a subsequence q ε k with ε k 0 and a BV solution q for (Q, d, E) such that q ε k (t) → q(t) as k → ∞ for all t ∈ [0, T ].
We conclude the section with the Proof of Lemma 4.1: Part A) is immediate. Part B) is an obvious extension of [11, 2.5.16], since each couple of points in Q can be connected by a geodesic. Notice that τ (V q (t)) = t and therefore τ ( ρ(σ(t))) = t if t ∈ C q . We thus get τ • ρ = t in C q .
In order to prove C) it is not restrictive to assume A = [0, T ] and ζ ∈ C 0 C ([0, T ]). Then, (64) follows from (62) and [11, 2.5.18(3)], observing that Let us now prove the first identity of (66): setting J q := t −1 (J q ) = [0, S] \ C q , we observe that t (s) = 0 for L-a.e. s ∈ J q . Since t is Lipschitz continuous and monotone, the change of variable formula and (64) yield, for every continuous function ζ with compact support in (0, T ), The second identity of (66) follows by a similar argument: The identity ρ •σ = | q |•σ follows from the property V q (t) = V q (σ(t)) for all t ∈ C q , so that V q (s) = ρ(s) for all s ∈ C q .

Other solution concepts.
Here we discuss other notions of solutions for rateindependent systems (Q, d, E), namely energetic solutions, local and approximable solutions, and Φ-minimal solutions.
5.1. Energetic solutions. The concept of energetic solutions provides the most general setting, in the sense that it does not even rely on a differentiability structure like the Finsler metric R, but only uses the distance d. In such a framework it is even possible to consider quasi-metrics (i.e. unsymmetric and allowed to take the value ∞), cf. [20].
We refer to [21,24] for the origins of this theory and to [20] for a survey. In analogy with (77) we have the jump relations for all t ∈ J q . Here they are easily obtained by considering the energy identity E(s, q(s)) + Var(q, [r, s]) = E(r, q(r)) + s r ∂ τ E(τ, q(τ )) dτ , which follows immediately from (E), for the intervals [t−h, t], [t, t+h], and [t−h, t+h], respectively, and taking the limit as h 0.

ALEXANDER MIELKE, RICCARDA ROSSI AND GIUSEPPE SAVARÉ
To compare energetic and BV solutions, we introduce the global slope G[E(t, ·)] : for all q ∈ D .

Remark 7.
It is well known that (S) implies the lower energy estimate The following result essentially states that every energetic solution q is a BV solution outside its jump set. Moreover, if the jump relations (77) and (82) are both satisfied, then an energetic solution is also a BV solution. Conversely, if a BV solution additionally satisfies G[E(t, ·)](q(t)) ≤ 1, then it is an energetic solution as well. Proof. To prove (A), we first note that the necessity of (76d) is obvious as it is part of the definition of BV solutions. To establish the sufficiency in (A), we observe that (76d) yields (77), so that the dissipation term Σ 1 of Proposition 5 satisfies Hence, (78) follows from (85). Thus, statement (A) is established.
The necessity of the additional condition on G[E(t, ·)](q(t)) is clear, since energetic solutions have to satisfy the stronger stability condition (S). To show the sufficiency we observe that the additional condition yields so that (85) follows from (78).

Remark 8. The additional condition in Proposition 7(B) is implied by the general condition
If (86) To establish (86) for F note that for each q and ε > 0 we have d( q, q) −ε for some q. Moreover, for each n ∈ N, choosing θ = 1/n we find q n with d( q, q n ) = 1 n d( q, q) and d(q n , q) = n−1 n d( q, q). Applying (87) we obtain In this way, part (B) of Proposition 7 is a generalization to the metric setting of Theorem 3.5 in [22], which states that for a Banach space Q, a convex energy functional E, and a translation invariant metric d the subdifferential formulation and the energetic formulation for (Q, d, E) are equivalent.

5.2.
Local and approximable solutions. As we have already mentioned in the introduction, energetic solutions have the disadvantage that the stability condition (S) is global, so that solutions tend to jump earlier than expected, see Example 1.
To avoid these early jumps, the vanishing viscosity method was employed in [10,6,27,14]. When avoiding parametrization and studying the limits of the viscous solutions q ε : [0, T ] → Q directly, one obtains an energy inequality and a local stability condition. Hence, we next introduce the notions of local solution and of approximable solution, generalizing the definitions given in [27] to the metric setting. (a1) |∂ q E| (t, q(t)) ≤ 1 for a.a. t ∈ [0, T ]; (a2) for all r, s ∈ [0, T ] with r < s we have E(s, q(s)) + Var(q, [r, s]) ≤ E(r, q(r)) + s r ∂ t E(τ, q(τ )) dτ. We will see in the examples of Section 6 that the notion of local solution is very general. Using (83), it is clear that all energetic solutions are local solutions. Similarly, all BV solutions are local solutions. To see this, we use (76c) and (76d) to obtain (a1), since there are at most a countable number of jump points, and we conclude Σ 0 (q, [r, s]) ≥ Var(q, [r, s]) for 0 ≤ r ≤ s ≤ T , which gives (a2).
On the other hand, note that, unlike the case of energetic solutions, the combination of the local stability condition with the energy inequality does not provide full information on the solution q. In particular, the behavior of the solution at jumps is poorly described by relations (a1) and (a2). This also highlights the role of the term Σ 0 (q, ·), here missing, in the energy identity for BV solutions. As a consequence there are many more local solutions, see also Example 1.
Indeed, the vanishing viscosity method turns out to provide a selection criterion for local solutions. Among local solutions, we thus distinguish the following ones: Definition 5.3. A mapping q : [0, T ] → Q is called approximable solution, if there exists a sequence (ε k ) k∈N with ε k 0 and solutions q ε k ∈ AC([0, T ], Q) of (26) with ψ = ψ ε k such that for all t ∈ [0, T ] we have q ε k (t) → q(t).
It follows from Corollary 1 that, under the assumptions (3.Q), (3.R), and (3.E), any approximable solution is a BV solution as well.
The notion of approximable solutions suffers from two drawbacks. First of all, there is no direct characterization of the limits in terms of a subdifferential inclusion or variational inequality, unlike for parametrized/BV solutions, recall Proposition 3. Secondly, since the solution set is defined through a limit procedure, it is not upper semicontinuous with respect to small perturbations, as shown in Example 3. This is in contrast with the stability properties of the set of parametrized/BV solutions, see Remark 6.
Then, the order relation is given by Like for energetic solutions, this solution notion appears particularly suitable to handle nonsmooth energy functionals, since no derivatives/slopes of E with respect to the variable q occur in the definition of the functional Φ.
We now show that, in a smooth setting, Φ-minimal solutions are parametrized metric solutions. Using suitably chosen test functions, it can be shown that a necessary condition for Φ-minimality is the local condition d ds Φ[(t, q)](s) ≤ N(t(s), q(s)) for a.a. s ∈ (s 0 , s 1 ), where N : Q T → R is defined via A simple calculation gives N(t, q) = 0 for |∂ q E| (t, q) ≤ 1, 1 − |∂ q E| (t, q) for |∂ q E| (t, q) ≥ 1 ∀ (t, q) ∈ Q T .
Since t + |q | = 1 a.e. and d ds Φ[(t, q)](s) = d ds E(t(s), q(s)) + |q |(s) − ∂ t E(t(s), q(s))t (s) for a.a. s ∈ (s 0 , s 1 ), we conclude that all Φ-minimal solutions satisfy a.e. in (s 0 , s 1 ) d ds E(t(s), q(s)) − ∂ t E(t(s), q(s))t (s) ≤ − M t (s), |q |(s), |∂ q E| (t(s), q(s)) , together with the constraint t (s) + |q |(s) = 1, where M (α, ν, ξ) = ν + (ξ − 1) + . We have thus proved that any Φ-minimal solution (t, q) on [s 0 , s 1 ] is a parametrized metric solution, and hence a BV solution (up to a parametrization). The opposite is in general not true, see Example 2. Further, Example 3 shows that the set of Φ-minimal solutions is not stable with respect to perturbations. 6. Examples. Many of the differences between the various solution concepts discussed above manifest themselves already in the case in which the state space is the real line. Hence, we discuss the very simple model with