Abstract
In this paper, we investigate the origin of the balanced viscosity solution concept for rate-independent evolution, in the setting of a finite-dimensional space. Namely, given a family of dissipation potentials \((\Psi _n)_{n\in {\mathbb {N}}}\) with superlinear growth at infinity and suitably converging to a 1-positively homogeneous potential \(\Psi _0\), and a smooth energy functional \({\mathcal {E}}\), we provide sufficient conditions on them ensuring that the solutions of the associated (generalized) gradient systems\((\Psi _n,{\mathcal {E}})\) converge as \(n\rightarrow \infty \) to a Balanced Viscosity solution of the rate-independent system driven by \(\Psi _0\) and \({\mathcal {E}}\). In specific cases, we also obtain results on the reverse approximation of balanced viscosity solutions by means of solutions to gradient systems. Our approach is based on the key observation that solutions to gradient systems/that Balanced Viscosity solutions to rate-independent systems can be characterized as (null-)minimizers of suitable trajectory functionals, for which we indeed prove Mosco-convergence. As particular cases, our analysis encompasses both the vanishing-viscosity approximation of rate-independent systems from Mielke et al. (ESAIM Control Optim Calc Var 18(1):36–80, 2012, J Eur Math Soc 18(9):2107–2165, 2016), and their stochastic derivation developed in Bonaschi and Peletier (Contin Mech Thermodyn 28:1191–1219, 2016).
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1 Introduction
Over the last years, rate-independent systems have been the object of intensive mathematical investigations. This is undoubtedly due to their vast range of applicability. Indeed, this kind of processes seems to be ubiquitous in continuum mechanics, ranging from shape-memory alloys to crack propagation, from elastoplasticity to damage and delamination. They also occur in fields such as ferromagnetism and ferroelectricity. We refer to [32, 39] for a thorough survey of all these problems.
Besides its applicative relevance, though, rate-independent evolution has an own, intrinsic, mathematical interest. This is apparent already in the context of a finite-dimensional rate-independent system, to which we shall confine the analysis developed within this paper. In general, such a system is driven by a dissipation potential\(\Psi _0: {\mathbb {R}}^d \rightarrow [0,+\infty )\) (non-degenerate), convex, and positively homogeneous of degree 1, and an energy functional\({\mathcal {E}}: [0,T]\times {\mathbb {R}}^d \rightarrow {\mathbb {R}}\); in particular, throughout the paper, we will consider a smooth energy \({\mathcal {E}}\) such that the power function \(\partial _t {\mathcal {E}}\) is controlled by \({\mathcal {E}}\) itself, namely
The pair \((\Psi _0,{\mathcal {E}}) \) gives rise to the simplest example of rate-independent evolution, namely the gradient system
where \(\partial \Psi _0: {\mathbb {R}}^d \rightrightarrows {\mathbb {R}}^d \) is the subdifferential of \(\Psi _0\) in the sense of convex analysis, whereas \(\mathrm {D}{\mathcal {E}}\) is the differential of the map \(u \mapsto {\mathcal {E}}(t, u)\). Due to the 0-homogeneity of \(\partial \Psi _0\), (1.1) is invariant for time rescalings, i.e., it is rate-independent. Now, it is well known that, even in the case of a smooth energy \({\mathcal {E}}\), if \(u\mapsto {\mathcal {E}}(t, u)\) fails to be strictly convex, then absolutely continuous solutions to (1.1) need not exist. In the last two decades, this has motivated the development of various weak solvability concepts for (1.1) and, in general, for rate-independent systems in infinite-dimensional Banach spaces, or even topological spaces.
While referring to [33, 39] for a survey of all weak notions of rate-independent evolution, in the following lines we shall focus on the concepts of energetic and balanced viscosity solutions. The study of these notions poses several interesting problems already in the finite-dimensional context and motivates the analysis developed in this paper.
Energetic and Balanced Viscosity solutions to rate-independent systems. The concept of energetic solution was first proposed in [43] and fully analyzed in [44]; an analogous notion, referred to as quasistatic evolution, was in parallel developed within the realm of brittle fracture, cf. [19, 22].
It consists of the global stability condition, holding at every \(t\in [0,T]\),
and of the \((\Psi _0,{\mathcal {E}})\)-energy-dissipation balance
Indeed, (\(\mathrm {E}_{\Psi _0,{\mathcal {E}}}\)) involves the dissipated energy \(\mathrm {Var}_{\Psi _0}(u; [0,t])\) (where \({\mathrm {Var}}_{\Psi _0}\) denotes the notion of total variation induced by \(\Psi _0\)), the stored energy \({\mathcal {E}}(t, u(t))\) at the process time t, the initial energy \({\mathcal {E}}(0, u(0))\), and the work of the external forces. Since the energetic formulation (\(\mathrm {S}\))–(\(\mathrm {E}_{\Psi _0,{\mathcal {E}}}\)) only features the (assumedly smooth) power of the external forces \(\partial _t {\mathcal {E}}\), and no other derivatives, it is particularly suited to solutions with discontinuities in time. It is also considerably flexible and can be indeed given for rate-independent processes in general topological spaces, cf. [35]. That is why, it has been exploited in a great variety of applicative contexts, ranging from fracture, damage, and delamination to plasticity, shape-memory alloys, ferroelectricity, to name a few; we refer to [39] for a comprehensive survey.
Nonetheless, over the years it has become apparent that, in the very case of a nonconvex dependence \(u\mapsto {\mathcal {E}}(t, u)\), the global stability (\(\mathrm {S}\)) fails provide a truthful description of the system behavior at jumps, leading to solutions jumping ‘too early’ and ‘too long’ (i.e., into very far-apart energetic configurations), as shown for instance by the example [31, Ex. 6.1], and by the characterization of energetic solutions to (one-dimensional) rate-independent systems in [48].
This circumstance has led to the introduction of alternative weak solvability concepts for (1.1) and its generalizations. In particular, [20] first set forth the vanishing-viscosity regularization of the rate-independent system as a selection criterion for mechanically feasible weak solutions. The vanishing-viscosity approach has in fact proved to be a robust method in diverse applications, ranging from plasticity (cf., e.g., [11, 17, 18]), to fracture (cf., e.g., [26, 30, 52]), and to damage (cf., e.g., [14, 28]) models. The solution concept obtained through this approach has been codified under the name of balanced viscosity solution in [41, 42]. We also refer to [46] for an alternative derivation of balanced viscosity solutions via time discretization, and to [36], where the notion was recovered with the so-called epsilon-neighborhood method.
Let us briefly illustrate the vanishing-viscosity approach: We “augment by viscosity” the dissipation potential \(\Psi _0\) and thus introduce
with \(\Vert \cdot \Vert \) a second norm on \({\mathbb {R}}^d\), which may or may not coincide with the norm associated with \(\Psi _0\). The corresponding (generalized) gradient system\((\Psi _\varepsilon ,{\mathcal {E}})\), namely the doubly nonlinear differential inclusion
thus provides an approximation of the rate-independent system (1.1). Since \(\Psi _\varepsilon \) has superlinear growth at infinity, (1.3) does admit absolutely continuous solutions. It is to be expected that, as the viscosity parameter \(\varepsilon \) vanishes, solutions \((u_\varepsilon )_\varepsilon \) to (1.3) will converge to a suitable weak solution to the rate-independent system (1.1). In [41], it was indeed shown that any limit curve \(u\in \mathrm {BV}([0,T];{\mathbb {R}}^d)\) of the functions \((u_\varepsilon )_\varepsilon \) complies with the stability condition
and with the energy-dissipation balance
which individuate the notion of balanced viscosity solution to the rate-independent system (\(\mathrm {S}\)). Although (\(\mathrm {S}_\mathrm {loc}\)) and (\(\mathrm {E}_{\Psi _0, {\mathfrak {p}},{\mathcal {E}}}\)) look similar to (\(\mathrm {S}\)) and (\(\mathrm {E}_{\Psi _0,{\mathcal {E}}}\)), they are in fact significantly different. First of all, (\(\mathrm {S}_\mathrm {loc}\)) is in fact a local version of the global stability (\(\mathrm {S}\)). Secondly, (\(\mathrm {E}_{\Psi _0, {\mathfrak {p}},{\mathcal {E}}}\)) shares the same structure with the energy-dissipation balance (\(\mathrm {E}_{\Psi _0,{\mathcal {E}}}\)), but it features a novel type of total variation functional, \({\mathrm {Var}}_{\Psi _0,{\mathfrak {p}},{\mathcal {E}}}\). While referring to Sect. 3 for its precise definition (cf. (3.23) ahead), we may mention here that, in addition to the dissipation potential \(\Psi _0\), \({\mathrm {Var}}_{\Psi _0,{\mathfrak {p}},{\mathcal {E}}}\) involves the vanishing-viscosity contact potential
which indeed lies at the core of the balanced viscosity concept. The second equality in (1.4) ensues from a direct calculation; \(K^*\) is the stable set from (\(\mathrm {S}_\mathrm {loc}\)) and \(\Vert \cdot \Vert _*\) the dual norm of the “viscosity-related” norm \(\Vert \cdot \Vert \). As we will see in Sec. 3 with more detail, the functional \({\mathfrak {p}}\) thus encodes how viscosity, neglected in the vanishing-viscosity limit, pops back into the description of the solution behavior at jumps, whereas in the continuous (‘sliding’) regime, the system is only governed by the 1-homogeneous dissipation \(\Psi _0\).
A characterization of balanced viscosity solutions, again for one-dimensional systems, has been provided in [48], showing that they model jumps more accurately than energetic solutions. On the other hand, as evident from (1.4), this notion seems to be strongly reminiscent of the vanishing-viscosity approximation (1.3).
It is thus natural to wonder whether there are ways, alternative to the vanishing viscosity [41, 42] and to the time-discretization [46] (cf. also [36]) approaches, to generate balanced viscosity solutions. The present paper aims to contribute in this direction.
The stochastic origin of balanced viscosity solutions. Recently, this question has been answered affirmatively in [12], investigating the role of stochasticity in the origin of rate independence, in the context of one-dimensional rate-independent systems (i.e., with (1.1) set in the ambient space \({\mathbb {R}}\)). The motivation for the analysis carried out in [12] stems from the argument that rate independence may arise through the interplay between thermal noise and a rough energy landscape, cf. [5, 9, 25, 47]. The approach to rate-independent evolution developed in [12] is akin to that in [1, 2], where a connection has been established between the evolution of a class of many-particle systems and Wasserstein gradient flows, through a large-deviations principle. We also refer to [29, 38], where deeper insight has been gained into the principles underlying this connection within the more general context of (generalized) gradient structures; in this mainstream, [12] has, however, been the first paper investigating rate-independent, in place of rate-dependent, evolution.
More specifically, [12] has focused on a continuous-time Markov jump process \(t\mapsto X_t^h\) on a one-dimensional lattice, with lattice spacing \(\frac{1}{h}\), \(h\in {\mathbb {N}}\). While referring to Sect. 2 for more details, we may mention here that this process models the evolution of a Brownian particle in a wiggly energy landscape, involving the energy \({\mathcal {E}}\), in the following way. If the particle is at the position x at time t, then it jumps in continuous time to its neighbors \(x\pm \frac{1}{h}\) with rates \(h r^{\pm }(x)\), where \(r^\pm (x) = \alpha \exp (\mp \beta \mathrm {D}{\mathcal {E}}(t, x))\). Here, \(\alpha \) and \(\beta \) are positive parameters, the former characterizing the rate of jumps, and thus the global timescale of the process, and the latter related to noise.
First of all, in [12] it was shown that the deterministic limit, in a ‘large-deviations’ sense, as \(h\rightarrow \infty \) and for \(\alpha \) and \(\beta \) fixed, of this stochastic process solves
Observe that the latter equation is a reformulation of the (generalized) gradient system
where the convex dissipation potential \(\Psi _{\alpha ,\beta }\) is such that its Fenchel–Moreau convex conjugate fulfills \(\partial \Psi _{\alpha ,\beta }^{-1}(\xi ) = \partial \Psi _{\alpha ,\beta }^* (\xi ) = \{ \mathrm {D}\Psi _{\alpha ,\beta }^* (\xi ) \} = \{2\alpha \sinh (\beta \xi )\}\). More precisely, in [12] it was proved that the process \(X^h\) satisfies a large-deviations principle (cf. Sec. 2), with rate function given by the functional of trajectories \(\tilde{{\mathscr {J}}}_{\Psi _{\alpha ,\beta }, {\mathcal {E}}}: \mathrm {BV}([0,T];{\mathbb {R}}^d) \rightarrow [0,+\infty ]\) defined by
if \(u\in \mathrm {AC}([0,T];{\mathbb {R}}^d)\), and \(+\infty \) else. In fact, the null-minimizers of \( \tilde{{\mathscr {J}}}_{\Psi _{\alpha ,\beta }, {\mathcal {E}}} \) are solutions to the gradient system (1.5), as shown in Lemma 3.1 ahead. Let us mention that this kind of argument, relating solutions of evolutionary equations to the (null-)minimization of functionals of entire trajectories, can be traced back to the seminal Brézis-Ekeland-Nayroles principle [7, 8, 45], cf. also [51], and the monograph [23] for a comprehensive overview.
Secondly, the variational limits of the functionals \(\tilde{{\mathscr {J}}}_{\Psi _{\alpha ,\beta }, {\mathcal {E}}} \) have been addressed in [12] under different scalings of the parameters \(\alpha \) and \(\beta \), leading to gradient flow or rate-independent evolution. To illustrate the result in the latter case, here and throughout the paper we will confine the discussion to the following choice of parameters: \(\alpha =\alpha _n := \frac{e^{-nA}}{2}\) and \(\beta : = \beta _n =n\), with \(n\in {\mathbb {N}}\). In this case, the associated dissipation potentials are given by
In [12, Thm. 4.2], it was proved that that the functionals of trajectories \({\mathscr {J}}_{\Psi _n,{\mathcal {E}}}: = \frac{1}{n} \tilde{{\mathscr {J}}}_{\Psi _n, {\mathcal {E}}}\) converge in the sense of Mosco with respect to the weak-strict topology in \( \mathrm {BV}([0,T];{\mathbb {R}}^d) \), to the functional \( {\mathscr {J}}_{\Psi _0,{\mathfrak {p}},{\mathcal {E}}} : \mathrm {BV}([0,T];{\mathbb {R}}^d) \rightarrow [0,+\infty ]\) defined by
with \(\Psi _0(v) = A|v|\), \({\mathfrak {p}}\) given by (1.4) and the associated total variation functional \( {\mathrm {Var}}_{\Psi _0,{\mathsf {p}},{\mathcal {E}}}\) defined in (3.23) ahead, and with \(I_{K^*} \) denoting the indicator function of the set stable set \(K^*=\partial \Psi _0(0)=[-A,A]\). While postponing the precise definition of Mosco-convergence in \( \mathrm {BV}([0,T];{\mathbb {R}}^d) \) to (3.17) ahead, let us only mention that \( \mathrm {BV}([0,T];{\mathbb {R}}^d) \) is the appropriate space for the solutions to rate-independent systems, liable to jump as functions of time.
The Mosco-convergence result in [12, Thm. 4.2] establishes a deep connection between the generalized gradient system (1.5) and the one-dimensional rate-independent system driven by \(\Psi _0(v) = A|v|\) and \({\mathcal {E}}\). In fact, (null-)minimizers of \( {\mathscr {J}}_{\Psi _0,{\mathfrak {p}},{\mathcal {E}}}\) are balanced viscosity solutions of the rate-independent system driven by \(\Psi _0\) and \({\mathcal {E}}\) (cf. Proposition 3.7 ahead). Therefore, on the one hand the \(\Gamma \)-\(\liminf \)-estimate verified within Mosco-convergence, joint with the observation that (null-)minimizers of \({\mathscr {J}}_{\Psi _n,{\mathcal {E}}}\), are solutions to the gradient system (1.5), ensures the convergence of the latter to balanced viscosity solutions of the rate-independent system. On the other hand, the \(\Gamma \)-\(\limsup \)-estimate yields a reverse approximation result.
Therefore, [12, Thm. 4.2] ultimately reveals the connection between the jump process \(X^h\) and the (one-dimensional) rate-independent system (1.1), understood in a balanced viscosity sense. Furthermore, observe that the functionals \(\Psi _n\) from (1.6) are not of form (1.2). Hence, this result provides a way, alternative to the vanishing viscosity approach, to generate balanced viscosity solutions.
1.1 Our results
Our first motivation for this work was to extend the ‘stochastic generation’ of balanced viscosity solutions unraveled
in [12] to the multi-dimensional rate-independent system (1.1), where now
The first obvious difference between the one- and the multi-dimensional contexts is that, even conjecturing that the viscosity contact potential defining the limiting Balanced Viscosity solution notion is of form (1.4), in this multi-dimensional context it is no longer obvious which choice of the viscous norm \(\Vert \cdot \Vert \) should enter into (1.4). With our main results, Theorem 5.2 (\(\liminf \)-estimate) and Theorem 5.9 (\(\limsup \)-estimate), we have shown that the multi-dimensional analogues of the functionals \(({\mathscr {J}}_{\Psi _n,{\mathcal {E}}})_n \)Mosco-converge, with respect to the weak-strict topology of \(\mathrm {BV}([0,T];{\mathbb {R}}^d)\), to the functional \( {\mathscr {J}}_{\Psi _0,{\mathfrak {p}},{\mathcal {E}}}\) featuring the total variation functional associated with the contact potential
It can be checked that \({\mathfrak {p}}\) from (1.9) is indeed of form (1.4). In this case, however, the ‘viscous’ norm \(\Vert \cdot \Vert \) in fact coincides with that associated with \(\Psi _0\) from (1.8), i.e., \(\Vert v\Vert = \Vert v\Vert _1 \). Therefore, the notion of balanced viscosity solution arising from the stochastic approximation coincides with the one in which the 1-homogeneous dissipation potential \(\Psi _0\) is perturbed by a (superlinear) function of \(\Psi _0\) itself. Referring to this case, we shall speak of vanishing \(\Psi _0\)-viscosity, cf. Example 5.5 ahead.
Our \(\Gamma \)-\(\liminf \) Theorem 5.2 has in fact a broader and deeper scope and indeed contributes to understanding the origin of rate-independent evolution in a Balanced Viscosity sense. More precisely, in this paper
- (1)
We will introduce an ‘extended’ notion of Balanced Viscosity solution, induced by a class of viscosity contact potential\({\mathsf {p}}: [0,+\infty ) \times {\mathbb {R}}^d \times {\mathbb {R}}^d \rightarrow [0+\infty ]\), \( {\mathsf {p}}= {\mathsf {p}} (\tau ,v,\xi )\), cf. Def. 3.2 ahead, in place of the vanishing-viscosity contact potential \({\mathfrak {p}}\) from, e.g., (1.4). We may understand the functional \( {\mathsf {p}}\) as obtained by augmenting \({\mathfrak {p}}\) with the time variable, in that the contact potentials \({\mathfrak {p}}(v,\xi ) \) from (1.4), for instance, stem from setting \(\tau =0\) in (specific choices of) \({\mathsf {p}}\), i.e., \({\mathsf {p}} (0,v,\xi )= {\mathfrak {p}}(v,\xi )\).
- (2)
We will provide a series of conditions under which a sequence \((\Psi _n)_n\) of general dissipation potentials with superlinear growth at infinity, not necessarily of form (1.2) (vanishing viscosity) or (1.6) (stochastic approximation), give rise to a viscosity contact potential. Such conditions will amount to requiring that the bipotentials \(b_{\Psi _n} : [0,+\infty ) \times {\mathbb {R}}^d \times {\mathbb {R}}^d \rightarrow [0+\infty ]\), associated with the functionals \(\Psi _n\), and defined by
$$\begin{aligned} {\mathsf {b}}_{\Psi _n}(\tau ,v,\xi ): = \tau \Psi _n(\frac{v}{\tau }) + \tau \Psi _n^*(\xi ) \qquad \text {for } \tau >0 \end{aligned}$$(1.10)(cf. (4.1) ahead), converge in a suitable variational sense to \({\mathsf {p}}\), as \(n\rightarrow \infty \). As we will see, such conditions are for instance verified in the case of the vanishing viscosity and of the stochastic approximations.
- (3)
It will turn out (cf. Theorem 5.2) that under this condition on the bipotentials \(b_{\Psi _n}\), jointly with a suitable uniform coercivity requirement for the functionals \((\Psi _n)_n\), a \(\Gamma \)-\(\liminf \) estimate for the associated trajectory functionals \(({\mathscr {J}}_{\Psi _n,{\mathcal {E}}})_n\) holds, with respect to the weak topology in \(\mathrm {BV}([0,T];{\mathbb {R}}^d)\).
Like previously illustrated for the \(\Gamma \)-\(\liminf \) result in [12], our Theorem 5.2 has the following outcome in terms of evolutionary systems: it implies that limit curves \(u\in \mathrm {BV}([0,T];{\mathbb {R}}^d)\) of sequences of solutions \((u_n)_n\) to the gradient systems \((\Psi _n,{\mathcal {E}})\) (where \((\Psi _n)_n\) are dissipation potentials with superlinear growth at infinity that give rise to a viscosity contact potential \({\mathsf {p}}\) in the aforementioned sense) are balanced viscosity solutions to the rate-independent system \((\Psi _0,{\mathsf {p}},{\mathcal {E}})\), cf. Theorem 5.3 ahead for a precise statement. In other words, we conclude that the gradient systems \((\Psi _n,{\mathcal {E}})_n\)Evolutionary \(\Gamma \)-converge to the rate-independent system \((\Psi _0,{\mathsf {p}},{\mathcal {E}})\), using a terminology recently popularized in [34] (cf. also, e.g., [13, 40, 49,50,51, 54] for limit passages in gradient systems driven by \(\Gamma \)-converging energy functionals and dissipation potentials).
The proof of the \(\limsup \)-estimate in Theorem 5.9 focuses on the specific cases of the vanishing-viscosity and the stochastic approximations. It could be generalized in some directions, cf. Remarks 5.10 and 6.2 ahead. Nonetheless, we have preferred to confine the discussion to the vanishing-viscosity and the stochastic cases, in order to make the calculations more explicit.
Let us, however, mention that in [12] the \(\Gamma \)-\(\limsup \) result was obtained in a much larger generality, for a broader class of dissipation potentials \(\Psi _n\) that encompass those considered here, with calculations tailored to the one-dimensional context therein. It is not clear to us how to fully extend such calculations in the frame of the approach we have adopted in our multi-dimensional setup, which, differently from [12], is based on careful computations involving the bipotentials \( {\mathsf {b}}_{\Psi _n}\) from (1.10). All the more, the proof of the \(\limsup \)-estimate in the fully general case, i.e., under the sole condition that the bipotentials \(b_{\Psi _n}\) from (1.10) variationally converge to \({\mathsf {p}}\), remains an open problem.
Still, we believe Theorem 5.9 to be relevant. Again, the key observation is that (null-)minimizers of the involved functionals of trajectories are solutions of the associated evolutionary problems. Hence, Theorem 5.9 yields a reverse approximation result, Theorem 5.12 ahead, of balanced viscosity solutions of the limiting rate-independent system, by means of solutions of the approximating gradient systems. Such a result (i) extends what was proved in [12] to the multi-dimensional case; (ii) is, to our knowledge, new in the case of the vanishing-viscosity approximation.
Plan of the paper. In Sect. 2, we present the multi-dimensional analogue of the stochastic model considered in [12] and (formally) derive the associated dissipation potential \(\Psi _n\) and the induced trajectory functional \({\mathscr {J}}_{\Psi _n,{\mathcal {E}}}\).
Section 3 revolves around an extended notion of balanced viscosity solution to a rate-independent system, based on the concept of viscosity contact potential \({\mathsf {p}}\) introduced in Definition 3.2. This concept of solution is defined in Definition 3.5; its properties are illustrated thereafter.
In Sect. 4, we address the generation of a viscosity contact potential \({\mathsf {p}}\) starting from a family \((\Psi _n)_n\) of dissipation potentials with superlinear growth at infinity, such that the associated bipotentials \( {\mathsf {b}}_{\Psi _n}\) from (1.10) variationally converge to \({\mathsf {p}}\).
Our main results, Theorems 5.2 and 5.9 , along with their consequences Theorems 5.3 and 5.12 , are stated in Sect. 5 and proved throughout Sect. 6.
2 The stochastic origin of rate-independent systems
In this section, we briefly describe the multi-dimensional extension of the one-dimensional stochastic model for rate-independent evolution considered in [12].
We consider a jump process \(t\mapsto X^h(t)\) on a d-dimensional lattice, with lattice spacing \(\frac{1}{h}\). The evolution of the process can be described as follows: Fix the origin as initial point. If the process is at the position x at time t, then it jumps in continuous time to its neighbors \((x\pm \frac{1}{m} {\mathbf {e}}_i)\) with rate \(mr_i^{\pm }\), for \(i=1, \ldots , d\), where \(({\mathbf {e}}_1, \ldots , {\mathbf {e}}_d)\) is the basis of \({\mathbb {R}}^d\), cf. Fig. 1. The jump rates depend on two parameters \(\alpha \) and \(\beta \), and on the partial derivatives \( \mathrm {D}_i {\mathcal {E}}: = \mathrm {D}_{x_i} {\mathcal {E}}\) of a smooth energy functional \({\mathcal {E}}: [0,T]\times {\mathbb {R}}^d \rightarrow {\mathbb {R}}\), namely
A sketch of the jump process on the lattice
The choice of the stochastic process (and thus of the jump rates \(r_i^\pm \)) reflects Kramers’ formula [10, 12, 27]. Given a particle evolving in a wiggly energy landscape with noise, this formula provides an estimate of the rate of jumps from one energy well to the next one.
We are interested in the continuum limit as \(h\rightarrow \infty \). With this aim, we apply the method developed by Feng & Kurtz, cf. [21], to prove large-deviations principles for Markov processes.
As in [12, Sec. 2.5], we will provisionally assume that the jump rates \(r^\pm \) are constant in space and time and thus derive the expression of the rate function, and then formally substitute (2.1) into it. Following [21], we consider the generator
of the continuous-time Markov process \(X^h\) and the nonlinear generator
According to the Feng–Kurtz method, if \(\mathrm {H}_h\) converges to some \(\mathrm {H}\) in a suitable sense, and if the limiting operator \(\mathrm {H}f\) depends locally on \(\mathrm {D}f\), we can then define the Hamiltonian\(H = H(x,\xi )\) through
and the Lagrangian as the Legendre transform of H, namely
Then, the Markov process satisfies a large-deviations principle, with rate function
We may roughly state this principle in the following form:
Namely, the probability of finding \( X^h \) close to some x is close, in a suitable sense, to \( \exp (-h {\mathscr {J}}(x))\) as \(h\rightarrow \infty \); in particular, large values of \({\mathscr {J}}(x)\) imply small probability. The purposefully vague notations \(\approxeq \) and \( \approx \) featuring in (2.3) are made precise in [12, Definition 2]. Let us also mention that (2.3) has to be understood as a relation holding in the Skorokhod space D([0, T]), namely the space of cadlag functions (right continuous and with limit from left). The Feng–Kurtz method applied in [12] to prove the large-deviations principle for the one-dimensional stochastic model extends to the present multi-dimensional setting.
In our context, it can be seen that
Then, L is given by
Substituting in (2.4) expression (2.1) for the jump rates, and choosing the parameters
we obtain
where \(\langle \cdot ,\cdot \rangle \) denotes the inner product in \({\mathbb {R}}^d\), and \(\Psi _n: {\mathbb {R}}^d \rightarrow [0,+\infty )\) is given by
and \(\Psi _n^*\) the Legendre transform of \(\Psi _n\). It can be easily checked that the structure \(\Psi _n(v) = \sum _{i=1}^d \psi _n(v_i)\) transfers to the conjugate, hence
In view of Lemma 3.1 ahead, we can see that, with choice (2.5) for L, the (positive) functional \({\mathscr {J}}\) from (2.2) is minimized by the solutions of the generalized gradient system
Taking into account that \( \mathrm {D}\Psi _n^* = (\mathrm {D}\Psi _n)^{-1}\), the latter rewrites as the ODE system
3 Viscosity contact potentials and balanced viscosity solutions to rate-independent systems
The central objective of this section is to introduce and illustrate the main properties of the notion of balanced viscosity solution to a rate-independent system induced by a general viscosity potential. Therefore, Definition 3.5 ahead shall (slightly) extend the definition given in [41], which was based on the concept of vanishing-viscosity contact potential.
3.1 Preliminaries
Before introducing the notion of balanced viscosity solution to a rate-independent system and fixing its main properties, we recall some properties of dissipation potentials and provide some basics of the theory of functions of bounded variation; for a more comprehensive tractation, the reader is referred to, e.g., [3, 37].
Dissipation potentials. Hereafter, we will call dissipation potential any function
It follows from the above conditions that the Fenchel–Moreau conjugate \(\Psi ^*\) then fulfills \(\Psi ^*(0)=0 \le \Psi ^*(\xi )\) for all \(\xi \in {\mathbb {R}}^d\). We will distinguish two cases:
Dissipation potentials with superlinear growth at infinity: Namely, \(\Psi \) fulfills
$$\begin{aligned} \lim _{\Vert v\Vert \rightarrow +\infty } \frac{\Psi (v)}{\Vert v \Vert } = +\infty \end{aligned}$$(3.2)for some norm \(\Vert \cdot \Vert \) on \({\mathbb {R}}^d\). For later use, we point out here that, as a consequence of (3.2) one has
$$\begin{aligned} \lim _{\tau \downarrow 0} \tau \Psi \left( \frac{v}{\tau }\right) = +\infty \qquad \text {for all } v \in {\mathbb {R}}^d {\setminus } \{0\}. \end{aligned}$$(3.3)1-homogeneous dissipation potentials: In what follows, we will denote by \(\Psi _0\) a dissipation potential
$$\begin{aligned}&\Psi _0: {\mathbb {R}}^d \rightarrow [0,+\infty ) \ \text {convex, 1-}{} positively homogenous ,\nonumber \\&\quad \text { and non-degenerate, viz. } \Psi _0(v) >0 \ \text { if } v\ne 0. \end{aligned}$$(3.4)Thus, for any norm \(\Vert \cdot \Vert \) on \({\mathbb {R}}^d\)
$$\begin{aligned} \exists \, \eta >0 \ \forall \, v\in {\mathbb {R}}^d \, : \ \eta ^{-1} \Vert v\Vert \le \Psi _0(v) \le \eta \Vert v\Vert \,. \end{aligned}$$(3.5)Its convex-analysis subdifferential \(\partial \Psi _0: {\mathbb {R}}^d \rightrightarrows {\mathbb {R}}^d\) at \(v\in {\mathbb {R}}^d\) can be characterized by
$$\begin{aligned} \zeta \in \partial \Psi _0(v) \ \Leftrightarrow \ \left\{ \begin{array}{ll} \langle \zeta , w\rangle \le \Psi _0(w) \text { for all } w\in {\mathbb {R}}^d, \\ \langle \zeta , v\rangle = \Psi _0(v). \end{array} \right. \end{aligned}$$(3.6)Throughout, we will use the notation
$$\begin{aligned} K^* \quad \text {for the stable set} \quad \partial \Psi _0(0)\,. \end{aligned}$$(3.7)Recall that \(\partial \Psi _0(v) \subset K^*\) for all \(v\in {\mathbb {R}}^d\) and that, indeed, \(\Psi _0\) is the support function of \(K^*\), namely
$$\begin{aligned} \Psi _0(v) = \sup _{\zeta \in K^*} \langle \zeta , v\rangle , \quad \text {whence } \quad \Psi _0^*(\xi ) = I_{K^*}(\xi ). \end{aligned}$$(3.8)
We conclude with the following Lemma 3.1 that fixes the observation that, in the case of a dissipation potential \(\Psi \) with superlinear growth, the solutions of the generalized gradient system \((\Psi ,{\mathcal {E}})\), i.e.,
can be characterized as the (null-)minimizers of the functional of trajectories \({\mathcal {J}}_{\Psi ,{\mathcal {E}}} : \mathrm {AC}([0,T];{\mathbb {R}}^d) \rightarrow [0,+\infty )\) given by
Lemma 3.1
Let \(\Psi \) have superlinear growth at infinity. Then, a curve \(u\in \mathrm {AC}([0,T];{\mathbb {R}}^d)\) is a solution to (3.9) if and only if
Proof
First of all, the positivity of \({\mathcal {J}}_{\Psi ,{\mathcal {E}}} \) is due to
where the first inequality follows from the elementary convex analysis inequality \(\Psi (v) + \Psi ^*(\xi ) \ge \langle \xi , v \rangle \), and the last equality by the chain rule. Suppose now that \( {\mathcal {J}}_{\Psi ,{\mathcal {E}}} (u)=0\). Then, again using the chain rule we conclude that
Since the integrand is positive, we infer that
which ultimately yields, again by convex analysis, that u solves (3.9). The very same arguments yield that a solution u of (3.9) fulfills \( {\mathcal {J}}_{\Psi ,{\mathcal {E}}} (u)=0\). \(\square \)
\(\mathrm {BV}\)functions Throughout, we will work with functions of bounded variation pointwise defined at every point \(t\in [0,T]\). We recall that a function u in \(\mathrm {BV}([0,T];{\mathbb {R}}^d)\) admits left and right limits at every \(t\in [0,T]\):
and its pointwise jump set \(\mathrm {J}_{u}\) is the at most countable set defined by
We also recall that the distributional derivative \(u'\) of u is a Radon vector measure that can be decomposed (cf. [3]) into the sum of the three mutually singular measures
Here, \(u'_{\mathscr {L}}\) is the absolutely continuous part with respect to the Lebesgue measure \({\mathscr {L}}^1\), whose Lebesgue density \(\dot{u}\) is the pointwise (and \({\mathscr {L}}^{1}\)-a.e. defined) derivative of u, \(u'_\mathrm {J}\) is a discrete measure concentrated on \( \mathop {\text {ess-J}}\nolimits _u \subset \mathrm {J}_{u}\), and \(u'_\mathrm{C}\) is the so-called Cantor part. We will use the notation \(u'_\mathrm {co}:=u'_{\mathscr {L}}+u'_\mathrm{C}\) for the diffuse part of the measure, which does not charge \( \mathrm {J}_{u}\).
Given a (non-degenerate) 1-homogeneous dissipation potential \(\Psi _0\), it induces a notion of (pointwise) total variation for a curve \(u\in \mathrm {BV}([0,T];{\mathbb {R}}^d)\) via
for any \([a,b]\subset [0,T]\). Therefore, with any \(u\in \mathrm {BV}([0,T];{\mathbb {R}}^d)\) we can associate the non-decreasing function \(V_{\Psi _0}: {\mathbb {R}}\rightarrow [0,+\infty )\) defined by
Its distributional derivative \(\mu _{\Psi _0}\) is in turn a Radon measure that can be decomposed into a jump part \(\mu _{\Psi _0,\mathrm {J}}\), concentrated on \(\mathrm {J}_{u}\) and given by
and a diffuse part
and \({\mathscr {L}}^{1}\) the 1-dimensional Lebesgue measure. There holds
with the jump contribution \( \mathrm {Jmp}_{\Psi _0}(u; [a,b])\) given by
Finally, for later use we recall that a sequence \((u_n)_n \)weakly converges in \( \mathrm {BV}([0,T];{\mathbb {R}}^d)\) to a curve u (we will write \(u_n \rightharpoonup u\)) if \(u_n(t) \rightarrow u(t)\) as \(n\rightarrow \infty \) for every \(t\in [0,T]\) and \(\sup _n \mathrm {Var}_{}(u_n; [0,T]) \le C<\infty \) (in what follows, we shall denote by \(\mathrm {Var}_{}(u; [0,T])\) the total variation of a curve u induced by a generic norm \(\Vert \cdot \Vert \) on \({\mathbb {R}}^d\)), whereas \((u_n)_n \)strictly converges in \( \mathrm {BV}([0,T];{\mathbb {R}}^d)\) to u (\(u_n\rightarrow u\)) if \(u_n \rightharpoonup u\) and \(\mathrm {Var}_{}(u_n; [0,T]) \rightarrow \mathrm {Var}_{}(u; [0,T]) \). Finally, a sequence of functionals \({\mathscr {G}}_n : \mathrm {BV}([0,T];{\mathbb {R}}^d) \rightarrow {\mathbb {R}}\)Mosco-converge to a functional \({\mathscr {G}} : \mathrm {BV}([0,T];{\mathbb {R}}^d) \rightarrow {\mathbb {R}}\) with respect to the weak-strict topology in \( \mathrm {BV}([0,T];{\mathbb {R}}^d)\) if
We refer to [4] for more details on Mosco-convergence.
Viscosity contact potentials. The notion we are going to introduce now lies at the core of the definition of balanced viscosity solution to a rate-independent system, driven by an energy functional \({\mathcal {E}}\) complying with (\(\textsc {E}\)). Indeed, the concept of viscosity contact potential encodes how viscosity enters into the description of the solution behavior at jumps, cf. (3.26) ahead. It is an extension of the notion of vanishing-viscosity contact potential introduced in [41], in that we are augmenting the contact potential defined therein by the time variable. That is why, we have used two different notations, \({\mathfrak {p}}\) and \({\mathsf {p}}\), respectively, to distinguish the contact potential from [41] from that introduced here. Furthermore, in referring to the contact potential \({\mathsf {p}}\), we will drop the word ‘vanishing’ in order to highlight that balanced viscosity solutions do not necessarily arise from a vanishing-viscosity approximation, cf. Sect. 5.2.
Definition 3.2
We call a lower semicontinuous function \({\mathsf {p}}: [0,+\infty ) \times {\mathbb {R}}^d \times {\mathbb {R}}^d \rightarrow [0,+\infty ] \) a (viscosity) contact potential if it satisfies the following properties:
- (1)
for every \(\tau \ge 0\) there holds \({\mathsf {p}} (\tau ,v,\xi ) \ge \langle v, \xi \rangle \) for all \((v,\xi ) \in {\mathbb {R}}^d \times {\mathbb {R}}^d\);
- (2)
for every \(\xi \in {\mathbb {R}}^d\) the map \((\tau ,v) \mapsto {\mathsf {p}} (\tau ,v,\xi )\) is convex and positively 1-homogeneous.
- (3)
for every \(\tau >0 \) and \(v\in {\mathbb {R}}^d\), the map \(\xi \mapsto {\mathsf {p}} (\tau ,v,\xi )\) is convex.
Moreover, we say that \({\mathsf {p}}\) is non-degenerate if
- (4)
for every \(\tau \ge 0\) there holds \({\mathsf {p}} (\tau ,v,\xi ) >0\) if \(v\ne 0\).
Finally, given a (non-degenerate) 1-homogeneous dissipation potential \(\Psi _0\) as in (3.4), we say that \({\mathsf {p}}\) is \(\Psi _0\)-non-degenerate if
- (5)
for all \((v,\xi ) \in {\mathbb {R}}^d \times {\mathbb {R}}^d\) there holds \({\mathsf {p}} (0,v,\xi ) \ge \Psi _0(v)\).
A crucial object related to a (viscosity) contact potential \({\mathsf {p}}\) is the set where the inequality in (1) holds as an equality. We will call it contact set and denote it by
whereas we will use the notation
Let us point out a first important consequence of the properties defining a contact potential:
Lemma 3.3
For fixed \((\tau ,\xi ) \in [0,+\infty ) \times {\mathbb {R}}^d\), denote by \(\partial _v {\mathsf {p}}(\tau ,\cdot , \xi )(v)\) the (convex analysis) subdifferential at \(v\) of the functional \(v\mapsto {\mathsf {p}} (\tau ,v,\xi )\). Then,
Proof
Since \(v\mapsto {\mathsf {p}} (\tau ,v,\xi )\) is convex and positively homogeneous of degree 1, we have (cf. (3.6)),
and the thesis follows. \(\square \)
Remark 3.4
Observe that, for fixed \(\tau \in [0,+\infty )\), the function \({\mathsf {p}} (\tau ,\cdot ,\cdot ): {\mathbb {R}}^d \times {\mathbb {R}}^d \rightarrow [0,+\infty )\) enjoys some of the properties of the notion of bipotential (cf., e.g., [6]), which is by definition a functional \({\mathsf {b}} : {\mathbb {R}}^d \times {\mathbb {R}}^d \rightarrow [0,+\infty ]\) convex and lower semicontinuous w.r.t. both variables, separately, and fulfilling \({\mathsf {b}}(v,\xi ) \ge \langle v,\xi \rangle \) for all \((v,\xi ) \in {\mathbb {R}}^d \times {\mathbb {R}}^d\), as well as a stronger version of (3.20), namely
where the contact set \(\Lambda _{{\mathsf {b}}} \) is defined similarly as in (3.18).
As discussed in [41], the conditions defining the notion of bipotential, however, seem to be too restrictive for the contact potentials arising in the vanishing-viscosity limit of viscous systems approximating rate-independent evolution. Nonetheless, in Sec. 4 we will see how viscosity contact potentials can in fact be generated, via \(\Gamma \)-convergence, by bipotentials (in the sense of [6]) associated with families of dissipation potentials.
3.2 BV solutions to rate-independent systems
We are now in a position to recall the preliminary definitions at the basis of the concept of balanced viscosity solution; notice that all of them involve the reduced contact potential \({\mathsf {p}} (0,\cdot ,\cdot )\) and the energy functional \({\mathcal {E}}\in \mathrm {C}^1 ([0,T]\times {\mathbb {R}}^d)\).
First of all, we introduce the (possibly asymmetric) Finsler distance coming into play in the description of the energetic behavior of a rate-independent system at a jump time: For a fixed \(t\in [0,T]\), the Finsler distance induced by \({\mathsf {p}}\) and \({\mathcal {E}}\) at the time t is defined for every \( u_0,\, u_1\in {\mathbb {R}}^d\) by
Observe that, if \({\mathsf {p}}\) is a \(\Psi _0\)-non-degenerate contact potential for some 1-positively homogeneous potential \(\Psi _0\), we clearly have \(\Delta _{{\mathsf {p}},{\mathcal {E}}}(t; u_0, u_1) \ge \Delta _{\Psi _0}(u_0,u_1): = \Psi _0(u_1-u_0)\). Mimicking notion (3.13) of \(\Psi _0\)-total variation, moving from (3.21) we introduce a notion of total variation that measures the dissipation of a \(\mathrm {BV}\)-curve at its jump points. Indeed, along the footsteps of [41, Def. 3.4] and in analogy with (3.16), for a given curve \(u\in \mathrm {BV}([0,T];{\mathbb {R}}^d)\) with jump set \(\mathrm {J}_{u} \), we define the jump variation of u induced by \(({\mathsf {p}},{\mathcal {E}})\) on an interval \([a,b]\subset [0,T]\) by
Finally, given a (non-degenerate) 1-positively homogeneous dissipation potential \(\Psi _0\) and a contact viscosity potential \({\mathsf {p}}\), the (pseudo-)total variation of a curve \(u \in \mathrm {BV}([0,T];{\mathbb {R}}^d)\) induced by \((\Psi _0, {\mathsf {p}}, {\mathcal {E}})\) is defined by (cf. (3.15))
with \( \mu _{\Psi _0,\mathrm {co}} \) from (3.14) the diffuse part of the total variation measure of the map \(t\mapsto \mathrm {Var}_{\Psi _0}(u; [0,t])\). Let us mention that the notation \(\mathrm {Var}_{\Psi _0,{\mathsf {p}},{\mathcal {E}}}\) is used here with slight abuse, since \(\mathrm {Var}_{\Psi _0,{\mathsf {p}},{\mathcal {E}}}\) does not enjoy all of the standard properties of total variation functionals, see [41, Rmk. 3.6] for further details. Also observe that, if \({\mathsf {p}}\) is \(\Psi _0\)-non-degenerate, then we have \(\mathrm {Var}_{\Psi _0,{\mathsf {p}},{\mathcal {E}}}(u; [a,b]) \ge \mathrm {Var}_{\Psi _0}(u; [a,b])\).
We are finally in a position to introduce the concept of balanced viscosity solution to the rate-independent system individuated by the triple \(( \Psi _0,{\mathsf {p}},{\mathcal {E}})\). This definition extends [41, Def. 4.1].
Definition 3.5
(Balanced viscosity solution) Given a (non-degenerate) 1-homogeneous dissipation potential \(\Psi _0\) and a (non-degenerate) viscosity contact potential \({\mathsf {p}}\), we say that a curve \(u\in \mathrm {BV}([0,T];{\mathbb {R}}^d)\) is a balanced viscosity \((\mathrm {BV})\) solution to the rate-independent system \(( \Psi _0,{\mathsf {p}},{\mathcal {E}})\) if it fulfills the local stability (\(\mathrm {S}_\mathrm {loc}\)) and the (\(\mathrm {E}_{\Psi _0, {\mathsf {p}},{\mathcal {E}}}\))-energy-dissipation balance
with \(K^* = \partial \Psi _0(0)\).
While referring to [41, Sec. 4] and [42, Sec. 3] for a detailed survey of the properties of \(\mathrm {BV}\) solutions, let us only mention here a few. Firstly, by the analogue of [41, Prop. 4.2], for a \(\mathrm {BV}\) solution the energy-dissipation balance (\(\mathrm {E}_{\Psi _0, {\mathsf {p}},{\mathcal {E}}}\)) indeed holds on every subinterval \([s,t] \subset [0,T]\), i.e.,
Furthermore, the concept of \(\mathrm {BV}\) solution yields a thorough description of the energetic behavior of the solution at jumps through the concept of optimal jump transition. For fixed \(t\in [0,T]\) and \(u_-, u_+ \in {\mathbb {R}}^d\), we call a curve \( \theta \in \mathrm {AC} ([0,1];{\mathbb {R}}^d)\) (up to a rescaling, we may indeed suppose the curves in (3.21) to be defined on [0, 1]), with \( \theta (0) = u_- \) and \(\theta (1) = u_+\), a \(({\mathsf {p}}, {\mathcal {E}}_t )\)-optimal transition between \(u_-\) and \(u_+\) if it has constant Finsler velocity \({\mathsf {p}} (0,\dot{\theta }(\cdot ),{-}\mathrm {D}{\mathcal {E}}(t,\theta (\cdot )))\) and there holds
The following result subsumes [41, Prop. 4.6, Thm. 4.7].
Proposition 3.6
Let \(u\in \mathrm {BV}([0,T];{\mathbb {R}}^d)\) be a balanced viscosity solution to the rate-independent system \(( \Psi _0,{\mathsf {p}},{\mathcal {E}})\). Then, at every jump time \(t\in \mathrm {J}_{u} \) there exists a \(({\mathsf {p}},{\mathcal {E}}_t)\)-optimal transition \(\theta ^t\) between the left and right limits \(u_-(t)\) and \(u_+(t)\), such that \(\theta ^t (r) = u(t)\) for some \(r\in [0,1]\). Moreover, any optimal jump transition \(\theta ^t\) between \(u_-(t)\) and \(u_+(t)\) complies with the contact contact condition
with the contact set \( \Lambda _{{\mathsf {p}},0} \) from (3.19).
A crucial consequence of (3.25) and of (3.20) from Lemma 3.3 is that any optimal jump transition \(\theta ^t\) complies with the subdifferential inclusion
The validity of this flow rule explicitly shows how the contact potential \({\mathsf {p}}\) enters into the description of the solution behavior at jumps.
With the last result of this section, we reformulate the \(\mathrm {BV}\) solution concept in terms of the null-minimization of a functional defined on \(\mathrm {BV}\)-trajectories; this will be crucial for the variational convergence analysis developed in Sec. 5. Namely, given a rate-independent system \(( \Psi _0,{\mathsf {p}},{\mathcal {E}})\), we define the trajectory functional \({\mathscr {J}}_{\Psi _0,{\mathsf {p}},{\mathcal {E}}}: \mathrm {BV}([0,T];{\mathbb {R}}^d) \rightarrow (-\infty ,+\infty ]\) by
where the second identity follows from (3.15), with \(\dot{u}\) the density of the absolutely continuous part of \(u'\) w.r.t. the Lebesgue measure \({\mathscr {L}}^1\). We then have the following
Proposition 3.7
A curve \(u\in \mathrm {BV}([0,T];{\mathbb {R}}^d)\) is a balanced viscosity solution to the rate-independent system \(( \Psi _0,{\mathsf {p}},{\mathcal {E}})\) if and only if
Proof
First of all, observe that conditions (\(\mathrm {S}_\mathrm {loc}\))–(\(\mathrm {E}_{\Psi _0, {\mathsf {p}},{\mathcal {E}}}\)) are indeed equivalent to (\(\mathrm {S}_\mathrm {loc}'\))–(\(\mathrm {E}_{\Psi _0, {\mathsf {p}},{\mathcal {E}}}\)), with
Indeed, if (\(\mathrm {S}_\mathrm {loc}'\)) holds, with a continuity argument one deduces \( -\mathrm {D}{\mathcal {E}}(t, u(t))\in K^*\) at all \(t\in [0,T]{\setminus } \mathrm {J}_{u}\).
Clearly, (\(\mathrm {S}_\mathrm {loc}'\))–(\(\mathrm {E}_{\Psi _0, {\mathsf {p}},{\mathcal {E}}}\)) are then equivalent to
Now, with an argument based on the chain rule for \({\mathcal {E}}\), one sees (cf. the proof of [42, Cor. 3.4] and of Lemma 3.1) that along a given curve \(v\in \mathrm {BV}([0,T];{\mathbb {R}}^d)\) the map \( {\mathscr {J}}_{\Psi _0,{\mathsf {p}},{\mathcal {E}}}(v) \ge 0\), so that (3.29) holds if and only if \( {\mathscr {J}}_{\Psi _0,{\mathsf {p}},{\mathcal {E}}}(u) \le 0\), i.e., \(u\in \mathrm {Argmin}_{v\in \mathrm {BV}([0,T];{\mathbb {R}}^d) } {\mathscr {J}}_{\Psi _0,{\mathsf {p}},{\mathcal {E}}}(v)\). This concludes the proof. \(\square \)
4 Generation of viscosity contact potentials via \(\Gamma \)-convergence
In this section, we provide a possible procedure to generate a viscosity contact potential via a \(\Gamma \)-convergence procedure, starting from a family \((\Psi _n)_n\) of dissipation potentials with superlinear growth at infinity (cf. (3.2)).
Preliminarily, given a convex dissipation potential \(\Psi \), we define the bipotential\( {\mathsf {b}}_{\Psi } : [0,+\infty ) \times {\mathbb {R}}^d \times {\mathbb {R}}^d \rightarrow [0,+\infty ]\) induced by \(\Psi \) via
It is immediate to check that
- (1)
for every \((v,\xi )\in {\mathbb {R}}^d \times {\mathbb {R}}^d\) the map \(\tau \mapsto {\mathsf {b}}_{\Psi }(\tau ,v,\xi )\) is convex;
- (2)
for every \(\tau \ge 0\) the functional \((v,\xi )\mapsto {\mathsf {b}}_{\Psi }(\tau ,v,\xi )\) is a bipotential in the sense of [6] (cf. Remark 3.4);
- (3)
for every \(v \ne 0\) and \(\xi \in {\mathbb {R}}^d\) with \(\Psi ^*(\xi ) \ne 0\), the set \(\mathrm {Argmin}_{\tau >0} {\mathsf {b}}_{\Psi }(\tau ,v,\xi )\) is non-empty,
where the latter property is due to the fact that \(\lim _{\tau \downarrow 0} {\mathsf {b}}_{\Psi }(\tau ,v,\xi ) =+\infty \) due to the superlinear growth of \(\Psi \), and \(\lim _{\tau \uparrow +\infty } {\mathsf {b}}_{\Psi }(\tau ,v,\xi ) =+\infty \).
Let us now be given a sequence \((\Psi _n)_n\) of dissipation potentials, and let \(({\mathsf {b}}_{\Psi _n})_n\) be the associated bipotentials. We assume the following.
Hypothesis 4.1
Let \( {\mathsf {p}}: [0,+\infty ) \times {\mathbb {R}}^d \times {\mathbb {R}}^d \rightarrow [0,+\infty ]\) be defined by
Then,
In Sect. 5.2 ahead, we will exhibit two classes of dissipation potentials \((\Psi _n)_n\), with superlinear growth at infinity, and associated functionals \({\mathsf {p}}\), complying with Hypothesis 4.1.
Observe that with (4.3) we are imposing a stronger condition than \({\mathsf {p}}= \Gamma \text {-}\limsup _{n\rightarrow \infty } {\mathsf {b}}_{\Psi _n}\), namely we are asking that
This property will play a key role in the proof of Lemma 4.3 below.
The main result of this section, Theorem 4.2, ensures that the functional \({\mathsf {p}}\) generated via (4.2)–(4.3) is a contact potential in the sense of Definition 3.2.
Theorem 4.2
Let \((\Psi _n)_n\) be a sequence of dissipation potentials on \({\mathbb {R}}^d\) complying with Hypothesis 4.1. Then, \({\mathsf {p}}\) is a viscosity contact potential according to Definition 3.2, and there exists a 1-homogeneous dissipation potential \(\Psi _0\) such that
Moreover, if the dissipation potentials \((\Psi _n)_n\) fulfill the uniform coercivity condition
with \(\Vert \cdot \Vert \) any norm in \({\mathbb {R}}^d\), then \(\Psi _0\) is non-degenerate and \({\mathsf {p}}\) is \(\Psi _0\)-non-degenerate.
We postpone the proof of Theorem 4.2 to the end of this section, after obtaining a series of preliminary lemmas on the structure that the functional \({\mathsf {p}}\) defined by Hypothesis 4.1 inherits from the potentials \(\Psi _n\).
Lemma 4.3
Assume Hypothesis 4.1. Then, for every \((\tau ,v,\xi ) \in [0,+\infty ) \times {\mathbb {R}}^d \times {\mathbb {R}}^d\) there holds
- (1)
\({\mathsf {p}} (\tau ,v,\xi )\ge \langle v,\xi \rangle \);
- (2)
the map \((\tau ,v) \mapsto {\mathsf {p}} (\tau ,v,\xi )\) is convex and positively homogeneous of degree 1.
Proof
Property (1) is an immediate consequence of (4.2), using that for every \(n\in {\mathbb {N}}\) there holds \({\mathsf {b}}_{\Psi _n}(\tau ,v,\xi ) \ge \langle v , \xi \rangle \) for every \((\tau ,v,\xi ) \in [0,+\infty ) \times {\mathbb {R}}^d \times {\mathbb {R}}^d\).
In order to show that the mapping \({\mathsf {p}} (\cdot ,\cdot ,\xi )\) is convex for fixed \(\xi \), let \((\xi _n)_n\) with \(\xi _n\rightarrow \xi \) fulfill (4.3). For fixed \((\tau _0,v_0)\) and \((\tau _1,v_1)\) let \((\tau _n^i,v_n^i)_n\), \(i=1,2\), be two associated recovery sequences for \({\mathsf {b}}_{\Psi _n}(\cdot ,\cdot ,\xi _n)\) as in (4.4). Then, for every \(\lambda \in [0,1]\) there holds
where (1) follows from (4.2), (2) from the convexity of the maps \({\mathsf {b}}_{\Psi _n}(\cdot ,\cdot ,\xi _n)\), and (3) from (4.3).
With an analogous argument, one proves that \({\mathsf {p}} (\cdot ,\cdot ,\xi )\) is 1-positively homogeneous. \(\square \)
We now show that, for \(\tau >0\) the functional \({\mathsf {p}} (\tau ,\cdot ,\cdot )\) has the same form (4.1) as \({\mathsf {b}}_{\Psi _n}(\tau ,\cdot ,\cdot )\), cf. (4.8).
Lemma 4.4
Assume Hypothesis 4.1. Let \(\Psi _0: {\mathbb {R}}^d \rightarrow [0+\infty )\) be defined by
Then, \(\Psi _0\) is a 1-positively homogeneous dissipation potential, the sequence \((\Psi _n)_n\)\(\Gamma \)-converges to \(\Psi _0\), and thus \((\Psi _n^*)_n\)\(\Gamma \)-converges to \(\Psi _0^*\). Furthermore,
Proof
Observe that \(\Psi _0\) from (4.7) is convex and 1-homogeneous thanks to item (2) in the statement Lemma 4.3, which obviously yields the convexity of \(v\mapsto {\mathsf {p}} (1,v,0)\),
It follows from (4.2), applied with the choices \(\tau =1\) and \(\xi =0\) and with the sequences \(\tau _n \equiv 1\) and \(\xi _n \equiv 1\), that \(\Psi _0\le \Gamma \text {-}\liminf _{n \rightarrow \infty } \Psi _n\). Conversely, applying (4.3), we deduce that \(\Gamma \text {-}\limsup _{n \rightarrow \infty } \Psi _n \le \Psi _0\). In fact, we use (4.4) with \(\tau =1\) and \(\xi =0\) to find a sequence \(\xi _n \rightarrow 0\) and sequences \(\tau _n \rightarrow 1\) (we may indeed suppose that \(\tau _n \uparrow 1\)) and \(v_n\rightarrow v\) such that
In turn, \( \limsup _{n\rightarrow \infty } \left( \tau _n \Psi _n(\frac{v_n}{\tau _n}) {+} \tau _n \Psi _n^*(\xi _n)\right) \ge \limsup _{n\rightarrow \infty } \tau _n \Psi _n(\frac{v_n}{\tau _n}) \ge \limsup _{n\rightarrow \infty } \Psi _n(v_n)\), where the latter inequality ensues from the fact that, for any dissipation potential \(\Psi \), the map \(\tau \mapsto \Psi \left( \tfrac{v}{\tau }\right) \) is non-increasing for all \(v\in {\mathbb {R}}^d\). Combining these two estimates yields \(\Gamma \text {-}\limsup _{n \rightarrow \infty } \Psi _n \le \Psi _0\). All in all, we conclude that \(\Psi _0= \Gamma \text {-}\lim _{n\rightarrow \infty } \Psi _n\). Then, \((\Psi _n^*)_n\)\(\Gamma \)-converges to \(\Psi _0^*\) by [4, Thm. 2.18, p. 495]. As a consequence of these convergences and of (4.1), we have (4.8). \(\square \)
Our next two results address the characterization of \({\mathsf {p}}\) for \(\tau =0\), providing a formula for \({\mathsf {p}} (0,v,w)\) in the two cases \(\Psi _0^*(\xi )<+\infty \) and \(\Psi _0^*(\xi )=+\infty \).
Lemma 4.5
Assume Hypothesis 4.1. If \(\Psi _0^*(\xi ) < +\infty \), then
Proof
It follows from (4.8) and the fact that \(\Psi _0^*(\xi ) < +\infty \) that
To prove the converse inequality, we again use that the map \(\tau \mapsto \Psi \left( \tfrac{v}{\tau }\right) \) is non-increasing for every \(v \in {\mathbb {R}}^d\). Therefore, for all \(0<\tau<\sigma <1\) we have
Now, let us fix a sequence \(\xi _n \rightarrow \xi \) for which (4.3) holds, and accordingly a sequence \((\tau _n,v_n) \rightarrow (0,v)\) such that \({\mathsf {p}} (0,v,\xi )= \liminf _{n \rightarrow \infty } ( \tau _n \Psi _n ({v_n}/{\tau _n}) + \tau _n \Psi _n^* (\xi _n) ) \). It follows from inequality (4.11) applied to the functionals \(\Psi _n\) that for every \(\sigma \in (0,1)\)
where we have also exploited the positivity of the functionals \(\Psi _n^*\). Therefore, in view of (4.3) we find
and conclude the converse of (4.10) passing to the limit as \(\sigma \rightarrow 0\). \(\square \)
Lemma 4.6
Assume Hypothesis 4.1. If \(\Psi _0^*(\xi ) =+ \infty \), then
Proof
Inequality \(\ge \) follows from the definition of \({\mathsf {p}}\). To prove the converse one, we may suppose that \(v \ne 0\), since \({\mathsf {p}} (0,0,\xi ) =0\). Take \((v_n,\xi _n)\rightarrow (v,\xi )\) that attains \(\Gamma \text {-}\liminf _{n \rightarrow \infty } \inf _{\tau >0} {\mathsf {b}}_{\Psi _n}(\tau ,v,\xi )\), i.e., \(\inf _{\tau>0} {\mathsf {b}}_{\Psi _n}(\tau ,v_n,\xi _n) \rightarrow \Gamma \text {-}\liminf _{n \rightarrow \infty } \inf _{\tau >0} {\mathsf {b}}_{\Psi _n}(\tau ,v,\xi )\). In particular, \(\liminf _{n\rightarrow \infty } \Psi _n^*(\xi _n)=+\infty \). Therefore, we may choose \(\bar{\tau }_n\) as
Since \(\liminf _{n\rightarrow \infty } \Psi _n^*(\xi _n)=+\infty \), it is clear that \(\bar{\tau }_n \rightarrow 0\), hence
thanks to (4.2). \(\square \)
We now prove a pseudo-monotonicity result for \({\mathsf {p}}\).
Lemma 4.7
Assume Hypothesis 4.1. Then, for every \(\tau ,\,{\bar{\tau }} \in [0,+\infty )\), \(v,\,{\bar{v}} \in {\mathbb {R}}^d\) and \(\xi ,\,\bar{\xi }\in {\mathbb {R}}^d\) we have that
Proof
Observe that (4.13) holds for the bipotentials \({\mathsf {b}}_{\Psi _n}\): indeed, in that case, it reduces to \( \tau \bar{\tau } (\Psi _n^*(\xi )-\Psi _n^*(\bar{\xi }))^2 \ge 0\).
Assume that \({\mathsf {p}} (\tau ,v,\xi )> {\mathsf {p}} (\tau ,v,\bar{\xi }) \) and choose \(\bar{\xi }_n\) as in (4.3) with \((\tau _n,v_n)\) such that
It follows from Definition (4.2) of \({\mathsf {p}}\) that \({\mathsf {p}} (\tau ,v,\xi ) \le \liminf _{n \rightarrow \infty } {\mathsf {b}}_{\Psi _n}(\tau _n,v_n,\xi _n)\) for every sequence \(\xi _n \rightarrow \xi \) in \({\mathbb {R}}^d\), and for \((\tau _n, v_n)\) as in (4.14). Then,
Therefore, for sufficiently big n, we have that
Now, again in view of (4.3), choose \(\xi _n \rightarrow \xi \) (notice that (4.15) holds for any sequence \(\xi _n\) converging to \(\xi \)) and \(\bar{\tau }_n\rightarrow {\bar{\tau }}\), \({\bar{v}}_n \rightarrow {{\bar{v}}}\) such that \(\limsup _{n \rightarrow \infty } {\mathsf {b}}_{\Psi _n}({{\bar{\tau }}}_n,{\bar{v}}_n,\xi _n) \le {\mathsf {p}} ({{\bar{\tau }}},{\bar{v}},\xi )\). Since \(\liminf _{n \rightarrow \infty } {\mathsf {b}}_{\Psi _n}({{\bar{\tau }}}_n,{\bar{v}}_n,\bar{\xi }_n) \ge {\mathsf {p}} ({\bar{\tau }},{\bar{v}},\bar{\xi })\) by (4.2), we conclude that
taking into account that \( {\mathsf {b}}_{\Psi _n}({{\bar{\tau }}}_n,{\bar{v}}_n,\xi _n) - {\mathsf {b}}_{\Psi _n}({{\bar{\tau }}}_n,{\bar{v}}_n,\bar{\xi }_n)\ge 0 \) for sufficiently big n thanks to (4.16) and the previously observed monotonicity property (4.13) for \({\mathsf {b}}_{\Psi _n}\). Thus, (4.13) follows. \(\square \)
Finally, let us consider contact sets associated with the bipotentials \({\mathsf {b}}_{\Psi _n}\), i.e.,
Observe that for every \(n\in {\mathbb {N}}\)
- (1)
\(\Lambda _{{\mathsf {b}}_{\Psi _n}} \cap \{0\} \times {\mathbb {R}}^d \times {\mathbb {R}}^d = \{0\} \times \{ 0\} \times {\mathbb {R}}^d\);
- (2)
for \(\tau >0\), if \((\tau ,v,\xi ) \in \Lambda _{{\mathsf {b}}_{\Psi _n}}\), then \(\tau \in \mathop {\mathrm {Argmin}}_{\sigma \in (0,+\infty )} ( \sigma \Psi _n (\tfrac{v}{\sigma }) + \sigma \Psi _n^*(\xi ))\).
The following closedness property may be easily derived from (4.2).
Lemma 4.8
Assume Hypothesis 4.1. Then,
We are now in a position to carry out the proof of Theorem 4.2 by verifying that \({\mathsf {p}}\) complies with properties (1)–(5) from Definition 3.2.
Properties (1) and (2) are guaranteed by Lemma 4.3, whereas (3) ensues from (4.8) in Lemma 4.4. Concerning property (5), observe that (4.5) ensues from (4.8) for \(\tau >0\). For \(\tau =0\), it directly follows from (4.9) in the case \(\Psi _0^*(\xi )<+\infty \), whereas for \(\Psi _0^*(\xi )=+\infty \) we use the monotonicity property (4.13), giving
Now, \( {\mathsf {p}} (1,v,\xi ) = \Psi _0(v) + \Psi _0^*(\xi ) = +\infty \), hence we deduce that \( {\mathsf {p}} (0,v,\xi )\ge {\mathsf {p}} (0,v,0) \ge \Psi _0(v)\) (here we have used that \(\Psi _0^*(0) =0\)).
Under additional (4.6), it is immediate to check that \(\Psi _0\) given by (4.7) is non-degenerate, whence the validity of property (4). This concludes the proof of Theorem 4.2. \(\square \)
5 Main results
Let us consider a sequence \((\Psi _n)_n\) of dissipation potentials on \({\mathbb {R}}^d\) with superlinear growth at infinity, namely fulfilling (3.2) for every \(n\in {\mathbb {N}}\). A straightforward extension of the by now classical results by Colli&Visintin for doubly nonlinear evolution equations (cf. [15, 16]) yields that for every \(n\in {\mathbb {N}}\) there exists at least a solution \(u \in \mathrm {AC}([0,T];{\mathbb {R}}^d)\) of (the Cauchy problem for) the generalized gradient system \((\Psi _n,{\mathcal {E}})\), with \({\mathcal {E}}\) complying with (\(\textsc {E}\)). Namely, u solves the doubly nonlinear differential inclusion
As we have seen with Lemma 3.1, the solutions to (5.1) coincide with the null-minimizers for the (positive) functional of trajectories \({\mathcal {J}}_{\Psi _n,{\mathcal {E}}} : \mathrm {AC}([0,T];{\mathbb {R}}^d) \rightarrow [0,+\infty )\) defined by
The main results of this paper, Theorems 5.2 and 5.9 ahead, concern the Mosco-convergence to the functional \({\mathscr {J}}_{\Psi _0,{\mathsf {p}},{\mathcal {E}}}\) from (3.27), with respect to the weak-strict topology of \(\mathrm {BV}([0,T];{\mathbb {R}}^d)\) (cf. (3.17)), of a family of functionals \(({\mathscr {J}}_{\Psi _n,{\mathcal {E}}})_n\) extending \( {\mathcal {J}}_{\Psi _n,{\mathcal {E}}} \) to \( \mathrm {BV}([0,T];{\mathbb {R}}^d)\). Namely, we define
More precisely, Sect. 5.1 is centered around the \(\Gamma \)-\(\liminf \) result, Theorem 5.2, which implies (cf. Theorem 5.3) the Evolutionary \(\Gamma \)-convergence of the gradient systems (5.1) to a limiting rate-independent system (understood in the balanced viscosity sense). Theorems 5.2 and 5.3 are valid under the condition that the bipotentials \(({\mathsf {b}}_{\Psi _n})_n\) associated with the functionals \((\Psi _n)_n\) comply with Hypothesis 4.1. In Sect. 5.2, we show that Hypothesis 4.1 is, in particular, verified by two classes of dissipation potentials approximating a 1-homogeneous one, namely for ‘vanishing-viscosity’ approximations and for the ‘stochastic’ approximation advanced in [12]. For these two cases (the vanishing-viscosity one further particularized), in Sect. 5.3 we state our \(\Gamma \)-\(\limsup \) result, Theorem 5.9, along with the reverse approximation Theorem 5.12.
With the exception of Proposition 5.8, the proofs of all the upcoming results in this section shall be carried out throughout Sect. 6.
5.1 The \(\Gamma \)-liminf result
First of all, let us fix the compactness properties of a sequence \((u_n)_n \subset \mathrm {BV}([0,T];{\mathbb {R}}^d)\) with \(\sup _{n} {\mathscr {J}}_{\Psi _n,{\mathcal {E}}}(u_n) \le C\), assuming that the potentials \(\Psi _n\) comply with a suitable coercivity property.
Proposition 5.1
Let \((\Psi _n)_n\) be a family of dissipation potentials with superlinear growth at infinity and assume that
for some norm \(\Vert \cdot \Vert \) on \({\mathbb {R}}^d\). Let \((u_n)_n \subset \mathrm {BV}([0,T];{\mathbb {R}}^d)\) fulfill \( \Vert u_n(0)\Vert + {\mathscr {J}}_{\Psi _n,{\mathcal {E}}}(u_n) \le C\) for some constant \(C>0\), uniform w.r.t. \(n\in {\mathbb {N}}\). Then, there exist a subsequence \(k\mapsto n_k\) and a curve u such that \(u_{n_k} \rightharpoonup u\) in \(\mathrm {BV}([0,T];{\mathbb {R}}^d)\).
We are now in a position to state the \(\Gamma \text {-}\liminf \) result for the sequence \(({\mathscr {J}}_{\Psi _n,{\mathcal {E}}})_n\).
Theorem 5.2
Let \((\Psi _n)_n\) be a family of dissipation potentials with superlinear growth at infinity such that the associated bipotentials \(({\mathsf {b}}_{\Psi _n})_n\) comply with Hypothesis 4.1, with limiting viscosity contact potential \({\mathsf {p}}\). Let \(\Psi _0\) be the 1-positively homogeneous dissipation potential defined by \(\Psi _0(v) := {\mathsf {p}} (1,v,0)\), and suppose that \(\Psi _0\) is non-degenerate.
Then, for every \((u_n)_n,\, u \in \mathrm {BV}([0,T];{\mathbb {R}}^d)\), we have that
More precisely, we have as \(n\rightarrow \infty \)
for every \(0\le s\le t \le T\).
A straightforward consequence of Theorem 5.2 is the following result.
Theorem 5.3
Under the assumptions of Theorem 5.2, let \((u_n)_n \subset \mathrm {AC}([0,T];{\mathbb {R}}^d)\) fulfill \({\mathscr {J}}_{\Psi _n}(u_n) \le \varepsilon _n \) for every \(n\in {\mathbb {N}}\), for some vanishing sequence \((\varepsilon _n)_n\).
Then, any limit point u of \((u_n)_n\) with respect to the weak-\(\mathrm {BV}([0,T];{\mathbb {R}}^d)\)-topology is a balanced viscosity solution to the rate-independent system \(( \Psi _0,{\mathsf {p}},{\mathcal {E}})\), and, up to a subsequence, convergences (5.6) and
hold for all \(0\le s \le t \le T\).
Remark 5.4
By virtue of Proposition 5.1, under the uniform coercivity condition (5.4) the set of the limit points of the sequence \((u_n)_n\) in the statement of Theorem 5.3 is non-empty (if, in addition, \(\sup _{n\in {\mathbb {N}}}\Vert u_n(0)\Vert \le C\)). If (5.4) is strengthened to (4.6), we also have that the limiting dissipation potential \(\Psi _0\) is non-degenerate.
5.2 Examples
We now focus on two classes of dissipation potentials \((\Psi _n)_n\), with superlinear growth at infinity, approximating a 1-positively homogeneous dissipation potential \(\Psi _0\). In the first case, the dissipation potentials \(\Psi _n\) are obtained by rescaling from a given dissipation potential \(\Psi \) with superlinear growth at infinity and suitably converge to a 1-homogeneous potential \(\Psi _0\). In the second case, we consider the stochastic model introduced in Sect. 2 and the associated potentials \(\Psi _n\) given by (2.6): the limiting potential is \( \Psi _0(v) = A \Vert v\Vert _1\), where \(\Vert \cdot \Vert _1\) denotes the \(L^1\)-norm on \({\mathbb {R}}^d\)
We will show that in both cases Hypothesis 4.1 is fulfilled.
5.2.1 The vanishing-viscosity approximation
We consider the dissipation potentials
with \(\Psi :{\mathbb {R}}^d \rightarrow [0,+\infty ) \) a fixed potential with superlinear growth at infinity. We suppose that there exists a 1-homogeneous dissipation potential \(\Psi _0\) such that
Example 5.5
In particular, we focus on these two cases (cf. [41, Ex. 2.3]):
- (1)
\(\Psi _0\)-viscosity: the superlinear dissipation potential \(\Psi \) is obtained augmenting \(\Psi _0\) by a superlinear function of \(\Psi _0\) itself. Namely, given a convex superlinear function \(F_V: [0,+\infty ) \rightarrow [0,+\infty )\), we set
$$\begin{aligned}&\Psi (v) := \Psi _0(v) + F_V(\Psi _0(v)), \quad \text {whence }\nonumber \\&\Psi _n(v)= \Psi _0(v) + \frac{1}{\varepsilon _n} F_V(\varepsilon _n \Psi _0(v)) \quad \text {for all } v \in {\mathbb {R}}^d. \end{aligned}$$(5.10)To fix ideas, we may think of \(\Psi _0(v) = A \Vert v\Vert _1\) and \(F_V(\rho ) = \frac{1}{2}\rho ^2\), giving rise to
$$\begin{aligned} \Psi _n(v) = A \Vert v\Vert _1 + \frac{\varepsilon _n}{2} A^2 \Vert v\Vert _1^2. \end{aligned}$$(5.11) - (2)
2-norm vanishing viscosity: Let us now consider a norm \(\Vert \cdot \Vert \) on \({\mathbb {R}}^d\), different from that associated with \(\Psi _0\). We set
$$\begin{aligned} \Psi (v) : = \Psi _0(v) + F_V( \Vert v\Vert ), \quad \text {whence } \Psi _n(v) = \Psi _0(v) + \frac{1}{\varepsilon _n} F_V(\varepsilon _n \Vert v\Vert )\quad \text {for all } v \in {\mathbb {R}}^d, \end{aligned}$$(5.12)with again \(F_V: [0,+\infty ) \rightarrow [0,+\infty )\) convex and superlinear. In this way, we generate, for example, the dissipation potentials
$$\begin{aligned} \Psi _n(v) = A \Vert v\Vert _1 + \frac{\varepsilon _n}{2} \Vert v\Vert _2^2, \end{aligned}$$(5.13)with \( \Vert v\Vert _2 : = \left( \sum _{i=1}^d |v_i|^2\right) ^{1/2}\) and, more in general,
$$\begin{aligned} \Psi _n(v) = A \Vert v\Vert _1 + \frac{\varepsilon _n^{p-1}}{p} \Vert v\Vert _p^p \qquad \text {with } \Vert v\Vert _p : = \left( \sum _{i=1}^d |v_i|^p\right) ^{1/p}\,. \end{aligned}$$(5.14)
This family of dissipation potentials comply with the hypotheses of Theorem 5.2, as shown by the following result.
Proposition 5.6
The dissipation potentials from (5.9) comply with (4.6) and with Hypothesis 4.1, where
The proof can be straightforwardly retrieved from the argument for [41, Lemma 6.1].
Example 5.7
(Example 5.5 continued) Following [41, Rem. 3.1], we explicitly calculate \({\mathsf {p}} (0,v,\xi )\), using formula (5.15), in the two cases of Example 5.5:
- (1)
\(\Psi _0\)-viscosity: We have
$$\begin{aligned} {\mathsf {p}} (0,v,\xi ) : = {\left\{ \begin{array}{ll} \Psi _0(v) &{} \text {if } \xi \in K^*, \\ \Psi _0(v) \sup \nolimits _{v\ne 0} \frac{\langle \xi ,v \rangle }{\Psi _0(v)} &{} \text {if } \xi \notin K^*. \end{array}\right. } \end{aligned}$$Therefore, in the particular case \(\Psi _0(v) = A \Vert v\Vert _1\), taking into account that
$$\begin{aligned} K^* = {\overline{B}}_A^{\infty } (0) := \{ \xi \in {\mathbb {R}}^d\, : \ \Vert \xi \Vert _\infty \le A \} \qquad \text {with } \Vert v \Vert _\infty : = \max _{i=1,\ldots , d} |v_i|\,, \end{aligned}$$we retrieve the formula
$$\begin{aligned} {\mathsf {p}} (0,v,\xi ) = \Vert v\Vert _1 (A \vee \Vert \xi \Vert _\infty ) \end{aligned}$$(5.16)(here and in what follows, we use the notation \(a\vee b\) for \(\max \{a,b\}\)).
- (2)
2-norm vanishing viscosity: In this case, we have
$$\begin{aligned} {\mathsf {p}} (0,v,\xi ) = \Psi _0(v) + \Vert v\Vert \min _{\zeta \in K^*} \Vert \xi -\zeta \Vert _*, \end{aligned}$$(5.17)where we have used the notation \(\Vert \zeta \Vert _* : = \sup _{v\ne 0} \tfrac{\langle \zeta , v \rangle }{\Vert v\Vert }\).
The stochastic approximation. We now consider the dissipation potentials \(\Psi _n\) from (2.6), i.e.,
Preliminarily, we observe that
In order to check the above statement, e.g., for \(\Psi _n(v)\), it is sufficient to recall that \(\Psi _n(v) = \sum _{i=1}^d \psi _n(v_i)\), and that the real functions \((\psi _n)_n\) pointwise and \(\Gamma \)-converge to the 1-positively homogeneous potential \(\psi _0:{\mathbb {R}}\rightarrow {\mathbb {R}}\) given by \(\psi _0(v) = A |v|\). We will now prove that the analogue of Proposition 5.6 holds for the potentials from (5.18).
Proposition 5.8
The dissipation potentials from (5.18) comply with (4.6) and with Hypothesis 4.1, with limiting viscosity contact potential
Proof
We will split the proof in several claims.
Claim 1: (5.20) holds for \(\tau >0\). It follows from the \(\Gamma \)-convergence properties in (5.19) that \({\mathsf {p}}= \Gamma \text {-}\liminf _{n\rightarrow \infty } {\mathsf {b}}_{\Psi _n}\) fulfills \({\mathsf {p}} (\tau ,v,\xi ) \ge \Psi _0(v) + I_{K^*}(\xi ) \) for all \((v,\xi )\in {\mathbb {R}}^d \times {\mathbb {R}}^d\), if \(\tau >0\). For the converse inequality, for every \(\xi \in {\mathbb {R}}^d\) we take the constant sequence \(\xi _n \equiv \xi \) and again choose for fixed \((\tau ,v) \in (0,+\infty ) \times {\mathbb {R}}^d\) the sequences \(\tau _n \equiv \tau \) and \(v_n \equiv v\). The pointwise convergences from (5.19) ensure that
Hence, we conclude that \({\mathsf {p}} (\tau ,v,\xi )=\Psi _0(v) + I_{K^*}(\xi )\), i.e., the validity of (5.20) for \(\tau >0\).
Claim 2: (5.20) holds for \(\tau =0\)and\(v=0\). In this case, we have to check that \({\mathsf {p}} (0,0,\xi ) =0\), which is equivalent to showing that \({\mathsf {p}} (0,0,\xi ) \le 0\) as the functional \({\mathsf {p}}\) is positive. To this aim, for every fixed \(\xi \in {\mathbb {R}}^d\), we observe that for any null sequence \(\tau _n\downarrow 0\)
and then we choose \((\tau _n)_n\) vanishing fast enough in such a way that the \(\limsup \) on the right-hand side equals zero.
Claim 3: (5.20) holds for \(\tau =0\)and \(v\ne 0\). We will split the proof in several (sub-)claims. In the following calculations, taking into account that \(\Psi _n = \sum _{i=1}^d \psi _n\) and \(\Psi _n^* = \sum _{i=1}^d \psi _n^*\) with \(\psi _n:{\mathbb {R}}\rightarrow {\mathbb {R}}\) and \(\psi _n^* : {\mathbb {R}}\rightarrow {\mathbb {R}}\)even functions, we will often confine the discussion to the case in which \(v = (v_1, \ldots , v_d)\) fulfills \(v_i \ge 0\) for all \(i=1, \ldots , d\), and analogously for \(\xi = (\xi _1, \ldots , \xi _d) \).
Moreover, we will need to work with the perturbed bipotentials \({\mathsf {b}}_{\Psi _n}^{\delta } : [0,+\infty ) \times {\mathbb {R}}^d \times {\mathbb {R}}^d \rightarrow [0,+\infty ]\) given by
with \(\delta >0\) fixed. We remark that \(\mathop {\mathrm {Argmin}}_{\tau >0} {\mathsf {b}}_{\Psi _n}^{\delta }(\tau ,v,\xi )\ne \emptyset \). Indeed, for every fixed \((v,\xi ) \in {\mathbb {R}}^d \times {\mathbb {R}}^d\) the functional \(\tau \mapsto {\mathsf {b}}_{\Psi _n}^{\delta }(\tau ,v,\xi )\) is convex on \((0,\infty )\) and, since \(v \ne 0\), it fulfills \(\lim _{\tau \downarrow 0} {\mathsf {b}}_{\Psi _n}^{\delta }(\tau ,v,\xi )= \lim _{\tau \uparrow _\infty } {\mathsf {b}}_{\Psi _n}^{\delta }(\tau ,v,\xi )=+\infty \) by the superlinear growth property of the functionals \(\Psi _n\), cf. also (3.3). A straightforward calculation also shows that for every \(\xi \in {\mathbb {R}}^d\) the map \(v\mapsto \min _{\tau >0} {\mathsf {b}}_{\Psi _n}^{\delta }(\tau ,v,\xi )\) is 1-positively homogeneous. Therefore, there exists a closed convex set \( K_{n,\delta }^*(\xi ) \) such that
Indeed, it turns out (cf. [41, Thm. A.17]) that
We need an intermediate estimate before proving the \(\ge \)-inequality in (5.20), i.e., (5.24) below.
Claim 3.1: there holds
This follows from
where (2) follows from the fact that \(\lim _{n\rightarrow \infty }\delta \tau _n=0\) for every vanishing sequence \((\tau _n)\) .
Claim 3.2: there holds
In view of (5.23), it is sufficient to prove that
Hence, we fix a sequence \((v_n,\xi _n) \rightarrow (v,\xi )\) and, for n sufficiently big such that \(\frac{1}{n} \log d < A\), define \(w_n \in {\mathbb {R}}^d\) by
Taking into account form (2.7) of \(\Psi _n^*\), we estimate
distinguishing the two cases \( \Vert \xi _n\Vert _{\infty }\le A\) and \( \Vert \xi _n\Vert _{\infty }> A\). In the former situation, it is sufficient to observe that \( \Vert w_n \Vert _{\infty }\le A, \) so that
for n sufficiently big. In the case \( \Vert \xi _n\Vert _{\infty }> A\), we use that
for n sufficiently big such that \(\frac{d^2-1}{2n } \le \delta \). All in all, (5.26) gives that
which implies that \(w_n \in K_{n,\delta }^*(\xi _n)\), for all n sufficiently big. Now, using the representation formula (5.22a) for \({\mathsf {b}}_{\Psi _n}^{\delta }({\bar{\tau }}_n^\delta ,\cdot ,\cdot )\), we find
where the last equality follows from the fact that \(v_n = (v_n^1,\ldots , v_n^d)\) fulfills \(v_n^i \ge 0\) for all \(i=1,\ldots , d\). Hence, \(\liminf _{n\rightarrow \infty } {\mathsf {b}}_{\Psi _n}^\delta ({\bar{\tau }}_n^\delta ,v_n,\xi _n) \ge \Vert v \Vert _1 (A \vee \Vert \xi \Vert _{\infty }) \) and, since the sequences \((v_n)_n\) and \((\xi _n)_n\) are arbitrary, we conclude (5.25), and thus (5.24).
In order to prove the converse of inequality (5.24), and thus conclude (5.20), we preliminarily need to investigate the properties of the sets \(K_{n,\delta }^*\).
Claim 3.3: there holds
Indeed, every \(w\in K_{n,\delta }^*(\xi )\) fulfills \(\Psi _n^*(w) \le \Psi _n^*(\xi ) + \delta \). Using the explicit formula for \(\Psi _n^*\), we obtain that
whereby
and thus
Now, doing some algebraic manipulations on the logarithmic term on the right-hand side, we find
where for (1) we have used that \(n\delta > d\) for n sufficiently big and for (2) we have estimated \(\log ( 1 + e^{n (\Vert \xi \Vert _{\infty }- A)_+}) = \log (e^{n (\Vert \xi \Vert _{\infty }- A)_+})+ \log (e^{-n (\Vert \xi \Vert _{\infty }- A)_+}+1) \le \log (e^{n (\Vert \xi \Vert _{\infty }- A)_+}) +1 \). Then, (5.27) ensues.
Claim 3.4: for every \((v,\xi ) \in {\mathbb {R}}^d \times {\mathbb {R}}^d\) and all sequences \((v_n)_n,\, (\xi _n)_n\) with \(v_n \rightarrow v \) and \(\xi _n \rightarrow \xi \), for every \({\bar{\tau }}_n^\delta \in \mathop {\mathrm {Argmin}}{\mathsf {b}}_{\Psi _n}^\delta (\cdot ,v_n,\xi _n)\) there holds
We distinguish two cases: (1) \(\Psi _0^*(\xi ) =+\infty \) and (2) \(\Psi _0^*(\xi ) =0\).
- (1)
In the first case, we have \(\liminf _{n\rightarrow \infty } \Psi _n^*(\xi _n) =+\infty \). Then, \({\bar{\tau }}_n^\delta \) must be vanishing to ‘cancel’ the \(\tau \Psi _n^*\)-contribution, cf. also the proof of Lemma 4.6.
- (2)
In the second case, to show (5.28) we will provide an estimate from above for \( {\bar{\tau }}_n^\delta \) by exploiting the Euler–Lagrange equation for the minimization problem \( \min _{\tau >0} {\mathsf {b}}_{\Psi _n}^\delta (\tau ,v,\xi ) \). Namely, observe that \( {\bar{\tau }}_n^\delta \) complies with
$$\begin{aligned} 0 \in \partial _\tau {\mathsf {b}}_{\Psi _n}^\delta (\cdot ,v_n,\xi _n)({\bar{\tau }}_n^\delta ) = \Psi _n\left( \frac{v_n}{{\bar{\tau }}_n^\delta } \right) - \left\langle \mathrm {D} \Psi _n \left( \frac{v_n}{{\bar{\tau }}_n^\delta } \right) , \frac{v_n}{{\bar{\tau }}_n^\delta } \right\rangle + \Psi _n^*(\xi _n) + \delta \,. \end{aligned}$$(5.29)Using the explicit formula (5.18) for \(\Psi _n\), we find
$$\begin{aligned} \Psi _n\left( \frac{v_n}{{\bar{\tau }}_n^\delta } \right) -\left\langle \mathrm {D} \Psi _n \left( \frac{v_n}{{\bar{\tau }}_n^\delta } \right) , \frac{v_n}{{\bar{\tau }}_n^\delta } \right\rangle = \frac{de^{-nA}}{n} - \sum _i \frac{1}{n} \sqrt{\frac{ (v_{n}^i)^2 }{ ({\bar{\tau }}_n^\delta )^2} + e^{-2nA}}\,. \end{aligned}$$Therefore, (5.29) yields
$$\begin{aligned}&n\delta + de^{-nA} + n\Psi _n^*(\xi _n)= \sum _i \sqrt{\frac{ (v_{n}^i)^2 }{({\bar{\tau }}_n^\delta )^2} + e^{-2nA}} \le d \sqrt{\frac{\Vert v_n \Vert _{\infty }^2}{({\bar{\tau }}_n^\delta )^2} + e^{-2nA}}, \nonumber \\&\qquad \text {whence} \quad ({\bar{\tau }}_n^\delta )^2 \le \frac{d^2 \Vert v_n \Vert _{\infty }^2}{n^2 \delta ^2 + n^2 \left( \Psi _n^*(\xi _n)\right) ^2} \rightarrow 0 \quad \text { for } n \rightarrow \infty . \end{aligned}$$(5.30)
We are now in a position to conclude the proof of (5.20).
Claim 3.5: there holds
We will in fact prove that
Taking into account that \({\mathsf {p}}= \Gamma \text {-}\liminf _{n\rightarrow \infty } {\mathsf {b}}_{\Psi _n}\), we will then conclude (5.31). To check (5.32), let us choose the constant recovery sequences \(\xi _n \equiv \xi \) and \(v_n \equiv v\), and let \(\tau _n: = {\bar{\tau }}_n^\delta \in \mathop {\mathrm {Argmin}}_{\tau >0} {\mathsf {b}}_{\Psi _n}^{\delta }(\tau ,v,\xi )\). By the previous Claim 3.4, we have that \(\tau _n \downarrow 0\). Now, in view of the representation formula (5.22a) for \(\min _{\tau >0} {\mathsf {b}}_{\Psi _n}^\delta (\tau ,v,\xi )\), we can construct a sequence \(( \tilde{\xi }_{n} )_n \subset K_{n,\delta }^*(\xi )\) such that
where the second estimate ensues from (5.27). Therefore, \( \limsup _{n\rightarrow \infty }{\mathsf {b}}_{\Psi _n}^{\delta }({\bar{\tau }}_n^\delta ,v,\xi ) \le \Vert v \Vert _1 (A \vee \Vert \xi \Vert _{\infty }) \). Since \(\limsup _{n\rightarrow \infty }{\mathsf {b}}_{\Psi _n}^{\delta }({\bar{\tau }}_n^\delta ,v,\xi ) = \limsup _{n\rightarrow \infty }{\mathsf {b}}_{\Psi _n}({\bar{\tau }}_n^\delta ,v,\xi )\) as the sequence \(({\bar{\tau }}_n^\delta )_n\) is vanishing, we conclude the desired claim (5.32), and thus (5.31).
This finishes the proof of Proposition 5.8. \(\square \)
5.3 The \(\Gamma \)-limsup result
For the \(\Gamma \)-\(\limsup \) counterpart to Theorem 5.2, where we now consider the strict topology in \(\mathrm {BV}([0,T];{\mathbb {R}}^d)\), we will focus on the 1-positively homogeneous potential
and the two following specific approximations of \(\Psi _0\):
vanishing viscosity: the dissipation potentials \(\Psi _n\) are obtained by augmenting \(\Psi _0\) by a quadratic term involving a (possibly) different norm \(\Vert \cdot \Vert \) (cf. 5.12), i.e.,
$$\begin{aligned}&\Psi _n(v) = A \Vert v\Vert _1 + \frac{\varepsilon _n}{2} \Vert v\Vert ^2 \text { with } \varepsilon _n \downarrow 0, \nonumber \\&\quad \text {with limiting viscosity contact potential } \nonumber \\&{\mathsf {p}} (\tau ,v,\xi ) ={\left\{ \begin{array}{ll} \Psi _0(v) + I_{K^*}(\xi ) &{} \text {if } \tau >0, \\ \Psi _0(v) + \Vert v\Vert \min \nolimits _{\zeta \in K^*} \Vert \xi -\zeta \Vert _* &{} \text {if } \tau =0; \end{array}\right. } \end{aligned}$$(5.33)stochastic approximation: the dissipation potentials \(\Psi _n\) are given by (5.18), with viscosity contact potential
$$\begin{aligned} {\mathsf {p}} (\tau ,v,\xi ) = {\left\{ \begin{array}{ll} \Psi _0(v) + I_{K^*}(\xi ) &{} \text { if }\tau >0, \\ \Vert v\Vert _1 (A \vee \Vert \xi \Vert _\infty ) &{} \text { if }\tau =0. \end{array}\right. } \end{aligned}$$(5.34)
Finally, let us mention in advance that, like in [12], for the \(\limsup \)-result we will need to impose some enhanced regularity for \({\mathcal {E}}(t,\cdot )\), namely
Theorem 5.9
Let \({\mathcal {E}}\) comply with (\(\textsc {E}\)) and with (5.35), and let the dissipation potentials \((\Psi _n)_n\) be given either by (5.33) or by (5.18), with associated limiting bipotential \({\mathsf {p}}\) from (5.33) or (5.34), respectively.
Then, for every \(u \in \mathrm {BV}([0,T];{\mathbb {R}}^d)\) there exists a sequence \((u_n)_n \subset \mathrm {AC}([0,T];{\mathbb {R}}^d)\), converging to u in the strict topology of \(\mathrm {BV}([0,T];{\mathbb {R}}^d)\), such that
Remark 5.10
In [12], which focused on one-dimensional rate-independent systems, the \(\Gamma \)-\(\limsup \) result was obtained in a much larger generality, for a class of dissipation potentials \(\Psi _n\) fulfilling suitable growth conditions and other properties. Such properties are satisfied in the two abovementioned particular cases (5.18) and (5.33).
We believe that, to some extent, the results in [12] could be extended to the present multi-dimensional context. Still, we have preferred to confine the discussion to the vanishing-viscosity and the stochastic approximations, in order to develop more explicit calculations than those in the proof of [12, Thm. 4.2], significantly exploiting the specific structure of these examples.
Nonetheless, we will briefly comment in Remark 6.2 ahead how the \(\Gamma \)-\(\limsup \) result in the vanishing-viscosity case in fact extends to the broader class of dissipation potentials
which still have the limiting viscosity contact potential \({\mathsf {p}}\) from (5.33).
Clearly, Theorems 5.2 and 5.9 yield the Mosco-convergence of the functionals \(( {\mathscr {J}}_{\Psi _n, {\mathcal {E}}})_n\) to \( {\mathscr {J}}_{\Psi _0,{\mathsf {p}},{\mathcal {E}}}\), in the vanishing-viscosity and stochastic cases.
Corollary 5.11
Let \({\mathcal {E}}\) comply with (\(\textsc {E}\)) and with (5.35), and let the dissipation potentials \((\Psi _n)_n\) be given either by (5.18), or by (5.33).
Then, the functionals \(( {\mathscr {J}}_{\Psi _n, {\mathcal {E}}})_n\)Mosco-converge to \( {\mathscr {J}}_{\Psi _0,{\mathsf {p}},{\mathcal {E}}}\) with respect to the weak-strict topology of \(\mathrm {BV}([0,T];{\mathbb {R}}^d)\).
In the spirit of Theorem 5.3, we also have the following straightforward consequence of Theorem 5.9, of Lemma 3.1, and of Proposition 3.7. Theorem 5.12, whose proof is omitted, is a reverse approximation result.
Theorem 5.12
Let \({\mathcal {E}}\) comply with (\(\textsc {E}\)) and with (5.35). Consider the 1-homogeneous potential \(\Psi _0(v) = A \Vert v \Vert _1 \).
Consider the viscosity contact potential \({\mathsf {p}}\) from (5.33). Then, for every balanced viscosity solution \(u \in \mathrm {BV}([0,T];{\mathbb {R}}^d) \) to the rate-independent system \(( \Psi _0,{\mathsf {p}},{\mathcal {E}})\), there exists a sequence \((u_n)_n \subset \mathrm {AC}([0,T];{\mathbb {R}}^d)\) of solutions to the gradient systems \((\Psi _n,{\mathcal {E}})\), with the dissipation potentials \((\Psi _n)_n\) given by \(\Psi _n(v) = A \Vert v\Vert _1 + \frac{\varepsilon _n}{2} \Vert v\Vert ^2\) for all \(n\in {\mathbb {N}}\), where \((\varepsilon _n)_n \subset (0,+\infty )\) is any vanishing sequence as \(n\rightarrow \infty \), such that \(u_n \rightarrow u\) as \(n\rightarrow \infty \)strictly in \( \mathrm {BV}([0,T];{\mathbb {R}}^d)\).
A completely analogous statement holds with the viscosity contact potential \({\mathsf {p}}\) from (5.34), and the dissipation potentials \((\Psi _n)_n\) from (5.18).
6 Proofs
In what follows, we will denote by C a generic positive constant independent of n, whose meaning may vary even within the same line.
We will just outline the argument for the proof of Proposition 5.1, referring to the argument for [41, Thm. 4.1] (see also [12, Thm. 4.2]) for all details. Combining the information that \({\mathscr {J}}_{\Psi _n,{\mathcal {E}}}(u_n) \le C\) with the power control condition from (\(\textsc {E}\)), we find that
where we have also used that \(\Vert u_n(0) \Vert \le C\), and thus \(\sup _n |{\mathcal {E}}(0, u_n(0))| \le C\). Taking into account that both \(\Psi _n\) and \(\Psi _n^*\) are positive, via the Gronwall Lemma we deduce from the above inequality that \(\sup _{t\in [0,T]}|{\mathcal {E}}(t, u_n(t))|\le C\), whence \(\sup _{t\in [0,T]}|\partial _t{\mathcal {E}}(t, u_n(t))|\le C\). Hence,
which implies, thanks to the coercivity (5.4), that \(\mathrm {Var}_{}(u_n; [0,T])\le C\). Then, the thesis readily follows from the Helly theorem. \( \square \)
Before developing the proof of Theorem 5.2, we preliminarily give the following lower semicontinuity result, in the spirit of [41, Lemma 4.3]).
Lemma 6.1
Let \(m,\, d \ge 1\) and \({\mathfrak {F}}_n,\, {\mathfrak {F}}_\infty : {\mathbb {R}}^m \times {\mathbb {R}}^d \rightarrow [0,+\infty )\) be normal integrands (i.e., for \(n\in {\mathbb {N}}{\cup }\{\infty \}\) the functionals \({\mathfrak {F}}_n\) are measurable, and for every \(v\in {\mathbb {R}}^m\) the mappings \(\xi \mapsto {\mathfrak {F}}_n(v, \xi )\) are lower semicontinuous) such that
- (1)
for fixed \(\xi \in {\mathbb {R}}^d\) the functionals \({\mathfrak {F}}_n(\cdot , \xi ) \) are convex for every \(n \in {\mathbb {N}}\cup \{ \infty \}\),
- (2)
there holds
$$\begin{aligned} \Gamma \text {-}\liminf _{n \rightarrow \infty } {\mathfrak {F}}_n \ge {\mathfrak {F}}_\infty \quad \text {in } {\mathbb {R}}^m \times {\mathbb {R}}^d. \end{aligned}$$(6.1)
Let I be a bounded interval in \({\mathbb {R}}\), and let \(w_n, \, w : I \rightarrow {\mathbb {R}}^m\) fulfill \(w_n \rightharpoonup w \) in \(L^1(I;{\mathbb {R}}^m)\), and \(\xi _n, \, \xi : I \rightarrow {\mathbb {R}}^d\) fulfill \(\xi _n(s) \rightarrow \xi (s)\) for almost all \(s\in I\). Then,
Proof
We introduce the functional
It follows from (6.1) that \(\overline{{\mathfrak {F}}}\) is lower semicontinuous on \( {\mathbb {N}}\cup \{ \infty \} \times {\mathbb {R}}^m \times {\mathbb {R}}^d\); hence, it is a positive normal integrand. Then, (6.2) follows from the Ioffe Theorem, cf. [24] and also, e.g., [53, Thm. 21]. \(\square \)
Proof of Theorem 5.2
Let \((u_n)_n \subset \mathrm {BV}([0,T];{\mathbb {R}}^d)\) be a sequence weakly converging to \(u \in \mathrm {BV}([0,T],{\mathbb {R}}^d)\). We may suppose that \(\liminf _{n\rightarrow \infty } {\mathscr {J}}_{\Psi _n,{\mathcal {E}}}(u_n)<+\infty \), as otherwise there is nothing to prove. Therefore, up to a subsequence we have \( {\mathscr {J}}_{\Psi _n,{\mathcal {E}}}(u_n) \le C\), in particular yielding that \(u_n \in \mathrm {AC}([0,T];{\mathbb {R}}^d)\) for every \(n\in {\mathbb {N}}\). With the very same arguments as in the proof of Prop. 5.1, also based on the power control (\(\textsc {E}\)), we see that each contribution to \({\mathscr {J}}_{\Psi _n,{\mathcal {E}}}(u_n) \) is itself bounded. Convergences (5.6) follow from the pointwise convergence of \((u_n)_n\), the fact that \({\mathcal {E}}\in \mathrm {C}^1 ([0,T]\times {\mathbb {R}}^d)\), and the Lebesgue dominated convergence theorem, recalling that \((u_n)_n\) is bounded in \(L^\infty (0,T;{\mathbb {R}}^d)\). Moreover, we have that \(\mathrm {D}{\mathcal {E}}(t, u_n(t))\rightarrow \mathrm {D}{\mathcal {E}}(t, u(t))\) for every \(t \in [0,T]\). Then, taking into account that the functionals \((\Psi _n^*)_n\)\(\Gamma \)-converge to \(\Psi _0^*\), we can apply Lemma 6.1 to the functionals \({\mathfrak {F}}_n(w,\xi ) := \Psi _n^*(\xi )\) and \({\mathfrak {F}}(w,\xi ):= \Psi _0^*(\xi )\) to obtain
Define the nonnegative finite measures on [0, T]
Up to extracting a subsequence, we can suppose that they \(\hbox {weakly}^*\) converge in duality with \(\mathrm {C}^0([0,T])\) to a positive measure
Let us now preliminarily show that
For this, we shall in fact observe that \(\mu \ge \Psi _0(\dot{u}) {\mathscr {L}}^{1} + \mu _{\Psi _0,\mathrm{C}}\). This will follow upon proving that
Indeed, let us fix a partition \(t_0=\alpha< t_1< \ldots < t_k = \beta \) of \([\alpha ,\beta ]\) and notice that
where (1) follows from the Jensen inequality, (2) from the fact that the potentials \((\Psi _n)_n\)\(\Gamma \)-converge to \(\Psi _0\) (cf. Lemma 4.4), and (3) from the 1-positive homogeneity of \(\Psi _0\). Since the partition of \([\alpha ,\beta ]\) is arbitrary, we conclude (6.5).
However, we need to improve (6.4) by obtaining a finer characterization for the jump part of \(\nu \). We will in fact prove that
by adapting the argument in the proof of [42, Prop. 7.3]. To this end, for fixed \(t\in \mathrm {J}_{u} \) let us pick two sequences \(h_n^-\uparrow t\) and \(h_n^+ \downarrow t\) such that \(u_n (h_n^-)\rightarrow u(t_-)\) and \(u_n (h_n^+)\rightarrow u(t_+)\). Define \({\mathsf {s}}_n : [ h_n^-, h_n^+] \rightarrow {\mathbb {R}}\) by
where the normalization constant \(c_n\) is chosen in such a way that \({\mathsf {s}}_n(h_n^+)=1\). Therefore, \({\mathsf {s}}_n\) takes values in [0, 1]. Observe that for every n the function \({\mathsf {s}}_n\) is strictly increasing and thus invertible, and let
There holds
Now, by the upper semicontinuity property of the \(\hbox {weak}^*\)-convergence of measures on closed sets, we have
where (1) follows from a change of variables, and (2) from the very Definition (4.1) of \({\mathsf {b}}_{\Psi _n}\). Now, it follows from (6.8) and from the fact that the range of \({\mathsf {t}}_n\) is \( [h_n^- , h_n^+] \) that there exists \(({\mathsf {t}},\vartheta ) \in \mathrm {C}_{\mathrm {lip}}^0 ([0,1]; [0,T]\times {\mathbb {R}}^d)\) such that, up to a not relabeled subsequence,
Therefore, applying Lemma 6.1 with the choices \(m=d+1\) and, for \(w=(\tau ,v) \in {\mathbb {R}}\times {\mathbb {R}}^d\), with \({\mathfrak {F}}_n(w, \xi ) = {\mathfrak {F}}_n(\tau ,v, \xi ) := {\mathsf {b}}_{\Psi _n}(\tau ,v,\xi )\) and \({\mathfrak {F}}_\infty (w,\xi ): = {\mathsf {p}} (\tau ,v,\xi )\) (where we still denote by \({\mathsf {b}}_{\Psi _n}\) and by \({\mathsf {p}}\) their extensions to \({\mathbb {R}}\times {\mathbb {R}}^d \times {\mathbb {R}}^d\) by infinity), and taking into account (4.2) from Hypothesis 4.1, which ensures the validity of condition (6.1) in Lemma 6.1, we conclude
Similarly, we prove that
Repeating the very same arguments as in the proof of [42, Prop. 7.3], we ultimately find that
whence (5.7) also in view of (6.3). This concludes the proof. \(\square \)
Proof of Theorem 5.3
Let \(u\in \mathrm {BV}([0,T];{\mathbb {R}}^d)\) be a limit point of the sequence \((u_n)_n \subset \mathrm {AC}([0,T];{\mathbb {R}}^d)\). It follows from the lower-semicontinuity property (5.5) that \( {\mathscr {J}}_{\Psi _0,{\mathsf {p}},{\mathcal {E}}}(u)=0\); hence, by Prop. 3.7u is a balanced viscosity solution to \(( \Psi _0,{\mathsf {p}},{\mathcal {E}})\). Moreover, for every \(0\le s \le t \le T\), we have that
where (1) follows from \({\mathscr {J}}_{\Psi _n}(u_n) \le \varepsilon _n \), (2) from convergences (5.6), and (3) from the fact that \( {\mathscr {J}}_{\Psi _0,{\mathsf {p}},{\mathcal {E}}}(u)=0\). Combining this with (5.7), we conclude the enhanced convergence properties (5.8). \(\square \)
Proof of Theorem 5.9
Given \(u \in \mathrm {BV}([0,T],{\mathbb {R}}^d)\), we will construct a sequence \((u_n)_n\subset \mathrm {AC}([0,T];{\mathbb {R}}^d)\) such that \( u_n\rightarrow u \) strictly in \(\mathrm {BV}([0,T];{\mathbb {R}}^d)\) and
We split the proof of in several steps; for Steps 1–4, we suitably adapt the arguments from the proof of [12, Thm. 4.2].
Step 1: reparameterization. First, we reparameterize the curve u, in terms of a new time-like parameter s on a domain [0, S]. The aim is to expand the jumps in u into smooth connections. Following [41, Prop. 6.9], we define
Then, there exists a Lipschitz parameterization \(({\mathsf {t}},{\mathsf {u}}):[0,S]\rightarrow [0,T]\times {\mathbb {R}}\;\) such that \({\mathsf {t}}\) is non-decreasing,
and such that
Moreover, it also holds that
Step 2: preliminary remarks. Since we will construct a sequence \((u_n)_n\) strictly (and in particular pointwise) converging to u in \(\mathrm {BV}([0,T];{\mathbb {R}}^d)\), thanks to the smoothness of \({\mathcal {E}}\) (cf. (\(\textsc {E}\))), we will have for the first three contributions to \({\mathcal {J}}_{\Psi _n,{\mathcal {E}}}(u_n)\)
as \(n\rightarrow \infty \). Therefore, in order to prove (6.11), it will be sufficient to focus on the other terms in \({\mathcal {J}}_{\Psi _n,{\mathcal {E}}}\) and \({\mathcal {J}}_{\Psi _0,{\mathsf {p}},{\mathcal {E}}}\). In view of (6.13), it will be sufficient to prove that
Step 3: definition of the new time \({\mathsf {t}}_n\) and of the recovery sequence \(u_n\). For the sake of simplicity, in what follows we construct a recovery sequence for a curve uwith jumps only at 0 and T, postponing to the end of the proof (cf. Step 7), the discussion of the case of a curve with countably many jumps. We define \(u_n\) by first perturbing the time variable \({\mathsf {t}}\): we fix \(\delta > 0\) and consider a selection
with \( {\mathsf {b}}_{\Psi _n}^\delta \) given by (5.21). We define \({\mathsf {t}}_n:[0,S]\rightarrow [0,T_{n}]\) as the solution of the differential equation
Observe that \(\dot{{\mathsf {t}}}(s) =0\) in \( [0,{\mathsf {s}}(0^+)] \cup [{\mathsf {s}}(T^-),S]\), but we can assume that \(|{\dot{{\mathsf {u}}}} (s)| > 0\) on \( [0,{\mathsf {s}}(0^+)] \cup \in [{\mathsf {s}}(T^-),S]\). This will be sufficient to guarantee that \( \mathop {\mathrm {Argmin}}_{\tau >0} {\mathsf {b}}_{\Psi _n}^\delta (\tau ,{\dot{{\mathsf {u}}}}(s), \mathrm {D}{\mathcal {E}}( {\mathsf {t}}(s), {\mathsf {u}}(s))) \ne \emptyset \) for \(s\in [0,{\mathsf {s}}(0^+)] \cup \in [{\mathsf {s}}(T^-),S]\). Thus, \(\tau _n^\delta \) shall be well defined on the latter set. On the other hand, for \(s \in [{\mathsf {s}}(0^+),{\mathsf {s}}(T^-)]\), we have \( {\dot{{\mathsf {t}}}} (s) = \left. \frac{1}{{\dot{{\mathsf {s}}}}(t)} \right| _{t={\mathsf {t}}(s)} > 0. \) All in all, \({\dot{{\mathsf {t}}}}_n(s)>0\) for all \(s \in [0,S]\). The range of \({\mathsf {t}}_n \) is \([0,T_n]\), with \(T_n\ge T\); since the recovery sequence \(u_n\) has to be defined on the interval [0, T], we rescale \({\mathsf {t}}_n\) by
and define our recovery sequence as follows:
Now we substitute the explicit formula for \(u_n\); we perform a change of variable and obtain
so that
Step 4: Strict convergence of \((u_n)_n\). Recall that we need to prove the pointwise convergence \(u_n(t) \rightarrow u(t)\) for all \(t \in [0,T]\) and the convergence of the variations. For this, it will be crucial to have the following property that shall be verified (even uniformly w.r.t. \(s\in [0,S]\)) both in the stochastic (cf. (6.24)) and in the vanishing-viscosity cases (cf. (6.39)):
This shall imply that \(\dot{{\mathsf {t}}}_{n}(s) \rightarrow \dot{{\mathsf {t}}}(s)\) for almost all \(s\in (0,S)\), and then it will hold
Moreover, \({\dot{{\mathsf {t}}}}_n(s) > 0\) implies that \({\mathsf {t}}_n^{-1} (0) = 0\) and \({\mathsf {t}}_n^{-1} (T_n) = S\), and so we will have the desired pointwise convergence
The convergence of the variations will be automatic, since by definition of \(u_{n}\) we will have
In view of the above observations, from now on we can concentrate on the proof of the \(\limsup \) estimate (6.15).
Step 5: strategy for (6.15). First of all, we will show the following pointwise\(\limsup \)-inequality
Secondly, we will apply the following version of the Fatou’s lemma
for measurable functions \((f_n)_n\) and f, in order to conclude that
whence (6.15) and ultimately (6.11). For the proof of (6.21) and (6.23), we will distinguish the stochastic and the vanishing-viscosity cases.
Step 6a: proof of (6.21) and (6.23) for \(\Psi _n\) given by (5.18) (stochastic approximation). Preliminarily, we observe that, with the very same calculations as for (5.30) (cf. Claim 3.4 in the proof of Proposition 5.8), one has
(with a slight abuse of notation, we use the symbol \(\sup \) also for an essential supremum) where we have exploited the Lipschitz continuity of u. In order to prove the pointwise estimate (6.21), we start with the following algebraic manipulation
and prove the following three claims for the terms on the right-hand side.
Claim 6.a.1: there holds
with \({\mathsf {p}}\) given by (5.20).
Indeed, the representation formula (5.22) for \( \min _{\tau >0}{\mathsf {b}}_{\Psi _n}^{\delta }(\tau _n^\delta (s),{\dot{{\mathsf {u}}}}(s) , - \mathrm {D}{\mathcal {E}}( {\mathsf {t}}(s), {\mathsf {u}}(s)))\) and estimate (5.27) (cf. Claim 3.3 in the proof of Proposition 5.8) yield
and we conclude sending \(n\rightarrow \infty \). Furthermore, we observe that \(\tau _n^\delta (s) \delta \rightarrow 0\) as \(n \rightarrow \infty \) thanks to the previously proved (6.24).
Claim 6.a.2: there holds
Indeed, from the assumed uniform Lipschitz continuity of \(\mathrm {D}{\mathcal {E}}(\cdot , u)\) (cf. (5.35)), we gather that
for all \(s\in [0,S]\) and all \( i =1, \ldots , d \). We now observe that
where we have used the fact that \(\lambda _n \ge 1\) and the definition of \({\mathsf {t}}_n\) from (6.17). We also have
again using Definition (6.17) of \({\mathsf {t}}_n\). Hence, combining estimate (6.29) with (6.30) and (6.31), we gather that
the latter estimate due to (6.24). Therefore, using now the explicit formula (2.7) for \(\Psi _n^*\) we get for almost all \(s\in (0,S)\) that
where (1) follows from the positivity of \(\Psi _n^*\) and from the trivial inequality \(\cosh (nx) - 1 \le \cosh (n|x|)\), (2) from (6.29), (3) from (6.32) and using the estimate \(\cosh (x) \le \frac{e^x+1}{2}\) for all \(x\ge 0\), and (4) is due to the fact that \(\Vert \mathrm {D}{\mathcal {E}}({\mathsf {t}}(s), {\mathsf {u}}(s))\Vert _{\infty }\le A \) for \(s\in [{\mathsf {s}}(0^+),{\mathsf {s}}(T^-)] \), and to an elementary inequality on \([0,{\mathsf {s}}(0^+)) \cup ({\mathsf {s}}(T^-),S]\). Clearly, \(\frac{C_1}{n} \rightarrow 0\); on the other hand, it follows again from (6.24) and the Lipschitz continuity of u that
Therefore, (6.28) ensues.
Claim 6.a.3: there holds
We use the explicit formula (5.18) for \(\Psi _n\), obtaining
for almost all \(s\in (0,S)\), whence
Observe that the right-hand side of (6.36) tends to zero as \(n\rightarrow \infty \) taking into account that \( \sup _{s\in [0,S]}\Vert {\dot{{\mathsf {u}}}}(s) \Vert _{\infty }\le C\), that \(\lambda _n \rightarrow 1\), and that \(\sup _{s\in [0,S]}\tau _n^\delta (s) \rightarrow 0\) by (6.24). This yields (6.35) and, ultimately, (6.21).
Finally, we conclude the integrated \(\limsup \)-estimate (6.23) by observing that the Fatou’s lemma (cf. (6.22)) applies: this can be checked combining (6.24), (6.25), (6.27) (taking into account that \(\sup _{s\in [0,S]} \Vert {\dot{{\mathsf {u}}}}(s) \Vert _1 \le C\)), (6.33), (6.34), and (6.36).
Step 6b: proof of (6.21) and (6.23) for \(\Psi _n\) given by (5.33) (vanishing-viscosity approximation).
To simplify the notation, in what follows we shall focus on the particular case in which the sequence \((\varepsilon _n)_n\) is given by
Preliminarily, we recall that, in the case (5.33),
where \(\Vert \cdot \Vert _*\) is the dual norm to \(\Vert \cdot \Vert \), and \(d_*(\cdot , K^*)\) denotes the induced distance from the set \(K^*\). Taking into account (6.37), we provide a bound for the parameter \(\tau _n^\delta \) from (6.16) again resorting to the Euler–Lagrange equation (5.29). In the present case, it rewrites as
where we have formally written \(\partial \Psi _0\) as a single-valued mapping. Taking into account that \(\langle \partial \Psi _0(\zeta ),\zeta \rangle = \Psi _0(\zeta )\) for all \(\zeta \in {\mathbb {R}}^d\), we ultimately conclude that
In what follows, we will take
but we will continue to write \(\delta \) in place of \(\delta _n\) for shorter notation.
In order to show (6.21), we start from the very same algebraic manipulation as in (6.25) and prove that the terms on the right-hand side converge to the desired limit. We observe that
where the last inequality follows from the fact that \(\mathrm {D}{\mathcal {E}}({\mathsf {t}}(s),{\mathsf {u}}(s)) \in K^*\) when \({\dot{{\mathsf {t}}}}(s) > 0\). Thus, we conclude the analogue of (6.26). Moreover, observe that, as a consequence of estimate (6.39) and of the scaling for \(\delta _n\) from (6.40), we have
We then proceed to show the counterpart to (6.28). The very same calculations as in (6.29) (cf. also (6.31)), and again (6.39) give for every \(s\in [0,S]\) and all \(i=1,\ldots , d\)
Resorting now to the explicit formula (6.37) for \(\Psi _n^*\) (and using \(\xi _n(s) \) and \(\xi (s)\) as place-holders for \( -\mathrm {D}{\mathcal {E}}( {\mathsf {t}}_n(s) \lambda _n^{-1},{\mathsf {u}}(s))\) and \( -\mathrm {D}{\mathcal {E}}( {\mathsf {t}}(s), {\mathsf {u}}(s))\), respectively, to avoid overburdening notation), we get
In (6.44), (1) and (2) are due to the fact that \(\mathrm {D}{\mathcal {E}}( {\mathsf {t}}(s), {\mathsf {u}}(s)) \in K^*\) for almost all \( s \in ({\mathsf {s}}(0^+),{\mathsf {s}}(T^-))\), so that \(\Psi _n^*( \xi (s))=0\), and (3) follows from estimate (6.43). To prove the analogue of (6.28), we will first treat the case in which \(s \in [0,{\mathsf {s}}(0^+)) \cup ({\mathsf {s}}(T^-),S]\) (where \(\dot{{\mathsf {t}}}_n(s) = \tau _n^\delta (s)\)). Here, we use the Lipschitz estimate (6.43) and the explicit formula for \(\Psi _n^*\). Thus, we find
Combining (6.44) and (6.45) we infer (6.28), since \(\delta =\delta _n \rightarrow \infty \) and \(\frac{\delta _n}{n} \rightarrow 0 \) as \(n\rightarrow \infty \). In order to prove the analogue of (6.35), we use the explicit formula (5.33) of \(\Psi _n\), obtaining
It follows from (6.39) and (6.40) that, for n sufficiently big,
hence we deduce that
Then, (6.35), and ultimately (6.21), ensue, since \(\lambda _n \rightarrow 1\).
It remains to verify the integrated inequality (6.23). For this, we apply the Fatou’s lemma by checking the validity of conditions (6.22). They can be verified using (6.25) and resorting to the uniform (w.r.t. \(s\in (0,S)\)) estimates (6.42), (6.44), (6.45), and (6.46), as well as to the following estimate
again by the general representation formula (5.22). We then conclude (6.22) by taking into account the boundedness of \(\Vert {\dot{{\mathsf {u}}}}(s) \Vert _1, \Vert \mathrm {D}{\mathcal {E}}({\mathsf {t}}(s),{\mathsf {u}}(s)) \Vert _{\infty }\) and sending \(n \rightarrow \infty \). Thus, (6.23) is proven.
Step 7: recovery sequence for a general curve u and conclusion of the proof. Now we construct a recovery sequence for a curve with countably many jumps, following the argument from the proof of [12, Thm. 4.2]. Given the jump set \(\mathrm {J}_{u}\), fix \(\varepsilon > 0\), consider a countable set \(\{ t^i \}_{i\in I} \subseteq \mathrm {J}_{u} \cup \{0,T\}\) (with \(t^i < t^{i+1}\) for all \(i\in I\)) such that
(recall the definition (3.22) of \(\mathrm {Jmp}_{{\mathsf {p}},{\mathcal {E}}}\)), and such that the interval [0, T] can be written as the union of disjoint subintervals
Then, we let \(t_n^i={\mathsf {t}}_n({\mathsf {s}}(t^i))\), and set
We define the recovery sequence by
so that
We have now that
Applying estimate (6.21) in every subinterval \([{\mathsf {s}}(t^i),{\mathsf {s}}(t^{i+1})]\) and Fatou’s lemma (cf. (6.22)) on the whole interval [0, S], we obtain inequality (6.15).
The convergence of the variations again follows by the definition of \(u_n\). The pointwise convergence \(u_n(t) \rightarrow u(t)\) for \(t \in [0,T] {\setminus } \mathrm {J}_{u} \) is again trivial. The following calculations show that, by construction, the convergence holds also in the points \(\{ t^i \} \subseteq \mathrm {J}_{u} \). Indeed,
while from (6.47) and the convergence of the variations we have that
In fact, the recovery sequence \(u_n\) has a hidden dependence on \(\varepsilon \). Then taking \(\varepsilon = n^{-1}\) we define a new recovery sequence, that we keep labeling \(u_n\), and sending \(n \rightarrow \infty \) (\(\varepsilon \) to zero) we conclude.
This finishes the proof of Theorem 5.9. \(\square \)
Remark 6.2
Let us briefly outline how the proof of Theorem 5.9 carries over to the case in which the dissipation potentials are given by (5.37). Clearly, it is sufficient to comment on the changes in the calculations carried out throughout Step 6b, starting from the Euler–Lagrange equation (5.29). In the case
and \(p' = \tfrac{p}{p-1}\) the conjugate exponent of p, (5.29) becomes
whence we deduce that
We now write the analogue of (6.41), taking into account formulae (6.49), and prove with straightforward algebra that
All the remaining calculations in Step 6b extend to the case of (5.37).
References
Adams, S., Dirr, N., Peletier, M.A., Zimmer, J.: From a large-deviations principle to the Wasserstein gradient flow: a new micro-macro passage. Commun. Math. Phys. 307(3), 791–815 (2011)
Adams, S., Drri, N., Peletier, M., Zimmer, J.:: Large deviations and gradient flows. J. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 371(2005), 20120341 (2013)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford (2005)
Attouch, H.: Variational Convergence for Functions and Operators. Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston, MA (1984)
Basinski, Z.S.: Thermally activated glide in face-centred cubic metals and its application to the theory of strain hardening. Philos. Mag. 4, 393–432 (1959)
Buliga, M., de Saxcé, G., Vallée, C.: Existence and construction of bipotentials for graphs of multivalued laws. J. Convex Anal. 15(1), 87–104 (2008)
Brézis, H., Ekeland, I.: Un principe variationnel associé à certaines équations paraboliques. Le cas dépendant du temps. C. R. Acad. Sci. Paris Sér. A-B 282((20):Ai), A1197–A1198 (1976)
Brézis, H., Ekeland, I.: Un principe variationnel associé à certaines équations paraboliques. Le cas indépendant du temps. C. R. Acad. Sci. Paris Sér. A-B 282((17):Aii), A971–A974 (1976)
Becker, R.: Über die plasticität amorpher und kristalliner fester körper. Phys. Z. 26, 919–925 (1925)
Berglund, N.: Kramers’ law: validity, derivations and generalisations. Markov Process. Relat. Fields 19(3), 459–490 (2013)
Babadjian, J.-F., Francfort, G., Mora, M.G.: Quasistatic evolution in non-associative plasticity–the cap model. SIAM J. Math. Anal. 44, 245–292 (2012)
Bonaschi, G.A., Peletier, M.A.: Quadratic and rate-independent limits for a large-deviations functional. Contin. Mech. Thermodyn. 28, 1191–1219 (2016)
Braides, A.: Local Minimization, Variational Evolution and \(\Gamma \)-convergence, Volume 2094 of Lecture Notes in Mathematics. Springer, Cham (2014)
Crismale, V., Lazzaroni, G.: Viscous approximation of quasistatic evolutions for a coupled elastoplastic-damage model. Calc. Var. Partial Differ. Equ. 55(1), 17 (2016)
Colli, P.: On some doubly nonlinear evolution equations in Banach spaces. Jpn. J. Indust. Appl. Math. 9(2), 181–203 (1992)
Colli, P., Visintin, A.: On a class of doubly nonlinear evolution equations. Commun. Partial Differ. Equ. 15(5), 737–756 (1990)
Dal Maso, G., DeSimone, A., Solombrino, F.: Quasistatic evolution for cam-clay plasticity: a weak formulation via viscoplastic regularization and time rescaling. Calc. Var. Partial Differ. Equ. 40, 125–181 (2011)
Dal Maso, G., DeSimone, A., Mora, M.G., Morini, M.: A vanishing viscosity approach to quasistatic evolution in plasticity with softening. Arch. Ration. Mech. Anal. 189(3), 469–544 (2008)
Dal Maso, G., Toader, R.: A model for the quasi-static growth of brittle fractures: existence and approximation results. Arch. Ration. Mech. Anal. 162(2), 101–135 (2002)
Efendiev, M., Mielke, A.: On the rate-independent limit of systems with dry friction and small viscosity. J. Convex Anal. 13(1), 151–167 (2006)
Feng, J., Kurtz, T.G.: Large Deviations for Stochastic Processes, Volume 131 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2006)
Francfort, G.A., Marigo, J.-J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46(8), 1319–1342 (1998)
Ghoussoub, N.: Self-Dual Partial Differential Systems and Their Variational Principles. Springer Monographs in Mathematics. Springer, New York (2009)
Ioffe, A.D.: On lower semicontinuity of integral functionals. I. SIAM J. Control Optim. 15(4), 521–538 (1977)
Krausz, A.S., Eyring, H.: Deformation Kinetics. Wiley, New York (1975)
Knees, D., Mielke, A., Zanini, C.: On the inviscid limit of a model for crack propagation. Math. Models Methods Appl. Sci. 18(9), 1529–1569 (2008)
Kramers, H.A.: Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7(4), 284–304 (1940)
Knees, D., Rossi, R., Zanini, C.: A vanishing viscosity approach to a rate-independent damage model. Math. Models Methods Appl. Sci. 23(4), 565–616 (2013)
Liero, M., Mielke, A., Peletier, M.A., Renger, D.R.M.: On microscopic origins of generalized gradient structures. Discrete Contin. Dyn. Syst. Ser. S 10(1), 1–35 (2017)
Lazzaroni, G., Toader, R.: A model for crack propagation based on viscous approximation. Math. Models Methods Appl. Sci. 21(10), 2019–2047 (2011)
Mielke, A.: Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Cont. Mech. Thermodyn. 15, 351–382 (2003)
Mielke, A.: Evolution in rate-independent systems (Ch.6). In: Dafermos, C.M., Feireisl, E. (eds.) Handbook of Differential Equations, Evolutionary Equations, vol. 2, pp. 461–559. Elsevier, Amsterdam (2005)
Mielke, A.: Differential, energetic, and metric formulations for rate-independent processes. In: Nonlinear PDE’s and Applications, Volume 2028 of Lecture Notes in Mathematics, pp. 87–170. Springer, Heidelberg (2011)
Mielke, A.: On evolutionary \(\Gamma \)-convergence for gradient systems. In: Macroscopic and Large Scale Phenomena: Coarse Graining. Mean Field Limits and Ergodicity, Volume 3 of Lecture Notes in Applied Mathematics and Mechanics, pp. 187–249. Springer, Cham (2016)
Mainik, A., Mielke, A.: Existence results for energetic models for rate-independent systems. Calc. Var. Partial Differ. Equ. 22, 73–99 (2005)
Mach Nguyet, M.: BV solutions constructed using the epsilon-neighborhood method. ESAIM Control Optim. Calc. Var. 22(1), 188–207 (2016)
Moreau, J.-J.: Bounded variation in time. In: Moreau, J.-J., Panagiotopoulos, P.D., Strang, G. (eds.) Topics in Nonsmooth Mechanics, pp. 1–74. Birkhäuser, Basel (1988)
Mielke, A., Peletier, M.A., Renger, D.R.M.: On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion. Potential Anal. 41(4), 1293–1327 (2014)
Mielke, A., Roubíček, T.: Rate-Independent Systems. Theory and Application, Volume 193 of Applied Mathematical Sciences. Springer, New York (2015)
Mielke, A., Roubíček, T., Stefanelli, U.: \(\Gamma \)-limits and relaxations for rate-independent evolutionary problems. Calc. Var. Partial Differ. Equ. 31, 387–416 (2008)
Mielke, A., Rossi, R., Savaré, G.: BV solutions and viscosity approximations of rate-independent systems. ESAIM Control Optim. Calc. Var. 18(1), 36–80 (2012)
Mielke, A., Rossi, R., Savaré, G.: Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems. J. Eur. Math. Soc. 18(9), 2107–2165 (2016)
Mielke, A., Theil, F.: A mathematical model for rate-independent phase transformations with hysteresis. In: Alber, H.-D., Balean, R.M., Farwig, R. (eds.) Proceedings of the Workshop on Models of Continuum Mechanics in Analysis and Engineering, pp. 117–129, Shaker, Aachen (1999)
Mielke, A., Theil, F.: On rate-independent hysteresis models. NoDEA Nonlinear Differ. Equ. Appl. 11(2), 151–189 (2004)
Nayroles, B.: Deux théorèmes de minimum pour certains systèmes dissipatifs. C. R. Acad. Sci. Paris Sér. A-B 282(17):Aiv, A1035–A1038 (1976)
Negri, M.: Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics. ESAIM Control Optim. Calc. Var. 20(4), 983–1008 (2014)
Orowan, E.: Problems of plastic gliding. Proc. Phys. Soc. 52, 8–22 (1940)
Rossi, R., Savaré, G.: A characterization of energetic and BV solutions to one-dimensional rate-independent systems. Discrete Contin. Dyn. Syst. Ser. S 6(1), 167–191 (2013)
Serfaty, S.: Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete Contin. Dyn. Syst. 31(4), 1427–1451 (2011)
Sandier, E., Serfaty, S.: Gamma-convergence of gradient flows with applications to Ginzburg-Landau. Commun. Pure Appl. Math. 57(12), 1627–1672 (2004)
Stefanelli, U.: The Brézis–Ekeland principle for doubly nonlinear equations. SIAM J. Control Optim. 47(3), 1615–1642 (2008)
Toader, R., Zanini, C.: An artificial viscosity approach to quasistatic crack growth. Boll. Unione Mat. Ital. 9(2), 1–35 (2009)
Valadier, M.: Young measures. In: Cellina, A. (ed.) Methods of nonconvex analysis (Varenna, 1989), pp. 152–188. Springer, Berlin (1990)
Visintin, A.: Variational formulation and structural stability of monotone equations. Calc. Var. Partial Differ. Equ. 47(1–2), 273–317 (2013)
Acknowledgements
We would like to thank Mark Peletier for suggesting this problem to us and for his support. We are deeply grateful to Giuseppe Savaré for several enlightening discussions, in particular on the results of Sect. 4.
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G.A.B. kindly acknowledges support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), VICI Grant 639.033.008.
R.R. has been partially supported by GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica).
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Bonaschi, G.A., Rossi, R. Generation of balanced viscosity solutions to rate-independent systems via variational convergence. Annali di Matematica 199, 81–126 (2020). https://doi.org/10.1007/s10231-019-00867-4
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DOI: https://doi.org/10.1007/s10231-019-00867-4
Keywords
- Gradient systems
- Rate-independent systems
- Balanced viscosity solutions
- Vanishing viscosity
- Large deviations
- Variational convergence