The notions of inertial balanced viscosity and inertial virtual viscosity solution for rate-independent systems
-
Filippo Riva
,
Giovanni Scilla
und
Francesco Solombrino
Abstract
The notion of inertial balanced viscosity (IBV) solution to rate-independent evolutionary processes is introduced. Such solutions are characterized by an energy balance where a suitable, rate-dependent, dissipation cost is optimized at jump times. The cost is reminiscent of the limit effect of small inertial terms. Therefore, this notion proves to be a suitable one to describe the asymptotic behavior of evolutions of mechanical systems with rate-independent dissipation in the limit of vanishing inertia and viscosity. It is indeed proved, in finite dimension, that these evolutions converge to IBV solutions. If the viscosity operator is neglected, or has a nontrivial kernel, the weaker notion of inertial virtual viscosity (IVV) solutions is introduced, and the analogous convergence result holds. Again in a finite-dimensional context, it is also shown that IBV and IVV solutions can be obtained via a natural extension of the minimizing movements algorithm, where the limit effect of inertial terms is taken into account.
1 Introduction
Rate-independent evolutions frequently occur in physics and mechanics when the problem under consideration presents such small rate-dependent effects, as inertia or viscosity, that can be neglected. Several applications can be (formally) modelled by the doubly nonlinear differential inclusion
where
If the driving energy
Condition (GS) actually turns out to be still too restrictive in the nonconvex case, where a local minimality condition would be preferable. Starting from this consideration, in [20, 19, 18] the notion of balanced viscosity solution has been introduced and analyzed; see also the recent paper [25]. The idea of Mielke, Rossi and Savaré [20, 19, 18] relies on the fact that physical solutions to (1.1) should arise as the vanishing-viscosity limit of a richer and more natural viscous problem
as the parameter
While in (EB) the classical total variation (actually,
in ((EB${}^{\mathbb{V}}$)) the jump part of the “viscous” variation involves a more complicated cost function (a Finsler distance) which takes into account the original presence of viscosity:
At time
where
In [20, 19], it is also shown that BV solutions can be obtained as the limit of time discrete approximations, solutions to the recursive discrete-in-time variational incremental scheme with time step τ:
when sending simultaneously ε and τ to 0 with also
Although the notion of BV solution turned out to be extremely powerful in applications (see, for instance, [2, 6, 12, 21]), it still lacks inertial terms, which are however essential in the description of real world phenomena, as stated by the second principle of dynamics.
In this paper, we thus present a novel notion of solution which takes into account this feature. Our starting point is augmenting (1.2) with an inertial term
and then sending
This approach has already been adopted for concrete models (in infinite dimension) in [7, 8, 14, 15, 17, 26, 29], all in the case of convex, or even quadratic, energies
In our nonconvex setting, the limiting evolution of (1.5) provides the novel notion of inertial balanced viscosity (IBV) solution (we refer to the discussion in Section 3 for the subtler notion of inertial virtual viscosity (IVV) solution, arising when the viscosity operator
for every
where
Notice that, differently than in the vanishing-viscosity case, the dissipation cost we consider is no longer invariant under a time-reparametrization, due to the presence of the second-order term
The proposed notion of solution is indeed a suitable one to extend the results of [20, 19] to a context where the limiting effect of inertial terms is taken into consideration. We show this in our main result, Theorem 3.10, which fulfills a twofold goal. First, we show convergence of the solutions
by sending both τ and ε to 0. Differently from (1.4), for technical reasons we need to strengthen the rate of convergence requiring
Plan of this paper.
In Section 2, we fix the main notation and list the main assumptions of this paper. We also recall some basic properties of functions of bounded
2 Notation and setting
Let
Given any symmetric positive-semidefinite linear operator
and we denote with a capital letter
We point out that such a constant Q exists, since in finite dimension any linear operator is necessarily continuous. The least Q that may be chosen here is the operator norm of
If
Furthermore, the inverse operator
which is dual to
We briefly recall some basic definitions in convex analysis (see, for instance, [27]). Given a proper, convex, lower semicontinuous function
Notice that if
The Fenchel conjugate of f is the convex, lower semicontinuous function
and for every
Given a subset
2.1 Main assumptions
Below, we list the main assumptions we will use throughout the paper.
In the dynamic problem (1.5), the inertial term is described by a
which represents a mass distribution.
The possible presence of viscosity is also considered by introducing the
In particular, in our analysis we also include the case
Both the rate-independent problem (1.1) and the dynamic problem (1.5)
are damped by a rate-independent dissipation potential
The function
Assumption (R1) implies subadditivity, namely
and the existence of two positive constants
This means that
Furthermore, since
By (2.4), we notice that there holds
It is also well known (see, e.g., [27]) that
We finally consider the driving potential energy
For a.e.
where
For every
We point out that the prototypical example of potential energy
with
As mentioned in [10], under these hypotheses one can prove that
Remark 2.1.
Thanks to (E4), it is easy to see that
where
Remark 2.2.
All applications that are presented in [10, Section 7], basically regarding masses connected with springs, are described by adding together elastic quadratic energies of the form
with
which is of course nonconvex. It is however straightforward to check that it satisfies (E1)–(E4) (and actually also (E3’) and (E5) below), and thus it can be included within the framework of this paper.
In Section 5, where we deal with the discrete approximation of IBV and IVV solutions, in addition to the previous assumptions, we need to require the following assumptions:
The energy
for some symmetric positive-definite linear operator
We notice that by (E3’) and Gronwall’s lemma, we can infer
whence
It is also easy to check that (E5) implies
Indeed, by using the mean value theorem, for some
whence (2.9) follows up to simplifying θ in both sides and then letting
We finally point out that an energy
2.2 Functions of bounded
ℛ
-variation
We recall here a suitable generalization of functions of bounded variation useful to deal with functions satisfying (R1).
Definition 2.3.
Let a function
We also set
We say that f is a function of bounded
Notice that, by virtue of (2.4), we have
with the convention
It is well known (see, e.g., [3]) that f can be uniquely decomposed as follows:
with
In (2.11),
and
We also remark that, for
where
It follows from (2.12) that the continuous part of the
We finally notice that, by dropping the pointwise value of f at jump points (by the subadditivity of
The term
3 Inertial balanced viscosity and inertial virtual viscosity solutions
In this section, we rigorously introduce the notions of inertial balanced viscosity and inertial virtual viscosity solution. We also state our main result, see Theorem 3.10, postponing its proof to the forthcoming sections.
As in the vanishing-viscosity approach of [19], the starting point consists in an alternative formulation of the dynamic problem (1.5) based on the so–called De Giorgi’s energy-dissipation principle (see the pioneering work [9] and other applications in [16, 28]). Roughly speaking, the idea is to keep together all dissipative terms appearing in the dynamic model, namely viscosity and rate-independent dissipation; we are thus led to consider the functional
It is then easy to check (see, e.g., [19, p. 47]) that the subdifferential of
so the dynamic problem (1.5) can be rewritten as
By using (2.1), and exploiting the classical chain-rule formula for
We point out that in our case of additive viscosity (3.1), the Fenchel conjugate
where
is the annihilator of
In particular, in the two extreme situations
where
denotes the distance from
3.1 Contact potentials
The energy balance (3.3) naturally leads to the introduction of a so-called (viscous) contact potential associated to
Definition 3.1.
The (viscous) contact potential related to the viscosity operator
defined by
In the two extreme situations
Therefore, in the positive-definite case we retrieve the vanishing-viscosity contact potential defined in [19].
By the explicit formula, we easily infer the following properties for the contact potential
Furthermore, we also observe the following property:
At this stage, a warning is mandatory: we point out that our potential
Definition 3.2.
We say that a continuous map
There exists a positive constant
Remark 3.3.
In the case that
Notice that in the above definition we are not requiring the convexity of p with respect to the variable v. Instead, the convexity in the second variable, i.e. property (ii), will be crucial in Proposition 3.11.
We also point out that the main property of regularized contact potentials, missing in general for
We finally observe that by (iii) and (iv), any
where we exploited (2.4). In particular, it holds
3.2 Parametrized families of regularized contact potentials
With the following result, we show that a whole family of regularized contact potentials can be constructed by means of a suitable version of the Yosida transform.
For every
Proposition 3.4.
Let
If in addition
Proof.
We first notice that
Now, we have to confirm properties (i)–(iv) of Definition 3.2. Property (i) follows by the one-homogeneity of
Property (ii) follows since the Yosida transform of a convex function is convex, and
The right inequality in (iii) is obtained by choosing
Property (iv) is again a straightforward consequence of the Yosida transform: one can choose
We are only left to prove that
Claim 1.
There exists a positive constant
Claim 2.
If
where
Claim 3.
There exists a positive constant
for every
From Claim 3, we easily deduce the continuity of
We start with Claim 1, and we observe that it is enough to prove it for
and Claim 1 is proved.
To prove Claim 2, it is enough to show that an infimizing sequence
Thus, by using (3.6), we infer
We now need to prove Claim 3. To this end, we take
By using Claim 2, from the above inequality we deduce
By interchanging the roles of
We notice that in the case that
Indeed, by means of the explicit formula (3.5), it holds
where the last inequality is a simple byproduct of the triangle inequality. This fact corroborates Remark 3.3.
In the opposite situation
Compare this last formula with (3.5), the case of
3.3 The inertial energy-dissipation cost
Once the notion of contact potential has been developed, we are in a position to rigorously introduce the cost function which will govern the jump transient of IBV and IVV solutions. The crucial difference with respect to the vanishing-viscosity cost of BV solutions [19] is its rate-dependent nature, caused by the term
Definition 3.5.
For every
where
denotes the class of the admissible curves and
We also define the inertial cost directly related to the viscosity operator
Remark 3.6.
We point out that in the case that
This is consistent with the vanishing-viscosity analysis performed in [19], in which the cost is (formally) equivalent to (3.12) by taking
A relevant feature of the inertial cost is that the value
Proposition 3.7.
For every
Proof.
Fix
and the assertion follows by the arbitrariness of v, N and p. ∎
With the notion of inertial energy-dissipation cost at hand, we can give the definition of inertial virtual viscosity and inertial balanced viscosity solutions.
Definition 3.8.
We say that a function
and the energy balance
for every
If
Remark 3.9.
By Proposition 3.7, we deduce that for any IVV solution there holds
and thus in (3.14) we can actually replace
It is however not clear whether we can replace it with the cost related to the contact potential
The term “virtual” in the definition of IVV solutions is motivated by the presence of the Yosida-type potentials (3.7) inside the set of regularized contact potentials
The main result of this paper can now be stated as follows.
Theorem 3.10.
Let
Assume in addition (E3’) and (E5) . Then, for every sequence
there exists a subsequence (not relabelled) along which the sequence of piecewise affine interpolants
In both cases, the limit function is an inertial balanced viscosity solution if
The proof of part (i) is carried out in Section 4, while part (ii) is proved in Section 5. The rest of this section is devoted to the main properties of the inertial cost.
Proposition 3.11.
Fix
Proof.
For a fixed
By the definition of
For the sake of clarity, we introduce the following notation:
and we notice that
By Definition 3.2 (ii) and (iv), we observe that the map
Since
which vanishes since
To obtain formula (3.15), we simply notice that the map
is nonincreasing. Indeed, if
With the following proposition, we prove that the cost function
Proposition 3.12.
For every
If in addition
Proof.
We start by proving (3.17). If
Hence, we get
and (3.17) is proved.
To show the validity of (3.18), we fix
belongs to
This yields (3.18).
To prove (3.19), we fix
This implies
and, by interchanging the roles of
4 Continuous slow-loading limit
The aim of this section is to show that inertial balanced (and virtual) viscosity solutions can be obtained as slow-loading limit (i.e. as
4.1 Known results
First, we briefly recall the known results proved in [10]. In particular, for the existence of solutions to (1.5), we refer to [10, Theorem 3.8], where the problem is considered under more general assumptions.
Theorem 4.1.
For every pair of initial data
for every
Proposition 4.2.
Let
Proof.
See [10, Corollary 3.4] for (4.2), (4.3) and (4.5). The uniform bound (4.4) follows by exploiting the differential inclusion solved by
With the a priori bounds of Proposition 4.2 at hand, an argument based on Helly’s selection theorem provides the existence of a convergent subsequence of dynamic solutions
Proposition 4.3.
Let
In addition, arguing as for [10, Propositions 6.2-6.3], it can be proven that the limit evolution u above complies with the local stability condition (3.13) and a suitable energy inequality. We highlight that a function fulfilling such properties is usually called an a.e. local solution to the rate-independent system (1.1) (see, for instance, [22, Chapter 3]).
Proposition 4.4.
Let
holds for every
or equivalently
Moreover, the energy inequality
holds for every
It is worth mentioning that, under the additional assumption of (uniform) convexity on the energy
4.2 Characterization of the energy loss at jumps
In the nonconvex setting, continuity of the limit function is no more reasonable, and hence the gap of the energy in (4.7) has to be characterized. This first proposition shows that, as expected, the peculiar behavior of the limit function u is restricted to its (essential) jump set.
Proposition 4.5.
Let
In particular,
Proof.
By reasoning as in [31, Theorem 5.4], it is easy to see that the map
is nonincreasing; it essentially follows from the energy balance (4.1) by dropping the dissipated energy (i.e. the terms with
This in particular yields that the distributional derivative of
On the other hand, by the chain-rule formula in BV (see, for instance, [5, Theorem 4.1] and recall Remark 2.1), it holds
where
By combining (4.9) and (4.10), we obtain
The latter equality yields
where we define
By choosing
By integrating the above inequality in
To obtain the reverse inequality, we combine (4.8) and (4.7), and use (2.12) and (2.14), to get
Since both μ and
In particular, we deduce
By joining (4.12) with (4.13), we finally obtain
and we conclude. ∎
Thanks to Proposition 3.7, we already know that the inertial cost
Proposition 4.6.
Let
Proof.
Let
as
By exploiting (3.3) and from the definition of the contact potential
We now take any
Then, by using the Lipschitzianity of p in the second variable (property (iv) in Definition 3.2), we notice that
If we denote by ω a modulus of continuity for
which vanishes as
This means that we can freeze the time t in
Following [31, Proposition 5.8], we now set
where we denoted by
Then, through the change of variables
We also notice that (4.15)–(4.16) can be re-read for
as
We now introduce the functions
and the competitor
For the sake of clarity, we denote by
Fix now an arbitrary
By means of (3.6) and (4.21), it is easy to see that both terms in the last line above vanish as
With the time translation
and by the arbitrariness of
5 Incremental minimization scheme
This last section is devoted to the proof of Theorem 3.10 (ii): namely, we show that IBV and IVV solutions can also be obtained as limits of time-discrete solutions when ε and the time step τ vanish simultaneously (with a certain rate). For this, in addition to the assumptions of the previous section, we need to require (E3’) and (E5).
Let
We construct a recursive sequence
(5.1)
where
and
with Λ and
The addition in the functional
We observe that the existence of a minimum in (5.1a) follows easily from the direct method. Furthermore, if
By defining
we notice that the Euler–Lagrange equation solved by
We also observe that by (5.1b), one has
In order to enlighten the notation, from now on we will drop the dependence on
As in the continuous counterpart developed in Section 4, the first step in the analysis consists in finding uniform a priori estimates, which usually follows by combining an energy inequality together with Grönwall’s lemma. In the discrete setting, we employ the following version of the discrete Grönwall’s inequality, whose proof can be found for instance in [13, Appendix A].
We want to stress that here and henceforth we adopt the convention that an empty sum is equal to 0.
Lemma 5.1 (Grönwall).
Let
then one has
Proposition 5.2.
For every
Furthermore, if
then there exists
for every
Proof.
Testing (5.3) by
we deduce
Subtracting
We now come to the proof of (5.6). We first notice that, defining
where
We indeed observe that the term
Thanks to (2.8), we now have
and thus from (5.7) we infer
Hence, by means of Lemma 5.1, we get
By defining
from (5.8) we finally obtain
Since
Corollary 5.3.
Assume that
Proof.
The bounds (5.10) and (5.12) can be easily inferred from (5.6).
We then prove (5.9). Let
Lastly, (5.11) can be obtained from the Euler-Lagrange equation (5.3) taking into account (2.6), (E2) and (5.10). ∎
5.1 The main interpolants
Once the discrete bounds are obtained, in order to retrieve the continuous framework we need to introduce suitable interpolants of the discrete-in-time sequence
respectively.
We denote by
Since in the definition of the inertial cost (3.11) a second derivative is present, we also need to keep track of its discrete counterpart
Notice indeed that
Thus, thanks to (5.11), the function
For any
Note that
We finally define a piecewise constant interpolant of the values
From assumptions (E1) and (E2), we deduce that, in the limit as
uniformly with respect to
In terms of interpolants, the energy inequality (5.4) can be rewritten as
for every
Corollary 5.4.
Assume that
for every
Moreover, up to enlarging the constant
The next proposition shows that the mismatch between the many interpolants defined above can be bounded by suitable ratios of the parameters τ and ε.
Proposition 5.5.
Assume that
Moreover, it holds
Proof.
We first notice that, by virtue of (5.10)
and since
Thus, let
In order to deal with the last term in (5.19), we observe that
and thus
Since
and assertion (5.19) follows from (5.21).
From (5.22), the bound (5.20) easily follows by means of (5.11). ∎
We are now in a position to prove the analogues of Propositions 4.3–4.5 for the sequence of piecewise affine interpolants
Proposition 5.6.
Let
For every
If in addition
For every
Proof.
We can argue as in [10, Theorem 6.1]. In view of the a priori bounds of Corollary 5.4 and (2.4), the sequence
By virtue of Corollary 5.4 and (2.4), we also have
whence (iii) follows, up to possibly passing to a further subsequence.
To obtain (ii’), it is enough to observe that
and to notice that the last two terms vanish as
and the same holds for the other one.
The proof of (iii’) follows exactly as in [10, Theorem 6.1], by using (5.26) and recalling that thanks to (5.19) we also have
Proposition 5.7.
Let
Then the inequality
holds for every
Moreover, if in addition
In particular,
Proof.
We only prove (5.23), the remaining assertions being as in [10, Propositions 6.2 and 6.3] and in Proposition 4.5, exploiting Proposition 5.6 (ii’) and (iii’). From (5.3) and (2.5), for every
By multiplying both sides by
namely for every
Passing to the limit as
From Corollary 5.4, (2.4) and the Cauchy–Schwarz inequality, we also have
and a similar argument shows that the last term in the inequality above vanishes as well as
We conclude observing that (5.17) and (5.19) allow us to use the dominated convergence theorem, getting
5.2 The convergence result
As already done in the time-continuous setting in (3.3), we now rephrase the energy inequality (5.4) in terms of De Giorgi’s principle. For simplicity, we set
Recalling the definitions of the interpolants
Then, by virtue of Corollary 5.4 and Proposition 5.5, if
Furthermore, thanks to the continuity of
From (5.3) and recalling that using (2.1) it holds
inequality (5.4) can be rewritten, arguing in a similar way, as
and thus, in terms of interpolants, as
for every
To conclude the proof of Theorem 3.10 (ii), we only have to confirm the validity of an analogue of Proposition 4.6 for the function u obtained with Proposition 5.6. For this, we will need to reinforce the assumption
Proposition 5.8.
Let
Then for every
Proof.
As already remarked in the proof of Proposition 4.6, it will suffice to prove (5.28) in the case
and
By exploiting (5.27), Proposition 5.5 yields as a byproduct
and
The continuity of
For a lighter exposition, with a little abuse of notation we denote by
Taking into account (3.9), we can continue the above inequality, getting
Thanks to (5.20), (5.24) and the assumption (5.27), we easily obtain
Thus, also taking any
By exploiting property (iv) of Definition 3.2 and using (5.18) and (5.25), we deduce
We indeed notice that the first term within the brackets is bounded (it actually vanishes) by arguing as in (5.33).
Therefore, we finally obtain
The rest of the proof follows closely the argument of Proposition 4.6 from (4.17) on, with
5.3 An enhanced version of the scheme
We conclude by proposing a slightly modified discrete algorithm which allows us to get rid of the assumption that
needed for Proposition 5.8. To describe it, we consider an additional parameter
For
Since the only change with respect to previous sections is the replacement of ε by
Theorem 5.9.
Let
there exists a subsequence (not relabelled) along which the sequence of piecewise affine interpolants
Furthermore, the limit function is an inertial balanced viscosity solution if
The advantage of condition (5.35) is that it is automatically satisfied by any sequence
and thus it permits to separate the vanishing rates of
We finally notice that the simplest choice of
Funding source: Ministero dell’Università e della Ricerca
Award Identifier / Grant number: 2017BTM7SN
Funding source: Sapienza Università di Roma
Award Identifier / Grant number: RM12117A7ABE51AF
Funding statement: The authors are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of INdAM. The authors have been supported by the Italian Ministry of Education, University and Research through the MIUR – PRIN project 2017BTM7SN “Variational methods for stationary and evolution problems with singularities and interfaces”. The second author acknowledges the support of Sapienza University of Rome through the project Progetti Medi 2021 – RM12117A7ABE51AF “Equazioni differenziali degeneri, singolari e frazionari”.
References
[1] V. Agostiniani, Second order approximations of quasistatic evolution problems in finite dimension, Discrete Contin. Dyn. Syst. 32 (2012), no. 4, 1125–1167. 10.3934/dcds.2012.32.1125Suche in Google Scholar
[2] S. Almi, G. Lazzaroni and I. Lucardesi, Crack growth by vanishing viscosity in planar elasticity, Math. Eng. 2 (2020), no. 1, 141–173. 10.3934/mine.2020008Suche in Google Scholar
[3] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., The Clarendon, Oxford, 2000. 10.1093/oso/9780198502456.001.0001Suche in Google Scholar
[4] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed., Lectures in Math. ETH Zürich, Birkhäuser, Basel, 2008. Suche in Google Scholar
[5] G. Crasta and V. De Cicco, A chain rule formula in the space BV and applications to conservation laws, SIAM J. Math. Anal. 43 (2011), no. 1, 430–456. 10.1137/100804462Suche in Google Scholar
[6] V. Crismale and R. Rossi, Balanced viscosity solutions to a rate-independent coupled elasto-plastic damage system, SIAM J. Math. Anal. 53 (2021), no. 3, 3420–3492. 10.1137/19M1303563Suche in Google Scholar
[7] G. Dal Maso and F. Sapio, Quasistatic limit of a dynamic viscoelastic model with memory, Milan J. Math. 89 (2021), no. 2, 485–522. 10.1007/s00032-021-00343-wSuche in Google Scholar
[8] G. Dal Maso and R. Scala, Quasistatic evolution in perfect plasticity as limit of dynamic processes, J. Dynam. Differential Equations 26 (2014), no. 4, 915–954. 10.1007/s10884-014-9409-7Suche in Google Scholar
[9] E. De Giorgi, A. Marino and M. Tosques, Problems of evolution in metric spaces and maximal decreasing curve, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 68 (1980), no. 3, 180–187. Suche in Google Scholar
[10] P. Gidoni and F. Riva, A vanishing-inertia analysis for finite-dimensional rate-independent systems with nonautonomous dissipation and an application to soft crawlers, Calc. Var. Partial Differential Equations 60 (2021), no. 5, Paper No. 191. 10.1007/s00526-021-02067-6Suche in Google Scholar
[11] C. Goffman and J. Serrin, Sublinear functions of measures and variational integrals, Duke Math. J. 31 (1964), 159–178. 10.1215/S0012-7094-64-03115-1Suche in Google Scholar
[12] D. Knees, R. Rossi and C. Zanini, Balanced viscosity solutions to a rate-independent system for damage, European J. Appl. Math. 30 (2019), no. 1, 117–175. 10.1017/S0956792517000407Suche in Google Scholar
[13] R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, Lecture Notes in Math. 2093, Springer, Cham, 2014. 10.1007/978-3-319-02231-4Suche in Google Scholar
[14] G. Lazzaroni and L. Nardini, On the quasistatic limit of dynamic evolutions for a peeling test in dimension one, J. Nonlinear Sci. 28 (2018), no. 1, 269–304. 10.1007/s00332-017-9407-0Suche in Google Scholar PubMed PubMed Central
[15] G. Lazzaroni, R. Rossi, M. Thomas and R. Toader, Rate-independent damage in thermo-viscoelastic materials with inertia, J. Dynam. Differential Equations 30 (2018), no. 3, 1311–1364. 10.1007/s10884-018-9666-ySuche in Google Scholar
[16] A. Mielke, On evolutionary Γ-convergence for gradient systems, Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Lect. Notes Appl. Math. Mech. 3, Springer, Cham (2016), 187–249. 10.1007/978-3-319-26883-5_3Suche in Google Scholar
[17] A. Mielke, A. Petrov and J. a. A. C. Martins, Convergence of solutions of kinetic variational inequalities in the rate-independent quasi-static limit, J. Math. Anal. Appl. 348 (2008), no. 2, 1012–1020. 10.1016/j.jmaa.2008.07.077Suche in Google Scholar
[18] A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete Contin. Dyn. Syst. 25 (2009), no. 2, 585–615. 10.3934/dcds.2009.25.585Suche in Google Scholar
[19] A. Mielke, R. Rossi and G. Savaré, BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var. 18 (2012), no. 1, 36–80. 10.1051/cocv/2010054Suche in Google Scholar
[20] A. Mielke, R. Rossi and G. Savaré, Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 9, 2107–2165. 10.4171/JEMS/639Suche in Google Scholar
[21] A. Mielke, R. Rossi and G. Savaré, Balanced-viscosity solutions for multi-rate systems, J. Phys. Conf. Ser. 727 (2016), Article ID 012010. 10.1088/1742-6596/727/1/012010Suche in Google Scholar
[22] A. Mielke and T. Roubíček, Rate-Independent Systems: Theory and Application, Appl. Math. Sci. 193, Springer, New York, 2015. 10.1007/978-1-4939-2706-7Suche in Google Scholar
[23] A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis, Proceedings of the Workshop on Models of Continuum Mechanics in Analysis and Engineering, Shaker, Aachen (1999), 117–129. Suche in Google Scholar
[24] A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl. 11 (2004), no. 2, 151–189. 10.1007/s00030-003-1052-7Suche in Google Scholar
[25] F. Rindler, S. Schwarzacher and J. J. L. Velázquez, Two-speed solutions to non-convex rate-independent systems, Arch. Ration. Mech. Anal. 239 (2021), no. 3, 1667–1731. 10.1007/s00205-020-01599-zSuche in Google Scholar
[26] F. Riva, On the approximation of quasistatic evolutions for the debonding of a thin film via vanishing inertia and viscosity, J. Nonlinear Sci. 30 (2020), no. 3, 903–951. 10.1007/s00332-019-09595-8Suche in Google Scholar
[27] R. T. Rockafellar, Convex Analysis, Princeton Math. Ser. 28, Princeton University, Princeton, 1970. Suche in Google Scholar
[28] R. Rossi and M. Thomas, Coupling rate-independent and rate-dependent processes: Existence results, SIAM J. Math. Anal. 49 (2017), no. 2, 1419–1494. 10.1137/15M1051567Suche in Google Scholar
[29] R. Scala, Limit of viscous dynamic processes in delamination as the viscosity and inertia vanish, ESAIM Control Optim. Calc. Var. 23 (2017), no. 2, 593–625. 10.1051/cocv/2016006Suche in Google Scholar
[30] G. Scilla and F. Solombrino, Multiscale analysis of singularly perturbed finite dimensional gradient flows: The minimizing movement approach, Nonlinearity 31 (2018), no. 11, 5036–5074. 10.1088/1361-6544/aad6acSuche in Google Scholar
[31] G. Scilla and F. Solombrino, A variational approach to the quasistatic limit of viscous dynamic evolutions in finite dimension, J. Differential Equations 267 (2019), no. 11, 6216–6264. 10.1016/j.jde.2019.06.018Suche in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
This work is licensed under the Creative Commons Attribution 4.0 International License.
Artikel in diesem Heft
- Frontmatter
- Harnack inequality for parabolic equations with coefficients depending on time
- Morse theory and the calculus of variations
- Relaxation of functionals with linear growth: Interactions of emerging measures and free discontinuities
- Fractional Poincaré and localized Hardy inequalities on metric spaces
- Higher order Ambrosio–Tortorelli scheme with non-negative spatially dependent parameters
- The notions of inertial balanced viscosity and inertial virtual viscosity solution for rate-independent systems
- Higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions
- On some variational problems involving capacity, torsional rigidity, perimeter and measure
- Sub-elliptic boundary value problems in flag domains
- On master test plans for the space of BV functions
- Regularity results for an optimal design problem with lower order terms
- On the behavior of the first eigenvalue of the p-Laplacian with Robin boundary conditions as p goes to 1
Artikel in diesem Heft
- Frontmatter
- Harnack inequality for parabolic equations with coefficients depending on time
- Morse theory and the calculus of variations
- Relaxation of functionals with linear growth: Interactions of emerging measures and free discontinuities
- Fractional Poincaré and localized Hardy inequalities on metric spaces
- Higher order Ambrosio–Tortorelli scheme with non-negative spatially dependent parameters
- The notions of inertial balanced viscosity and inertial virtual viscosity solution for rate-independent systems
- Higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions
- On some variational problems involving capacity, torsional rigidity, perimeter and measure
- Sub-elliptic boundary value problems in flag domains
- On master test plans for the space of BV functions
- Regularity results for an optimal design problem with lower order terms
- On the behavior of the first eigenvalue of the p-Laplacian with Robin boundary conditions as p goes to 1
