Open Access Published online by De Gruyter May 31, 2022

The notions of inertial balanced viscosity and inertial virtual viscosity solution for rate-independent systems

Filippo Riva , Giovanni Scilla and 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

(1.1) { ( u ˙ ( t ) ) + D x ( t , u ( t ) ) 0 in  X * ,  for a.e.  t [ 0 , T ] , u ( 0 ) = u 0 ,

where is a rate-independent dissipation potential, while is a time-dependent potential energy. In this paper, we limit ourselves to the case of a finite-dimensional normed space X, although we plan to extend the whole analysis to the infinite-dimensional context, where several difficulties concerning weak topologies and nonsmoothness of naturally arise (we refer to the monograph [22] for more details, also compare [19] with [20]).

If the driving energy is nonconvex, continuous solutions to (1.1) are not expected to exist, and thus in the past decades huge efforts have been spent to develop weak notions of solution capable of describing the behavior of the system at jumps. A first attempt can be found in the notion of energetic solution [23, 24] based on a global stability condition together with an energy balance which must hold for every t [ 0 , T ] :

(GS) ( t , u ( t ) ) ( t , x ) + ( x - u ( t ) ) for every  x X ,
(EB) ( t , u ( t ) ) + V ( u ; 0 , t ) = ( 0 , u 0 ) + 0 t t ( r , u ( r ) ) d r .

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

(1.2) { ε 𝕍 u ˙ ε ( t ) + ( u ˙ ε ( t ) ) + D x ( t , u ε ( t ) ) 0 in  X * ,  for a.e.  t [ 0 , T ] , u ε ( 0 ) = u 0 ε ,

as the parameter ε 0 . Here, 𝕍 denotes a symmetric positive-definite linear operator modelling viscosity. Actually, in [20, 19] more general viscous potentials are considered. The resulting evolution, called balanced viscosity (BV) solution, turns out to satisfy a local stability condition together with an augmented energy balance:

(LS) - D x ( t , u ( t ) ) ( 0 ) for a.e.  t [ 0 , T ] ,
(EB\${}^{\mathbb{V}}\$) ( t , u ( t ) ) + V 𝕍 ( u ; 0 , t ) = ( 0 , u 0 ) + 0 t t ( r , u ( r ) ) d r for all  t [ 0 , T ] .

While in (EB) the classical total variation (actually, -variation, see Definition 2.3) controls both the continuous part u co of the evolution and the jump part, as it holds (see also (2.13))

V ( u ; 0 , t ) = V ( u co ; 0 , t ) + r J u [ 0 , t ] ( ( u ( r ) - u - ( r ) ) + ( u + ( r ) - u ( r ) ) ) ,

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:

V 𝕍 ( u ; 0 , t ) = V ( u co ; 0 , t ) + r J u [ 0 , t ] ( c r 𝕍 ( u - ( r ) , u ( r ) ) + c r 𝕍 ( u ( r ) , u + ( r ) ) ) .

At time t [ 0 , T ] , the viscous cost function is obtained as

(1.3) c t 𝕍 ( u 1 , u 2 ) := inf { 0 1 p 𝕍 ( v ˙ ( r ) , - D x ( t , v ( r ) ) ) d r v W 1 , ( 0 , 1 ; X ) , v ( 0 ) = u 1 , v ( 1 ) = u 2 } ,

where p 𝕍 , called vanishing-viscosity contact potential, is a suitable density arising from both the viscous and the rate-independent dissipation (see also Definition 3.1).

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:

(1.4) u τ , ε k arg min x X { ε 2 τ x - u τ , ε k - 1 𝕍 2 + ( x - u τ , ε k - 1 ) + ( t k , x ) } , k = 1 , , T τ ,

when sending simultaneously ε and τ to 0 with also τ / ε 0 .

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

(1.5) { ε 2 𝕄 u ¨ ε ( t ) + ε 𝕍 u ˙ ε ( t ) + ( u ˙ ε ( t ) ) + D x ( t , u ε ( t ) ) 0 in  X * ,  for a.e.  t [ 0 , T ] , u ε ( 0 ) = u 0 ε , u ˙ ε ( 0 ) = u 1 ε ,

and then sending ε 0 , namely performing a vanishing-inertia and viscosity argument. The symmetric positive-definite linear operator 𝕄 appearing in (1.5) represents masses. Its presence allows to consider also the case of null viscosity, i.e. 𝕍 = 0 , or more generally of a positive-semidefinite linear operator. We point out that such a limit procedure is also known as slow-loading limit, since (1.5) comes from a dynamic problem with slow data after a reparametrization of time. We refer the interested reader to [10], or to [22, Chapter 5] for a more detailed explanation.

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 . An abstract analysis has been performed in [10], in finite dimension and always under convexity assumptions. Hence, the results contained in this paper on the one hand represent an extension to nonconvex energies of the ones presented in [10] (see Remark 2.2), and on the other hand put the basis for an abstract investigation in infinite dimension, where nonconvex problems are common in applications (we refer again to [2, 6, 12, 21, 22]).

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 𝕍 is not positive-definite) for the rate-independent system (1.1), namely a function satisfying the same local stability condition (LS) as BV solutions, together with an energy balance in which the cost at jump points is sensitive of the presence of inertia:

(LS) - D x ( t , u ( t ) ) ( 0 ) for a.e.  t [ 0 , T ] ,
(EB\${}^{\mathbb{M},\mathbb{V}}\$) ( t , u + ( t ) ) + V ( u co ; s , t ) + r J u e [ s , t ] c r 𝕄 , 𝕍 ( u - ( r ) , u + ( r ) ) = ( s , u - ( s ) ) + s t t ( r , u ( r ) ) d r ,

for every 0 s t T , where now the cost turns out to be

c t 𝕄 , 𝕍 ( u 1 , u 2 ) := inf { - N N p 𝕍 ( v ˙ ( r ) , - 𝕄 v ¨ ( r ) - D x ( t , v ( r ) ) ) d r N , v V u 1 , u 2 𝕄 , N } ,

where

V u 1 , u 2 𝕄 , N := { v W 2 , ( - N , N ; X ) v ( - N ) = u 1 , v ( N ) = u 2 , 𝕄 v ˙ ( ± N ) = 0 , ess sup r [ - N , N ] 𝕄 v ¨ ( r ) * C ¯ } .

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 𝕄 v ¨ ( r ) inside the integral. This prevents an easy generalization of the notion of parametrized BV solution, again introduced in [20, 19], to the inertial setting: such a notion is indeed build starting from a suitable viscous-reparametrization of the time variable. Furthermore, the rate-dependent nature of the cost forces one to consider minimization problems on an asymptotically infinite time horizon and take the infimum over them. We also refer to [1, 31] for a similar analysis with no rate-independent dissipation, namely considering = 0 , where an analogous notion of solution was developed. As it happened in [31], we prefer to consider a notion of solution which does not depend on the chosen representative of u in its Lebesgue class, which is done by considering left and right limits only in the energy balance.

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 u ε of (1.5) to an inertial balanced viscosity solution of (1.1), under a quite general set of assumptions on the energy , which includes the one considered in [10]. Secondly, we also prove that IBV solutions can be obtained via a natural extension of the minimizing movements algorithm (1.4), namely

(1.6) u τ , ε k arg min x X { ε 2 2 τ 2 x - 2 u τ , ε k - 1 + u τ , ε k - 2 𝕄 2 + ε 2 τ x - u τ , ε k - 1 𝕍 2 + ( x - u τ , ε k - 1 ) + ( t k , x ) } ,

by sending both τ and ε to 0. Differently from (1.4), for technical reasons we need to strengthen the rate of convergence requiring τ / ε 2 to be bounded. Furthermore, we have to require to be Λ-convex (assumption (E5) in Section 2). Such a condition, which amounts to require that the sum of with a suitably large quadratic perturbation is convex, is quite typical in the analysis of such approximation schemes (see [4]) and complies with many relevant applications. It actually allows one to have precise estimates on some rest terms in the energy balance, which arise from the iterative minimization schemes.

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 -variation (Section 2.2). In Section 3, we introduce the notions of inertial balanced viscosity and inertial virtual viscosity solution. We also define the contact potentials (Section 3.1) and the regularized contact potentials (Section 3.2), while in Section 3.3 we introduce the inertial cost function which will characterize the description of the jumps. Section 4 contains the first characterization of the IBV and IVV solutions as the slow-loading limit as ε 0 of dynamical solutions to (1.5). Finally, in Section 5, we derive these solutions as the limit of the time-discrete incremental variational scheme (1.6) as τ and ε go simultaneously to 0.

2 Notation and setting

Let ( X , ) be a finite-dimensional normed vector space. We denote by ( X * , * ) the topological dual of X and by w , v the duality product between w X * and v X . For R > 0 , we denote by B R the open ball in X of radius R centered at the origin, and by B R ¯ its closure.

Given any symmetric positive-semidefinite linear operator : X X * , we introduce the induced (Hilbertian) seminorm

| x | := x , x 1 2 ,

and we denote with a capital letter Q 0 a nonnegative constant satisfying

0 | x | 2 Q x 2 for every  x X .

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 , denoted by op .

If is positive-definite, the induced seminorm is actually a norm, denoted by , and, up to possibly enlarging the constant Q, there holds

1 Q x 2 x 2 Q x 2 for every  x X .

Furthermore, the inverse operator - 1 : X * X induces in X * the norm

w - 1 := w , - 1 w ,

which is dual to and thus satisfies

| w , v | v w - 1 for every  w X *  and  v X .

We briefly recall some basic definitions in convex analysis (see, for instance, [27]). Given a proper, convex, lower semicontinuous function f : X ( - , + ] , its (convex) subdifferential f : X X * at a point v X is defined by

f ( v ) = { w X * f ( z ) f ( v ) + w , z - v  for every  z X } .

Notice that if f ( v ) = + , then from the very definition it turns out that f ( v ) = .

The Fenchel conjugate of f is the convex, lower semicontinuous function

f * : X * ( - , + ] , defined by f * ( w ) := sup v X { w , v - f ( v ) } ,

and for every w X * and v X it satisfies

(2.1) f * ( w ) + f ( v ) w , v , with equality if and only if w f ( v ) .

Given a subset A X , we denote with χ A : X [ 0 , + ] its characteristic function, defined by

χ A ( x ) := { 0 if  x A , + if  x A .

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

(2.2) symmetric positive-definite linear operator  𝕄 : X X * ,

which represents a mass distribution.

The possible presence of viscosity is also considered by introducing the

(2.3) symmetric positive-semidefinite linear operator  𝕍 : X X * .

In particular, in our analysis we also include the case 𝕍 0 (for which | x | 𝕍 0 ), corresponding to the absence of viscous friction forces.

Both the rate-independent problem (1.1) and the dynamic problem (1.5) are damped by a rate-independent dissipation potential : X [ 0 , + ) , which models for instance dry friction. We make the following assumption:

1. The function is coercive, convex and positively homogeneous of degree one.

( v 1 + v 2 ) ( v 1 ) + ( v 2 ) for every  v 1 , v 2 X ,

and the existence of two positive constants α * α * > 0 for which

(2.4) α * v ( v ) α * v for every  v X .

This means that fails to be a norm only for the lack of symmetry.

Furthermore, since is one-homogeneous, for every v X its subdifferential ( v ) can be characterized by

(2.5) ( v ) = { w ( 0 ) w , v = ( v ) } ( 0 ) = : K * .

By (2.4), we notice that there holds

(2.6) K * B α * ¯ .

It is also well known (see, e.g., [27]) that K * coincides with the proper domain of the Fenchel conjugate * of ; indeed, it actually holds * = χ K * .

We finally consider the driving potential energy : [ 0 , T ] × X [ 0 , + ) , which we assume to possess the following properties:

1. ( , u ) is absolutely continuous in [ 0 , T ] for every u X .

2. ( t , ) is differentiable for every t [ 0 , T ] , and the differential D x is continuous from [ 0 , T ] × X to X * .

3. For a.e. t [ 0 , T ] and for every u X , it holds

| t ( t , u ) | a ( ( t , u ) ) b ( t ) ,

where a : [ 0 , + ) [ 0 , + ) is nondecreasing and continuous, while b L 1 ( 0 , T ) is nonnegative.

4. For every R > 0 , there exists a nonnegative function c R L 1 ( 0 , T ) such that for a.e. t [ 0 , T ] and for every u 1 , u 2 B R it holds

| t ( t , u 2 ) - t ( t , u 1 ) | c R ( t ) u 2 - u 1 .

We point out that the prototypical example of potential energy

(2.7) ( t , u ) = 𝒰 ( u ) - ( t ) , u ,

with 𝒰 C 1 ( X ) superlinear and W 1 , 1 ( 0 , T ; X * ) , fulfils all previous assumptions.

As mentioned in [10], under these hypotheses one can prove that is a continuous map, and that t ( t , u ( t ) ) is absolutely continuous (resp. of bounded variation) if u is absolutely continuous (resp. of bounded variation).

Remark 2.1.

Thanks to (E4), it is easy to see that D x ( , u ) is absolutely continuous in [ 0 , T ] for every u X :

D x ( t , u ) - D x ( s , u ) * = D x ( t , u ) - D x ( s , u ) , v
= lim h 0 ( t , u + h v ) - ( t , u ) - ( s , u + h v ) + ( s , u ) h
lim inf h 0 1 h s t | t ( r , u + h v ) - t ( r , u ) | d r
s t c R ( r ) d r ,

where v X is a suitable unit vector at which the dual norm is attained, and R can be chosen for instance equal to u + 1 . This last property will be used in Proposition 4.5 in order to apply a chain-rule formula for functions of bounded variation (see [5]).

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

( t , u ) = k 2 ( u - ( t ) ) 2 ,

with k > 0 and W 1 , 1 ( 0 , T ; ) . This specific form simply is the second order expansion of the real elastic potential energy of the springs

( t , u ) = k ( 1 - cos ( u - ( t ) ) ) ,

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:

1. The energy fulfils (E3) with the particular choice a ( y ) = y + a 1 for some a 1 0 .

1. ( t , ) is Λ-convex for every t [ 0 , T ] ; i.e., there exists Λ > 0 such that for every t [ 0 , T ] , u 1 , u 2 X and every θ ( 0 , 1 ) it holds

( t , ( 1 - θ ) u 1 + θ u 2 ) ( 1 - θ ) ( t , u 1 ) + θ ( t , u 2 ) + Λ 2 θ ( 1 - θ ) u 1 - u 2 𝕀 2

for some symmetric positive-definite linear operator 𝕀 : X X * .

We notice that by (E3’) and Gronwall’s lemma, we can infer

( t , u ) + a 1 ( ( s , u ) + a 1 ) e s t b ( r ) d r for every  0 s t T ,

whence

(2.8) | t ( t , u ) | ( ( s , u ) + a 1 ) b ( t ) e s t b ( r ) d r for every  0 s t T .

It is also easy to check that (E5) implies

(2.9) D x ( t , u 1 ) , u 2 - u 1 ( t , u 2 ) - ( t , u 1 ) + Λ 2 u 1 - u 2 𝕀 2 for every  t [ 0 , T ] , u 1 , u 2 X .

Indeed, by using the mean value theorem, for some ζ [ 0 , 1 ] we have

θ [ ( t , u 2 ) - ( t , u 1 ) + Λ 2 ( 1 - θ ) u 1 - u 2 𝕀 2 ] ( t , ( 1 - θ ) u 1 + θ u 2 ) - ( t , u 1 )
= θ D x ( t , u 1 + ζ θ ( u 1 - u 2 ) ) , u 2 - u 1 ,

whence (2.9) follows up to simplifying θ in both sides and then letting θ 0 .

We finally point out that an energy as in (2.7) always complies with (E3’), while it fulfils (E5) if in addition 𝒰 is Λ-convex.

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 f : [ a , b ] X be given. Then we define the pointwise -variation of f in [ s , t ] , with a s < t b , by

V ( f ; s , t ) := sup { k = 1 n ( f ( t k ) - f ( t k - 1 ) ) s = t 0 < t 1 < < t n - 1 < t n = t } .

We also set V ( f ; t , t ) := 0 for every t [ a , b ] .

We say that f is a function of bounded -variation in [ a , b ] , and we write f BV ( [ a , b ] ; X ) , if its -variation in [ a , b ] is finite; i.e., V ( f ; a , b ) < + .

Notice that, by virtue of (2.4), we have f BV ( [ a , b ] ; X ) if and only if f BV ( [ a , b ] ; X ) in the classical sense. In particular, f BV ( [ a , b ] ; X ) is regulated, i.e., it admits left and right limits at every t [ a , b ] :

f + ( t ) := lim t j t f ( t j ) and f - ( t ) := lim t j t f ( t j ) ,

with the convention f - ( a ) := f ( a ) and f + ( b ) := f ( b ) . Moreover, its pointwise jump set J f is at most countable.

It is well known (see, e.g., [3]) that f can be uniquely decomposed as follows:

(2.10) f = f + f Ca + f J ,

with f being an absolutely continuous function, f Ca a continuous Cantor-type function, and f J a jump function. If we denote by f the distributional derivative of f BV ( [ a , b ] ; X ) , and recall that f is a Radon vector measure with finite total variation | f | , it follows that f can be decomposed into the sum of the three mutually singular measures

(2.11) f = f + f Ca + f J , f = f ˙ 1 , f co := f + f Ca .

In (2.11), f is the absolutely continuous part with respect to the Lebesgue measure 1 , whose Lebesgue density f ˙ is the usual pointwise ( 1 -a.e. defined) derivative, f J is the jump part concentrated on the essential jump set of f, i.e.

J f e := { t [ a , b ] f + ( t ) f - ( t ) } J f ,

and f Ca is the Cantor part such that f Ca ( { t } ) = 0 for every t [ 0 , T ] . The measure f co is the diffuse part of the measure, and does not charge J f . The functions f , f Ca and f J in (2.10) are exactly the distributional primitives of the measures f , f Ca and f J in (2.11). We will use the notation f co to denote the continuous part of f, that is, f co = f + f Ca .

We also remark that, for a s t b , the function V ( f ; s , t ) is monotone in both entries, and hence it makes sense to consider the limits V ( f ; s - , t + ) . The following formula (see, for instance, [19, Section 2]) relates V ( f ; s - , t + ) with the distributional derivative of f, up to the jump part which is depending on the pointwise behavior of f. Setting λ = 1 + | f Ca | , it namely holds

(2.12) V ( f ; s - , t + ) = s t ( d f co d λ ( r ) ) d λ ( r ) + r J f [ s , t ] ( ( f + ( r ) - f ( r ) ) + ( f ( r ) - f - ( r ) ) ) ,

where d f co d λ is the Radon-Nikodym derivative. Observe that, by the positive one-homogeneity of , actually any measure ν such that f co ν can replace λ in the integral term on the right-hand side.

It follows from (2.12) that the continuous part of the -variation of f agrees with the -variation of f co and satisfies

(2.13) V ( f co ; s , t ) = s t ( d f co d λ ( r ) ) d λ ( r ) .

We finally notice that, by dropping the pointwise value of f at jump points (by the subadditivity of ), and only considering the essential jumps, we are led to the so-called essential -variation

(2.14) ( f ) ( [ s , t ] ) := V ( f co ; s , t ) + r J f e [ s , t ] ( f + ( r ) - f - ( r ) ) V ( f ; s - , t + ) .

The term ( f ) actually defines a Radon measure (see [11]), which generalizes the concept of total variation | f | (corresponding to the particular choice ( ) = ).

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

(3.1) ε ( v ) := ( v ) + ε 2 | v | 𝕍 2 .

It is then easy to check (see, e.g., [19, p. 47]) that the subdifferential of ε is explicitly given by

ε ( v ) = ( v ) + ε 𝕍 v ,

so the dynamic problem (1.5) can be rewritten as

(3.2) ε ( u ˙ ε ( t ) ) - ε 2 𝕄 u ¨ ε ( t ) - D x ( t , u ε ( t ) ) = : w ε ( t ) for a.e.  t [ 0 , T ] .

By using (2.1), and exploiting the classical chain-rule formula for , one obtains that the dynamic problem (3.2) is actually equivalent to the augmented energy balance

ε 2 2 u ˙ ε ( t ) 𝕄 2 + ( t , u ε ( t ) ) + s t ε ( u ˙ ε ( r ) ) + ε * ( w ε ( r ) ) d r
(3.3) = ε 2 2 u ˙ ε ( s ) 𝕄 2 + ( s , u ε ( s ) ) + s t t ( r , u ε ( r ) ) d r for every  0 s t T .

We point out that in our case of additive viscosity (3.1), the Fenchel conjugate ε * can be explicitly computed by means of the inf-sup convolution formula (see, e.g., [27, Section 12]) and turns out to be

(3.4) ε * ( w ) = { 1 2 ε inf z K * w - z ( ker 𝕍 ) w - z , 𝕍 ( w - z ) if  w K * + ( ker 𝕍 ) , + otherwise,

where

( ker 𝕍 ) = { w X * w , v = 0  for every  v ker 𝕍 }

is the annihilator of ker 𝕍 and 𝕍 : ( ker 𝕍 ) X is the inverse of the operator 𝕍 restricted to (the identification of) ( ker 𝕍 ) (in X).

In particular, in the two extreme situations 𝕍 = 0 and being 𝕍 positive-definite, we get respectively

ε * ( w ) = * ( w ) = χ K * ( w ) , ε * ( w ) = 1 2 ε dist 𝕍 - 1 2 ( w , K * ) ,

where

dist 𝕍 - 1 ( w , K * ) := inf z K * w - z 𝕍 - 1 ,

denotes the distance from K * , measured with respect to the norm 𝕍 - 1 .

3.1 Contact potentials

The energy balance (3.3) naturally leads to the introduction of a so-called (viscous) contact potential associated to 𝕍 . In the spirit of [19] and taking into account (3.4), we thus give the following definition.

Definition 3.1.

The (viscous) contact potential related to the viscosity operator 𝕍 is the map

p 𝕍 : X × X * [ 0 , + ]

defined by

p 𝕍 ( v , w ) := inf ε > 0 ( ε ( v ) + ε * ( w ) ) = { ( v ) + | v | 𝕍 inf z K * w - z ( ker 𝕍 ) | w - z | 𝕍 if  w K * + ( ker 𝕍 ) , + otherwise .

In the two extreme situations 𝕍 = 0 and 𝕍 being positive-definite, we get respectively

(3.5) p 0 ( v , w ) = ( v ) + χ K * ( w ) , p 𝕍 ( v , w ) = ( v ) + v 𝕍 dist 𝕍 - 1 ( w , K * ) .

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 p 𝕍 :

1. p 𝕍 ( , w ) is positively one-homogeneous and convex for every w X * .

2. p 𝕍 ( v , ) is convex for every v X .

3. p 𝕍 ( v , w ) max { ( v ) , w , v } for every v X and w X * .

4. p 𝕍 ( 0 , w ) = χ K * + ( ker 𝕍 ) ( w ) and p 𝕍 ( v , 0 ) = ( v ) .

Furthermore, we also observe the following property:

1. p 𝕍 ( , w ) is symmetric for every w X * if and only if is symmetric.

At this stage, a warning is mandatory: we point out that our potential p 𝕍 in general can take the value + , due to the semidefiniteness of the viscosity operator 𝕍 . This feature does not appear in [19], where indeed a full viscosity is always present and the contact potential is continuous and finite. This difference will create serious issues in the forthcoming analysis, leading to the original notion of inertial virtual viscosity; we are thus led to couple p 𝕍 with a “regularized” contact potential p, as follows.

Definition 3.2.

We say that a continuous map p : X × X * [ 0 , + ) is a regularized contact potential with respect to p 𝕍 , and we write p RCP 𝕍 , if the following conditions hold:

1. p ( , w ) is positively one-homogeneous for every w X * .

2. p ( v , ) is convex for every v X .

3. max { ( v ) , w , v } p ( v , w ) p 𝕍 ( v , w ) for every v X and w X * .

4. There exists a positive constant L > 0 such that

| p ( v , w 1 ) - p ( v , w 2 ) | L v w 1 - w 2 * for every  v X  and  w 1 , w 2 X * .

Remark 3.3.

In the case that 𝕍 is positive-definite, the contact potential p 𝕍 itself belongs to RCP 𝕍 . This easily follows by the explicit form (3.5). Observe that in this case p 𝕍 takes only finite values.

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 p 𝕍 , is the weighted Lipschitzianity (iv) with respect to the second variable: this will be heavily used in Proposition 4.6.

We finally observe that by (iii) and (iv), any p RCP 𝕍 satisfies

(3.6) p ( v , w ) | p ( v , w ) - p ( v , 0 ) | + p ( v , 0 ) L v w * + p 𝕍 ( v , 0 ) = L v w * + ( v ) ( α * + L w * ) v ,

where we exploited (2.4). In particular, it holds

p ( 0 , w ) = 0 for every  w X * .

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 λ 1 and every symmetric positive-definite linear operator 𝕌 : X X * , we define the function p 𝕍 λ , 𝕌 : X × X * [ 0 , + ) by

(3.7) p 𝕍 λ , 𝕌 ( v , w ) := inf η X * { p 𝕍 ( v , η ) + λ v 𝕌 w - η 𝕌 - 1 } , v X , w X * .

Proposition 3.4.

Let λ 1 and let U be a symmetric positive-definite linear operator. Then p V λ , U RCP V . Furthermore, for every v 0 , one has

(3.8) p 𝕍 ( v , w ) = sup λ 1 p 𝕍 λ , 𝕌 ( v , w ) = lim λ + p 𝕍 λ , 𝕌 ( v , w ) .

If in addition R is symmetric, then for every w X * the function p V λ , U ( , w ) is symmetric as well.

Proof.

We first notice that p 𝕍 λ , 𝕌 has nonnegative finite values since p 𝕍 is not identically + and it is nonnegative. Moreover, (3.8) is a standard property of the Yosida transform (notice that λ v 𝕌 > 0 if v 0 ). Also, if is symmetric, the symmetry of p 𝕍 λ , 𝕌 ( , w ) is a straightforward byproduct of (3.7) since in this case p 𝕍 ( , w ) is symmetric.

Now, we have to confirm properties (i)–(iv) of Definition 3.2. Property (i) follows by the one-homogeneity of p 𝕍 ( , η ) and of the norm.

Property (ii) follows since the Yosida transform of a convex function is convex, and p 𝕍 ( v , ) is convex.

The right inequality in (iii) is obtained by choosing η = w in the definition of p 𝕍 λ , 𝕌 , while the left one follows from the fact that p 𝕍 ( v , ) ( v ) combined with the simple inequality

w , v = η , v + w - η , v p 𝕍 ( v , η ) + λ v 𝕌 w - η 𝕌 - 1 .

Property (iv) is again a straightforward consequence of the Yosida transform: one can choose

L = λ 𝕌 op 𝕌 - 1 op .

We are only left to prove that p 𝕍 λ , 𝕌 is continuous. We first observe that, thanks to (iv), it is enough to prove that p 𝕍 λ , 𝕌 ( , w ) is continuous for every fixed w X * . The continuity in v = 0 follows easily by (3.6); if v 0 , we need to do more work. We make the following claims.

Claim 1.

There exists a positive constant C 1 > 0 such that

(3.9) p 𝕍 ( v 1 , w ) p 𝕍 ( v 2 , w ) + C 1 ( 1 + w * ) v 1 - v 2 for every  v 1 , v 2 X  and  w X * .

Claim 2.

If v 0 , then there exists a positive constant C 2 > 0 such that

p 𝕍 λ , 𝕌 ( v , w ) = inf η X * η * C 2 ρ ( v , w * ) { p 𝕍 ( v , η ) + λ v 𝕌 w - η 𝕌 - 1 } ,

where ρ ( v , w * ) := 1 + w * + 1 / v .

Claim 3.

There exists a positive constant C 3 > 0 such that

| p 𝕍 λ , 𝕌 ( v 1 , w ) - p 𝕍 λ , 𝕌 ( v 2 , w ) | C 3 max { ρ ( v 1 , w * ) , ρ ( v 2 , w * ) } v 1 - v 2

for every v 1 , v 2 X { 0 } and w X * .

From Claim 3, we easily deduce the continuity of p 𝕍 λ , 𝕌 ( , w ) in v 0 , and thus we only need to prove its validity.

We start with Claim 1, and we observe that it is enough to prove it for w K * + ( ker 𝕍 ) . By exploiting the subadditivity of together with (2.4) and (2.6), we easily obtain

p 𝕍 ( v 1 , w ) = ( v 1 ) + | v 1 | 𝕍 inf z K * w - z ( ker 𝕍 ) | w - z | 𝕍
( v 2 ) + α * v 1 - v 2 + ( | v 2 | 𝕍 + V v 1 - v 2 ) inf z K * w - z ( ker 𝕍 ) | w - z | 𝕍
( v 2 ) + | v 2 | 𝕍 inf z K * w - z ( ker 𝕍 ) | w - z | 𝕍 + ( α * + V V w * + V V α * ) v 1 - v 2
= p 𝕍 ( v 2 , w ) + ( α * + V V w * + V V α * ) v 1 - v 2 ,

and Claim 1 is proved.

To prove Claim 2, it is enough to show that an infimizing sequence { η j } j for p 𝕍 λ , 𝕌 ( v , w ) is uniformly bounded by ρ ( v , w * ) , up to a multiplicative constant. Being an infimizing sequence, η j satisfies

1 + p 𝕍 λ , 𝕌 ( v , w ) p 𝕍 ( v , η j ) + λ v 𝕌 w - η j 𝕌 - 1 v 𝕌 w - η j 𝕌 - 1 .

Thus, by using (3.6), we infer

η j * w * + C w - η j 𝕌 - 1
w * + C 1 + p 𝕍 λ , 𝕌 ( v , w ) v
w * + C 1 + ( α * + L w * ) v v
C ρ ( v , w * ) .

We now need to prove Claim 3. To this end, we take η X * such that η * C 2 ρ ( v 2 , w * ) and, exploiting Claim 1, we estimate

p 𝕍 λ , 𝕌 ( v 1 , w ) p 𝕍 ( v 1 , η ) + λ v 1 𝕌 w - η 𝕌 - 1
p 𝕍 ( v 2 , η ) + C 1 ( 1 + w * ) v 1 - v 2 + λ v 2 𝕌 w - η 𝕌 - 1 + λ w - η 𝕌 - 1 v 1 - v 2 𝕌
p 𝕍 ( v 2 , η ) + λ v 2 𝕌 w - η 𝕌 - 1 + C ( 1 + w * + η * ) v 1 - v 2
p 𝕍 ( v 2 , η ) + λ v 2 𝕌 w - η 𝕌 - 1 + C ρ ( v 2 , w * ) v 1 - v 2 .

By using Claim 2, from the above inequality we deduce

p 𝕍 λ , 𝕌 ( v 1 , w ) p 𝕍 λ , 𝕌 ( v 2 , w ) + C ρ ( v 2 , w * ) v 1 - v 2 .

By interchanging the roles of v 1 and v 2 , we thus complete the proof of Claim 3, and we conclude the proof of the proposition. ∎

We notice that in the case that 𝕍 is positive-definite, the contact potential p 𝕍 coincides with its Yosida approximation if we choose 𝕌 = 𝕍 :

p 𝕍 λ , 𝕍 ( v , w ) = p 𝕍 ( v , w ) for every  λ 1 , v X  and  w X * .

Indeed, by means of the explicit formula (3.5), it holds

p 𝕍 ( v , w ) p 𝕍 λ , 𝕍 ( v , w )
p 𝕍 1 , 𝕍 ( v , w )
= inf η X * { p 𝕍 ( v , η ) + v 𝕍 w - η 𝕍 - 1 }
= inf η X * { ( v ) + v 𝕍 ( dist 𝕍 - 1 ( η , K * ) + w - η 𝕍 - 1 ) }
= ( v ) + v 𝕍 inf η X * { dist 𝕍 - 1 ( η , K * ) + w - η 𝕍 - 1 }
( v ) + v 𝕍 dist 𝕍 - 1 ( w , K * )
= p 𝕍 ( v , w ) ,

where the last inequality is a simple byproduct of the triangle inequality. This fact corroborates Remark 3.3.

In the opposite situation 𝕍 = 0 , it is not difficult to see that the Yosida transform takes a more explicit form:

(3.10) p 0 λ , 𝕌 ( v , w ) = ( v ) + λ v 𝕌 dist 𝕌 - 1 ( w , K * ) .

Compare this last formula with (3.5), the case of 𝕍 being positive-definite.

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 𝕄 v ¨ inside the integral which is reminiscent of the original inertial effects.

Definition 3.5.

For every t [ 0 , T ] and u 1 , u 2 X , we define the inertial energy-dissipation cost related to p RCP 𝕍 { p 𝕍 } by

(3.11) c t 𝕄 , p ( u 1 , u 2 ) := inf { - N N p ( v ˙ ( r ) , - 𝕄 v ¨ ( r ) - D x ( t , v ( r ) ) ) d r N , v V u 1 , u 2 𝕄 , N } ,

where

V u 1 , u 2 𝕄 , N := { v W 2 , ( - N , N ; X ) v ( - N ) = u 1 , v ( N ) = u 2 , 𝕄 v ˙ ( ± N ) = 0 , ess sup r [ - N , N ] 𝕄 v ¨ ( r ) * C ¯ }

denotes the class of the admissible curves and C ¯ is the constant of Proposition 4.2 and Corollary 5.4 (depending only on the data of the problem).

We also define the inertial cost directly related to the viscosity operator 𝕍 by taking the supremum among the costs over all regularized contact potentials:

c t 𝕄 , 𝕍 ( u 1 , u 2 ) := sup p RCP 𝕍 c t 𝕄 , p ( u 1 , u 2 ) .

Remark 3.6.

We point out that in the case that 𝕍 is positive-definite, Remark 3.3 yields

(3.12) c t 𝕄 , 𝕍 ( u 1 , u 2 ) = c t 𝕄 , p 𝕍 ( u 1 , u 2 ) .

This is consistent with the vanishing-viscosity analysis performed in [19], in which the cost is (formally) equivalent to (3.12) by taking 𝕄 0 (see (1.3)).

A relevant feature of the inertial cost is that the value c t 𝕄 , p ( u 1 , u 2 ) provides an upper bound for the energy gap ( t , u 1 ) - ( t , u 2 ) for every p RCP 𝕍 , as shown with the following proposition.

Proposition 3.7.

For every t [ 0 , T ] and u 1 , u 2 X , we have

( t , u 1 ) - ( t , u 2 ) inf p RCP 𝕍 c t 𝕄 , p ( u 1 , u 2 ) .

Proof.

Fix p RCP 𝕍 and let N and v V u 1 , u 2 𝕄 , N . From the fundamental theorem of calculus and property (iii) of regularized contact potentials, we deduce

( t , u 1 ) - ( t , u 2 ) = ( t , v ( - N ) ) - ( t , v ( N ) ) + 1 2 v ˙ ( - N ) 𝕄 2 - 1 2 v ˙ ( N ) 𝕄 2
= - N N - 𝕄 v ¨ ( r ) - D x ( t , v ( r ) ) , v ˙ ( r ) d r
- N N p ( v ˙ ( r ) , - 𝕄 v ¨ ( r ) - D x ( t , v ( r ) ) ) d r ,

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 u BV ( [ 0 , T ] ; X ) is an inertial virtual viscosity (IVV) solution to the rate-independent system (1.1), related to 𝕄 and 𝕍 , if it complies both with the local stability condition

(3.13) - D x ( t , u ( t ) ) K * for every  t [ 0 , T ] J u ,

and the energy balance

(3.14) ( t , u + ( t ) ) + V ( u co ; s , t ) + r J u e [ s , t ] c r 𝕄 , 𝕍 ( u - ( r ) , u + ( r ) ) = ( s , u - ( s ) ) + s t t ( r , u ( r ) ) d r

for every 0 s t T .

If 𝕍 is positive-definite, in which case (3.14) is satisfied with c 𝕄 , p 𝕍 in place of c 𝕄 , 𝕍 (see (3.12)), we say that u is an inertial balanced viscosity (IBV) solution.

Remark 3.9.

By Proposition 3.7, we deduce that for any IVV solution there ho