Analysis of a rate-and-state friction problem with viscoelastic materials

Abstract

We consider a mathematical model which describes the frictional contact between a viscoelastic body and a foundation. The contact is modelled with normal compliance associated to a rate-and-state version of Coulombโ€™s law of dry friction. We start by presenting a description of the friction law, together with some examples used in geophysics. Then we state the classical formulation of the problem, list the assumptions on the data and derive a variational formulation of the model. It is in a form of a differential variational inequality in which the unknowns are the displacement field and the surface state variable. Next, we prove the unique weak solvability of the problem. The proof is based on arguments of history-dependent variational inequalities and nonlinear implicit integral equations in Banach spaces.

Authors

Flavius Patrulescu
(Tiberiu Popoviciu Institute of Numerical Analysis,
Romanian Academy)

Mircea Sofonea
(Laboratoire de Mathรฉmatiques et Physique, Universitรฉ de Perpignan)

Keywords

viscoelastic material; frictional contact; normal compliance; rate-and-state friction; differential variational inequality; history-dependent operator; weak solution

Cite this paper as

F. Patrulescu, M. Sofonea, Analysis of a rate-and-state friction problem with viscoelastic materials, Electron. J. Differential Equations, vol. 2017 (2017), no. 299, pp. 1-17.

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Southwest Texas State University, Department of Mathematics, San Marcos, TX; North Texas State University, Department of Mathematics, Denton, TX

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1072-6691

MR

3748017

ZBL

1386.74104

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ANALYSIS OF A RATE-AND-STATE FRICTION PROBLEM WITH VISCOELASTIC MATERIALS

FLAVIUS Pฤ‚TRULESCU, MIRCEA SOFONEA
Communicated by Vicentiu D. Rฤƒdulescu
Abstract

We consider a mathematical model which describes the frictional contact between a viscoelastic body and a foundation. The contact is modelled with normal compliance associated to a rate-and-state version of Coulombโ€™s law of dry friction. We start by presenting a description of the friction law, together with some examples used in geophysics. Then we state the classical formulation of the problem, list the assumptions on the data and derive a variational formulation of the model. It is in a form of a differential variational inequality in which the unknowns are the displacement field and the surface state variable. Next, we prove the unique weak solvability of the problem. The proof is based on arguments of history-dependent variational inequalities and nonlinear implicit integral equations in Banach spaces.

Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 299, pp. 1-17. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

1. Introduction

Phenomena of contact between deformable bodies abound in industry and everyday life. Usually, they give rise to additional phenomena like friction, wear, adhesion, damage and heat generation. Among these additional effects, friction represents the main ingredient on most of the contact problems. Due to their inherent complexity, contact phenomena lead to strongly nonlinear boundary value problems and their mathematical analysis requires tools of nonsmooth functional analysis, including results on variational inequalities and nonlinear differential equations.

Frictional contact is usually modelled with the Coulomb law of dry friction or a version thereof. According to this law, the tangential traction ฯƒฯ„\sigma_{\tau} can reach a bound HH, the so-called friction bound, which is the maximal frictional resistance that the surfaces can generate, and once it has been reached, a relative slip motion commences. Thus,

โ€–๐ˆฯ„โ€–โ‰คH,โˆ’๐ˆฯ„=H๐ฎห™ฯ„โ€–๐ฎห™ฯ„โ€– if ๐ฎห™ฯ„โ‰ ๐ŸŽ\left\|\boldsymbol{\sigma}_{\tau}\right\|\leq H,\quad-\boldsymbol{\sigma}_{\tau}=H\frac{\dot{\mathbf{u}}_{\tau}}{\left\|\dot{\mathbf{u}}_{\tau}\right\|}\quad\text{ if }\quad\dot{\mathbf{u}}_{\tau}\neq\mathbf{0} (1.1)

Here, ๐ฎห™ฯ„\dot{\mathbf{u}}_{\tau} is the relative tangential velocity or slip rate, and once slip starts, the frictional resistance has magnitude HH and is opposing the motion. The bound HH

00footnotetext: 2010 Mathematics Subject Classification. 74M15, 74M10, 74G25, 74G30, 49J40.
Key words and phrases. Viscoelastic material; frictional contact; normal compliance; rate-and-state friction; differential variational inequality; history-dependent operator; weak solution.
ยฉ 2017 Texas State University.
Submitted September 21, 2017. Published December 5, 2017.

depends on the process variables and, often, especially in engineering publications, is chosen as

H=ฮผ|ฯƒฮฝ|,H=\mu\left|\sigma_{\nu}\right|, (1.2)

where ฮผ\mu is the friction coefficient and ฯƒฮฝ\sigma_{\nu} denotes the normal stress on the contact surface.

We observe that the friction coefficient ฮผ\mu is not an intrinsic thermodynamic property of a material, a body or its surface, since it depends on the contact process and the operating conditions. It is defined as the ratio between the normal stress and the modulus of the tangential stress on the contact surface when sliding commences, and there is no theoretical reason for this ratio to be a well defined function. This may explain the difficulties in the experimental measurements of the friction coefficient. The issue is considerably complicated by the following facts. Engineering surfaces are not mathematically smooth surfaces, but contain asperities and various irregularities. Moreover, very often they contain some or all of the following: moisture, lubrication oils, various debris, wear particles, oxide layers, and chemicals and materials that are different from those of the parent body. Therefore, it is not surprising that the friction coefficient is found to depend on the surface characteristics, on the surface geometry and structure, on the relative velocity between the contacting surfaces, on the surface temperature, on the wear or rearrangement of the surface and, therefore, on its history, and other factors which we skip here. A very thorough description of these issues can be found in [18] (see also the survey [26]). However, and it is somewhat surprising, the concept of a friction coefficient is found to be sufficiently useful to be employed almost universally in frictional contact problems. Indeed, there seems to be no generally accepted current alternative to it.

Until recently, mathematical models for frictional contact used a constant friction coefficient, mainly for mathematical reasons. This is rapidly changing, and the dependence of ฮผ\mu on the process parameters has been incorporated into the models in recent publications. The dependence of the friction coefficient ฮผ\mu on the location ๐ฑ\mathbf{x} on the contacting surface, when the surface is not homogeneous, is easy to incorporate into the mathematical models, but is rarely made explicit, except for possibly mentioning it in passing. On the other hand, it is well documented that such dependence may be very pronounced. Indeed, in experiments on axisymmetric stretch forming in [27, 28] the friction coefficient was found to vary steeply from a value close to zero at the center to about 0.3 at the edge, with a very sharp transition region in between which was found to depend on the forming speed.

General models which take into consideration the dependence of the coefficient of friction on the process can be obtained by considering that

ฮผ(t)=ฮผ(โ€–๐ฎห™ฯ„(t)โ€–,ฮฑ(t)),ฮฑห™(t)=G(ฮฑ(t),โ€–๐ฎห™ฯ„(t)โ€–)\mu(t)=\mu\left(\left\|\dot{\mathbf{u}}_{\tau}(t)\right\|,\alpha(t)\right),\quad\dot{\alpha}(t)=G\left(\alpha(t),\left\|\dot{\mathbf{u}}_{\tau}(t)\right\|\right) (1.3)

where GG is an appropriate function and ฮฑ\alpha represents an internal state surface variable. Note that in such laws, the coefficient of friction depends both the rate of the slip, denoted โ€–๐ฎห™ฯ„โ€–\left\|\dot{\mathbf{u}}_{\tau}\right\|, and on the state variable ฮฑ\alpha. For this reason, the literature refers to friction laws of the form (1.1)-(1.3) as rate-and-state friction laws. References in the field are [15, 16, 17, 20].

Contact models constructed by using equalities of the form (1.3) have been used in most geophysical publications dealing with earthquakes. A first example is the
so-called Dieterich-Ruina model (see, e.g., 14) in which

ฮผ=ฮผ0โˆ’Alnโก(1+โ€–๐ฎห™ฯ„(t)โ€–vโˆž)+Blnโก(1+ฮฑ(t)ฮฑ0).\mu=\mu_{0}-A\ln\left(1+\frac{\left\|\dot{\mathbf{u}}_{\tau}(t)\right\|}{v_{\infty}}\right)+B\ln\left(1+\frac{\alpha(t)}{\alpha_{0}}\right). (1.4)

Here ฮผ0\mu_{0} is the static friction coefficient, vโˆžv_{\infty} is the maximal slip velocity in the system, and ฮฑ\alpha is an internal state variable describing the surface, and whose equation of evolution is given by

ฮฑห™(t)=1โˆ’โ€–๐ฎห™ฯ„(t)โ€–Lโˆ—ฮฑ(t)\dot{\alpha}(t)=1-\frac{\left\|\dot{\mathbf{u}}_{\tau}(t)\right\|}{L^{*}}\alpha(t) (1.5)

where Lโˆ—,A,BL^{*},A,B are adjusted system parameters. More elaborate expressions can be found in [6, 14, 15, 16, and we refer the reader there and the references therein. A second example is obtained by taking

ฮผ=ฮผ(ฮฑ),ฮฑห™(t)=โ€–๐ฎห™ฯ„(t)โ€–.\mu=\mu(\alpha),\quad\dot{\alpha}(t)=\left\|\dot{\mathbf{u}}_{\tau}(t)\right\|. (1.6)

In this case state variable is the total slip rate, i.e., ฮฑ(t)=โˆซ0tโ€–๐ฎห™ฯ„(s)โ€–๐‘‘s\alpha(t)=\int_{0}^{t}\left\|\dot{\mathbf{u}}_{\tau}(s)\right\|ds. The dependence on the process history via this parameter takes into account the morphological changes undergone by the contacting surfaces as the process goes on. Finally, the slip rate dependence ฮผ=ฮผ(โ€–๐ฎห™ฯ„โ€–)\mu=\mu\left(\left\|\dot{\mathbf{u}}_{\tau}\right\|\right) is also an example of (1.3), in which ฮฑ\alpha is a constant and GG vanishes.

A friction coefficient which depends on the slip rate has been employed in dynamic cases in [8, 12, 13] where the non-uniqueness of the solution and possible solutions with shocks were investigated in a special setting. A result on quasistatic contact with slip rate or total slip rate dependent friction coefficient can be found in 1]. The modelling of dynamic contact problems with rate-and-state friction of the form (1.3) have been considered recently in [15, 16, associated to Kelvin-Voingt viscoelastic materials. An algorithm for the numerical simulation of these problems was considered in 17. There, numerical simulations were provided and compared with experimental results made to a laboratory scale. However, the well-posedness of models with such friction conditions is, as yet, an unsolved problem. The reason arises in the coupling between the rate and the state variables in the friction law.

The aim of this paper is to present a rigorous analysis of a contact model with rate-and-state friction. In contrast with the models considered in 15, 16, in this paper we consider only quasistatic process of contact but we assume a more general viscoelastic constitutive law. Considering a dependence of the form (1.3) for the coefficient of friction leads to a new and nonstandard mathematical model which couples a variational inequality for the displacement field with an ordinary differential equation for the surface state variable. The analysis of this model represents the main trait of novelty of this paper.

The rest of the manuscript is structured as follows. In Section 2 we present the notation we shall use as well as some preliminary material. In Section 3 we describe the model of the contact process and list the assumptions on the data. Then, in Section 4 we derive the variational formulation of the problem and state our main existence and uniqueness result, Theorem 4.1. The proof of the theorem is provided in Section 5, based on arguments on history-dependent variational inequalities and nonlinear implicit integral equations in Banach spaces.

2. Notation and preliminaries

As already mentioned in the previous section, we start by introducing the notion we use everywhere in this paper together with some preliminary results.

General notation. Everywhere in this paper dโˆˆ{1,2,3}d\in\{1,2,3\} and ๐•Šd\mathbb{S}^{d} represents the space of second order symmetric tensors on โ„d\mathbb{R}^{d} or, equivalently, the space of symmetric matrices of order dd. The zero element of the spaces โ„d\mathbb{R}^{d} and ๐•Šd\mathbb{S}^{d} will be denoted by ๐ŸŽ\mathbf{0}. The inner product and norm on โ„d\mathbb{R}^{d} and ๐•Šd\mathbb{S}^{d} are defined by

๐ฎโ‹…๐ฏ=uivi,โ€–๐ฏโ€–=(๐ฏโ‹…๐ฏ)1/2โˆ€๐ฎ=(ui),๐ฏ=(vi)โˆˆโ„d๐ˆโ‹…ฯ„=ฯƒijฯ„ij,โ€–ฯ„โ€–=(ฯ„โ‹…ฯ„)1/2โˆ€๐ˆ=(ฯƒij),ฯ„=(ฯ„ij)โˆˆ๐•Šd\begin{array}[]{cll}\mathbf{u}\cdot\mathbf{v}=u_{i}v_{i},&\|\mathbf{v}\|=(\mathbf{v}\cdot\mathbf{v})^{1/2}&\forall\mathbf{u}=\left(u_{i}\right),\mathbf{v}=\left(v_{i}\right)\in\mathbb{R}^{d}\\ \boldsymbol{\sigma}\cdot\tau=\sigma_{ij}\tau_{ij},&\|\tau\|=(\tau\cdot\tau)^{1/2}&\forall\boldsymbol{\sigma}=\left(\sigma_{ij}\right),\tau=\left(\tau_{ij}\right)\in\mathbb{S}^{d}\end{array}

where the indices i,ji,j run between 1 and dd and, unless stated otherwise, the summation convention over repeated indices is used.

The norm on the space XX will be denote by โˆฅโ‹…โˆฅX\|\cdot\|_{X}, and 0X0_{X} will represent the zero element of XX. Moreover, we denote by X=X1ร—X2ร—โ€ฆร—XmX=X_{1}\times X_{2}\times\ldots\times X_{m} the product of the normed spaces X1,X2,โ€ฆ,XmX_{1},X_{2},\ldots,X_{m}, endowed with the canonical product norm

โ€–๐ฎโ€–X=โ€–u1โ€–X12+โ€ฆ+โ€–umโ€–Xm2,\|\mathbf{u}\|_{X}=\sqrt{\left\|u_{1}\right\|_{X_{1}}^{2}+\ldots+\left\|u_{m}\right\|_{X_{m}}^{2}}, (2.1)

for all ๐ฎ=(u1,โ€ฆ,um)โˆˆX\mathbf{u}=\left(u_{1},\ldots,u_{m}\right)\in X. For a Hilbert space XX we denote by (โ‹…,โ‹…)X(\cdot,\cdot)_{X} its inner product. In addition, if XiX_{i} are real Hilbert spaces with the inner products (โ‹…,โ‹…)Xi(\cdot,\cdot)_{X_{i}} and associated norms โˆฅโ‹…โˆฅXi,i=1,โ€ฆ,m\|\cdot\|_{X_{i}},i=1,\ldots,m, then the product space X=X1ร—X2ร—โ€ฆร—XmX=X_{1}\times X_{2}\times\ldots\times X_{m} will be endowed with with the canonical inner product (โ‹…,โ‹…)X(\cdot,\cdot)_{X} defined by

(๐ฎ,๐ฏ)X=(u1,v1)X1+โ€ฆ+(um,vm)Xm,(\mathbf{u},\mathbf{v})_{X}=\left(u_{1},v_{1}\right)_{X_{1}}+\ldots+\left(u_{m},v_{m}\right)_{X_{m}}, (2.2)

for all ๐ฎ=(u1,โ€ฆ,um),๐ฏ=(v1,โ€ฆ,vm)โˆˆX\mathbf{u}=\left(u_{1},\ldots,u_{m}\right),\mathbf{v}=\left(v_{1},\ldots,v_{m}\right)\in X.
Below in this paper II will represent either a bounded interval of the form [0,T][0,T] with T>0T>0, or the unbounded interval โ„+=[0,+โˆž)\mathbb{R}_{+}=[0,+\infty). We denote by C(I;X)C(I;X) the space of continuous functions on II with values in XX. In the case I=[0,T]I=[0,T], the space C(I;X)C(I;X) will be equipped with the norm

โ€–vโ€–C([0,T];X)=maxtโˆˆ[0,T]โกโ€–v(t)โ€–X.\|v\|_{C([0,T];X)}=\max_{t\in[0,T]}\|v(t)\|_{X}. (2.3)

It is well known that if XX is a Banach space, then C([0,T];X)C([0,T];X) is also a Banach space. Assume now that I=โ„+I=\mathbb{R}_{+}. It is well known that if XX is a Banach space, then C(I;X)C(I;X) can be organized in a canonical way as a Frรฉchet space, i.e., a complete metric space in which the corresponding topology is induced by a countable family of seminorms. The convergence of a sequence {vk}k\left\{v_{k}\right\}_{k} to the element vv, in the space C(โ„+;X)C\left(\mathbb{R}_{+};X\right), can be described as follows: vkโ†’vv_{k}\rightarrow v in C(โ„+;X)C\left(\mathbb{R}_{+};X\right) as kโ†’โˆžk\rightarrow\infty if and only if

maxrโˆˆ[0,n]โกโ€–vk(r)โˆ’v(r)โ€–Xโ†’0 as kโ†’โˆž for all nโˆˆโ„•.\max_{r\in[0,n]}\left\|v_{k}(r)-v(r)\right\|_{X}\rightarrow 0\text{ as }k\rightarrow\infty\text{ for all }n\in\mathbb{N}. (2.4)

In other words, the sequence {vk}k\left\{v_{k}\right\}_{k} converges to the element vv in the space C(โ„+;X)C\left(\mathbb{R}_{+};X\right) if and only if it converges to vv in the space C([0,n];X)C([0,n];X) for all nโˆˆโ„•n\in\mathbb{N}. In addition, we denote by C1(I;X)C^{1}(I;X) the space of continuously differentiable functions on II with values in XX. Therefore, vโˆˆC1(I;X)v\in C^{1}(I;X) if and only if vโˆˆC(I;X)v\in C(I;X) and vห™โˆˆC(I;X)\dot{v}\in C(I;X) where, here and below, vห™\dot{v} represents the time derivative of the function vv.

History-dependent variational inequalities. We proceed with an abstract existence and uniqueness result for a special class of time-dependent variational inequalities. To this end, we consider a real Hilbert space XX and a normed space YY. Moreover, we consider the operators A:Xโ†’X,โ„›:C(I;X)โ†’C(I;Y)A:X\rightarrow X,\mathcal{R}:C(I;X)\rightarrow C(I;Y), the
functional ฯ†:Yร—Xร—Xโ†’โ„\varphi:Y\times X\times X\rightarrow\mathbb{R} and the function f:Iโ†’Xf:I\rightarrow X, and we assume that the following conditions hold.
(a) There exists mA>0m_{A}>0 such that

(Au1โˆ’Au2,u1โˆ’u2)Xโ‰ฅmAโ€–u1โˆ’u2โ€–X2โˆ€u1,u2โˆˆX.\left(Au_{1}-Au_{2},u_{1}-u_{2}\right)_{X}\geq m_{A}\left\|u_{1}-u_{2}\right\|_{X}^{2}\quad\forall u_{1},u_{2}\in X. (2.5)

(b) There exists MA>0M_{A}>0 such that

โ€–Au1โˆ’Au2โ€–Xโ‰คMAโ€–u1โˆ’u2โ€–Xโˆ€u1,u2โˆˆX.\left\|Au_{1}-Au_{2}\right\|_{X}\leq M_{A}\left\|u_{1}-u_{2}\right\|_{X}\quad\forall u_{1},u_{2}\in X.

For any compact JโŠ‚IJ\subset I, there exists LJ>0L_{J}>0 such that

โ€–โ„›u1(t)โˆ’โ„›u2(t)โ€–Yโ‰คLJโˆซ0tโ€–u1(s)โˆ’u2(s)โ€–X๐‘‘s\left\|\mathcal{R}u_{1}(t)-\mathcal{R}u_{2}(t)\right\|_{Y}\leq L_{J}\int_{0}^{t}\left\|u_{1}(s)-u_{2}(s)\right\|_{X}ds (2.6)

for all u1,u2โˆˆC(I;X)u_{1},u_{2}\in C(I;X) and all tโˆˆJt\in J.
(a) For all yโˆˆYy\in Y and uโˆˆX,ฯ†(y,u,โ‹…):Xโ†’โ„u\in X,\varphi(y,u,\cdot):X\rightarrow\mathbb{R} is convex and lower semicontinuous on XX.
(b) There exist c1โ‰ฅ0c_{1}\geq 0 and c2โ‰ฅ0c_{2}\geq 0 such that

ฯ†(y1,u1,v2)โˆ’ฯ†(y1,u1,v1)+ฯ†(y2,u2,v1)โˆ’ฯ†(y2,u2,v2)\displaystyle\varphi\left(y_{1},u_{1},v_{2}\right)-\varphi\left(y_{1},u_{1},v_{1}\right)+\varphi\left(y_{2},u_{2},v_{1}\right)-\varphi\left(y_{2},u_{2},v_{2}\right) (2.7)
โ‰คc1โ€–y1โˆ’y2โ€–Yโ€–v1โˆ’v2โ€–X+c2โ€–u1โˆ’u2โ€–Xโ€–v1โˆ’v2โ€–X\displaystyle\leq c_{1}\left\|y_{1}-y_{2}\right\|_{Y}\left\|v_{1}-v_{2}\right\|_{X}+c_{2}\left\|u_{1}-u_{2}\right\|_{X}\left\|v_{1}-v_{2}\right\|_{X}

for all y1,y2โˆˆY,u1,u2,v1,v2โˆˆXy_{1},y_{2}\in Y,u_{1},u_{2},v_{1},v_{2}\in X.

fโˆˆC(I;X).f\in C(I;X). (2.8)

Note that assumption (2.5) shows that AA is a Lipschitz continuous strongly monotone operator. Moreover, following the terminology introduced in 22, condition (2.6), shows that the operator โ„›\mathcal{R} is a history-dependent operator. Such kind of operators arise both in Functional Analysis and Solid Mechanics, as explained in the recent book 23 . We have the following existence and uniqueness result for variational inequalities with history-dependent operators, the so-called historydependent variational inequalities.

Theorem 2.1. Assume that 2.5-(2.8) hold. Moreover, assume that

c2โ‰ฅmAc_{2}\geq m_{A} (2.9)

where mAm_{A} and c2c_{2} are the constants in (2.5) and (2.7), respectively. Then, there exists a unique function uโˆˆC(I;X)u\in C(I;X) such that, for all tโˆˆIt\in I, it holds

(Au(t),vโˆ’u(t))X+ฯ†(โ„›u(t),u(t),v)โˆ’ฯ†(โ„›u(t),u(t),u(t))\displaystyle(Au(t),v-u(t))_{X}+\varphi(\mathcal{R}u(t),u(t),v)-\varphi(\mathcal{R}u(t),u(t),u(t))
โ‰ฅ(f(t),vโˆ’u(t))Xโˆ€vโˆˆX.\displaystyle\geq(f(t),v-u(t))_{X}\quad\forall v\in X. (2.10)

This theorem represents a particular case of a more general result presented in 23, pag 58]. Its proof is based on arguments of time-dependent quasivariational inequalities and a fixed point result for history-dependent operators defined on the Frรฉchet space C(I;X)C(I;X). A version of Theorem 2.1 could be found in 25.

A nonlinear implicit equation. Assume in what follows that ( X,โˆฅโ‹…โˆฅXX,\|\cdot\|_{X} ) is a normed space and ( Y,โˆฅโ‹…โˆฅYY,\|\cdot\|_{Y} ) is a Banach space. Moreover, assume that the operators A:Xโ†’YA:X\rightarrow Y and ๐’ข:Iร—Xร—Yโ†’Y\mathcal{G}:I\times X\times Y\rightarrow Y satisfy the following conditions.

There exists LA>0L_{A}>0 such that

โ€–Ax1โˆ’Ax2โ€–Yโ‰คLAโ€–x1โˆ’x2โ€–Xโˆ€x1,x2โˆˆX.\left\|Ax_{1}-Ax_{2}\right\|_{Y}\leq L_{A}\left\|x_{1}-x_{2}\right\|_{X}\quad\forall x_{1},x_{2}\in X. (2.11)

(a) There exists LG>0L_{G}>0 such that

โ€–๐’ข(t,x1,y1)โˆ’๐’ข(t,x2,y2)โ€–Yโ‰คL๐’ข(โ€–x1โˆ’x2โ€–X+โ€–y1โˆ’y2โ€–Y)\left\|\mathcal{G}\left(t,x_{1},y_{1}\right)-\mathcal{G}\left(t,x_{2},y_{2}\right)\right\|_{Y}\leq L_{\mathcal{G}}\left(\left\|x_{1}-x_{2}\right\|_{X}+\left\|y_{1}-y_{2}\right\|_{Y}\right) (2.12)

for all x1,x2โˆˆX,y1,y2โˆˆY,tโˆˆIx_{1},x_{2}\in X,y_{1},y_{2}\in Y,t\in I.
(b) The mapping tโ†ฆ๐’ข(t,x,y):Iโ†’Yt\mapsto\mathcal{G}(t,x,y):I\rightarrow Y is continuous for all xโˆˆX,yโˆˆYx\in X,y\in Y.
The following result will be used in the proof of Lemma 5.1 below.
Theorem 2.2. Assume that (2.11)-(2.12) hold. Then:
(1) For each function xโˆˆC(I;X)x\in C(I;X), there exists a unique function yโˆˆC(I;Y)y\in C(I;Y) such that

y(t)=Ax(t)+โˆซ0t๐’ข(s,x(s),y(s))๐‘‘sโˆ€tโˆˆIy(t)=Ax(t)+\int_{0}^{t}\mathcal{G}(s,x(s),y(s))ds\quad\forall t\in I (2.13)

(2) There exists a history-dependent operator โ„›:C(I;X)โ†’C(I;Y)\mathcal{R}:C(I;X)\rightarrow C(I;Y) (i.e., an operator which satisfies condition (2.6) such that for all functions xโˆˆC(I;X)x\in C(I;X) and yโˆˆC(I;Y)y\in C(I;Y), equality (2.13) holds if and only if

y(t)=Ax(t)+โ„›x(t)โˆ€tโˆˆI.y(t)=Ax(t)+\mathcal{R}x(t)\quad\forall t\in I. (2.14)

Note that this theorem describes the history-dependence feature of the solution of the implicit integral equation (2.13). Its proof can be found in [23, pag 52]. A versions of this theorem was previously obtained in [24], in the case when I=[0,T]I=[0,T] with T>0T>0.

Function spaces. Everywhere in this paper ฮฉ\Omega denotes a bounded domain of โ„d\mathbb{R}^{d} with a Lipschitz continuous boundary ฮ“\Gamma and ฮ“1,ฮ“2,ฮ“3\Gamma_{1},\Gamma_{2},\Gamma_{3} will represent a partition of ฮ“\Gamma into three measurable parts such that meas (ฮ“1)>0\left(\Gamma_{1}\right)>0. We use ๐ฑ=(xi)\mathbf{x}=\left(x_{i}\right) for the generic point in ฮฉโˆชฮ“\Omega\cup\Gamma. An index that follows a comma will represent the partial derivative with respect to the corresponding component of the spatial variable ๐ฑโˆˆฮฉโˆชฮ“\mathbf{x}\in\Omega\cup\Gamma, e.g. f,i=โˆ‚f/โˆ‚xif_{,i}=\partial f/\partial x_{i}. Moreover, ๐‚=(ฮฝi)\boldsymbol{\nu}=\left(\nu_{i}\right) denotes the outward unit normal at ฮ“\Gamma.

We use standard notation for Sobolev and Lebesgue spaces associated to ฮฉ\Omega and ฮ“\Gamma. In particular, we use the spaces L2(ฮฉ)d,L2(ฮ“2)d,L2(ฮ“3)L^{2}(\Omega)^{d},L^{2}\left(\Gamma_{2}\right)^{d},L^{2}\left(\Gamma_{3}\right) and H1(ฮฉ)dH^{1}(\Omega)^{d}, endowed with their canonical inner products and associated norms. Moreover, we recall that for an element ๐ฏโˆˆH1(ฮฉ)d\mathbf{v}\in H^{1}(\Omega)^{d} we sometimes write ๐ฏ\mathbf{v} for the trace ฮณ๐ฏโˆˆL2(ฮ“)d\gamma\mathbf{v}\in L^{2}(\Gamma)^{d} of ๐ฏ\mathbf{v} to ฮ“\Gamma. In addition, we consider the following spaces:

V={๐ฏโˆˆH1(ฮฉ)d:๐ฏ=๐ŸŽ on ฮ“1}Q={๐ˆ=(ฯƒij):ฯƒij=ฯƒjiโˆˆL2(ฮฉ)}\begin{gathered}V=\left\{\mathbf{v}\in H^{1}(\Omega)^{d}:\mathbf{v}=\mathbf{0}\text{ on }\Gamma_{1}\right\}\\ Q=\left\{\boldsymbol{\sigma}=\left(\sigma_{ij}\right):\sigma_{ij}=\sigma_{ji}\in L^{2}(\Omega)\right\}\end{gathered}

The spaces VV and QQ are real Hilbert spaces endowed with the canonical inner products

(๐ฎ,๐ฏ)V=โˆซฮฉ๐œบ(๐ฎ)โ‹…๐œบ(๐ฏ)๐‘‘x,(๐ˆ,ฯ„)Q=โˆซฮฉ๐ˆโ‹…ฯ„๐‘‘x(\mathbf{u},\mathbf{v})_{V}=\int_{\Omega}\boldsymbol{\varepsilon}(\mathbf{u})\cdot\boldsymbol{\varepsilon}(\mathbf{v})dx,\quad(\boldsymbol{\sigma},\tau)_{Q}=\int_{\Omega}\boldsymbol{\sigma}\cdot\tau dx (2.15)

Here and below ฮต\varepsilon and Div represent the deformation and the divergence operators, respectively, i.e.,

ฮต(๐ฎ)=(ฮตij(๐ฎ)),ฮตij(๐ฎ)=12(ui,j+uj,i),Divโก๐ˆ=(ฯƒij,j).\varepsilon(\mathbf{u})=\left(\varepsilon_{ij}(\mathbf{u})\right),\quad\varepsilon_{ij}(\mathbf{u})=\frac{1}{2}\left(u_{i,j}+u_{j,i}\right),\quad\operatorname{Div}\boldsymbol{\sigma}=\left(\sigma_{ij,j}\right). (2.16)

The associated norms on these spaces are denoted by โˆฅโ‹…โˆฅV\|\cdot\|_{V} and โˆฅโ‹…โˆฅQ\|\cdot\|_{Q}, respectively. Also, recall that the completeness of the space VV follows from the assumption meas (ฮ“1)>0\left(\Gamma_{1}\right)>0 which allows the use of Kornโ€™s inequality.

For any element ๐ฏโˆˆV\mathbf{v}\in V we denote by vฮฝv_{\nu} and ๐ฏฯ„\mathbf{v}_{\tau} its normal and tangential components on ฮ“\Gamma given by vฮฝ=๐ฏโ‹…๐‚v_{\nu}=\mathbf{v}\cdot\boldsymbol{\nu} and ๐ฏฯ„=๐ฏโˆ’vฮฝ๐‚\mathbf{v}_{\tau}=\mathbf{v}-v_{\nu}\boldsymbol{\nu}, respectively. For a regular function ๐ˆ:ฮฉโ†’๐•Šd\boldsymbol{\sigma}:\Omega\rightarrow\mathbb{S}^{d} we denote by ฯƒฮฝ\sigma_{\nu} and ๐ˆฯ„\boldsymbol{\sigma}_{\tau} the normal and tangential stress on ฮ“\Gamma, that is ฯƒฮฝ=(๐ˆ๐‚)โ‹…๐‚\sigma_{\nu}=(\boldsymbol{\sigma}\boldsymbol{\nu})\cdot\boldsymbol{\nu} and ๐ˆฯ„=๐ˆ๐‚โˆ’ฯƒฮฝ๐‚\boldsymbol{\sigma}_{\tau}=\boldsymbol{\sigma}\boldsymbol{\nu}-\sigma_{\nu}\boldsymbol{\nu}, and we recall that the following Greenโ€™s formula holds:

โˆซฮฉ๐ˆโ‹…๐œบ(๐ฏ)๐‘‘x+โˆซฮฉDivโก๐ˆโ‹…๐ฏdx=โˆซฮ“๐ˆ๐‚โ‹…๐ฏ๐‘‘a for all ๐ฏโˆˆH1(ฮฉ)d\int_{\Omega}\boldsymbol{\sigma}\cdot\boldsymbol{\varepsilon}(\mathbf{v})dx+\int_{\Omega}\operatorname{Div}\boldsymbol{\sigma}\cdot\mathbf{v}dx=\int_{\Gamma}\boldsymbol{\sigma}\boldsymbol{\nu}\cdot\mathbf{v}da\quad\text{ for all }\mathbf{v}\in H^{1}(\Omega)^{d} (2.17)

We also recall that there exists c0>0c_{0}>0 which depends on ฮฉ,ฮ“1\Omega,\Gamma_{1} and ฮ“3\Gamma_{3} such that

โ€–๐ฏโ€–L2(ฮ“3)dโ‰คc0โ€–๐ฏโ€–V for all ๐ฏโˆˆV.\|\mathbf{v}\|_{L^{2}\left(\Gamma_{3}\right)^{d}}\leq c_{0}\|\mathbf{v}\|_{V}\quad\text{ for all }\mathbf{v}\in V. (2.18)

Inequality (2.18) represents a consequence of the Sobolev trace theorem.
Finally, we denote by ๐โˆž\mathbf{Q}_{\infty} the space of fourth order tensor fields given by

๐โˆž={โ„ฐ=(โ„ฐijkl):โ„ฐijkl=โ„ฐjikl=โ„ฐklijโˆˆLโˆž(ฮฉ),1โ‰คi,j,k,lโ‰คd}.\mathbf{Q}_{\infty}=\left\{\mathcal{E}=\left(\mathcal{E}_{ijkl}\right):\mathcal{E}_{ijkl}=\mathcal{E}_{jikl}=\mathcal{E}_{klij}\in L^{\infty}(\Omega),\quad 1\leq i,j,k,l\leq d\right\}.

The space ๐โˆž\mathbf{Q}_{\infty} is a real Banach space with the norm

โ€–โ„ฐโ€–๐โˆž=max1โ‰คi,j,k,lโ‰คdโกโ€–โ„ฐijklโ€–Lโˆž(ฮฉ).\|\mathcal{E}\|_{\mathbf{Q}_{\infty}}=\max_{1\leq i,j,k,l\leq d}\left\|\mathcal{E}_{ijkl}\right\|_{L^{\infty}(\Omega)}.

Moreover, a simple calculation shows that

โ€–โ„ฐฯ„โ€–Qโ‰คdโ€–โ„ฐโ€–๐โˆžโ€–ฯ„โ€–Qโˆ€โ„ฐโˆˆ๐โˆž,ฯ„โˆˆQ.\|\mathcal{E}\tau\|_{Q}\leq d\|\mathcal{E}\|_{\mathbf{Q}_{\infty}}\|\tau\|_{Q}\quad\forall\mathcal{E}\in\mathbf{Q}_{\infty},\tau\in Q. (2.19)

In addition to the spaces V,Q,๐โˆžV,Q,\mathbf{Q}_{\infty}, whose properties will be used in various places in the next section, we shall use the space of vectorial functions C(I;X)C(I;X) and C1(I;X)C^{1}(I;X) where XX denotes one of the spaces V,Q,๐โˆžV,Q,\mathbf{Q}_{\infty} and, recall, II represents the time interval of interest.

3. The model

The classical formulation of the rate-and-state frictional contact problem we consider in this paper is the following.

Problem ๐’ซ\mathcal{P}. Find a displacement field ๐ฎ:ฮฉร—Iโ†’โ„d\mathbf{u}:\Omega\times I\rightarrow\mathbb{R}^{d}, a stress field ๐ˆ:ฮฉร—Iโ†’๐•Šd\boldsymbol{\sigma}:\Omega\times I\rightarrow\mathbb{S}^{d} and a surface state variable ฮฑ:ฮ“3ร—Iโ†’โ„\alpha:\Gamma_{3}\times I\rightarrow\mathbb{R} such that

๐ˆ(t)=๐’œ๐œบ(๐ฎห™(t))+โ„ฌ๐œบ(๐ฎ(t))+โˆซ0t๐’ฆ(tโˆ’s)๐œบ(๐ฎห™(s))๐‘‘s in ฮฉ,\displaystyle\boldsymbol{\sigma}(t)=\mathcal{A}\boldsymbol{\varepsilon}(\dot{\mathbf{u}}(t))+\mathcal{B}\boldsymbol{\varepsilon}(\mathbf{u}(t))+\int_{0}^{t}\mathcal{K}(t-s)\boldsymbol{\varepsilon}(\dot{\mathbf{u}}(s))ds\quad\text{ in }\Omega, (3.1)
Divโก๐ˆ(t)+๐Ÿ0(t)=๐ŸŽ in ฮฉ,\displaystyle\operatorname{Div}\boldsymbol{\sigma}(t)+\mathbf{f}_{0}(t)=\mathbf{0}\quad\text{ in }\Omega, (3.2)
๐ฎ(t)=๐ŸŽ on ฮ“1,\displaystyle\mathbf{u}(t)=\mathbf{0}\quad\text{ on }\Gamma_{1}, (3.3)
๐ˆ(t)๐‚=๐Ÿ2(t) on ฮ“2,\displaystyle\boldsymbol{\sigma}(t)\boldsymbol{\nu}=\mathbf{f}_{2}(t)\quad\text{ on }\Gamma_{2}, (3.4)
โˆ’ฯƒฮฝ(t)=p(uฮฝ(t)) on ฮ“3,\displaystyle-\sigma_{\nu}(t)=p\left(u_{\nu}(t)\right)\quad\text{ on }\Gamma_{3}, (3.5)
โ€–๐ˆฯ„(t)โ€–โ‰คฮผ(โ€–๐ฎห™ฯ„(t)โ€–;ฮฑ(t))|ฯƒฮฝ(t)|\displaystyle\left\|\boldsymbol{\sigma}_{\tau}(t)\right\|\leq\mu\left(\left\|\dot{\mathbf{u}}_{\tau}(t)\right\|;\alpha(t)\right)\left|\sigma_{\nu}(t)\right| (3.6)
โˆ’๐ˆฯ„(t)=ฮผ(โˆฅ๐ฎห™ฯ„(t)โˆฅ;ฮฑ(t))|ฯƒฮฝ(t)|๐ฎห™ฯ„(t)โ€–๐ฎห™ฯ„(t)โ€– if ๐ฎห™ฯ„(t)โ‰ ๐ŸŽ} on ฮ“3,\displaystyle\left.-\boldsymbol{\sigma}_{\tau}(t)=\mu\left(\left\|\dot{\mathbf{u}}_{\tau}(t)\right\|;\alpha(t)\right)\left|\sigma_{\nu}(t)\right|\frac{\dot{\mathbf{u}}_{\tau}(t)}{\left\|\dot{\mathbf{u}}_{\tau}(t)\right\|}\quad\text{ if }\dot{\mathbf{u}}_{\tau}(t)\neq\mathbf{0}\right\}\quad\text{ on }\Gamma_{3}, (3.7)
ฮฑห™(t)=G(ฮฑ(t),โ€–๐ฎห™ฯ„(t)โ€–) on ฮ“3,\displaystyle\dot{\alpha}(t)=G\left(\alpha(t),\left\|\dot{\mathbf{u}}_{\tau}(t)\right\|\right)\quad\text{ on }\Gamma_{3},

for all tโˆˆIt\in I and, in addition,

๐ฎ(0)=๐ฎ0,ฮฑ(0)=ฮฑ0 on ฮ“3.\mathbf{u}(0)=\mathbf{u}_{0},\quad\alpha(0)=\alpha_{0}\quad\text{ on }\Gamma_{3}. (3.8)

Problem ๐’ซ\mathcal{P} describes the evolution of a viscoelastic body under the action of body forces and surface tractions. In the reference configuration the body occupies the domain ฮฉ\Omega and is in contact with a foundation on the part ฮ“3\Gamma_{3} of its boundary.

For more details on the physical setting and the mathematical modeling of contact phenomena we send the reader to the monographs [7, 19, 23,

We now provide a description of the equations and the conditions (3.1)-(3.8) and introduce the assumptions on the data. Note that, here and below, to simplify the notation, we do not mention explicitly the dependence of various functions on the spatial variable ๐ฑโˆˆฮฉโˆชฮ“\mathbf{x}\in\Omega\cup\Gamma.

First, equation (3.1) represents the viscoelastic constitutive law, in which ๐’œ\mathcal{A} is the viscosity operator, โ„ฌ\mathcal{B} is the elasticity operator, ๐’ฆ\mathcal{K} represents the relaxation tensor and ฮต(๐ฎ)\varepsilon(\mathbf{u}) denotes the linearized strain tensor, see (2.16). Various results, examples and mechanical interpretations in the study of viscoelastic materials of the form (3.1), can be found in 5) and the references therein. Such kind of constitutive laws were used in the literature in order to model the behavior of real materials like rubbers, rocks, metals, pastes and polymers. In particular, equation (3.1) was employed in [3, 4] in order to model the hysteresis damping in elastomers. Moreover, incorporating it into equation of motion results in integro-partial differential equation which is computationally challenging both in simulation and control design balance, as mentioned in [5]. Note that when ๐’ฆ\mathcal{K} vanishes (3.1) becomes the wellknown Kelvin-Voigt constitutive law, used in 15, 16, for instance. The analysis of various mathematical models of contact problems with viscoelastic materials of the form (3.1) was provided in [21, 23, 25, for instance. Below in this paper we assume that the viscosity operator, the elasticity operator and the relaxation tensor in the constitutive law (3.1) satisfy the following conditions.
(a) ๐’œ:ฮฉร—๐•Šdโ†’๐•Šd\mathcal{A}:\Omega\times\mathbb{S}^{d}\rightarrow\mathbb{S}^{d}.
(b) There exists L๐’œ>0L_{\mathcal{A}}>0 such that

โ€–๐’œ(๐ฑ,๐œบ1)โˆ’๐’œ(๐ฑ,๐œบ2)โ€–โ‰คL๐’œโ€–๐œบ1โˆ’๐œบ2โ€–\left\|\mathcal{A}\left(\mathbf{x},\boldsymbol{\varepsilon}_{1}\right)-\mathcal{A}\left(\mathbf{x},\boldsymbol{\varepsilon}_{2}\right)\right\|\leq L_{\mathcal{A}}\left\|\boldsymbol{\varepsilon}_{1}-\boldsymbol{\varepsilon}_{2}\right\|

for all ฮต1,ฮต2โˆˆ๐•Šd\varepsilon_{1},\varepsilon_{2}\in\mathbb{S}^{d}, a.e. ๐ฑโˆˆฮฉ\mathbf{x}\in\Omega.
(c) There exists m๐’œ>0m_{\mathcal{A}}>0 such that

(๐’œ(๐ฑ,ฮต1)โˆ’๐’œ(๐ฑ,ฮต2))โ‹…(ฮต1โˆ’ฮต2)โ‰ฅm๐’œโ€–ฮต1โˆ’ฮต2โ€–2\left(\mathcal{A}\left(\mathbf{x},\varepsilon_{1}\right)-\mathcal{A}\left(\mathbf{x},\varepsilon_{2}\right)\right)\cdot\left(\varepsilon_{1}-\varepsilon_{2}\right)\geq m_{\mathcal{A}}\left\|\varepsilon_{1}-\varepsilon_{2}\right\|^{2} (3.9)

for all ฮต1,ฮต2โˆˆ๐•Šd\varepsilon_{1},\varepsilon_{2}\in\mathbb{S}^{d}, a.e. ๐ฑโˆˆฮฉ\mathbf{x}\in\Omega.
(d) The mapping ๐ฑโ†ฆ๐’œ(๐ฑ,ฮต)\mathbf{x}\mapsto\mathcal{A}(\mathbf{x},\varepsilon) is measurable on ฮฉ\Omega, for any ฮตโˆˆ๐•Šd\varepsilon\in\mathbb{S}^{d}.
(e) The mapping ๐ฑโ†ฆ๐’œ(๐ฑ,๐ŸŽ)\mathbf{x}\mapsto\mathcal{A}(\mathbf{x},\mathbf{0}) belongs to QQ.
(a) โ„ฌ:ฮฉร—๐•Šdโ†’๐•Šd\mathcal{B}:\Omega\times\mathbb{S}^{d}\rightarrow\mathbb{S}^{d}.
(b) There exists Lโ„ฌ>0L_{\mathcal{B}}>0 such that

โ€–โ„ฌ(๐ฑ,๐œบ1)โˆ’โ„ฌ(๐ฑ,๐œบ2)โ€–โ‰คLโ„ฌโ€–๐œบ1โˆ’๐œบ2โ€–\left\|\mathcal{B}\left(\mathbf{x},\boldsymbol{\varepsilon}_{1}\right)-\mathcal{B}\left(\mathbf{x},\boldsymbol{\varepsilon}_{2}\right)\right\|\leq L_{\mathcal{B}}\left\|\boldsymbol{\varepsilon}_{1}-\boldsymbol{\varepsilon}_{2}\right\| (3.10)

for all ฮต1,ฮต2โˆˆ๐•Šd\varepsilon_{1},\varepsilon_{2}\in\mathbb{S}^{d}, a.e. ๐ฑโˆˆฮฉ\mathbf{x}\in\Omega.
(c) The mapping ๐ฑโ†ฆโ„ฌ(๐ฑ,ฮต)\mathbf{x}\mapsto\mathcal{B}(\mathbf{x},\varepsilon) is measurable on ฮฉ\Omega, for any ฮตโˆˆ๐•Šd\varepsilon\in\mathbb{S}^{d}.
(d) The mapping ๐ฑโ†ฆโ„ฌ(๐ฑ,๐ŸŽ)\mathbf{x}\mapsto\mathcal{B}(\mathbf{x},\mathbf{0}) belongs to QQ.

๐’ฆโˆˆC(I;๐โˆž).\mathcal{K}\in C\left(I;\mathbf{Q}_{\infty}\right). (3.11)

Next, equation (3.2) represents the equation of equilibrium in which ๐Ÿ0\mathbf{f}_{0} represents the density of body forces, assumed to have the regularity

๐Ÿ0โˆˆC(I;L2(ฮฉ)d).\mathbf{f}_{0}\in C\left(I;L^{2}(\Omega)^{d}\right). (3.12)

We use this equation in the statement of Problem ๐’ซ\mathcal{P} since we assume that the mechanical process is quasistatic and, therefore, the inertial terms in the equation of motion are neglected.

Conditions 3.3 and (3.4) are the displacement and the traction boundary condition, respectively, in which ๐Ÿ2\mathbf{f}_{2} represents the density of surface tractions, assumed to have the regularity

๐Ÿ2โˆˆC(I;L2(ฮ“2)d).\mathbf{f}_{2}\in C\left(I;L^{2}\left(\Gamma_{2}\right)^{d}\right). (3.13)

These conditions show that the body is held fixed on the part ฮ“1\Gamma_{1} on his boundary and is acted upon by time-dependent forces on the part ฮ“2\Gamma_{2}.

Condition (3.5) is the normal compliance contact condition on ฮ“3\Gamma_{3} in which ฯƒฮฝ\sigma_{\nu} denotes the normal stress, uฮฝu_{\nu} is the normal displacement and pp is a given normal compliance function. This condition models the contact with a deformable foundation. It was first introduced in [1] and used in may publications see, e.g., [7, 19, 23 and the references therein. Moreover, the term normal compliance was first used in 9. 10. Below in this paper we assume that the function pp satisfies the following condition
(a) p:ฮ“3ร—โ„โ†’โ„+p:\Gamma_{3}\times\mathbb{R}\rightarrow\mathbb{R}_{+}.
(b) There exists Lp>0L_{p}>0 such that

|p(๐ฑ,r1)โˆ’p(๐ฑ,r2)|โ‰คLp|r1โˆ’r2|\left|p\left(\mathbf{x},r_{1}\right)-p\left(\mathbf{x},r_{2}\right)\right|\leq L_{p}\left|r_{1}-r_{2}\right| (3.14)

for all r1,r2โˆˆโ„r_{1},r_{2}\in\mathbb{R}, a.e. ๐ฑโˆˆฮ“3\mathbf{x}\in\Gamma_{3}.
(c) The mapping ๐ฑโ†ฆp(๐ฑ,r)\mathbf{x}\mapsto p(\mathbf{x},r) is measurable on ฮ“3\Gamma_{3} for all rโˆˆโ„r\in\mathbb{R}.
(d) p(๐ฑ,r)=0p(\mathbf{x},r)=0 for all rโ‰ค0r\leq 0, a.e. ๐ฑโˆˆฮ“3\mathbf{x}\in\Gamma_{3}.
(e) There exists pโˆ—>0p^{*}>0 such that p(๐ฑ,r)โ‰คpโˆ—p(\mathbf{x},r)\leq p^{*} for all rโˆˆโ„r\in\mathbb{R}, a.e. ๐ฑโˆˆฮ“3\mathbf{x}\in\Gamma_{3}.

A typical example of such function is

p(๐ฑ,r)={ฮทr+if r<r0ฮทr0 if rโ‰ฅr0p(\mathbf{x},r)=\begin{cases}\eta r^{+}&\text{if }r<r_{0}\\ \eta r_{0}&\text{ if }r\geq r_{0}\end{cases}

for all ๐ฑโˆˆฮ“3\mathbf{x}\in\Gamma_{3}, where r+r^{+}denotes the positive part of r,r0>0r,r_{0}>0 is a given bound and ฮท>0\eta>0 represents the stiffness coefficient of the foundation.

Condition (3.6) represents the rate-and-state friction law, introduced in Section 1 . It is obtained by using the Coulomb law of dry friction (1.1), with the friction bound (1.2) in which the coefficient of friction depends on the relative slip rate โ€–๐ฎห™ฯ„โ€–\left\|\dot{\mathbf{u}}_{\tau}\right\| and the internal state variable ฮฑ\alpha, as shown in (1.3). For the coefficient of friction we assume that
(a) ฮผ:ฮ“3ร—โ„ร—โ„โ†’โ„+\mu:\Gamma_{3}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}_{+}.
(b) There exists Lฮผ>0L_{\mu}>0 such that

|ฮผ(โ‹…,r1,a1)โˆ’ฮผ(โ‹…,r2,a2)|โ‰คLฮผ(|r1โˆ’r2|+|a1โˆ’a2|)\left|\mu\left(\cdot,r_{1},a_{1}\right)-\mu\left(\cdot,r_{2},a_{2}\right)\right|\leq L_{\mu}\left(\left|r_{1}-r_{2}\right|+\left|a_{1}-a_{2}\right|\right) (3.16)

for all r1,r2,a1,a2โˆˆโ„r_{1},r_{2},a_{1},a_{2}\in\mathbb{R}, a.e. ๐ฑโˆˆฮ“3\mathbf{x}\in\Gamma_{3}.
(c) The mapping ๐ฑโ†ฆฮผ(๐ฑ,r,a)\mathbf{x}\mapsto\mu(\mathbf{x},r,a) is measurable on ฮ“3\Gamma_{3}, for all r,aโˆˆโ„r,a\in\mathbb{R}.
(d) There exists ฮผโˆ—>0\mu^{*}>0 such that ฮผ(๐ฑ,r,a)โ‰คฮผโˆ—\mu(\mathbf{x},r,a)\leq\mu^{*} for all r,aโˆˆโ„r,a\in\mathbb{R}, a.e. ๐ฑโˆˆฮ“3\mathbf{x}\in\Gamma_{3}.
This assumption shows that ฮผ\mu is a Lipschitz continuous function of its arguments, which seems very reasonable in many applications. However, there are cases when
the transition from the static to the dynamic value is rather sharp, and a graph may better describe the situation.

Next, (3.7) represents the differential equation which describes the evolution of the surface state variable. Here GG is a given function assumed to satisfy
(a) G:ฮ“3ร—โ„ร—โ„โ†’โ„G:\Gamma_{3}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}.
(b) There exists LG>0L_{G}>0 such that

|G(๐ฑ,ฮฑ1,r1)โˆ’G(๐ฑ,ฮฑ2,r2)|โ‰คLG(|ฮฑ1โˆ’ฮฑ2|+|r1โˆ’r2|)\left|G\left(\mathbf{x},\alpha_{1},r_{1}\right)-G\left(\mathbf{x},\alpha_{2},r_{2}\right)\right|\leq L_{G}\left(\left|\alpha_{1}-\alpha_{2}\right|+\left|r_{1}-r_{2}\right|\right) (3.17)

for all ฮฑ1,ฮฑ1,r1,r2โˆˆโ„\alpha_{1},\alpha_{1},r_{1},r_{2}\in\mathbb{R}, a.e. ๐ฑโˆˆฮฉ\mathbf{x}\in\Omega.
(c) The mapping ๐ฑโ†ฆG(๐ฑ,ฮฑ,r)\mathbf{x}\mapsto G(\mathbf{x},\alpha,r) is measurable on ฮฉ\Omega, for all ฮฑ,rโˆˆโ„\alpha,r\in\mathbb{R}.
(d) The mapping ๐ฑโ†ฆG(๐ฑ,0,0)\mathbf{x}\mapsto G(\mathbf{x},0,0) belongs to L2(ฮ“3)L^{2}\left(\Gamma_{3}\right).

Note that condition (3.17) is satisfied in the case of the total slip rate friction law (1.6) but is not satisfied for the Dietrich-Ruina model, see (1.5). Nevertheless, several regularized version of the differential equations (1.5) can be considered, in which the corresponding function GG satisfies assumption (3.17). These regularizations are obtained by truncation, as explained in 15.

Finally, (3.8) represents the initial conditions in which ๐ฎ0\mathbf{u}_{0} and ฮฑ0\alpha_{0} denote the initial displacement and the initial surface state variable, respectively, supposed to have the regularity

๐ฎ0โˆˆV,ฮฑ0โˆˆL2(ฮ“3).\mathbf{u}_{0}\in V,\quad\alpha_{0}\in L^{2}\left(\Gamma_{3}\right). (3.18)

We end this section with the remark that Problem ๐’ซ\mathcal{P} represents the classical formulation of the rate-and-state friction problem we consider in this paper. In general, this problem does not have classical solution, i.e., solution which have all the necessary classical derivatives. For this reason, as usual in the analysis of frictional contact problems, there is a need to associate to Problem ๐’ซ\mathcal{P} a new problem, the so called variational formulation.

4. Variational Formulation

In this section we derive the variational formulation of Problem ๐’ซ\mathcal{P} and state our main existence and uniqueness result, Theorem 4.1. To this end, we start by using use the Riesz representation theorem to define the function ๐Ÿ:Iโ†’V\mathbf{f}:I\rightarrow V by equality

(๐Ÿ(t),๐ฏ)V=โˆซฮฉ๐Ÿ0(t)โ‹…๐ฏ๐‘‘x+โˆซฮ“๐Ÿ2(t)โ‹…๐ฏ๐‘‘a(\mathbf{f}(t),\mathbf{v})_{V}=\int_{\Omega}\mathbf{f}_{0}(t)\cdot\mathbf{v}dx+\int_{\Gamma}\mathbf{f}_{2}(t)\cdot\mathbf{v}da (4.1)

for all ๐ฏโˆˆV\mathbf{v}\in V and tโˆˆIt\in I. The regularities (3.12), (3.13) imply that

๐ŸโˆˆC(I;V).\mathbf{f}\in C(I;V). (4.2)

Next, we assume ( ๐ฎ,๐ˆ,ฮฑ\mathbf{u},\boldsymbol{\sigma},\alpha ) are sufficiently regular functions which satisfies (3.1)(3.8). Let ๐ฏโˆˆV\mathbf{v}\in V and tโˆˆIt\in I be given. We use the Green formula (2.17) and the equilibrium equation (3.2) to deduce that

โˆซฮฉ๐ˆ(t)โ‹…(๐œบ(๐ฏ)โˆ’๐œบ(๐ฎห™(t)))๐‘‘x\displaystyle\int_{\Omega}\boldsymbol{\sigma}(t)\cdot(\boldsymbol{\varepsilon}(\mathbf{v})-\boldsymbol{\varepsilon}(\dot{\mathbf{u}}(t)))dx
=โˆซฮฉ๐Ÿ0(t)โ‹…(๐ฏโˆ’๐ฎห™(t))๐‘‘x+โˆซฮ“๐ˆ(t)๐‚โ‹…(๐ฏโˆ’๐ฎห™(t))๐‘‘a\displaystyle=\int_{\Omega}\mathbf{f}_{0}(t)\cdot(\mathbf{v}-\dot{\mathbf{u}}(t))dx+\int_{\Gamma}\boldsymbol{\sigma}(t)\boldsymbol{\nu}\cdot(\mathbf{v}-\dot{\mathbf{u}}(t))da

Then, we split the surface integral over ฮ“1,ฮ“2\Gamma_{1},\Gamma_{2} and ฮ“3\Gamma_{3}, use equalities ๐ฏโˆ’๐ฎห™(t)=๐ŸŽ\mathbf{v}-\dot{\mathbf{u}}(t)=\mathbf{0} on ฮ“1\Gamma_{1} and ๐ˆ(t)ฮฝ=๐Ÿ2(t)\boldsymbol{\sigma}(t)\nu=\mathbf{f}_{2}(t) on ฮ“2\Gamma_{2} and definition (4.1) to deduce that

(๐ˆ(t),๐œบ(๐ฏ)โˆ’๐œบ(๐ฎห™(t)))Q=(๐Ÿ(t),๐ฏโˆ’๐ฎห™(t))V+โˆซฮ“3๐ˆ(t)๐‚โ‹…(๐ฏโˆ’๐ฎห™(t))๐‘‘a(\boldsymbol{\sigma}(t),\boldsymbol{\varepsilon}(\mathbf{v})-\boldsymbol{\varepsilon}(\dot{\mathbf{u}}(t)))_{Q}=(\mathbf{f}(t),\mathbf{v}-\dot{\mathbf{u}}(t))_{V}+\int_{\Gamma_{3}}\boldsymbol{\sigma}(t)\boldsymbol{\nu}\cdot(\mathbf{v}-\dot{\mathbf{u}}(t))da (4.3)

On the other hand, the boundary conditions (3.5), (3.6) combined with the positivity of the function pp yield

ฯƒฮฝ(t)(vฮฝโˆ’uห™ฮฝ(t))=โˆ’p(uฮฝ(t))(vฮฝโˆ’uห™ฮฝ(t))๐ˆฯ„(t)โ‹…(๐ฏฯ„โˆ’๐ฎห™ฯ„(t))โ‰ฅฮผ(โ€–๐ฎห™ฯ„(t)โ€–;ฮฑ(t))p(uฮฝ(t))(โ€–๐ฎห™ฯ„(t)โ€–โˆ’โ€–๐ฏฯ„โ€–)\begin{gathered}\sigma_{\nu}(t)\left(v_{\nu}-\dot{u}_{\nu}(t)\right)=-p\left(u_{\nu}(t)\right)\left(v_{\nu}-\dot{u}_{\nu}(t)\right)\\ \boldsymbol{\sigma}_{\tau}(t)\cdot\left(\mathbf{v}_{\tau}-\dot{\mathbf{u}}_{\tau}(t)\right)\geq\mu\left(\left\|\dot{\mathbf{u}}_{\tau}(t)\right\|;\alpha(t)\right)p\left(u_{\nu}(t)\right)\left(\left\|\dot{\mathbf{u}}_{\tau}(t)\right\|-\left\|\mathbf{v}_{\tau}\right\|\right)\end{gathered}

on ฮ“3\Gamma_{3}. Therefore, since

๐ˆ(t)๐‚โ‹…(๐ฏโˆ’๐ฎห™(t))=ฯƒฮฝ(t)(vฮฝโˆ’uห™ฮฝ(t))+๐ˆฯ„(t)โ‹…(๐ฏฯ„โˆ’๐ฎห™ฯ„(t)) on ฮ“3,\boldsymbol{\sigma}(t)\boldsymbol{\nu}\cdot(\mathbf{v}-\dot{\mathbf{u}}(t))=\sigma_{\nu}(t)\left(v_{\nu}-\dot{u}_{\nu}(t)\right)+\boldsymbol{\sigma}_{\tau}(t)\cdot\left(\mathbf{v}_{\tau}-\dot{\mathbf{u}}_{\tau}(t)\right)\quad\text{ on }\Gamma_{3},

we deduce that

โˆซฮ“3๐ˆ(t)๐‚โ‹…(๐ฏโˆ’๐ฎห™(t))๐‘‘aโ‰ฅ\displaystyle\int_{\Gamma_{3}}\boldsymbol{\sigma}(t)\boldsymbol{\nu}\cdot(\mathbf{v}-\dot{\mathbf{u}}(t))da\geq โˆ’(p(uฮฝ(t)),vฮฝโˆ’uห™ฮฝ(t))L2(ฮ“3)\displaystyle-\left(p\left(u_{\nu}(t)\right),v_{\nu}-\dot{u}_{\nu}(t)\right)_{L^{2}\left(\Gamma_{3}\right)}
+(ฮผ(โ€–๐ฎห™ฯ„(t)โ€–;ฮฑ(t))p(uฮฝ(t)),โ€–๐ฎห™ฯ„(t)โ€–โˆ’โ€–๐ฏฯ„โ€–)L2(ฮ“3)\displaystyle+\left(\mu\left(\left\|\dot{\mathbf{u}}_{\tau}(t)\right\|;\alpha(t)\right)p\left(u_{\nu}(t)\right),\left\|\dot{\mathbf{u}}_{\tau}(t)\right\|-\left\|\mathbf{v}_{\tau}\right\|\right)_{L^{2}\left(\Gamma_{3}\right)}

We now combine this inequality with (4.3) to obtain

(๐ˆ(t),๐œบ(๐ฏ)โˆ’๐œบ(๐ฎห™(t)))Q+(p(uฮฝ(t)),vฮฝโˆ’uห™ฮฝ(t))L2(ฮ“3)\displaystyle(\boldsymbol{\sigma}(t),\boldsymbol{\varepsilon}(\mathbf{v})-\boldsymbol{\varepsilon}(\dot{\mathbf{u}}(t)))_{Q}+\left(p\left(u_{\nu}(t)\right),v_{\nu}-\dot{u}_{\nu}(t)\right)_{L^{2}\left(\Gamma_{3}\right)}
+(ฮผ(โ€–๐ฎห™ฯ„(t)โ€–;ฮฑ(t))p(uฮฝ(t)),โ€–๐ฏฯ„โ€–โˆ’โ€–๐ฎห™ฯ„(t)โ€–)L2(ฮ“3)\displaystyle+\left(\mu\left(\left\|\dot{\mathbf{u}}_{\tau}(t)\right\|;\alpha(t)\right)p\left(u_{\nu}(t)\right),\left\|\mathbf{v}_{\tau}\right\|-\left\|\dot{\mathbf{u}}_{\tau}(t)\right\|\right)_{L^{2}\left(\Gamma_{3}\right)}
โ‰ฅ(๐Ÿ(t),๐ฏโˆ’๐ฎห™(t))V\displaystyle\geq(\mathbf{f}(t),\mathbf{v}-\dot{\mathbf{u}}(t))_{V}

Finally, we substitute the constitutive law (3.1) in the previous inequality and gather the resulting inequality with the differential equation (3.7) and the initial conditions (3.8) to obtain the following variational formulation of Problem ๐’ซ\mathcal{P}.

Problem ๐’ซV\mathcal{P}^{V}. Find a displacement field ๐ฎ:Iโ†’V\mathbf{u}:I\rightarrow V and an surface state variable ฮฑ:Iโ†’L2(ฮ“3)\alpha:I\rightarrow L^{2}\left(\Gamma_{3}\right) such that ๐ฎ(0)=๐ฎ0,ฮฑ(0)=ฮฑ0\mathbf{u}(0)=\mathbf{u}_{0},\alpha(0)=\alpha_{0} and, for any tโˆˆIt\in I, the following hold:

(๐’œ๐œบ(๐ฎห™(t))+โ„ฌ๐œบ(๐ฎ(t))+โˆซ0t๐’ฆ(tโˆ’s)๐œบ(๐ฎห™(s))๐‘‘s,๐œบ(๐ฏ)โˆ’๐œบ(๐ฎห™(t)))Q\displaystyle\left(\mathcal{A}\boldsymbol{\varepsilon}(\dot{\mathbf{u}}(t))+\mathcal{B}\boldsymbol{\varepsilon}(\mathbf{u}(t))+\int_{0}^{t}\mathcal{K}(t-s)\boldsymbol{\varepsilon}(\dot{\mathbf{u}}(s))ds,\boldsymbol{\varepsilon}(\mathbf{v})-\boldsymbol{\varepsilon}(\dot{\mathbf{u}}(t))\right)_{Q}
+(p(uฮฝ(t)),vฮฝโˆ’uห™ฮฝ(t))L2(ฮ“3)\displaystyle+\left(p\left(u_{\nu}(t)\right),v_{\nu}-\dot{u}_{\nu}(t)\right)_{L^{2}\left(\Gamma_{3}\right)}
+(ฮผ(โ€–๐ฎห™ฯ„(t)โ€–;ฮฑ(t))p(uฮฝ(t)),โ€–๐ฏฯ„โ€–โˆ’โ€–๐ฎห™ฯ„(t)โ€–)L2(ฮ“3)\displaystyle+\left(\mu\left(\left\|\dot{\mathbf{u}}_{\tau}(t)\right\|;\alpha(t)\right)p\left(u_{\nu}(t)\right),\left\|\mathbf{v}_{\tau}\right\|-\left\|\dot{\mathbf{u}}_{\tau}(t)\right\|\right)_{L^{2}\left(\Gamma_{3}\right)}
โ‰ฅ(๐Ÿ(t),๐ฏโˆ’๐ฎห™(t))Vโˆ€vโˆˆV,\displaystyle\geq(\mathbf{f}(t),\mathbf{v}-\dot{\mathbf{u}}(t))_{V}\quad\forall v\in V,
ฮฑห™(t)=G(ฮฑ(t),โ€–๐ฎห™ฯ„(t)โ€–)\displaystyle\dot{\alpha}(t)=G\left(\alpha(t),\left\|\dot{\mathbf{u}}_{\tau}(t)\right\|\right)

Note that Problem ๐’ซV\mathcal{P}^{V} represents a system which couples a differential equation for the surface state variable with a variational inequality for displacement field. Therefore, following the notion introduced in [2], it represents a differential variational inequality. In the study of this problem we have the following existence and uniqueness result.

Theorem 4.1. Assume that (3.9)-(3.18) hold and, moreover, assume that

c02pโˆ—Lฮผโ‰คm๐’œ.c_{0}^{2}p^{*}L_{\mu}\leq m_{\mathcal{A}}. (4.4)

Then, Problem ๐’ซV\mathcal{P}^{V} has a unique solution with regularity

๐ฎโˆˆC1(I;V),ฮฑโˆˆC1(I;L2(ฮ“3)).\mathbf{u}\in C^{1}(I;V),\quad\alpha\in C^{1}\left(I;L^{2}\left(\Gamma_{3}\right)\right). (4.5)

A solution ( ๐ฎ,ฮฑ\mathbf{u},\alpha ) of Problem ๐’ซV\mathcal{P}^{V} is called a weak solution to the contact problem ๐’ซ\mathcal{P}. We conclude that Theorem 4.1 states the unique weak solvability of Problem ๐’ซ\mathcal{P}, under the smallness assumption (4.4) on the normal compliance function and the coefficient of friction.

5. Proof of Theorem 4.1

The proof of Theorem 4.1 is carried out in several steps. Everywhere below we assume that (3.9)-(3.18) hold and we consider the operator ๐’ฎ:C(I;V)โ†’C(I;V)\mathcal{S}:C(I;V)\rightarrow C(I;V) defined by

๐’ฎ๐ฐ(t)=โˆซ0t๐ฐ(s)๐‘‘s+๐ฎ0\mathcal{S}\mathbf{w}(t)=\int_{0}^{t}\mathbf{w}(s)ds+\mathbf{u}_{0} (5.1)

for all ๐ฐโˆˆC(I;V)\mathbf{w}\in C(I;V) and tโˆˆIt\in I. Note that

โ€–๐’ฎ๐ฐ1(t)โˆ’๐’ฎ๐ฐ2(t)โ€–Vโ‰คโˆซ0tโ€–๐ฐ1(s)โˆ’๐ฐ2(s)โ€–V๐‘‘s\left\|\mathcal{S}\mathbf{w}_{1}(t)-\mathcal{S}\mathbf{w}_{2}(t)\right\|_{V}\leq\int_{0}^{t}\left\|\mathbf{w}_{1}(s)-\mathbf{w}_{2}(s)\right\|_{V}ds (5.2)

for all ๐ฐ1,๐ฐ2โˆˆC(I;V)\mathbf{w}_{1},\mathbf{w}_{2}\in C(I;V) and tโˆˆIt\in I, and, therefore the operator ๐’ฎ\mathcal{S} is a historydependent operator. The first step in the proof of Theorem 4.1 is the following.

Lemma 5.1. (1) For each function ๐ฐโˆˆC(I;V)\mathbf{w}\in C(I;V), there exists a unique function ฮฑโˆˆC1(I;L2(ฮ“3))\alpha\in C^{1}\left(I;L^{2}\left(\Gamma_{3}\right)\right) such that

ฮฑห™(t)=G(ฮฑ(t),โ€–๐ฐฯ„(t)โ€–)โˆ€tโˆˆI,\displaystyle\dot{\alpha}(t)=G\left(\alpha(t),\left\|\mathbf{w}_{\tau}(t)\right\|\right)\quad\forall t\in I, (5.3)
ฮฑ(0)=ฮฑ0.\displaystyle\alpha(0)=\alpha_{0}. (5.4)

(2) There exists a history-dependent operator โ„›1:C(I;V)โ†’C(I;L2(ฮ“3))\mathcal{R}_{1}:C(I;V)\rightarrow C\left(I;L^{2}\left(\Gamma_{3}\right)\right) such that for all functions ๐ฐโˆˆC(I;V)\mathbf{w}\in C(I;V) and ฮฑโˆˆC(I;L2(ฮ“3))\alpha\in C\left(I;L^{2}\left(\Gamma_{3}\right)\right), the following statements are equivalent:
(a) ฮฑโˆˆC1(I;L2(ฮ“3))\alpha\in C^{1}\left(I;L^{2}\left(\Gamma_{3}\right)\right) and equalities (5.3)-(5.4) hold;
(b) ฮฑ(t)=ฮฑ0+โ„›1๐ฐ(t)\alpha(t)=\alpha_{0}+\mathcal{R}_{1}\mathbf{w}(t) for all tโˆˆIt\in I.

Proof. Let ๐ฐโˆˆC(I;V)\mathbf{w}\in C(I;V). Then, using assumptions (3.17), 3.18) it is easy to see that the function ฮฑ\alpha is a solution to the Cauchy problem (5.3)-(5.4) with regularity ฮฑโˆˆC1(I,L2(ฮ“3))\alpha\in C^{1}\left(I,L^{2}\left(\Gamma_{3}\right)\right) if and only if ฮฑโˆˆC(I,L2(ฮ“3))\alpha\in C\left(I,L^{2}\left(\Gamma_{3}\right)\right) and

ฮฑ(t)=ฮฑ0+โˆซ0tG(ฮฑ(s);โ€–๐ฐฯ„(s)โ€–)๐‘‘s.\alpha(t)=\alpha_{0}+\int_{0}^{t}G\left(\alpha(s);\left\|\mathbf{w}_{\tau}(s)\right\|\right)ds. (5.5)

Then Lemma 5.1 is a direct consequence of Theorem 2.2 applied with X=VX=V, Y=L2(ฮ“3)Y=L^{2}\left(\Gamma_{3}\right) and

A๐ฐโ‰กฮฑ0,๐’ข(t,๐ฐ,ฮฑ)=G(ฮฑ;โ€–๐ฐฯ„โ€–),A\mathbf{w}\equiv\alpha_{0},\quad\mathcal{G}(t,\mathbf{w},\alpha)=G\left(\alpha;\left\|\mathbf{w}_{\tau}\right\|\right), (5.6)

for all ๐ฐโˆˆV,ฮฑโˆˆL2(ฮ“3)\mathbf{w}\in V,\alpha\in L^{2}\left(\Gamma_{3}\right) and tโˆˆIt\in I.

We now state the following equivalence result whose proof is a direct consequence of Lemma 5.1 and definition (5.1).

Lemma 5.2. The couple ( ๐ฎ,ฮฑ\mathbf{u},\alpha ) is a solution of Problem ๐’ซV\mathcal{P}^{V} with regularity (4.5) if and only if there exists a function ๐ฐโˆˆC(I;V)\mathbf{w}\in C(I;V) such that

๐ฎ(t)=๐’ฎ๐ฐ(t)\displaystyle\mathbf{u}(t)=\mathcal{S}\mathbf{w}(t) (5.7)
ฮฑ(t)=ฮฑ0+โ„›1๐ฐ(t)\displaystyle\alpha(t)=\alpha_{0}+\mathcal{R}_{1}\mathbf{w}(t) (5.8)

and, moreover, for all tโˆˆIt\in I, the inequality below holds:

(๐’œ๐œบ(๐ฐ(t))+โ„ฌ๐œบ((๐’ฎ๐ฐ)(t))+โˆซ0t๐’ฆ(tโˆ’s)๐œบ(๐ฐ(s))๐‘‘s,๐œบ(๐ฏ)โˆ’๐œบ(๐ฐ(t)))Q\displaystyle\left(\mathcal{A}\boldsymbol{\varepsilon}(\mathbf{w}(t))+\mathcal{B}\boldsymbol{\varepsilon}((\mathcal{S}\mathbf{w})(t))+\int_{0}^{t}\mathcal{K}(t-s)\boldsymbol{\varepsilon}(\mathbf{w}(s))ds,\boldsymbol{\varepsilon}(\mathbf{v})-\boldsymbol{\varepsilon}(\mathbf{w}(t))\right)_{Q}
+(p((๐’ฎ๐ฐ)ฮฝ(t)),vฮฝโˆ’wฮฝ(t))L2(ฮ“3)\displaystyle+\left(p\left((\mathcal{S}\mathbf{w})_{\nu}(t)\right),v_{\nu}-w_{\nu}(t)\right)_{L^{2}\left(\Gamma_{3}\right)} (5.9)
+(ฮผ(โ€–๐ฐฯ„(t)โ€–;ฮฑ0+โ„›1๐ฐ(t))p((๐’ฎ๐ฐ)ฮฝ(t)),โ€–๐ฏฯ„โ€–โˆ’โ€–๐ฐฯ„(t)โ€–)L2(ฮ“3)\displaystyle+\left(\mu\left(\left\|\mathbf{w}_{\tau}(t)\right\|;\alpha_{0}+\mathcal{R}_{1}\mathbf{w}(t)\right)p\left((\mathcal{S}\mathbf{w})_{\nu}(t)\right),\left\|\mathbf{v}_{\tau}\right\|-\left\|\mathbf{w}_{\tau}(t)\right\|\right)_{L^{2}\left(\Gamma_{3}\right)}
โ‰ฅ(๐Ÿ(t),๐ฏโˆ’๐ฐ(t))Vโˆ€vโˆˆV\displaystyle\geq(\mathbf{f}(t),\mathbf{v}-\mathbf{w}(t))_{V}\quad\forall v\in V

Note that in (5.9) and below, (๐’ฎ๐ฐ)ฮฝ(t)(\mathcal{S}\mathbf{w})_{\nu}(t) represents the normal component of the element (๐’ฎ๐ฐ)(t)โˆˆV(\mathcal{S}\mathbf{w})(t)\in V. The next step in the proof of Theorem 4.1 consists to obtain the unique solvability of the variational inequality (5.9) for the velocity field ๐ฐ=๐ฎห™\mathbf{w}=\dot{\mathbf{u}}. We have the following existence and uniqueness result.

Lemma 5.3. There exists a unique solution ๐ฐ\mathbf{w} of (5.9). Moreover, the solution satisfies

๐ฐโˆˆC(I;V).\mathbf{w}\in C(I;V). (5.10)

Proof. We consider the product Hilbert space ฮ›=L2(ฮ“3)ร—Qร—L2(ฮ“3)\Lambda=L^{2}\left(\Gamma_{3}\right)\times Q\times L^{2}\left(\Gamma_{3}\right) and the set KK defined by

K={zโˆˆL2(ฮ“3):0โ‰คzโ‰คpโˆ— a.e. on ฮ“3}K=\left\{z\in L^{2}\left(\Gamma_{3}\right):0\leq z\leq p^{*}\text{ a.e. on }\Gamma_{3}\right\} (5.11)

We note that KK is a nonempty closed subset of the space L2(ฮ“3)L^{2}\left(\Gamma_{3}\right) and we denote by PK:L2(ฮ“3)โ†’KP_{K}:L^{2}\left(\Gamma_{3}\right)\rightarrow K the projection map on KK. Next, we define the operators A:Vโ†’V,โ„›2:C(I;V)โ†’C(I;Q),โ„›3:C(I;V)โ†’C(I;L2(ฮ“3))A:V\rightarrow V,\mathcal{R}_{2}:C(I;V)\rightarrow C(I;Q),\mathcal{R}_{3}:C(I;V)\rightarrow C\left(I;L^{2}\left(\Gamma_{3}\right)\right) and โ„›:C(I;V)โ†’C(I;ฮ›)\mathcal{R}:C(I;V)\rightarrow C(I;\Lambda) by equalities

(๐’œ๐ฎ,๐ฏ)V=(๐’œ๐œบ(๐ฎ),๐œบ(๐ฏ))Q\displaystyle(\mathcal{A}\mathbf{u},\mathbf{v})_{V}=(\mathcal{A}\boldsymbol{\varepsilon}(\mathbf{u}),\boldsymbol{\varepsilon}(\mathbf{v}))_{Q} (5.12)
โ„›2๐ฐ(t)=โ„ฌ๐œบ(๐’ฎ๐ฐ(t))+โˆซ0t๐’ฆ(tโˆ’s)๐œบ(๐ฐ(s))๐‘‘s\displaystyle\mathcal{R}_{2}\mathbf{w}(t)=\mathcal{B}\boldsymbol{\varepsilon}(\mathcal{S}\mathbf{w}(t))+\int_{0}^{t}\mathcal{K}(t-s)\boldsymbol{\varepsilon}(\mathbf{w}(s))ds (5.13)
โ„›3๐ฐ(t)=p((๐’ฎ๐ฐ)ฮฝ(t))\displaystyle\mathcal{R}_{3}\mathbf{w}(t)=p\left((\mathcal{S}\mathbf{w})_{\nu}(t)\right) (5.14)
โ„›๐ฐ(t)=(ฮฑ0+โ„›1๐ฐ(t),โ„›2๐ฐ(t),โ„›3๐ฐ(t))\displaystyle\mathcal{R}\mathbf{w}(t)=\left(\alpha_{0}+\mathcal{R}_{1}\mathbf{w}(t),\mathcal{R}_{2}\mathbf{w}(t),\mathcal{R}_{3}\mathbf{w}(t)\right) (5.15)

for all ๐ฎ,๐ฏโˆˆV,๐ฐโˆˆC(I;V)\mathbf{u},\mathbf{v}\in V,\mathbf{w}\in C(I;V) where, recall, โ„›1\mathcal{R}_{1} is the operator defined in Lemma 5.1. We also define the functional ฯ†:ฮ›ร—Vร—Vโ†’โ„\varphi:\Lambda\times V\times V\rightarrow\mathbb{R} by equality

ฯ†(๐€,๐ฐ,๐ฏ)=(๐ฒ,๐œบ(๐ฏ))Q+(z,vฮฝ)L2(ฮ“3)+(ฮผ(โ€–๐ฐฯ„โ€–;x)PKz,โ€–๐ฏฯ„โ€–)L2(ฮ“3)\varphi(\boldsymbol{\lambda},\mathbf{w},\mathbf{v})=(\mathbf{y},\boldsymbol{\varepsilon}(\mathbf{v}))_{Q}+\left(z,v_{\nu}\right)_{L^{2}\left(\Gamma_{3}\right)}+\left(\mu\left(\left\|\mathbf{w}_{\tau}\right\|;x\right)P_{K}z,\left\|\mathbf{v}_{\tau}\right\|\right)_{L^{2}\left(\Gamma_{3}\right)} (5.16)

for all ๐€=(x,๐ฒ,z)โˆˆฮ›\boldsymbol{\lambda}=(x,\mathbf{y},z)\in\Lambda and ๐ฐ,๐ฏโˆˆV\mathbf{w},\mathbf{v}\in V. With these data we consider the problem of finding a function ๐ฐ:Iโ†’V\mathbf{w}:I\rightarrow V such that, for all tโˆˆIt\in I, the following inequality holds:

(A๐ฐ(t),๐ฏโˆ’๐ฐ(t))V+ฯ†(โ„›๐ฐ(t),๐ฐ(t),๐ฏ)โˆ’ฯ†(โ„›๐ฐ(t),๐ฐ(t),๐ฐ(t))\displaystyle(A\mathbf{w}(t),\mathbf{v}-\mathbf{w}(t))_{V}+\varphi(\mathcal{R}\mathbf{w}(t),\mathbf{w}(t),\mathbf{v})-\varphi(\mathcal{R}\mathbf{w}(t),\mathbf{w}(t),\mathbf{w}(t))
โ‰ฅ(๐Ÿ(t),๐ฏโˆ’๐ฐ(t))Vโˆ€vโˆˆV\displaystyle\geq(\mathbf{f}(t),\mathbf{v}-\mathbf{w}(t))_{V}\quad\forall v\in V (5.17)

We use the bound (3.14) (e) to see that for any function ๐ฐโˆˆC(I;V)\mathbf{w}\in C(I;V) we have 0โ‰คp((๐’ฎ๐ฐ)ฮฝ(t))โ‰คpโˆ—0\leq p\left((\mathcal{S}\mathbf{w})_{\nu}(t)\right)\leq p^{*} a.e. on ฮ“3\Gamma_{3} for all tโˆˆIt\in I. Therefore, using definition (5.11) of the set KK it follows that PKp((๐’ฎ๐ฐ)ฮฝ(t))=p((๐’ฎ๐ฐ)ฮฝ(t))P_{K}p\left((\mathcal{S}\mathbf{w})_{\nu}(t)\right)=p\left((\mathcal{S}\mathbf{w})_{\nu}(t)\right) for all tโˆˆIt\in I. Using
this equality and the definitions (5.12)-(5.16) it is easy to see that a function ๐ฐโˆˆC(I;V)\mathbf{w}\in C(I;V) is a solution of (5.9) if and only if ๐ฐ\mathbf{w} is a solution of the inequality (5.17). For this reason, our aim in what follows is to prove the unique solvability of this problem and, to this end, we check the assumptions of Theorem 2.1 with X=VX=V and Y=ฮ›Y=\Lambda.

First, we use assumptions (3.9) to deduce that AA satisfies (2.5) with

mA=m๐’œ and MA=L๐’œ.m_{A}=m_{\mathcal{A}}\quad\text{ and }\quad M_{A}=L_{\mathcal{A}}. (5.18)

Let JโŠ‚I,tโˆˆJJ\subset I,t\in J and let ๐ฎ,๐ฏโˆˆC(I;V)\mathbf{u},\mathbf{v}\in C(I;V). Lemma 5.1 (2) guarantees that โ„›1\mathcal{R}_{1} is a history dependent operator and, therefore, there exists LJ1>0L_{J}^{1}>0 such that

โ€–โ„›1๐ฎ(t)โˆ’โ„›1๐ฏ(t)โ€–L2(ฮ“3)โ‰คLJ1โˆซ0tโ€–๐ฎ(s)โˆ’๐ฏ(s)โ€–V๐‘‘s\left\|\mathcal{R}_{1}\mathbf{u}(t)-\mathcal{R}_{1}\mathbf{v}(t)\right\|_{L^{2}\left(\Gamma_{3}\right)}\leq L_{J}^{1}\int_{0}^{t}\|\mathbf{u}(s)-\mathbf{v}(s)\|_{V}ds (5.19)

On the other hand, definition (5.13), assumptions (3.10), (3.11) and inequalities (5.2), 2.19) imply that

โ€–โ„›2๐ฎ(t)โˆ’โ„›2๐ฏ(t)โ€–Qโ‰ค(Lโ„ฌ+dmaxrโˆˆJโกโ€–๐’ฆ(r)โ€–๐โˆž)โˆซ0tโ€–๐ฎ(s)โˆ’๐ฏ(s)โ€–V๐‘‘s\left\|\mathcal{R}_{2}\mathbf{u}(t)-\mathcal{R}_{2}\mathbf{v}(t)\right\|_{Q}\leq\left(L_{\mathcal{B}}+d\max_{r\in J}\|\mathcal{K}(r)\|_{\mathbf{Q}_{\infty}}\right)\int_{0}^{t}\|\mathbf{u}(s)-\mathbf{v}(s)\|_{V}ds (5.20)

Finally, we use again inequality (5.2), assumption (3.14) and inequality (2.18) to deduce that

โ€–โ„›3๐ฎ(t)โˆ’โ„›3๐ฏ(t)โ€–L2(ฮ“3)โ‰คc0Lpโˆซ0tโ€–๐ฎ(s)โˆ’๐ฏ(s)โ€–V๐‘‘s\left\|\mathcal{R}_{3}\mathbf{u}(t)-\mathcal{R}_{3}\mathbf{v}(t)\right\|_{L^{2}\left(\Gamma_{3}\right)}\leq c_{0}L_{p}\int_{0}^{t}\|\mathbf{u}(s)-\mathbf{v}(s)\|_{V}ds (5.21)

We now combine inequalities (5.19)-(5.21) to obtain that

โ€–โ„›๐ฎ(t)โˆ’โ„›๐ฏ(t)โ€–ฮ›\displaystyle\|\mathcal{R}\mathbf{u}(t)-\mathcal{R}\mathbf{v}(t)\|_{\Lambda}
โ‰ค(LJ1+Lโ„ฌ+dmaxrโˆˆJโกโ€–๐’ฆ(r)โ€–๐โˆž+c0Lp)โˆซ0tโ€–๐ฎ(s)โˆ’๐ฏ(s)โ€–V๐‘‘s\displaystyle\leq\left(L_{J}^{1}+L_{\mathcal{B}}+d\max_{r\in J}\|\mathcal{K}(r)\|_{\mathbf{Q}_{\infty}}+c_{0}L_{p}\right)\int_{0}^{t}\|\mathbf{u}(s)-\mathbf{v}(s)\|_{V}ds (5.22)

which shows that the operator โ„›\mathcal{R} satisfies condition with

LJ=LJ1+Lโ„ฌ+dmaxrโˆˆJโกโ€–๐’ฆ(r)โ€–๐โˆž+c0Lp.L_{J}=L_{J}^{1}+L_{\mathcal{B}}+d\max_{r\in J}\|\mathcal{K}(r)\|_{\mathbf{Q}_{\infty}}+c_{0}L_{p}.

On the other hand, it is easy to see that that the functional ฯ†\varphi satisfies condition 2.7) (a). To satisfy condition 2.7) (b) let ๐€1=(x1,๐ฒ1,z1),๐€2=(x2,๐ฒ2,z2)โˆˆฮ›\boldsymbol{\lambda}_{1}=\left(x_{1},\mathbf{y}_{1},z_{1}\right),\boldsymbol{\lambda}_{2}=\left(x_{2},\mathbf{y}_{2},z_{2}\right)\in\Lambda and ๐ฐ1,๐ฐ2,๐ฏ1,๐ฏ2โˆˆV\mathbf{w}_{1},\mathbf{w}_{2},\mathbf{v}_{1},\mathbf{v}_{2}\in V. We use definition (5.16) to deduce that

ฯ†(๐€1,๐ฐ1,๐ฏ2)โˆ’ฯ†(๐€1,๐ฐ1,๐ฏ1)+ฯ†(๐€2,๐ฐ2,๐ฏ1)โˆ’ฯ†(๐€2,๐ฐ2,๐ฏ2)\displaystyle\varphi\left(\boldsymbol{\lambda}_{1},\mathbf{w}_{1},\mathbf{v}_{2}\right)-\varphi\left(\boldsymbol{\lambda}_{1},\mathbf{w}_{1},\mathbf{v}_{1}\right)+\varphi\left(\boldsymbol{\lambda}_{2},\mathbf{w}_{2},\mathbf{v}_{1}\right)-\varphi\left(\boldsymbol{\lambda}_{2},\mathbf{w}_{2},\mathbf{v}_{2}\right)
=(๐ฒ1โˆ’๐ฒ2,ฮต(๐ฏ2)โˆ’ฮต(๐ฏ1))Q+(z1โˆ’z2,v2ฮฝโˆ’v1ฮฝ)L2(ฮ“3)\displaystyle=\left(\mathbf{y}_{1}-\mathbf{y}_{2},\varepsilon\left(\mathbf{v}_{2}\right)-\varepsilon\left(\mathbf{v}_{1}\right)\right)_{Q}+\left(z_{1}-z_{2},v_{2\nu}-v_{1\nu}\right)_{L^{2}\left(\Gamma_{3}\right)} (5.23)
+(ฮผ(โ€–๐ฐ1ฯ„โ€–;x1)PKz1โˆ’ฮผ(โ€–๐ฐ2ฯ„โ€–;x2)PKz2,โ€–๐ฏ2ฯ„โ€–โˆ’โ€–๐ฏ1ฯ„โ€–)L2(ฮ“3).\displaystyle\quad+\left(\mu\left(\left\|\mathbf{w}_{1\tau}\right\|;x_{1}\right)P_{K}z_{1}-\mu\left(\left\|\mathbf{w}_{2\tau}\right\|;x_{2}\right)P_{K}z_{2},\left\|\mathbf{v}_{2\tau}\right\|-\left\|\mathbf{v}_{1\tau}\right\|\right)_{L^{2}\left(\Gamma_{3}\right)}.

Next, using the definition of the norm in the product space ฮ›\Lambda and the trace inequality (2.18), it is easy to see that

(๐ฒ1โˆ’๐ฒ2,๐œบ(๐ฏ2)โˆ’๐œบ(๐ฏ1))Qโ‰คโ€–๐€1โˆ’๐€2โ€–ฮ›โ€–๐ฏ1โˆ’๐ฏ2โ€–V,\displaystyle\left(\mathbf{y}_{1}-\mathbf{y}_{2},\boldsymbol{\varepsilon}\left(\mathbf{v}_{2}\right)-\boldsymbol{\varepsilon}\left(\mathbf{v}_{1}\right)\right)_{Q}\leq\left\|\boldsymbol{\lambda}_{1}-\boldsymbol{\lambda}_{2}\right\|_{\Lambda}\left\|\mathbf{v}_{1}-\mathbf{v}_{2}\right\|_{V}, (5.24)
(z1โˆ’z2,v2ฮฝโˆ’v1ฮฝ)L2(ฮ“3)โ‰คc0โ€–๐€1โˆ’๐€2โ€–ฮ›โ€–๐ฏ1โˆ’๐ฏ2โ€–V.\displaystyle\left(z_{1}-z_{2},v_{2\nu}-v_{1\nu}\right)_{L^{2}\left(\Gamma_{3}\right)}\leq c_{0}\left\|\boldsymbol{\lambda}_{1}-\boldsymbol{\lambda}_{2}\right\|_{\Lambda}\left\|\mathbf{v}_{1}-\mathbf{v}_{2}\right\|_{V}. (5.25)

We denote

ฮผ(โ€–๐ฐ1ฯ„โ€–;x1)=ฮผ1,ฮผ(โ€–๐ฐ2ฯ„โ€–;x2)=ฮผ2.\mu\left(\left\|\mathbf{w}_{1\tau}\right\|;x_{1}\right)=\mu_{1},\quad\mu\left(\left\|\mathbf{w}_{2\tau}\right\|;x_{2}\right)=\mu_{2}.

Then, using inequalities |ฮผ1|โ‰คฮผโˆ—,0โ‰คPKz2โ‰คpโˆ—\left|\mu_{1}\right|\leq\mu^{*},0\leq P_{K}z_{2}\leq p^{*} a.e. on ฮ“3\Gamma_{3}, guaranteed by (3.16) (d) and (5.11), respectively, combined with the nonexpansivity of the projection map and assumption 3.16) (b), it is easy to see that

(\displaystyle( ฮผ(โˆฅ๐ฐ1ฯ„โˆฅ;x1)PKz1โˆ’ฮผ(โˆฅ๐ฐ2ฯ„โˆฅ;x2)PKz2,โˆฅ๐ฏ2ฯ„โˆฅโˆ’โˆฅ๐ฏ1ฯ„โˆฅ)L2(ฮ“3)\displaystyle\left.\mu\left(\left\|\mathbf{w}_{1\tau}\right\|;x_{1}\right)P_{K}z_{1}-\mu\left(\left\|\mathbf{w}_{2\tau}\right\|;x_{2}\right)P_{K}z_{2},\left\|\mathbf{v}_{2\tau}\right\|-\left\|\mathbf{v}_{1\tau}\right\|\right)_{L^{2}\left(\Gamma_{3}\right)}
=\displaystyle= (ฮผ1(PKz1โˆ’PKz2),โ€–๐ฏ2ฯ„โ€–โˆ’โ€–๐ฏ1ฯ„โ€–)L2(ฮ“3)\displaystyle\left(\mu_{1}\left(P_{K}z_{1}-P_{K}z_{2}\right),\left\|\mathbf{v}_{2\tau}\right\|-\left\|\mathbf{v}_{1\tau}\right\|\right)_{L^{2}\left(\Gamma_{3}\right)}
+((ฮผ1โˆ’ฮผ2)PKz2,โ€–๐ฏ2ฯ„โ€–โˆ’โ€–๐ฏ1ฯ„โ€–)L2(ฮ“3)\displaystyle\quad+\left(\left(\mu_{1}-\mu_{2}\right)P_{K}z_{2},\left\|\mathbf{v}_{2\tau}\right\|-\left\|\mathbf{v}_{1\tau}\right\|\right)_{L^{2}\left(\Gamma_{3}\right)}
โ‰ค\displaystyle\leq ฮผโˆ—(|PKz1โˆ’PKz2|,โ€–๐ฏ1โˆ’๐ฏ2โ€–)L2(ฮ“3)\displaystyle\mu^{*}\left(\left|P_{K}z_{1}-P_{K}z_{2}\right|,\left\|\mathbf{v}_{1}-\mathbf{v}_{2}\right\|\right)_{L^{2}\left(\Gamma_{3}\right)}
+pโˆ—(|ฮผ1โˆ’ฮผ2|,โ€–๐ฏ1โˆ’๐ฏ2โ€–)L2(ฮ“3)\displaystyle\quad+p^{*}\left(\left|\mu_{1}-\mu_{2}\right|,\left\|\mathbf{v}_{1}-\mathbf{v}_{2}\right\|\right)_{L^{2}\left(\Gamma_{3}\right)}
โ‰ค\displaystyle\leq ฮผโˆ—โ€–PKz1โˆ’PKz2โ€–L2(ฮ“3)โ€–๐ฏ1โˆ’๐ฏ2โ€–L2(ฮ“3)d\displaystyle\mu^{*}\left\|P_{K}z_{1}-P_{K}z_{2}\right\|_{L^{2}\left(\Gamma_{3}\right)}\left\|\mathbf{v}_{1}-\mathbf{v}_{2}\right\|_{L^{2}\left(\Gamma_{3}\right)^{d}}
+pโˆ—Lฮผ(โ€–๐ฐ1โˆ’๐ฐ2โ€–+|x1โˆ’x2|,โ€–๐ฏ1โˆ’๐ฏ2โ€–)L2(ฮ“3)\displaystyle\quad+p^{*}L_{\mu}\left(\left\|\mathbf{w}_{1}-\mathbf{w}_{2}\right\|+\left|x_{1}-x_{2}\right|,\left\|\mathbf{v}_{1}-\mathbf{v}_{2}\right\|\right)_{L^{2}\left(\Gamma_{3}\right)}
โ‰ค\displaystyle\leq ฮผโˆ—โ€–z1โˆ’z2โ€–L2(ฮ“3)โ€–๐ฏ1โˆ’๐ฏ2โ€–L2(ฮ“3)d\displaystyle\mu^{*}\left\|z_{1}-z_{2}\right\|_{L^{2}\left(\Gamma_{3}\right)}\left\|\mathbf{v}_{1}-\mathbf{v}_{2}\right\|_{L^{2}\left(\Gamma_{3}\right)^{d}}
+pโˆ—Lฮผ(โ€–๐ฐ1โˆ’๐ฐ2โ€–L2(ฮ“3)d+โ€–x1โˆ’x2โ€–L2(ฮ“3))โ€–๐ฏ1โˆ’๐ฏ2โ€–L2(ฮ“3)d\displaystyle\quad+p^{*}L_{\mu}\left(\left\|\mathbf{w}_{1}-\mathbf{w}_{2}\right\|_{L^{2}\left(\Gamma_{3}\right)^{d}}+\left\|x_{1}-x_{2}\right\|_{L^{2}\left(\Gamma_{3}\right)}\right)\left\|\mathbf{v}_{1}-\mathbf{v}_{2}\right\|_{L^{2}\left(\Gamma_{3}\right)^{d}}

Therefore, using again the definition of the norm in the product space ฮ›\Lambda and the trace inequality (2.18) yields

(ฮผ(โ€–๐ฐ1ฯ„โ€–;x1)PKz1โˆ’ฮผ(โ€–๐ฐ2ฯ„โ€–;x2)PKz2,โ€–๐ฏ2ฯ„โ€–โˆ’โ€–๐ฏ1ฯ„โ€–)L2(ฮ“3)\displaystyle\left(\mu\left(\left\|\mathbf{w}_{1\tau}\right\|;x_{1}\right)P_{K}z_{1}-\mu\left(\left\|\mathbf{w}_{2\tau}\right\|;x_{2}\right)P_{K}z_{2},\left\|\mathbf{v}_{2\tau}\right\|-\left\|\mathbf{v}_{1\tau}\right\|\right)_{L^{2}\left(\Gamma_{3}\right)}
โ‰คc0ฮผโˆ—โ€–๐€1โˆ’๐€2โ€–ฮ›โ€–๐ฏ1โˆ’๐ฏ2โ€–V\displaystyle\leq c_{0}\mu^{*}\left\|\boldsymbol{\lambda}_{1}-\boldsymbol{\lambda}_{2}\right\|_{\Lambda}\left\|\mathbf{v}_{1}-\mathbf{v}_{2}\right\|_{V} (5.26)
+c02pโˆ—Lฮผโ€–๐ฐ1โˆ’๐ฐ2โ€–Vโ€–๐ฏ1โˆ’๐ฏ2โ€–V+c0pโˆ—Lฮผโ€–๐€1โˆ’๐€2โ€–ฮ›โ€–๐ฏ1โˆ’๐ฏ2โ€–V.\displaystyle\quad+c_{0}^{2}p^{*}L_{\mu}\left\|\mathbf{w}_{1}-\mathbf{w}_{2}\right\|_{V}\left\|\mathbf{v}_{1}-\mathbf{v}_{2}\right\|_{V}+c_{0}p^{*}L_{\mu}\left\|\boldsymbol{\lambda}_{1}-\boldsymbol{\lambda}_{2}\right\|_{\Lambda}\left\|\mathbf{v}_{1}-\mathbf{v}_{2}\right\|_{V}.

We now combine equality (5.23) with inequalities (5.24)-(5.26) to find that

ฯ†(๐€1,๐ฐ1,๐ฏ2)โˆ’ฯ†(๐€1,๐ฐ1,๐ฏ1)+ฯ†(๐€2,๐ฐ2,๐ฏ1)โˆ’ฯ†(๐€2,๐ฐ2,๐ฏ2)\displaystyle\varphi\left(\boldsymbol{\lambda}_{1},\mathbf{w}_{1},\mathbf{v}_{2}\right)-\varphi\left(\boldsymbol{\lambda}_{1},\mathbf{w}_{1},\mathbf{v}_{1}\right)+\varphi\left(\boldsymbol{\lambda}_{2},\mathbf{w}_{2},\mathbf{v}_{1}\right)-\varphi\left(\boldsymbol{\lambda}_{2},\mathbf{w}_{2},\mathbf{v}_{2}\right)
โ‰ค(1+c0+c0ฮผโˆ—+c0pโˆ—Lฮผ)โ€–๐€1โˆ’๐€2โ€–ฮ›โ€–๐ฏ1โˆ’๐ฏ2โ€–V\displaystyle\leq\left(1+c_{0}+c_{0}\mu^{*}+c_{0}p^{*}L_{\mu}\right)\left\|\boldsymbol{\lambda}_{1}-\boldsymbol{\lambda}_{2}\right\|_{\Lambda}\left\|\mathbf{v}_{1}-\mathbf{v}_{2}\right\|_{V} (5.27)
+c02pโˆ—Lฮผโˆฅ๐ฐ1โˆ’๐ฐ2โˆฅVโˆฅโˆ’๐ฏ1๐ฏ2โˆฅV.\displaystyle\quad+c_{0}^{2}p^{*}L_{\mu}\left\|\mathbf{w}_{1}-\mathbf{w}_{2}\right\|_{V}\left\|{}_{\mathbf{v}_{1}}-\mathbf{v}_{2}\right\|_{V}.

This inequality shows that the functional ฯ†\varphi satisfies condition (2.7) (b) with

c1=1+c0+c0ฮผโˆ—+c0pโˆ—Lฮผ and c2=c02pโˆ—Lฮผ.c_{1}=1+c_{0}+c_{0}\mu^{*}+c_{0}p^{*}L_{\mu}\quad\text{ and }\quad c_{2}=c_{0}^{2}p^{*}L_{\mu}. (5.28)

Therefore, it follows from (5.18), (5.28) and (4.4) that the smallness condition (2.9) holds. Finally, taking into account the regularity (4.2) we find that (2.8) holds, too. We are now in a position to apply Theorem 2.1 and we deduce in this way that inequality 5.17 has a unique solution ๐ฐโˆˆC(I;V)\mathbf{w}\in C(I;V), which completes the proof.

We now have all the ingredients to provide the proof of Theorem 4.1.
Proof of Theorem 4.1. Let ๐ฐ\mathbf{w} denote the unique solution of inequality (5.9) obtained in Lemma 5.3 and let ๐ฎ=๐’ฎ๐ฐ,ฮฑ=ฮฑ0+โ„›1๐ฐ\mathbf{u}=\mathcal{S}\mathbf{w},\alpha=\alpha_{0}+\mathcal{R}_{1}\mathbf{w}. Then, Lemma 5.2 implies that ( ๐ฎ,ฮฑ\mathbf{u},\alpha ) is a solution of Problem ๐’ซV\mathcal{P}^{V}. This proves the existence part of the theorem. The uniqueness of the solution is now a consequence of the unique solvability of the variational inequality (5.9), guaranteed by Lemma 5.3, combined with the equivalence result in Lemma 5.2.

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Flavius Pฤƒtrulescu
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O. Box 68-1, 400110 Cluj-Napoca, Romania

E-mail address: fpatrulescu@ictp.acad.ro
Mircea T. Sofonea
Laboratoire de Mathรฉmatiques et Physique, Universitรฉ de Perpignan Via Domitia, 52 Avenue de Paul Alduy, 66860 Perpignan, France

E-mail address: sofonea@univ-perp.fr

2017

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