Solvability of a class of thermal dynamical contact problems with subdifferential conditions

We study a class of dynamic thermal sub-differential contact 
problems with friction, for long memory visco-elastic materials, 
which can be put into a general model of 
 system defined by a second order evolution inequality, 
coupled with a first order evolution equation. We present and 
establish an existence and uniqueness result, by using general 
results on first order evolution inequality, with monotone 
operators and fixed point methods.


1.
Introduction. Despite of numerous recent progress, contact mechanics still remains a rich domain of various new problems, and the literature devoted to the subject is more and more extensive. An early attempt at the study of contact problems for elastic viscoelastic materials within the mathematical analysis framework was introduced in the pioneering reference works [5,7,13,11,15]. More recently, the dynamic Signorini's problem for a cracked viscoelastic body is studied in [6]. Mathematical analysis of unilateral contact problems involving static, quasi-static and dynamic process may be found in the recent self-contained book [8]. Further extensions to non convex contact conditions with non-monotone and possible multivalued constitutive laws led to the active domain of non-smooth mechanics within the framework of the so-called hemivariational inequalities, for a mathematical as well as mechanical treatment we refer to [9,16]. Quasi-static contact problems, by taking into account the parameter of the temperature field, were analyzed in [1], where the friction is described by a general normal damped response condition, there the existence and uniqueness of weak solution has been established. The existence of solutions for thermal viscoelastic dynamic contact problems with Coulomb friction law were treated in [8]. This work is a companion paper of the results obtained in [4]. In [4] we studied a class of dynamical long memory viscoelastic problems, where the contact is bilateral and frictional obeying the Tresca's law. The variational formulation of the problem led to a second order evolution inequality on the displacement field. Then an existence and uniqueness result of "weak" solution on displacement has been proved. Here we consider a class of dynamical long memory viscoelastic problems, the new feature in this paper is that the contact is defined by a general sub-differential condition on the velocity, and with heat exchange, putting then the problem into a coupled system, defined by a second order evolution inequality on the displacement field, and a differential equation which governs the evolution of the temperature field. On the other hand we investigate an existence and uniqueness result of "strong" solution on displacement and temperature fields, i.e. solution with more regularity. For this proposal we specify stronger assumptions on the data and operators, and use a new method based on a version of first order evolution inequality with monotonicity, convexity and fixed point theory.
The paper is organized as follows. In Section 2 we describe the mechanical problem, specify the assumptions on the data to derive the variational formulation, and then we state our main existence and uniqueness result. In Section 3, we give the proof of the claimed result.
2. Statement of the problem. In this section we study a class of thermal contact problems with sub-differential conditions, for long memory visco-elastic materials. We describe the mechanical problems, list the assumptions on the data and derive the corresponding variational formulations. Then we state an existence and uniqueness result on displacement and temperature fields, which we will prove in the next section. The physical setting is as follows. A visco-elastic body occupies the domain Ω with surface Γ that is partionned into three disjoint measurable parts Γ 1 , Γ 2 and Γ 3 , such that meas(Γ 1 ) > 0. Let [0, T ] be the time interval of interest, where T > 0. The body is clamped on Γ 1 × (0, T ) and therefore the displacement field vanishes there. We also assume that a volume force of density f 0 acts in Ω × (0, T ) and that surface tractions of density f 2 act on Γ 2 × (0, T ). The body may come into contact with an obstacle, the foundation, over the potential contact surface Γ 3 . The model of the contact is specified by a general sub-differential boundary condition, where thermal effects may occur in the frictional contact with the basis. We are interested in the dynamic evolution of the body. Let us recall now some classical notations, see e.g. [7,14] for further details. We denote by S d the space of second order symmetric tensors on R d (d = 2, 3), while " · " and | · | will represent the inner product and the Euclidean norm on S d and R d .
Let Ω ⊂ R d be a bounded domain with a Lipschitz boundary Γ and let ν denote the unit outer normal on Γ. Everywhere in the sequel the indexes i and j run from 1 to d, summation over repeated indices is implied and the index that follows a comma represents the partial derivative with respect to the corresponding component of the independent variable. We also use the following notation: Here ε : H 1 −→ H and Div : H 1 −→ H are the deformation and the divergence operators, respectively, defined by : The spaces H, H, H 1 and H 1 are real Hilbert spaces endowed with the canonical inner products given by : 10) Here, (2.1) is the Kelving Voigt's long memory thermo-visco-elastic constitutive law of the body, σ the stress tensor, A is the viscosity operator, G for the elastic operator, C e := (c ij ) represents the thermal expansion tensor, and B is the so called tensor of relaxation which defines the long memory of the material, as an important particular case, when B ≡ 0, we find again the usual visco-elasticity of short memory. In (2.2) is the dynamic equation of motion where the mass density ̺ ≡ 1. On the contact surface, the general relation (2.5) is a sub-differential boundary condition, where D(Ω) d ⊂ U ⊂ H 1 represents the set of contact admissible test functions, σν denotes the Cauchy stress vector on the contact boundary and ϕ : Γ 3 × R d −→ R is a given function. Various situations may be modelled by such a condition, and some concrete examples will be recalled below. The differential equation (2.6) describes the evolution of the temperature field, where K c := (k ij ) represents the thermal conductivity tensor, q(t) the density of volume heat sources. The associated temperature boundary condition is given by (2.7), where θ R is the temperature of the foundation, and k e is the heat exchange coefficient between the body and the obstacle. Finally, u 0 , v 0 , θ 0 represents the initial displacement, velocity and temperature, respectively. To derive the variational formulation of the mechanical problems (2.1)-(2.10) we need additional notations. Thus, let V denote the closed subspace of H 1 defined by Since meas Γ 1 > 0, Korn's inequality holds : there exists C K > 0 which depends only on Ω and Γ 1 such that A proof of Korn's inequality may be found in [14], p.79. On V we consider the inner product given by It follows that · H1 and · V are equivalent norms on V and therefore (V, · V ) is a real Hilbert space. Moreover, by the Sobolev's trace theorem, we have a constant C 0 > 0 depending only on Ω, Γ 1 , and Γ 3 such that In the study of the mechanical problem (2.1)-(2.10), we assume that the viscosity operator A : is linear on the second variable and satisfies the usual properties of ellipticity and symmetry, i.e.
The elasticity operator G : We suppose the body forces and surface tractions satisfy For the thermal tensors and the heat sources density, we suppose that The boundary thermic data satisfy The thermal conductivity tensor verifies the usual symmetry and ellipticity : for some c k > 0 and for all (ξ i ) ∈ R d , Finally we have to put technical assumptions on the initial data and the subdifferential condition on the contact surface, in order to use classical results on first order set valued evolution equations. Many various possibilities could be considered (see e.g. ( [2,3,12]). Here we use a general theorem taken in ( [12]) p. 46, in a simplified case, which is enough for our proposal and applications. We assume that the initial data satisfy the conditions On the contact surface, the following frictional contact function (iii) for all sequence (w n ) and w in W 1,2 (0, T ; V ) such that Here ψ ′ n (v) denotes the Fréchet derivative of ψ n at v. To continue, using Green's formula, we obtain the variational formulation of the mechanical problem Q in abstract form as follows.
Here, the operators and functions A, Before giving the proof, we show in the following two typical examples of subdifferential conditions. Example 1. Tresca's friction law.
Here g represents the friction bound, i.e., the magnitude of the limiting friction traction at which slip begins, with g ∈ L ∞ (Γ 3 ), g ≥ 0 a.e. on Γ 3 . We deduce the admissible displacement space: V := {w ∈ H 1 ; with w = 0 on Γ 1 ; w ν = 0 on Γ 3 }, and the sub-differential contact function: We have then is well defined on V with the property: for some c > 0, Let us show that assumptions in (2.19) are verified.
Firstly it is clear that ψ : V −→ R is convex. By using the continuous embedding from V into L 2 (Γ 3 ) d and the last inequality, we find that ψ is Lipschitz continuous on V.
This gives (2.19)(i). To approximate the function ψ, we use the sequence Now we claim that ψ n ( n ∈ N * ) is Frechet differentiable and the Frechet derivative of ψ n is given by Indeed. Let fix n ∈ N * and v ∈ V . For any h ∈ V : We obtain as h −→ 0 V : where O(1) denotes some scalar function of h bounded in a neighborhood of 0 V .

SAMIR ADLY, OANH CHAU AND MOHAMED ROCHDI
This ends the proof of the claim as is a linear continuous functional.
To continue, we deduce from the Frechet derivative's formula that ψ n is of class C 1 . Direct algebraic computations show that for all α ≥ 0, β ≥ 0 such that α + β = 1, and for all reals x and y, n ≥ 1: Then ψ n is convex for all n ≥ 1. The convergence property in (2.19)(ii) follows from Lebesgue's dominated convergence theorem.
To justify (2.19)(iii), consider a sequence (w n ) and w in W 1,2 (0, T ; V ) satisfying Now using the compact embedding theorem, we obtain Because of the continuous embedding from V into L 2 (Γ 3 ) d , we deduce that As ψ is Lipschitz continuous on V , we have that the functional is well defined and is continuous. Consequently which gives the property (2.19)(iii). Finally (2.19)(iv) is immediate from the formula stated for the Frechet derivative of ψ n (see (2.21)).

Example 2. Thermal contact problem with normal damped response and Tresca's friction law
The normal damped response contact condition with Tresca's friction law (see e.g. [1]) is defined by: Here 0 < r < 1 and g, k 0 ∈ L ∞ (Γ 3 ), g ≥ 0, k 0 ≥ 0. The coefficient k 0 represents the hardness of the foundation, and g the friction threshold. The admissible displacement space is given by: and the sub-differential contact function Then denoting by p := r + 1, we have the contact function well defined on V by Here 1 < p < 2. Then the mapping x ≥ 0 −→ x p is convex, which implies that ψ is convex on V . Using the continuous embeddings from V into L 2 (Γ 3 ) d , and from L 2 (Γ 3 ) into L p (Γ 3 ), we verify also that ψ is Lipschitz continuous on V.
To approximate the function ψ we use the sequence We verify that the Frechet derivative of ψ n is given by From the fact that the mapping x ≥ 0 −→ x p is convex increasing, and using (2.22), we verify that ψ n is convex for all n ≥ 1. Similarly we have the conditions (ii)-(iv) in (2.19) by the same arguments in the previous example. Then we conclude that the assumptions in (2.19) are also satisfied in this example.
3. Proof of Theorem 1. The idea is to bring the second order inequality to a first order inequality, using monotone operator, convexity and fixed point arguments, and will be carried out in several steps.
Let us introduce the velocity variable v =u.

SAMIR ADLY, OANH CHAU AND MOHAMED ROCHDI
The system in Problem QV is then written for a.e. t ∈ (0, T ): Various abstract formulations concerning the existence and uniqueness result on parabolic variational inequalities of the second kind, could be found in the literature, depending on the assumptions on the operators and data (see e.g. [3,7,12]). We will use a version taken in ( [7]) p. 46, which is sufficient for our proposal, and which we recall as follows.
Here and below, we denote by c > 0 a generic constant, whose value may change from lines to lines. Lemma 2. For all η ∈ W, there exists an unique Moreover, ∃c > 0 such that ∀η 1 , η 2 ∈ W: Proof. The existence and uniqueness result verifying (3.3) follows from classical result on first order evolution equation, which can be seen as a particular case of Theorem 2 applied to the Gelfand evolution triple (see e.g. [17] p . 416).