Coercive limits for a subclass of monotone constitutive equations in the theory of inelastic

material behaviour of metals

Krzysztof Chełminski^{∗} Technical University of Darmstadt

Abstract

We prove existence and uniqueness of strong global in time solutions for a subclass of monotone constitutive equations in the theory of inelastic material behaviour of metals without the coercivity assumption for the free energy function. We approximate noncoercive models by a sequence of coercive problems and prove the convergence result.

AMS classification: 35Q72, 73F99

1 Introduction

Systems of equations describing inelastic deformation of metals consist of partial differential equations resulting from mechanical balance laws and of a set of so called constitutive equations, which are always dependent of the material under consideration. One way to derive the constitutive relations is to assume that there exists a finite set of internal variables, whose evolution uniquely determine the state of the material. Therefore in the literature we find constitutive equations in the form of ordinary (often nonlinear) differential equations or differential inclusions for the internal variables. Thus we study here a linear system of partial differential equations coupled with a nonlinear system of ordinary differential equations (or general differential inclusions). Mathematical analysis of such systems of equations began in the seventies with works of J.J. Moreau [21], B. Halphen and Nguyen Quoc Son [14], and G. Duvant and J.L. Lions [12]. Then the research was continued by many authors, for example P. Suquet [25], R. Temam [27], P. LeTallec [18]. All the above mentioned articles or books deal with special constitutive equations or with a “small” class thereof. H.D. Alber in [3] was the first to try to classify constitutive equations applied in the theory of inelastic material behaviour of metals. The author defined the class of constitutive relations “of monotone type”, for which the natural mathematical tool to study such equations is the theory of evolution equations for monotone operators. With the theory of H. Brezis [6] H.D. Alber proved existence and uniqueness of strong, global in time solutions for all constitutive equations of monotone type provided the coercivity assumption for the free energy function. However there are many examples of constitutive equations for which the above assumption fails. This is the starting point of this article. We describe here a subclass of constitutive equations of monotone type without the coercivity assumption, which we can approximate by coercive problems of monotone type.

∗Supported by Sonderforschungsbereich 298

In Section 2 using the notations from [3] we formulate the initial boundaryvalue problem and define the constitutive equations of monotone type. Section 3 presents the existence theory for models with constitutive equations of monotone type, provided that the free energy function is positive definite. All the results are obtained in [3]. In Section 4 we study a noncoercive example – the monotone model of Bodner-Partom. Using the ideas from Section 4 we define in Section 5 the class of self-controlling models and prove existence of global strong solutions for this class. In section 6 we define another coercive approximation process, which we use in Section 7 to study a noncoercive and nonself-controlling example. In Section 8 we give ideas to extend the method from Section 5.

2 Formulation of the initial-boundary value problem

In this section we give a precise formulation of the inital-boundary value-problem, which we are studying, and define the class of constitutive equations of monotone type.

Let a solid body occupy a bounded domain Ω ⊂ ℝ^{3} with the smooth boundary ∂Ω . We denote by u(x, t) ∈ ℝ^{3} the displacement vector of the mass point x ∈ Ω at time t ≥ 0 . Let us denote by S^{3} the set of symmetric 3 × 3 matrices. The Cauchy stress tensor T (x, t) at the point x ∈ Ω at time t ≥ 0 belongs to S^{3}. The balance of momentum in the homogeneous case (with vanishing external forces) yields

ρu_{tt}(x, t) = div_{x} T (x, t) ,  (2.1)

where ρ > 0 is the density, which we assume to be constant. The subscripts t and x denote the partial derivatives with respect to t or x. The fundamental idea to describe an inelastic deformation of a solid body with the theory of internal variables is to assume that there exists the inelastic strain tensor ε^{p}(x, t) ∈ S^{3}, such that the whole strain tensor ε(x, t) ∈ S^{3} is given as the sum

ε(x, t) = ε(x, t) − ε^{p}(x, t) + ε^{p}(x, t) ,  (2.2)

where the difference ε(x, t) − ε^{p}(x, t) should be represented only by the pure elastic deformation. Thus for the elastic part of the strain tensor we assume that the following linear constitutive relation holds

T (x, t) = D(ε(x, t) − ε^{p}(x, t)) ,  (2.3)

where D : S^{3} → S^{3} is the elasticity tensor, which we assume to be constant, symmetric and positive definite. Moreover we use in this article the assumption of small deformations, which allows us to write that

ε(x, t) = \frac{1}{2}(∇u(x, t) + ∇^{T} u(x, t)) .  (2.4)

The partial differential equation (2.1) with the constitutive equation (2.3), where ε is given by (2.4), describe with given ε^{p} the classical theory of linear elasticity. Obviously the inelastic strain tensor is an unkown function, and to close our system of equations we should additionally derive the evolution of ε^{p}. Thus let us denote by z(x, t) = (ε^{p}(x, t), z̃(x, t)) ∈ S^{3} × ℝ^{N−9} the vector of internal variables, where the first nine components are defined by the inelastic strain tensor and the other components of the vector z describe various properties of the material under consideration, for example, isotropic or directional hardening of the metal. The evolution of the vector z is given by the inelastic, or sometimes called plastic, constitutive equations

z_{t}(x, t) ∈ f(ε(x, t), z(x, t)) ,  (2.5)

where the multivalued function f : D(f) ⊂ S^{3} × ℝ^{N} → P(ℝ^{N}) with domain D(f) = {(ε, z) ∈ S^{3} × ℝ^{N} : f(ε, z) ≠ ∅} is given. Let us denote by B the linear operator B : ℝ^{N} → S^{3} defined by the formula

Bz = \frac{1}{2}(z_{p} + z_{p}^{T}) ,  (2.6)

where z_{p} is the 3 × 3 matrix formed by the first nine components of the vector z. Thus the system of equations in the theory of inelastic material behaviour of metals has the form

\begin{aligned}ρu_{tt}(x, t) & = & div_{x} T (x, t) , \\ T (x, t) & = & D(ε(x, t) − Bz(x, t)) , \\ z_{t}(x, t) & ∈ & f(ε(x, t), z(x, t)) . \end{aligned}  (GS)

Moreover we assume that the homogeneous Dirichlet boundary condition

u(x, t) = 0 for x ∈ ∂Ω and t ≥ 0 ,  (2.7)

or the homogeneous Neumann boundary condition

T (x, t) n(x) = 0 for x ∈ ∂Ω and t ≥ 0  (2.8)

is satisfied, where n(x) denotes the exterior unit normal vector to ∂Ω at x ∈ ∂Ω . Finally the initial conditons are

\begin{aligned}u(x, 0) & = & u^{0}(x) , \\ u_{t}(x, 0) & = & u^{1}(x) , \\ z(x, 0) & = & z^{0}(x) \end{aligned}  (IC)

with given functions u^{0}, u^{1} : Ω → ℝ^{3} , z^{0} : Ω → ℝ^{N}.

Before we start with mathematical analysis of the system (GS) we define a class of admissible inelastic constitutive equations. We require that there exists a function ψ : D(f) ⊂ S^{3} × ℝ^{N} → ℝ_{+} satisfying for all (ε, z) ∈ D(f) :

ρ∇_{ε}ψ(ε, z) = T ,  (2.9)
∀z^{∗} ∈ f(ε, z) ρ∇_{z}ψ(ε, z) · z^{∗} ≤ 0 .  (2.10)

The function ψ is called free energy and the existence of such a function follows from thermodynamic considerations. The inequality (2.10) restricts the class of multivalued functions f and, as we will see below, yields dissipation to the problem (GS). Therefore in the literature the inequality (2.10) is called the dissipation inequality. Let us assume that u : Ω × (0, ∞) → ℝ^{3} , z : Ω × (0, ∞) → ℝ^{3} is a sufficiently smooth (so that the integrals and derivatives below are well defined) solution of the system (GS) satisfying the boundary condition (2.7) or (2.8) with the free energy ψ satisfying (2.9) and (2.10). Then we have

\frac{d}{dt} ∫_{Ω} ρ\{\frac{1}{2}|u_{t}(x, t)|^{2} + ψ(ε(x, t), z(x, t))\} dx

= ∫_{Ω} {ρu_{t}(x, t)u_{tt}(x, t) + ρ∇_{ε}ψ(ε(x, t), z(x, t)) ε_{t}(x, t)

+ ρ∇_{z}ψ(ε(x, t), z(x, t)) z_{t}(x, t)} dx

= ∫_{Ω} {ρu_{t}(x, t)u_{tt}(x, t) + T (x, t) · ∇u_{t}(x, t)

+ ρ∇_{z}ψ(ε(x, t), z(x, t)) z_{t}(x, t)} dx

= ∫_{∂Ω} T (x, t) · n(x) · u_{t}(x, t) dS(x)
+ ∫_{Ω} ρ∇_{z}ψ(ε(x, t), z(x, t)) z_{t}(x, t) dx ≤ 0  (2.11)

which denotes that the sum of the kinetic energy and the free energy do not increase in time. This dissipation effect is naturally a consequence of existence of a function ψ satisfying (2.9) and (2.10). Therefore we should describe all such functions. The requirement (2.9), the constitutive equation (GS2) and the properties of D yield

ρψ(ε, z) = \frac{1}{2}[D(ε − Bz)](ε − Bz) + ψ_{1}(z) .  (2.12)

Thus necessarily ψ must be quadratic in the variable ε − Bz, and the function ψ_{1} should be chosen so, that (2.10) holds. But it is not so easy to describe all functions ψ, which satisfy the dissipation inequality (2.10). Therefore in this article we restrict our considerations only to a class of multivalued functions f for which (2.10) is always satisfied.

Let us assume that there exists a monotone multifunction g : D(g) ⊂ ℝ^{N} → P(ℝ^{N}) with 0 ∈ g(0) , such that

f(ε, z) = g(−ρ∇_{z}ψ(ε, z))  (2.13)

where ψ is the free energy. To see that such functions f satisfy (2.10), we obtain from the monotonicity of g

∀_{z^{∗}∈f(ε,z)} ρ∇_{z}ψ(ε, z) · z^{∗} = −(−ρ∇_{z}ψ(ε, z) − 0) · (z^{∗} − 0) ≤ 0

while z^{∗} ∈ g(−ρ∇_{z}ψ(ε, z)) and 0 ∈ g(0) .

Next we want to obtain a similar inequality to (2.11) (dissipation of energy) for the difference of two solutions of the problem (GS) with f satisfying (2.13). This can be done if we additionally assume that the free energy function ψ is a positive semi-definite quadratic form in all variables. Thus ψ must be of the form

ψ(ε, z) = \frac{1}{2}[D(ε − Bz)] · (ε − Bz) + \frac{1}{2}(Lz) · z ,  (2.14)

where L is a symmetric positive semi-definite N × N matrix. Then the gradient of ψ with respect to z is linear and has the form

−ρ∇_{z}ψ(ε, z) = B^{T} D(ε − Bz) − Lz =^{df} \overline{L}ε − Mz

with linear mappings \overline{L} = B^{T} D and M = B^{T} DB + L. Note that the assumption (2.14) restricts again the class of functions f , which we consider here.

Let us assume that (u, z) and (\overline{u}, \overline{z}) are two sufficiently smooth solutions of the problem (GS) for the same initial data satisfying the boundary condition (2.7) or (2.8) with the multivalue function f , for which (2.13) holds and with the free energy ψ satisfying (2.9), (2.10) and (2.14). By an analogous computation to the one that led to the inquality (2.11) we obtain

\frac{d}{dt} ∫_{Ω} ρ(\frac{1}{2}|u_{t} − \overline{u}_{t}|^{2} + ψ(ε − \overline{ε}, z − \overline{z})) dx

= − ∫_{Ω}[−ρ∇_{z}ψ(ε, z) + ρ∇_{z}ψ(\overline{ε}, \overline{z})] · (z_{t} − \overline{z}_{t}) dx ≤ 0  (2.15)

where we used the linearity of the gradient ∇_{z}ψ and the monotonicity of the function g. The free energy function ψ is positive semi-definite, hence (2.15) yields

∫_{Ω} |(u_{t} − \overline{u}_{t})|^{2} dx = 0 .  (2.16)

This denotes the uniqueness of the velocity vector field u_{t} = \overline{u}_{t} and consequently the uniqueness of the displacement u = \overline{u} and of the strain ε = \overline{ε}. With this information we use again (2.15) and have

0 = ∫_{Ω} ρψ(0, z − \overline{z}) dx = ∫_{Ω} \frac{1}{2}[(B^{T} DB + L)(z − \overline{z})] · (z − \overline{z}) dx
= ∫_{Ω} \frac{1}{2}[M(z − \overline{z})] · (z − \overline{z}) dx .

Thus we see that the uniqueness of the vector z can be proved, provided that the N × N matrix M is positive definite. This statement will give reasons for the definition of the constitutive equations of monotone type. We note additionally that the system (GS) can be studied with a general linear operator B : ℝ^{N} → S^{3}. Then we are ready to define the class of constitutive equations of monotone type.

Definition 2.1

a) We say that a pair (f, B) of a function f : D(f) ⊂ S^{3} × ℝ^{N} → P(ℝ^{N}) and a linear mapping B : ℝ^{N} → S^{3} is of pre-monotone type, if there exists a symmetric, positive definite N × N matrix M with the property that the symmetric N × N matrix L =^{df} M − B^{T} DB is positive semi-definite, and there exists a function g : D(g) ⊂ ℝ^{N} → P(ℝ^{N}) such that

f(ε, z) = g(−ρ∇_{z}ψ(ε, z)) = g(B^{T} Dε − Mz)

for all (ε, z) ∈ D(f) , with the positive semi-definite quadratic form

ρψ(ε, z) = \frac{1}{2}[D(ε − Bz)] · (ε − Bz) + \frac{1}{2}(Lz) · z .

b) If the function g is monotone and satisfies

∀_{z∈D(g)} ∀_{w∈g(z)} w · z ≥ 0 ,

then we say that the pair (f, B) is of monotone type.

c) We say that the constitutive equations

T = D(ε − Bz) ,
z_{t} ∈ f(ε, z)

are of pre-monotone type (monotone type) if the pair (f, B) is of pre-monotone type (monotone type).

Hence we have proved above, that smooth solutions of the problem (GS) with constitutive equations of monotone type satisfying the boundary condition (2.7) or (2.8) are unique.

3 Existence theory for constitutive equations of monotone type

In this section we show that for constitutive equations of monotone type not only uniqueness, but also existence of solutions can be proved. The proof of the existence theorem is based on the theory of evolution equations for monotone operators ([6]), and uses stronger assumptions for the free energy function ψ and for the monotone multivalued function g. Namely, in this section we require that the quadratic form ψ is positive definite and the function g is maximal monotone with D(g) = ℝ^{N}. Note that a monotone operator A : H → P(H) on a Hilbert space H is called maximal monotone if it does not have a proper monotone extension ([6]). Thus we will write our initial-boundary value problem as an evolution problem to a maximal monotone operator. Let us set v = u_{t}. Then the initial-boundary value problem with constitutive equations of monotone type has the form

\begin{aligned}v_{t}(x, t) & = & div \frac{1}{ρ} D(ε(x, t) − Bz(x, t)) , \\ ε_{t}(x, t) & = & \frac{1}{2}(∇v(x, t) + ∇^{T} v(x, t)) , \\ z_{t}(x, t) & ∈ & g(−ρ∇_{z}ψ(ε(x, t), z(x, t))) , \end{aligned}  (MP)

with the Dirichlet boundary condition

v(x, t) = 0 for x ∈ ∂Ω

or with the Neumann boundary condition

D(ε(x, t) − Bz(x, t)) · n(x) = 0 for x ∈ ∂Ω

and with the initial conditions

v(x, 0) = v^{(0)}(x) , ε(x, 0) = ε^{0}(x) , z(x, 0) = z^{0}(x) ,

where v^{0}(x) = u^{1}(x) and ε^{0}(x) = \frac{1}{2}(∇u^{0}(x) + ∇^{T} u^{0}(x)). By ℍ_{m}(Ω; ℝ^{k}) we denote the usual Sobolev space of all functions in 𝕃^{2}(Ω; ℝ^{k}) , which have weak derivatives in 𝕃^{2}(Ω; ℝ^{k}) up to order m. The norm and the scalar product in ℍ_{m}(Ω; ℝ^{k}) we denote by ∥·∥_{m} and (·, ·)_{m} respectively. For m = 0 we use for simplicity the notation ∥·∥_{0} = ∥·∥ and (·, ·)_{0} = (·, ·) .

Let us define W = ℝ^{3} × S^{3} × ℝ^{N} and an operator

C : 𝕃^{2}(Ω; W) → P(𝕃^{2}(Ω; W))

by C(v, ε, z) = ∅ if v ∈ 𝕃^{2}(Ω; ℝ^{3})\ℍ_{1}(Ω; ℝ^{3}) and

C(v, ε, z) = \{w = (w_{1}, w_{2}, w_{3}) ∈ 𝕃^{2}(Ω; W) | w_{1} = \frac{1}{ρ} div D(ε − Bz) ,

w_{2} = \frac{1}{2}(∇v + ∇^{T} v) , w_{3}(x) ∈ g(−ρ∇_{z}ψ(ε(x), z(x))) a.e. in Ω\}

if (v, ε, z) ∈ ℍ_{1}(Ω; ℝ^{3}) × 𝕃^{2}(Ω; S^{3} × ℝ^{N}) .

Additionally for the Dirichlet boundary value problem we assume that

v ∈ ℍ^{∘}_{1}(Ω; ℝ^{3})

and for the Neumann boundary value problem, that the weak divergence div D(ε − Bz) satisfies

(div D(ε − Bz), u) = −(D(ε − Bz), ∇u) ∀_{u∈ℍ_{1}(Ω;ℝ^{3})} .

Thus the problem (MP) can be written in the form

w_{t}(t) ∈ C(w(t)) , w(0) = w^{(0)} = (v^{0}, ε^{0}, z^{0}) .  (3.1)

The goal of the section is to show that the operator −C is maximal monotone. This is the statement of the theorem below.

Theorem 3.1 Let the quadratic form

ρψ(ε, z) = \frac{1}{2}[D(ε − Bz)] · (ε − Bz) + \frac{1}{2}(Lz) · z

be positive definite on S^{3} × ℝ^{N} and let g : ℝ^{N} → P(ℝ^{N}) be maximal monotone with 0 ∈ g(0) . Let us assume that, for the Dirichlet boundary condition, Ω ⊆ ℝ^{3} be an open bounded set, and for the Neumann boundary condition, Ω ⊆ ℝ^{3} be an open bounded set with Lipschitz boundary. Then the operator −C : 𝕃^{2}(Ω; W) → P(𝕃^{2}(Ω; W)) is maximal monotone if the space 𝕃^{2}(Ω; W) is equipped with the scalar product

⟨(v, ε, z), (\overline{v}, \overline{ε}, \overline{z})⟩ = ∫_{Ω} \{ρv · \overline{v} + [D(ε − Bz)] · (\overline{ε} − B\overline{z}) + (Lz) · \overline{z}\} dx . (3.2)

The proof of the Theorem 3.1 is the main result of chapter 4 in the book [3]. Theorem 3.1 shows that the problem (MP) has a natural monotonicity property if we define the scalar product in 𝕃^{2}(Ω; S^{3}×ℝ^{N}) by the form ψ. This scalar product is equivalent to the standard scalar product, hence the topology defined by the product ⟨·, ·⟩ is the standard strong 𝕃^{2}-toplogy. It is clear that if ψ is only positive semi-definite then the monotonicity property of the operator −C fails.

Definition 3.2 We say that the problem (MP) is coercive if and only if the free energy function ψ connected with the problem is a positive definite quadratic form.

Thus Theorem 3.1 says that the coercive problems with constitutive equations of monotone type can be written in the form (3.1) with a monotone operator −C, and if the right hand side of the inelastic constitutive equations is maximal monotone, then the operator −C is maximal monotone too. Then by the theory presented in the books [4], [6] we have the following global in time existence of large solutions:

Theorem 3.3 Let the assumptions of Theorem 3.1 be satisfied. Then to every w^{0} = (v^{0}, ε^{0}, z^{0}) ∈ D(C) there exists a unique function w ∈ 𝕎^{1,∞}((0, ∞); 𝕃^{2}(Ω; W)) such that

(i) w(t) ∈ D(C) for all t ≥ 0 , (ii) w_{t}(t) ∈ C(w(t)) a.e. in [0, ∞) and w(0) = w^{0},

(iii) w has everywhere a t-derivative from the right,

(iv) if w and \overline{w} are two solutions satisfying (i) – (ii), then

∥w(t) − \overline{w}(t)∥_{⟨·,·⟩} ≤ ∥w(0) − \overline{w}(0)∥_{⟨·,·⟩}

for all t ≥ 0, where ∥·∥_{⟨·,·⟩} denotes the norm in 𝕃^{2}(Ω; W) generated by the scalar product ⟨·, ·⟩.

Theorem 3.3 is the consequence of Theorem 3.1 and the general theory of evolution equations for monotone operators, which is presented, for example, in [4], [6]. All results of Sections 2 and 3 are obtained in the book [3]. In the next sections we want to give an answer to the question:

Have noncoercive problems global in time large solutions?

4 A noncoercive example. The monotone model of Bodner-Partom

In this and in the next section we study only single-valued functions g : ℝ^{N} → ℝ^{N}. Thus the inelastic constitutive equations have now the form of ordinary differential equations. Note that if a single-valued function g^{∗} : ℝ^{N} → ℝ^{N} is monotone and continuous, then the single-valued operator g(z) = {g^{∗}(z)} generated by g^{∗} is maximal monotone. Before we start with the monotone model of Bodner-Partom we give a general approximation lemma used in this and in the next sections. Let us again state the problem (GS). We want to find functions v : Ω × [0, T) → ℝ^{3}, ε : Ω × [0, T) → S^{3}, z : Ω × [0, T) → ℝ^{N} satisfying the system of equations

\begin{aligned}v_{t} & = & div \frac{1}{ρ}(ε − Bz) , \\ ε_{t} & = & \frac{1}{2}(∇v + ∇^{T} v) , \\ z_{t} & = & f(ε, z) \end{aligned}  (GS)

and the boundary condition

v(x, t)_{|_{∂Ω}} = 0 or D(ε(x, t) − Bz(x, t))_{|_{∂Ω}} = 0 ,  (4.1)

and the initial conditions

v(0) = v^{0} , ε(0) = ε^{0} , z(0) = z^{0} .  (4.2)

We shall approximate this problem with a family of approximate problems

\begin{aligned}v_{t}^{k} & = & div \frac{1}{ρ}(D(ε^{k} − Bz^{k})) , \\ ε_{t}^{k} & = & \frac{1}{2}(∇v^{k} + ∇^{T} v^{k}) , \\ z_{t}^{k} & = & f^{k}(ε^{k}, z^{k}) , \end{aligned}  (AS)

v^{k}_{|_{∂Ω}} = 0 or D(ε^{k} − Bz^{k})_{|_{∂Ω}} = 0 ,  (4.3)
v^{k}(0) = v_{k}^{0} , ε^{k}(0) = ε_{k}^{0} , z^{k}(0) = z_{k}^{0} ,  (4.4)

where k ∈ ℕ and the family of functions {f^{k}} approximate pointwise the function f , which means that for all sequences (ε^{k}, z^{k}) which converge pointwise for almost every (x, t) ∈ Ω × [0, T) to (ε, z), we have

f^{k}(ε^{k}(x, t), z^{k}(x, t)) ⟶^{pointwise a.e. in Ω × [0, T)} f(ε(x, t), z(x, t) .

Moreover let us suppose that (v_{k}^{0}, ε_{k}^{0}, z_{k}^{0}) → (v^{0}, ε^{0}, z^{0}) in 𝕃^{2}(Ω; W) .

Lemma 4.1 Let Ω ⊂ ℝ^{3} be a bounded open set with smooth boundary ∂Ω , (v^{0}, ε^{0}, z^{0}) ∈ 𝕃^{2}(Ω; W) such that (div D(ε^{0} − Bz^{0}) , \frac{1}{2}(∇v^{0} + ∇^{T} v^{0}) , f(ε^{0}, z^{0})) ∈ 𝕃^{2}(Ω; W) and v^{0} ∈ ℍ^{∘}_{1}(Ω) for the Dirichlet boundary value problem and with (div D(ε^{0} − Bz^{0}), u) = −(D(ε^{0} − Bz^{0}), ∇u) ∀u ∈ ℍ_{1}(Ω; ℝ^{3}) for the Neumann boundary condition. Let us assume that a sequence (v^{k}, ε^{k}, z^{k}) of solutions for the problem (AS) with boundary condition (4.3) and initial conditions (4.4) satisfies

ess sup_{t∈[0,T)} {∥v_{t}^{k}(t)∥ , ∥ε_{t}^{k}(t)∥ , ∥z_{t}^{k}(t)∥} ≤ C ,  (4.5)

where the positive constant C does not depend on k. Moreover suppose that for all γ > 0 there exists a natural number N(γ) such that

ess sup_{t∈[0,T)} {∥v^{k}(t) − v^{m}(t)∥ + ∥ε^{k}(t) − ε^{m}(t)∥ + ∥z^{k}(t) − z^{m}(t)∥} ≤ γ for k, m ≥ N(γ) .

(4.6)

Then there exists a solution (v, ε, z) ∈ 𝕎^{1,∞}((0, T); 𝕃^{2}(Ω; W)) of the problem (GS) satisfying the boundary condition (4.1) and the initial conditions (4.2).

Proof: By (4.6) we have that the sequence (v^{k}, ε^{k}, z^{k}) converges in the strong topology in 𝕃^{∞}((0, T); 𝕃^{2}(Ω; W)) to the limit (v, ε, z) . Moreover (4.5) yields that there exists a subsequence (v_{t}^{k_{ℓ}}, ε_{t}^{k_{ℓ}}, z_{t}^{k_{ℓ}}) of the sequence (v_{t}^{k}, ε_{t}^{k}, z_{t}^{k}) which converges weak– ∗ in 𝕃_{w}^{∞}((0, T); 𝕃^{2}(Ω; W)) to the limes (v_{t}, ε_{t}, z_{t})^{1} where (v, ε, z) was defined above. This subsequence we denote again with (v_{t}^{k}, ε_{t}^{k}, z_{t}^{k}) . Thus we obtain

\begin{aligned}v_{t} ↼^{∗} v_{t}^{k} & = & \frac{1}{ρ} div D(ε^{k} − Bz^{k}) & ⇀^{∗} & \frac{1}{ρ} div D(ε − Bz) \\ ε_{t} ↼^{∗} ε_{t}^{k} & = & \frac{1}{2}(∇v^{k} + ∇^{T} v^{k}) & ⇀^{∗} & \frac{1}{2}(∇v + ∇^{T} v) \\ z_{t} ↼^{∗} z_{t}^{k} & = & f^{k}(ε^{k}, z^{k}) & ⇀^{∗} & χ \end{aligned}

in the space 𝕃_{w}^{∞}((0, T); 𝕃^{2}(Ω; W)). Next we would like to show that χ = f(ε, z) . We can assume that the sequence (ε^{k}, z^{k}) converges pointwise for almost every (x, t) ∈ Ω × [0, T) to (ε, z) . (If it is not the case then there exists a subsequence with this property.) Hence using the approximation f^{k} → f we obtain

f^{k}(ε^{k}, z^{k}) ⟶^{a.e.} f(ε, z)
f^{k}(ε^{k}, z^{k}) ⇀ χ in 𝕃^{2}((0, T) × Ω)

while the weak-∗ convergence in 𝕃_{w}^{∞}((0, T); 𝕃^{2}(Ω; W)) implies weak convergence in 𝕃^{2}((0, T) × Ω)). Thus the sequence f^{k}(ε^{k}, z^{k}) is bounded in 𝕃^{2}((0, T) × Ω) and uniformly integrable in 𝕃^{p}((0, T) × Ω) for 1 < p < 2 :

1y^{n} ⇀^{+} y in 𝕃_{w}^{∞}((0, T; 𝕃^{2}(Ω)) ⇔ ∀_{ϕ∈L^{1}(0,T)} ∀_{ψ∈L^{2}(Ω)}

∫_{0}^{T} ( ∫_{Ω} y^{n}ψdx)ϕ dt → ∫_{0}^{T} ( ∫_{Ω} yψ dx)ϕ dt

Let E ⊂ (0, T) × Ω be measurable, then

sup_{k} ∫_{E}|f^{k}(ε^{k}, z^{k})|^{p} dx dt ≤ |E|^{\frac{2−p}{2}} sup_{k} ( ∫_{Ω}^{T} ∫_{Ω} (f^{k}(ε^{k}, z^{k}))^{2}dx dt)^{p/2} → 0

for |E| → 0 . Hence by the Vitali theorem we conclude

f^{k}(ε^{k}, z^{k}) → f(ε, z) in 𝕃^{p}((0, T) × Ω) for 1 < p < 2 .

Moreover f^{k}(ε^{k}, z^{k}) ⇀ χ in 𝕃^{p}((0, T) × Ω) for 1 < p < 2 , then χ = f(ε, z) a.e. in Ω × [0, T) . To end the proof we must show that the limit (v, ε, z) satisfies the boundary condition (4.1) and the initial conditions (4.2). For the Dirichlet boundary value problem by the Korn inequality we have v^{k}(t) ∈ ℍ^{∘}_{1}(Ω) for a.e. t ∈ [0, T) . Moreover the sequence v^{k}(t) is uniformly bounded with respect to k in ℍ^{∘}_{1}(Ω) , hence we can assume v^{k}(t) ⇀ v(t) in ℍ_{1}-topology for a.e. t ∈ [0, T) . The space ℍ^{∘}_{1}(Ω) is weakly closed in ℍ_{1}(Ω) , thus v(t) ∈ ℍ^{∘}_{1}(Ω) for a.e. t ∈ [0, T) . For the Neumann boundary condition we have for a.e. t ∈ [0, T) and for all k

(div D(ε^{k}(t) − Bz^{k}(t)), u) = −(D(ε^{k}(t) − Bz^{k}(t)), ∇u) ∀_{u∈ℍ_{1}(Ω;ℝ^{3})} .

The sequence div D(ε^{k}(t) − Bz^{k}(t)) converges weakly to div D(ε(t) − Bz(t)) in

𝕃^{2}(Ω; ℝ^{3}) for a.e. t ∈ [0, T) and the sequence D(ε^{k}(t) − Bz^{k}(t)) converges in the

strong topology to D(ε(t) − Bz(t)) in 𝕃^{2}(Ω; S^{3}) for a.e. t ∈ [0, T) . Thus for a.e. t ∈ [0, T)

(div D(ε(t) − Bz(t)), u) = −(D(ε(t) − Bz(t)), ∇u) ∀_{u∈ℍ_{1}(Ω;ℝ^{3})} .

We see that the limit (v, ε, z) ∈ 𝕎^{1,∞}((0, T); 𝕃^{2}(Ω; W)), hence the trace operator

(v, ε, z) → (v_{|_{t=0}} ε_{|_{t=0}}, z_{|_{t=0}}) is well defined. The sequence (v^{k}, ε^{k}, z^{k}) ⇀^{∗} (v, ε, z) in

𝕎^{1,∞}((0, T); 𝕃^{2}(Ω; W)) then the weak continuity of the trace operator yields

(v^{k}_{|_{t=0}}, ε^{k}_{|_{t=0}}, z^{k}_{|_{t=0}}) ⇀ (v_{|_{t=0}}, ε_{|_{t=0}}, z_{|_{t=0}})

in 𝕃^{2}(Ω; W) . The sequence (v^{k}_{|_{t=0}}, ε^{k}_{|_{t=0}}, z^{k}_{|_{t=0}}) converges to (v^{0}, ε^{0}, z^{0}) in 𝕃^{2}(Ω; W) then (v_{|_{t=0}}, ε_{|_{t=0}}, z_{|_{t=0}}) = (v^{0}, ε^{0}, z^{0}) .

Now we are ready to study noncoercive problems with constitutive equations of monotone type. The main idea is to approximate such problems by a family of coercive problems, for which Theorem 3.1 holds. We start with the monotone model of Bodner-Partom. In this model the vector z of internal variables consists of the six components of the inelastic strain tensor ε^{p}, and of one positive variable y, which describes isotropic hardening. Thus N = 7 and z(x, t) = (ε^{p}(x, t), y(x, t)) with y(x, t) > 0 . The inelastic constitutive equations are in the form

ε_{t}^{p} = F (\frac{|σ|}{y}) \frac{σ}{|σ|} ,
y_{t} = \frac{|σ|}{y} F (\frac{|σ|}{y}) ,

where f : ℝ_{+} → ℝ_{+} , is a given function and σ = T − \frac{1}{3} (tr T)·I is the stress deviator. It is proved in [3], that the model is of monotone type only for F (s) = ds^{β}, d > 0, β > 1 . Therfore the system of equations for the monotone model of Bodner-Partom has the form

\begin{aligned}ρv_{t} & = & div D(ε − ε^{p}), \\ ε_{t} & = & \frac{1}{2}(∇v + ∇^{T} v) , \\ ε_{t}^{p} & = & d(\frac{|σ|}{y})^{β} \frac{σ}{|σ|} , \\ y_{t} & = & d(\frac{|σ|}{y})^{β+1} \end{aligned}  (MBP)

with the free energy

ρψ(ε, ε^{p}, y) = \frac{1}{2} [D(ε − ε^{p})] · (ε − ε^{p}) + \frac{β}{2(β + 1)} y^{2} .

The simple calculation

ρ∇_{(ε^{p},y)} ψ(ε, ε^{p}, y) · (ε_{t}^{p}, y_{t})

= −D(ε − ε^{p}) · d(\frac{|σ|}{y})^{β} \frac{σ}{|σ|} + \frac{β}{β + 1} y · d(\frac{|σ|}{y})^{β+1}

= −d(\frac{|σ|}{y})^{β} |σ| + \frac{β}{β + 1} d(\frac{|σ|}{y})^{β} · |σ| ≤ 0

shows that the dissipation inequality is satisfied. Moreover the operator L(ε^{p}, y) = (0, \frac{β}{β+1} y) is only positive semi-definite and the operator M(ε^{p}, y) = (B^{T} DB + L)(ε^{p}, y) = (Dε^{p}, \frac{β}{β+1} y) is positive definite. Thus the problem (MBP) is noncoercive. We want now to approximate this noncoercive model with a sequence of coercive models. First of course, we should choose a “good” approximation process. Here the mechanical analysis helps us to solve the problem. The system (MBP) is connected with the so-called rheological diagram:

[figure: ..........................................................................................................................................................................................................................................................................................................................................................................................................; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; ε; ε_{p}; T; σ; •; •; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .]

ψ-positive semi-definite

which shows that the properties of the metal under consideration are described by a connection in series of a pure elastic element (the spring) with an inelastic element (the dapshot). The coercivity of the problem fails because the inelastic deformation is fully irreversible. This statement yields the main idea in the theory of coercive limits. We will try to approximate the problem (MBP) with coercive models, for which the rheological diagram is given by the figure below:

[figure: ..........................................................................................................................................................................................................................................................................................................................................................................................................; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; ..........................................................................................................................................................................................................................................................................................................................................................................................................; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; •; •; •; ε; ε_{p}; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; T; σ − \frac{1}{k} σ^{p,k}; \frac{1}{k} Dε^{p,k}]

ψ-positive definite

The diagram shows that the stress deviator σ, responsible for the inelastic deformation, is now “reduced” by the stress from the inserted spring. Therefore we require that the approximation satisfies the system:

\begin{aligned}ρv_{t}^{k} & = & div D(ε^{k} − ε^{p,k}) , \\ ε_{t}^{p,k} & = & \frac{1}{2}(∇v^{k} + ∇^{T} v^{k}) , \\ ε_{t}^{p,k} & = & d(\frac{|σ^{k} − \frac{1}{k} σ^{p,k}|}{y^{k}})^{β} \frac{σ^{k} − \frac{1}{k} σ^{p,k}}{|σ^{k} − \frac{1}{k} σ^{p,k}|} , \\ y_{t}^{k} & = & d(\frac{|σ^{k} − \frac{1}{k}σ^{p,k}|}{y^{k}})^{β+1} \end{aligned}  (CBP)

(here σ^{p,k} = Dε^{p,k} − \frac{1}{3} (tr Dε^{p,k}) · I) and the homogeneous Dirichlet or Neumann boundary condition, and initial conditions

v^{k}(0) = v_{k}^{0} , ε^{k}(0) = ε_{k}^{0} , ε^{p,k}(0) = ε_{k}^{p,0} , y^{k}(0) = y_{k}^{0} ,  (4.7)

where the inital data should be suitably chosen.

Definition 4.2 We say that the initial data (v^{0}, ε^{0}, ε^{p,0}, y^{0}) ∈ 𝕃^{2}(Ω; W) is admissible for the system (MBP) ⇔

tr ε^{p,0} = 0 a.e. in Ω , y^{0} ≥ α > 0 a.e. in Ω ,  (4.8)
(div D(ε^{0} − ε^{p,0}) , \frac{1}{2}(∇v^{0} + ∇^{T} v^{0}) , (\frac{|σ^{0}|}{y^{0}})^{β} \frac{σ^{0}}{|σ^{0}|} , (\frac{|σ^{0}|}{y^{0}})^{β+1}) ∈ 𝕃^{2}(Ω; W)(4.9)

where σ^{0} = T^{0} − \frac{1}{3}(tr T^{0})I and T^{0} = D(ε^{0} − ε^{p,0}) , and additionally for the Dirichlet boundary condition it holds that:

v^{0} ∈ ℍ^{∘}_{1}(Ω; ℝ^{3}) ,  (4.10)

and for the Neumann boundary condition we require that:

∀_{u∈ℍ_{1}(Ω;ℝ^{3})} (div D(ε^{0} − ε^{p,0}), u) = −(D(ε^{0} − ε^{p,0}), ∇u) .  (4.11)

Remark: We say that the initial data (4.7) is admissible for the system (CBP) if and only if the data belongs to the domain of the operator on the right hand side of the system (CBP) (a similar condition to (4.9)) and satisfies (4.8) and (4.10) or (4.11), respectively.

Lemma 4.3 Let the initial data (v^{0}, ε^{0}, ε^{p,0}, y^{0}) ∈ 𝕃^{2}(Ω; W) be admissible for the system (MBP), div Dε^{0} ∈ 𝕃^{2}(Ω; ℝ^{3}) and for the Neumann boundary value problem Dε^{0}(x) · n(x)_{|_{x∈∂Ω}} = 0 , then (v^{0}, ε^{0}, \frac{k}{k+1} ε^{p,0}, y^{0}) is admissible for the system (CBP).

Proof: Let us denote by P the linear operator P : S^{3} → S^{3} defined by P T = T − \frac{1}{3} (tr T) · I. Hence using this notation we have σ^{k} = P D(ε^{k} − ε^{p,k}) and σ^{p,k} = P D(ε^{p,k}) . Then for the initial data the difference

σ^{k}(0) − \frac{1}{k} σ^{p,k}(0) = P D(ε^{k}(0) − ε^{p,k}(0)) − \frac{1}{k} P D(ε^{p,k}(0))
= P D(ε^{0} − ((1 + \frac{1}{k}) · \frac{k}{k + 1} ε^{p,0}) = P D(ε^{0} − ε^{p,0})

does not depend on k, and the right hand side of the inelastic constitutive equation in the system (CBP) is well defined for the inital data (v^{0}, ε^{0}, \frac{k}{k+1}, ε^{p,0}, y^{0}) . Moreover the weak divergence

div D(ε^{k}(0) − ε^{p,k}(0)) = div (ε^{0} − \frac{k}{k + 1} ε^{p,0}) =
= \frac{k}{k + 1} div D(ε^{0} − ε^{p,0}) + \frac{1}{k + 1} div Dε^{0}

belongs to 𝕃^{2}(Ω; ℝ^{3}) while div Dε^{0} ∈ 𝕃^{2}(Ω; ℝ^{3}) . Finally for the Neumann boundary condition the assumption Dε^{0} · n_{|_{∂Ω}} = 0 yields (4.11) for the data (ε^{0}, \frac{k}{k+1} ε^{p,0}) .

Lemma 4.4 Let (v^{0}, ε^{0}, ε^{p,0}, y^{0}) ∈ 𝕃^{2}(Ω; W) be admissible for the system (MBP), div Dε^{0} ∈ 𝕃^{2}(Ω; ℝ^{3}) and for the Neumann boundary problem Dε^{0} · n_{|_{∂Ω}} = 0 . Then for all k ∈ ℕ the problem (CBP) has a unique, strong, global in time solution (v^{k}, ε^{k}, ε^{p,k}, y^{k}) to the initial data (v^{0}, ε^{0}, \frac{k}{k+1} ε^{p,0}, y^{0}) .

Proof: It is easy to see that the problem (CBP) possesses the free energy function of the form

ρψ^{k}(ε^{k}, ε^{p,k}, y^{k}) = \frac{1}{2}[D(ε^{k} − ε^{p,k})] · (ε^{k} − ε^{p,k}) + \frac{β}{2(β + 1)} (y^{k})^{2} + \frac{1}{2k} (Dε^{p,k}, ε^{p,k}) .

Thus the quadratic form ψ^{k} is positive definite and the operator L is defined by

L(ε^{p,k}, y^{k}) = (\frac{1}{k} D ε^{p,k} , \frac{β}{β + 1} y^{k}) .

Moreover the operator

M(ε^{p,k}, y^{k}) = (B^{T} DB + L)(ε^{p,k}, y^{k}) = (D(1 + \frac{1}{k})ε^{p,k} , \frac{β}{β + 1} y^{k})

is positive definite. Then the problem (CBP) is coercive and of monotone type (compare with the monotone problem (MBP)). The inelastic constitutive equations are of the form g(−ρ∇_{z}ψ(ε, z)) with a monotone function g. This function has a maximal monotone extention g̃ with the property 0 ∈ g̃(0). Moreover for admissible initial data all solutions must belong to D(g) and tr ε^{p}(x, t) = 0 for all t and for a.e. x ∈ Ω. If D(g̃) = ℝ^{N} then we can use Theorem 3.1 and if D(g̃) ≠ ℝ^{N} then the results from Section 8 allow us to use Theorem 3.3. (4.2) yields that the initial data (v^{0}, ε^{0}, \frac{k}{k+1} ε^{p,0}, y^{0}) ∈ D(C) , where C is the operator from Theorem 3.1. Theorem 3.1 and Theorem 3.3 complete the proof.

Next we will study the limit case k → ∞. This denotes that the inserted spring becomes weaker and weaker, and we can expect that on finite time intervals the sequence (v^{k}, ε^{k}, ε^{p,k}, y^{k}) converges to a solution of the problem (MBP).

Theorem 4.5 Let T > 0 be fixed and (v^{0}, ε^{0}, ε^{p,0}, y^{0}) ∈ 𝕃^{2}(Ω; W) be admissible for (MBP) with div Dε^{0} ∈ 𝕃^{2}(Ω; ℝ^{3}) and for the Neumann boundary problem Dε^{0} · n_{|_{∂Ω}} = 0 . The approximate sequence (v^{k}, ε^{k}, ε^{p,k}, y^{k}) from Lemma 4.4 satisfies

(i) ess sup_{t∈[0,T)} \{∥vk t(t)∥ , ∥εk t(t)∥ , ∥ε{p,t \atop t}(t)∥ , ∥yk t(t)∥\} < C  (4.12)

and the positive constant C does not depend on k and T ,

(ii) \begin{matrix}∀_{γ>0} ∃_{N(γ)} ess sup_{t∈[0,T)}\{∥v^{k}(t) − v^{m}(t)∥ \\ + ∥ε^{k}(t) − ε^{p,k}(t) − ε^{m}(t) + ε^{p,m}(t)∥ \\ + ∥y^{k}(t) − y^{m}(t)∥\} ≤ γ for k, m ≥ N(γ) . \end{matrix} (4.13)

Proof: By the existence theorem for coercive monotone problems we have for all k > 0

(v^{k}, ε^{k}, ε^{p,k}, y^{k}) ∈ 𝕎^{1,∞}((0, T); H_{k}) ,

where H_{k} denotes the space 𝕃^{2}(Ω, W) equipped with the scalar product defined by the usual 𝕃^{2}-scalar product for the first component v and by the free energy ψ^{k} for the other components. Let us denote by C^{k} the operator from Theorem 3.1 associated with the system (MBP), then Theorem 3.3 yields

ess sup_{t∈[0,T]} ∥(v_{t}^{k}(t), ε_{t}^{k}(t), ε^{p,k}(t), y_{t}^{k}(t))∥_{H_{k}} ≤ ∥C^{k}(v^{0}, ε^{0}, \frac{k}{k + 1} ε^{p,0}, y^{0})∥_{H_{k}} . (4.14)

Inequality (4.14) says that the left hand side of the system (MBP) is bounded in the space 𝕃^{∞}((0, T), H_{k}) by the right hand side evaluated for the initial data. This is the classical result from the theory of evolution equations for monotone operators. First we will prove that the right hand side of (4.14) is bounded:

∥C^{k}(v^{0}, ε^{0}, \frac{k}{k + 1} ε^{p,0}, y^{0})∥_{H_{k}} ≤ C\{∥div D(ε^{0} − \frac{k}{k + 1} ε^{p,0})∥

+ ∥\frac{1}{2}(∇v^{0} + ∇^{T} v^{0}) − d(\frac{|σ^{0}|}{y^{0}})^{β}∥

+ ∥d (\frac{|σ^{0}|}{y^{0}})^{β+1}∥ + \frac{1}{\sqrt{k}} ∥d(\frac{|σ^{0}|}{y^{0}})^{β}∥\} , (4.15)

where σ^{0} = P D(ε^{0} − ε^{p,0}) and P is the projector from Lemma 4.3. It is easy to see that for the initial data (v^{0}, ε^{0}, ε^{p,0}, y^{0}) admissible for the system (MBP) with div Dε^{0} ∈ 𝕃^{2}(Ω; ℝ^{3}), there exists a positive constant C independent of k and T such that

ess sup_{t∈[0,T]} \{∥v_{t}^{k}(t)∥ , ∥ε_{t}^{k}(t) − ε_{t}^{p,k}(t)∥ , y_{t}^{k}(t)∥ , \frac{1}{\sqrt{k}} ∥ε_{t}^{p,k}(t)∥\} ≤ C .

To end the proof of (i) we must only show that the sequence ε_{t}^{p,k}(t) is bounded in 𝕃^{∞}((0, T); 𝕃^{2}(Ω; S^{3})). We can do this using the information that the sequence y_{t}^{k}(t)

is bounded in 𝕃^{∞}((0, T); 𝕃^{2}(Ω; R)) and the special form of the system (MBP).

∥ε_{t}^{p,k}(t)∥^{2} = d^{2} ∫_{Ω} (\frac{|σ^{k}(t) − \frac{1}{k}σ^{p,k}(t)|}{y^{k}(t)})^{2β}dx
≤ d^{2}\{ ∫_{Ω} (\frac{|σ^{k}(t) − \frac{1}{k}σ^{p,k}(t)|}{y^{k}(t)})^{2β+2}dx\}^{\frac{β}{β+1}} · |Ω|^{\frac{1}{β+1}}
= d^{\frac{2}{β+1}} ∥y_{t}^{k}(t)∥^{\frac{2β}{β+1}} · |Ω|^{\frac{1}{β+1}} .  (4.16)

Inequality (4.16) shows that we can control the right hand side of the equation for the inelastic deformation by the right hand side of the equation for the isotropic hardening. Then (4.16) completes the proof of (i).

To prove (ii) we use the energy method. Let us define by

E^{k,m}(t) = \frac{ρ}{2} ∥v^{k}(t) − v^{m}(t)∥^{2} + \frac{1}{2} ∫_{Ω} [D(ε^{k}(t) − ε^{m}(t) − ε^{p,k}(t) + ε^{p,m}(t))]

·(ε^{k}(t) − ε^{m}(t) − ε^{p,k}(t) + ε^{p,m}(t)) dx + \frac{β}{2(β + 1)} ∥y^{k}(t) − y^{m}(t)∥^{2}

the energy for the difference (v^{k}, ε^{k}, ε^{p,k}, y^{k}) − (v^{m}, ε^{m}, ε^{p,m}, y^{m}) . Then for the time derivative of the function E^{k,m}(t) (the derivative is well defined) we have

\frac{d}{dt}E^{k,m}(t) = ∫_{Ω} ρ(v^{k}(t) − v^{m}(t)) · (v_{t}^{k}(t) − v_{t}^{m}(t)) dx

+ ∫_{Ω} [D(ε^{k}(t) − ε^{m}(t) − ε^{p,k}(t) + ε^{p,m}(t))]

·(ε_{t}^{k}(t) − ε_{t}^{m}(t) − ε_{t}^{p,k}(t) + ε_{t}^{p,m}(t)) dx

+\frac{β}{β + 1} ∫_{Ω} (y^{k}(t) − y^{m}(t)) · (y_{t}^{k}(t) − y_{t}^{m}(t)) dx

= ∫_{Ω} ρ(v^{k}(t) − v^{m}(t)) · (v_{t}^{k}(t) − v_{t}^{m}(t)) dx

− ∫_{Ω} (div D(ε^{k}(t) − ε^{m}(t) − ε^{p,k}(t) + ε^{p,m}(t))) · (v^{k}(t) − v^{m}(t)) dx

− ∫_{Ω} (σ^{k}(t) − σ^{m}(t)) (ε_{t}^{p,k}(t) − ε_{t}^{p,m}(t)) dx

+\frac{β}{β + 1} ∫_{Ω} (y^{k}(t) − y^{m}(t)) (y_{t}^{k}(t) − y_{t}^{m}(t)) dx .

We have used here that tr ε_{t}^{p,m}(t) = tr ε_{t}^{p,m}(t) = 0 and that (v^{k}, ε^{k}, ε^{p,k}, y^{t}) and (v^{m}, ε^{m}, ε^{p,m}, y^{m}) satisfy homogeneous boundary conditions. By equation (CBP1) we obtain

\frac{d}{dt}E^{k,m}(t) = − ∫_{Ω} (σ^{k}(t) − \frac{1}{k}σ^{p,k}(t) − σ^{m}(t) + \frac{1}{m} σ^{p,m}(t)) (ε_{t}^{p,k}(t) − ε_{t}^{p,m}(t)) dx

− ∫_{Ω} (\frac{1}{k} Dε^{p,k}(t) − \frac{1}{m} Dε^{p,m}(t)) · (ε_{t}^{p,k}(t) − ε_{t}^{p,m}(t)) dx

+\frac{β}{β + 1} ∫_{Ω} (y^{k}(t) − y^{m}(t))(y_{t}^{k}(t) − y_{t}^{m}(t)) dx  (4.17)

(monotonicity of (BP))

≤ − ∫_{Ω} (\frac{1}{k} Dε^{p,k}(t) − \frac{1}{m} Dε^{p,m}(t)) (ε_{t}^{p,k}(t) − ε_{t}^{p,m}(t)) dx
≤ C · (\frac{1}{m} + \frac{1}{k} )(∥ε^{p,k}(t)∥^{2} + ∥ε^{p,m}(t)∥^{2} + ∥ε_{t}^{p,k}(t)∥^{2} + ∥ε_{t}^{p,m}(t)∥^{2}) .

The uniformly boundedness of the sequence ∥ε_{t}^{p,k}(t)∥ implies for fixed T, ∥ε^{p,k}(t)∥ ≤ C(T) where the constant C(T) does not depend on k. (Note that the sequence of initial data ε^{p,k}(0) = \frac{k}{k+1} ε^{p,0} is bounded in 𝕃^{2}(Ω; S^{3}) .) Then there exists a positive constant C′(T) independent of k and m such that

E^{k,m}(t) ≤ E^{k,m}(0) + C′(T)(\frac{1}{m} + \frac{1}{k} ) · T
≤ C · (\frac{1}{k + 1} + \frac{1}{m + 1}) ∥ε^{p,0}∥^{2} + C′(T)(\frac{1}{m} + \frac{1}{k} ) · T .  (4.18)

Thus the right hand side of (4.18) for fixed T can be arbitrarily small for sufficiently large k and m.

Corollary 4.6 For initial data (v^{0}, ε^{0}, ε^{p,0}, y^{0}) ∈ 𝕃^{2}(Ω; W) satisfying assumptions of Theorem 4.5 there exists a subsequence (v^{k_{ℓ}}, ε^{k_{ℓ}}, ε^{p,k_{ℓ}}, y^{k_{ℓ}}) of the sequence (v^{k}, ε^{k}, ε^{p,k}, y^{k}) from Lemma 4.4, which converges weak-∗ in the space 𝕎_{w}^{1,∞}((0, T); 𝕃^{2}(Ω; W)) to a solution (v, ε, ε^{p}, y) of the system (MBP). Moreover the sequences (v^{k_{ℓ}}, ε^{k_{ℓ}} − ε^{p,k_{ℓ}}, y^{k_{ℓ}}) converge in the strong 𝕃^{2}-topology to (v, ε − ε^{p}, y) .

Proof: From Theorem 4.5 we conclude that (for a selected subsequence)

(v^{k}, ε^{k}, ε^{p,k}, y^{k}) ⇀^{∗} (v, ε, ε^{p}, y) in 𝕎_{w}^{1,∞}((0, T); 𝕃^{2}(Ω; W))

and moreover

(v^{k}, ε^{k} − ε^{p,k}, y^{k}) → (v, ε − ε^{p}, y) in 𝕃^{∞}((0, T); 𝕃^{2}(Ω; ℝ^{3} × S^{3} × ℝ)) .

Thus the sequence σ^{k} − \frac{1}{k} σ^{p,k} = P D(ε^{k} − ε^{p,k}) − \frac{1}{k} P Dε^{p,k} → P D(ε − ε^{p}) =^{df} σ in

𝕃^{∞}((0, T); 𝕃^{2}(Ω; S^{3})) while the sequence ∥ε^{p,k}∥ is bounded. Then we can assume that σ^{k} − \frac{1}{k} σ^{p,k} → σ pointwise for a.e. (x, t) ∈ Ω × [0, T) . Lemma 4.1 ends this proof.

Remark 4.7 With the energy method presented in Section 2 we can show that strong solutions of the problem (MBP) are unique. (The operator M(ε^{p}, y) = (Dε^{p}, \frac{β}{β+1} y) is positive definite.)

Remark 4.8 We have proved existence of strong, global in time solutions for the problem (MBP) using additionally that div Dε^{0} exists and belongs to 𝕃^{2}(Ω; ℝ^{3}) . This is naturally equivalent to div Dε^{p,0} ∈ 𝕃^{2}(Ω; ℝ^{3}) . This assumption is a consequence of the type of approximation process we have used. We can try to approximate the inital data ε^{0} ∈ 𝕃^{2}(Ω; S^{3}) by a sequence of smooth data ε_{n}^{0} for which div Dε_{n}^{0} ∈ 𝕃^{2}(Ω; ℝ^{3}). (4.15) shows that such approximations must have the following property: the sequence

∥(\frac{|σ_{n}^{0}|}{y^{0}})^{β+1}∥ should be bounded. This can be done, for example, for ε^{0} ∈ 𝕃^{2β+2}(Ω) .

Remark 4.9 For the Neumann boundary value problem the condition Dε^{0} · n_{|_{∂Ω}} = 0 can be omitted if we require that the approximation process satisfies the nonhomogeneous boundary condition

D(ε^{k}(x, t) − ε^{p,k}(x, t)) · n(x)_{|_{x∈∂Ω}} = \frac{1}{k + 1} Dε^{0}(x) · n(x)_{|_{x∈∂Ω}} .

This condition is independent of t and we can try to prove Theorem 3.1 with the inhomogeneous, constant in time Neumann boundary data.

5 Self-controlling models

In this section we want to generalize the phenomenon of the monotone Bodner– Partom model. Let us consider a system with general constitutive equations of monotone type

\begin{aligned}v_{t} & = & div \frac{1}{ρ} D(ε − Bz) , \\ ε_{t} & = & \frac{1}{2} (∇v + ∇^{T} v) , \\ z_{t} & = & g(−ρ∇_{z}ψ(ε, z)) \end{aligned}  (MP)

with a continuous maximal monotone function g : ℝ^{N} → ℝ^{N} and with the free energy function

ρψ(ε, z) = \frac{1}{2} [D(ε − Bz)] · (ε − Bz) + \frac{1}{2} (Lz) · z .

We assume that the homogeneous Dirichlet boundary condition v(x, t)_{|_{x∈∂Ω}} = 0 or the Neumann boundary condition D(ε(x, t) − Bz(x, t))_{|_{x∈∂Ω}} = 0 is satisfied and the initial conditions are

v(x, 0) = v^{0}(x) , ε(x, 0) = ε^{0}(x) , z(x, 0) = z^{0}(x) .  (5.1)

Moreover suppose that the linear operator

M = B^{T} DB + L is positive definite  (5.2)

and the operator L is only positive semi–definite and our model is noncoercive. We want to approximate the problem (MP) by a sequence of coercive problems. First we define the class of admissible initial data for the system (MP).

Definition 5.1 We say that the initial data (v^{0}, ε^{0}, z^{0}) ∈ 𝕃^{2}(Ω; W) is admissible for the problem (MP) if and only if:

(div D(ε^{0} − Bz^{0}) , \frac{1}{2} (∇v^{0} + ∇^{T} v^{0}) , g(−ρ∇_{z}ψ(ε^{0}, z^{0}))) ∈ 𝕃^{2}(Ω; W)

and additional v^{0} ∈ ℍ^{∘}_{1}(Ω; ℝ^{3}) for the Dirichlet boundary problem or

∀_{u∈ℍ^{1}(Ω;ℝ^{3})} (div D(ε^{0} − Bz^{0}), u) = −(D(ε^{0} − Bz^{0}), ∇u)

for the Neumann boundary condition.

We define now a coercive approximation of the problem (MP): for all ℕ ∋ k > 0 . Let (v^{k}, ε^{k}, z^{k}) be the unique solution of the system

\begin{aligned}v_{t}^{k} & = & div \frac{1}{ρ} D(ε^{k} − Bz^{k}) , \\ ε_{t}^{k} & = & \frac{1}{2} (∇v^{k} + ∇^{T} v^{k}) , \\ z_{t}^{k} & = & g(−ρ∇_{z}ψ^{k}(ε^{k}, z^{k})) \end{aligned}  (CMP)

with the same funtion g : ℝ^{N} → ℝ^{N} and with the free energy in the form

ρψ^{k}(ε^{k}, z^{k}) = \frac{1}{2} [D(ε^{k} − Bz^{k})] · (ε^{k} − Bz^{k}) + \frac{1}{2} (Lz^{k}) · z^{k} + \frac{1}{2k} (DB_{L}z^{k}, B_{L}z^{k}) (5.3)

where the linear operator B_{L} : ℝ^{N} → S^{3} is defined by

B_{L}z = \{ \begin{matrix}Bz & if & z ∈ ker L, , \\ 0 & if & z ∈ (ker L)^{⊥} , \end{matrix}  (5.4)

satisfying the homogeneous Dirichlet or Neumann boundary condition and initial conditions

v^{k}(x, 0) = v^{0}(x) , ε^{k}(x, 0) = ε^{0}(x) , z^{k}(x, 0) = z_{k}^{0}(x)  (5.5)

where Lz_{k}^{0}(x) = Lz^{0}(x) and B_{L}z_{k}^{0}(x) = \frac{k}{k + 1} B_{L}z^{0}(x) . Thus we change the initial

conditions only for πz where π : ℝ^{N} → ℝ^{N} is the orthogonal projection on ker L . For all other components of the vector (v, ε, z) the initial conditions are independent of k and coincide with conditions 5.1. The definition of the operator B_{L} and assumption (5.2) yield that the free energy function (5.3) is positive definite and the problem (CMP) is coercive.

Remark: We say that the initial data are admissible for the system (CMP) if the right hand side of the system (CMP) evaluated for the initial data belongs to 𝕃^{2}(Ω; W) and the data satisfies the boundary condition.

Lemma 5.2 Let (v^{0}, ε^{0}, z^{0}) ∈ 𝕃^{2}(Ω; W) be admissible for the system (MP), div DB_{L}z^{0} ∈ 𝕃^{2}(Ω; ℝ^{3}) and for the Neumann boundary problem DB_{L}z^{0} · n_{|_{∂Ω}} = 0 ,

then the initial data (v^{0}, ε^{0}, z_{k}^{0}) where z_{k}^{0} = \frac{k}{k + 1} πz^{0} + (I − π)z^{0} is admissible for

the system (CMP). (Here π : ℝ^{N} → ℝ^{N} is the orthogonal projection on ker L).

Proof: The initial data z_{k}^{0} is so chosen, that the argument of the nonlinear function g for t = 0 is independent of k :

−ρ∇_{z}ψ^{k}(ε^{k}(0), z^{k}(0)) = −ρ∇_{z}ψ^{k}(ε^{0}, z_{k}^{0})
= B^{T} Dε^{0} − (B^{T} DB + L + \frac{1}{k}B_{L}^{T}DB_{L})(\frac{k}{k + 1} πz^{0} + (I − π)z^{0})
= B^{T} Dε^{0} − Lz^{0} − B^{T} DB((I − π)z^{0})

− B^{T} DB(\frac{k}{k + 1} πz^{0}) − \frac{1}{k + 1} B_{L}^{T}DB_{L}(πz^{0})

= B^{T} Dε^{0} − Lz^{0} − B^{T} DBz^{0} = −ρ∇_{z}ψ(ε^{0}, z^{0}) .

Moreover for the weak divergence div D(ε^{k}(0) − Bz^{k}(0)) we have

div D(ε^{0} − Bz_{k}^{0}) = div D(ε^{0} − Bz^{0}) + \frac{1}{k + 1} div DBπz^{0}
= div D(ε^{0} − Bz^{0}) + \frac{1}{k + 1} div DB_{L}z^{0} ∈ 𝕃^{2}(Ω; ℝ^{3}).

For the Neumann boundary problem the assumption DB_{L}z^{0} · n_{|_{∂Ω}} = 0 yields that the initial data (ε^{0}, z_{k}^{0}) satisfies the homogeneous Neumann boundary condition.

Lemma 5.3 For the initial data (v^{0}, ε^{0}, z^{0}) ∈ 𝕃^{2}(Ω; W) satisfying the assumption of Lemma 5.2 for all N ∋ k > 0 there exists the unique, strong, global in time solution for the problem (CMP).

Proof: This lemma is a consequence of Lemma 5.2, Theorem 3.1 and Theorem 3.3.

Before we study the limit case k → ∞ , we define a subclass of constitutive equations having a similar property to (4.16).

Definition 5.4 We say that the model (MP) is self-controlling if and only if there exists a function F : ℝ_{+}^{2} → ℝ_{+} mapping sets, bounded in at least one variable, into bounded sets such that for all functions y ∈ 𝕃^{2}(Ω; ℝ^{N}) with |y(x)| ≥ 1 for a.e. x ∈ Ω it holds that

∥B_{L}g(y)∥ ≤ F(∥Lg(y)∥, ∥y∥) ,  (5.6)

where the operator B_{L} is defined by (5.4).

Inequality (5.6) denotes that for self-controlling models we can control the right hand side of the inelastic constitutive equation for the variables B_{L}z by the right hand side for the variables Lz or by the argument ∇_{z}ψ of the nonlinear function g . Note that for the monotone Bodner–Partom model the function F is given by (4.16):

F(a, b) = ca^{\frac{β}{β+1}}

where c is a positive constant.

Theorem 5.5 Let us assume that the model (MP) is self-controlling and the initial data (v^{0}, ε^{0}, z^{0}) ∈ 𝕃^{2}(Ω; W) is admissible for the system (MP) with div DB_{L}z^{0} ∈ 𝕃^{2}(Ω; ℝ^{3}) and for the Neumann boundary condition with DB_{L}z^{0} · n_{|_{∂Ω}} = 0 . Then from the sequence (v^{k}, ε^{k}, z^{k}) defined in Lemma 5.3 we can select a subsequence (v^{k_{ℓ}}, ε^{k_{ℓ}}, z^{k_{ℓ}}) , which converges weak-∗ in W_{w}^{1,∞}([0, T); 𝕃^{2}(Ω; W)) for arbitrary T > 0 to a solution (v, ε, z) of the problem (MP). Moreover it holds that

(v^{k_{ℓ}}, ε^{k_{ℓ}} − Bz^{k_{ℓ}}, Lz^{k_{ℓ}}) → (v, ε − Bz, Lz) in the 𝕃^{2}-strong topology.

Proof: Similar to Theorem 4.5 we obtain

ess sup_{t∈[0,T]} ∥(v_{t}^{k}(t), ε_{t}^{k}(t), z_{t}^{k}(t)∥_{H_{k}} ≤ ∥C^{k}(v^{0}, ε^{0}, z_{k}^{0})∥_{H_{k}}  (5.7)

where the Hilbert space H_{k} and the operator C^{k} are defined similar to those in the proof of Theorem 4.5. For the right hand side of (5.7) we have

∥C^{k}(v^{0}, ε^{0}, z_{k}^{0})∥_{H_{k}} ≤ C\{∥div D(ε^{0} − Bz_{k}^{0})∥ + ∥ \frac{1}{2} (∇v^{0} + ∇^{T} v^{0}) − Bg(−ρ∇_{z}ψ^{k}(ε^{0}, z_{k}^{0}))∥

+ ∥Lg(−ρ∇_{z}ψ^{k}(ε^{0}, z_{k}^{0}))∥ + \frac{1}{\sqrt{k}} ∥B_{L}g(−ρ∇_{z}ψ^{k}(ε^{0}, z_{k}^{0}))∥\}

≤ C\{∥div D(ε^{0} − Bz^{0})∥ + \frac{1}{k + 1} ∥div DB_{L}z^{0}∥

+ ∥\frac{1}{2} (∇v^{0} + ∇^{T} v^{0}) − Bg(−ρ∇_{z}ψ(ε^{0}, z^{0}))∥ + ∥Lg(−ρ∇_{z}ψ(ε^{0}, z^{0}))∥

+ \frac{1}{\sqrt{k}} ∥B_{L}g(−ρ∇_{z}ψ(ε^{0}, z^{0}))∥\}.

Therefore there exists a positive constant C independent of k such that

ess sup_{t∈[0,T]} \{∥v_{t}^{k}(t)∥, ∥ε_{t}^{k}(t) − Bz_{t}^{k}(t)∥, ∥Lz_{t}^{k}(t)∥, \frac{1}{\sqrt{k}} ∥B_{L}z_{t}^{k}(t)∥\} ≤ C .  (5.8)

Now using that the model (MP) is self-controlling we obtain

∥B_{L}z_{t}^{k}(t)∥^{2} = ∥B_{L}g(−ρ∇_{z}ψ^{k}(ε^{k}(t), z^{k}(t)))∥^{2}
= ∫_{Ω∩{|y^{k}(t)|≤1}}|B_{L}g (−ρ∇_{z}ψ^{k}(ε^{k}(x, t), z^{k}(x, t)))|^{2}dx
+ ∫_{Ω∩{|y^{k}(t)|>1}}|B_{L}g(−ρ∇_{z}ψ^{k}(ε^{k}(x, t), z^{k}(x, t))}|^{2}dx
≤ C_{∗}^{2}|Ω| + F^{2}(∥Lg(y^{k}(t))∥, ∥y^{k}(t)∥)

where y^{k}(t) = −ρ∇_{z}ψ^{k}(ε^{k}(t), z^{k}(t)) , C_{∗} = sup_{y∈B_{N} (1)} |B_{L}g(y)| and B_{N}(1) denotes the unit ball in ℝ^{N} . Inequality (5.8) yields that the arguments ∥Lg(y^{k}(t))∥ and

∥y^{k}(t)∥ are bounded for a.e. t ∈ [0, T] . Then we can remove the factor \frac{1}{\sqrt{k}} in

inequality (5.8). Next we show with the energy method that the sequence (v^{k}, ε^{k} − Bz^{k}, Lz^{k}) is a Cauchy sequence. Let us denote by

E^{k,m}(t) = \frac{ρ}{2} ∥v^{k}(t) − v^{m}(t)∥^{2} + \frac{1}{2} ∫_{Ω} [D(ε^{k}(t) − Bz^{k}(t) − ε^{m}(t) + Bz^{m}(t))]

· (ε^{k}(t) − Bz^{k}(t) − ε^{m}(t) − Bz^{m}(t)) dx

+ \frac{1}{2} ∫_{Ω} [L(z^{k}(t) − z^{m}(t))] · (z^{k}(t) − z^{m}(t)) dx

the energy for the difference (v^{k}, ε^{k} − Bz^{k}, Lz^{k}) − (v^{m}, ε^{m} − Bz^{m}, Lz^{m}) . Then for the time derivative of the function E^{k,m}(t) we obtain similarly to (4.17)

\frac{d}{dt} E^{k,m}(t) = ∫_{Ω} ρ(v^{k}(t) − v^{m}(t))(v_{t}^{k}(t) − v_{t}^{m}(t)) dx

+ ∫_{Ω} [D(ε^{k}(t) − Bz^{k}(t) − ε^{m}(t) − Bz^{m}(t))]

· (ε_{t}^{k}(t) − Bz_{t}^{k}(t) − ε_{t}^{m}(t) + Bz_{t}^{m}(t)) dx

+ ∫_{Ω} [L(z^{k}(t) − z^{m}(t))] · (z_{t}^{k}(t) − z_{t}^{m}(t)) dx

(using the equation CMP1)

= − ∫_{Ω} [D(ε^{k}(t) − Bz^{k}(t) − ε^{m}(t) + Bz^{m}(t))] · (Bz_{t}^{k}(t) − Bz_{t}^{m}(t)) dx

+ ∫_{Ω} [L(z^{k}(t) − z^{m}(t))] · (z_{t}^{k}(t) − z_{t}^{m}(t)) dx

= − ∫_{Ω} [B^{T} D(ε^{k}(t) − Bz^{k}(t) − ε^{m}(t) + Bz^{m}(t))

+ B_{L}^{T}D(\frac{1}{k}B_{L}z^{k}(t) − \frac{1}{m}B_{L}z^{m}(t))] · (z_{t}^{k}(t) − z_{t}^{m}(t)) dx

+ ∫_{Ω} [L(z^{k}(t) − z^{m}(t))] · (z_{t}^{k}(t) − z_{t}^{m}(t)) dx

+ ∫_{Ω} [B_{L}^{T}D(\frac{1}{k}B_{L}z^{k}(t) − \frac{1}{m}B_{L}z^{m}(t))] · (z_{t}^{k}(t) − z_{t}^{m}(t)) dx

= − ∫_{Ω} [g(−ρ∇_{z}ψ^{k}(ε^{k}(t), z^{k}(t))) − g(−ρ∇_{z}ψ^{m}(ε^{m}(t), z^{m}(t)))]

· (−ρ∇_{z}ψ^{k}(ε^{k}(t), z^{k}(t)) + ρ∇_{z}ψ^{m}(ε^{m}(t), z^{m}(t))) dx

+ ∫_{Ω} [D(\frac{1}{k}B_{L}z^{k}(t) − \frac{1}{m}B_{L}z^{m}(t))] · (B_{L}z_{t}^{k}(t) − B_{L}z_{t}^{m}(t)) dx

(the monotonicity of g)

≤ C · (\frac{1}{k} + \frac{1}{m}) (∥B_{L}z^{k}(t)∥^{2} + ∥B_{L}z^{m}(t)∥^{2} + ∥B_{L}z_{t}^{k}(t)∥^{2} + ∥B_{L}z_{t}^{m}(t)∥^{2}) .

The boundedness of the sequence ∥B_{L}z^{k}(t)∥^{2} + ∥B_{L}z_{t}^{k}(t)∥^{2} uniformly for a.e. t ∈ [0, T) and the following estimate for the initial value of the function E^{k,m}(t)

E^{k,m}(0) = \frac{1}{2} ∫_{Ω} [D(Bz_{k}^{0} − Bz_{m}^{0})] · (Bz_{k}^{0} − Bz_{m}^{0}) dx
= \frac{1}{2} ∫_{Ω} [D(\frac{1}{k + 1}B_{L}z^{0} − \frac{1}{m + 1}B_{L}z^{0})]

· (\frac{1}{k + 1}B_{L}z^{0} − \frac{1}{m + 1}B_{L}z^{0}) dx

≤ C · (\frac{1}{k + 1} + \frac{1}{m + 1})^{2}∥B_{L}z^{0}∥^{2}

yields that the function E^{k,m}(t) is arbitrarily small for sufficiently large k, m . Finally we can select from the sequence (v^{k}, ε^{k}, z^{k}) a subsequence, which we again denote by (v^{k}, ε^{k}, z^{k}) satisfying

(v^{k}, ε^{k}, z^{k}) ⇀^{∗} (v, ε, z) , (v_{t}^{k}, ε_{t}^{k}, z_{t}^{k}) ⇀^{∗} (v_{t}, ε_{t}, z_{t})

in 𝕃_{w}^{∞}((0, T); 𝕃^{2}(Ω; W)) and

(v^{k}, ε^{k} − Bz^{k}, Lz^{k}) → (v, ε − Bz, Lz) in the space 𝕃^{∞}((0, T); 𝕃^{2}(Ω; W)) .

Moreover, perhaps again passing to a subsequence, we can assume that

∇_{z}ψ^{k}(ε^{k}(x, t), z^{k}(x, t)) → ∇_{z}ψ(ε(x, t), z(x, t))

pointwise for almost every (x, t) ∈ Ω × [0, T) . Thus using Lemma 4.1 the limit (v, ε, z) is a strong solution to the noncoercive problem (MP).

Similar to section 4 we can show using the energy method that strong solutions of the problem (MP) are unique. From the definition of the self-controlling models it is clear that for Lipschitz operators B_{L} we can prove convergence of the coercive approximation process (CMP) to the solution of the problem (MP).

6 Another coercive approximation

In this section we define and study another type of coercive approximation. The idea of the approximation process is again a consequence of mechanical considerations. Let us consider again a general noncoercive model of monotone type. We want to approximate this model by a sequence of coercive models, for which the defintition is motivated by the following rheological diagram:

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................  ............................................................................................................................................................................................................................................................................................................................................................................................................................... ........................................................................................................................................................................................................ general noncoercive model coercive approximation \frac{1}{k} Dε^{k} ..........................................................................................................................................................................................................................................................................................................................................................................................................

Let k be a natural number greater than zero. Thus the approximate system of equations is of the form

\begin{aligned}v_{t}^{k} & = & div \frac{1}{ρ}D(ε^{k} − Bz^{k} + \frac{1}{k}ε^{k}) , \\ ε_{t}^{k} & = & \frac{1}{2}(∇v^{k} + ∇^{T} v^{k}) , \\ z_{t}^{k} & ∈ & g(−ρ∇_{z}ψ^{k}(ε^{k}, z^{k})) \end{aligned}  (ACA)

with the homogenous Dirichlet boundary condition

v^{k}_{|_{x∈∂Ω}} = 0 ,

or with the homogeneous Neumann boundary condition

D(ε^{k} − Bz^{k} + \frac{1}{k}ε^{k}) · n_{|_{x∈∂Ω}} = 0 ,

and with the initial conditions

v(0) = v^{0} , ε(0) = ε^{0} , z(0) = z^{0} .

Here g : ℝ^{N} → P(ℝ^{N}) is a maximal montone mapping with 0 ∈ g(0) , and the free energy function ψ^{k} is defined by

ρψ^{k}(ε^{k}, z^{k}) = \frac{1}{2}[D(ε^{k} − Bz^{k})] · (ε^{k} − Bz^{k}) + \frac{1}{2}(Lz^{k})z^{k} + \frac{1}{2k} [D(ε^{k})] · ε^{k} (6.1)

with the given linear operators B : ℝ^{N} → S^{3} and L : ℝ^{N} → ℝ^{N} . If the operator L is symmetric and positive semi–definite and the operator M = B^{T} DB + L is positive definite then the free energy ψ^{k} is a positive definite quadratic form on S^{3} × ℝ^{N} . Thus we see that the problem (ACA) is coercive, but this system of equations is not of the form which we studied in Section 3. The first equation (ACA 1) contains the perturbation term \frac{1}{k} div \frac{1}{ρ} Dε^{k} . Therefore we must prove an analogous theorem to Theorem 3.1 for the problem (ACA). Using the notation of Section 3 let us define the operator

C^{k} : 𝕃^{2}(Ω; W) → P(𝕃^{2}(Ω; W)); W = ℝ^{3} × S^{3} × ℝ^{N}

by C^{k}(v, ε, z) = ∅ if v ∈ 𝕃^{2}(Ω; ℝ^{3})\ℍ_{1}(Ω; ℝ^{3}) and

C^{k}(v, ε, z) = \{w = (w_{1}, w_{2}, w_{3}) ∈ 𝕃^{2}(Ω; W) | w_{1} = div \frac{1}{ρ} D(ε − Bz + \frac{1}{k} ε)

w_{2} = \frac{1}{2} (∇v − ∇^{T} v) , w_{3}(x) ∈ g(−ρ∇_{z}ψ^{k}(ε^{k}(x), z^{k}(x))) a.e. in Ω\} (6.2)

if v ∈ ℍ_{1}(Ω; ℝ^{3}) . Moreover for the Dirichlet boundary condition we require additionally that v ∈ ℍ^{∘}_{1}(Ω; ℝ^{3}) and for the Neumann boundary value problem that the weak divergence satisfies

(div D(ε − Bz + \frac{1}{k} ε), u) = −(D(ε − Bz + \frac{1}{k} ε) , ∇u) for all u ∈ ℍ_{1}(Ω; ℝ^{3}) .

(6.3)

Theorem 6.1 Let Ω ⊂ ℝ^{3} be an open bounded set, and for the Neumann boundary value problem let additionally ∂Ω be Lipschitz. Then the operator −C^{k} is maximal monotone, if the space 𝕃^{2}(Ω; W) is equipped with the scalar product

⟨(v, ε, u) , (\overline{v}, \overline{ε}, \overline{z})⟩ = ∫_{Ω} (ρv\overline{v} + [D(ε−Bz)](\overline{ε}−B\overline{z})+ \frac{1}{k}[Dε]·\overline{ε}+(Lz)·\overline{z})dx . (6.4)

Remark: The proof of Theorem 3.1 presented in [3] uses that the operator L is positive definite. In the problem (ACA) we have that L is only positive semi–definite, but the first equation (ACA1) yields noncoerciveness of the problem. Therefore the proof presented below is only a modification of the proof of Theorem 3.1 from [3].

Proof: First we show that the operator −C^{k} is monotone in the space 𝕃^{2}(Ω; W) with the scalar product defined by (6.4). Let (v, ε, z) , (\overline{v}, \overline{ε}, \overline{z}) ∈ D(C^{k}) and w = (w_{1}, w_{2}, w_{3}) ∈ C^{k}(v, ε, z) , \overline{w} = (\overline{w}_{1}, \overline{w}_{2}, \overline{w}_{3}) ∈ C^{k}(\overline{v}, \overline{ε}, \overline{z}) . Then we compute that

⟨w − \overline{w}, (v, ε, u) − (\overline{v}, \overline{ε}, \overline{z})⟩ = ∫_{Ω} \{ρ(w_{1} − \overline{w}_{1})(v − \overline{v})

+ [D(w_{2} − \overline{w}_{2} − Bw_{3} + B\overline{w}_{3})](ε − \overline{ε} − Bz − B\overline{z})

+ \frac{1}{k} [D(w_{2} − \overline{w}_{2})](ε − \overline{ε}) + [L(w_{3} − \overline{w}_{3})] · (z − \overline{z})\}dx

(using definiton (6.2))

= ∫_{Ω} \{div D(ε − \overline{ε} − Bz + B\overline{z} + \frac{1}{k}\overline{ε}) · (v − \overline{v})

+ [D(\frac{1}{2}(∇(v − \overline{v}) + ∇^{T} (v − \overline{v}))) − B(w_{3} − \overline{w}_{3})](ε − \overline{ε} − Bz − B\overline{z})

+ \frac{1}{k} [D(\frac{1}{2}(∇(v − \overline{v}) + ∇^{T} (v − \overline{v})))](ε − \overline{ε}) + [L(w_{3} − \overline{w}_{3})](z − \overline{z})\}dx

= ∫_{Ω} \{−[DB(w_{3} − \overline{w}_{3})] · (ε − \overline{ε} − Bz + B\overline{z}) + L(w_{3} − \overline{w}_{3})(z − \overline{z})\}dx
= ∫_{Ω}(w_{3} − \overline{w}_{3}) · [B^{T} D(ε − \overline{ε} − B(z − \overline{z})) − L(z − \overline{z})]dx
= − ∫_{Ω}(w_{3} − \overline{w}_{3}) · [ − ρ∇_{z}ψ^{k}(ε, z) + ρ∇_{z}ψ^{k}(\overline{ε}, \overline{z})]dx ≤ 0

Since w_{3}(x) ∈ g(−ρ∇_{z}ψ^{k}(ε(x), z(x)) and \overline{w}_{3}(x) ∈ g(−ρ∇_{z}ψ^{k}(\overline{ε}(x), \overline{z}(x))) for a.e. x ∈ Ω .

To prove that the monotone operator −C^{k} is maximal monotone we use the theorem of Minty. We show that for all λ > 0 the operator λI −C^{k} is surjective. Let r = (r_{1}, r_{2}, r_{3}) ∈ 𝕃^{2}(Ω; W) . We should prove that there exists w = (v, ε, z) ∈ D(C^{k}) such that

−div \frac{1}{ρ} D(ε − Bz + \frac{1}{k}ε) + λv = r_{1} ,  (6.5)
−\frac{1}{2}(∇v + ∇^{T} v) + λε = r_{2} ,  (6.6)
−g(−ρ∇_{z}ψ^{k}(ε, z)) + λz ∋ r_{3} .  (6.7)

First we study the differential inclusion (6.7). We assume that ε is given and solve (6.7) for z. Using the relation −ρ∇_{z}ψ^{k}(ε, z) = B^{T} Dε − Mz we obtain

− r_{3} + λM^{−1}B^{T} Dε ∈ g(B^{T} Dε − Mz) + λM^{−1}(B^{T} Dε − Mz) .  (6.8)

Thus the solvability of (6.8) is ensured by the maximal monotonicity of g. Moreover the solution z(x) is unique and depends continuously on r_{3}(x) and ε(x) . Let us denote by G(ε, r_{3}) the solution z of (6.8). We consider now the case z = G(ε, r_{3}) and \overline{z} = G(\overline{ε}, r_{3}) . Then we have

−r_{3} + λz ∈ g(B^{T} Dε − Mz) and − r_{3} + λ\overline{z} ∈ g(B^{T} D\overline{ε} − M\overline{z}) .

By the monotonicity of g we obtain

0 ≤ [(B^{T} Dε − Mz) − (B^{T} D\overline{ε} − M\overline{z})] · λ(z − \overline{z})
= −λ[M(z − \overline{z})](z − \overline{z}) + λ[D(ε − \overline{ε})] · B(z − \overline{z}) .

Then the assumption λ > 0 yields

[M(z − \overline{z})] · (z − \overline{z}) = [B^{T} DB(z − \overline{z})](z − \overline{z}) + [L(z − \overline{z})](z − \overline{z})
≤ [D(ε − \overline{ε})] · B(z − \overline{z}) .  (6.9)

(6.9) and L ≥ 0 imply

∫_{Ω} [DB(z − \overline{z})[ B(z − \overline{z}) dx ≤ ∫_{Ω} [D(ε − \overline{ε})] B(z − \overline{z}) dx

≤ ( ∫_{Ω} [D(ε − \overline{ε})] · (ε − \overline{ε}) dx)^{1/2} ( ∫_{Ω} [D(Bz − B\overline{z})] · B(z − \overline{z}) dx)^{1/2} Let us denote by ∥ε∥_{D} = ( ∫_{Ω}(Dε) · ε dx)^{1/2} and by ∥ε∥_{D_{k}} = ( ∫_{Ω} (1 + \frac{1}{k} )(Dε) ·

ε dx)^{1/2} . Then using the notation we have

∥BG(ε, r_{3}) − BG(\overline{ε}, r_{3})∥_{D} ≤ ∥ε − \overline{ε}∥_{D} ≤ (\frac{k}{k + 1})^{1/2} ∥ε − \overline{ε}∥_{D_{k}} .  (6.10)

Next we solve (6.5) and (6.6) for all (r_{1}, r_{2}) ∈ 𝕃^{2}(Ω; ℝ^{3} × S^{3}), assuming that z ∈ 𝕃^{2}(Ω; ℝ^{N}) is given. We solve (6.6) for ε

ε = \frac{1}{λ}(\frac{1}{2}(∇v + ∇^{T}) + r_{2})

and insert the result into (6.5). Thus we must find v ∈ ℍ_{1}(Ω; ℝ^{3}) , which satisfies

\begin{matrix}∫_{Ω} \{\frac{1}{λρ} [(1 + \frac{1}{k} ) D(\frac{1}{2}(∇v + ∇^{T} v)) · (\frac{1}{2}(∇u + ∇^{T} u)) + λv · u\} dx \\ = ∫_{Ω} \{\frac{1}{ρ}[D(Bz − \frac{1}{λ}(1 + \frac{1}{k} )r_{2})] · ∇u + r_{1} · u\} dx \end{matrix} (6.11)

for all u ∈ ℍ^{∘}_{1}(Ω; ℝ^{3}) in the case of the Dirichlet boundary condition, and for all u ∈ ℍ_{1}(Ω; ℝ^{3}) in the case of the Neumann boundary value problem. The solvability of (6.11) follows from the theorem of Lax-Milgram. Moreover for (r_{1}, r_{2}) = 0 we obtain from (6.11) setting u = v

\begin{matrix}λ ∫_{Ω} \{\frac{1}{ρ} [(1 + \frac{1}{k} )Dε] · ε + |v|^{2}\} dx \\ = ∫_{Ω} \frac{1}{ρ}(DBz) · ∇v dx = λ ∫_{Ω} \frac{1}{ρ} (DBz) · ε dx . \end{matrix}  (6.12)

Let us define the linear operator R : 𝕃^{2}(Ω; ℝ^{N} × ℝ^{3} × S^{3}) → 𝕃^{2}(Ω; S^{3}) by R(z, r_{1}, r_{2}) = ε, where ε is the solution of (6.5) and (6.6) with given functions z, r_{1}, r_{2} . Then (6.12) implies

∥R(z, 0, 0)∥_{D_{k}} ≤ ∥Bz∥_{D} .  (6.13)

Now for given (r_{1}, r_{2}, r_{3}) ∈ 𝕃^{2}(Ω; W) we define the mapping P : 𝕃(Ω; S^{3}) → 𝕃^{2}(Ω; S^{3}) by

P(ε) = R(G(ε, r_{3}), r_{1}, r_{2}) .

The inequalities (6.10) and (6.13) and the linearity of the operator R yield

∥P(ε) − P(\overline{ε})∥_{D_{k}} = ∥R(G(ε, r_{3}) − G(\overline{ε}, r_{3}), 0, 0)∥_{D_{k}}
≤ ∥BG(ε, r_{3}) − BG(\overline{ε}, r_{3})∥_{D} ≤ (\frac{k}{k + 1})^{1/2} ∥ε − \overline{ε}∥_{D_{k}} .

Therefore the mapping P is a contraction in 𝕃^{2}(Ω; S^{3}) equipped with the norm ∥·∥_{D_{k}} . Then P has a unique fixed point ε^{∗} ∈ 𝕃^{2}(Ω; S^{3}) . If we define now z = G(ε^{∗}, r_{3}) and (v, ε) as the solution of (6.5) and (6.6) with z = G(ε^{∗}, r_{3}) then

ε = R(G(ε^{∗}, r_{3}), r_{1}, r_{2}) = P(ε^{∗}) = ε^{∗} .

Thus w = (v, ε^{∗}, z) is the solution of (6.5), (6.6) and (6.7) and the proof is complete.

Corollary 6.2 Let the initial data (v^{0}, ε^{0}, z^{0}) be admissible for the problem (MP) and additionally div Dε^{0} ∈ 𝕃^{2}(Ω; ℝ^{3}) and for the Neumann boundary condition Dε^{0} · n_{|_{x∈∂Ω}} = 0 . Then the sequence of solutions (v^{k}, ε^{k}, z^{k}) of the problem (ACA)

converges weak-∗ in the space 𝕎^{1,∞}((0, T); 𝕃^{2}(Ω; W)) to a solution of the problem (MP) for all T > 0 provided the model (MP) is self-controlling.

Proof: The proof of this result is similar to the proof of Theorem 5.5. Only the estimation for differences of two solutions (v^{k}, ε^{k}, z^{k})−(v^{m}, ε^{m}, z^{m}) is different. Thus with the notation

E^{k,m}(t) = \frac{ρ}{2}∥v^{k}(t) − v^{m}(t)∥^{2}

+ \frac{1}{2} ∫_{Ω} [D(ε^{k}(t) − Bz^{k}(t) − ε^{m}(t) + Bz^{m}(t))]

· (ε^{k}(t) − Bz^{k}(t) − ε^{m}(t) + Bz^{m}(t)) dx

+ \frac{1}{2} ∫_{Ω} [L(z^{k}(t) − z^{m}(t))] · (z^{k}(t) − z^{m}(t)) dx

we obtain

\frac{d}{dt} E^{k,m}(t) = − ∫_{Ω} [D(ε^{k}(t) − Bz^{k} − ε^{m}(t) + Bz^{m}(t))] · (Bz_{t}^{k}(t) − Bz_{t}^{m}(t)) dx

+ ∫_{Ω} [L(z^{k}(t) − z^{m}(t))] · (z_{t}^{k}(t) − z_{t}^{m}(t)) dx

− ∫_{Ω} (\frac{1}{k} ε^{k}(t) − \frac{1}{m} ε^{m}(t)) (ε_{t}^{k}(t) − ε_{t}^{m}(t)) dx

= − ∫_{Ω} [g(−ρ∇_{z}ψ^{k}(ε^{k}(t), z^{k}(t))) − g(−ρ∇_{z}ψ^{m}(ε^{m}(t), z^{m}(t)))]

· (−ρ∇_{z}ψ(ε^{k}(t), z^{k}(t)) + ρ∇_{z}ψ^{m}(ε^{m}(t), z^{m}(t))) dx

− ∫_{Ω} (\frac{1}{k} ε^{k}(t) − \frac{1}{m} ε^{m}(t))(ε_{t}^{k}(t) − ε_{t}^{m}(t)) dx

(the monotonicity of g)

≤ C · (\frac{1}{k} + \frac{1}{m})(∥ε^{k}(t)∥^{2} + ∥ε^{m}(t)∥^{2} + ∥ε_{t}^{k}(t)∥^{2} + ∥ε_{t}^{m}(t)∥^{2}) .

For the self-controlling models we have

∥ε^{k}(t)∥^{2} + ∥ε_{t}^{k}(t)∥^{2} ≤ ∥ε^{k}(t) − Bz^{k}(t)∥^{2} + ∥ε_{t}^{k}(t) − Bz_{t}^{k}(t)∥^{2}
+ ∥Bz^{k}(t)∥^{2} + ∥Bz_{t}^{k}(t)∥^{2}
≤ C + ∥B_{L}^{k}z^{k}(t)∥^{2} + ∥B_{L}z_{t}^{k}(t)∥^{2} ≤ C(T) .

Therefore the function E^{k,m}(t) is arbitrarily small for sufficiently large k, m.

7 Another noncoercive example

We want to apply the approximation process (ACA) to a model which is not self– controlling. The system of equations is

\begin{aligned}v_{t} & = & div \frac{1}{ρ}(D(ε − ε^{p})) , \\ ε_{t} & = & \frac{1}{2}(∇v + ∇^{T} v) , \\ ε_{t}^{p} & = & d|σ|^{β}σ \end{aligned}  (NSCP)

where σ = P D(ε − ε^{p}) = D(ε − ε^{p}) − \frac{1}{3} tr (D(ε − ε^{p})) · I and β > 0 . As usual we assume that the homogeneous Dirichlet or Neumann boundary condition is satisfied, and the initial conditions are

v(0) = v^{0} , ε(0) = ε^{0} , ε^{p}(0) = ε^{p,0} with tr ε^{p,0}(x) = 0 for a.e. x ∈ Ω . (7.1)

In this model the vector z of internal variables contains only ε^{p}. Then the free energy function is defined by

ρψ(ε, ε^{p}) = \frac{1}{2}[D(ε − ε^{p})] · (ε − ε^{p})  (7.2)

and the operator M : S^{3} → S^{3} is given by Mε^{p} = Dε^{p} , since Bε^{p} = ε^{p} and Lε^{p} = 0 . Thus the function ψ is only positive semi–definite and the model (NSCS) is noncoercive. Moreover the right hand side of the equation (NSCS 3) is not controlled by the argument σ in the 𝕃^{2}–topology (note β > 0) , and the operator L ≡ 0 . This denotes that the model (NSCP) does not belong to the class of self–controlling models. We approximate the problem (NSCP) using the process (ACA)

\begin{aligned}v_{t}^{k} & = & div \frac{1}{ρ} (D(ε^{k} − ε^{p,k} + \frac{1}{k}ε^{k})) , \\ ε_{t}^{k} & = & \frac{1}{2} (∇v^{k} + ∇^{T} v^{k}) , \\ ε_{t}^{p,k} & = & d|σ^{k}|^{β}σ^{k} \end{aligned}  (ACA)

where σ^{k} = P D(ε^{k} − ε^{p,k}) with the homogeneous Dirichlet or Neumann boundary condition and with the initial conditions given by (7.1) for all natural numbers k > 0 . The free energy function

ρψ^{k}(ε^{k}, ε^{p,k}) = \frac{1}{2}[D(ε^{k} − ε^{p,k})] · (ε^{k} − ε^{p,k}) + \frac{1}{2k} (Dε^{k}) · ε^{k} .

is connected with this approximate problem. This function is positive definite on S^{3} × S^{3} and by Theorem 6.1 there exists the unique global in time strong solution (v^{k}, ε^{k}, ε^{p,k}) , provided that the initial data (v^{0}, ε^{0}, ε^{p,0}) satisfies:

A1. (div (D(ε^{0} − ε^{p,0} + \frac{1}{k} ε^{0})) , \frac{1}{2} (∇v^{0} + ∇^{T} v^{0}) , |σ^{0}|^{β}σ^{0}) ∈ 𝕃^{2}(Ω; W) , where W = ℝ^{3} × S^{3} × S^{3} and σ = P D(ε^{0} − ε^{p,0}) , A2. for the Dirichlet–boundary condition v^{0} ∈ ℍ^{∘}_{1}(Ω; ℝ^{3}) and for the Neumann

boundary value problem it holds that

∀_{u∈ℍ_{1}(Ω;ℝ^{3})} (div D(ε^{0} − ε^{p,0} + \frac{1}{k} ε^{0}), u) = −(D(ε^{0} − ε^{p,0} + \frac{1}{k} ε^{0}), ∇u) .

Now we want to study the limit case of k → ∞ . Thus we assume additionally that the initial data (v^{0}, ε^{0}, ε^{p,0}) satisfies

A3. div Dε_{0} ∈ 𝕃^{2}(Ω; ℝ^{3}) and for the Neumann boundary condition Dε^{0}·n_{|_{x∈∂Ω}} = 0 .

Theorem 7.1 Let the initial data (v^{0}, ε^{0}, ε^{p,0}) ∈ 𝕃^{2}(Ω; W) satisfy A1 – A3 . Then the sequence (v^{k}, ε^{k}, ε^{p,k}) of solutions for the problem (ACA) contains a subsequence which converges to the solution (v, ε, ε^{p}) of the problem (NSCP) in the following topology:

v^{k} ⇀^{∗} v in 𝕎^{1,∞}((0, T); 𝕃^{2}(Ω; ℝ^{3})) ,
ε^{k} ⇀^{∗} ε in 𝕎^{1,∞}((0, T); 𝕃^{p}(Ω; S^{3})) ,
ε^{p,k} ⇀^{∗} ε^{p} in 𝕎^{1,∞}((0, T); 𝕃^{p}(Ω; S^{3})) ,
σ^{k} ⇀^{∗} σ in 𝕃^{∞}((0, T); 𝕃^{β+2}(Ω; S^{3}))

where p = \frac{β+2}{β+1} is the conjugate exponent to β + 2 .

Proof: We start similarly as in the proof of Theorem 4.5 and obtain

ess sup_{t∈[0,T]} \{∥v_{t}^{k}(t)∥ , ∥ε_{t}^{k} − ε_{t}^{p,k}∥ , \frac{1}{\sqrt{k}} ∥ε_{t}^{k}∥\} ≤ C ,  (7.3)

where C is a positive constant, which does not depend on k and T . This result allows us to prove that the sequence (v^{k}, ε^{k} −ε^{p,k}) contains a subsequence, which converges weak-∗ in the space 𝕎_{w}^{1,∞}((0, T); 𝕃^{2}(Ω; ℝ^{3} × S^{3})) to a solution (v, ε − ε^{p}) of the first equation (NSCP 1). (Here ε−ε^{p} is only a symbol and does not denote the difference of the limits of the sequences {ε^{k}} and {ε^{p,k}}.) To prove a similar result for the equations (NSCP 2) and (NSCP 3) we need some estimates for the sequence {ε_{t}^{k}} or for the sequence {ε_{t}^{p,k}} , which are independent of k . For the self–controlling models this additional information gave us the special form of the inelastic constitutive equations. For the system (NSCP) we must use another method. If we define by

E^{k}(t) = \frac{ρ}{2} ∥v^{k}(t)∥^{2} + \frac{1}{2} ∫_{Ω} [D(ε^{k} − ε^{βk})] · (ε^{k} − ε^{p,k})dx + \frac{1}{2k} ∫_{Ω}(Dε^{k}) · ε^{k}dx

the energy function for the problem (ACA), then we obtain for the time derivative

\frac{d}{dt} E^{k}(t) = ∫_{Ω} ρv^{k}v_{t}^{k}dx + ∫_{Ω} [D(ε^{k} − ε^{p,k})] (ε_{t}^{k} − ε_{t}^{p,k})dx + \frac{1}{k} ∫_{Ω}(Dε^{k}) · ε_{t}^{k}dx
= − ∫_{Ω} [D(ε^{k} − ε^{p,k})] · ε_{t}^{p,k}dx = − ∫_{Ω} d|σ^{k}|^{β+2}dx .  (7.4)

Integrating (7.4) we have

E^{k}(t) + ∫_{0}^{t} ∫_{Ω} d|σ^{k}|^{β+2} dxdτ = E^{k}(0)
= \frac{ρ}{2} ∥v^{0}∥^{2} + \frac{1}{2} ∫_{Ω} [D(ε^{0} − ε^{p,0})] · (ε^{0} − ε^{p,0})dx + \frac{1}{2k} ∫_{Ω}(Dε^{0}) · ε^{0}dx . (7.5)

Thus the sequence {σ^{k}} is bounded in the space 𝕃^{β+2}((0, t)×Ω; S^{3}) for all t ∈ [0, T] . Moreover from (7.4) we conclude that

ess sup_{t∈[0,T]} ∫_{Ω} |σ^{k}|^{β+2}dx ≤ c · ess sup_{t∈[0,T]} {∥v^{k}∥ · ∥v_{t}^{k}∥

+ ∥ε^{k} − ε^{p,k}∥ · ∥ε_{t}^{k} − ε_{t}^{p,k}∥ + \frac{1}{\sqrt{k}} ∥ε^{k}∥ · \frac{1}{\sqrt{k}} ∥ε_{t}^{k}∥\}

≤ (by (7.3)) ≤ C(T)  (7.6)

where C(T) is a positive constant undependent of k. This yields that the sequence {σ^{k}} is bounded in the space 𝕃^{∞}((0, T); 𝕃^{β+2}(Ω; S^{3})) . Now for the right hand side of the equation (ACA 3) holds

ess sup_{t∈[0,T]} ∫_{Ω} |ε_{t}^{p,k}|^{\frac{β+2}{β+1}} dx = ess sup_{t∈[0,T]} ∫_{Ω} d^{\frac{β+2}{β+1}} |σ^{k}|^{β+2} ≤ C(T) · d^{\frac{β+2}{β+1}} .  (7.7)

Inequality (7.7) denotes that the sequence {ε_{t}^{p,k}} is bounded in 𝕃^{∞}((0, T); 𝕃^{p}(Ω; S^{3})) where p = \frac{β+2}{β+1} and by (7.3) the same result holds for the sequence \{ε_{t}^{k}\} . Therefore the sequence (v^{k}, ε^{k}, ε^{p,k}) contains a subsequence (v^{k_{i}}, ε^{k_{i}}, ε^{p,k_{i}}) , which we denote again by (v^{k}, ε^{k}, ε^{p,k}) satisfying

v_{t} ↼^{∗} v_{t}^{k} = div \frac{1}{ρ} D(ε^{k} − ε^{p,k} + \frac{1}{k} ε^{k}) ⇀^{∗} div \frac{1}{ρ} D(ε − ε^{p}) in 𝕃^{∞}((0, T); 𝕃^{2}(Ω; ℝ^{3})) ,
ε_{t} ↼^{∗} ε_{t}^{k} = \frac{1}{2} (∇v^{k} + ∇^{T} v^{k}) ⇀^{∗} \frac{1}{2} (∇v + ∇^{T} v) in 𝕃^{∞}((0, T); 𝕃^{p}(Ω; S^{3})) ,
ε_{t}^{p} ↼^{∗} ε_{t}^{p,k} = d|σ^{k}|σ^{k} ⇀^{∗} χ in 𝕃^{∞}((0, T); 𝕃^{p}(Ω; S^{3}))

and moreover σ^{k} ⇀^{∗} σ in 𝕃^{∞}((0, T); 𝕃^{β+2}(Ω; S^{3})) . Here v = w^{∗} − lim v^{k} in 𝕃^{∞}((0, T); 𝕃^{2}(Ω; ℝ^{3})) , ε = w^{∗}−lim ε^{k} in 𝕃^{∞}((0, T); 𝕃^{p}(Ω; S^{3})) and ε^{p} = w^{∗}−lim ε^{p,k} in 𝕃^{∞}((0, T); 𝕃^{p}(Ω; S^{3})) . To end the proof we must show that χ = d · |σ|^{β}σ . For self–controlling models we have proved that the argument −ρ∇_{z}ψ^{k}(ε^{k}, z^{k}) of the nonlinear function g converges in the strong 𝕃^{2}–topology. Here we have only the weak convergence of the sequence {σ^{k}} in 𝕃^{β+2}(Ω; S^{3}) and the weak convergence of the nonlinearity |σ^{k}|^{β} · σ^{k} in 𝕃^{p}(Ω; S^{3}) . To prove that in fact the sequence |σ^{k}|^{p} · σ^{k} converges weakly to |σ|^{β} · σ we use the method presented by J. L. Lions in [19]. In the book [13] this method is called the Minty–Browder method. The limit functions (v, ε, ε^{p}) satisfy the system

\begin{aligned}v_{t} & = & div \frac{1}{ρ} D(ε − ε^{p}) & ∈ 𝕃^{2}(Ω; ℝ^{3}) , \\ ε_{t} & = & \frac{1}{2} (∇v + ∇^{T} v) & ∈ 𝕃^{p}(Ω; S^{3}) , \\ ε_{t}^{p} & = & χ & ∈ 𝕃^{p}(Ω; S^{3}) . \end{aligned}  (7.8)

Moreover for the trace of the strain velocity we obtain tr ε_{t} = tr (ε_{t}−ε_{t}^{p}) ∈ 𝕃^{2}(Ω; ℝ) . Therefore the integral ∫_{Ω} D(ε−ε^{p})ε_{t}dx = ∫_{Ω} P D(ε−ε^{p})P ε_{t}dx+\frac{1}{3} ∫_{Ω} trD(ε−ε^{p})·trε_{t}dx is well defined. (Note P D(ε − ε^{p}) = σ ∈ 𝕃^{β+2}(Ω; S^{3})) . Thus if we denote by

E(t) = \frac{ρ}{2}∥v(t)∥^{2} + \frac{1}{2} ∫_{Ω} [D(ε(t) − ε^{p}(t))] · (ε(t) − ε^{p}(t))dx

the energy function for the system (7.8) then similar to (7.4) and (7.5) we obtain

E(t) + ∫_{0}^{t} ∫_{Ω} χ · σdxdτ = E(0) = \frac{ρ}{2}∥v^{0}∥^{2} + \frac{1}{2} ∫_{Ω} [D(ε^{0} − ε^{p,0})](ε^{0} − ε^{p,0})dx . (7.9)

The equalities (7.9) and (7.5) yield

\frac{ρ}{2}∥v^{k}(t)∥^{2} + \frac{1}{2} ∫_{Ω} [D(ε^{k}(t) − ε^{p,k}(t))] · (ε^{k}(t) − ε^{p,k}(t))dx

+ ∫_{0}^{t} ∫_{Ω} d|σ^{k}|^{β+2}dxdt − \frac{1}{2k} ∫_{Ω}(Dε^{0}) · ε^{0}dx  (7.10)

≤ E(t) + ∫_{0}^{t} ∫_{Ω} χ · σdx · dτ .

The convexity of the function E(t) implies

lim inf_{k→∞} (\frac{ρ}{2}∥v^{k}(t)∥^{2} + \frac{1}{2} ∫_{Ω} [D(ε^{k}(t) − ε^{p,k}(t))] · (ε^{k}(t) − ε^{p,k})(t))dx) ≥ E(t) .

Then from inequality (7.10) we obtain for all t ∈ [0, T]

lim inf_{k→∞} ( ∫_{0}^{t} ∫_{Ω} d|σ^{k}|^{β+2}dxdτ − \frac{1}{2k} ∫_{Ω}(Dε^{0}) · ε^{0}dx)

= lim inf_{k→∞} ( ∫_{0}^{t} ∫_{Ω} d|σ^{k}|^{β+2}dxdτ ) ≤ ∫_{0}^{t} ∫_{Ω} χ · σdxdτ .  (7.11)

Let us choose a function α ∈ 𝕃^{β+2}((0, t) × Ω; S^{3}) . Then the monotonicity of the nonlinear operator |σ^{k}|^{β}σ^{k} yields

lim inf_{k→∞} ∫_{0}^{t} ∫_{Ω}(d|σ^{k}|^{β}σ^{k} − d|α|^{β}α)(σ^{k} − α)dxdτ ≥ 0 .

Using (7.11) and the weak convergence d|σ^{k}|^{β}σ^{k} ⇀ χ in 𝕃^{p}((0, t)×Ω; S^{3}) and σ^{k} ⇀ σ in 𝕃^{β+2}((0, t) × Ω; S^{3}) we have

∫_{0}^{t} ∫_{Ω}(χ − d|α|^{β}α)(σ − α)dxdτ ≥ 0 .  (7.12)

If we now choose α in the form σ − λw with λ > 0 and w ∈ 𝕃^{β+2}((0, t) × Ω; S^{3}) then from (7.12) follows

λ ∫_{0}^{t} ∫_{Ω} (χ − d|σ − λw|^{β}(σ − λw)) · wdxdτ ≥ 0 .

Upon canceling λ we pass to the limit λ → 0^{+} and obtain

∫_{0}^{t} ∫_{Ω}(χ − d|σ|^{β}σ) · wdxdτ ≥ 0 .

Replacing w by −w yields χ = d|σ|^{β}σ for a.e. (x, t′) ∈ Ω × (0, t) .

Corollary 7.2 The sequence (v^{k}, ε^{k}, ε^{p,k}) of solutions for the problem (ACA) contains a subsequence (v^{k_{l}}, ε^{k_{l}}, ε^{p,k_{l}}) such that for all t ∈ [0, T]

\begin{matrix}v^{k_{l}}(t) & ⟶ & v(t) & in 𝕃^{2}(Ω; ℝ^{3}) \\ ε^{k_{l}}(t) − ε^{p,k_{l}}(t) & ⟶ & ε(t) − ε^{p}(t) & in 𝕃^{2}(Ω; S^{3}) \end{matrix}

where (v, ε, ε^{p}) is the solution from Theorem 7.1.

Proof: In the proof of Theorem 7.1 we have obtained that

|σ^{k}|^{β}σ^{k} ⇀^{∗} |σ|^{β}σ in 𝕃^{∞}((0, T); 𝕃^{p}(Ω; S^{3}))

(here σ = P D(ε−ε^{p}) and p = \frac{β+2}{β+1}). The convexity of the norm in 𝕃^{β+2}((0, t)×Ω; S^{3}) yields

lim inf_{k→∞} ( ∫_{0}^{t} ∫_{Ω} d|σ^{k}|^{β+2}dxdτ ) ≥ ∫_{0}^{t} ∫_{Ω} |σ|^{β+2} dxdτ .  (7.13)

Then (7.11) and (7.13) imply for all t ∈ [0, T]

lim inf_{k→∞} ( ∫_{0}^{t} ∫_{Ω} d|σ^{k}|^{β+2}dxdτ ) = ∫_{0}^{t} ∫_{Ω} |σ|^{β+2} dxdτ .

Thus passing to the limit inferior for k → ∞ in inequality (7.10) and using the convexity of E(t) we obtain

lim inf_{k→∞} (\frac{ρ}{2}∥v^{k}(t)∥^{2} + \frac{1}{2} ∫_{Ω} [D(ε^{k}(t) − ε^{p,k}(t))] · (ε^{k}(t) − ε^{p,k}(t))dx) = E(t) .

The last result and the weak convergence v^{k}(t) ⇀ v(t) in 𝕃^{2}(Ω; ℝ^{3}) and ε^{k}(t) − ε^{p,k}(t) ⇀ ε(t) − ε^{p}(t) in 𝕃^{2}(Ω; S^{3}) complete the proof.

8 End remarks

In this section we discuss briefly the case g : D(g) ⊂ ℝ^{N} → P(ℝ^{N}). We assume that the mapping g is maximal monotone with 0 ∈ g(0) . g generates a maximal monotone operator in the space 𝕃^{2}(Ω; ℝ^{N}) by w ∈ g(v) iff v(x) ∈ D(g) and w(x) ∈ g(v(x)) for a.e. x ∈ Ω (see [4] p.146). We denote the operator by g again. Let us rewrite the general problem (MP) from Section 3.

\begin{aligned}v_{t}(x, t) & = & div \frac{1}{ρ} D(ε(x, t) − Bz(x, t)) , \\ ε_{t}(x, t) & = & \frac{1}{2}(∇v(x, t) + ∇^{T} v(x, t)) , \\ z_{t}(x, t) & ∈ & g(−ρ∇_{z}ψ(ε(x, t), z(x, t))) \end{aligned}  (MP)

with the homogeneous Dirichlet or Neumann boundary condition and with the initial conditions

v(x, 0) = v^{(0)}(x) , ε(x, 0) = ε^{0}(x) , z(x, 0) = z^{0}(x) .

The free energy function ψ is defined by

ρψ(ε, z) = \frac{1}{2}[D(ε − Bz)] · (ε − Bz) + \frac{1}{2}(Lz) · z  (8.1)

where B : ℝ^{N} → S^{3} , L : ℝ^{N} → ℝ^{N} are given linear operators (L is symmetric and semi-positive definite), for which the operator M = B^{T} DB + L is positive definite. Let us denote by A_{λ} the Yosida approximation of the operator g. A_{λ} is generated by the pointwise Yosida approximation for the mapping g : D(g) ⊂ ℝ^{N} → P(ℝ^{N}). We want to approximate the problem (MP) by a sequence of problems, for which we can use Theorem 3.1

\begin{aligned}v_{t}^{λ}(x, t) & = & div \frac{1}{ρ} D(ε^{λ}(x, t) − Bz^{λ}(x, t)) , \\ ε_{t}^{λ}(x, t) & = & \frac{1}{2}(∇v^{λ}(x, t) + ∇^{T} v^{λ}(x, t)) , \\ z_{t}^{λ}(x, t) & = & A_{λ}(−ρ∇_{z}ψ(ε^{λ}(x, t), z^{λ}(x, t))) − A_{λ}(0) \end{aligned}  (YA)

with the same boundary and initial conditions as in the problem (MP). Moreover the function ψ is defined by (8.1) with the same linear operators B and L. Since A_{λ} − A_{λ}(0) : ℝ^{N} → ℝ^{N} is a maximal monotone operator with (A_{λ} − A_{λ}(0))(0) = 0 then this operator satisfies the assumptions of Theorem 3.1. (Note that for λ → 0^{+} , A_{λ}(0) → 0, provided 0 ∈ g(0).)

A. The case ψ - positive definite

Let us assume that the function ψ is positive definite. If we denote by H the space 𝕃^{2}(Ω; ℝ^{3} × S^{3} × ℝ^{N}) with the scalar product from Theorem 3.1

⟨(v, ε, z), (\overline{v}, \overline{ε}, \overline{z})⟩ = ∫_{Ω} \{ρv · \overline{v} + [D(ε − Bz)] · (\overline{ε} − B\overline{z}) + (Lz) · \overline{z}\} dx .

then we obtain

ess sup_{t∈[0,T]} ∥(v_{t}^{λ}(t), ε_{t}^{λ}(t), z_{t}^{λ}(t)∥_{H} ≤ ∥C^{λ}(v^{0}, ε^{0}, z^{0})∥_{H}

where C^{λ} is the operator defined by the right hand side of the system (YA). Thus we obtain

\begin{aligned}∥C^{λ}(v^{0}, ε^{0}, z^{0})∥_{H} & ≤ & C\{∥div D(ε^{0} − Bz^{0})∥ + ∥LA_{λ}(−ρ∇_{z}ψ(ε^{0}, z^{0}))∥ \\ & & ∥\frac{1}{2}(∇v^{0} + ∇^{T} v^{0}) − B(A_{λ}(−ρ∇_{z}ψ(ε^{0}, z^{0})) − A_{λ}(0))∥\} . \end{aligned} (8.2)

If we assume that −ρ∇_{z}ψ(ε^{0}, z^{0}) ∈ D(g) then there exists a positive constant C independent of λ such that

ess sup_{t∈[0,T]} (∥(v_{t}^{λ}(t)∥ , ∥ε_{t}^{λ}(t)∥ , ∥z_{t}^{λ}(t)∥) ≤ C  (8.3)

(note that the operator L is positive definite and the boundedness of ∥Lz_{t}^{λ}∥ implies boundedness of ∥z_{t}^{λ}∥). The second important step is an estimate for the difference of two approxomation steps. Let us define the energy for such differences by:

E^{λ,μ}(t) = \frac{ρ}{2} ∥v^{λ}(t) − v^{μ}(t)∥^{2} + \frac{1}{2} ∫_{Ω} [D(ε^{λ}(t) − Bz^{λ}(t) − ε^{μ}(t) + Bz^{μ}(t)]

· (ε^{λ}(t) − Bz^{λ}(t) − ε^{μ}(t) − Bz^{μ}(t)) dx

+ \frac{1}{2} ∫_{Ω} [L(z^{λ}(t) − z^{μ}(t))] · (z^{λ}(t) − z^{μ}(t)) dx .

Then for the time derivative of the function E^{λ,μ}(t) we have

\frac{d}{dt} E^{λ,μ}(t) = − ∫_{Ω} (−ρ∇_{z}ψ(ε^{λ}(t), z^{λ}(t)) + ρ∇_{z}ψ(ε^{μ}(t), z^{μ}(t))) ·

(A_{λ}(−ρ∇_{z}ψ(ε^{λ}(t), z^{λ}(t))) − A_{μ}(−ρ∇_{z}ψ(ε^{μ}(t), z^{μ}(t)))) dx

− ∫_{Ω} (−ρ∇_{z}ψ(ε^{λ}(t), z^{λ}(t)) + ρ∇_{z}ψ(ε^{μ}(t), z^{μ}(t))) · (A_{λ}(0) − A_{μ}(0)) dx

(similar to the proof of Theorem 3.1 from [6] p. 54)

≤ \frac{λ}{4}∥A_{λ}(−ρ∇_{z}ψ(ε^{λ}(t), z^{λ}(t)))∥^{2} + \frac{μ}{4}∥A_{μ}(−ρ∇_{z}ψ(ε^{μ}(t), z^{μ}(t)))∥^{2}

+∥−ρ∇_{z}ψ(ε^{λ}(t), z^{λ}(t)) + ρ∇_{z}ψ(ε^{μ}(t), z^{μ}(t))∥ · ∥A_{λ}(0) − A_{μ}(0)∥ .

Therefore using (8.3) we conclude that E^{λ,μ}(t) is arbitrarily small for λ , μ → 0^{+} . This implies that the sequence −ρ∇_{z}ψ(ε^{λ}, z^{λ}) ⟶ −ρ∇_{z}ψ(ε, z) in 𝕃^{∞}((0, T); 𝕃^{2}(Ω; ℝ^{N})) and the right hand side

A_{λ}(−ρ∇_{z}ψ(ε^{λ}, z^{λ})) ⇀^{∗} χ ∈ g(−ρ∇_{z}ψ(ε, z))

in 𝕃^{∞}((0, T); 𝕃^{2}(Ω; ℝ^{N})). Thus we have proved that the approximation process (YA) converges to a solution of the problem (MP).

B. The case ψ - semi-positive definite

Now let us suppose that the free energy ψ is only semi-positive definite, which is equivalent to the operator L being only semi-positive definite. In this case we can try to apply the approximation process (ACA) from Section 6:

\begin{aligned}v_{t}^{λ,k}(x, t) & = & div \frac{1}{ρ} D(ε^{λ,k}(x, t) − Bz^{λ,k}(x, t)) , \\ ε_{t}^{λ,k}(x, t) & = & \frac{1}{2}(∇v^{λ,k}(x, t) + ∇^{T} v^{λ,k}(x, t)) , \\ z_{t}^{λ,k}(x, t) & = & A_{λ}(−ρ∇_{z}ψ^{k}(ε^{λ,k}(x, t), z^{λ,k}(x, t))) − A_{λ}(0) \end{aligned}  (AYA)

with the same boundary and initial conditions as in the problem (MP). The free energy function is now defined by

ρψ(ε^{λ,k}, z^{λ,k}) = \frac{1}{2}[D(ε^{λ,k} − Bz^{λ,k})] · (ε^{λ,k} − Bz^{λ,k}) + \frac{1}{2}(Lz^{λ,k}) · z^{λ,k} + \frac{1}{2k} D(ε^{λ,k}) · ε^{λ,k} .

To prove that the process (AYA) converges to a solution of the problem (YA) we use only that the Yosida approximation is a family of lipschitz operators but then to obtain that the process (YA) converges to a solution of the problem (MP) we need a control of the right hand side B_{L}A_{λ}(−ρ∇_{z}ψ^{k}(ε^{λ,k}, z^{λ,k})), where the operator B_{L} was defined in Section 5. Hence we try to extend the definition of the self-controlling models:

there exists a positive constant λ^{∗} such that for all λ ≤ λ^{∗} and for all functions y ∈ 𝕃^{2}(Ω; ℝ^{N}) with |y(x)| ≥ 1 for a.e. x ∈ Ω it holds that

∥B_{L}A_{λ}(y)∥ ≤ F(∥LA_{λ}(y)∥, ∥y∥)  (8.4)

where F is a function satisfying the assumptions of Definition 5.4. If we now suppose that our model satisfies (8.4), then in the same manner as in Section 6 we can prove that for fixed λ ≤ λ^{∗} the process (AYA) converges to a solution of the problem (YA), provided that the initial data ε^{0} satiesfies

div Dε^{0} ∈ 𝕃^{2}(Ω; ℝ^{3}) and for the Neumann boundary condition Dε^{0} · n_{|_{∂Ω}} = 0 .

(8.5)

Using that the domain of A_{λ} is equal to 𝕃^{2}(Ω; ℝ^{N}) and A_{λ} is continuous we can omit the assumption (8.5) (see Remark 4.7). Also similar to the case A we show that the process (YA) converges to a solution of the problem (MP).

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Addresses: Krzysztof Chełmiński Fachbereich Mathematik Technische Hochschule Darmstadt Schloßgartenstr. 7 64289 Darmstadt, Germany e-mail: chelminski@mathematik.th-darmstadt.de