Energy estimates and global in time results for a problem from nonlinear viscoelasticity

Krzysztof Chełmiński

Technical University of Darmstadt and Institute of Applied Mathematics and Mechanics Warsaw University

Abstract

This paper proves global in time existence to large solutions for a problem in nonlinear viscoelasticity in the three–dimensional case with a bounded constitutive function. The idea of the proof is to show energy estimates for a Faedo–Galerkin sequence of approximate solutions.

AMS classifications: 35Q72, 73F99

1 Introduction and statement of the problem

We will study a system of equations modelling the nonelastic deformation of metals. We assume that spatial Lagrangian coordinates occupy a bounded region Ω ⊂ ℝ^{3} with smooth boundary ∂Ω . From the balance of momentum we obtain the following system of partial differential equations

ρ∂_{t}^{2}u(x, t) = div S(x, t) ,  (1.1)

where u(x, t) = (u_{1}(x, t), u_{2}(x, t), u_{3}(x, t)) ist the displacement field, S(x, t) is the stress field, and ρ > 0 is the density, which we assume to be constant. Moreover we assume that the linear stress – elastic strain relation (the constitutive equation in the continuum mechanics) holds

S(x, t) = D(E(x, t) − E^{n}(x, t)) .  (1.2)

Here E(x, t) is the linear part of the strain field

E(x, t) = \frac{1}{2}(∇u(x, t) + ∇^{T} u(x, t)) ,  (1.3)

E^{n}(x, t) is a field of inner variables, which describes the nonelastic part of the strain field and D = (d_{ijkl})_{i,j,k,l=1,2,3} is the elasticity tensor, which we assume to be constant and to satisfy

\begin{aligned}∀_{ijkl=1,2,3} & d_{ijkl} = d_{klij} \\ & d_{ijkl} = d_{jikl} \\ ∃_{d>0}∀_{η∈ℝ_{sym}^{3×3}} & ∑_{i,j,k,l=1}^{3} d_{ijkl}η_{ij}η_{kl} ≥ d |η|^{2} \end{aligned}  (D)

where |η|^{2} = ∑_{i,j=1}^{3} η_{ij}η_{ji} . When we fix the field E^{n}(x, t) then (1.1) – (1.3) are the ordinary equations of the linear elasticity theory. To determinate the nonelastic strain field E^{n}(x, t) we need some more information about our material. We will study a phenomenological model proposed by S.R. Bodner and Y. Partom. E^{n} is assumed to be a function, which depends on the stress deviator and a scalar inner variable called hardening

E^{n}(x, t) = E^{n}(s(x, t), z(x, t))

where s(x, t) = S(x, t) − \frac{1}{3}tr S(x, t) · I (I – the identity matrix) and z(x, t) is the hardening function. The evolution of all inner variables is given by the system of nonlinear ordinary differential equations

∂_{t}E^{n}(x, t) = G (\frac{|s(x, t)|}{z(x, t)}) \frac{s(x, t)}{|s(x, t)|}
∂_{t}z(x, t) = m(z^{1} − z(x, t))G (\frac{|s(x, t)|}{z(x, t)}) |s(x, t)| .  (1.4)

Here m and z^{1} are positive material parameters. The function G : ℝ_{+} → ℝ_{+}(ℝ_{+} = [0, ∞)) is called the constitutive function. In their paper [4] S.R. Bodner and Y. Partom have proposed the constitutive function to be of the form

G_{BP} (p) = A exp (−\frac{B}{p^{2n}} ) , A, B, ∈ ℝ_{+}, n ∈ ℕ .

In this paper we assume that the constitutive function G satisfies:

Assumption

1. G(p), \frac{G(p)}{p} are ℂ^{∞}(ℝ_{+}, ℝ_{+}) functions, (A)

2. ∃_{g>0} ∀_{p∈ℝ_{+}} G(p) < g , 3. ∀_{p∈ℝ_{+}} G′(p) ≥ 0

4. ∃_{g′>0} ∀_{p∈ℝ_{+}} G′(p)p^{2} < g′ .

An easy computation shows that the function G_{BP} satisfies (A) . Thus we can now formulate our problem: To find a displacement vector u : [0, T)×Ω → ℝ^{3} , a nonelastic strain field E^{n} : [0, T) × Ω → ℝ_{sym}^{3×3} and a hardening function z : [0, T) × Ω → ℝ_{+}, which satisfy the following system of differential equations

\begin{aligned}ρ∂_{t}^{2}u(x, t) & = & div S(x, t) \\ S(x, t) & = & D(E(x, t) − E^{n}(x, t)) \\ E(x, t) & = & \frac{1}{2}(∇u(x, t) + ∇^{T} u(x, t)) \\ s(x, t) & = & S(x, t) − \frac{1}{3}tr S(x, t) · I \\ ∂_{t}E^{n}(x, t) & = & G(\frac{|s(x, t)|}{z(x, t)}) \frac{s(x, t)}{|s(x, t)|} \\ ∂_{t}z(x, t) & = & m(z^{1} − z(x, t))G(\frac{|s(x, t)|}{z(x, t)})|s(x, t)| , \end{aligned}  (P)

the homogeneous Dirichlet boundary condition

u(x, t)_{|∂Ω} = 0  (B)

and the initial conditions

\begin{aligned}u(x, 0) & = & u^{0}(x) , \\ ∂_{t}u(x, 0) & = & u^{1}(x) , \\ E^{n}(x, 0) & = & E^{n,0}(x) , \\ z(x, 0) & = & z^{0}(x) . \end{aligned}  (I)

For the initial data z^{0} we assume that

∃_{z^{∗}>0} ∀_{x∈Ω} z^{∗} ≤ z^{0}(x) ≤ z^{1} .  (A5)

By (A5) the hardening function z(x, t) is a positive 𝕃^{∞}–function, while from (P 6) we have,

z(x, t) = z^{1} + (z^{0}(x) − z^{1}) exp{−m ∫_{0}^{t} |s(x, τ)| |∂_{t}E^{n}(x, τ)|dτ} ,  (1.5)

and for all (x, t)

z^{∗} ≤ z(x, t) ≤ z^{1} .

Problem (P) + (B) + (I) in the quasistatic case (without the term ρ∂_{t}^{2}u(x, t)) has been studied in [7]. The global in time existence result for large initial data is proved with the energy estimates method. The crucial fact of the estimates was to obtain 𝕃^{2}–bounds for the spatial derivatives of the stress tensor. This proof was based on the regularity property of solutions of linear elliptic operators.

The basic idea of this work is to use the method from [7] with apriori 𝕃^{2}-estimates for the term ρ∂_{t}^{2}u . We denote by ℍ^{s}(Ω; ℝ^{k}) the Sobolev space of functions defined on Ω with values in ℝ^{k} , with the norm ∥ · ∥_{s} . We denote by ∥ · ∥ and (· , ·) the norm and scalar product in ℍ^{0}(Ω; ℝ^{k}) = 𝕃^{2}(Ω, ℝ^{k}) . Moreover let us denote by ∥ · ∥_{∞} the usual norm in the space 𝕃^{∞}(Ω; ℝ^{k}) .

2 The main theorem

The object of this paper is to prove global in time existence of strong solutions of problem (P) + (B) + (I) with assumptions (A1) − (A5) .

Main Theorem

Assume that the initial data satisfy:

u^{0} ∈ ℍ^{2}(Ω; ℝ^{3}) , u^{1} ∈ ℍ^{1}(Ω; ℝ^{3})  (2.1)
E^{n,0} ∈ ℍ^{1} (Ω; ℝ_{sym}^{3×3}) , z^{0} ∈ ℍ^{1}(Ω; ℝ_{+})

with tr E^{n,0}(x) = 0 for almost all x ∈ Ω . Moreover suppose that the assumptions (A1) − (A5) and compatibility conditions

u^{0}(x) , u^{1}(x)_{|∂Ω} = 0  (2.2)

hold. Then for all T > 0 there exists a solution

(u, E^{n}, z) : [0, T] × Ω → ℝ^{3} × ℝ_{sym}^{3×3} × ℝ_{+}

of problem (P) + (B) + (I) with

u ∈ ℍ^{2}([0, T] × Ω; ℝ^{3}) ,
E^{n} ∈ ℍ^{1}([0, T] × Ω; ℝ_{sym}^{3×3}) ,
z ∈ ℍ^{1}([0, T] × Ω; ℝ_{+}) .

To prove this result we construct first with the Faedo–Galerkin method a sequence of smooth approximate solutions. Then for this sequence the energy estimates may be proved in almost the same way as in [7].

3 The sequence of approximate solutions

In this section we will approximate a solution of our problem by a sequence of smooth functions. First we regularize the singular behavior of the right hand side of the equations (P 5) and (P 6) at the point s = 0 . Let χ ∈ ℂ^{∞}(ℝ) be a cut–off function with χ(p) = 0 for p < \frac{1}{2} , χ(p) = 1 for p > 1 , 0 ≤ χ ≤ 1 , χ′ ≥ 0 and for all p χ′(p) ≤ χ_{0} (χ_{0} is a positive constant). We define a new constitutive function G_{k} : ℝ_{+} → ℝ_{+} for k ∈ ℕ as follows

G_{k}(p) = χ(kp)G(p) .

We note that for all k ∈ ℕ the function G_{k} satisfies the assumptions (A1) − (A4) . To prove this we must check only (A4)

G′_{k}(p)p^{2} = kχ′(kp)G(p)p^{2} + χ(p)G′(p)p^{2}
= kpχ′(kp)G(p)p + χ(g)G′(p)p^{2}
≤ χ_{0}G (\frac{1}{k} ) \frac{1}{k} + g′ =: g′_{k} → g′ for k → ∞ .

To construct a Faedo–Galerkin approximate sequence we choose a basis in the space 𝕃^{2}(Ω; ℝ^{3}) . We select a basis as follows: Let us solve the problem

Lw(x) = −div D\frac{1}{2} (∇w(x) + ∇^{T} w(x)) = f(x) f ∈ 𝕃^{2}(Ω; ℝ^{3})
w(x)_{|∂Ω} = 0 .

The operator 𝕃^{2}(Ω; ℝ^{3}) ∋ f → 𝕀(Lf) ∈ 𝕃^{2}(Ω; ℝ^{3}) where 𝕀 is the injection ℍ^{2}(Ω; ℝ^{3}) in 𝕃^{2}(Ω; ℝ^{3}) , is compact and symmetric (see the symmetry property of the elasticity tensor D) . Let {λ_{j}}_{j=1}^{∞} be the sequence of eigenvalues, and {w_{j}}_{j=1}^{∞} be a sequence of eigenfunctions of the operator 𝕀L . We can assume that for all j ∈ ℕ Lw_{j} = λ_{j}w_{j} , the sequence {λ}_{j=1}^{∞} is nondecreasing, λ_{j} → ∞ for j = ∞ and the eigenfunctions w_{j} are orthonormal in the space 𝕃^{2}(Ω; ℝ^{3}) . The standard regularity property of solutions of elliptic systems implies that w_{j} ∈ ℂ^{∞}(Ω; ℝ^{3}) . We now want to approximate the solution (u, E^{n}, z) of the problem (P) + (B) + (I) by a sequence {(u_{k}, E_{k}^{n}, z_{k})}_{k=1}^{∞} where the function u_{k} is a linear combination

u_{k}(x, t) = ∑_{j=0}^{k} α_{kj}(t)w_{j}(x)  (3.1)

and the coefficients α_{kj}(t) and the other components of the approximate sequence satisfy the following system of ordinary differential equations:

(ρ∂_{t}^{2}u_{k}(t), w_{j}) + (S_{k}(t), ∇w_{j}) = 0 j = 1, …, k
S_{k}(x, t) = D(E_{k}(x, t) − E_{k}^{n}(x, t))

E_{k}(x, t) = \frac{1}{2}(∇u_{k}(x, t) + ∇^{T} u_{k}(x, t))
s_{k}(x, t) = S_{k}(x, t) − \frac{1}{3}(tr S_{k}(x, t))I  (AP)
∂_{t}E_{k}^{n}(x, t) = G_{k} (\frac{|s_{k}(x, t)|}{z_{k}(x, t)}) \frac{s_{k}(x, t)}{|s_{k}(x, t)|}

∂_{t}z_{k}(x, t) = m(z^{1} − z_{k}(x, t))G_{k} (\frac{|s_{k}(x, t)|}{z_{k}(x, t)}) |s_{k}(x, t)| .

with the initial conditions

\begin{aligned}α_{kj}(0) & = & (u^{0}, w_{j}) , \\ \frac{d}{dt}α_{kj}(0) & = & (u^{1}, w_{j}) j = 1, …, k , \\ E_{k}^{n}(0, x) & = & E^{n,0}(x) , \\ z_{k}(0, x) & = & z^{0}(x) . \end{aligned}  (AI)

Existence and uniqueness of solutions of problem (AP)+(AI) are proved in the next lemma.

Lemma 3.1.

Assume that the initial data satisfy:

u^{0} ∈ ℍ^{2}(Ω; ℝ^{3}) , u^{1} ∈ ℍ^{1}(Ω; ℝ^{3}) ,
E^{n,0} ∈ ℂ^{∞} (Ω; ℝ_{sym}^{3×3}) , z^{0} ∈ ℂ^{∞}(Ω; ℝ_{+}) ,  (3.2)

with tr E^{n,0}(x) = 0 for all x ∈ Ω . Moreover assume that (A1) − (A5) , and the compatibility conditions (2.2) hold.

Then for all k ∈ ℕ there exists a positive number T_{k} and a unique smooth solution (u_{k}, E_{k}^{n}, z_{k}) : Ω×[0, T_{k}) → ℝ^{3}×ℝ_{sym}^{3×3}×ℝ_{+} of problem (AP)+(AI) , with tr E_{k}^{n}(t, x) = 0 for all (x, t) ∈ Ω × [0, T_{k}) .

Proof. Let β_{kj}(t) =: α′_{kj}(t) for j = 1, …, k , and

A =: {(α_{k1}, …, α_{kk}, β_{k1}, …, β_{kk}, E_{k}^{n}, z_{k}) ∈ ℝ^{k} × ℝ^{k} × ℂ^{∞}(Ω; ℝ_{sym}^{3×3})
× ℂ^{∞}(Ω; (\frac{1}{2}z^{∗}, ∞))\} .

We transform system (AP) into the first order system of ordinary differential equations

\frac{d}{dt}α_{kj}(t) = β_{kj}(t)
ρ ∑_{l=1}^{k}(w_{l}, w_{j})\frac{d}{dt}β_{kl}(t) = − ∑_{l=1}^{k} (\frac{1}{2}D(∇w_{l} + ∇^{T} w_{l}), ∇w_{j}) α_{kl}(t)
+ (DE_{k}^{n}(t), ∇w_{j}) j = 1, …, k  (3.3)
\frac{d}{dt}E_{k}^{n}(t) = G_{k} (\frac{|s_{k}(t)|}{z_{k}(t)}) \frac{s_{k}(t)}{|s_{k}(t)|}
\frac{d}{dt}z_{k}(t) = m(z^{1} − z_{k}(t))|s_{k}(t)| G_{k} (\frac{|s_{k}(t)|}{z_{k}(t)})

where s_{k}(t) = S_{k}(t) − \frac{1}{3}(tr S_{k}(t))I and

S_{k}(t) = ∑_{l=1}^{k} \frac{1}{2}D(∇w_{l} + ∇^{T} w_{l})α_{kl}(t) − DE_{k}^{n}(t) .

We can write system (3.3) in the following form

\frac{d}{dt}W_{k}(t) = F(W_{k}(t))

where w_{k}(t) = (α_{k1}(t), …, α_{kk}(t), β_{k1}(t), …, β_{kk}(t), E_{k}^{n}(t), z_{k}(t)) and F : A → ℝ^{k} × ℝ^{k} × ℂ^{∞}(Ω; ℝ_{sym}^{3×3}) × ℂ^{∞}(Ω; ℝ_{+}) is a infinitely differentiable function (χ vanishes in the neighborhood of zero). From assumption (3.2), W_{k}(0) ∈ A . The usual theory of ordinary differential equations in Banach spaces implies that there exists T_{k} > 0 and a unique solution W_{k} ∈ ℂ^{∞}([0, T_{k}); A) . Moreover we can conclude that the solution W_{k} can be continued as long as it satisfies an inequality

sup_{t} ∥W_{k}(t)∥_{∞} < ∞ .

The properties E_{k}^{n}(x, t) = (E_{k}^{n}(x, t))^{T} and tr E_{k}^{n}(x, t) = 0 are a simple consequence of the symmetry of the elasticity tensor D and of the definition of the deviator stress field s_{k}(x, t) . □

4 Energy estimates

To obtain a solution of problem (P) + (B) + (I) we will show that the sequence {(u_{k}, E_{k}^{n}, z_{k})}_{k=1}^{∞} contains a subsequence, which converges strongly in 𝕃^{2}(Ω) to (u, E^{n}, z) and the limit satisfies (P) + (B) + (I) . To do this, we use the standard energy method. Let us define by E_{k}(t) the “energy function”

E_{k}(t) = \frac{1}{2}(ρ ∂_{t}u_{k}(t), ∂_{t}u_{k}(t)) + \frac{1}{2}(S_{k}(t), E_{k}(t) − E_{k}^{n}(t)) .  (4.1)

Assumptions (D) imply

E_{k}(t) ≥ \frac{ρ}{2}∥∂_{t}u_{k}(t)∥^{2} + \frac{d}{2}∥E_{k}(t) − E_{k}^{n}(t)∥^{2} .

Lemma 4.1

Let (u_{k}, E_{k}^{n}, z_{k}) be a smooth solution of problem (AP) + (AI) : then

E_{k}(t) ≤ E_{k}(0) .

Proof:

\frac{d}{dt}E_{k}(t) = (ρ∂_{t}^{2}u_{k}(t), ∂_{t}u_{k}(t)) + (S_{k}(t), ∂_{t}E(t)) − (S_{k}(t), ∂_{t}E_{k}^{n}(t))
= (ρ∂_{t}^{2}u_{k}(t), ∂_{t}u_{k}(t)) + (S_{k}(t), ∇∂_{t}u_{k}(t)) − (s_{k}(t), ∂_{t}E_{k}^{n}(t))
= ∑_{j=1}^{k} α′_{kj}(t) \{(ρ∂_{t}^{2}u_{k}(t), w_{j}) + (S_{k}(t), ∇w_{j})\} − ∫_{Ω} G_{k}(\frac{|s_{k}(t)|}{z_{k}(t)})|s_{k}(t)|dx .

Integrating the last equality we obtain

E_{k}(t) + ∫_{0}^{t} ∫_{Ω} G_{k}(x, τ)|s_{k}(x, τ)|dxdt = E_{k}(0) .  (4.2)
□

From the last lemma we can conclude that the sequence {(u_{k}, ∂_{t}u_{k}, S_{k}, E_{k}^{n}, z_{k})}_{k=1}^{∞} of approximate solutions is 𝕃^{2}–bounded. Therefore we can select a subsequence, which converges weakly in 𝕃^{2} . But this is not sufficient to show that the 𝕃^{2}–weak limit is a solution of our problem. (The nonlinear function G is not weakly continuous.) To solve this problem we can try and estimate all derivatives of the functions (u_{k}, ∂_{t}u_{k}, S_{k}, E_{k}, z_{k}) . We start with the time derivatives.

5 Estimates of t– derivatives

In the same way as in the last section we define first an “energy function” for the t–derivatives:

E_{k}^{t}(t) = \frac{1}{2}(ρ∂_{t}^{2}u_{k}(t), ∂_{t}^{2}u_{k}(t)) + \frac{1}{2}(∂_{t}S_{k}(t), ∂_{t}E_{k}(t) − ∂_{t}E_{k}^{n}(t)) .  (5.1)

This function is obviously positive define and satisfies the next

Lemma 5.1

Let (u_{k}, E_{k}^{n}, z_{k}) be a smooth solution of problem (AP) + (AI) : then

∀_{0<T <T_{k}} ∃_{C(T)>0} ∀_{0≤t≤T} E_{k}^{t}(t) ≤ E_{k}^{t}(0) + C(T) · E_{k}(0)  (5.2)

and the constant C(T) does not depend on k . Proof: A simple calculation yields

\frac{d}{dt}E_{k}^{t}(t) = (ρ∂_{t}^{3}u_{k}(t), ∂_{t}^{2}u_{k}(t)) + (∂_{t}S_{k}(t), ∂_{t}^{2}E_{k}(t)) − (∂_{t}S_{k}(t), ∂_{t}^{2}E_{k}^{n}(t))
= (ρ∂_{t}^{3}u_{k}(t), ∂_{t}^{2}u_{k}(t)) + (∂_{t}S_{k}(t), ∂_{t}^{2}∇u_{k}(t)) − (∂_{t}s_{k}(t), ∂_{t}^{2}E_{k}^{n}(t)) (5.3)
= ∑_{j=1}^{k} α′′_{kj}(t) {(ρ∂_{t}^{3}u_{k}(t), w_{j}) + (∂_{t}S_{k}(t), ∇w_{j})} − (∂_{t}s_{k}(t), ∂_{t}^{2}E_{k}^{n}(t))
= − (∂_{t}s_{k}(t), ∂_{t}^{2}E_{k}^{n}(t)) .

We now compute ∂_{t}^{2}E_{k}^{n} from equation (AP 5)

∂_{t}^{2}E_{k}^{n} = ∂_{t} (G_{k} (\frac{|s_{k}|}{z_{k}}) \frac{s_{k}}{|s_{k}|}) = G′_{k} (\frac{|s_{k}|}{z_{k}}) \frac{z_{k}∂_{t}|s_{k}| − |s_{k}|∂_{t}z_{k}}{z_{k}^{2}}

+ G_{k} (\frac{|s_{k}|}{z_{k}}) \frac{|s_{k}|∂_{t}s_{k} − s_{k}∂_{t}|s_{k}|}{|s_{k}|^{2}}

and insert the result into (5.3):

\frac{d}{dt}E^{t}(t) = − ∫_{Ω} G′_{k} (\frac{|s_{k}(t)|}{z_{k}(t)}) \frac{(∂_{t}|s_{k}(t)|)^{2}}{z_{k}(t)} dx
− ∫_{Ω} G_{k} (\frac{|s_{k}(t)|}{z_{k}(t)}) \{\frac{|∂_{t}s_{k}(t)|^{2} − (∂_{t}|s_{k}(t)|)^{2}}{|s_{k}(t)|} \} dx  (5.4)

+ ∫_{Ω} G′_{k} (\frac{|s_{k}(t)|}{z_{k}(t)}) \frac{∂_{t}z_{k}(t)}{z_{k}^{2}(t)}|s_{k}(t)| ∂_{t}|s_{k}(t)|dx .

The two first terms of the right hand side of (5.4) are negative. The last term does not have a sign. From equation (AP 6) we obtain

∫_{Ω} G′_{k} (\frac{|s_{k}(t)|}{z_{k}(t)}) \frac{∂_{t}z_{k}(t)}{z_{k}^{2}(t)}|s_{k}(t)| ∂_{t}|s_{k}(t)|dx

= ∫_{Ω} G′_{k} (\frac{|s_{k}(t)|}{z_{k}(t)}) \frac{m(z^{1} − z_{k}(t))}{z_{k}^{2}(t)}G_{k} (\frac{|s_{k}(t)|}{z_{k}(t)}) |s_{k}(t)|^{2}∂_{t}|s_{k}(t)|dx  (5.5)
≤ \frac{1}{2} ∫_{Ω} G′_{k} (\frac{|s_{k}(t)|}{z_{k}(t)}) \frac{(∂_{t}|s_{k}(t)|)^{2}}{z_{k}(t)} dx + \frac{1}{2}g′_{k}g^{2} \frac{m^{2}(z^{1} − z^{∗})^{2}}{z^{∗}} ∫_{Ω} |s_{k}(t)|^{2}dx .

Inequality (5.5) and (5.4) imply

∃_{C>0} \frac{d}{dt}E_{k}^{t}(t) ≤ CE_{k}(t) . (5.6)

Lemma 4.1 completes this proof. □

6 Estimates of x–derivatives.

Unfortunately we can not repeat for the x–derivatives the same “energy method” as in the last section. But from lemma 5.1 we obtain 𝕃^{2}–bound for the acceleration and we can try and apply the method from [7]. We start with the 𝕃^{2}–boundness for the spatial derivative of the hardening function.

Lemma 6.1

Let the initial data z^{0} satisfy assumptions (A5) and (3.2) . Moreover let (u_{k}, E_{k}^{n}, z_{k}) be a smooth solution of problem (AP) + (AI) . Then

∀_{0<T <T_{k}} ∃_{Z(T)>0} ∀_{0<t≤T} ∥∂_{x_{i}}z_{k}(t)∥^{2} ≤ Z(T) {∥∂_{x_{i}}z^{0}∥^{2} + ∫_{0}^{t} ∥∇S_{k}(τ)∥^{2}dτ }

for i = 1, 2, 3 . Here the constant Z(T) does not depend on k .

Proof: We calculate the derivative ∂_{x_{i}}z_{k} from equation 1.5

∂_{x_{i}}z_{k} = −m(z^{1} − z^{0}) exp {−m ∫_{0}^{t} |s_{k}| |∂_{t}E_{k}^{n}|dτ }
× {∫_{0}^{t} (∂_{x_{i}}|s_{k}| |∂_{t}E_{k}^{n}| + |s_{k}|∂_{x_{i}}|∂_{t}E_{k}^{n}|)dτ }
+ ∂_{x_{i}}z^{0} exp {−m ∫_{0}^{t} |s_{k}| |∂_{t}E_{k}^{n}|dτ } .

Thus for the 𝕃^{2}–norm we have

∥∂_{x_{i}}z_{k}∥^{2} ≤ 2∥∂_{x_{i}}z^{0}∥^{2} + 2(z^{1} − z^{∗})^{2}m^{2}

× ∫_{Ω} \{∫_{0}^{t} (| ∂_{x_{i}}|s_{k}| | G_{k} (\frac{|s_{k}|}{z_{k}}) + |s_{k}||∂_{x_{i}}G_{k} (\frac{|s_{k}|}{z_{k}}) |) dτ \}^{2} dx

Recalling assumptions (A) we find

∥∂_{x_{i}}z_{k}∥^{2} ≤ 2∥∂_{x_{i}}z^{0}∥^{2}

+ 4(z^{1} − z^{∗})^{2}m^{2} · t( sup_{p≥0} G′_{k}(p)p^{2})^{2} ∫_{0}^{t} ∥∂_{x_{i}}z_{k}∥^{2}dτ

+ 4(z^{1} − z^{∗})^{2}m^{2} · t [( sup_{p≥0} G_{k}(p))^{2} + ( sup_{p≥0} G′_{k}(p)p)^{2}] ∫_{0}^{t} ∥∇S_{k}∥^{2}dτ .

The conclusion of the lemma follows now from Gronwall’s lemma.

□

The main idea of the method described in [7] was to use the regularity property of solutions of linear elliptic equations. In our case a solution of the problem (P) must satisfy the following inequality

∥u∥_{2} ≤ C(∥div DE^{n}∥ + ∥∂_{t}^{2}u∥)  (6.1)

where C is a positive constant, which does not depend on u . We should prove this type of regularity for our sequence of approximate solutions.

Lemma 6.2

Let (u_{k}, E_{k}^{n}, z_{k}) be a smooth solution of problem (AP) + (AI) : then

∃_{C>0} ∥u_{k}(t)∥_{2} ≤ C(∥div DE_{k}^{n}(t)∥ + ∥∂_{t}^{2}u_{k}(t)∥)

and the constant C does not depend on k .

Proof: First we multiply equation (AP 1) by λ_{j} and use the definition of the functions w_{j} .

(∂_{t}^{2}u_{k}(t), λ_{j}w_{j}) + (S_{k}(t), λ_{j}∇w_{j}) = 0
(∂_{t}^{2}u_{k}(t), Lw_{j}) + (s_{k}(t), ∇Lw_{j}) = 0 j = 1, …, k .

In the second term we integrate by parts and the boundary integral vanishes because Lw_{j|∂Ω} = 0 for all j .

(∂_{t}^{2}u_{k}(t), Lw_{j}) + (Lu_{k}(t), Lw_{j}) = −(div DE_{k}^{n}(t), Lw_{j}) j = 1, …, k . (6.2)

Now multiplying (6.2) by α_{kj}(t) and summing for j = 1, …, k we obtain

∥Lu_{k}(t)∥ ≤ ∥div DE_{k}^{n}(t)∥ + ∥∂_{t}^{k}u_{k}(t)∥ .

The operator L is strongly elliptic and u_{k}(t)_{|∂Ω} = 0 : hence the conclusion of the lemma.

□

The object of this method of the estimation of x–derivatives is now to show that the 𝕃^{2}–norm of the gradient of the stress tensor is bounded in time. This is described by the next lemma.

Lemma 6.3

With the same assumptions as in the last lemma, the gradient of the stress tensor satisfies the following estimate:

∀_{0<T <T_{k}} ∃_{C_{0}^{s}(T),C_{1}^{s}(T),C_{2}^{s}(T)>0} ∀_{0<t≤T}

∥∇S_{k}(t)∥^{2} ≤ C_{0}^{s}(T) {∥∇z^{0}∥^{2} + ∥∇E^{n,0}∥^{2} + ∥∂_{t}^{2}u_{k}(t)∥^{2}}
+ C_{1}^{s}(T) ∫_{0}^{t} ∥∇S_{k}(τ)∥^{2}dτ + C_{2}^{s}(T) ∫_{0}^{t} ∫_{0}^{τ} ∥∇S_{k}(ρ)∥^{2}dρdτ .

Proof: Assumptions (D) imply that there exists a constant D > 0 such that

∥∇S_{k}(t)∥^{2} ≤ D(∇S_{k}(t), ∇(E_{k}(t) − E_{k}^{n}(t)))

≤ D\{\frac{1}{4D}∥∇S_{k}(t)∥^{2} + D∥u_{k}(t)∥_{2}^{2}  (6.3)

+ \frac{1}{4D}∥∇S_{k}(t)∥^{2} + D∥∇E_{k}^{n}(t)∥^{2}\} .

With lemma (6.2) we can write (6.3) in the other form

∥∇S_{k}(t)∥^{2} ≤ 2D^{2} {∥∇E_{k}^{n}(t)∥^{2} + 2C^{2}∥div DE_{k}^{n}(t)∥^{2} + 2C^{2}∥∂_{t}^{2}u_{k}(t)∥^{2}} .

For i = 1, 2, 3 we obtain

∥∂_{x_{i}}E_{k}^{n}∥^{2} = ∫_{Ω} | ∫_{0}^{t} ∂_{x_{i}}∂_{t}E_{k}^{n}dτ + ∂_{x_{i}}E^{n,0} |^{2} dx
≤ 2t ∫_{0}^{t} ∫_{Ω} | ∂_{x_{i}}∂_{t}E_{k}^{n} |^{2} dxdτ + 2∥∂_{x_{i}}E^{n,0}∥^{2} .  (6.4)

We now calculate ∂_{x_{i}}∂_{t}E_{k}^{n} from equation (AP 5) and insert the result into (6.4); thus we have

∥∂_{x_{i}}E_{k}^{n}∥^{2} ≤ 8t ∫_{0}^{t} {((z^{∗})^{−1} sup_{p≥0} G′_{k}(p))^{2}∥∂_{x_{i}}s_{k}∥^{2}
+ ((z^{∗})^{−2} sup_{p≥0} G′_{k}(p)p)^{2}∥∂_{x_{i}}z_{k}∥^{2}  (6.5)

+ 2( sup_{p≥0} \frac{G_{k}(p)}{p})^{2}∥∂_{x_{i}}s_{k}∥^{2}\}dτ + 2∥∂_{x_{i}}E^{n,0}∥^{2} .

The lemma follows now from assumptions (A) and lemma 6.1.

□

7 Proof of the Main Theorem

The energy estimates imply that T_{k} = +∞ ( α_{kj}, β_{kj} are uniformly bounded and E_{k}^{n}, z_{k} are 𝕃^{∞}–functions). Lemma (6.3) with Gronwall’s inequality gives that the 𝕃^{2}–norm of the gradient of the stress tensor is uniformly bounded in time. Moreover we have that there exists a constant K(T) > 0 , which does not depend on k such that

∥u_{k}(t)∥_{2}, ∥∂_{t}u_{k}(t)∥_{1}, ∥∂_{t}^{2}u_{k}(t)∥,
∥E_{k}^{n}(t)∥_{1}, ∥∂_{t}E_{k}^{n}(t)∥,
∥z_{k}(t)∥_{1}, ∥∂_{t}z_{k}(t)∥ ≤ K(T) .

Thus the approximate sequence {(u_{k}, E_{k}^{n}, z_{k})}_{k=1}^{∞} contains a subsequence {(u_{k_{j}}, E_{k_{j}}^{n}, z_{k_{j}})}_{j=1}^{∞}, which converges to (u, E^{n}, z) strongly in the space ℍ^{1}([0, T] × Ω; ℝ^{3}) × 𝕃^{2}([0, T] × Ω; ℝ_{sym}^{3×3}) × 𝕃^{2}([0, T] × Ω; ℝ_{+}) and weakly in the space ℍ^{2}([0, T]×Ω; ℝ^{3}) × ℍ^{1}([0, T]×Ω; ℝ_{sym}^{3×3}) × ℍ^{1}([0, T]×Ω; ℝ_{+}) . This convergence is now sufficent to show that the limit (u, E^{n}, z) is a solution of problem (P) + (B) + (I) . (For more details we refer to [1], [5].) We have proved the Main Theorem for initial conditions which satisfy assumption (3.2). For initial data E^{n,0} ∈ ℍ^{1}(Ω; ℝ_{sym}^{3×3}) , z^{0} ∈ ℍ^{1}(Ω; ℝ_{+}) we approximate these functions by a sequence {E_{i}^{n,0}, z_{i}^{0}}_{i=1}^{∞} of smooth functions and repeat exactly the same approximation process described in this paper. The proof of the Main Theorem is complete.

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