Stress L^{∞}-Estimates and the Uniqueness Problem for the Equations to the Model of Bodner-Partom

in the Two Dimensional Case

Krzysztof Chełmiński ^{∗}

Technical University of Darmstadt

Abstract

This paper is a continuation of the work [9]. We prove the uniqueness result for global in time large solutions of dynamic equations to an inelastic model of material behaviour of metals in the two dimensional case, provided a higher regularity of the solutions. Moreover the 𝕃^{p}-stability for p < 2 of the solutions in the case of homogeneous boundary data is established.

AMS classification: 35Q72, 73F99

1 Introduction

We continue to study the uniqueness problem of the equations of the model of Bodner-Partom. In [9] we have proved the uniqueness result to the quasistatic problem. The main idea of the work [9] was to replace the uniqueness problem for the equations under consideration with the question about regularity. For the quasistatic case it was enough to prove a 𝕎^{1,q}-estimation for the Cauchy stress tensor for q larger as the space dimension. The method presented in [9] is based on the 𝕃^{q}- regularity theorem for elliptic systems. Unfortunately we cannot apply the method to the dynamical case. Therefore in this paper we try to prove a higher regularity result for the solutions of the dynamic system with the energy estimates for the second derivatives of the solutions. We are offering here this type of estimates only in the two dimensional case. Thus the uniqueness problem in the three dimensional case is still open. We state first the problem: Let Ω ⊆ ℝ^{2} be an open set with smooth boundary ∂Ω . Let u(x, t) ∈ ℝ^{3} be the displacement of the masspoint x ∈ Ω at time t ≥ 0 , and let T (x, t) ∈ ℝ_{sym}^{3×3} (the set of symmetric 3 × 3 matrices) be the Cauchy stress tensor. We consider the problem

∗Supported by Sonderforschungsbereich 298.

\begin{aligned}ρu_{tt}(x, t) & = & div_{x} T (x, t) \\ T (x, t) & = & D(ε(∇_{x} u(x, t)) − ε_{p}(x, t)) , \\ z_{t}(x, t) & = & f(ε(∇_{x} u(x, t)), z(x, t)) \end{aligned}  (P)

where ρ > 0 is the density, which we assume to be constant in space and time, ε(∇_{x} u(x, t)) = \frac{1}{2} (∇_{x} u(x, t) + (∇_{x} u(x, t))^{T} ) is the linear strain tensor, ε_{p}(x, t) ∈

ℝ_{sym}^{3×3} is the plastic strain tensor, D is the elasticity tensor, which we assume to be positive definite, independent of x and t and z(x, t) = (ε_{p}(x, t), z̃(x, t)) ∈ ℝ_{sym}^{3×3} × ℝ^{N−9} ⊂ ℝ^{N} (where N > 9 denotes the quantity of ordinary differential equations) is the vector of internal variables. Finally the given function f defines the model under consideration. In this paper we study the model of Bodner-Partom. In the model N = 10 , and the vector z consists of nine components of the plastic strain tensor ε_{p}, and of one positive variable y describing isotropic hardening. The system of ordinary differential equations (P3) proposed in [5] by S.R. Bodner and Y. Partom has the following form

\begin{aligned}\frac{∂}{∂t} ε_{p} & = & G(\frac{|σ|}{y}) \frac{σ}{|σ|} \\ \frac{∂}{∂t} y & = & m(y_{1} − y) G(\frac{|σ|}{y}) |σ| , \end{aligned}  (C)

where σ = T − \frac{1}{3} (tr T) · I is the stress deviator, G : ℝ_{+} → ℝ_{+} is a given function, and m > 0 , y_{1} > 0 are given constants. In [5] the function G has been proposed to be of the form

G_{BP} (p) = a exp (−\frac{b}{p^{2k}} ) , a, b ∈ ℝ_{+} , k > \frac{1}{2} .

By the assumptions

1.  G(p) , \frac{G(p)}{p} are C^{∞}(ℝ_{+}, R_{+}) functions,  (A1)
2.  ∃_{g} ∀_{p}  G(p) < p (G is bounded),  (A2)
3.  ∀_{p∈ℝ_{+}}  G′(p) ≥ 0 (G is nondecreasing),  (A3)
4.  ∃_{g′} ∀_{p∈ℝ_{+}}  G′(p)p^{2} < g′  (A4)

it can be proved the global in time existence of large solutions (u, T, ε_{p}, y) in the space ℍ^{2}(Ω × [0, t^{∗}]; ℝ^{3}) × ℍ^{1}(Ω × [0, t^{∗}]; ℝ_{sym}^{3×3})^{2} × ℍ^{1}(Ω × [0, t^{∗}]; ℝ_{+}) for all t^{∗} > 0, to initial data (u_{|_{t=0}} , u_{t}_{|_{t=0}} , ε_{p}_{|_{t=0}} , y_{|_{t=0}}) of the space ℍ^{2}(Ω; ℝ^{3}) × ℍ^{1}(Ω; ℝ^{3}) × ℍ^{1}(Ω; ℝ_{sym}^{3×3}) × ℍ^{1}(Ω; ℝ_{+}) (see [8]). (By ℍ^{s}(Ω) we denote the usual Sobolev space over 𝕃^{2}(Ω) ). Furthermore in this paper we assume that

5.  \frac{G(p)}{p^{2}} is a C^{∞}(ℝ_{+}, ℝ_{+}) function,  (A5)
6.  ∃_{g′′>0} ∀_{p∈R_{+}} |G′′(p)|p^{3} < g′′.  (A6)

It is easy to see that the function G_{BP} satisfies all assumptions (A1) – (A6). Thus we study the following system of equations:

\begin{matrix}ρu_{tt}(x, t) = div_{x} T (x, t) \\ T (x, t) = D(ε(∇_{x}u(x, t)) − ε_{p}(x, t)) \\ ε(∇_{x}(u(x, t))) = \frac{1}{2} (∇_{x}u(x, t) + (∇_{x}u(x, t))^{T} ) \\ σ(x, t) = T (x, t) − \frac{1}{3} (tr T (x, t)) I \\ \frac{∂}{∂t} ε_{p}(x, t) = G (\frac{|σ(x, t)|}{y(x, t)}) \frac{σ(x, t)}{|σ(x, t)|} \\ \frac{∂}{∂t} y(x, t) = m(y_{1} − y(x, t)) G(\frac{|σ(x, t)|}{y(x, t)}) |σ(x, t)| \end{matrix}  (BP)

with the homogeneous Dirichlet boundary condition

u(x, t)_{|_{∂Ω}} = 0

and with the initial conditions

\begin{aligned}u(x, 0) & = & u^{(0)}(x) , & u_{t}(x, 0) & = & u^{(1)}(x) , \\ ε_{p}(x, 0) & = & ε_{p}^{(0)}(x) , & y(x, 0) & = & y^{(0)}(x) , \end{aligned}

where u^{(0)}, u^{(1)} : Ω → ℝ^{3} , ε_{p}^{(0)} : Ω → ℝ_{sym}^{3×3} , y^{(0)} : Ω → ℝ_{+} , are given functions. Moreover, similarly as in [6, 7, 8, 9], we assume that the initial function y^{(0)} satisfies

∃_{y^{∗}} ∀_{x∈Ω} y^{∗} ≤ y^{(0)}(x) ≤ y_{1} . (A7)

This assumption implies that y is a positive 𝕃^{∞}-function with

y^{∗} ≤ y(x, t) ≤ y_{1} for all (x, t) ∈ Ω × [0, ∞) .

The object of this work is to prove the uniqueness theorem for the global in time large solutions of the system (BP), of which existence has been proved in [8]. In the last section of this work we prove a stability result for the global in time solutions of the problem (BP).

2 Uniqueness theorem and the energy method

In this section we present the uniqueness theorem for the problem (BP), provided an a priori 𝕃^{∞}-estimation for the stress deviator σ. This theorem is similar to Theorem 1 presented in the paper [9].

Theorem 1. Assume that the initial data

(u^{(0)}, u^{(1)}, ε_{p}^{(0)}, y^{(0)}) ∈ ℍ^{2}(Ω; ℝ^{3}) × ℍ^{1}(Ω; ℝ^{3}) × ℍ^{1}(Ω; ℝ_{sym}^{3×3}) × ℍ^{1}(Ω; ℝ_{+})

with tr ε_{p}^{(0)}(x) = 0 for almost all x ∈ Ω . Moreover suppose that the assumptions (A1) – (A4), (A7) and the compatibility conditions

u^{(0)}(0), u^{(1)}(0)_{|_{∂Ω}} = 0  (CC)

hold. Let (u^{i}, T^{i}, ε_{p}^{i}, y^{i}) , i = 1, 2 , be two solutions of the problem (BP) to the same initial data u^{(0)}, u^{(1)}, ε_{p}^{(0)}, y^{(0)} . If the estimations

∀_{t^{∗}>0} ∃_{C_{i}(t^{∗})} sup_{t≤t^{∗}} ∥σ^{i}(t)∥_{L^{∞}(Ω)} ≤ C_{i}(t^{∗})  (∗)

are valid for i = 1, 2, then (u^{1}, T^{1}, ε_{p}^{1}, y^{1}) = (u^{2}, T^{2}, ε_{p}^{2}, y^{2}) .

The proof of this theorem can be done similarly as the one of Theorem 1 in [9]. Thus Theorem 1 replaces the uniqueness problem for the system (BP) by a regularity question. Now we need only an 𝕃^{∞}-estimation for the stress deviator. To prove (∗) we use the energy method. Let us define the energy function

E(u, T, ε_{p})(t) = ((ρu_{t}(t), u_{t}(t)) + \frac{1}{2}(T (t), ε(∇_{x}u(t)) − ε_{p}(t)) = \frac{1}{2}ρ ∫_{Ω} u_{t}^{2}(x, t) dx

+ \frac{1}{2} ∫_{Ω} T (x, t) · (ε(∇_{x}u(x, t)) − ε_{p}(x, t)) dx .

By the assumptions for the elasticity tensor D we have

∃_{d>0} E(u, T, ε_{p})(t) ≥ \frac{ρ}{2} ∥u_{t}(t)∥^{2} + \frac{d}{2} ∥ε(∇_{x}u(t)) − ε_{p}(t)∥^{2}

where we denote by (·, ·) and by ∥ · ∥ the scalar product and the norm in the space 𝕃^{2} respectively. The paper [8] yields the following

Lemma 1. Let (u, T, ε_{p}, y) ∈ ℍ^{2}(Ω; ℝ^{3}) × (ℍ^{1}(Ω; ℝ_{sym}^{3×3}))^{2} × ℍ^{1}(Ω; ℝ_{+}) be a global in time solution of the problem (BP) with the homogeneous Dirichlet boundary condition and with initial conditions

(u^{(0)}, u^{(1)}, ε_{p}^{(0)}, y^{(0)}) ∈ ℍ^{2}(Ω; ℝ^{3}) × ℍ^{1}(Ω; ℝ^{3}) × ℍ^{1}(Ω; ℝ_{sym}^{3×3}) × ℍ^{1}(Ω; ℝ_{+})

satisfying compatibility conditions (CC) and assumption (A7). Moreover assume that the assumptions (A1) – (A4) hold. Then

(i) E(u, T, ε_{p})(t) + ∫_{0}^{t} ∫_{Ω} G(\frac{|σ(x, τ)|}{y(x, τ)}) |σ(x, t)| dx dτ = E(u, T, ε_{p})(0)

where T (x, 0) = D(ε(∇_{x}u^{(0)}(x) − ε_{p}(0)(x)) ,

(ii) \begin{matrix}∀_{t^{∗}>0} ∃_{C_{t}(t^{∗})>0} ∀_{0≤t≤t^{∗}} \\ E(∂_{t}u, ∂_{t}T, ∂_{t}ε_{p})(t) ≤ E(∂_{t}u, ∂_{t}T, ∂_{t}ε_{p})(0) + C_{t}(t^{∗}) · E(u, T, ε_{p})(0) , \end{matrix}

where

\begin{aligned}∂_{t}T (x, 0) & = & D(ε(∇_{x}u^{(1)}(x) − ∂_{t}ε_{p}(x, 0)) , \\ ∂_{t}ε_{p}(x, 0) & = & G(\frac{|σ(x, 0)|}{y^{(0)}(x)}) \frac{σ(x, 0)}{|σ(x, 0)|} , \\ σ(x, 0) & = & T (x, 0) − \frac{1}{3} (tr T (x, 0)) · I . \end{aligned}

The last lemma yields 𝕃^{2}-estimates of global solutions to the problem (BP) and of their first time derivatives. Similarly for the first space derivatives it has been proved in [8]:

Lemma 2. With the same assumptions as in the last lemma the 𝕃^{2}-norm of the gradient of the Cauchy stress tensor T does not blow up in finite time.

Thus the same is true for the ℍ^{1}-norm of the stress tensor T . Note that it is not sufficient to prove an 𝕃^{∞}-estimation for the tensor T . Therefore, in this work we try to prove energy estimations of order two.

3 Energy estimates of order two

As well known it is a great problem to prove energy estimates for the second derivatives of global in time solutions to a nonlinear system of differential equations. First we want to present the main difficulty of such type of estimates. Let H : ℝ^{k} → ℝ^{k} be a smooth function with bounded first and second derivatives, and u be a function from the space ℍ^{2}(M; ℝ^{k}) , where M ⊂ ℝ^{n} is an open bounded set with smooth boundary ∂M . Moreover let the function u be a priori placed in a bounded set in the space ℍ^{1}(M; ℝ^{k}) with respect to the the strong topology.

Question: For which space dimensions the following estimation is true

∃_{C>0} ∥H(u)∥_{ℍ^{2}} ≤ C ∥u∥_{ℍ^{2}} ?  (Q)

For the second derivatives of the function H(u) we have

∥D^{2}H(u)∥ ≤ sup |D^{2}H| ∥ |Du|^{2}∥ + sup |DH| · ∥D^{2}u∥ .

Thus the second term in the last inequality has a good form. For the first term we obtain by the Gagliardo-Nirenberg inequality

∃_{C>0} ∥Du∥_{𝕃^{4}}^{2} ≤ C ∥u∥_{ℍ^{2}}^{n/2} ∥u∥_{ℍ^{1}}^{2−n/2} .

Therefore only for n ≤ 2 we can give a positive answer to the question (Q) . For this reason we study in this paper only the two-dimensional case of the problem (BP). Let us start with the energy estimates of order two.

Theorem 2. Let for initial data (u^{(0)}, u^{(1)}, ε_{p}^{(0)}, y^{(0)}) ∈ ℍ^{3}(Ω; ℝ^{3}) × ℍ^{2}(Ω; ℝ^{3}) × ℍ^{2}(Ω, ℝ_{sym}^{3×3}) × ℍ^{1}(Ω; ℝ_{+}) (u, T, ε_{p}, y) ∈ ℍ^{3}(Ω × [0, t^{∗}]) × (ℍ^{2}(Ω × [0, t^{∗}]; ℝ_{sym}^{3×3}))^{2} × H^{1}(Ω × [0, t^{∗}]; ℝ_{+}) be a local solution for some t^{∗} > 0 . Moreover let all assumptions (A1) – (A7) hold. Then

∃_{C_{2}(t^{∗})>0} ∀_{t≤t^{∗}} E(∂_{t}^{2}u, ∂_{t}^{2}T, ∂_{t}^{2}ε_{p})(t) ≤ C_{2}(t^{∗})

Proof: Similarly as for the energy estimates of order zero or one we calculate the time derivative of the function E(∂_{t}^{2}u, ∂_{t}^{2}T, ∂_{t}^{2}ε_{p}) .

\frac{d}{dt} E(∂_{t}^{2}u, ∂_{t}^{2} T, ∂_{t}^{2}ε_{p})(t) = (∂_{t}^{3}u(t), ∂_{t}^{4}u(t)) + (∂_{t}^{2} T (t), ε(∇_{x}∂_{t}^{3}u(t)) − ∂_{t}^{3}ε_{p}(t))
= (∂_{t}^{3}u(t), ∂_{t}^{4}u(t)) + (∂_{t}^{2} T (t), ∇∂_{t}^{3}u(t)) − (∂_{t}^{2}σ(t), ∂_{t}^{3}ε_{p}(t))
= (∂_{t}^{3}u(t), ∂_{t}^{4}u(t)) − (div ∂_{t}^{2} T (t), ∂_{t}^{3}u(t)) − (∂_{t}^{2}σ(t), ∂_{t}^{3}ε_{p}(t))
= −(∂_{t}^{2}σ(t), ∂_{t}^{3}ε_{p}(t)) .

(3.1)

In the last calculation there is required a higher regularity as assumed. However we can prove this theorem first for a smooth sequence of approximate solutions to the problem (BP) (for example for the sequence constructed in [8]) and then in the limit for the exact solution. From the ordinary differential equation for the plastic strain tensor we obtain by assumptions (A1) – (A7)

∂_{t}^{3}ε_{p} = G′′(\frac{|σ|}{y})(\frac{|σ|_{t}}{y} − \frac{|σ|y_{t}}{y^{2}} )^{2} \frac{σ}{|σ|} +

+ G′(\frac{|σ|}{y})(\frac{|σ|_{tt}}{y} − 2 \frac{|σ|_{t}y_{t}}{y^{2}} − \frac{|σ|y_{tt}}{y^{2}} + 2 \frac{|σ|(y_{t})^{2}}{y_{3}} ) \frac{σ}{|σ|} +

+ G′(\frac{|σ|}{y})(\frac{|σ|_{t}}{y} − \frac{|σ|y_{t}}{y^{2}} )(\frac{σ_{t}}{|σ|} − \frac{1}{|σ|^{2}} |σ|_{t} σ) +

+ G(\frac{|σ|}{y})(\frac{σ_{tt}}{|σ|} − 2 \frac{σ_{t}}{|σ|^{2}} |σ|_{t} − \frac{|σ|_{tt}}{|σ|^{2}} · σ + \frac{2(|σ|_{t})^{2}}{|σ|^{3}} · σ) .

Thus the right hand side of (3.1) can be written in the form

− (∂_{t}^{2}σ, ∂_{t}^{3}ε_{p}) = − ∫_{Ω} A σ · ∂_{t}^{2}σ dx − ∫_{Ω} B∂_{t}σ · ∂_{t}^{2}σ dx − ∫_{Ω} C · |∂_{t}^{2}σ|^{2} dx , (3.2)

where

A = \underbrace{G′′(\frac{|σ|}{y})(\frac{|σ|_{t}}{y} − \frac{|σ|y_{t}}{y^{2}} ) \frac{1}{|σ|}}_{A_{1}} +

+ \underbrace{G′(\frac{|σ|}{y})(\frac{|σ|_{tt}}{y} − 2 \frac{|σ|_{t}y_{t}}{y^{2}} − \frac{|σ|y_{tt}}{y^{2}} + 2 \frac{|σ|(y_{t})^{2}}{y^{3}} )\frac{1}{|σ|}}_{A_{2}} +

+ \underbrace{G′(\frac{|σ|}{y})(\frac{|σ|_{t}}{y} − \frac{|σ|y_{t}}{y^{2}} ) \frac{|σ|_{t}}{|σ|^{2}}}_{A_{3}} + \underbrace{G(\frac{|σ|}{y})(\frac{2(|σ|_{t})^{2}}{|σ|^{3}} − \frac{|σ|_{tt}}{|σ|^{2}} )}_{A_{4}} ,

B = G′(\frac{|σ|}{y})(\frac{|σ|_{t}}{y} − \frac{|σ|y_{t}}{y^{2}} ) \frac{1}{|σ|} − 2G(\frac{|σ|}{y}) \frac{|σ|_{t}}{|σ|^{2}} ,

C = G(\frac{|σ|}{y}) \frac{1}{|σ|} .

Now we estimate all terms of the right hand side of (3.2) in sequence.

A_{1} : I_{A_{1}} := |− ∫_{Ω} G′′(\frac{|σ|}{y})(\frac{|σ|_{t}}{y} − \frac{|σ|y_{t}}{y^{2}} )^{2} \frac{1}{|σ|} σ · σ_{tt} dx|
≤ 2 sup_{p∈ℝ_{+}} |G′′(p)|(y^{∗})^{−2} ∫_{Ω} |σ_{t}|^{2} · |σ_{tt}| dx

+2 ∫_{Ω}|G′′(\frac{|σ|}{y}) | \frac{|σ|^{2}|y_{t}|^{2}}{y^{4}} |σ_{tt}| dx

≤ 2 sup_{p∈ℝ_{+}} |G′′(p)|(y^{∗})^{−2} ∫_{Ω} |σ_{t}|^{2} · |σ_{tt}| dx

2 ∫_{Ω} |G′′(\frac{|σ|}{y})| (\frac{|σ|}{y})^{4}m^{2}(y_{1} − y)^{2}G^{2}(\frac{|σ|}{y}) |σ_{tt}| dx

≤ 2 sup_{p∈ℝ_{+}} |G′′(p)|(y^{∗})^{−2} ∫_{Ω} |σ_{t}|^{2} · |σ_{tt}| dx

2 \frac{m^{2}(y_{1} − y^{∗})^{2}}{y^{∗}} sup_{p∈ℝ}(|G′′(p)|p^{3}G(p)) ∫_{Ω} |σ| |σ_{tt}| dx .

We have used here the ordinary differential equation for the function y. Thus by assumptions (A2) and (A6) we have that

∃_{C_{1}>0} I_{A_{1}} ≤ C_{1}( ∥σ_{t}∥_{𝕃^{4}}^{4} + ∥σ_{tt}∥^{2} + ∥σ∥^{2}) .
A_{2} : I_{A_{2}} := | − ∫_{Ω} G′(\frac{|σ|}{y})(\frac{|σ|_{tt}}{y} − 2 \frac{|σ|_{t} y_{t}}{y^{2}} − \frac{|σ|y_{tt}}{y^{2}} + 2 \frac{|σ|(y_{t})^{2}}{y^{3}} ) \frac{1}{|σ|} σ · σ_{tt} dx|

≤ ∫_{Ω} G′(\frac{|σ|}{y}) \frac{1}{y} |σ_{tt}| · ||σ|_{tt}| dx + 2 ∫_{Ω} G′(\frac{|σ|}{y}) \frac{| |σ|_{t}| y_{t} |σ_{tt}|}{y^{2}} dx +

+ ∫_{Ω} G′(\frac{|σ|}{y}) \frac{|σ| |y_{tt}| |σ_{tt}|}{y^{2}} dx + 2 ∫_{Ω} G′(\frac{|σ|}{y}) \frac{|σ|(y_{t})^{2}|σ_{tt}|}{y^{3}} dx

=: I_{A_{21}} + I_{A_{22}} + I_{A_{23}} + I_{A_{24}} .

In I_{A_{21}} we must estimate | |σ|_{tt} | in terms of |σ_{tt}| and |σ_{t}|. To do this we observe that

σ · σ_{tt} = (|σ|_{t})^{2} − |σ_{t}|^{2} + |σ| · |σ|_{tt}

and this equality yields

|σ|_{tt}|σ| ≤ |σ_{tt}| |σ| + 2|σ_{t}|^{2} .  (3.3)

Consequently for I_{A_{21}} we have

I_{A_{21}} ≤ \frac{1}{y^{∗}} sup_{p∈ℝ_{+}} G′(p) ∥σ_{tt}∥^{2} + \frac{1}{(y^{∗})^{2}} sup_{p∈ℝ_{+}} \frac{G′(p)}{p} (∥σ_{t}∥_{𝕃^{4}}^{4} + ∥σ_{tt}∥^{2}) .

In the estimation for the integral I_{A_{22}} we use the equation (BP6) to obtain

I_{A_{22}} ≤ 2 ∫_{Ω} G′(\frac{|σ|}{y}) \frac{|σ_{t}| |σ_{tt|}}{y^{2}} · m(y_{1} − y) G(\frac{|σ|}{y}) |σ| dx
≤ m(y_{1} − y^{∗}) \frac{1}{y^{∗}} sup_{p∈ℝ_{+}} (G′(p)p · G(p)) ( ∥σ_{t}∥^{2} + ∥σ_{tt}∥^{2}) .

Now we calculate the second time derivative y_{tt} from (BP6) and put the result into I_{A_{23}}:

I_{A_{23}} ≤ ∫_{Ω} G′(\frac{|σ|}{y}) \frac{|σ| |σ_{tt}|}{y^{2}} \{y_{t} G(\frac{|σ|}{y})|σ| + m(y_{1} − y) G′(\frac{|σ|}{y})(\frac{|σ_{t}|}{y} + \frac{|σ|y_{t}}{y^{2}} ) |σ| +

+ m(y_{1} − y) G(\frac{|σ|}{y}) |σ_{t}|\} dx

≤ \frac{1}{y^{∗}} sup_{p∈ℝ_{+}} G′(p)p ∫_{Ω} |σ_{tt}| \{m(y_{1} − y) G^{2}(\frac{|σ|}{y})|σ|^{2} + m(y_{1} − y) G′(\frac{|σ|}{y}) \frac{|σ_{t}|}{y} |σ| +

+ m^{2}(y_{1} − y)^{2} G′(\frac{|σ|}{y}) G(\frac{|σ|}{y}) |σ|^{3} + m(y_{1} − y) G(\frac{|σ|}{y}) |σ_{t}|\} dx .

Thus by assumptions (A1) – (A7) there exists a positive constant C_{I_{23}} such that

I_{A_{23}} ≤ C_{I_{23}} ( ∥σ_{tt}∥^{2} + ∥σ_{t}∥^{2} + ∥σ∥^{2}) .

Finally for I_{A_{24}} we have

I_{A_{24}} ≤ 2 ∫_{Ω} G′(\frac{|σ|}{y}) \frac{|σ| |σ_{tt|}}{y^{3}} m^{2}(y_{1} − y)^{2} G^{2}(\frac{|σ|}{y}) |σ|^{2} dx
≤ m^{2}(y_{1} − y^{∗})^{2} \frac{1}{y^{∗}} sup_{p∈ℝ_{+}} (G′(p)p^{2} · G^{2}(p)) ( ∥σ_{tt}∥^{2} + ∥σ∥^{2}) .

Summation of all the estimations for I_{A_{2i}}, i = 1, 2, 3, 4 finally yields

I_{A_{2}} ≤ C_{2}( ∥σ_{t}∥_{𝕃^{4}}^{4} + ∥σ∥^{2} + ∥σ_{t}∥^{2} + ∥σ_{tt}∥^{2})

with a positive constant C_{2} .

A_{3} : I_{A_{3}} := | ∫_{Ω} G′(\frac{|σ|}{y}) (\frac{|σ|_{t}}{y} − \frac{|σ|y_{t}}{y^{2}} ) \frac{|σ|_{t}}{|σ|^{2}} σ · σ_{tt} dx|

≤ \frac{1}{(y^{∗})^{2}} sup_{p∈ℝ_{+}} G′(p)/p ∫_{Ω} |σ_{t}|^{2} |σ_{tt}| dx +

+ m(y_{1} − y^{∗}) \frac{1}{y^{∗}} sup_{p∈ℝ_{+}} (G′(p)p G(p)) ∫_{Ω} |σ_{t}| |σ_{tt}| dx

≤ C_{3}( ∥σ_{t}∥_{𝕃^{4}}^{4} + ∥σ_{t}∥^{2} + ∥σ_{tt}∥^{2}) ,

where C_{3} > 0 is a constant.

A_{4} : I_{A_{4}} := | − ∫_{Ω} G(\frac{|σ|}{y})(\frac{2(|σ|_{t})^{2}}{|σ|^{3}} − \frac{|σ|_{tt}}{|σ|^{2}} ) σ · σ_{tt} dx| .

In the estimation of this integral we use again the inequality (3.3)

I_{A_{4}} ≤ ∫_{Ω} G(\frac{|σ|}{y})\{\frac{1}{|σ|} |σ_{tt}|^{2} + \frac{4|σ_{t}|^{2}}{|σ|^{2}} |σ_{tt}|\} dx
≤ y_{1} sup_{p∈ℝ_{+}} G(p)/p ∥σ_{tt}∥^{2} + 2(y_{1})^{2} sup_{p∈ℝ_{+}} G(p)/p^{2} ( ∥σ_{t}∥_{𝕃^{4}}^{4} + ∥σ_{tt}∥^{2})

B : I_{B} := | − ∫_{Ω} \{G′(\frac{|σ|}{y})(\frac{|σ|_{t}}{y} − \frac{|σ|y_{t}}{y^{2}} ) \frac{1}{|σ|} − 2G(\frac{|σ|}{y}) \frac{|σ|_{t}}{|σ|^{2}} \} σ_{t} · σ_{tt} dx| .

Similarly as above we show that there exists a positive constant C_{B} such that

I_{B} ≤ C_{B}( ∥σ_{t}∥_{𝕃^{4}}^{4} + ∥σ_{t}∥^{2} + ∥σ_{tt}∥^{2})
C : I_{C} := − ∫_{Ω} G(\frac{|σ|}{y})\frac{1}{|σ|} |σ_{tt}|^{2} dx

This integral is negative, and so it needs not to be estimated. To end up proof we need an estimation for the norm of the time derivative of the stress deviator in the space 𝕃^{4}(Ω; ℝ_{sym}^{3×3}) . In two dimensional case, we obtain by the Gagliardo-Nirenberg inequality

∥σ_{t}∥_{𝕃^{4}}^{4} ≤ C_{GN} {∥∇_{x}σ_{t}∥^{2} ∥σ_{t}∥^{2} + ∥σ_{t}∥^{4}} .

On the right side of the last inequality by (ii) of Lemma 1 the 𝕃^{2}-norm of the function σ_{t} does not blow up in finite time. Thus we should prove the boundedness of the mixed derivatives of the stress deviator in the space 𝕃^{2}(Ω; ℝ_{sym}^{3×3}) . The constitutive equation (BP2) yields

∥∂_{x_{i}}∂_{t} T (t)∥ ≤ D( ∥∂_{t}u(t)∥_{ℍ^{2}} + ∥∂_{t}ε_{p}(t)∥_{ℍ_{1}})

with a constant D > 0 . The velocity vector u_{t} satisfies the following equation

div_{x} Dε(∇_{x}u_{t}(x, t)) = div_{x} D∂_{t}ε_{p}(x, t) + u_{ttt}(x, t) ,

and by the regularity theorem for elliptic systems we conclude that

∥u_{t}(t)∥_{ℍ^{2}} ≤ C_{E}(∥u_{ttt}(t)∥ + ∥∂_{t}ε_{p}(t)∥_{ℍ^{1}}) .

Finally we obtain

\frac{d}{dt} E(∂_{t}^{2}u, ∂_{t}^{2}T, ∂_{t}^{2}ε_{p})(t)

≤ C( ∥σ_{tt}(t)∥^{2} + ∥u_{ttt}(t)∥^{2} + ∥σ_{t}(t)∥^{2} + ∥σ(t)∥^{2} + ∥∂_{t}ε_{p}(t)∥_{ℍ^{1}}^{2})

≤ C(E(∂_{t}^{2}u, ∂_{t}^{2}T, ∂_{t}^{2}ε_{p})(t) + E(∂_{t}u, ∂_{t}T, ∂_{t}ε_{p})(t) + E(u, T, ε_{p})(t) + ∥∂_{t}ε_{p}(t)∥_{ℍ^{1}}^{2}) .

In the paper [8] it has been proved that the term ∥∂_{t}ε_{p}(t)∥_{ℍ^{1}} is bounded by the ℍ^{1}-norms of the function y and the Cauchy stress tensor T . Thus ∥∂_{t}ε_{p}(t)∥_{ℍ^{1}} does not blow up in finite time. The Gronwall inequality now completes the proof.

The next step in this section is to prove the same result as above for the second space derivatives of solutions of the problem (BP). To do this we need first an 𝕃^{2}- estimation for the second space derivatives of the isotropic hardening function y. This is described in:

Lemma 3. With the same assumptions as in Theorem 2 the following estimates hold for t ≤ t^{∗}

a) ∥∂_{x_{i}}y(t)∥_{𝕃^{4}}^{4} ≤ C_{y}(t^{∗})(∥∂_{x_{i}}y^{(0)}∥_{𝕃^{4}}^{4} + ∫_{0}^{t} ∥∂_{x_{i}}σ(τ)∥_{𝕃^{4}}^{4}dτ ) , i = 1, 2

b) \begin{matrix}∑_{i,j=1}^{2} ∥∂_{x_{i}x_{j}}^{2}y(t)∥^{2} & ≤ & C_{y}(t^{∗})(∥y^{(0)}∥_{ℍ^{2}}^{2} \\ & & + ∑_{i,j=1}^{2} \{ ∫_{0}^{t} ∥∂_{x_{i}}σ(τ)∥_{𝕃^{4}}^{4} dτ + ∫_{0}^{t} ∥∂_{x_{i}x_{j}}^{2}σ(τ)∥^{2} dτ \}) . \end{matrix}

Here C_{y}(t^{∗}) is a positive constant independent of y and σ. Proof: The estimation a) has been proved in [9]. From the equation (BP6) we have

y(x, t) = y_{1} + (y^{(0)}(x) − y_{1}) exp \underbrace{\{−m ∫_{0}^{t} G(\frac{|σ(x, τ)|}{y(x, τ)}) |σ(x, τ)| dτ\}}_{Y (x,t)} .

For the second derivatives of the function y we obtain

y_{x_{i}x_{j}}(x, t) = y_{x_{i}x_{j}}^{(0)}(x) exp Y (x, t) + y_{x_{i}}^{(0)}(x) exp Y (x, t) · Y_{x_{j}}(x, t) + (y^{(0)}(x) − y_{1}) ·
· exp Y (x, t) · Y_{x_{i}}(x, t) · Y_{x_{j}}(x, t) + (y^{(0)}(x) − y_{1}) ·
· exp Y (x, t) · Y_{x_{i}x_{j}}(x, t) .

The function exp Y (x, t) is bounded by one, and thus

∥y_{x_{i}x_{j}}(t)∥^{2} ≤ C_{Y} (∥y_{x_{i}x_{j}}^{(0)}∥^{2} + ∥y_{x_{i}y_{i}}^{(0)}∥_{𝕃^{4}}^{4} + ∥Y_{x_{i}}(t)∥_{𝕃^{4}}^{4} + ∥Y_{x_{j}}(t)∥_{𝕃^{4}}^{4} + ∥Y_{x_{i}x_{j}}(t)∥^{2}) ,
i, j = 1, 2 , where C_{Y} > 0 .  (3.4)

The term ∥Y_{x_{i}}(t)∥_{𝕃^{4}}^{4} can be estimated similar as in the proof of the inequality a):

∥Y_{x_{i}}(t)∥_{𝕃^{4}}^{4} ≤ C′_{Y} (t^{∗})( ∫_{0}^{t} ∥∂_{x_{i}}σ(τ)∥_{𝕃^{4}}^{4} dτ + ∫_{0}^{t} ∥y_{x_{i}}(t)∥_{𝕃^{4}}^{4} dτ ) , i = 1, 2 and by the Gagliardo-Nirenberg inequality

∥Y_{x_{i}}(t)∥_{𝕃^{4}}^{4} ≤ C′′_{Y} (t^{∗})(1 + ∫_{0}^{t} ∥∂_{x_{i}}σ(t)∥_{𝕃^{4}}^{4} dτ + ∫_{0}^{t} ∥∇_{x}y_{x_{i}}(t)∥^{2} dτ ) , i = 1, 2 . To complete the proof we need a similar estimation as above for the last term in (3.4).

∥Y_{x_{i}x_{j}}∥^{2} ≤ C_{Y}^{∗} (t^{∗})( ∫_{0}^{t} ∫_{Ω}|G(\frac{|σ|}{y})|σ|_{x_{i}x_{j}} + |σ|_{x_{i}} G_{x_{j}} (\frac{|σ|}{y}) + |σ|_{x_{j}} G_{x_{i}}(\frac{|σ|}{y}) +

+|σ| G_{x_{i}x_{j}} (\frac{|σ|}{y})|^{2} dx dτ ) , i, j = 1, 2 .

The equality

G_{x_{i}x_{y}}(\frac{|σ|}{y}) = G′′(\frac{|σ|}{y})(\frac{|σ|_{x_{i}}}{y} − \frac{|σ| y_{x_{i}}}{y^{2}} )(\frac{|σ|_{x_{j}}}{y} − \frac{|σ| y_{x_{j}}}{y^{2}} +

+ G′(\frac{|σ|}{y})(\frac{|σ|_{x_{i}x_{j}}}{y} − \frac{|σ|_{x_{i}}y_{x_{j}}}{y^{2}} − \frac{y|σ|_{x_{j}}y_{x_{i}} + y|σ| y_{x_{i}x_{j}} − 2|σ| · y_{x_{i}}y_{x_{j}}}{y^{3}} ) ,

i, j = 1, 2 , and the fact that

|σ|_{x_{i}x_{j}}|σ| ≤ |σ_{x_{i}x_{j}}| |σ| + 2|σ_{x_{i}}||σ_{x_{j}}| i, j = 1 , 2  (3.5)

finally yield

∥Y_{x_{i}x_{j}}∥^{2} ≤ C_{Y}^{∗∗}(t^{∗})( ∫_{0}^{t} ∥σ_{x_{i}x_{j}}∥^{2} dτ + ∫_{0}^{t}( ∥σ_{x_{i}}∥_{𝕃^{4}}^{4} + ∥σ_{x_{j}}∥_{𝕃^{4}}^{4}) dτ +
+ ∫_{0}^{t}( ∥y_{x_{i}}∥_{𝕃^{4}}^{4} + ∥y_{x_{j}}∥_{𝕃^{4}}^{4}) dτ + ∫_{0}^{t} ∥y_{x_{i}x_{j}}∥^{2} dτ ) , i, j = 1, 2 .

The Gagliardo-Nirenberg inequality and the Gronwall lemma complete the proof.

The last lemma and the Gagliardo-Nirenberg inequality yield

∥y_{x_{i}}(t)∥_{𝕃^{4}}^{4} + ∥y_{x_{i}x_{j}}(t)∥^{2} ≤ C_{Y} ∫_{0}^{t} ∑_{i,j=1,2}^{2} ∥σ_{x_{i}x_{j}}(τ)∥^{2} i, j = 1 , 2  (3.6)

where the positive constant C_{Y} depends only on the time interval length t^{∗} . Thus the gradient of the isotropic hardening function ∇y(t) belongs to the space 𝕃^{4}(Ω)∩ℍ^{1}(Ω) if the second space derivatives of the stress deviator do not blow up in finite time, which is the object of the section.

Theorem 3. With the same assumptions as in Theorem 2 the stress tensor T satisfies

∃_{C_{T} (t^{∗})>0} ∀_{t≤t^{∗}}∥T_{x_{i}x_{j}}(t)∥^{2} ≤ C_{2}(t^{∗}) i, j = 1, 2 .

Proof: For boundary value problems it is more difficul to estimate space derivatives of solutions as time derivatives, for which the energy method works. We start with the constitutive equation (P2)

T (x, t) = D(ε(∇_{x} u(x, t)) − ε_{p}(x, t))

which clearly forces for i, j = 1, 2

∥∂_{x_{i}x_{j}}^{2}T (t)∥ ≤ C(∥u(t)∥_{ℍ^{3}} + ∥∂_{x_{i}x_{j}}^{2}ε_{p}(t)∥) .  (3.7)

where C is a positive constant. From the partial differential equation

div_{x}Dε(∇_{x} u(x, t)) = div_{x}Dε_{p}(x, t) + u_{tt}(x, t)

we conclude that

∥u(t)∥_{ℍ^{3}} ≤ C_{E}(∥u_{tt}(t)∥_{ℍ^{1}} + ∥div_{x}ε_{p}(t)∥_{ℍ^{1}}) .  (3.8)

To estimate the first term of the right hand side of (3.8) we apply Theorem 2. This yields boudedness in time of the norm

∥T_{tt}(t)∥ = ∥Dε(∇_{x} u(t)) − D∂_{t}^{2}ε_{p}(t)∥ .

Finally by the Korn inequality (u_{tt}_{|_{∂Ω}} = 0) we have

∥∇u_{tt}(t)∥ ≤ C′(∥T_{tt}(t)∥ + ∥∂_{t}^{2}ε_{p}(t)∥)  (3.9)

with a suitable positive constant C′ . Substituting (3.9) and (3.8) into (3.7) we obtain for i, j = 1, 2

∥∂_{x_{i}x_{j}}^{2}T (t)∥ ≤ C(∥u_{tt}(t)∥ + ∥T_{tt}(t)∥ + ∥∂_{t}^{2}ε_{p}(t)∥ + ∑_{i,j=1,2}^{2} ∥∂_{x_{i}x_{j}}^{2}ε_{p}(t)∥) . (3.10)

The first term on the right hand side of (3.10) is bounded in time by Lemma 1 i), the second by Theorem 2 and boundedness of the third follows from the equation (BP5):

∂_{t}^{2}ε_{p} = G′(\frac{|σ|}{y})(\frac{|σ|_{t}}{y} − \frac{|σ|y_{t}}{y^{2}} ) \frac{σ}{|σ|}G(\frac{|σ|}{y})(\frac{σ_{t}}{|σ|} − \frac{1}{|σ|^{2}} |σ|_{t} σ)

(assumptions (A1)-(A4)+(A7) imply boundedness of the norm ∥∂_{t}^{2}ε_{p}(t)∥). We end up with an estimate of the last term on the right hand side of (3.10). From the equation (BP5) we conclude

∥∂_{x_{i}x_{j}}^{2}ε_{p}(t)∥^{2} ≤ C_{ε_{p}}(∥∂_{x_{i}x_{j}}^{2}ε_{p}^{(0)}∥^{2} + ∫_{0}^{t} ∥∂_{t}∂_{x_{i}x_{j}}^{2}ε_{p}(τ)∥^{2}dτ ) i, j = 1 , 2 .

Similar as in the proof of theorem 2 we obtain

∂_{t}∂_{x_{i}x_{j}}^{2}ε_{p} = G′′(\frac{|σ|}{y})(\frac{|σ|_{x_{i}}}{y} − \frac{|σ|y_{x_{i}}}{y^{2}} )(\frac{|σ|_{x_{j}}}{y} − \frac{|σ|y_{x_{j}}}{y^{2}} ) \frac{σ}{|σ|} +

+ G′(\frac{|σ|}{y})(\frac{|σ|_{x_{i}x_{j}}}{y} − \frac{|σ|_{x_{i}}y_{x_{j}} + |σ|_{x_{j}}y_{x_{i}}}{y^{2}} − \frac{|σ|y_{x_{i}x_{j}}}{y^{2}} + 2 \frac{|σ|y_{x_{i}}y_{x_{j}}}{y_{3}} ) \frac{σ}{|σ|} +

+ G′(\frac{|σ|}{y})(\frac{|σ|_{x_{i}}}{y} − \frac{|σ|y_{x_{i}}}{y^{2}} )(\frac{σ_{x_{j}}}{|σ|} − \frac{1}{|σ|^{2}} |σ|_{x_{j}} σ) +

+ G′(\frac{|σ|}{y})(\frac{|σ|_{x_{j}}}{y} − \frac{|σ|y_{x_{j}}}{y^{2}} )(\frac{σ_{x_{i}}}{|σ|} − \frac{1}{|σ|^{2}} |σ|_{x_{i}} σ) +

+ G(\frac{|σ|}{y})(\frac{σ_{x_{i}x_{j}}}{|σ|} − \frac{σ_{x_{i}}|σ|_{x_{j}} + σ_{x_{j}}|σ|_{x_{i}}}{|σ|^{2}} − \frac{|σ|_{x_{i}x_{j}}}{|σ|^{2}} · σ + \frac{2 |σ|_{x_{i}}|σ|_{x_{j}}}{|σ|^{3}} · σ)

=: E_{1} + E_{2} + E_{3} + E_{4} + E_{5} .

Now all terms ∫_{0}^{t} |E_{i}|^{2} dx i = 1, …, 5 are estimated separately. For the first we find

I_{E_{1}} := ∫_{0}^{t} |E_{1}|^{2} dx ≤ 2 (y^{∗})^{−4} { sup_{p∈ℝ_{+}} |G′′(p)|^{2} ∫_{Ω}(|σ_{x_{i}}|^{4} + |σ_{x_{j}}|^{4}) dx+

+ sup_{p∈ℝ_{+}} |G′′(p)p|^{2} ∫_{Ω}(|σ_{x_{i}}|^{4} + |σ_{x_{j}}|^{4} + |y_{x_{i}}|^{4} + |y_{x_{j}}|^{4}) dx +

+ sup_{p∈ℝ_{+}} |G′′(p)p^{2}|^{2} ∫_{Ω}(|y_{x_{i}}|^{4} + |y_{x_{j}}|^{4}) dx}

≤ C_{E_{1}}(∥σ_{x_{i}}∥_{𝕃^{4}}^{4} + ∥σ_{x_{j}}∥_{𝕃^{4}}^{4} + ∥y_{x_{i}}∥_{𝕃^{4}}^{4} + ∥y_{x_{j}}∥_{𝕃^{4}}^{4})

where C_{E_{1}} > 0 . In the next estimation we use (3.5) and obtain

I_{E_{2}} := ∫_{0}^{t} |E_{2}|^{2} dx ≤ 5 (y^{∗})^{−2} { sup_{p∈ℝ_{+}} (G′(p))^{2}( ∫_{Ω} 2 |σ_{x_{i}x_{j}}|^{2} dx +

1/2 (y^{∗})^{−2} ∫_{Ω}(|σ_{x_{i}}|^{4} + |σ_{x_{j}}|^{4} + |y_{x_{i}}|^{4} + |y_{x_{j}}|^{4}) dx) +

sup_{p∈ℝ_{+}} (G′(p)p)^{2}( ∫_{Ω} |y_{x_{i}x_{j}}|^{2} dx + 2 (y^{∗})^{−2} ∫_{Ω}(|y_{x_{i}}|^{4} + |y_{x_{j}}|^{4}) dx) +

4 (y^{∗})^{−2} sup_{p∈ℝ_{+}} (G′(p)/p)^{2} ∫_{Ω}(|σ_{x_{i}}|^{4} + |σ_{x_{j}}|^{4}) dx}

≤ C_{E_{2}}(∥σ_{x_{i}x_{j}}∥^{2} + ∥σ_{x_{i}}∥_{𝕃^{4}}^{4} + ∥σ_{x_{j}}∥_{𝕃^{4}}^{4} + ∥y_{x_{i}x_{j}}∥^{2} + ∥y_{x_{i}}∥_{𝕃^{4}}^{4} + ∥y_{x_{j}}∥_{𝕃^{4}}^{4})

where C_{E_{2}} is a positive constant.

I_{E_{3}+E_{4}} := ∫_{0}^{t}(|E_{3}|^{2} + |E_{4}|^{2}) dx ≤ 8 (y^{∗})^{−4} { sup_{p∈ℝ_{+}} (G′(p)/p)^{2}( ∫_{Ω}(|σ_{x_{i}}|^{4} + |σ_{x_{j}}|^{4} dx) +

+ 1/2 sup_{p∈ℝ_{+}} (G′(p))^{2} ∫_{Ω}(|σ_{x_{i}}|^{4} + |σ_{x_{j}}|^{4} + |y_{x_{i}}|^{4} + |y_{x_{j}}|^{4}) dx}
≤ C_{E_{3}+E_{4}}(∥σ_{x_{i}}∥_{𝕃^{4}}^{4} + ∥σ_{x_{j}}∥_{𝕃^{4}}^{4} + ∥y_{x_{i}}∥_{𝕃^{4}}^{4} + ∥y_{x_{j}}∥_{𝕃^{4}}^{4})

(C_{E_{3}+E_{4}} > 0) . Finally using again (3.5) we have

I_{E_{5}} := ∫_{0}^{t} |E_{5}|^{2} dx ≤ 5 (y^{∗})^{−2} { sup_{p∈ℝ_{+}} (G(p)/p)^{2}( ∫_{Ω}(|σ_{x_{i}x_{j}}|^{2} dx)+

5 (y^{∗})^{−2} sup_{p∈ℝ_{+}} (G(p)/p)^{2}( ∫_{Ω}(|σ_{x_{i}}|^{4} + |σ_{x_{j}}|^{4} dx)}

≤ C_{E_{5}}(∥σ_{x_{i}x_{j}}∥^{2} + ∥σ_{x_{i}}∥_{𝕃^{4}}^{4} + ∥σ_{x_{j}}∥_{𝕃^{4}}^{4})

(C_{E_{5}} > 0) . Summing I_{E_{1}} + I_{E_{2}} + I_{E_{3}+E_{4}} + I_{E_{5}} we conclude that there exists a new positive constant C_{ε_{p}} such that

∑_{i,j=1,2}^{2} ∥∂_{x_{i}x_{j}}^{2}ε_{p}(t)∥^{2} ≤ C_{ε_{p}}(∥∂_{x_{i}x_{j}}^{2}ε_{p}^{(0)}∥^{2} + ∫_{0}^{t} ∑_{i=1,2}^{2}(∥σ_{x_{i}}(τ)∥_{𝕃^{4}}^{4} + ∥y_{x_{i}}(τ)∥_{𝕃^{4}}^{4}) dτ +

∫_{0}^{t} ∑_{i,j=1,2}^{2}(∥σ_{x_{i}x_{j}}(τ)∥^{2} + ∥y_{x_{i}x_{j}}(τ)∥^{2}) dτ .

Thus the Gagliardo–Nirenberg inequality, Lemma 3 and the Gronwall inequality complete the proof.

4 𝕃^{p}–Stability for p < 2

In this section we study stability in time of solutions to the Bodner–Partom system (BP) with homogeneous boundary data. The nonelastic deformation is described by the deviator part of the stress tensor T. If we assume that boundary data are equal to zero, then for t ⟶ ∞ σ should go to zero in a suitable topology. A proof of the conjecture is our object in this section. From Lemma 1 i) we observe that

∫_{0}^{∞} ∫_{Ω} G(\frac{|σ(x, τ)|}{y(x, τ)}) |σ(x, t)| dx dτ ≤ E(u, T, ε_{p})(0) .

This implies that the function F (t) := ∫_{Ω} G (\frac{|σ(t)|}{y(t)}) |σ(t)| dx belongs to 𝕃^{1}(ℝ_{+}) . Our goal is to prove that F (t) ⟶ 0 for t ⟶ ∞ . To do this we reinforce assumption (A4):

∃_{g′} ∀_{p∈ℝ_{+}}  G′(p)p^{3} < g′  (A4’)

By this assumption we can prove now a uniformly (independent of time) energy estimation for the first time derivatives of the solutions to the problem (BP).

Lemma 4. Let (u, T, ε_{p}, y) ∈ ℍ^{2}(Ω; ℝ^{3}) × (ℍ^{1}(Ω; ℝ_{sym}^{3×3}))^{2} × ℍ^{1}(Ω; ℝ_{+}) be a global in time solution of the problem (BP) with the homogeneous Dirichlet boundary condition to initial data, satisfying compatibility conditions (CC) and assumption (A7). Moreover assume that the assumptions (A1) – (A3)+(A4’) hold. Then

∃_{C_{t}>0} E(∂_{t}u, ∂_{t}T, ∂_{t}ε_{p})(t) ≤ E(∂_{t}u, ∂_{t}T, ∂_{t}ε_{p})(0) + C_{t} · E(u, T, ε_{p})(0)

where the constant C_{t} does not depend on t .

Proof: Similarly as in Theorem 1 we calculate the time derivative of the energy function E(∂_{t}u, ∂_{t}T, ∂_{t}ε_{p})(t) to obtain

\frac{d}{dt}E(∂_{t}u, ∂_{t}T, ∂_{t}ε_{p})(t) = −(∂_{t}σ(t), ∂_{t}^{2}ε_{p}(t)) .

By the expression for the derivative ∂_{t}^{2}ε_{p}(t) (see the proof of Theorem 3) we have

−(∂_{t}σ, ∂_{t}^{2}ε_{p}) = ∫_{Ω} G′(\frac{|σ|}{y})\frac{(|σ|_{t})^{2}}{y} dx − ∫_{Ω} G(\frac{|σ|}{y})(\frac{|σ_{t}|^{2} − (|σ|_{t})^{2}}{σ} dx) +

+ ∫_{Ω} G′(\frac{|σ|}{y}) |σ| |σ|_{t} \frac{y_{t}}{y^{2}} dx

≤ ∫_{Ω} G′(\frac{|σ|}{y})\frac{(|σ|_{t})^{2}}{y} dx + ∫_{Ω} G′(\frac{|σ|}{y}) |σ| |σ|_{t} \frac{m(y_{1} − y)}{y^{2}}G(\frac{|σ|}{y}) |σ| dx

≤ ∫_{Ω} G′(\frac{|σ|}{y})\frac{(|σ|_{t})^{2}}{y} dx + ∫_{Ω} G′(\frac{|σ|}{y})\frac{m^{2}(y_{1} − y)^{2}}{y^{3}} G^{2}(\frac{|σ|}{y}) |σ|^{4} dx

≤ C_{t} ∫_{Ω} G(\frac{|σ|}{y}) |σ| dx .

After integration of the last inequality an application of Lemma 1 i) implies

E(∂_{t}u, ∂_{t}T, ∂_{t}ε_{p})(t) ≤ E(∂_{t}u, ∂_{t}T, ∂_{t}ε_{p})(0) + C_{t} · E(u, T, ε_{p})(0) .

Lemma 4 yields boundedness of the derivative of the function F:

\frac{d}{dt} F = ∫_{Ω} (G′(\frac{|σ|}{y}) (\frac{|σ|_{t}y − |σ|y_{t}}{y^{2}} dx) |σ| + G(\frac{|σ|}{y}) |σ|_{t}) dx
≤ sup_{p∈ℝ_{+}} (G′(p)p + G(p)) ∫_{Ω} |σ_{t}| dx +

+\frac{m(y_{1} − y^{∗})}{y^{∗}} sup_{p∈ℝ_{+}} G(p)G′(p)p ∫_{Ω} |σ|^{2} dx

≤ C_{F} ( ∫_{Ω} |σ_{t}|^{2} dx + ∫_{Ω} |σ|^{2} dx) .

The last inequality gives that F (t) ⟶ 0 for t ⟶ ∞ . Let us define for ε > 0 and t > 0 A_{ε}(t) = {x ∈ Ω : |σ(x, t)| < ε } . By assumption (A7) we obtain

|Ω \ A_{ε}(t)| · G(\frac{ε}{y^{∗}} )ε ≤ ∫_{Ω\A_{ε}(t)} G(\frac{|σ|}{y}) |σ| dx ≤ ∫_{Ω} G(\frac{|σ|}{y}) |σ| dx ⟶ 0

for t ⟶ ∞ . Therefore we see that for all ε > 0

|Ω \ A_{ε}(t)| ⟶ 0 for t ⟶ ∞ .  (4.1)

Let p < 2 and denote by χ(A) the characteristic function to the set A . Thus for the 𝕃^{p}–norm of σ we conclude

∫_{Ω} |σ(t)|^{p} dx = ∫_{A_{ε}(t)} |σ(t)|^{p} dx + ∫_{Ω\A_{ε}(t)} |σ(t)|^{p} dx

≤ |Ω|ε^{p} + ∫_{Ω} χ(Ω \ A_{ε}(t))|σ(t)|^{p} dx

≤ |Ω|ε^{p} + ( ∫_{Ω} χ(Ω \ A_{ε}(t)) dx)^{1− \frac{p}{2}} ( ∫_{Ω} |σ(t)|^{2} dx)^{\frac{p}{2}} .

Since from the energy estimates (Lemma 1 i)) it follows that the integral ∫_{Ω} |σ(t)|^{2} dx is bounded in time we find by using (4.1)

∫_{Ω} |σ(t)|^{p} dx ⟶ 0 for t ⟶ ∞ .

References

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[8] Chełmiński, K. : “Energy estimates and global in time results for a problem from nonlinear viscoelasticity ” – Bull. of the Pol. Acad. of Sci.: Math. v. 44 p. 465-477, 1996

[9] Chełmiński, K. : “Stress L^{∞}– estimates and the uniqueness problem for the quasistatic equations to the model of Bodner–Partom ” – submitted to Math. Meth. in the App. S.

[10] Ionescu, I.R., Sofonea, M. : “Functional and numerical methods in viscoplasticity” – Oxford, New York Tokio: Oxford University Press, 1993 [11] Klaus, F. : “Existenz– und Eindeutigkeitssatz für die Millerschen Gleichungen

zur nichtlinearen Viskoelastizität mit Härtung” – Dissertation, FB Mathematik, TH Darmstadt 1994

[12] Le Tallec, P. : “Numerical analysis of viscoelastic problems” – Paris: Masson. Berlin, Heidelberg, New York : Springer 1990

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