数理解析研究所講究録
1197 巻 2001 年 145-162

145

THERMOELASTICITY SYSTEM IN
SHAPE MEMORY PROBLEMSANTONI OCHOWSKI\dagger

IRENA PAWLOW\dagger$\dot{\mathrm{Z}}$\dagger Systems Research Institute of the Polish Academy of Sciences,$\mathrm{u}1$$\mathrm{e}$. Newelska 6, 01-447 Warszawa, Poland,zochowsk@ibspan.waw.pl-mail: pawlow@ibspan.waw.pl,(ポーランド科学アカデミー)

Abstract

thermoelastic system, which arises as a
We are concerned with a nonlinear
model of dynamical processes in shape memory materials. The system has the formof a fourth order viscoelasticity equation coupled with the temperature equation.
For such a system we have proved global in time existence for arbitrary data, as
well as the uniqueness of the solution. In addition, we have studied stability ofsolutions with respect to distributed body forces and heat sources.
$3-\mathrm{D}$1

Thermoelasticity system$n=2$ or 3, be a bounded domain with a smooth boundaryLet
the following problem (P):$\Omega\subset \mathrm{B}^{n},$$\mathrm{u}_{t}\iota-l^{\text{ノ}}\mathrm{Q}\mathrm{u}_{t}+\frac{\kappa}{4}\mathrm{Q}\mathrm{Q}\mathrm{u}=\nabla\cdot F/\epsilon(\epsilon, \theta)+\mathrm{b}$$c(\epsilon, \theta)\theta t-k\triangle\theta=\theta F/\theta\epsilon(\epsilon, \theta)$:$\partial\Omega$. We consider
(1.1)

,

:

$\epsilon_{t}+\iota \text{ノ}(\mathrm{A}\epsilon_{t})$$\epsilon_{t}+g$in$Q_{T}$,(1.2)

with initial$\mathrm{u}(0, \mathrm{x})=\mathrm{u}_{0}(\mathrm{x})$,$\mathrm{u}_{t}(0, \mathrm{x})=\mathrm{u}_{1}(\mathrm{x})$$\theta(0, \mathrm{x})=\theta_{0}(\mathrm{x})$in$\Omega$

,

,

(1.3)
(1.4)

and boundary conditions$\mathrm{u}=0$,

Qu $=0$ ,$\nabla\theta\cdot \mathrm{n}=0$on$S_{T}$

,

(1.5)
(1.6)

146
where
$c(\epsilon, \theta)=Cv-\theta F/\theta\theta(\epsilon, \theta)$

.

(1.7)

Here refers to the region in a reference configuration.
The subscript denotes differentiation with respect to time, . is the divergence withrespect tois the unit outward normal toanddenote derivativesof.
with respect to
The meaning of the quantities in problem (P) is as follows::is thedisplacement vector, :is the absolute temperature,with, is the linearized strain tensor,
is the strain rate tensor,
is the elastic energy,
is the specific heat coefficient.
, l ノ and correspond to thermal specific heat, heat conducThe positive constants
tivity, viscosity and interface energy.
The vector is a distributed external force and is a distributed heat source.
The linear map(1.8),$\Omega$$t$$\mathrm{x},$$\nabla$$\partial\Omega,$$\mathrm{n}$$F(\epsilon, \theta)$$F/\theta(\epsilon, \theta)$$F/\epsilon(\epsilon, \theta)$$\theta$$\epsilon,$$\mathrm{u}$$\theta$$Q_{T}arrow \mathrm{B}_{+}$$Q_{T}arrow 1\mathrm{R}^{n}$$\epsilon=(\epsilon_{ij})$$\frac{1}{2}(u_{i/j}+u_{j/i})$$\epsilon_{ij}(\mathrm{u})=$$F(\epsilon, \theta)$$\epsilon_{t}=\epsilon(\mathrm{u}_{t})$$c(\epsilon, \theta)$$k$$c_{v},$$\kappa$$\mathrm{b}$$g$$\mathrm{u}-+\mathrm{A}\epsilon(\mathrm{u})=\lambda \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\epsilon(\mathrm{u})\mathrm{I}+2\mu\epsilon(\mathrm{u})$where $\mu>0$ are Lam\’e constants andis the unit matrix, represents Hooke’s
law for the homogeneous isotropic material. Herewith$\mathrm{I}=(\delta_{ij})$$\lambda,$$\mathrm{A}=(A_{ijkl})$$A_{iji}k\iota=\lambda\delta j\delta kl+\mu(\delta ik\delta j\iota+\delta il\delta jk)$,is the fourth order elasticity tensor satisfying the following symmetry conditions:$A_{ijkl}=A_{jikl}=A_{klij}$

The second order differential operator$\mathrm{Q}$defined by$\mathrm{u}\vdasharrow \mathrm{Q}\mathrm{u}=\nabla\cdot(\mathrm{A}\epsilon(\mathrm{u}))$

(1.9)

,

is known as operator of linearized elasticity. By (1.8) it admits the representation
Qu

In the divergence$\nabla$$\nabla$$=\mu\triangle \mathrm{u}+(\lambda+\mu)\nabla(\nabla\cdot \mathrm{u})$.

(1.10). we use the convention of the contraction over the last index, i.e.

. (A$\epsilon(\mathrm{u})$)$=\partial_{j}(A_{ijl}k\epsilon k\iota(\mathrm{u}))=A_{i}jk\iota\partial j\epsilon k\iota(\mathrm{u})=$AV$\epsilon(\mathrm{u})$.

Moreover, throughout the summation convention is used, and the following notation: forvectors a $=(a_{i}),$, A $=(A_{ijkl})$ we write
and tensors, Ba, etc.
To problem (P) corresponds the free energy functional of the Ginzburg-Landau form$\mathrm{B}=(B_{ij}),$$\mathrm{b}=(b_{i})$$\mathrm{a}\cdot \mathrm{b}=a_{i}b_{i},$$\mathrm{B}:\mathrm{C}=B_{i}jC_{ij},$$\mathrm{a}\mathrm{B}=a_{i}B_{ij}$$=B_{ij}a_{j},$$\mathrm{C}=(C_{ij})$$\mathrm{B}\mathrm{A}=B_{ij}A_{ijk}l$$f( \epsilon, \nabla\epsilon, \theta)=-C_{v}\theta\log\theta+F(\epsilon(\mathrm{u}), \theta)+\frac{\kappa}{8}|$Qu$|^{2}$,(1.11)

with the subsequent terms representing thermal energy, elastic energy and interfacial

energy.
The main characteristic feature of (1.11) as a model of shape memory materials is thenonlinearity in
is a multiple-well in with the shape changing qualitatively
with . The second characteristic feature is the presence of higher order term with$\epsilon:F(\epsilon, \theta)$$\theta$$\epsilon$

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coefficient , which accounts for interaction effects on phase interfaces. Terms of thistype are known in the so called multiscale approach to modelling of phase transitions.
The particular form of -term in (1.11) can be interpreted as a resultant of mechanicalforces acting on a layer element of interface.
A typical example of the elastic energy is the Falk-Konopka model [10] in the formof sixth order polynomial in terms of :$\kappa$$\kappa$$\epsilon_{ij}$,$F( \epsilon, \theta)=\sum i=13F^{2}i(\theta)J_{i}^{2}(\epsilon)+\sum_{i=1}^{5}F^{4}i(\theta)J_{i}^{4}(\epsilon)+\sum_{i=1}^{2}F^{6}i(\theta)J_{i}^{6}(\epsilon)$(1.12)

where, , are k-th order crystallographical invariants, that is appro$i=1,$
priate combinations of the strain tensor components
andare corresponding
temperature-dependent coefficients.
The form (1.12) represents a generalization of the well known 1-D Landau-Devonshireenergy $[8],[9]$,(1.13)
where
are constant parameters, andis a critical temperature.
$i^{k}$$\sqrt{}^{k}(i\epsilon),$$\ldots$$F_{i}^{k}(\theta)$$\epsilon_{ij}$$F(\epsilon, \theta)=\alpha_{1}(\theta-\theta C)\epsilon 2-\alpha_{2}\epsilon 4+\alpha_{3}\epsilon^{6}$$\alpha_{i}>0$2$\theta_{c}>0$

Motivation and known results

The equations $(1.1),(1.2)$ express balance laws of linear momentum and energy (underassumption of constant material density)$\mathrm{u}_{tt}-\nabla\cdot\sigma=\mathrm{b}$$e_{t}+\nabla\cdot \mathrm{q}-\sigma$

with constitutive equations for stress tensor
effects:$\sigma=\frac{\delta f}{\delta\epsilon}+\sigma^{v}$$\sigma$:$\epsilon t=g$

(2.1)
(2.2)

and energy flux,$\mathrm{q}^{=-k}\nabla\theta+\mathrm{P}$,,$\mathrm{q}$accounting for interfacial$\sigma^{v}=\nu \mathrm{A}\epsilon_{t}$,,$\mathrm{p}=-\epsilon_{t}f/\mathrm{D}\epsilon$

(2.3)

.

(2.4)

Here
denotes the first variation of the functional , and the internal energy is
given by Gibbs relation$e=f+\theta s$ ,$s=-f/\theta$ ,
(2.5)
$f$

$\delta f/\delta\epsilon$

where

$s$

$e$

is the entropy. In case of free energy (1.11) expressions (2.3), (2.4) become$\sigma=F/\epsilon-\frac{\kappa}{4}\mathrm{A}\epsilon(\mathrm{Q}\mathrm{u})+\nu \mathrm{A}\epsilon_{t}$$\mathrm{q}=-k\nabla\theta-\frac{\kappa}{4}\epsilon t(\mathrm{A}\mathrm{Q}\mathrm{u})$

.

,

(2.6)
(2.7)

The stress tensor
contains three contributions: elastic stress, interfacial stress andviscous stress according to Hooke’s-like law.
The energy flux is composed of the usuall heat flux expressed by Fourier law, andnonequilibrium phase interface flux . The latter, called interstitial work flux corresponding to working of phase interfaces, appears in Dunn-Serrin [6] thermodynamical
theory of higher grade thermoelastic materials.
$\sigma$$\mathrm{p}$148free energy models describing phase
In the literature there are known severaltransitions in crystals, in particular model of Falk-Konopka [10], model of Ericksen [7],gradient-type model of Barschwhich has been analysed numerically in [4], [14],model for noncrystalline shape memory material [23].
Krumhansl [2], andIn 1-D case the dynamical model describing martensitic phase transitions of the shear
type has been developed by Falk $[8],[9]$ . There the shear strain plays the role of the orderparameter, that is quantity characterizing different phases. The corresponding free energy
has the form of Ginzburg-Landau functional, dependent on strain, absolute temperatureand strain gradient.
The resulting 1-D nonlinear thermoelasticity system has been the subject of intensive
mathematical studies, see e.g. references in the monograph of Brokate and Sprekels [3],and [1].
dynamical model corresponding to Falk-Konopka energy has been derived inA[17]. In a special case it constitutes an analog of 1-D Falk’s dynamical model. The
governing constitutive laws $(2.3),(2.4)$ (imposed by entropy principle) are characteristic
for thermoelastic materials of higher grade.
evolution model is due to Fr\’emond [11]. The free energy
A conceptually different
depends on volumetric proportions of phases, strain tensor, absolute temperature and. For the existing mathematical analysis of this model we
gradient of strain tensor
refer e.g. to [3].
We mention also the model due to Fried-Gurtin [12], with energy dependent on strain
tensor, multicomponent order parameter and its gradient.
$3-\mathrm{D}$$2-\mathrm{D}$$2-\mathrm{D}$$3-\mathrm{D}$$3-\mathrm{D}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$3Assumptions and main results

We assume that(D) the boundary$\partial\Omega$is of class$C^{2}$.

Further assumptions concern the elastic energy:
denotes the set of, whereon
is of class(FE-1) Structure:. We assume the splitting
symmetric tensors of second order in$C^{3}$$F(\epsilon, \theta)$$S^{2}$$S^{2}\cross[0, \infty)$$l\mathrm{R}^{n}$$F(\epsilon, \theta)=F_{1}(\epsilon, \theta)+F_{2}(\epsilon)$

wherevalues of .

$F_{1}(.\epsilon, \theta)$

is linear in$\theta$over certain interval$[0, \theta_{1})$,and satisfies (FE-2) for large$\theta$(FE-2) Growth conditions: LetandA such that for$|\epsilon|\geq\epsilon_{1}$$\epsilon_{1}$andbe certain constants. There exists a constantthe following conditions are satisfied:$\theta_{1}$$\theta\geq\theta_{1}$$|F_{1/\epsilon\epsilon}(\epsilon, \theta)|\leq\Lambda+\Lambda\theta^{r}|\epsilon|^{q-}1$$|F_{1/\epsilon\theta}(\epsilon, \theta)|\leq\Lambda+\Lambda\theta^{r-1}|\epsilon|q$,,$|F_{2/\epsilon\epsilon}(\epsilon)|\leq\Lambda+\Lambda|\epsilon|^{\overline{q}}-1$,$|F_{1/\theta\theta}(\epsilon, \theta)|\leq\Lambda+\Lambda\theta^{r-2}|\epsilon|^{q}+1$

where$0<r< \frac{2}{p_{n}}$,,$1<q \leq q_{n}(\frac{1}{p_{n}}-\frac{r}{2})$,$1< \overline{q}\leq\frac{q_{n}}{p_{n}}$

149
into
is the Sobolev exponent for which the imbedding of$n\geq3$
and is any finite number
is continuous, that is $q_{n}=2n/(n-2)$ for$p_{n}=n+2$ , and$L_{q_{n}}(\Omega)$$W_{2}^{1}(\Omega)$$q_{n}$$q_{n}$

for $n=2$ .

We note that the above conditions imply the following growth of,$|F_{1}(\epsilon, \theta)|\leq\Lambda+\Lambda\theta^{r}|\epsilon|^{q}+1$:$F(\epsilon, \theta)$$|F_{2}(\epsilon)|\leq\Lambda+\Lambda|\epsilon|^{\overline{q}}+1$.

The next two assumptions are physically natural.

(FE-3) Concavity with respect to$\theta$(thermal stability):for$F_{1/\theta\theta}(\epsilon, \theta)\leq 0$$(\epsilon, \theta)\in S^{2}\cross[0, \infty)$.

This implies the lower bound for the specific heat coefficientfor$0<c_{v}\leq c(\epsilon, \theta)$$(\epsilon, \theta)\in S^{2}\cross[0, \infty)$.

(FE-4) Lower bound for the internal energy:for$-\Lambda\leq(F_{1}(\epsilon, \theta)-\theta F_{1/}\theta(\epsilon, \theta))+F_{2}(\epsilon)$$(\epsilon, \theta)\in S^{2}\cross[0, \infty)$.

The most restrictive is the assumption on -growth exponent $r<1./2$ and the assumption
case.
inon -growth exponentIncase the latter assumption is not active, since and are then any large numbers.
Hence our assumptions admit the form of sixth order polynomial (1.12) only incase they require the growth with respect to close to quadratic. The
case. In
(as in 1-D model (1.13)).
temperature dependence is restricted to quadratic terms
case.
in
2-and
both
neededison
The growth condition
We are looking for the solution in the anisotropic Sobolev space$\theta$$\overline{q}\leq 6/5$$\epsilon$$3-\mathrm{D}$$2-\mathrm{D}$$q$$\overline{q}$$2-\mathrm{D}$$3-\mathrm{D}$$\epsilon$$F_{i}^{2}(\theta)$$3-\mathrm{D}$$r$$V(p)=\{(\mathrm{u}, \theta)\in \mathrm{W}_{p}^{4}’ 2(Q_{\tau)}\mathrm{x}W_{p}^{2,1}(Q\tau)\}$,with a parameter related to the -integrability. The assumptions on the initial data
and the source terms correspond to this space.
$p$$L_{p}$(BV-p) Let $\delta>0,$ $p>1,$ $p_{1}=p+\delta$ . The initial conditions satisfy the inclusions$\mathrm{u}_{0}\in \mathrm{w}_{p}4-2/p(\Omega)$,$\mathrm{u}_{1}\in \mathrm{w}_{p}2-2/p(\Omega)$$0\leq\theta_{0}\in W_{p_{1^{-}}}^{22/p_{1}}(\Omega)$,,

and the compatibility relations. The source terms satisfy$\mathrm{b}\in \mathrm{L}_{p}(Q_{\tau})$,$g\in L_{p_{1}}(QT)$,$g\geq 0\mathrm{a}.\mathrm{e}$.

The first main result concerns the existence of solutions to problem (P).

150
Theorem 3.1 Existence.
Under assumptions $(D)$ , (FE-l) $-(FE-\mathit{4})$ , (BV-p) and the condition$0<\sqrt{\kappa}<\nu$there exists fora solution
the following a priori estimates hold,$(\mathrm{u}, \theta)\in V(p)$$p\geq p_{n}$$||\mathrm{u}||_{\mathrm{w}_{p}^{4}’(Q\tau)}2\leq\Lambda$,

to problem $(P)$ for any $T>0$ . Moreover,$||\theta||_{W_{\mathrm{p}}^{2,1}}(QT)\leq\Lambda$

with a constant A dependent on the data of the problem,

,

(3.1)

and time$\Omega$$T$.

The condition between and is needed for parabolic decomposition of elasticity equation (1.1).
This theorem has several consequences concerning regularity of the solution:$\nu$$\kappa$

Corollary 3.1 For a solution to problem $(P)$ the following holds: the functions
are continuous inand$\nabla^{2}\mathrm{u},$$\theta$$\mathrm{u}$, Vu,$Q_{T\mathrm{z}}$$\mathrm{u}_{t},$$|\mathrm{u}|,$$|\nabla \mathrm{u}|,$$|\nabla^{2}\mathrm{u}|,$$|\mathrm{u}_{t}|\leq\Lambda$,

in$0\leq\theta\leq\Lambda$

forA$||\nabla^{3}\mathrm{u}||_{\mathrm{L}_{p}(Q)}T’||\nabla \mathrm{u}_{t}||_{\mathrm{L}_{p}(Q)}T’||\nabla\theta||_{\mathrm{L}_{p}(Q)}T\leq$$c_{v}\leq c(\epsilon, \theta)\leq cmaxcmax(=\Lambda)$$Q_{T}$,$p_{n}\leq p<\infty$,.

The second result concerns the uniqueness of the solution. The proof requires anadditional regularity, which holds, provided $p>p_{n}$ . Moreover, stronger assumptions on
and have to be imposed.
$F(\epsilon, \theta)$$g$(FE-5) The function
satisfies$F_{1}(\epsilon, \theta)$

is of class$C^{4}$$g\in L_{\infty}(Q_{T})$on the set
and$S^{2}\cross[0, \infty)$$g\geq 0$$a.e$, and the heat source.

Theorem 3.2 Uniqueness.
Let the assumptions of Theorem 3.1 and (FE-5) be satisfied, and $p>p_{n}$ . Then thesolution to the problem $(P)$ is unique for any $T>0$ .
We note that in case of uniqueness the solution has additional continuity properties:
Corollary 3.2 For a solution to problem $(P)$ the following holds in case $p>p_{n}$ :, and satisfy the bounds
are continuous in$\nabla^{3}\mathrm{u},$$\nabla \mathrm{u}_{t},$$\nabla\theta$$Q_{T}$$|\nabla^{3}\mathrm{u}|,$Then, in particular, the stress tensor$|\nabla \mathrm{u}_{t}|,$$\sigma$$|\nabla\theta|\leq\Lambda$

is continuous in.
$Q_{T}$.

The third main result establishes the stability of solution.

Theorem 3.3 Stability.
Under the assumptions of Theorem 3.2 the solutions$i=1,2$ , satisfy the inequality
hand sides$(u^{i}, \theta^{i})$corresponding to the right-$(\mathrm{b}^{i}, g^{i}),$$||(\mathrm{u}^{2}-\mathrm{u}^{1}, \theta 2-\theta^{1})||_{V}(p)\leq\Lambda(||\mathrm{b}^{2}-\mathrm{b}^{1}||_{\mathrm{L}(}pQ\tau)+||g2-g1||_{L(}\mathrm{p}Q\tau))$

for any finite $p>p_{n}$ and $T>0_{f}$ whereproblem,$\Omega$and time$T$.

$\Lambda$(3.2)

is a constant dependent on the data of the151

Existence theorem–outline of the proof

4

The proof is based on the parabolic decomposition of equation (1.1) and the applicationof the Leray-Schauder fixed point theorem. Here we present the main steps, the details
are given in [19].
Further on denotes a constant, depending only on the data of the problem, domain
and time horizon.
so that
Step 1. Parabolic decomposition. Chosing numbers$\Lambda$$\alpha,$$\alpha+\beta=\nu$,$\alpha\beta=\frac{\kappa}{4}$$\beta$,(4.1)(1.1) with initial conditions (1.3) and boundary conditions (1.5) decomposes into thefollowing systems of BVP’s for a vector field :$\mathrm{w}$

in
in

,

$\mathrm{w}_{t}-\beta \mathrm{Q}\mathrm{W}=\nabla\cdot F/\epsilon(\epsilon, \theta)+\mathrm{b}$$\mathrm{w}(\mathrm{O}, \mathrm{X})=\mathrm{u}_{1}(\mathrm{x})-\alpha \mathrm{Q}\mathrm{u}_{0}(\mathrm{X})$$\mathrm{w}=0$

and the displacement$\mathrm{u}$on$S_{T}$$Q_{T}$$\Omega$

,

,

(4.2)

,

:

in

,

$\mathrm{u}_{t}-\alpha \mathrm{Q}\mathrm{u}=\mathrm{w}$$Q_{T}$$\mathrm{u}(\mathrm{O}, \mathrm{x})=\mathrm{u}_{0}(\mathrm{x})$inon$S_{T}$$\mathrm{u}=0$,$\Omega$

(4.3)

,

,

$\Re\beta>0$ . System $(4.2),(4.3)$
The condition between parameters and assures thatis coupled with the BVP (1.2), $(1.4),(1.6)$ for .
Step 2.
The solution map. We use the Leray-Schauder theorem in the following
formulation [5]:$\Re\alpha,$$\nu$$\kappa$$\theta$

Theorem 4.1 Let be a Banach space. Assume thatwith the following properties:$B$

For any fixed(i)$\lambda\in[0,1]$

the map$T$$B$(iii) There is a bounded subset, is contained in .

$C$

$0\leq\lambda\leq 1$

(iv)

Then

$T(0, \cdot)$$T(1, \cdot)$

of

$B$

$[0,1]\cross Barrow B$

is a map: $Barrow B$ is completely continuous.

$T(\lambda, \cdot)$(ii) For every bounded subset of , the family of maps
is uniformly equicontinuous.
$C$:: $[0,1]arrow B$ ,$T(\cdot, \chi)$, such that any fixed point in

$B$

of

$\chi\in C$,$T(\lambda, \cdot)$,$C$

has precisely one fixed point in$B$

has at least one fixed point in

$B$

.

.

To define the corresponding operator we extend the definition of, and thatof in 1R in such a way that it is of class$C^{3}$$\theta$$F_{1/\theta\theta}(\epsilon, \theta)\geq 0$

for$(\epsilon, \theta)\in S^{2}\cross(-\infty, 0)$.

$F_{1}(\epsilon, \theta)$

to all values

152
remains bounded
We note that with such extension the specific heat coefficient$V(p)$
frominto $V(p)$ ,. We define now the map
from below for all$c(\epsilon, \theta)$$(\epsilon, \theta)\in S^{2}\cross 1\mathrm{R}$$T_{\lambda}$:$T_{\lambda}$$(\overline{\mathrm{u}},\overline{\theta})arrow(\mathrm{u}, \theta)$,$\lambda\in[0,1]$

by means of the following three problems: BVP for$\mathrm{w}$inin$\mathrm{w}_{t}-\beta \mathrm{Q}_{\mathrm{W}}=\lambda[\nabla\cdot F/\epsilon(\overline{\epsilon},\overline{\theta})+\mathrm{b}]$$\mathrm{w}(\mathrm{O}, \mathrm{X})=\lambda[\mathrm{u}_{1}(\mathrm{x})-\alpha \mathrm{Q}\mathrm{u}_{0}(\mathrm{X})]$on$\mathrm{w}=0$

where$\overline{\epsilon}=\epsilon(\overline{\mathrm{u}})$, BVP for

$S_{T}$$Q_{T}$$\Omega$

,

(4.4)

,

,

$\mathrm{u}$in$\mathrm{u}_{t}-\alpha \mathrm{Q}\mathrm{u}=\mathrm{w}$$Q_{T}$

on$\mathrm{u}=0$

,

in
,

$\mathrm{u}(\mathrm{O}, \mathrm{x})=\lambda \mathrm{u}_{0}(\mathrm{x})$and BVP for,$\Omega$(4.5),$S_{T}$$\theta$$c_{\lambda}(\epsilon,\overline{\theta})\theta_{t}-k\Delta\theta=\lambda[\overline{\theta}F/\theta\epsilon(\epsilon,\overline{\theta}) :$\theta(0, \mathrm{x})=\lambda\theta_{0}(\mathrm{x})$$\nabla\theta\cdot \mathrm{n}=0$

on

in
,

$\Omega$in\epsilon_{t}+\nu(\mathrm{A}\epsilon_{t}):\epsilon_{t}+g]$$Q_{T}$

,

(4.6)

,

$S_{T}$

where$c_{\lambda}(\epsilon,\overline{\theta})=Cv-\lambda\overline{\theta}F/\theta\theta(\epsilon,\overline{\theta})$.

inof theClearly a fixed point ofinis equivalent to a solution.
decomposed system and thus constitutes a solution to problem (P) in
In further steps of the proof we shall verify assumptions of Theorem 4.1.
belongs
Step 3. First we show thatis well defined in $V(p)$ , i.e. the image
to $V(p)$ . Here we use the fact that the systems $(4.2),(4.3)$ are parabolic in the general
Solonnikov sense [19], and therefore the following result applies $[22],[21]$ .
$T_{1}$$(\mathrm{u}, \theta)$$\mathrm{V}(\mathrm{p})$$\mathrm{V}(\mathrm{p})$$\mathrm{V}(\mathrm{p})$$T_{\lambda}(V(p))$$T_{\lambda}$

Lemma 4.1 For a domain with$\partial\Omega\in C^{2}$, the solution of the system$\mathrm{u}_{t}-\mathrm{Q}\mathrm{u}=\mathrm{f}$$\mathrm{u}(\mathrm{O}, \mathrm{x})=\mathrm{u}_{0}$$\mathrm{u}=0$inin$Q_{T}$

on$\Omega$$S_{T}$,,,

satisfies the following inequality for $1<p<\infty$ ,.

$||\mathrm{u}||_{\mathrm{w}_{p}^{2,1}}(Q_{T})\leq\Lambda\{||\mathrm{f}||_{\mathrm{L}}p(QT)+||\mathrm{u}_{0}||_{\mathrm{W}_{p}^{2}}-2/p(\Omega)\}$

The application of Lemma 4.1 to BVP (4.4) implies
in $V(p)$ -norm. Furthermore, from the classicaland the corresponding bounds by
and is bounded by the same norm. Therefore
parabolic theory [15],.
$\mathrm{w}\in \mathrm{W}_{p}^{2,1}(Q_{\tau)},$$(\overline{\mathrm{u}},\overline{\theta})$$\theta\in W_{p}^{2,1}(Q_{T})$$T_{\lambda}(\overline{\mathrm{u}},\overline{\theta})\in V(p)$$\mathrm{u}\in \mathrm{W}_{p}^{4,2}(Q_{\tau)}$

153
Step 4. Equicontinuity of . A direct comparison of solutions, using Lemma 4.1, shows that
corresponding to$T_{\lambda}$$(\mathrm{w}^{i}, \mathrm{u}^{i}, \theta^{i}),$$i=1,2$ ,$\lambda^{i}\in[0,1]$$||\mathrm{w}^{1}-\mathrm{w}2||\mathrm{w}^{2,1}\mathrm{p}(Q_{T})$’$||\mathrm{u}^{1}-\mathrm{u}^{2}||\mathrm{W}_{p}^{4,2}(Q\tau)\leq\Lambda|\lambda^{1}-\lambda^{2}|$

.

(4.7)

By the classical parabolic theory, using (4.7), we get estimate$||\theta^{1}-\theta^{2}||_{\mathrm{W}^{2,1}}p(Q_{T})\leq\Lambda|\lambda^{1}-\lambda^{2}|$

(4.8)

.

Thus assumption (ii) of the Leray-Schauder theorem is satisfied.
.
for
Step 5. Uniqueness of the fixed point of
the system $(4.4)-(4.6)$ has the unique solution
By the regularity of the problem, for$V(p)\ni(\mathrm{u},\theta)=(0,0)$ is the unique fixed point of.
. Therefore
The essential part of the proof is the verification of assumption (iii) of the LeraySchauder theorem.
andStep 6. A priori bound for a fixed point. Without loss of generality we set
is a fixed point of .
assume that
Step 6.1 First we show that temperature is nonnegative. With this we prove energy
estimates, and then improve them reccursively.
$\lambda=0$$T_{\lambda}$$\lambda=0$$T_{0}(\cdot)$$(\mathrm{w}, \mathrm{u}, \theta)=(0,0, \mathrm{o})$$\lambda=1$$(\mathrm{u}, \theta)\in V(p)$Lemma 4.2 If$(P)$ in $V(p)$ , then$(\mathrm{u}, \theta)$

Proof. Let
problem for :$T_{1}$

is a fixed point of.
in$T_{1}$, that is it constitutes a solution to problem$Q_{T}$$\theta\geq 0$$(\mathrm{u}, \theta)\in V(p)$for $p=n+2$ , and $g\in L_{p}(Q_{T})$ . We consider the parabolic$\eta$

in$\eta_{t}-\nabla\cdot(\overline{k}\nabla\eta)+\mathrm{d}\cdot\nabla\eta-a\eta=f$$\eta(0, \mathrm{x})=\theta_{0}(\mathrm{X})$on$\nabla\eta\cdot \mathrm{n}=0$$Q_{T}$.

’

in$S_{T}$$\Omega$

(4.9)

,

,

where
$\overline{k}=\frac{k}{c(\epsilon,\theta)}$$a= \frac{F/\theta\epsilon(\epsilon,\theta).\epsilon_{t}}{c(\epsilon,\theta)}.$,,$\mathrm{d}=\nabla\overline{k}$,$f= \frac{\nu(\mathrm{A}\epsilon_{t}).\epsilon_{t}+g}{c(\epsilon,\theta)}.$.

areare continuous in
By assumption and the imbedding theorems,
for any $t\in[0, T]$ , the coefficient is bounded from below and frombounded inabove by positive constants, and the right-hand side is nonnegative.
With this we check that the assumptions of the existence theorem [15], Thm III 5.1,
are satisfied. Therefore there exists the unique generalized solution to problem (4.9), with
. Next we apply the modified version [19] of the stability result [15], Thm
III 4.5, to the solutions of the problem (4.9). To this purpose we take smooth functions$f,$in appropriate norms. By maximum$f^{m}\geq 0,$
converging toto problem (4.9) with smooth data
principle [15], Thm I 2.2, the classical solution. By stability theorem
insatisfies$\theta,$$Q_{T},$$\nabla\epsilon$$\epsilon,$$\overline{k}$$\mathrm{L}_{p}(\Omega)$$f$$\eta\in V_{2}^{1,1/2}(Q_{\tau})$$a^{m},$$a,$$\theta_{0}^{m}\geq 0$$d_{i}^{m},$$d_{i},$$\theta_{0}$$\eta^{m}$$\eta^{m}\geq 0$$Q_{T}$$\eta^{m}arrow\eta$strongly in$V_{2}^{1,0}(Q_{\tau})$.

$\epsilon_{t},\nabla\theta$

154
For the definition of spaces
following: For any nonnegative smooth function$V_{2}^{1,1/2}(Q_{\tau}),$$V_{2}^{1,0}(Q_{\tau})$$\phi$

see [15]. With this we conclude theand any$t\in(\mathrm{O}, T)$,$0 \leq\int_{\Omega}\phi\eta^{m}dx=\int_{\Omega}\phi(\eta^{m}-\eta)d_{X}+\int_{\Omega}\phi\eta dx$

where the first integral on the right-hand side converges to . Therefore,$0$$\int_{\Omega}\phi\eta d_{X}\geq 0$

and consequently. in. It is now enough to observe that coincides withsince (4.9) is equivalent to $(1.2),(1.4),(1.6)$ .
Step 6.2 Energy estimates. We multiply equation (1.1) by , equation (1.2) by 1, andintegrate over
performing integration by parts and using boundary conditions. Thenin view of the nonnegativity of temperature and boundedness from below of the internalenergy, using Gronwall’s inequality we arive at$Q_{T}$$\eta\geq 0\mathrm{a}.\mathrm{e}$$\eta$’$\theta\square$$\mathrm{u}_{t}$$Q_{t}$

Lemma 4.3

$A$

fixed point of$T_{1}$

satisfies, for almost any $t\in(0, T)$ ,(4.10),$\int_{\Omega}[\frac{1}{2}|\mathrm{u}_{t}|^{2}+C_{v}\theta+(F_{1}(\epsilon, \theta)-\theta F_{1/}\theta(\epsilon, \theta))+F_{2}(\epsilon)+\frac{\kappa}{8}|\mathrm{Q}\mathrm{u}|^{2}]dX\leq\Lambda$

with a constant A dependent only on the initial data, the sources$\mathrm{b},$and time

$g$

$T$

.

Step 6.3 Improvement of estimates. The energy estimates allow to obtain more refinedbounds for the fixed point. The essential tool is the following ellipticity property of theoperator , see Ne\v{c}as [16] p. 260:$\mathrm{Q}$

Lemma 4.4 Assume$\partial\Omega\in C^{2}$

Qu. Then for the solution of the problem$=\mathrm{f}$in$\Omega$,on$\mathrm{u}=0$$\partial\Omega$,

it holds$\mathrm{f}\in L_{2}(\Omega)\Rightarrow \mathrm{u}\in \mathrm{W}_{2}^{2}(\Omega)$.

Due to this property, and (4.10), we obtain the following chain of implications$||\mathrm{u}||_{L_{\infty}(\tau;\mathrm{W}_{2(\Omega}^{2}}0,))\leq\Lambda$$\Rightarrow$$||\epsilon||_{L_{\infty}}(0,\tau;\mathrm{W}21(\Omega))\leq\Lambda$$\Rightarrow$$||\epsilon||L_{\infty}(0,T;\mathrm{L}qn(\Omega))\leq\Lambda$.

(4.11)

The further procedure consists in obtaining more estimates from the decomposed system.
To this purpose we use the regularity property of parabolic systems given by the following
generalization [19] of Friedman-Ne\v{c}as Lemma [13]:Lemma 4.5 Let$\mathrm{u}$be a solution ofin$\mathrm{u}_{t}-\mathrm{Q}\mathrm{u}=\mathrm{f}+\nabla\cdot\sigma$$\mathrm{u}(\mathrm{O}, \mathrm{x})=0$$\mathrm{u}=0$If$\mathrm{f},$$\sigma\in \mathrm{L}_{p}(Q_{T})$inon$\Omega$$Q_{T}$,,$S_{T}$.

for $1<p<\infty$ , then$||\nabla \mathrm{u}||_{\mathrm{L}_{p}(Q\tau)}\leq\Lambda(p, T, \Omega)(||\mathrm{f}||_{\mathrm{L}_{\mathrm{P}}(Q)}T+||\sigma||_{\mathrm{L}_{\mathrm{p}}(Q)}T)$.

155
Using growth conditions (FE-2) and Lemma 4.5 we get$||F/\epsilon||_{\mathrm{L}_{pn}(}Q_{T})\leq\Lambda+\Lambda||\theta||_{L}2/2p_{n}(Q_{T})$.

$||\nabla \mathrm{W}||\mathrm{L}_{p}n(Q_{T})\leq\Lambda+\Lambda||\theta||_{L}2/2p_{n}(Q_{T})$$\Rightarrow$(4.12)

Consequently, by Lemma 4.1,.

(4.13)$||\nabla \mathrm{u}||_{\mathrm{W}(}p_{n}2,1Q_{T})\leq\Lambda+\Lambda||\theta||2L_{2}^{/}p(n_{T}Q)$

With these estimates we are able to prove the temperature bounds. To this end we
multiply the temperature equation by and integrate overusing boundary conditions.
Due to growth conditions and Gronwall’s inequality we get$Q_{t}$$\theta$

Lemma 4.6 There exists a constant , such that$\Lambda$.

$||\theta||_{L_{\infty}}(0,\tau;L2(\Omega))+||\nabla\theta||_{\mathrm{L}_{2}(Q_{T})}\leq\Lambda$

By this lemma it follows from (4.13) by imbedding theorems that$||\epsilon||_{\mathrm{W}_{pn}^{2,1}(Q)}T\leq\Lambda$$\Rightarrow$$|\epsilon|\leq\Lambda$

in$Q_{T}$,

for$||\nabla\epsilon||_{\mathrm{L}(Q)}pT\leq\Lambda$$p\geq p_{n}$.

(4.14)

Continuity of , together with (FE-1), (FE-2) implies$\epsilon$in$c_{v}\leq c(\epsilon, \theta)\leq c_{\max}(\Lambda)$.

$Q_{T}$(4.15)

We apply now the classical parabolic theory to BVP for . Estimating the right-handnorm we conclude
side in$\theta$$L_{p_{n}/2}(Q_{\tau})$$||\theta||_{W_{pn}^{2,1}(}/2Q_{T})\leq\Lambda$$||\nabla\theta||_{\mathrm{L}}pn(Q_{T}),$$\Rightarrow$for$||\theta||_{L(Q)}pT\leq\Lambda$$p \geq\frac{1}{2}p_{n}$.

(4.16)

Next we perform an iterative improvement of a priori bounds. Going back to the decomposed system we conclude the following:$||\nabla\cdot F/\epsilon||_{\mathrm{L}_{pn}}(Q_{T})\leq\Lambda$$\Rightarrow \mathrm{w}\in \mathrm{w}_{p_{n}}^{2,1}(Q_{\tau)}$$\mathrm{u}\in \mathrm{w}_{p_{n}}^{4,2}(Q_{\tau)}$$\Rightarrow$.

(4.17)

Hence, by the imbedding,$||\epsilon_{t}||\mathrm{L}(\mathrm{p}QT)\leq\Lambda$

for$p\geq p_{n}$.

(4.18)

Cosidering again the temperature equation gives the following:$||\theta||_{W^{2,1}(Q\tau)}p\leq\Lambda$

for$p\geq p_{n}$$\Rightarrow$$|\theta|\leq\Lambda$

in$Q_{T},$$||\nabla\theta||_{\mathrm{L}_{p}(Q_{T})}\leq\Lambda$

In the last iteration, using the above bounds, we get$||\nabla\cdot F/\epsilon||_{\mathrm{L}_{p}(Q\tau)}\leq\Lambda$

for$p\geq p_{n}$

,

hence
$||\mathrm{w}||_{\mathrm{w}_{p}^{2,1}}(Q\tau)\leq\Lambda\Rightarrow$$||\mathrm{u}||_{\mathrm{W}_{p}}4,2(Q\tau)\leq\Lambda$

for$p\geq p_{n}$This completes the derivation of a priori bounds for a fixed point.

.

for$p\geq p_{n}$.

156
Step 7. Complete continuity. Let$(\overline{\mathrm{u}}^{n},\overline{\theta}^{n})$$(\overline{\mathrm{u}}^{n},\overline{\theta}^{n})arrow(\overline{\mathrm{u}},\overline{\theta})$

We shall show that for the images ofbe a bounded sequence in $V(p)$ , such thatweakly in$V(p)$

for$narrow\infty$.

$T_{\lambda}$(4.19)$(\mathrm{u}^{n}, \theta^{n})=T_{\lambda(\overline{\mathrm{u}}^{n}},\overline{\theta}^{n})$it holds

for$narrow\infty$$\mathrm{u}^{n}arrow \mathrm{u}$strongly in$\mathrm{W}_{p}^{4,2}(Q_{\tau)}$$\theta^{n}arrow\theta$

strongly in$W_{p}^{2,1}(Q_{T})$(4.20),(4.21), where$(\mathrm{u}, \theta)=T_{\lambda}(\overline{\mathrm{u}},\overline{\theta})$(4.22),

maps the bounded subsets into precompact subsets in $V(p)$ . The argumentsthat isare based on the Aubin compactness theorem, which implies strong convergences$T_{\lambda}$$\overline{\mathrm{u}}^{n}arrow\overline{\mathrm{u}}$in$L_{p}(0, T, \mathrm{w}^{3}(p\Omega))$,$\overline{\theta}^{n}arrow\overline{\theta}$in$L_{p}(0, T, W_{p}^{1}(\Omega))$,and therefore$\nabla\cdot F/\epsilon(\overline{\epsilon}^{n},\overline{\theta}^{n})arrow\nabla\cdot F/\epsilon(\overline{\epsilon},\overline{\theta})$where$\overline{\epsilon}^{n}=\epsilon(\overline{\mathrm{u}}^{n})$,$\overline{\epsilon}=\epsilon(\overline{\mathrm{u}})$strongly in$\mathrm{L}_{p}(Q_{T})$,. Consequently, an application of Lemma 4.1 implies that$\mathrm{w}^{n}arrow \mathrm{w}$strongly in$\mathrm{W}_{p}^{2,1}(Q_{\tau)}$.

Thereby, the convergence (4.20) results.
To prove the convergence (4.21) we consider the BVP for the difference
By exploiting the additional regularity of datain (BV-p), which implies that
norm, and, we show that the right-hand side converges to in.
strongly in$\eta^{n}=\theta^{n}-\theta$$g,$$\theta_{0}$$0$$W_{p_{1}}^{2,1}(Q_{\tau})$.

$\theta^{n}\in$$L_{p}(Q_{T})$$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{f}_{0}\mathrm{r}_{\square }\mathrm{e}$$\eta^{n}arrow 0$5$W_{p}^{2,1}(Q_{T})$Uniqueness theorem–outline of the proof

The proof consists in the direct comparison of two solutions by means of energy estimates
and the application of Gronwall’s inequality. Letandbe the solutions
corresponding to the same data. To simplify notation we set for $i=1,2$$(\mathrm{u}^{1}, \theta^{1})$$\mathrm{v}=\mathrm{u}^{2}-\mathrm{u}^{1}$$c^{i}=c(\epsilon^{i}, \theta^{i})$$\eta=\theta^{2}-\theta^{1}$$\gamma^{i}=\frac{1}{c^{i}}$$\epsilon^{i}=\epsilon(\mathrm{u}^{i})$$F^{i}=F/\epsilon/\epsilon(\epsilon^{i}, \theta^{i})$$(\mathrm{u}^{2}, \theta^{2})$$\epsilon_{t}^{i}=\epsilon(\mathrm{u}_{t}^{i})$,$F^{i}=/\theta\epsilon F\theta(/\epsilon\epsilon^{i}, \theta^{i})$.

157
The difference$(\mathrm{v}, \eta)\in V(p)$satisfies the following BVP:$\mathrm{v}_{tt}-U\mathrm{Q}_{\mathrm{V}}t+\frac{\kappa}{4}\mathrm{Q}\mathrm{Q}\mathrm{v}=\nabla\cdot(F_{/}2-F1)\epsilon/\epsilon$

(5.1)

’

:

:

$\eta t-k\gamma^{1}\triangle\eta=\gamma\theta^{2}2F2-\theta^{1}$$\epsilon^{2}\gamma^{1}F\epsilon/\theta t/^{1}\theta\epsilon$$+\nu\gamma^{2}(\mathrm{A}\epsilon^{2})t$:$\epsilon_{t}^{1}$$\epsilon_{t}^{2}-\nu\gamma^{1}(\mathrm{A}\epsilon^{1})t$:$\epsilon_{t}^{1}$$+(\gamma^{2}-\gamma)1g$$+k(\gamma^{2}-\gamma^{1})\triangle\theta^{2}$

in$=:R_{1}+R_{2}+R_{3}+R_{4}$$\mathrm{v}(\mathrm{O}, X)=0,$$\mathrm{v}_{t}(0, X)=0,$$\eta(0, X)=0$

in$\Omega$$\mathrm{v}=\mathrm{Q}_{\mathrm{V}}=0,$$\nabla\eta\cdot \mathrm{n}=0$on$S_{T}$$Q_{T}$

(5.2)
(5.3)
(5.4)

,

,

.

In the first step we obtain energy estimates for the mechanical part. To this end weand integrate over . Integration by parts, and the use of initial
multiply (5.1) byand boundary conditions yield for almost any $t\in(0, T)$ the identity$Q_{t}$$\mathrm{v}_{t}$$\int_{\Omega}$($\frac{1}{2}|\mathrm{v}_{t}|^{2}+\frac{\kappa}{8}|$Qv$=- \int_{Q_{t}}(F_{/}^{2}-F^{1})\epsilon/\epsilon$$|^{2}$:)$dx+ \nu\int_{Q_{t}}(\mathrm{A}\epsilon(\mathrm{v}_{t}))$$\epsilon(\mathrm{V}_{t})dxdt’$:$\epsilon(\mathrm{v}_{t})dxdt^{l}$

(5.5)

.

Besides, we have the following identity,$\frac{1}{2}\int_{\Omega}|\epsilon(_{\mathrm{V}})|^{2}d_{X}=\frac{1}{2}\int_{Q_{t}}\frac{d}{dt}|\epsilon(\mathrm{V})|^{2}dXdt’=\int_{Q_{t}}\epsilon(\mathrm{v}):\epsilon(\mathrm{V}t)dxdt’$

.

(5.6)

Combining (5.5) and (5.6) and using the estimate

(5.7)

,

$|F^{2}-/\epsilon F/^{1}\epsilon|\leq\Lambda(|\epsilon(\mathrm{V})|+|\eta|)$which follows from the regularity assumption (FE-5) and the uniform bounds on, by Young’s and Gronwall’s inequalities, we arrive at$\epsilon^{i},$$\theta^{i}$in$Q_{T}$$||\mathrm{v}_{\iota}||L_{\infty}(0,T;\mathrm{L}2(\Omega))+||\epsilon(\mathrm{V})||L\infty(0,T;\mathrm{L}2(\Omega))+||\mathrm{Q}\mathrm{V}||_{L_{\infty}(}0,T;\mathrm{L}2(\Omega))+||\epsilon(\mathrm{v}t)||_{\mathrm{L}_{2(Q)}}T\leq\Lambda||\eta||_{L(Q)}2T^{\cdot}$(5.8)
Hence, due to Lemma 4.4,$||\mathrm{v}||_{L_{\infty}}(0,T;\mathrm{w}2(2\Omega))\leq\Lambda||\eta||L2(QT)$

.

(5.9)

The energy estimates for the thermal part are obtained from multiplying equation (5.2). In view of the boundby and integrating over$Q_{t}$$\eta$$0< \frac{k}{c_{\max}}\leq k\gamma^{1}$,

it follows that$\frac{1}{2}\int_{\Omega}\eta^{2}d_{X}+\frac{k}{c_{\max}}\int_{Qt}|\mathrm{v}_{\eta}|^{2}dXdt^{;}\leq-\int_{Q_{t}}k\eta\nabla\eta\cdot\nabla\gamma^{1}dxdt’+\sum_{i=1}^{4}\int_{Qt}\hslash\eta dxdt’$. (5.10)

158
Further procedure consists in majorizing the right-hand side of (5.10) and the subsequent
application of Gronwall’s inequality. To this end we use the uniform bounds on,, which imply uniform bounds on andand, as well as the estimatesin$\epsilon^{i},$$\nabla\epsilon^{i},$$\epsilon_{t}^{i}$$c^{i}$$\nabla\theta$$|R_{1}|$,$|R_{2}|$,$\nabla\gamma^{i}$$\theta^{i}$$Q_{T}$$|R_{3}|\leq\Lambda(|\eta|+|\epsilon(\mathrm{V})|+|\epsilon(\mathrm{V}_{t})|)$(5.11).

All integrals in (5.10), except the last one, are estimated directly by applying Young’s
inequality and (5.8). The -term is first integrated by parts, and majorized by using
the bounds$R_{4}$$|\gamma^{2}-\gamma^{1}|\leq\Lambda(|\eta|+|\epsilon(\mathrm{V})|)$(5.12)(5.13),$|\nabla(\gamma^{2}-\gamma^{1})|\leq\Lambda(|\eta|+|\nabla\eta|+|\epsilon(\mathrm{v})|+|\nabla\epsilon(\mathrm{v})|)$,and estimate (5.8). In consequence we arrive at$\int_{\Omega}\eta^{2}(t, \mathrm{x})dX+\frac{k}{2c_{\max}}\int_{Q_{t}}|\nabla\eta|dXdt’\leq\Lambda\int_{Q_{t}}\eta^{2}d_{X}dt’$.

(5.14)

Hence, via Gronwall’s inequality, $\eta=0$ in. Simultaneously, by inequality (5.9),
in. This completes the proof of uniqueness.
$Q_{T}$$\mathrm{v}=0\square$$Q_{T}$

6

Stability theorem–outline of the proof

The proof uses similar arguments as the existence theorem, utilizing:$\bullet$$\bullet$$\bullet$$\bullet$parabolic decomposition;

Solonnikov theory of parabolic systems, Lemma 4.1;
ellipticity property of operator$\mathrm{Q}$, Lemma 4.4;

regularity property of parabolic systems, Lemma 4.5.

Here we present the main steps of the proof, the details are given in [20].
The starting point are the energy estimates for, whereare solutions to the problem (P) corresponding to. We use the same simplifying
satisfy BVP $(5.1)-(5.4)$ ,notation as in the uniqueness proof. Functions
where in addition the right-hand side of (5.1) contains
and -term in (5.2) has
the form.
$\mathrm{v}=\mathrm{u}^{2}-\mathrm{u}^{1},$$\eta=\theta^{2}-\theta^{1}$$(\mathrm{u}^{i}, \theta^{i})$$(\mathrm{b}^{i}, g^{i})$$(\mathrm{v}, \eta)\in V(p)$$\mathrm{b}^{2}-\mathrm{b}^{1}$$R_{3}$$R_{3}=\gamma^{2}g^{2}-\gamma g11$

By repeating the arguments of the uniqueness proof, we get the following energy inequality:$||\mathrm{v}_{t}||_{L_{\infty}(0,T\mathrm{L}_{2(\Omega}};))+||\mathrm{v}||_{L_{\infty}(0},T;\mathrm{w}_{2}2(\Omega))+||\eta||_{L_{\infty}(0},T;L2(\Omega))+||\epsilon(\mathrm{V}_{t})||_{\mathrm{L}_{2}(Q\tau)}+||\nabla\eta||_{\mathrm{L}_{2}(Q\tau)}\leq$$\leq\Lambda(||\mathrm{b}^{2}-\mathrm{b}^{1}||_{\mathrm{L}_{2}(Q\tau)}+||g^{2}-g^{1}||_{L_{2}(Q_{T})})$

,

(6.1)

which leads to more refined estimates. By virtue of imbedding and parabolic estimates
it follows from (6.1) that$||\epsilon(\mathrm{v})||_{L_{\infty}(}0,T;\mathrm{L}_{q_{n}}(\Omega))\leq\Lambda(||\mathrm{b}^{2}-\mathrm{b}^{1}||_{\mathrm{L}(Q_{T})}2+||g^{2}-g^{1}||_{L(}2Q\tau))$

,

(6.2)

159

$||\eta||L_{2p}/n(nQ_{T})\leq\Lambda(||\mathrm{b}^{2}-\mathrm{b}^{1}||_{\mathrm{L}_{2}(Q)}T+||g^{2}-g^{1}||_{L_{2}(Q_{T})})$

Let$(\mathrm{w}^{1}, \mathrm{u}^{1}),$$(\mathrm{w}^{1}, \mathrm{u}^{1})$

.

(6.3)

denote the corresponding solutions of the decomposed problems$(4.2),(4.3)$ and$\mathrm{z}=\mathrm{w}^{2}-\mathrm{w}1$

The functions$(\mathrm{z}, \mathrm{v})$.

satisfy the following BVP’s:$\mathrm{z}_{t}-\beta \mathrm{Q}\mathrm{z}=\nabla\cdot(F^{2}-/\epsilon F^{1})/\epsilon+\mathrm{b}^{2}-\mathrm{b}^{1}$$\mathrm{z}(\mathrm{O}, X)=0$$\mathrm{z}=0$

inon$\Omega$in$Q_{T}$

,

(6.4)

,

$S_{T}$,

and$\mathrm{v}_{t}-\alpha \mathrm{Q}\mathrm{V}=\mathrm{z}$$\mathrm{v}(\mathrm{O}, X)=0$$\mathrm{v}=0$

in
in

on

$Q_{T}$$\Omega$

,

(6.5)

,

$S_{T}$.

By using Lemma 4.5 and estimate (5.7) we get(6.6)$||\nabla \mathrm{z}||_{\mathrm{L}_{p}(Q_{T})}\leq\Lambda(||\epsilon(\mathrm{v})||_{\mathrm{L}_{\mathrm{p}}(Q\tau)}+||\eta||_{L(Q_{T})}\mathrm{P}+||\mathrm{b}^{2}-\mathrm{b}^{1}||_{\mathrm{L}_{p}(Q_{T})})$

for $1<p<\infty$ . Consequently, the application of (6.2) and (6.3) yields
for$||\nabla \mathrm{z}||_{\mathrm{L}_{p}(Q_{T})}\leq\Lambda(||\mathrm{b}^{2}-\mathrm{b}^{1}||_{\mathrm{L}_{p}(Q_{T})}+||g^{2}-g^{1}||_{L(}2Q\tau))$$p \leq\frac{2p_{n}}{n}$

.

(6.7)

We note that in case $2\leq n\leq 4$ ,$2 \leq\frac{p_{n}}{2}\leq\frac{2p_{n}}{n}<q_{n}$.

By virtue of Lemma 4.1, (6.7) implies$||\nabla \mathrm{v}||_{\mathrm{W}_{\mathrm{p}}}2,1(Q_{T})\leq\Lambda(||\mathrm{b}^{2}-\mathrm{b}^{1}||_{\mathrm{L}(Q_{T})}\mathrm{p}+||g^{2}-g^{1}||_{L(}2Q\tau))$for$p \leq\frac{2p_{n}}{n}$

.

(6.8)

Hence, in particular (provided $n\leq 4$ ),$||\epsilon(\mathrm{v})||_{\mathrm{w}_{p}(}2,1n/2Q_{T})\leq\Lambda(||\mathrm{b}^{2}-\mathrm{b}^{1}||_{\mathrm{L}}p_{n}/2(Q\tau)+||g^{2}-g^{1}||_{L(}2Q_{T}))$

.

(6.9)

The application of the imbedding theorem implies that$||\nabla\epsilon(\mathrm{V})||_{\mathrm{L}_{pn}(Q_{T})}$,$||\epsilon(\mathrm{v})||_{\mathrm{L}_{p(Q)}}T\leq\Lambda(||\mathrm{b}^{2}-\mathrm{b}^{1}||_{\mathrm{L}_{pn/2}}(Q_{\tau})+||g^{2}-g^{1}||_{L(Q\tau)}2)$(6.10). The above estimates allow us to obtain further bounds on by applying
foragain the classical parabolic theory. To this end let us rewrite -equation (5.2) in theform$p \geq\frac{p_{n}}{2}$$\eta$$\eta$$c^{1} \eta-k\triangle\eta=\sum_{i=1}^{4}R_{i}^{*}$

where$R_{i}^{*}=c^{1}R_{i}$.

(6.11)

160
We start with limiting the right-hand side in$L_{2}(Q_{T})$-norm by.

$\Lambda(||\mathrm{b}^{2}-\mathrm{b}^{1}||_{\mathrm{L}(}2Q\tau)+||g2-g1||_{L(}2Q\tau))$are estimated directly by using bounds $(5.11),(6.3),(6.8)$ . The
The terms
term is estimated by applying H\"older inequality, and using bounds $(5.12),(6.2),(6.3)$ .
Since the coefficient in (6.11) is positive and bounded, the parabolic theory gives(6.12).
$R_{1}^{*},$$R_{2}*,$$R_{4}^{*}-$$R_{3}^{*}$$c^{1}$$||\eta||_{W_{2}}2,1(QT)\leq\Lambda(||\mathrm{b}^{2}-\mathrm{b}^{1}||_{\mathrm{L}_{2}(Q)}T+||g^{2}-g^{1}||_{L_{2}(}Q\tau))$

Hence, by imbedding theorem,$||\eta||_{L(Q)}pT\leq\Lambda(||\mathrm{b}^{2}-\mathrm{b}^{1}||_{\mathrm{L}(Q_{T}}2)+||g^{2}-g^{1}||_{L(Q_{T})}2)$for$2 \leq p\leq\frac{q_{n}p_{n}}{n}$.

(6.13)

We note that in case $n>2$ ,$\frac{q_{n}p_{n}}{n}=q_{n}\frac{n+2}{n}>q_{n}$.

With estimates $(6.10),(6.13)$ we can bound the right-hand side of -equation (6.11) in. To this purpose we use-norm byarriveatsimilar arguments as above. Consequently, we$\eta$$\Lambda(||\mathrm{b}^{2}-\mathrm{b}^{1}||_{\mathrm{L}_{pn/2}(Q)}\tau+||g^{2}-g^{1}||_{L_{pn/\tau)}}2(Q)$$L_{p_{n}/2}(Q_{\tau})$$||\eta||_{W_{pn}^{2,1}(}/2Q\tau)\leq\Lambda(||\mathrm{b}^{2}-\mathrm{b}^{1}||_{\mathrm{L}_{\mathrm{p}}}n/2(Q\tau)+||g^{2}-g^{1}||_{L_{p/T}}n2(Q))$(6.14),

and by the imbedding,$||\nabla\eta||_{\mathrm{L}(}\mathrm{p}nQ_{T})$,$||\eta||_{L(Q)}pT\leq\Lambda(||\mathrm{b}^{2}-\mathrm{b}^{1}||_{\mathrm{L}}pn/2(Q\tau)+||g^{2}-g^{1}||_{L}\mathrm{p}_{n}/2(QT))$(6.15)

This concludes the first round of estimation. In order to obtain furtherforimprovements, we return again to the system (6.4), (6.5). Using bound(6.16),$p\geq Ln2^{\cdot}$$|\nabla\cdot(F_{/\epsilon/^{1}\epsilon}^{2}-F)|\leq\Lambda(|\epsilon(\mathrm{V})|+|\eta|+|\nabla\epsilon(\mathrm{v})|+|\nabla\eta|)$which holds owing to the continuity of$(6.10),(6.15)$ , Lemma 4.5 gives$\epsilon^{i},$$\theta^{i},$$\nabla\epsilon^{i}$and$\nabla\theta^{i}$, together with estimates.

$||\mathrm{z}||_{\mathrm{W}_{pn}(}2,1Q_{T})’||\mathrm{v}||_{\mathrm{W}_{pn}(}4,2Q_{T})\leq\Lambda(||\mathrm{b}2-\mathrm{b}^{1}||_{\mathrm{L}}pn(QT)+||g^{2}-g^{1}||_{L}/2(QT))pn$(6.17)

Hence, due to imbedding,$||\nabla\epsilon(\mathrm{v})||_{\mathrm{L}_{p(Q)}}T$’$||\epsilon(\mathrm{v}_{t})||\mathrm{L}p(QT)\leq\Lambda(||\mathrm{b}^{2}-\mathrm{b}^{1}||_{\mathrm{L}_{pn}(}Q_{T})+||g^{2}-g^{1}||_{L_{\mathrm{p}}(}n/2QT))$(6.18)

for $p\geq p_{n}$ . Now we can return again to -equation and bound its right-hand side in. In the analysis
by-norm for anyof the -term by H\"older inequality we take advantage of the fact, thatfor $p_{1}=p+\delta$ , $\delta>0$ .
Similarly as above, the parabolic theory gives$\eta$$p\geq p_{n}$$L_{p}(Q_{T})$$\Lambda(||\mathrm{b}^{2}-\mathrm{b}^{1}||_{\mathrm{L}(Q)}p_{n}/2\tau+||g^{2}-g^{1}||_{L(}pQ\tau))$$\theta^{i}\in W_{p_{1}}^{2,1}(Q_{\tau})$$R_{4}^{*}$$||\eta||W_{p}^{2,1}(Q\tau)\leq\Lambda(||\mathrm{b}^{2}-\mathrm{b}^{1}||_{\mathrm{L}}pn(Q\tau)+||g^{2}-g^{1}||_{L(Q_{T})}p)$for$p\geq p_{n}$.

(6.19)

for$p\geq p_{n}$.

(6.20)

By the imbedding,$||\nabla\eta||_{\mathrm{L}(Q)}pT\leq\Lambda(||\mathrm{b}^{2}-\mathrm{b}^{1}||_{\mathrm{L}}pn(QT)+||g^{2}-g^{1}||_{L(Q\tau)}p)$

Thanks to (6.10), (6.15), (6.18), (6.20) we can estimate the right-hand side of system. The subsequent application of Solonnikov theory allows-norm for(6.4) into obtain the final estimates(6.21),$\mathrm{L}_{p}(Q_{T})$$p\geq p_{n}$$||\mathrm{z}||\mathrm{w}_{p}^{2,1}(Q_{T})’||\mathrm{v}||\mathrm{w}_{\mathrm{p}}^{4}’(2QT)\leq\Lambda(||\mathrm{b}^{2}-\mathrm{b}^{1}||_{\mathrm{L}(Q)}pT+||g^{2}-g|1|_{L_{p}(Q_{T})})$

This completes the proof.

$\square$161

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