# Existence of large-data global weak solutions
to a model of a strain-limiting viscoelastic body

###### Abstract.

We prove the existence of a unique large-data global-in-time weak solution to
a class of models of the form $\boldsymbol{u}_{tt}=\mbox{div
}\mathbb{T}+\boldsymbol{f}$ for viscoelastic bodies exhibiting strain-limiting
behaviour, where the constitutive equation, relating the linearised strain
tensor $\boldsymbol{\varepsilon}(\boldsymbol{u})$ to the Cauchy stress tensor
$\mathbb{T}$, is assumed to be of the form
$\boldsymbol{\varepsilon}(\boldsymbol{u}_{t})+\alpha\boldsymbol{\varepsilon}(\boldsymbol{u})=F(\mathbb{T})$,
where we define $F(\mathbb{T})=(1+|\mathbb{T}|^{a})^{-\frac{1}{a}}\mathbb{T}$,
for constant parameters $\alpha\in(0,\infty)$ and $a\in(0,\infty)$, in any
number $d$ of space dimensions, with periodic boundary conditions. The Cauchy
stress $\mathbb{T}$ is shown to belong to $L^{1}(Q)^{d\times d}$ over the
space-time domain $Q$. In particular, in three space dimensions, if
$a\in(0,\frac{2}{7})$, then in fact $\mathbb{T}\in L^{1+\delta}(Q)^{d\times
d}$ for a $\delta>0$, the value of which depends only on $a$.

###### Key words and phrases:

Keywords: nonlinear viscoelasticity, strain-limiting theory, evolutionary
problem, global existence, weak solution, regularity

###### 1991 Mathematics Subject Classification:

AMS 2020 Subject Classification: 35M13, 35K99, 74D10, 74H20

MIROSLAV BULÍČEK1, VICTORIA PATEL2,111Corresponding author.

, YASEMIN ŞENGÜL3, AND ENDRE SÜLI2

1Faculty of Mathematics and Physics, Charles University Prague, Sokolovská 83,
186 75 Prague 8, Czech Republic.

2Mathematical Institute, University of Oxford, Andrew Wiles Building,
Woodstock Road, Oxford OX2 6GG, UK.

3Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla 34956,
Istanbul, Turkey.

##  1\. Introduction

The development of a thermodynamically consistent implicit constitutive theory
for continuum mechanics was initiated by Rajagopal in [16]. While, as was
noted in [16], implicit constitutive theories within a purely mechanical
format had been studied earlier, “these early studies do not recognize the
important role thermodynamics has to play in the response of materials, nor do
they provide a systematic framework that allows one to incorporate issues
concerning constraints and material symmetry.” In contrast, the theoretical
framework proposed in [16] is a systematic and thermodynamically consistent
way of justifying implicit and nonlinear relationships between stress, strain
and their derivatives for elastic and viscoelastic materials. In classical
models of such bodies, the strain is given as a function of the stress, which
leads to an explicit, usually linear, relationship between the Cauchy stress
tensor $\mathbb{T}$ and the linearised strain tensor
$\boldsymbol{\varepsilon}$ (and the derivative $\boldsymbol{\varepsilon}_{t}$
in the viscoelastic case) under the small displacement gradient assumption.
However, it is well-known that linear models are insufficient to describe all
physical phenomena in the small strain range; see, for example, [24], [13] and
[8], to name just a few, thus justifying the need for alternative models. One
of the key contributions in [19] was to use linearisations of nonlinear
constitutive models to justify the presence of small strain in nonlinear
models.

We are interested in a particular subclass of these implicit constitutive
models, which represent a generalisation of the classical Kelvin–Voigt model
to strain-limiting viscoelastic solids. Mathematically, the term strain-
limiting refers to the fact that the strain is a priori bounded. As we study
viscoelastic solids rather than elastic ones, a linear combination of the
linearised strain and its time derivative will be taken to be a bounded
function of the stress. In this paper, we will focus on the rigorous
mathematical analysis of the initial-boundary-value problem that results from
such models.

Strain-limiting behaviour has been exhibited in experimental procedures by
certain biological matter [7], for example. Models of strain-limiting
materials are also of importance in the study of fracture mechanics and crack
propagation. Fracture of a brittle material can occur in the small-strain
range and is caused by a large stress. However, if we use a linear
constitutive relation to model a body, the strain will behave like
$O(r^{-\frac{1}{2}})$, where $r$ is the distance to the crack tip [20]. This
contradicts the small displacement gradient assumption under which the model
is derived. Hence, a better model might ensure that the strain is a priori
bounded, as in the case of strain-limiting models. Another class of problems
in which strain-limiting models are of relevance are those that concern bodies
that undergo a small local deformation in the presence of a concentrated load.
Concentration of load causes a large stress in a small subregion of the body.
Thus, in order to ensure that the assumptions under which the model has been
derived are not violated, one should ensure that the strain remains small. We
refer to [25] and the references therein for further discussion on such
models.

###  1.1. Basic kinematics and derivation of the problem

We begin by discussing how the problem of interest, stated as system (2.1) in
the next section, can be deduced from standard balance equations in continuum
mechanics under appropriate assumptions.

Let $\Omega\subset\mathbb{R}^{3}$ be the initial configuration of a given body
and $\Omega_{t}\subset\mathbb{R}^{3}$ the current configuration of the body at
time $t\in(0,\infty)$. The initial configuration is assumed to be a stress-
free state. For a point $\mathbf{X}\in\Omega$, we denote by
$\mathbf{x}=\chi(t,\mathbf{X})\in\Omega_{t}$ the position at time $t$ of that
point. The displacement is given by
$\boldsymbol{u}(t,x):=\mathbf{x}-\mathbf{X}$. The velocity $\boldsymbol{v}$
and deformation gradient $\mathbb{F}$ are defined by

$\boldsymbol{v}:=\frac{\partial\chi}{\partial
t},\quad\mathbb{F}:=\frac{\partial\chi}{\partial\mathbf{X}}.$

The left Cauchy–Green stretch tensor $\mathbb{B}$ is defined by

$\mathbb{B}:=\mathbb{F}\mathbb{F}^{\mathrm{T}}.$

The velocity gradient $\nabla_{\mathbf{x}}\boldsymbol{v}$ is denoted by
$\mathbb{L}$ with the linearised strain $\boldsymbol{\varepsilon}$ and
symmetric part of the velocity gradient $\mathbb{D}$ given, respectively, by

$\boldsymbol{\varepsilon}=\boldsymbol{\varepsilon}(\boldsymbol{u}):=\frac{1}{2}\big{(}\nabla_{\mathbf{X}}\boldsymbol{u}+(\nabla_{\mathbf{X}}\boldsymbol{u})^{\mathrm{T}}\big{)},\quad\mathbb{D}:=\frac{1}{2}\big{(}\mathbb{L}+\mathbb{L}^{\mathrm{T}}\big{)}.$

With respect to the initial configuration, the system of balance equations for
mass, linear momentum and angular momentum can be expressed in the following
way:

$\displaystyle\rho$ $\displaystyle=\rho_{0}\mathrm{det}(\mathbb{F}),$ (1.1)
$\displaystyle\rho_{0}\frac{\partial^{2}\chi}{\partial t^{2}}$
$\displaystyle=\mathrm{div}_{\mathbf{X}}\mathbb{S}+\rho_{0}\boldsymbol{f},$
$\displaystyle\mathbb{S}\mathbb{F}^{\mathrm{T}}$
$\displaystyle=\mathbb{F}\mathbb{S}^{\mathrm{T}},$

where $\rho$ is the current density, $\rho_{0}$ is the initial density,
$\boldsymbol{f}$ is the density of body forces, and $\mathbb{S}$ is the first
Piola–Kirchhoff stress tensor. To close the system, a constitutive relation is
required. Following Rajagopal and Saccomandi in [21], we consider the
following class of implicit constitutive relations:

$\mathcal{F}(\mathbb{T},\mathbb{B},\mathbb{D})=\mathbf{0}.$ (1.2)

Taking account of representation theorems for isotropic functions, as well as
the classical Kelvin–Voigt model, we consider the following subclass of
constitutive relations:

$\tilde{\nu}\mathbb{D}+\tilde{\alpha}\mathbb{B}=\tilde{\beta}_{0}\mathbb{I}+\tilde{\beta}_{1}\mathbb{T}+\tilde{\beta}_{2}\mathbb{T}^{2},$
(1.3)

where $\tilde{\beta}_{i}=\tilde{\beta}_{i}(I_{1},I_{2},I_{3})$ for
$i\in\\{1,2,3\\}$, are the material moduli,
$I_{1}=\operatorname{tr}\mathbb{T}$,
$I_{2}=\frac{1}{2}\operatorname{tr}\mathbb{T}^{2}$, $I_{3}\
=\frac{1}{3}\operatorname{tr}\mathbb{T}^{3}$ are the principle invariants, and
$\tilde{\nu}$, $\tilde{\alpha}$ are positive constants.

Linearising (1.1) and (1.3) under the assumption that

$\max_{\mathbf{X}\in\Omega,\,t\in[0,\infty)}|\nabla_{\mathbf{X}}\boldsymbol{u}|=O(\delta),$
(1.4)

for some $\delta\ll 1$, the constitutive relation reduces to

$\nu\boldsymbol{\varepsilon}(\boldsymbol{u}_{t})+\alpha\boldsymbol{\varepsilon}(\boldsymbol{u})=\beta_{0}\mathbb{I}+\beta_{1}\mathbb{T}+\beta_{2}\mathbb{T}^{2},$
(1.5)

where $\beta_{i}=\beta_{i}(I_{1},I_{2},I_{3})$, $i\in\\{1,2,3\\}$, and $\nu$,
$\alpha$ are positive constants. We use $\boldsymbol{\varepsilon}$ to denote
the symmetric gradient operator
$\frac{1}{2}((\nabla\,\cdot\,)+(\nabla\,\cdot\,)^{\mathrm{T}})$ with
derivatives taken with respect to $\mathbf{X}$, and for
$\mathbb{A}\in\mathbb{R}^{d\times d}$ we shall denote by
$|\mathbb{A}|:=(\mathbb{A}:\mathbb{A})^{\frac{1}{2}}$ the Frobenius norm of
$\mathbb{A}$. The same notation will be used to denote the Euclidean norm of a
$d$-component vector and the absolute value of a real number; it will be clear
from the context which of these interpretations of the symbol $|\cdot|$ is
intended. We note that under the assumption (1.4), derivatives with respect to
$\mathbf{X}$ and $\mathbf{x}$ may be used interchangeably. In particular, we
have that $\boldsymbol{\varepsilon}(\boldsymbol{u}_{t})\approx\mathbb{D}$.

The balance equations (1.1) thereby reduce to

$\displaystyle\rho_{0}$
$\displaystyle=\rho(1+\operatorname{tr}\boldsymbol{\varepsilon}),$ (1.6)
$\displaystyle\rho_{0}\boldsymbol{u}_{tt}$
$\displaystyle=\mathrm{div}_{\mathbf{X}}\mathbb{T}+\rho_{0}\boldsymbol{f},$
$\displaystyle\mathbb{T}$ $\displaystyle=\mathbb{T}^{\mathrm{T}},$

up to an error of order $\delta^{2}$. For the sake of simplicity we will
assume henceforth that $\rho_{0}\equiv 1$ and $\nu=1$. If $\nu=0$, problem
(1.6), (1.5) models a strain-limiting elastic body. The form of (1.5) that we
are particularly interested in is

$\boldsymbol{\varepsilon}(\boldsymbol{u}_{t})+\alpha\boldsymbol{\varepsilon}(\boldsymbol{u})=\frac{\mathbb{T}}{(1+|\mathbb{T}|^{a})^{\frac{1}{a}}}=:F(\mathbb{T}),$
(1.7)

for a positive parameter $a\in(0,\infty)$, corresponding to
$\beta_{0}=\beta_{2}=0$, and $\beta_{1}=(1+|\mathbb{T}|^{a})^{-\frac{1}{a}}$.
As $a$ is treated as a fixed parameter we do not explicitly indicate the
dependence of $F$ on $a$. Furthermore, we note that $F$ is a bounded function
on $\mathbb{R}^{d\times d}$, encapsulating the strain-limiting property.
Indeed, it follows that the linearised strain
$\boldsymbol{\varepsilon}(\boldsymbol{u})$ is bounded by the following
reasoning. Multiplying both sides of (1.7) by $\mathrm{e}^{\alpha t}$ and
integrating with respect to the time variable, we deduce that

$\displaystyle|\boldsymbol{\varepsilon}(\boldsymbol{u}(t))|=\bigg{|}\mathrm{e}^{-\alpha
t}\boldsymbol{\varepsilon}(\boldsymbol{u}_{0})+\int_{0}^{t}\mathrm{e}^{\alpha(s-t)}F(\mathbb{T}(s))\,\mathrm{d}s\bigg{|}\leq\mathrm{e}^{-\alpha
t}\|\boldsymbol{\varepsilon}(\boldsymbol{u}_{0})\|_{\infty}+\frac{1}{\alpha}(1-\mathrm{e}^{-\alpha
t}),\qquad t>0,$

where $\boldsymbol{u}_{0}=\boldsymbol{u}(0)$. In particular, we see that if
$\|\boldsymbol{\varepsilon}(\boldsymbol{u}_{0})\|_{\infty}\leq\alpha^{-1}$,
then $\|\boldsymbol{\varepsilon}(\boldsymbol{u}(t))\|_{\infty}\leq\alpha^{-1}$
for all $t\in(0,\infty)$. We note that this particular choice of $F$ has
already been studied in the one-dimensional case, for example, in [9] and
[11].

For the sake of completeness we shall discuss the thermodynamic motivation for
this choice of the function $F$. In particular, we will show that the total
energy associated with the model is decreasing in time, provided that the body
force $\boldsymbol{f}\equiv\mathbf{0}$. In order to simplify the presentation
of the following analysis, we divide (1.7) through by $\alpha$. Thus we are
motivated to define the function $f_{\alpha}:\mathbb{R}^{d\times
d}\rightarrow[0,\infty)$ by

$\displaystyle
f_{\alpha}(\mathbb{T}):=\frac{1}{\alpha}\int_{0}^{|\mathbb{T}|}\frac{t}{(1+t^{a})^{\frac{1}{a}}}\,\mathrm{d}t,$

for $\mathbb{T}\in\mathbb{R}^{d\times d}$. In particular, we have that
$\frac{\partial
f_{\alpha}}{\partial\mathbb{T}}(\mathbb{T})=\frac{1}{\alpha}F(\mathbb{T})=:F_{\alpha}(\mathbb{T})$.
By direct calculation, $\nabla^{2}_{\mathbb{T}}f_{\alpha}$ is positive
definite and continuous. Thus we see that $f_{\alpha}$ is a convex $C^{2}$
function on $\mathbb{R}^{d\times d}$. We define the convex conjugate
$f_{\alpha}^{*}$ on $\mathbb{R}^{d\times d}$ by

$f^{*}_{\alpha}(\mathbb{S})=\sup_{\mathbb{T}\in\mathbb{R}^{d\times
d}}\big{(}\,\mathbb{S}:\mathbb{T}-f_{\alpha}(\mathbb{T})\,\big{)}.$

By the Fenchel–Young inequality and the differentiability properties of
$f_{\alpha}$, we have that the supremum is either infinity or it is attained
when $\mathbb{S}=F_{\alpha}(\mathbb{T})$, and therefore

$f_{\alpha}(\mathbb{T})+f_{\alpha}^{*}(F_{\alpha}(\mathbb{T}))=F_{\alpha}(\mathbb{T}):\mathbb{T},\qquad\mathbb{T}\in\mathbb{R}^{d\times
d}.$

Noting that $F_{\alpha}^{-1}$ is well-defined on the set of matrices with
Frobenius norm strictly less than $\alpha^{-1}$, we also have that

$f_{\alpha}(F^{-1}_{\alpha}(\boldsymbol{\varepsilon}))+f^{*}_{\alpha}(\boldsymbol{\varepsilon})=\boldsymbol{\varepsilon}:F_{\alpha}^{-1}(\boldsymbol{\varepsilon}),$

for every $\boldsymbol{\varepsilon}\in\mathbb{R}^{d\times d}$,
$|\boldsymbol{\varepsilon}|<\alpha^{-1}$. With this in mind, we can write

$\displaystyle\mathbb{T}:\boldsymbol{\varepsilon}(\boldsymbol{u}_{t})$
$\displaystyle=\Big{(}\mathbb{T}-\frac{\partial
f^{*}_{\alpha}}{\partial\boldsymbol{\varepsilon}}(\boldsymbol{\varepsilon}(\boldsymbol{u}))\Big{)}\boldsymbol{\varepsilon}(\boldsymbol{u}_{t})+\frac{\partial
f^{*}_{\alpha}}{\partial\boldsymbol{\varepsilon}}(\boldsymbol{\varepsilon}(\boldsymbol{u})):\boldsymbol{\varepsilon}(\boldsymbol{u}_{t})$
$\displaystyle=\Big{(}\mathbb{T}-F_{\alpha}^{-1}(\boldsymbol{\varepsilon}(\boldsymbol{u}))\Big{)}:\Big{(}F(\mathbb{T})-\alpha\boldsymbol{\varepsilon}(\boldsymbol{u})\Big{)}+\frac{\partial}{\partial
t}\Big{(}f_{\alpha}^{*}(\boldsymbol{\varepsilon}(\boldsymbol{u}))\Big{)}$
$\displaystyle=\alpha\Big{(}\mathbb{T}-\mathbb{T}_{0,\alpha}\Big{)}:\Big{(}F_{\alpha}(\mathbb{T})-F_{\alpha}(\mathbb{T}_{0,\alpha})\Big{)}+\frac{\partial}{\partial
t}\Big{(}f_{\alpha}^{*}(\boldsymbol{\varepsilon}(\boldsymbol{u}))\Big{)},$

where $\mathbb{T}_{0,\alpha}$ is the unique element of $\mathbb{R}^{d\times
d}$ with
$\mathbb{T}_{0,\alpha}=F_{\alpha}^{-1}(\boldsymbol{\varepsilon}(\boldsymbol{u}))$.
Using (1.6)2 with $\rho_{0}\equiv 1$ and assuming that the boundary conditions
are chosen so that when integrating by parts all boundary integrals vanish, we
get

$\displaystyle 0$
$\displaystyle=\int_{\Omega}\boldsymbol{u}_{tt}\cdot\boldsymbol{u}_{t}+\mathbb{T}:\boldsymbol{\varepsilon}(\boldsymbol{u}_{t})\,\mathrm{d}x$
$\displaystyle=\int_{\Omega}\frac{\partial}{\partial
t}\bigg{(}\frac{|\boldsymbol{u}_{t}|^{2}}{2}+f_{\alpha}^{*}(\boldsymbol{\varepsilon}(\boldsymbol{u}))\bigg{)}+\alpha(\mathbb{T}-\mathbb{T}_{0,\alpha}):(F_{\alpha}(\mathbb{T})-F_{\alpha}(\mathbb{T}_{0,\alpha}))\,\mathrm{d}x$
$\displaystyle=\int_{\Omega}\frac{\partial}{\partial
t}\bigg{(}\frac{|\boldsymbol{u}_{t}|^{2}}{2}+f_{\alpha}^{*}(\boldsymbol{\varepsilon}(\boldsymbol{u}))\bigg{)}+(\mathbb{T}-\mathbb{T}_{0,\alpha}):(F(\mathbb{T})-F(\mathbb{T}_{0,\alpha}))\,\mathrm{d}x.$

This yields

$\displaystyle\int_{\Omega}\frac{|\boldsymbol{u}_{t}(t)|^{2}}{2}+f_{\alpha}^{*}(\boldsymbol{\varepsilon}(\boldsymbol{u}(t)))\,\mathrm{d}x+\int_{0}^{t}\int_{\Omega}(\mathbb{T}-\mathbb{T}_{0,\alpha}):(F(\mathbb{T})-F(\mathbb{T}_{0,\alpha}))\,\mathrm{d}x\,\mathrm{d}s$
(1.8)
$\displaystyle\quad\quad=\int_{\Omega}\frac{|\boldsymbol{u}_{t}(0)|^{2}}{2}+f_{\alpha}^{*}(\boldsymbol{\varepsilon}(\boldsymbol{u}(0)))\,\mathrm{d}x,$

for every $t\in(0,\infty)$. The first term on the left-hand side represents
the total energy (i.e., the sum of the kinetic and the stored energy) and the
second term is a nonnegative, increasing function of $t$ by the monotonicity
of $F$. Thus the total energy is a decreasing function of $t$. Furthermore, we
note that

$\frac{\partial
f_{\alpha}^{*}}{\partial\mathbb{T}}(\mathbb{T})=F^{-1}_{\alpha}(\mathbb{T})=\frac{\alpha\mathbb{T}}{(1-\alpha^{a}|\mathbb{T}|^{a})^{\frac{1}{a}}}.$

In particular, we have $f_{\alpha}^{*}(\mathbb{T})=\infty$ if
$|\boldsymbol{\varepsilon}(\boldsymbol{u})|>\alpha^{-1}$. (See [1] and
references therein for details of the reasoning from convex analysis.) Thus we
must have $|\boldsymbol{\varepsilon}(\boldsymbol{u})|\leq\alpha^{-1}$ a.e. in
$(0,\infty)\times\Omega$, provided that the right-hand of (1.8) is finite.

###  1.2. Statement of the model problem.

We will make some mathematical simplifications in order to make the rigorous
mathematical analysis of the problem more manageable. Neglecting (1.6)1 and
assuming that $\rho_{0}\equiv 1$, we obtain the following system of equations:

$\displaystyle\boldsymbol{u}_{tt}$
$\displaystyle=\mathrm{div}_{\mathbf{X}}\mathbb{T}+\boldsymbol{f},$ (1.9)
$\displaystyle\boldsymbol{\varepsilon}(\boldsymbol{u}_{t})+\alpha\boldsymbol{\varepsilon}(\boldsymbol{u})$
$\displaystyle=F(\mathbb{T}),$ $\displaystyle\mathbb{T}$
$\displaystyle=\mathbb{T}^{\mathrm{T}},$

in $[0,\infty)\times\Omega$, with suitable initial and boundary conditions. We
note that if, more generally, $\rho_{0}\in
W^{1,\infty}(\Omega;\mathbb{R}_{\geq 0})$ is such that $\rho_{0}$ is uniformly
bounded away from $0$, then it is not much more difficult to include a
variable density.

A further simplification that we will make is to take $\Omega=(0,1)^{d}$, with
dimension $d\geq 2$, and to supplement the problem with a periodic boundary
condition. This allows us to work with Fourier basis functions in a finite-
dimensional approximation of the problem, leading to a sequence of semi-
discrete numerical approximations. Because of this simplification, we can view
the work here as a time-dependent analogue of the problem studied in [5],
dealing with viscoelastic solids rather than elastic ones. We note that
despite its geometrical simplicity, one can still use the framework presented
here to study the effects of concentrated loads that are active in the
neighbourhood of the centre point of the periodic cell, assuming that the
side-length of the cell is large enough so that the effects of concentrations
are not effective in the neighbourhood of the boundary of the cell. Trivially,
the results of the paper extend to any axiparallel parallelepiped
$\Omega\subset\mathbb{R}^{d}$, $d\geq 2$, independent of the edge-lengths of
$\Omega$.

The form of $F$ studied in this paper has already been considered in the
purely elastic case, for example, in [5]. For elastic solids, it has been
shown in [5] that a unique weak solution to the static problem in the periodic
setting exists for every $a\in(0,\frac{2}{d})$ and a renormalised solution
exists for every $a\in(0,\infty)$, with coincidence of these two notions of
solution for the range $a\in(0,\frac{2}{d})$; and in the case of a homogeneous
Dirichlet boundary condition the existence of a unique weak solution was shown
for $a\in(0,\frac{1}{d})$ in [3]. When considering mixed Dirichlet–Neumann
boundary conditions, the authors of [1] were able to show existence of a weak
solution up to a penalisation on the Neumann part of the boundary. The
extension of this work to the time-dependent problem, as well as further open
problems, will be discussed at the end of the paper.

The constitutive relation proposed by Rajagopal in [18] and [19] is in fact

$F(\mathbb{T})=f_{0}(\operatorname{tr}\mathbb{T},\operatorname{tr}\mathbb{T}^{2})\,\mathbb{I}+\frac{\mathbb{T}}{\mu_{0}(1+|\mathbb{T}|^{a})^{\frac{1}{a}}}.$
(1.10)

However, we note that under suitable, physically reasonable, structural
assumptions on the real-valued-function $f_{0}$ the extension of the results
in this paper to a problem with a constitutive relation given by (1.10) is
fairly straightforward. Hence, for the sake of simplicity of the exposition,
we shall assume in what follows that $f_{0}\equiv 0$ and $\mu_{0}=1$ and focus
on the key difficulty in the analysis: dealing with the second summand on the
right-hand side of (1.10).

A similar problem to (1.9) is discussed in [14] and [15]; the authors
considered a quasi-static problem in both papers, by which we mean that the
term $\boldsymbol{u}_{tt}$ was omitted from (1.9). The function $F$ there was
given by (1.10) with $f_{0}\equiv 0$. It was shown that a solution to the
problem exists, using an elliptic regularisation technique similar to the one
employed in this paper, combined with a fixed-point argument. Appropriate
bounds were then found in order to allow the regularisation parameter to go to
$0$. A marked difference here compared to the analysis in those papers is that
the stress $\mathbb{T}$ was only shown there to belong to the space
$C([0,T];\mathcal{M}(\overline{\Omega})^{d\times d})$, where
$\mathcal{M}(\overline{\Omega})$ is the space of Radon measures on
$\overline{\Omega}$. In contrast, we shall be able to ensure that the stress
is at least in the space $L^{1}(0,T;L^{1}(\Omega)^{d\times d})$ in any number
of space dimensions $d$; and, in three space dimensions, if
$a\in(0,\frac{2}{7})$, then $\mathbb{T}\in
L^{1+\delta}(0,T;L^{1+\delta}(\Omega))$ for a $\delta>0$, the value of which
depends only on $a$. Furthermore, we are able to show that the constitutive
relation holds in a pointwise sense rather than in a variational sense, as was
the case in [14] and [15]. This then enables us to prove the validity of the
constitutive relation by making use of the fact that the stress tensor is an
integrable function.

There are some important differences between the multi-dimensional problem
(2.1) and its one-dimensional counterpart. In one space dimension the
symmetric gradient reduces to simply the first derivative in the spatial
variable. The one-dimensional time-dependent problem has been studied in [9]
in the context of travelling wave solutions. A substitution was used in order
to reduce the problem in two dependent variables to a problem in a single
dependent variable. This work was expanded upon in [11], where the existence
of a strong solution to the one-dimensional time-dependent problem was proved
on the domain $\Omega=\mathbb{R}$. The proof relied on a substitution argument
that is specific to the one-dimensional case. Although the regularity of the
solutions is much stronger in [11], the authors were only able to prove local-
in-time existence. Here, we shall prove global-in-time existence of weak
solutions in any number of space dimensions.

The paper [17], with corresponding mathematical analysis contained in [2] and
[4], deals with a different generalisation of the Kelvin–Voigt model, where
the stress is decomposed into an elastic component $\mathbb{T}_{e}$ and a
dissipative component $\mathbb{T}_{f}$ associated with a viscous fluid.
Linearising under the assumption (1.4), the constitutive relations are
$\mathbb{T}_{e}=h_{1}(\boldsymbol{\varepsilon})$ and
$\mathbb{T}_{f}=h_{2}(\boldsymbol{\varepsilon}_{t})$. However, a key point is
that the functions $h_{1}$ and $h_{2}$ are not assumed to be bounded. In
particular, the problem is not strain-limiting, and some of the key technical
difficulties that we encounter here for the strain-limiting problem (2.1) are
therefore absent.

We are now ready to proceed with our analysis of the strain-limiting problem
(2.1). The paper is structured as follows. Section 2 introduces the
mathematical problem alongside the definition of a weak solution and the
statement of the regularised problem, which forms the basis of the weak
compactness argument that is at the heart of our proof of the existence of
global-in-time large-data weak solutions. Furthermore, some useful auxiliary
results that will be used in the proof are given. In Section 3, we focus on
proving the existence of a solution to the regularised problem (2.5) by a
Galerkin method using a basis of trigonometric polynomials. We also prove
various bounds on the sequence of solutions that are independent of the
regularisation parameter. Certain bounds may only be proved under stronger
restrictions on the data. In Section 4 we take the limit in the regularisation
parameter and show that the accumulation point of the sequence of solutions is
in fact the unique weak solution of (2.1). A key difficulty is showing that
the sequence of stress tensors converges pointwise a.e. and then using this to
prove that the weak form of the PDE does indeed hold. Then, under further
restrictions on the dimension $d$ and the parameter $a$, we show that a
stronger convergence result can in fact be proved for the sequence of stress
tensors. The main difficulty is showing that the sequence of approximate
stress tensors is bounded in a reflexive Lebesgue space, from which we are
able to deduce strong convergence in $L^{1}((0,T)\times\Omega)$. Finally, in
Section 5, we discuss future work and related open problems.

##  2\. Formulation of problem and auxiliary results

Let $\Omega:=(0,1)^{d}\subset\mathbb{R}^{d}$, $d\geq 2$. For a fixed final
time $T>0$, we define the time-space cylinder $Q:=(0,T)\times\Omega$. Given a
space $\mathcal{F}$ of real-valued functions on $\mathbb{R}^{d}$, let
$\mathcal{F}_{\\#}$ denote the subspace of functions $f\in\mathcal{F}$ that
are $1$-periodic, i.e., periodic with respect to each of the $d$ co-ordinate
directions. Let $\mathcal{F}_{*}$ be the subspace of functions from
$\mathcal{F}_{\\#}$ whose integral over $\Omega$ is equal to $0$. For example,
$L^{p}_{\\#}(\Omega)$ consists of $1$-periodic functions $f$ such that
$|f|^{p}$ is integrable over $\Omega$. Let $\|\cdot\|_{p}$ denote the usual
norm on $L^{p}(\Omega)$ for $p\in[1,\infty]$. When considering the norm in
$L^{p}(\Omega_{0})$ where $\Omega_{0}\neq\Omega$, we will state it explicitly.
Furthermore, $\mathcal{F}^{d}$ denotes the space of $d$-component vector
functions such that each component is an element of $\mathcal{F}$. The space
$\mathcal{F}^{d\times d}$ is defined analogously. If $\|\cdot\|_{\mathcal{F}}$
is the usual norm on $\mathcal{F}$, the norms on $\mathcal{F}^{d}$ and
$\mathcal{F}^{d\times d}$ will be given by
$\|\cdot\|_{\mathcal{F}}:=\||\cdot|\|_{\mathcal{F}}$ where $|\cdot|$ denotes
the absolute value on $\mathbb{R}$, the Euclidean norm on $\mathbb{R}^{d}$, or
the Frobenius norm on $\mathbb{R}^{d\times d}$, as the case may be. Let
$C^{\infty}_{\\#}(\overline{\Omega})$ denote the set of smooth, real-valued
functions on $\mathbb{R}^{d}$ that are $1$-periodic and
$C^{\infty}_{*}(\overline{\Omega})$ the subspace of
$C^{\infty}_{\\#}(\overline{\Omega})$ consisting of functions that have
integral over $\Omega$ equal to $0$. For $p\in[1,\infty)$ and
$k\in\mathbb{N}$, we define the Sobolev space $W^{k,p}_{\\#}(\Omega)$ to be
the closure of $C^{\infty}_{\\#}(\overline{\Omega})$ with respect to the norm

$\|f\|_{k,p}=\|f\|_{W^{k,p}(\Omega)}:=\bigg{(}\sum_{i=0}^{k}\sum_{|\alpha|=k}\|\partial^{\alpha}f\|_{p}^{p}\bigg{)}^{\frac{1}{p}},$

where $\alpha$ is taken from the set of multi-indices in $\mathbb{N}_{0}^{d}$.
The space $W^{k,p}_{*}(\Omega)$ is defined analogously. Let
$W^{k,\infty}(\Omega)$ denote the set of functions $f\in W^{k,1}(\Omega)$ such
that $\partial^{\alpha}f\in L^{\infty}(\Omega)$ for every
$\alpha\in\mathbb{N}_{0}^{d}$ such that $|\alpha|\leq k$. For a Banach space
$X$, we let $X^{\prime}$ denote the dual space of $X$ with duality pairing
$\langle\cdot,\cdot\rangle$. Furthermore, we will use Einstein’s summation
convention throughout.

Given a vector field $\boldsymbol{f}:Q\rightarrow\mathbb{R}^{d}$, initial data
$\boldsymbol{u}_{0}$, $\boldsymbol{v}_{0}:\Omega\rightarrow\mathbb{R}^{d}$,
and model parameters $\alpha>0$, $a>0$, we seek a unique couple
$(\boldsymbol{u},\mathbb{T}):Q\rightarrow\mathbb{R}^{d}\times\mathbb{R}^{d\times
d}$ such that

$\displaystyle\boldsymbol{u}_{tt}$
$\displaystyle=\mathrm{div}(\mathbb{T})+\boldsymbol{f},$ $\displaystyle\text{
in }Q,$ (2.1)
$\displaystyle\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}+\alpha\boldsymbol{u})$
$\displaystyle=\frac{\mathbb{T}}{(1+|\mathbb{T}|^{a})^{\frac{1}{a}}}=:F(\mathbb{T}),$
$\displaystyle\text{ in }Q,$ $\displaystyle\boldsymbol{u}(0,\cdot)$
$\displaystyle=\boldsymbol{u}_{0},$ $\displaystyle\text{ in }\Omega,$
$\displaystyle\boldsymbol{u}_{t}(0,\cdot)$ $\displaystyle=\boldsymbol{v}_{0},$
$\displaystyle\text{ in }\Omega.$

We wish to prove the existence of a unique weak solution to (2.1) in the
following sense. The choice of function spaces is the natural choice according
to bounds that we derive in the proof of existence. For the sake of
simplicity, from now on we will use the superscript $\dot{}$ rather than the
subscript $t$ in order to denote differentiation with respect to the time
variable.

######  Definition 2.1.

Let $\boldsymbol{u}_{0}$, $\boldsymbol{v}_{0}\in L^{2}_{*}(\Omega)^{d}$ and
$\boldsymbol{f}\in L^{2}(0,T;L^{2}_{*}(\Omega)^{d})$. The couple
$(\boldsymbol{u},\mathbb{T})$ is a weak solution of the strain-limiting
problem (2.1) if

  * •

$\boldsymbol{u}$, $\dot{\boldsymbol{u}}\in
L^{\infty}(0,T;L^{2}_{*}(\Omega)^{d})$,

  * •

$\dot{\boldsymbol{u}}+\alpha\boldsymbol{u}\in
L^{p}(0,T;W^{1,p}_{*}(\Omega)^{d})$ for every $p\in[1,\infty)$ with
$\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}+\alpha\boldsymbol{u})\in
L^{\infty}(Q)^{d\times d}$,

  * •

$\ddot{\boldsymbol{u}}\in L^{2}(0,T;L^{2}_{*}(\Omega)^{d})$,

  * •

$\mathbb{T}\in L^{1}(0,T;L^{1}_{\\#}(\Omega)^{d\times d})$,

and, for every $\boldsymbol{v}\in W^{1,2}_{*}(\Omega)^{d}$ such that
$\boldsymbol{\varepsilon}(\boldsymbol{v})\in L^{\infty}(\Omega)^{d\times d}$,

$\int_{\Omega}\ddot{\boldsymbol{u}}(t)\cdot\boldsymbol{v}+\mathbb{T}(t):\boldsymbol{\varepsilon}(\boldsymbol{v})\,\mathrm{d}x=\int_{\Omega}\boldsymbol{f}(t)\cdot\boldsymbol{v}\,\mathrm{d}x,$
(2.2)

for a.e. $t\in(0,T)$, where

$\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}+\alpha\boldsymbol{u})=F(\mathbb{T})\quad\text{a.e.
in }Q.$ (2.3)

Furthermore, the initial conditions must hold in the following sense:

$\lim_{t\rightarrow
0+}\bigg{(}\|\boldsymbol{u}(t)-\boldsymbol{u}_{0}\|_{2}+\|\dot{\boldsymbol{u}}(t)-\boldsymbol{v}_{0}\|_{2}\bigg{)}=0.$
(2.4)

We note that $\boldsymbol{u}$, $\dot{\boldsymbol{u}}\in
C([0,T];L^{2}_{*}(\Omega)^{d})$ by the regularity assumptions of Definition
2.1. (See [12, Chapter 5], for example.) Thus (2.4) is well-defined.

The map $F$ is a continuous, bounded and injective function. We will also
require the following properties of $F$. The proof of these results can be
found in [5].

######  Lemma 2.1.

For any $y\geq 0$ and $a>0$,

$\displaystyle\min\\{1,2^{-1+\frac{1}{a}}\\}(1+y)\leq(1+y^{a})^{\frac{1}{a}}\leq\max\\{1,2^{-1+\frac{1}{a}}\\}(1+y).$

######  Lemma 2.2.

Let $a>0$. For any $\mathbb{T}$, $\mathbb{S}\in\mathbb{R}^{d\times d}$,

$\displaystyle(\mathbb{T}-\mathbb{S}):(F(\mathbb{T})-F(\mathbb{S}))\geq\max\\{1,2^{\frac{1}{a}-a}\\}\cdot\frac{|\mathbb{T}-\mathbb{S}|^{2}}{(1+|\mathbb{T}|+|\mathbb{S}|)^{1+a}}.$

From Lemma 2.2, we deduce that $F$ is a monotonic function. However, it is not
a bijection from $\mathbb{R}^{d\times d}$ onto $\mathbb{R}^{d\times d}$ since
it is bounded. In order to consider finite-dimensional approximations, we
would like $F$ to be invertible on the whole of $\mathbb{R}^{d\times d}$.
Thus, in the spirit of [3] and [1], for every $n\in\mathbb{N}$ we first
consider the following regularised problem:

$\displaystyle\ddot{\boldsymbol{u}}$
$\displaystyle=\mathrm{div}(\mathbb{T})+\boldsymbol{f},$ $\displaystyle\text{
in }Q,$ (2.5)
$\displaystyle\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}+\alpha\boldsymbol{u})$
$\displaystyle=\frac{\mathbb{T}}{(1+|\mathbb{T}|^{a})^{\frac{1}{a}}}+\frac{\mathbb{T}}{n(1+|\mathbb{T}|^{1-\frac{1}{n}})}=:F_{n}(\mathbb{T}),$
$\displaystyle\text{ in }Q,$ $\displaystyle\boldsymbol{u}(0,\cdot)$
$\displaystyle=\boldsymbol{u}_{0},$ $\displaystyle\text{ in }\Omega,$
$\displaystyle\dot{\boldsymbol{u}}(0,\cdot)$
$\displaystyle=\boldsymbol{v}_{0},$ $\displaystyle\text{ in }\Omega.$

For the sake of simplicity, for the moment at least, we shall not indicate the
dependence of $\boldsymbol{u}$ and $\mathbb{T}$ on $n$. However, we shall
explicitly indicate the dependence of these functions on $n$ in Section 4
where we let $n\rightarrow\infty$. To see that $F_{n}$ is a bijection from
$\mathbb{R}^{d\times d}$ to itself, we use the Browder–Minty theorem noting
the monotonicity result from Lemma 2.2. Thus (2.5)2 is equivalent to

$\mathbb{T}=F_{n}^{-1}(\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}+\alpha\boldsymbol{u})),\quad\text{
in }Q.$

In analogy with Definition 2.1, we define a weak solution of (2.5) as follows.

######  Definition 2.2.

For a given $n\in\mathbb{N}$ and $\boldsymbol{u}_{0}$, $\boldsymbol{v}_{0}$,
$\boldsymbol{f}$ as in Definition 2.2, the couple
$(\boldsymbol{u},\mathbb{T})$ is a weak solution of (2.5) if

  * •

$\boldsymbol{u}$, $\dot{\boldsymbol{u}}\in
L^{\infty}(0,T;L^{2}_{*}(\Omega)^{d})$,

  * •

$\dot{\boldsymbol{u}}+\alpha\boldsymbol{u}\in
L^{n+1}(0,T;W^{1,n+1}_{*}(\Omega)^{d})$ with
$\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}+\alpha\boldsymbol{u})\in
L^{n+1}(0,T;L^{n+1}_{\\#}(\Omega)^{d\times d})$,

  * •

$\ddot{\boldsymbol{u}}\in L^{2}(0,T;L^{2}_{*}(\Omega)^{d})$,

  * •

$\mathbb{T}\in L^{1+\frac{1}{n}}(0,T;L^{1+\frac{1}{n}}_{\\#}(\Omega)^{d\times
d})$,

and, for every $\boldsymbol{v}\in W^{1,n+1}_{*}(\Omega)^{d}$,

$\int_{\Omega}\ddot{\boldsymbol{u}}(t)\cdot\boldsymbol{v}+\mathbb{T}(t):\boldsymbol{\varepsilon}(\boldsymbol{v})\,\mathrm{d}x=\int_{\Omega}\boldsymbol{f}(t)\cdot\boldsymbol{v}\,\mathrm{d}x,$
(2.6)

for a.e. $t\in(0,T)$ with

$\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}+\alpha\boldsymbol{u})=F_{n}(\mathbb{T})\quad\text{a.e.
in }Q.$ (2.7)

Furthermore, the initial conditions must hold in the following sense:

$\lim_{t\rightarrow
0+}\bigg{(}\|\boldsymbol{u}(t)-\boldsymbol{u}_{0}\|_{2}+\|\dot{\boldsymbol{u}}(t)-\boldsymbol{v}_{0}\|_{2}\bigg{)}=0.$
(2.8)

In order to prove the existence of a weak solution of (2.1), we begin by
proving the existence of a weak solution of (2.5). We will require the
following auxiliary results. For details of the proofs of Lemmas 2.3 and 2.4,
we refer to [5].

######  Lemma 2.3.

Let $\boldsymbol{v}\in W^{1,2}_{*}(\Omega)^{d}$ be such that
$\boldsymbol{\varepsilon}(\boldsymbol{v})\in L^{\infty}_{\\#}(\Omega)^{d}$.
Then there exists an approximating sequence $(\boldsymbol{v}^{n})_{n}\subset
C^{\infty}_{*}(\overline{\Omega})^{d}$ such that

  * •

$\boldsymbol{v}^{n}\rightarrow\boldsymbol{v}$ strongly in
$L^{2}_{*}(\Omega)^{d}$, and

  * •

$\boldsymbol{\varepsilon}({\boldsymbol{v}}^{n})\overset{\ast}{\rightharpoonup}\boldsymbol{\varepsilon}(\boldsymbol{v})$
weakly-* in $L^{\infty}_{\\#}(\Omega)^{d\times d}$.

######  Lemma 2.4 (Korn’s inequality in $L^{p}$).

Let $p\in(1,\infty)$, $d\geq 2$ and $\Omega=(0,1)^{d}$. There exists a
positive constant $c_{p}$ such that

$\|\boldsymbol{v}\|_{1,p}\leq
c_{p}\|\boldsymbol{\varepsilon}(\boldsymbol{v})\|_{p}\quad\forall\boldsymbol{v}\in
W^{1,p}_{*}(\Omega)^{d}.$

The following result is from [23] and is needed to show that the function
$F_{n}$ defined in the next section is a $C^{1}$-diffeomorphism.

######  Lemma 2.5.

A $C^{1}$-map $f:\mathbb{R}^{l}\rightarrow\mathbb{R}^{l}$ is a
$C^{1}$-diffeomorphism if and only if the Jacobian $\mathrm{det}(Df)$ never
vanishes and $|f(\boldsymbol{v})|\rightarrow\infty$ whenever
$|\boldsymbol{v}|\rightarrow\infty$.

##  3\. Existence of a solution to the approximate problem

First we will show that there exists a unique weak solution of (2.5). From
this proof, we will also obtain $n$-independent bounds on the solution. These
will be used when we consider the limit as $n\rightarrow\infty$.

######  Theorem 3.1.

Let $n\in\mathbb{N}$, $\alpha>0$ and $a>0$. Suppose that $\boldsymbol{u}_{0}$,
$\boldsymbol{v}_{0}\in L^{2}_{*}(\Omega)^{d}$ are such that
$\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}\in W^{k+1,2}_{*}(\Omega)^{d}$ for
some $k>\frac{d}{2}$ with

$\|\boldsymbol{\varepsilon}(\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0})\|_{\infty}\leq
C_{*}<1,$ (3.1)

for a constant $C_{*}\in(0,1)$. Furthermore, let $\boldsymbol{f}\in
L^{2}(0,T;L^{2}_{*}(\Omega)^{d})$ be given. Then, there exists a unique weak
solution $(\boldsymbol{u},\mathbb{T})$ of the regularised problem (2.5) in the
sense of Definition 2.2. In addition, the following bound holds:

$\displaystyle\sup_{t\in[0,T]}\|\boldsymbol{u}(t)\|_{2}+\sup_{t\in[0,T]}\|\dot{\boldsymbol{u}}(t)\|_{2}+\|\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}+\alpha\boldsymbol{u})\|_{L^{n+1}(Q)}+\int_{Q}|\mathbb{T}|\,\mathrm{d}x\,\mathrm{d}t$
(3.2)
$\displaystyle\quad\quad+\sup_{t\in[0,T]}\bigg{(}\int_{\Omega}|\mathbb{T}(t)|^{1-a}\chi_{\\{|\mathbb{T}(t)|\geq
1\\}}\bigg{)}\leq C,$

where $C$ is a positive constant that is independent of $n$ and $\chi_{A}$ is
the indicator function of any measurable set $A$.

The regularity requirements on the initial data are higher than one might
expect. It is for a technical reason that we demand
$\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}\in W^{k+1,2}_{*}(\Omega)^{d}$ for
a $k>\frac{d}{2}$. In particular, by the Sobolev embedding theorem
$W^{k+1,2}_{*}(\Omega)$ is continuously embedded into
$W^{1,\infty}_{*}(\Omega)$ . This will allow us to deduce the strong
convergence in $W^{1,\infty}_{*}(\Omega)$ of a certain sequence of
approximations of $\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}$. However, the
demand for
$\boldsymbol{\varepsilon}(\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0})$ to be
bounded in $L^{\infty}(\Omega)^{d\times d}$ is natural since we eventually
want to take the limit as $n\rightarrow\infty$ and thus we would like
$F^{-1}(\boldsymbol{\varepsilon}(\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}))$
to be well-defined. The weakening of the conditions on
$\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}$ will be discussed in the first
remark in Section 4.

###### Proof.

We will construct a weak solution of (2.5) by use of the Galerkin
approximation method. Let $(\phi_{i})_{i=1}^{\infty}$ be a sequence of
trigonometric polynomials from $C^{\infty}_{*}(\overline{\Omega})$ such that
they form an orthonormal basis of $L^{2}_{*}(\Omega)$ and for every
$m\in\mathbb{N}$ there exists an $M_{m}\in\mathbb{N}$ such that the linear
span of $(\phi_{i})_{i=1}^{M_{m}}$ is the vector space of trigonometric
polynomials of degree at most $m$ with integral over $\Omega$ equal to $0$.
Let $V_{m}=(\mathrm{span}\\{\phi_{1},...,\phi_{M_{m}}\\})^{d}$. We note that a
basis of $V_{m}$ that is orthogonal with respect to the inner product in
$L^{2}(\Omega)^{d}$ is $(\phi_{i}\boldsymbol{e}_{j})_{i,j=1}^{M_{,}d}$ where
$\boldsymbol{e}_{j}$ is the $j$-th standard basis vector in $\mathbb{R}^{d}$.

For each $m\in\mathbb{N}$, we want to find a function $\boldsymbol{u}^{m}\in
W^{2,2}([0,T];L^{2}_{*}(\Omega)^{d})$ of the form

$\boldsymbol{u}^{m}(t,x)=\sum_{i=1}^{M_{m}}\sum_{j=1}^{d}\beta^{m}_{i,j}(t)\phi_{i}(x)\boldsymbol{e}_{j},$

such that, for every $k\in\\{1,\dots,M_{m}\\}$ and $l\in\\{1,\dots,d\\}$, we
have

$\int_{\Omega}\ddot{\boldsymbol{u}}^{m}(t)\cdot(\phi_{k}\boldsymbol{e}_{l})+\mathbb{T}^{m}(t):\boldsymbol{\varepsilon}(\phi_{k}\boldsymbol{e}_{l})\,\mathrm{d}x=\int_{\Omega}\boldsymbol{f}(t)\cdot(\phi_{k}\boldsymbol{e}_{l})\,\mathrm{d}x,$
(3.3)

for every $t\in(0,T)$ with $\mathbb{T}^{m}$ is defined by

$\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})=F_{n}(\mathbb{T}^{m})\quad\text{
a.e. in }Q.$ (3.4)

The initial conditions are

$\displaystyle\boldsymbol{u}^{m}(0)=P^{m}\boldsymbol{u}_{0},\quad\dot{\boldsymbol{u}}^{m}(0)=P^{m}\boldsymbol{v}_{0},$

where $P^{m}$ is the orthogonal projection operator from
$L^{2}_{\\#}(\Omega)^{d}$ to the space of $\mathbb{R}^{d}$-valued
trigonometric polynomials of degree at most $m$. We note that the restriction
of $P^{m}$ to $L^{2}_{*}(\Omega)^{d}$ coincides with the orthogonal projection
operator from $L^{2}_{*}(\Omega)^{d}$ to $V_{m}$. We will refer to this
finite-dimensional problem as the Galerkin approximation from $V_{m}$.

For every $\boldsymbol{v}\in L^{2}_{*}(\Omega)^{d}$ we may identify
$P^{m}\boldsymbol{v}$ with a vector $(v_{ij})_{i,j=1}^{M_{m},d}$ such that

$P^{m}\boldsymbol{v}=\sum_{i=1}^{M_{m}}\sum_{j=1}^{d}v_{ij}\phi_{i}\boldsymbol{e}_{j}.$

With this in mind, the Galerkin approximation from $V_{m}$ can be rewritten as
the following system of first order ODEs:

$\displaystyle(\dot{\boldsymbol{\beta}}^{m}(t),\dot{\boldsymbol{\gamma}}^{m}(t))$
$\displaystyle=(\boldsymbol{\gamma}^{m}(t),\mathbf{g}(t,\boldsymbol{\beta}^{m}(t),\boldsymbol{\gamma}^{m}(t))),$
(3.5) $\displaystyle\boldsymbol{\beta}^{m}(0)$
$\displaystyle=P^{m}\boldsymbol{u}_{0},$
$\displaystyle\boldsymbol{\gamma}^{m}(0)$
$\displaystyle=P^{m}\boldsymbol{v}_{0},$

where the function $\mathbf{g}=(g_{kl})_{k,l=1}^{M_{m},d}$ is defined by

$g_{kl}(t,\boldsymbol{\beta},\boldsymbol{\gamma})=-\int_{\Omega}F_{n}^{-1}\bigg{(}\boldsymbol{\varepsilon}\Big{(}\sum_{i=1}^{M_{m}}\sum_{j=1}^{d}(\gamma_{ij}+\alpha\beta_{ij})\phi_{i}\boldsymbol{e}_{j}\Big{)}\bigg{)}:\boldsymbol{\varepsilon}(\phi_{k}\boldsymbol{e}_{l})\,\mathrm{d}x+\int_{\Omega}\boldsymbol{f}(t)\cdot(\phi_{k}\boldsymbol{e}_{l})\,\mathrm{d}x.$

The first term is independent of $t$ and is continuous in
$(\boldsymbol{\beta},\boldsymbol{\gamma})$ by the continuity of $F_{n}^{-1}$.
The second term is independent of $(\boldsymbol{\beta},\boldsymbol{\gamma})$
and is measurable with respect to $t$ thanks to the assumptions on
$\boldsymbol{f}$. Thus we may apply standard Carathéodory theory, for example
Theorem 2.4.1 from [26], to deduce that a solution exists to (3.5) on
$[0,T_{*})$ for some positive $T_{*}\leq T$ that may depend on $m$ and $n$.
Thus a solution exists to the Galerkin approximation from $V_{m}$ on a,
possibly small, time interval $[0,T_{*})$. In order to extend the existence
result to $[0,T]$ we will deduce an energy inequality and combine this with
the fact that

$\|\boldsymbol{u}^{m}(t)\|_{2}^{2}+\|\dot{\boldsymbol{u}}^{m}(t)\|_{2}^{2}=\sum_{i=1}^{M_{m}}\sum_{j=1}^{d}\Big{(}|\beta^{m}_{ij}(t)|^{2}+|\dot{\beta}^{m}_{ij}(t)|^{2}\Big{)},$
(3.6)

which follows from the orthonormality of the basis. We note that
$\boldsymbol{f}$ can be extended to the interval $(0,T+{\epsilon})$ and the
above reasoning shows that the solution can then be extended onto the interval
$(0,T+{\epsilon})$. This fact is need in the proof of Theorem 4.1.

We multiply (3.3) by $(\dot{\beta}^{m}_{kl}+\alpha\beta^{m}_{kl})(t)$ and sum
over $k\in\\{1,\dots,M_{m}\\}$ and $l\in\\{1,\dots,d\\}$ to deduce that

$\displaystyle 0$
$\displaystyle=\int_{\Omega}\ddot{\boldsymbol{u}}^{m}\cdot(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})+\mathbb{T}^{m}:\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})-\boldsymbol{f}\cdot(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})\,\mathrm{d}x$
$\displaystyle=\int_{\Omega}\frac{\partial}{\partial
t}\Big{(}\frac{|\dot{\boldsymbol{u}}^{m}|^{2}}{2}+\alpha\dot{\boldsymbol{u}}^{m}\cdot\boldsymbol{u}^{m}\Big{)}-\alpha|\dot{\boldsymbol{u}}^{m}|^{2}+\mathbb{T}^{m}:F_{n}(\mathbb{T}^{m})-\boldsymbol{f}\cdot(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})\,\mathrm{d}x.$

For arbitrary $t\in(0,T_{*})$ we integrate over $(0,t)$ to find that

$\displaystyle\frac{\|\dot{\boldsymbol{u}}^{m}(t)\|_{2}^{2}}{2}+\int_{0}^{t}\int_{\Omega}\frac{|\mathbb{T}^{m}|^{2}}{(1+|\mathbb{T}^{m}|^{a})^{\frac{1}{a}}}+\frac{|\mathbb{T}^{m}|^{2}}{n(1+|\mathbb{T}|^{1-\frac{1}{n}})}\,\mathrm{d}x\,\mathrm{d}s$
(3.7)
$\displaystyle\quad\quad=\frac{\|\dot{\boldsymbol{u}}^{m}(0)\|_{2}^{2}}{2}+\alpha\int_{\Omega}\dot{\boldsymbol{u}}^{m}(0)\cdot\boldsymbol{u}^{m}(0)-\dot{\boldsymbol{u}}^{m}(t)\cdot\boldsymbol{u}^{m}(t)\,\mathrm{d}x+\alpha\int_{0}^{t}\int_{\Omega}|\dot{\boldsymbol{u}}^{m}|^{2}\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\quad\quad\quad\quad+\int_{0}^{t}\int_{\Omega}\boldsymbol{f}\cdot(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\quad\quad\leq\Big{(}\frac{\alpha+1}{2}\Big{)}\|P^{m}\boldsymbol{v}_{0}\|_{2}^{2}+\Big{(}\frac{\alpha}{2}+2\alpha^{2}\Big{)}\|P^{m}\boldsymbol{u}_{0}\|_{2}^{2}+(2\alpha^{2}+\alpha+1)\int_{0}^{t}\int_{\Omega}|\dot{\boldsymbol{u}}^{m}|^{2}\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\quad\quad\quad\quad+\frac{\|\dot{\boldsymbol{u}}^{m}(t)\|_{2}^{2}}{4}+\alpha^{2}\int_{0}^{t}\int_{\Omega}|\boldsymbol{u}^{m}(t)|^{2}\,\mathrm{d}x\,\mathrm{d}s+\frac{1}{2}\int_{0}^{t}\int_{\Omega}|\boldsymbol{f}|^{2}\,\mathrm{d}x\,\mathrm{ds}.$

However, we have that

$\boldsymbol{u}^{m}(t)=P^{m}\boldsymbol{u}_{0}+\int_{0}^{t}\dot{\boldsymbol{u}}^{m}(s)\,\mathrm{d}s,$
(3.8)

for every $t\in(0,T_{*})$. Thus we get

$\displaystyle\int_{0}^{t}\int_{\Omega}|\boldsymbol{u}^{m}|^{2}\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\leq
2\int_{0}^{t}\int_{\Omega}|P^{m}\boldsymbol{u}_{0}|^{2}\,\mathrm{d}x\,\mathrm{d}s+2\int_{0}^{t}\int_{\Omega}\Big{|}\int_{0}^{s}\dot{\boldsymbol{u}}^{m}(\tau,x)\,\mathrm{d}\tau\Big{|}^{2}\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\leq
2t\|P^{m}\boldsymbol{u}_{0}\|_{2}^{2}+2\int_{0}^{t}\int_{\Omega}\Big{(}s\int_{0}^{s}|\dot{\boldsymbol{u}}^{m}(\tau)|^{2}\,\mathrm{d}\tau\Big{)}\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\leq
2t\|P^{m}\boldsymbol{u}_{0}\|_{2}^{2}+2t^{2}\int_{0}^{t}\int_{\Omega}|\dot{\boldsymbol{u}}^{m}(s)|^{2}\,\mathrm{d}x\,\mathrm{d}s.$

Substituting this into the right-hand side of (3.7), we deduce that

$\displaystyle\frac{\|\dot{\boldsymbol{u}}^{m}(t)\|_{2}^{2}}{4}+\int_{0}^{t}\int_{\Omega}\frac{|\mathbb{T}^{m}|^{2}}{(1+|\mathbb{T}^{m}|^{a})^{\frac{1}{a}}}+\frac{|\mathbb{T}^{m}|^{2}}{n(1+|\mathbb{T}|^{1-\frac{1}{n}})}\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\quad\quad\leq\Big{(}\frac{\alpha+1}{2}\Big{)}\|P^{m}\boldsymbol{v}_{0}\|_{2}^{2}+\Big{(}\frac{\alpha}{2}+2\alpha^{2}+2T_{*}\Big{)}\|P^{m}\boldsymbol{u}_{0}\|_{2}^{2}$
$\displaystyle\quad\quad\quad\quad+(2\alpha^{2}+\alpha+2T_{*}^{2}+1)\int_{0}^{t}\int_{\Omega}|\dot{\boldsymbol{u}}^{m}|^{2}\,\mathrm{d}x\,\mathrm{d}s+\frac{1}{2}\int_{0}^{t}\int_{\Omega}|\boldsymbol{f}|^{2}\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\quad\quad\leq\Big{(}\frac{\alpha+1}{2}\Big{)}\|\boldsymbol{v}_{0}\|_{2}^{2}+\Big{(}\frac{\alpha}{2}+2\alpha^{2}+2T+1\Big{)}\|\boldsymbol{u}_{0}\|_{2}^{2}$
$\displaystyle\quad\quad\quad\quad+4(2\alpha^{2}+\alpha+2(T+1)^{2}+1)\int_{0}^{t}\int_{\Omega}\frac{|\dot{\boldsymbol{u}}^{m}|^{2}}{4}\,\mathrm{d}x\,\mathrm{d}s+\frac{1}{2}\int_{Q}|\boldsymbol{f}|^{2}\,\mathrm{d}x\,\mathrm{d}t.$

Applying Grönwall’s inequality, together with Lemma 2.1 to the second term on
the left-hand side, and taking the supremum over $[0,T_{*})$, we deduce that

$\displaystyle\sup_{t\in[0,T_{*})}\|\dot{\boldsymbol{u}}^{m}(t)\|_{2}^{2}+\int_{0}^{T_{*}}\int_{\Omega}|\mathbb{T}^{m}|+\frac{|\mathbb{T}^{m}|^{1+\frac{1}{n}}}{n}\,\mathrm{d}x\,\mathrm{d}t$
(3.9) $\displaystyle\quad\quad\leq
C(a,\alpha,T)\Big{(}\|\boldsymbol{u}_{0}\|_{2}^{2}+\|\boldsymbol{v}_{0}\|_{2}^{2}+\|\boldsymbol{f}\|_{L^{2}(Q)}^{2}\Big{)},$

where $C$ is a finite constant depending only on $a$, $\alpha$ and $T$. Using
(3.8), (3.9) and (3.6), we deduce that

$\displaystyle\sup_{t\in[0,T_{*})}\max_{i,j}|\beta^{m}_{ij}(t)|^{2}+\sup_{t\in[0,T_{*})}\max_{i,j}|\dot{\beta}^{m}_{ij}(t)|^{2}\leq
C,$

where $C$ is the right-hand side of (3.9). Thus we may repeatedly apply the
Carathéodory existence theorem to deduce the existence of a solution
$(\boldsymbol{u}^{m},\mathbb{T}^{m})$ to the Galerkin approximation from
$V_{m}$ on the whole of $[0,T]$. Repeating the above reasoning and using
(3.8), we deduce that

$\displaystyle\sup_{t\in[0,T]}\|{\boldsymbol{u}}^{m}(t)\|_{2}^{2}+\sup_{t\in[0,T]}\|\dot{\boldsymbol{u}}^{m}(t)\|_{2}^{2}+\int_{Q}|\mathbb{T}^{m}|+\frac{|\mathbb{T}^{m}|^{1+\frac{1}{n}}}{n}\,\mathrm{d}x\,\mathrm{d}t$
(3.10) $\displaystyle\quad\quad\leq
C(a,\alpha,T)\Big{(}\|\boldsymbol{u}_{0}\|_{2}^{2}+\|\boldsymbol{v}_{0}\|_{2}^{2}+\|\boldsymbol{f}\|_{L^{2}(Q)}^{2}\Big{)}.$

From (3.10), we can immediately deduce a further estimate. Using (3.4), we get

$\displaystyle\Big{(}\int_{Q}|\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})|^{n+1}\,\mathrm{d}x\,\mathrm{d}t\Big{)}^{\frac{1}{n+1}}$
$\displaystyle\leq|Q|^{\frac{1}{n+1}}+\Big{(}\int_{Q}\frac{|\mathbb{T}^{m}|^{1+\frac{1}{n}}}{n^{n+1}}\,\mathrm{d}x\,\mathrm{d}t\Big{)}^{\frac{1}{n+1}}\leq
C,$ (3.11)

where $C$ is a positive constant depending only on $a$, $\alpha$, $T$, $d$ and
the data. Applying Theorem 2.4, it follows that
$(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})_{m}$ is bounded in
$L^{n+1}(0,T;W^{1,n+1}_{*}(\Omega)^{d})$, independent of $m$.

The bound on $\mathbb{T}^{m}$ in (3.10) allows us to deduce a bound on
$\ddot{\boldsymbol{u}}^{m}$ in
$L^{n+1}(0,T;(W^{1,n+1}_{*}(\Omega)^{d})^{\prime})$ by use of (3.3). However,
since eventually we will let $n\rightarrow\infty$ it would be preferable to
obtain a bound in a space that does not depend on $n$. To this end, we now
multiply (3.3) by $(\ddot{\beta}^{m}_{ij}+\alpha\dot{\beta}^{m}_{ij})(t)$ and
sum over $k\in\\{1,\dots,d\\}$ and $l\in\\{1,\dots,d\\}$ to deduce that

$0=\int_{\Omega}|\ddot{\boldsymbol{u}}^{m}|^{2}+\frac{\partial}{\partial
t}\left(\frac{\alpha}{2}|\dot{\boldsymbol{u}}^{m}|^{2}+\frac{h_{n}(|\mathbb{T}^{m}|^{2})}{2}\right)-\boldsymbol{f}\cdot(\ddot{\boldsymbol{u}}^{m}+\alpha\dot{\boldsymbol{u}}^{m})\,\mathrm{d}x,$
(3.12)

where $h_{n}:[0,\infty)\rightarrow[0,\infty)$ is defined by

$h_{n}(s)=\int_{0}^{s}\frac{1}{(1+t^{\frac{a}{2}})^{1+\frac{1}{a}}}+\frac{1}{n^{2}(1+t^{\frac{1}{2}-\frac{1}{2n}})^{2}}+\left(1-\frac{1}{n}\right)\frac{1}{n(1+t^{\frac{1}{2}-\frac{1}{2n}})^{2}}\,\mathrm{d}t,$

where we have made use of the following reasoning:

$\displaystyle\mathbb{T}^{m}:\frac{\partial}{\partial
t}\Big{(}F_{n}(\mathbb{T}^{m})\Big{)}$
$\displaystyle=\mathbb{T}^{m}:\bigg{[}\frac{\dot{\mathbb{T}}^{m}}{(1+|\mathbb{T}^{m}|^{a})^{\frac{1}{a}}}+\frac{\dot{\mathbb{T}}^{m}}{n(1+|\mathbb{T}^{m}|^{1-\frac{1}{n}})}-\frac{(\mathbb{T}^{m}:\dot{\mathbb{T}}^{m})|\mathbb{T}^{m}|^{a-2}\mathbb{T}^{m}}{(1+|\mathbb{T}^{m}|^{a})^{1+\frac{1}{a}}}$
$\displaystyle\quad\quad\quad\quad-\bigg{(}1-\frac{1}{n}\bigg{)}\frac{(\mathbb{T}^{m}:\dot{\mathbb{T}}^{m})|\mathbb{T}^{m}|^{-1-\frac{1}{n}}\mathbb{T}^{m}}{n(1+|\mathbb{T}^{m}|^{1-\frac{1}{n}})^{2}}\bigg{]}$
$\displaystyle=\frac{\mathbb{T}^{m}:\dot{\mathbb{T}}^{m}}{(1+|\mathbb{T}^{m}|^{a})^{1+\frac{1}{a}}}+\frac{\mathbb{T}^{m}:\dot{\mathbb{T}}^{m}}{n^{2}(1+|\mathbb{T}^{m}|^{1-\frac{1}{n}})}+\bigg{(}1-\frac{1}{n}\bigg{)}\frac{\mathbb{T}^{m}:\dot{\mathbb{T}}^{m}}{n(1+|\mathbb{T}^{m}|^{1-\frac{1}{n}})^{2}}$
$\displaystyle=h_{n}^{\prime}(|\mathbb{T}^{m}|^{2})\mathbb{T}^{m}:\dot{\mathbb{T}}^{m}$
$\displaystyle=h_{n}^{\prime}(|\mathbb{T}^{m}|^{2})\frac{\partial}{\partial
t}\left(\frac{|\mathbb{T}^{m}|^{2}}{2}\right)$
$\displaystyle=\frac{\partial}{\partial
t}\left(\frac{h_{n}(|\mathbb{T}^{m}|^{2})}{2}\right).$

Using Lemma 2.1, there exist positive constants $c_{a}$, $C_{a}$ depending
only on $a$ such that

$c_{a}\Big{(}s^{\frac{1}{2}-\frac{a}{2}}\chi_{\\{s\geq 1\\}}-1\Big{)}\leq
h_{n}(s)\leq
s^{\frac{1}{n}}+C_{a}\Big{(}s^{\frac{1}{2}-\frac{a}{2}}\chi_{\\{s\geq
1\\}}+1\Big{)}.$ (3.13)

We require the indicator functions since we do not necessarily have
$a\in(0,1]$. Integrating (3.12) over $(0,t)$ for an arbitrary $t\in(0,T)$ and
using (3.13), we deduce that

$\displaystyle\int_{0}^{t}\int_{\Omega}|\ddot{\boldsymbol{u}}^{m}|^{2}\,\mathrm{d}x\,\mathrm{d}s+\frac{\alpha}{2}\|\dot{\boldsymbol{u}}^{m}(t)\|_{2}^{2}+\int_{\Omega}c_{a}|\mathbb{T}^{m}(t)|^{1-a}\chi_{\\{|\mathbb{T}^{m}(t)|\geq
1\\}}\,\mathrm{d}x$
$\displaystyle\quad\quad\leq\frac{\alpha}{2}\|\dot{\boldsymbol{u}}^{m}(0)\|_{2}^{2}+\int_{\Omega}C_{a}|\mathbb{T}^{m}(0)|^{1-a}\chi_{\\{|\mathbb{T}^{m}(0)|\geq
1\\}}+|\mathbb{T}^{m}(0)|^{\frac{2}{n}}\,\mathrm{d}x$
$\displaystyle\quad\quad\quad\quad+\int_{0}^{t}\int_{\Omega}\boldsymbol{f}\cdot(\ddot{\boldsymbol{u}}^{m}+\alpha\dot{\boldsymbol{u}}^{m})\,\mathrm{d}x\,\mathrm{d}t+C(a,\Omega)$
$\displaystyle\quad\quad\leq\frac{\alpha}{2}\|\boldsymbol{v}_{0}\|_{2}^{2}+(C_{a}+1)\int_{\Omega}|\mathbb{T}^{m}(0)|^{2}\,\mathrm{d}x+\frac{1}{2}\int_{0}^{t}\int_{\Omega}|\ddot{\boldsymbol{u}}^{m}|^{2}\,\mathrm{d}x\,\mathrm{d}s+\int_{0}^{t}\int_{\Omega}|\boldsymbol{f}|^{2}\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\quad\quad\quad\quad+\frac{\alpha^{2}}{2}\int_{0}^{t}\int_{\Omega}|\dot{\boldsymbol{u}}^{m}|^{2}\,\mathrm{d}x\,\mathrm{d}s+C(a,\alpha,\Omega).$

Rearranging, applying (3.10) and taking the supremum over $t\in(0,T)$, it
follows that

$\displaystyle\frac{1}{2}\int_{Q}|\ddot{\boldsymbol{u}}^{m}|^{2}\,\mathrm{d}x\,\mathrm{d}t+\sup_{t\in[0,T]}\Big{(}\int_{\Omega}C_{a}|\mathbb{T}^{m}(t)|^{1-a}\chi_{\\{|\mathbb{T}^{m}(t)|\geq
1\\}}\,\mathrm{d}x\Big{)}$ (3.14) $\displaystyle\quad\quad\leq
C(a,\alpha,\Omega,T)\Big{(}\|\boldsymbol{u}_{0}\|_{2}^{2}+\|\boldsymbol{v}_{0}\|_{2}^{2}+\|\boldsymbol{f}\|_{L^{2}(Q)}^{2}+\int_{\Omega}|\mathbb{T}^{m}(0)|^{2}\,\mathrm{d}x\Big{)},$

where $C$ is a positive constant that is independent of $m$ and $n$. To show
that the sequence $(\mathbb{T}^{m}(0))_{m}$ is bounded in
$L^{2}(\Omega)^{d\times d}$, independent of $m$ and $n$, we first note that

$\displaystyle\boldsymbol{\varepsilon}(P^{m}(\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}))=\frac{\mathbb{T}^{m}(0)}{(1+|\mathbb{T}^{m}(0)|^{a})^{\frac{1}{a}}}+\frac{\mathbb{T}^{m}(0)}{n(1+|\mathbb{T}^{m}(0)|^{1-\frac{1}{n}})}.$

Suppose that there exist $m_{0}\in\mathbb{N}$ and $C_{1}\in(0,1)$ such that
$\|\boldsymbol{\varepsilon}(P^{m}(\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}))\|_{\infty}\leq
C_{1}$ for every $m\geq m_{0}$. Since
$|F^{-1}(\mathbb{T})|\geq|F_{n}^{-1}(\mathbb{T})|$ for every
$\mathbb{T}\in\mathbb{R}^{d\times d}$ with $|\mathbb{T}|<1$ and $F^{-1}$ is a
radial function that increases in absolute value as $|\mathbb{T}|$ increases,
we deduce that

$|F_{n}^{-1}(\boldsymbol{\varepsilon}(P^{m}(\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0})))|\leq|F^{-1}(\boldsymbol{\varepsilon}(P^{m}(\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0})))|\leq
f^{-1}(C_{1})<\infty,$

a.e. in $\Omega$, where $f$ is a function from $[0,\infty)$ to $[0,1)$ defined
by

$f(s)=\frac{s}{(1+s^{a})^{\frac{1}{a}}}.$

It follows that $(\mathbb{T}^{m}(0))_{m\geq m_{0}}$ is uniformly bounded in
$L^{\infty}(\Omega)^{d\times d}$ independent of $m$ and $n$.

It remains to prove the existence of such an $m_{0}$ and $C_{1}$. From
standard properties of projection operators, for every $\boldsymbol{v}\in
L^{2}_{\\#}(\Omega)^{d}$ we have
$P^{m}\boldsymbol{v}\rightarrow\boldsymbol{v}$ strongly in
$L^{2}_{\\#}(\Omega)^{d}$ as $m\rightarrow\infty$. Furthermore, the projection
operator commutes with derivation [6], i.e., for every $\boldsymbol{v}\in
W^{1,2}_{\\#}(\Omega)^{d}$ we have
$\nabla(P^{m}\boldsymbol{v})=P^{m}(\nabla\boldsymbol{v})$ (assuming that
$P^{m}$ acts component-wise on matrix-valued functions). Thus, for every
$\boldsymbol{v}\in W^{k,2}_{*}(\Omega)^{d}$ and $k\in\mathbb{N}$, the
following holds:

$\displaystyle\lim_{m\rightarrow\infty}\|P^{m}\boldsymbol{v}-\boldsymbol{v}\|_{k,2}=0.$

Applying the Sobolev embedding theorem, if $\boldsymbol{v}\in
W^{k+1,2}_{*}(\Omega)^{d}$ for $k\geq\frac{d}{2}$, we have

$\displaystyle\|\boldsymbol{\varepsilon}(P^{m}\boldsymbol{v})-\boldsymbol{\varepsilon}(\boldsymbol{v})\|_{\infty}$
$\displaystyle\leq
C\|\boldsymbol{\varepsilon}(P^{m}\boldsymbol{v})-\boldsymbol{\varepsilon}(\boldsymbol{v})\|_{k,2}$
$\displaystyle\leq C\|P^{m}\boldsymbol{v}-\boldsymbol{v}\|_{k+1,2},$

where the right-hand side vanishes in the limit as $m\rightarrow\infty$ and
$C$ is independent of $\boldsymbol{v}$. Setting
$\boldsymbol{v}=\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}$, we deduce that
there exists an $m_{0}\in\mathbb{N}$ such that

$\|\boldsymbol{\varepsilon}(P^{m}(\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}))-\boldsymbol{\varepsilon}(\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0})\|_{\infty}\leq\frac{1-C_{*}}{2},$

for every $m\geq m_{0}$. The right-hand side is positive since
$C_{*}\in(0,1)$. Using (3.1) it follows that

$\|\boldsymbol{\varepsilon}(P^{m}(\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}))\|_{\infty}\leq\frac{1+C_{*}}{2}=:C_{1}<1,$

for every $m\geq m_{0}$. Substituting this into (3.14) we get

$\displaystyle\int_{Q}|\ddot{\boldsymbol{u}}^{m}|^{2}\,\mathrm{d}x\,\mathrm{d}t+\sup_{t\in[0,T]}\Big{(}\int_{\Omega}|\mathbb{T}^{m}(t)|^{1-a}\chi_{\\{|\mathbb{T}^{m}(t)|\geq
1\\}}\,\mathrm{d}x\Big{)}$ (3.15) $\displaystyle\quad\quad\leq
C(a,\alpha,\Omega,T)\Big{(}\|\boldsymbol{u}_{0}\|_{2}^{2}+\|\boldsymbol{v}_{0}\|_{2}^{2}+\|\boldsymbol{f}\|_{L^{2}(Q)}^{2}+f^{-1}(C_{1})\Big{)}$
$\displaystyle\quad\quad\leq
C(a,\alpha,\Omega,T,\boldsymbol{u}_{0},\boldsymbol{v}_{0},\boldsymbol{f}),$

for every $m\geq m_{0}$, where $C$ is a positive constant that is independent
of $m$ and $n$.

Putting together (3.10), (3.11), (3.15) and using Korn’s inequality, we deduce
the following convergence results up to a subsequence that we do not relabel:

  * •

$\boldsymbol{u}^{m}\overset{\ast}{\rightharpoonup}\boldsymbol{u}$ weakly-* in
$L^{\infty}(0,T;L^{2}(\Omega)^{d})$;

  * •

$\dot{\boldsymbol{u}}^{m}\overset{\ast}{\rightharpoonup}\dot{\boldsymbol{u}}$
weakly-* in $L^{\infty}(0,T;L^{2}(\Omega)^{d})$;

  * •

$\ddot{\boldsymbol{u}}^{m}\rightharpoonup\ddot{\boldsymbol{u}}$ weakly in
$L^{2}(0,T;L^{2}_{*}(\Omega)^{d})$;

  * •

$\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m}\rightharpoonup\dot{\boldsymbol{u}}+\alpha\boldsymbol{u}$
weakly in $L^{n+1}(0,T;W^{1,n+1}_{*}(\Omega)^{d})$;

  * •

$\mathbb{T}^{m}\rightharpoonup\mathbb{T}$ weakly in
$L^{1+\frac{1}{n}}(0,T;L^{1+\frac{1}{n}}_{\\#}(\Omega)^{d\times d})$.

We claim that $(\boldsymbol{u},\mathbb{T})$ is a weak solution of (2.8).

By standard regularity results, we know that $\boldsymbol{u}$,
$\dot{\boldsymbol{u}}\in C([0,T];L^{2}(\Omega)^{d})$ up to redefinition almost
everywhere. In particular, we have

$\lim_{t\rightarrow
0+}\Big{(}\|\boldsymbol{u}(t)-\boldsymbol{u}(0)\|_{2}+\|\dot{\boldsymbol{u}}(t)-\dot{\boldsymbol{u}}(0)\|_{2}\Big{)}=0.$
(3.16)

However, by the Aubin–Lions lemma, we know that the sequences
$(\boldsymbol{u}^{m})_{m}$, $(\dot{\boldsymbol{u}}^{m})_{m}$ converge strongly
in $C([0,T];(W^{1,2}_{*}(\Omega)^{d})^{\prime})$. Thus we have

$\displaystyle 0$
$\displaystyle=\lim_{m\rightarrow\infty}\|\boldsymbol{u}(0)-\boldsymbol{u}^{m}(0)\|_{-1,2}+\|\dot{\boldsymbol{u}}(0)-\dot{\boldsymbol{u}}^{m}(0)\|_{-1,2}$
$\displaystyle=\lim_{m\rightarrow\infty}\|\boldsymbol{u}(0)-P^{m}\boldsymbol{u}_{0}\|_{-1,2}+\|\dot{\boldsymbol{u}}(0)-P^{m}\boldsymbol{v}_{0}\|_{-1,2}$
$\displaystyle=\|\boldsymbol{u}(0)-\boldsymbol{u}_{0}\|_{-1,2}+\|\dot{\boldsymbol{u}}(0)-\boldsymbol{v}_{0}\|_{-1,2},$

where $\|\cdot\|_{-1,2}$ denotes the norm in
$(W^{1,2}_{*}(\Omega)^{d})^{\prime}$. By combining this with (3.16), the
assertion (2.8) immediately follows.

To show that (2.6) holds, first note that for every $\boldsymbol{v}\in
C^{\infty}_{*}(\overline{\Omega})^{d}$ and $t\in(0,T)$ we have

$\int_{\Omega}\ddot{\boldsymbol{u}}^{m}(t)\cdot
P^{m}\boldsymbol{v}+\mathbb{T}^{m}(t):\boldsymbol{\varepsilon}(P^{m}\boldsymbol{v})\,\mathrm{d}x=\int_{\Omega}\boldsymbol{f}(t)\cdot
P^{m}\boldsymbol{v}\,\mathrm{d}x.$ (3.17)

For an arbitrary but fixed $\psi\in C([0,T])$, we multiply (3.17) by $\psi(t)$
and integrate over $(0,T)$. Noting that $P^{m}\boldsymbol{v}$ converges
strongly in $W^{k,2}_{*}(\Omega)^{d}$ to $\boldsymbol{v}$ for every
$k\in\mathbb{N}$, letting $m\rightarrow\infty$ we get

$\displaystyle 0$
$\displaystyle=\int_{Q}\ddot{\boldsymbol{u}}\cdot(\psi\boldsymbol{v})+\mathbb{T}:\boldsymbol{\varepsilon}(\psi\boldsymbol{v})-\boldsymbol{f}\cdot(\psi\boldsymbol{v})\,\mathrm{d}x\,\mathrm{dt}$
$\displaystyle=\int_{0}^{T}\psi(t)\cdot\Big{(}\int_{\Omega}\ddot{\boldsymbol{u}}(t)\cdot\boldsymbol{v}+\mathbb{T}(t):\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{f}(t)\cdot\boldsymbol{v}\,\mathrm{d}x\Big{)}\,\mathrm{d}t.$

Since $\psi$ is arbitrary, the second factor is integrable over $(0,T)$ and
$C^{\infty}_{*}(\overline{\Omega})^{d}$ is dense in $W^{1,p}_{*}(\Omega)^{d}$
for every $p\in[1,\infty)$, we deduce that (2.6) holds. Furthermore, we note
that $(\dot{\boldsymbol{u}}+\alpha\boldsymbol{u})(t)$ is a valid test function
in (2.6) for a.e. $t\in(0,T)$.

To show that (2.7) holds, we will use a variant of Minty’s method. First note
that the sequence $(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})_{m}$
converges weakly in $L^{2}(0,T;W^{1,2}_{*}(\Omega)^{d})$ and
$(\ddot{\boldsymbol{u}}^{m}+\alpha\dot{\boldsymbol{u}}^{m})_{m}$ converges
weakly in $L^{2}(0,T;L^{2}_{*}(\Omega)^{d})$. Thus by the Aubin–Lions lemma,
the sequence $(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})_{m}$
converges strongly in the space $L^{2}(0,T;L^{2}_{*}(\Omega)^{d})$ as
$m\rightarrow\infty$. Testing (3.3) against
$(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})(t)$ and integrating over
$(0,T)$ we deduce that

$\displaystyle\lim_{m\rightarrow\infty}\int_{Q}\mathbb{T}^{m}:\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle=\lim_{m\rightarrow\infty}\int_{Q}-\ddot{\boldsymbol{u}}^{m}\cdot(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})+\boldsymbol{f}\cdot(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})\,\mathrm{d}x\,\mathrm{dt}$
(3.18)
$\displaystyle=\int_{Q}-\ddot{\boldsymbol{u}}\cdot(\dot{\boldsymbol{u}}+\alpha\boldsymbol{u})+\boldsymbol{f}\cdot(\dot{\boldsymbol{u}}+\alpha\boldsymbol{u})\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle=\int_{Q}\mathbb{T}:\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}+\alpha\boldsymbol{u})\,\mathrm{d}x\,\mathrm{d}t,$

using that $(\dot{\boldsymbol{u}}+\alpha\boldsymbol{u})(t)$ is a valid test
function in (2.6) for the transition to the final line. Let $\mathbb{S}\in
L^{1+\frac{1}{n}}(Q)^{d\times d}$ be arbitrary but fixed. Using the
monotonicity of $F_{n}$, (3.18) and the convergence results, we have that

$\displaystyle 0$
$\displaystyle\leq\lim_{m\rightarrow\infty}\int_{Q}(\mathbb{T}^{m}-\mathbb{S}):(F_{n}(\mathbb{T}^{m})-F_{n}(\mathbb{S}))\,\mathrm{d}x\,\mathrm{d}t$
(3.19)
$\displaystyle=\int_{Q}(\mathbb{T}-\mathbb{S}):(\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}+\alpha\boldsymbol{u})-F_{n}(\mathbb{S}))\,\mathrm{d}x\,\mathrm{d}t.$

We replace $\mathbb{S}$ by $\mathbb{T}\pm\gamma\mathbb{U}$ for an arbitrary
$\gamma>0$ and $\mathbb{U}\in L^{\infty}(Q)^{d\times d}$ to obtain

$\displaystyle
0\leq\mp\int_{Q}\gamma\mathbb{U}:(\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}+\alpha\boldsymbol{u})-F_{n}(\mathbb{T}\pm\gamma\mathbb{U}))\,\mathrm{d}x\,\mathrm{d}t.$

We divide through by $\gamma$ and use Lebesgue’s dominated convergence theorem
when letting $\gamma\rightarrow 0+$ in order to deduce that

$\displaystyle
0\leq\mp\int_{Q}\mathbb{U}:(\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}+\alpha\boldsymbol{u})-F_{n}(\mathbb{T}))\,\mathrm{d}x\,\mathrm{d}t.$

Setting
$\mathbb{U}=\frac{\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}+\alpha\boldsymbol{u})-F_{n}(\mathbb{T})}{1+|\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}+\alpha\boldsymbol{u})-F_{n}(\mathbb{T})|}$,
it immediately follows that (2.7) holds. Hence $(\boldsymbol{u},\mathbb{T})$
is a weak solution of (2.5).

To show that (3.2) holds, we only need to show that
$\mathbb{T}^{m}\rightarrow\mathbb{T}$ converges pointwise a.e. on $Q$ as
$m\rightarrow\infty$ and $\mathbb{T}^{m}(t)\rightarrow\mathbb{T}(t)$ pointwise
a.e. on $\Omega$ for a.e. $t\in(0,T)$. Combining this with (3.10), (3.11),
(3.15) and Fatou’s lemma with the weak lower semi-continuity of norms, the
bound (3.2) will follow.

We can in fact prove the stronger result that
$\mathbb{T}^{m}\rightarrow\mathbb{T}$ strongly in $L^{1}(Q)^{d\times d}$ as
$m\rightarrow\infty$ by mimicking an argument contained in [5]. For each
$k>0$, we define the set

$\displaystyle Q^{m}_{k}=\\{(t,x)\in Q:1+|\mathbb{T}|+|\mathbb{T}^{m}|>k\\}.$

Using that $\mathbb{T}^{m}\rightharpoonup\mathbb{T}$ weakly in
$L^{1+\frac{1}{n}}(Q)^{d\times d}$ and the bound from (3.10), there exists a
positive constant $C=C(n)$ independent of $m$ such that

$\displaystyle\int_{Q}|\mathbb{T}|^{1+\frac{1}{n}}+|\mathbb{T}^{m}|^{1+\frac{1}{n}}\,\mathrm{d}x\,\mathrm{d}t\leq
C(n).$

It follows that $|Q^{m}_{k}|\leq C(n)k^{-(1+\frac{1}{n})}$. With this in mind,
we have that

$\displaystyle\Big{(}\int_{Q}|\mathbb{T}^{m}-\mathbb{T}|\,\mathrm{d}x\,\mathrm{d}t\Big{)}$
$\displaystyle\quad\quad\leq
C\|\mathbb{T}^{m}-\mathbb{T}\|_{L^{1+\frac{1}{n}}(Q^{m}_{k})}|Q^{m}_{k}|^{\frac{2}{n+1}}+Ck^{1+a}\int_{Q\setminus
Q^{m}_{k}}\frac{|\mathbb{T}^{m}-\mathbb{T}|^{2}}{(1+|\mathbb{T}^{m}|+|\mathbb{T}|)^{1+a}}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\quad\quad\leq
Ck^{-\frac{2}{n}}+Ck^{1+a}\int_{Q}(\mathbb{T}^{m}-\mathbb{T}):(F(\mathbb{T}^{m})-F(\mathbb{T}))\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\quad\quad\leq
Ck^{-\frac{2}{n}}+Ck^{1+a}\int_{Q}(\mathbb{T}^{m}-\mathbb{T}):(\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})-\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}+\alpha\boldsymbol{u}))\,\mathrm{d}x\,\mathrm{d}t,$

where $C$ is a positive constant that is independent of $k$ and $m$. In the
limit as $m\rightarrow\infty$, the second term on the right-hand side will
vanish, recalling (3.18). It follows that, for every $k>0$,

$\displaystyle\lim_{m\rightarrow\infty}\Big{(}\int_{Q}|\mathbb{T}^{m}-\mathbb{T}|\,\mathrm{d}x\,\mathrm{d}t\Big{)}^{2}\leq
Ck^{-\frac{2}{n}}.$

Since $k$ is arbitrary, we deduce that $\mathbb{T}^{m}\rightarrow\mathbb{T}$
strongly in $L^{1}(Q)^{d\times d}$ as $m\rightarrow\infty$. Taking a further
subsequence if necessary, we get that $\mathbb{T}^{m}\rightarrow\mathbb{T}$
pointwise a.e. in $Q$.

To prove that $\mathbb{T}^{m}(t)\rightarrow\mathbb{T}(t)$ converges pointwise
a.e. on $\Omega$ for a.e. $t\in(0,T)$, suppose otherwise. That is, assume that
there exists a measurable set $A\subset(0,T)$ of positive measure such that,
for each $t\in A$, there exists a measurable set $B(t)\subset\Omega$ of
positive measure such that $(\mathbb{T}^{m}(t,x))_{m}$ does not converge to
$\mathbb{T}(t,x)$, for every $x\in B(t)$. Let $M=\\{(t,x):t\in A,x\in
B(t)\\}$, a measurable subset of $Q$ such that

$|M|=\int_{A}\int_{B(t)}1\,\mathrm{d}x\,\mathrm{d}t=\int_{A}|B(t)|\,\mathrm{d}t>0.$

However, $\mathbb{T}^{m}\not\to\mathbb{T}$ pointwise on $M$. This contradicts
the fact that $\mathbb{T}^{m}\rightarrow\mathbb{T}$ pointwise a.e. on $Q$.
Thus our original claim holds and we deduce that (3.2) holds. This concludes
the existence part of the proof.

To prove uniqueness, suppose that $(\boldsymbol{u}_{1},\mathbb{T}_{1})$,
$(\boldsymbol{u}_{2},\mathbb{T}_{2})$ are weak solutions of (2.5) in the sense
of Definition 2.2 with respect to the same initial data. Let
$\boldsymbol{v}:=\boldsymbol{u}_{1}-\boldsymbol{u}_{2}$ and
$\mathbb{S}:=\mathbb{T}_{1}-\mathbb{T}_{2}$. Testing in (2.6) for
$(\boldsymbol{u}_{i},\mathbb{T}_{i})$, $i\in\\{1,2\\}$, with test function
$(\dot{\boldsymbol{v}}+\alpha\boldsymbol{v})(t)$, integrating over $(0,t)$ for
an arbitrary $t\in(0,T)$ and subtracting the result, we get

$\displaystyle 0$
$\displaystyle=\int_{0}^{t}\int_{\Omega}\ddot{\boldsymbol{v}}\cdot(\dot{\boldsymbol{v}}+\alpha\boldsymbol{v})+\mathbb{S}:\boldsymbol{\varepsilon}(\dot{\boldsymbol{v}}+\alpha\boldsymbol{v})\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle=\int_{\Omega}\frac{|\dot{\boldsymbol{v}}(t)|^{2}}{2}+\alpha\dot{\boldsymbol{v}}(t)\cdot\boldsymbol{v}(t)\,\mathrm{d}x+\int_{0}^{t}\int_{\Omega}\mathbb{S}:\boldsymbol{\varepsilon}(\dot{\boldsymbol{v}}+\alpha\boldsymbol{v})-\alpha|\dot{\boldsymbol{v}}|^{2}\,\mathrm{d}x\,\mathrm{d}s.$

We used the fact that $\boldsymbol{v}(t)\rightarrow\mathbf{0}$ and
$\dot{\boldsymbol{v}}(t)\rightarrow\mathbf{0}$ strongly in $L^{2}(\Omega)^{d}$
as $t\rightarrow 0+$, as well as $\boldsymbol{v}$, $\dot{\boldsymbol{v}}\in
C([0,T];L^{2}(\Omega)^{d})$ with Lemma 7.3 of [22]. Using the monotonicity of
$F_{n}$, it follows that

$\displaystyle 0$
$\displaystyle\leq\int_{\Omega}\frac{|\dot{\boldsymbol{v}}(t)|^{2}}{2}\,\mathrm{d}x+\int_{0}^{t}\int_{\Omega}(\mathbb{T}_{1}-\mathbb{T}_{2}):(F_{n}(\mathbb{T}_{1})-F_{n}(\mathbb{T}_{2}))\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\leq\int_{0}^{t}\int_{\Omega}(2\alpha^{2}+\alpha)|\dot{\boldsymbol{v}}|^{2}\,\mathrm{d}x\,\mathrm{d}s+\int_{\Omega}\frac{|\dot{\boldsymbol{v}}(t)|^{2}}{4}\,\mathrm{d}x.$

Applying Grönwall’s inequality, we deduce that

$\int_{\Omega}\frac{|\dot{\boldsymbol{v}}(t)|^{2}}{4}\,\mathrm{d}x+\int_{0}^{t}\int_{\Omega}(\mathbb{T}_{1}-\mathbb{T}_{2}):(F_{n}(\mathbb{T}_{1})-F_{n}(\mathbb{T}_{2}))\,\mathrm{d}x\,\mathrm{d}s=0,$

for a.e. $t\in(0,T)$. Thus $\dot{\boldsymbol{u}}_{1}=\dot{\boldsymbol{u}}_{2}$
a.e. in $Q$. Since $\boldsymbol{u}_{1}(0)=\boldsymbol{u}_{2}(0)$, it follows
that $\boldsymbol{u}_{1}=\boldsymbol{u}_{2}$ and
$F_{n}(\mathbb{T}_{1})=F_{n}(\mathbb{T}_{2})$ a.e. in $Q$. Noting that $F_{n}$
is a bijection, we must have $\mathbb{T}_{1}=\mathbb{T}_{2}$ a.e. in $Q$. In
particular,
$(\boldsymbol{u}_{1},\mathbb{T}_{1})=(\boldsymbol{u}_{2},\mathbb{T}_{2})$ and
we have uniqueness of weak solutions of (2.5). ∎

We would like to use this approximation in order to show that a weak solution
exists to the strain-limiting problem (2.1). To do this, we must obtain
further a priori estimates. Since we are working in the periodic setting, we
will be able to do this when working with the Galerkin approximation
$(\boldsymbol{u}^{m},\mathbb{T}^{m})$ of the regularised problem, rather than
the weak solution of (2.5) itself.

######  Lemma 3.1.

Suppose that the hypotheses of Theorem 3.1 hold and that additionally we have
$\boldsymbol{u}_{0}$, $\boldsymbol{v}_{0}\in W^{1,2}_{*}(\Omega)^{d}$ and
$\boldsymbol{f}\in L^{2}(0,T;W^{1,2}_{*}(\Omega)^{d})$. Let
$(\boldsymbol{u}^{m},\mathbb{T}^{m})$ be the solution of the Galerkin
approximation from $V_{m}$ as in the proof of Theorem 3.1. There exists a
constant $C$ independent of $m$ and $n$ such that

$\displaystyle\sup_{t\in[0,T]}\|\nabla\boldsymbol{u}^{m}(t)\|_{2}^{2}+\sup_{t\in[0,T]}\|\nabla\dot{\boldsymbol{u}}^{m}(t)\|_{2}^{2}+\int_{Q}\frac{|\nabla\mathbb{T}^{m}|^{2}}{(1+|\mathbb{T}^{m}|)^{1+a}}\,\mathrm{d}x\,\mathrm{d}t$
(3.20) $\displaystyle\quad\quad\leq
C(a,\alpha,\Omega,T)\Big{(}\|\nabla\boldsymbol{u}_{0}\|_{2}^{2}+\|\nabla\boldsymbol{v}_{0}\|_{2}^{2}+\|\nabla\boldsymbol{f}\|_{L^{2}(Q)}^{2}\Big{)}.$

Moreover, if we also have that $\boldsymbol{f}\in
W^{1,2}([0,T+\tilde{\epsilon});L^{2}_{*}(\Omega)^{d})$ for some
$\tilde{\epsilon}>0$, then

$\displaystyle\sup_{t\in[0,T]}\|\ddot{\boldsymbol{u}}^{m}(t)\|_{2}^{2}+\int_{Q}\frac{|\dot{\mathbb{T}}^{m}|^{2}}{(1+|\mathbb{T}^{m}|)^{1+a}}\,\mathrm{d}x$
(3.21) $\displaystyle\quad\quad\leq
C(a,\alpha,\Omega,T)\Big{(}\|\boldsymbol{v}_{0}\|_{2}^{2}+\|\dot{\boldsymbol{f}}\|_{L^{2}(Q)}^{d}+\|\boldsymbol{f}(0)\|_{L^{2}(\Omega)}^{2}+(1+f^{-1}(C_{1}))\|\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}\|_{2,2}^{2}\Big{)},$

for every $m\geq m_{0}$, where $C_{1}\in(0,1)$ and $m_{0}\in\mathbb{N}$ are
the constants from the proof of Theorem 3.1 and $f$ is defined on $[0,\infty)$
by $f(t)=(1+t^{a})^{-\frac{1}{a}}t$. The constant $C$ on the right-hand side
of (3.21) is independent of $m$ and $n$.

###### Proof.

We start with (3.20). Derivation does not increase the degree of a
trigonometric polynomial. In particular,
$\nabla\cdot\nabla(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})$ is a
trigonometric polynomial of degree at most $m$. Furthermore it has integral
over $\Omega$ equal to $0$ as a result of periodicity. Thus
$\nabla\cdot\nabla(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})$ is a
valid test function in (3.3). It follows that

$\displaystyle 0$
$\displaystyle=-\int_{\Omega}\ddot{\boldsymbol{u}}^{m}\cdot(\nabla\cdot\nabla(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m}))+\mathbb{T}^{m}:\boldsymbol{\varepsilon}(\nabla\cdot\nabla(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m}))$
(3.22)
$\displaystyle\quad\quad-\boldsymbol{f}\cdot(\nabla\cdot\nabla(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m}))\,\mathrm{d}x$
$\displaystyle=\int_{\Omega}\nabla\ddot{\boldsymbol{u}}^{m}:\nabla(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})+(\nabla\cdot\mathbb{T}^{m})\cdot(\nabla\cdot\nabla(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m}))$
$\displaystyle\quad\quad-\nabla\boldsymbol{f}:\nabla(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})\,\mathrm{d}x$
$\displaystyle=\int_{\Omega}\frac{\partial}{\partial
t}\Big{(}\frac{|\nabla\dot{\boldsymbol{u}}^{m}|^{2}}{2}+\alpha\nabla\dot{\boldsymbol{u}}^{m}:\nabla\boldsymbol{u}^{m}\Big{)}-\alpha|\nabla\dot{\boldsymbol{u}}^{m}|^{2}+\nabla\mathbb{T}^{m}\,\vdots\,\nabla
F_{n}(\mathbb{T}^{m})$
$\displaystyle\quad\quad-\nabla\boldsymbol{f}:\nabla(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})\,\mathrm{d}x.$

We use $\nabla\mathbb{S}$ to denote the third order tensor
$(\partial_{k}S_{ij})_{i,j,k}$ and if $\mathcal{S}_{1}$, $\mathcal{S}_{2}$ are
third order tensors, we let “$\vdots$” denote the triple scalar product

$\mathcal{S}_{1}\,\vdots\,\mathcal{S}_{2}=\sum_{i,j,k=1}^{d}(\mathcal{S}_{1})_{ijk}(\mathcal{S}_{2})_{ijk}.$

We justify the transition to the last line of (3.22) as follows. For ease of
notation, we write $\mathbb{S}=\mathbb{T}^{m}$ and
$\boldsymbol{v}=\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m}$ in the
following calculation. Using integration by parts and the periodic boundary
conditions, we have

$\displaystyle\int_{\Omega}(\nabla\cdot\mathbb{S})\cdot(\nabla\cdot\nabla\boldsymbol{v})\,\mathrm{d}x$
$\displaystyle=\int_{\Omega}\frac{\partial S_{ij}}{\partial
x_{j}}\frac{\partial^{2}v_{i}}{\partial x_{k}^{2}}$
$\displaystyle=\int_{\Omega}\frac{\partial S_{ij}}{\partial
x_{k}}\frac{\partial^{2}v_{i}}{\partial x_{k}\partial x_{j}}\,\mathrm{d}x$
$\displaystyle=\int_{\Omega}\frac{\partial S_{ij}}{\partial
x_{k}}\frac{\partial}{\partial x_{k}}\Big{(}\frac{1}{2}\Big{(}\frac{\partial
v_{i}}{\partial x_{j}}+\frac{\partial v_{j}}{\partial
x_{i}}\Big{)}\Big{)}\,\mathrm{d}x$
$\displaystyle=\int_{\Omega}\nabla\mathbb{S}\,\vdots\,\nabla\boldsymbol{\varepsilon}(\boldsymbol{v})\,\mathrm{d}x.$

Manipulating (3.22) in an almost identical way as in the calculation that
resulted (3.10), we deduce that

$\displaystyle\sup_{t\in[0,T]}\|\nabla\boldsymbol{u}^{m}(t)\|_{2}^{2}+\sup_{t\in[0,T]}\|\nabla\dot{\boldsymbol{u}}^{m}(t)\|_{2}^{2}+\int_{Q}\frac{|\nabla\mathbb{T}^{m}|^{2}}{(1+|\mathbb{T}^{m}|)^{1+a}}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\quad\quad\leq
C(a,\alpha,T)\Big{(}\|\nabla\boldsymbol{u}_{0}\|_{2}^{2}+\|\nabla\boldsymbol{v}_{0}\|_{2}^{2}+\|\nabla\boldsymbol{f}\|_{L^{2}(Q)}^{2}\Big{)},$

where $C$ is a constant that is independent of $n$ and $m$. Thus (3.20) holds.

For (3.21), we recall that the weak solution
$(\boldsymbol{u}^{m},\mathbb{T}^{m})$ can be extended to the larger interval
$(0,T+\epsilon)$ for some $\epsilon=\epsilon_{m,n}>0$. We define the undivided
difference quotient with increment $h$ in the time variable by

$\Delta^{h}_{t}g(t,x)=g(t+h,x)-g(t,x),$

for any function $g$. Provided that $h>0$ is sufficiently small, we deduce
that $\Delta^{h}_{t}\boldsymbol{u}^{m}$, $\Delta^{h}_{t}\mathbb{T}^{m}$ and
$\Delta^{h}_{t}\boldsymbol{f}$ are well-defined on $(0,T]$. From now on, we
will assume that $h>0$ is small enough so that this is true.

From (3.3) we have that

$\int_{\Omega}\Delta^{h}_{t}\ddot{\boldsymbol{u}}^{m}(t)\cdot\boldsymbol{v}+\Delta^{h}_{t}\mathbb{T}^{m}(t):\boldsymbol{\varepsilon}(\boldsymbol{v})\,\mathrm{d}x=\int_{\Omega}\Delta^{h}_{t}\boldsymbol{f}(t)\cdot\boldsymbol{v}\,\mathrm{d}x,$
(3.23)

for every $\boldsymbol{v}\in V_{m}$ and $t\in(0,T)$. Setting
$\boldsymbol{v}=\Delta^{h}_{t}(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})(t)$
in (3.23), we deduce that

$\displaystyle 0$ $\displaystyle=\int_{\Omega}\bigg{[}\frac{\partial}{\partial
t}\Big{(}\frac{|\Delta^{h}_{t}\dot{\boldsymbol{u}}^{m}|^{2}}{2}+\alpha\Delta^{h}_{t}\dot{\boldsymbol{u}}^{m}\cdot\Delta^{h}_{t}\boldsymbol{u}^{m}\Big{)}-\alpha|\Delta^{h}_{t}\dot{\boldsymbol{u}}^{m}|^{2}+\Delta^{h}_{t}\mathbb{T}^{m}:\Delta^{h}_{t}F_{n}(\mathbb{T}^{m})$
$\displaystyle\quad\quad-\Delta^{h}_{t}\boldsymbol{f}\cdot\Delta^{h}_{t}(\dot{\boldsymbol{u}}^{m}+\alpha\boldsymbol{u}^{m})\bigg{]}\,\mathrm{d}x.$

Integrating over $(0,t)$ for arbitrary $t\in(0,T)$, manipulating in the usual
way and dividing through by $h^{2}$ yields

$\displaystyle\frac{\|\Delta^{h}_{t}\dot{\boldsymbol{u}}^{m}(t)\|_{2}^{2}}{h^{2}}+\int_{0}^{t}\int_{\Omega}\frac{\Delta^{h}_{t}\mathbb{T}^{m}}{h}:\frac{\Delta^{h}_{t}F(\mathbb{T}^{m})}{h}\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\quad\quad\leq
C(a,\alpha,T)\Big{(}\frac{\|\Delta^{h}_{t}\dot{\boldsymbol{u}}^{m}(0)\|_{2}^{2}}{h^{2}}+\frac{\|\Delta^{h}_{t}\boldsymbol{u}^{m}(0)\|_{2}^{2}}{h^{2}}+\frac{\|\Delta^{h}_{t}\boldsymbol{f}\|_{L^{2}(Q)}^{2}}{h^{2}}\Big{)}.$

Using the regularity properties of $(\phi_{i})_{i=1}^{\infty}$ and the
coefficients $\boldsymbol{\beta}^{m}$ of the finite-dimensional solution, we
may use Lebesgue’s dominated convergence theorem when taking the limit as
$h\rightarrow 0+$ to deduce that

$\displaystyle\sup_{t\in[0,T]}\|\ddot{\boldsymbol{u}}^{m}(t)\|_{2}^{2}+\int_{Q}\frac{|\dot{\mathbb{T}}^{m}|^{2}}{(1+|\mathbb{T}^{m}|)^{1+a}}\,\mathrm{d}x\,\mathrm{d}t$
(3.24) $\displaystyle\quad\quad\leq
C(a,\alpha,T)\Big{(}\|\ddot{\boldsymbol{u}}^{m}(0)\|_{2}^{2}+\|\dot{\boldsymbol{u}}^{m}(0)\|_{2}^{2}+\|\dot{\boldsymbol{f}}\|_{L^{2}(Q)}^{2}\Big{)}$
$\displaystyle\quad\quad\leq
C(a,\alpha,T)\Big{(}\|\ddot{\boldsymbol{u}}^{m}(0)\|_{2}^{2}+\|\boldsymbol{v}_{0}\|_{2}^{2}+\|\dot{\boldsymbol{f}}\|_{L^{2}(Q)}^{2}\Big{)}.$

To bound $(\ddot{\boldsymbol{u}}^{m}(0))_{m}$ in $L^{2}(\Omega)^{d}$,
independent of $m$, we note that by the structure of the finite-dimensional
problem the following holds:

$\displaystyle\|\ddot{\boldsymbol{u}}^{m}(0)\|_{2}^{2}$
$\displaystyle=\int_{\Omega}\ddot{\boldsymbol{u}}^{m}(0)\cdot\ddot{\boldsymbol{u}}^{m}(0)\,\mathrm{d}x$
(3.25)
$\displaystyle=\int_{\Omega}-\mathbb{T}^{m}(0):\boldsymbol{\varepsilon}(\ddot{\boldsymbol{u}}^{m}(0))+\boldsymbol{f}(0)\cdot\ddot{\boldsymbol{u}}^{m}(0)\,\mathrm{d}x$
$\displaystyle=\int_{\Omega}\Big{(}\mathrm{div}(F_{n}^{-1}(\boldsymbol{\varepsilon}(P^{m}(\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}))))+\boldsymbol{f}(0)\Big{)}\cdot\ddot{\boldsymbol{u}}^{m}(0)\,\mathrm{d}x$
$\displaystyle\leq\Big{(}\|\mathrm{div}(F_{n}^{-1}(\boldsymbol{\varepsilon}(P^{m}(\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}))))\|_{2}+\|\boldsymbol{f}(0)\|_{2}\Big{)}\|\ddot{\boldsymbol{u}}^{m}(0)\|_{2}.$

To bound the first term on the right-hand side of (3.25), we first note that

$\displaystyle\|\mathrm{div}(F_{n}^{-1}(\boldsymbol{\varepsilon}(P^{m}(\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}))))\|_{2}$
(3.26)
$\displaystyle\quad\quad\leq\|D(F_{n}^{-1})(\boldsymbol{\varepsilon}(P^{m}(\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0})))\|_{\infty}\|D(\boldsymbol{\varepsilon}(P^{m}(\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0})))\|_{2}$
$\displaystyle\quad\quad\leq\|D(F_{n}^{-1})(\boldsymbol{\varepsilon}(P^{m}(\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0})))\|_{\infty}\|\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}\|_{2,2},$

where $F_{n}^{-1}$ is continuously differentiable by Lemma 2.5. By properties
of symmetric, positive definite matrices, we are also able to show that there
exists a constant $C_{a}$ depending only on the parameter $a$ such that

$|D(F_{n}^{-1})(\mathbb{S})|=|(DF_{n})^{-1}(F_{n}^{-1}(\mathbb{S}))|\leq
C_{a}(1+|F_{n}^{-1}(\mathbb{S})|^{a+1}),$ (3.27)

for every $\mathbb{S}\in\mathbb{R}^{d\times d}$. The first equality comes from
use of the inverse function theorem. Using (3.27) and (3.26) in (3.25) yields

$\displaystyle\|\ddot{\boldsymbol{u}}^{m}(0)\|_{2}$
$\displaystyle\leq\|\boldsymbol{f}(0)\|_{2}+\|(DF_{n})^{-1}(\mathbb{T}^{m}(0))\|_{\infty}\|\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}\|_{2,2}$
$\displaystyle\leq\|\boldsymbol{f}(0)\|_{2}+C_{a}(1+f^{-1}(C_{1})^{a+1})\|\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}\|_{2,2}$
$\displaystyle\leq\|\boldsymbol{f}\|_{L^{\infty}(0,T;L^{2}_{*}(\Omega))}+C_{a}(1+f^{-1}(C_{1})^{a+1})\|\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}\|_{2,2},$

for every $m\geq m_{0}$, where $m_{0}$ and $C_{1}$ are the constants from the
proof of Theorem 3.1. Substituting this into (3.24), we deduce that

$\displaystyle\sup_{t\in[0,T]}\|\ddot{\boldsymbol{u}}^{m}(t)\|_{2}^{2}+\int_{Q}\frac{|\dot{\mathbb{T}}^{m}|^{2}}{(1+|\mathbb{T}^{m}|)^{1+a}}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\quad\quad\leq
C(a,\alpha,T)\Big{(}\|\boldsymbol{f}\|_{L^{\infty}(0,T;L^{2}_{*}(\Omega))}+(1+f^{-1}(C_{1})^{a+1})\|\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}\|_{2,2}+\|\boldsymbol{v}_{0}\|_{2}^{2}+\|\dot{\boldsymbol{f}}\|_{L^{2}(Q)}^{2}\Big{)}$
$\displaystyle\quad\quad\leq
C(a,\alpha,T,\boldsymbol{u}_{0},\boldsymbol{v}_{0},\boldsymbol{f}),$

for every $m\geq m_{0}$, where $C$ is a constant that is independent of $m$
and $n$. Thus (3.20) is satisfied and the proof is complete. ∎

##  4\. Existence of a solution to the strain-limiting problem

######  Theorem 4.1.

Let $\alpha>0$ and $a>0$. Assume that $\boldsymbol{u}_{0}$,
$\boldsymbol{v}_{0}\in W^{1,2}_{*}(\Omega)^{d}$ are given such that
$\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}\in W^{k+1,2}_{*}(\Omega)^{d}$ for
some $k>\frac{d}{2}$ with

$\|\boldsymbol{\varepsilon}(\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0})\|_{\infty}\leq
C_{*}<1,$

for a constant $C_{*}\in(0,1)$. Let $\boldsymbol{f}$ be an element of
$W^{1,2}(0,T+\epsilon;L^{2}_{*}(\Omega)^{d})\cap
L^{2}(0,T;W^{1,2}_{*}(\Omega)^{d})$, for an $\epsilon>0$. Then, there exists a
unique weak solution $(\boldsymbol{u},\mathbb{T})$ of the strain-limiting
problem (2.1) in the sense of Definition 2.1.

Furthermore, if $(\boldsymbol{u}^{n},\mathbb{T}^{n})$ denotes the weak
solution of (2.5) for $n\in\mathbb{N}$, the following convergence results
hold:

  * •

$\boldsymbol{u}^{n}\overset{\ast}{\rightharpoonup}\boldsymbol{u}$ weakly-* in
$L^{\infty}(0,T;W^{1,2}_{*}(\Omega)^{d})$;

  * •

$\dot{\boldsymbol{u}}^{n}\overset{\ast}{\rightharpoonup}\dot{\boldsymbol{u}}$
weakly-* in $L^{\infty}(0,T;W^{1,2}_{*}(\Omega)^{d})$;

  * •

$\ddot{\boldsymbol{u}}^{n}\overset{\ast}{\rightharpoonup}\ddot{\boldsymbol{u}}$
weakly-* in $L^{\infty}(0,T;L^{2}_{*}(\Omega)^{d})$;

  * •

$\dot{\boldsymbol{u}}^{n}+\alpha\boldsymbol{u}^{n}\rightharpoonup\dot{\boldsymbol{u}}+\alpha\boldsymbol{u}$
weakly in $L^{p}(0,T;W^{1,p}_{*}(\Omega)^{d})$ for every $p\in[1,\infty)$;

  * •

$\mathbb{T}^{n}\rightarrow\mathbb{T}$ pointwise a.e. in $Q$.

###### Proof.

The proof of uniqueness is almost identical to that of Theorem 3.1 so we only
prove existence here. First, we would like to show that $(\mathbb{T}^{n})_{n}$
converges pointwise a.e. on $Q$. To this end, we use an argument similar to
one in [5] but adapted to the time-dependent problem under consideration here.
We let $(\boldsymbol{u}^{n,m},\mathbb{T}^{n,m})$ denote the solution to the
Galerkin approximation from $V_{m}$ of the regularised problem with the
parameter $n$, and $(\boldsymbol{u}^{n},\mathbb{T}^{n})$ the weak solution of
(2.5). We define sequences $(\mathbb{S}^{n,m})_{m}$ and $(s^{n,m})_{m}$ by

$\displaystyle\mathbb{S}^{n,m}=\frac{\mathbb{T}^{n,m}}{(1+|\mathbb{T}^{n,m}|)^{a+1}}\quad\text{
and }\quad s^{n,m}=\frac{1}{(1+|\mathbb{T}^{n,m}|)^{a+1}}.$

Trivially, we have

$\|\mathbb{S}^{n,m}\|_{L^{\infty}(Q)}+\|s^{n,m}\|_{L^{\infty}(Q)}\leq 2.$

From direct computation, we get

$|\dot{\mathbb{S}}^{n,m}|+|\dot{s}^{n,m}|\leq
C\frac{|\dot{\mathbb{T}}^{n,m}|}{(1+|\mathbb{T}^{n,m}|)^{\frac{1}{2}+\frac{a}{2}}},$

for a constant $C$ depending only on $a$. An analogous result holds for the
spatial derivatives. Using (3.20) and (3.21), it follows that

$\displaystyle\int_{Q}|\dot{\mathbb{S}}^{n,m}|^{2}+|\dot{s}^{n,m}|^{2}+|\nabla\mathbb{S}^{n,m}|^{2}+|\nabla
s^{n,m}|^{2}\,\mathrm{d}x\,\mathrm{d}t\leq C,$

for every $m\geq m_{0}$, where $C$ is a constant that is independent of $m$
and $n$. By weak compactness, for each fixed $n$, the sequence
$(\mathbb{S}^{n,m})_{m}$ converges weakly in
$W^{1,2}(0,T;L^{2}_{\\#}(\Omega)^{d\times d})$ and
$L^{2}(0,T;W^{1,2}_{\\#}(\Omega)^{d\times d})$. However, by the pointwise
convergence of $(\mathbb{T}^{n,m})_{m}$ to $\mathbb{T}^{n}$, it follows that
the limit of $(\mathbb{S}^{n,m})_{m}$ in the above spaces is
$\mathbb{S}^{n}=\frac{\mathbb{T}^{n}}{(1+|\mathbb{T}^{n}|)^{a+1}}$. An
analogous result holds for $(s^{n,m})_{m}$ and
$s^{n}=(1+|\mathbb{T}^{n}|)^{-a-1}$. Furthermore, by weak lower semi-
continuity of the norm we have

$\displaystyle\int_{Q}|\mathbb{S}^{n}|^{2}+|\dot{\mathbb{S}}^{n}|^{2}+|\nabla\mathbb{S}^{n}|^{2}+|s^{n}|^{2}+|\dot{s}^{n}|^{2}+|\nabla
s^{n}|^{2}\,\mathrm{d}x\,\mathrm{d}t\leq C,$

where $C$ is independent of $n$. Applying the Aubin–Lions lemma, we deduce
that $(\mathbb{S}^{n})_{n}$ and $(s^{n})_{n}$ converge strongly in
$L^{2}(0,T;L^{2}_{\\#}(\Omega)^{d\times d})$ and
$L^{2}(0,T;L^{2}_{\\#}(\Omega))$, respectively. Thus, up to a subsequence that
we do not relabel, the sequences converge pointwise a.e. on $Q$. Let us call
the limits $\overline{\mathbb{S}}$ and $\overline{s}$, respectively.

However, by Fatou’s lemma and (3.2), we know that $\overline{s}^{-1-a}$ is an
element of $L^{1}(Q)^{d\times d}$. Thus $\overline{s}>0$ a.e. on $Q$ and we
deduce that $\mathbb{T}^{n}=S^{n}(s^{n})^{-1}$ converges pointwise a.e. on $Q$
as $n\rightarrow\infty$. We denote the limit $\mathbb{T}$. By Fatou’s lemma,
we have $\mathbb{T}\in L^{1}(Q)^{d\times d}$.

Thanks to the bounds stated in (3.2), (3.20) and (3.21), we deduce that the
following convergence results hold:

  * •

$\boldsymbol{u}^{n}\overset{\ast}{\rightharpoonup}\boldsymbol{u}$ weakly-* in
$L^{\infty}(0,T;W^{1,2}_{*}(\Omega)^{d})$;

  * •

$\dot{\boldsymbol{u}}^{n}\overset{\ast}{\rightharpoonup}\dot{\boldsymbol{u}}$
weakly-* in $L^{\infty}(0,T;W^{1,2}_{*}(\Omega)^{d})$;

  * •

$\ddot{\boldsymbol{u}}^{n}\overset{\ast}{\rightharpoonup}\ddot{\boldsymbol{u}}$
weakly-* in $L^{\infty}(0,T;L^{2}_{*}(\Omega)^{d})$;

  * •

$\dot{\boldsymbol{u}}^{n}+\alpha\boldsymbol{u}^{n}{\rightharpoonup}\dot{\boldsymbol{u}}+\alpha\boldsymbol{u}$
weakly in $L^{p}(0,T;W^{1,p}_{*}(\Omega)^{d})$ for every $p\in[1,\infty)$.

The attainment of the initial data (3.1) follows immediately by reasoning in
much the same way as we did in the proof of Theorem 3.1. As a matter of fact,
we have the following stronger result:

$\displaystyle\lim_{t\rightarrow
0+}\Big{(}\|\boldsymbol{u}(t)-\boldsymbol{u}_{0}\|_{1,2}+\|\dot{\boldsymbol{u}}(t)-\boldsymbol{v}_{0}\|_{1,2}\Big{)}=0.$

For (2.3), we note that $(F(\mathbb{T}^{n}))_{n}$ converges pointwise a.e. on
$Q$ to $F(\mathbb{T})$. Also
$(\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}^{n}+\alpha\boldsymbol{u}^{n}))_{n}$
converges weakly in $L^{2}(Q)^{d\times d}$ to
$\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}+\alpha\boldsymbol{u})$. Since
pointwise and weak limits in $L^{2}(Q)$ must coincide, (2.3) is immediate if
we can show that
$(\frac{\mathbb{T}^{n}}{n(1+|\mathbb{T}^{n}|^{1-\frac{1}{n}})})_{n}$ converges
pointwise a.e. in $Q$ to $\mathbf{0}$. However, we have

$\displaystyle\int_{Q}\bigg{|}\frac{\mathbb{T}^{n}}{n(1+|\mathbb{T}^{n}|^{1-\frac{1}{n}})}\bigg{|}^{2}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\leq\int_{Q}\frac{1}{n^{2}}+\frac{|\mathbb{T}^{n}|^{\frac{2}{n}}}{n^{2}}\chi_{\\{|\mathbb{T}^{n}|\geq
1\\}}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\leq\frac{|Q|}{n^{2}}+\frac{1}{n}\int_{Q}\frac{|\mathbb{T}^{n}|^{1+\frac{1}{n}}}{n}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\leq\frac{C}{n},$

where $C$ is a constant that is independent of $n$. Hence the sequence
converges strongly in $L^{2}(Q)^{d\times d}$ (and thus pointwise a.e. in $Q$,
up to subsequence) to $\mathbf{0}$ as $n\rightarrow\infty$ as required.

It remains to show that (2.2) holds. We use similar reasoning to that in [1]
but we repeat all the details for completeness. Let $\tau\in
C^{1}_{c}(\mathbb{R})$, $\boldsymbol{v}\in C^{1}_{*}(\overline{\Omega})^{d}$
and $\chi\in C([0,T])$. Since $\ddot{\boldsymbol{u}}^{n}(t)$ and
$\boldsymbol{f}(t)$ are elements of $L^{2}_{*}(\Omega)^{d}$, we can use any
$\boldsymbol{v}\in W^{n+1}_{\\#}(\Omega)^{d}$ as a test function in (2.6),
rather than only those with integral over $\Omega$ equal to $0$. Thus
$\tau(|\mathbb{T}^{n}|)\boldsymbol{v}\chi$ is a valid test function in (2.6).
We integrate the result over $(0,T)$ to get

$\displaystyle 0$
$\displaystyle=\int_{Q}\ddot{\boldsymbol{u}}^{n}\cdot\chi\boldsymbol{v}\tau(|\mathbb{T}^{n}|)+\mathbb{T}^{n}:\boldsymbol{\varepsilon}(\chi\boldsymbol{v}\tau(|\mathbb{T}^{n}|))-\boldsymbol{f}\cdot(\chi\boldsymbol{v}\tau(|\mathbb{T}^{n}|)\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle=\int_{Q}\ddot{\boldsymbol{u}}^{n}\cdot\chi\boldsymbol{v}\tau(|\mathbb{T}^{n}|)+\tau(|\mathbb{T}^{n}|)\chi\mathbb{T}^{n}:\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{f}\cdot(\chi\boldsymbol{v}\tau(|\mathbb{T}^{n}|)+\chi\mathbb{T}^{n}:\nabla\tau(|\mathbb{T}^{n}|)\otimes\boldsymbol{v}\,\mathrm{d}x\,\mathrm{d}t.$

Using the compactness of the support of $\tau$ and the pointwise convergence
of $(\mathbb{T}^{n})_{n}$, we know that
$\tau(|\mathbb{T}^{n}|)\rightarrow\tau(|\mathbb{T}|)$ and
$\tau(|\mathbb{T}^{n}|)\mathbb{T}^{n}\rightarrow\tau(|\mathbb{T}|)\mathbb{T}$
strongly in $L^{p}(Q)$ for every $p\in[1,\infty)$. Thus we have

$\displaystyle\int_{Q}\ddot{\boldsymbol{u}}\cdot\chi\boldsymbol{v}\tau(|\mathbb{T}|)+\chi\tau(|\mathbb{T}|)\mathbb{T}:\boldsymbol{\varepsilon}(\boldsymbol{v})-\chi\tau(|\mathbb{T}|)\boldsymbol{f}\cdot\boldsymbol{v}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\quad\quad=\lim_{n\rightarrow\infty}\int_{Q}\ddot{\boldsymbol{u}}^{n}\cdot\chi\boldsymbol{v}\tau(|\mathbb{T}^{n}|)+\tau(|\mathbb{T}^{n}|)\chi\mathbb{T}^{n}:\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{f}\cdot(\chi\boldsymbol{v}\tau(|\mathbb{T}^{n}|)\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\quad\quad=-\lim_{n\rightarrow\infty}\int_{Q}\chi\mathbb{T}^{n}:\nabla\tau(|\mathbb{T}^{n}|)\otimes\boldsymbol{v}\,\mathrm{d}x\,\mathrm{d}t.$

We replace $\tau$ by $\tau_{k}\in C^{1}_{c}(\mathbb{R})$ where $\tau_{k}$ is
such that

$\displaystyle\tau_{k}(s)=\begin{cases}1,&\quad|s|\leq k,\\\ 0,&\quad|s|\geq
2k,\end{cases}$

and $|\tau_{k}^{\prime}(s)|\leq\frac{C}{k}$ for a constant $C$ that is
independent of $k$. Letting $k\rightarrow\infty$ and using Lebesgue’s
dominated convergence theorem, we deduce that

$\int_{Q}\ddot{\boldsymbol{u}}\cdot\chi\boldsymbol{v}+\chi\mathbb{T}:\boldsymbol{\varepsilon}(\boldsymbol{v})-\chi\boldsymbol{f}\cdot\boldsymbol{v}\,\mathrm{d}x\,\mathrm{d}t=-\lim_{k\rightarrow\infty}\lim_{n\rightarrow\infty}\int_{Q}\chi\mathbb{T}^{n}:\nabla\tau_{k}(|\mathbb{T}^{n}|)\otimes\boldsymbol{v}\,\mathrm{d}x\,\mathrm{d}t.$
(4.1)

The term inside the limit on the right-hand side can be rewritten as

$\displaystyle\int_{Q}\chi\mathbb{T}^{n}:\nabla\tau_{k}(|\mathbb{T}^{n}|)\otimes\boldsymbol{v}\,\mathrm{d}x\,\mathrm{d}t=\int_{Q}\chi
b(|\mathbb{T}^{n}|)F(\mathbb{T}^{n})_{ij}\frac{\partial}{\partial
x_{j}}\Big{(}\tau_{k}(|\mathbb{T}^{n}|)\Big{)}v_{i}\,\mathrm{d}x\,\mathrm{d}t,$

where $b$ is defined on $[0,\infty)$ by $b(t)=(1+t^{a})^{\frac{1}{a}}$. Now
define $B_{k}$ on $[0,\infty)$ by

$B_{k}(t)=\int_{0}^{t}b(s)\tau_{k}^{\prime}(s)\,\mathrm{d}s.$

With this is mind, we get

$\displaystyle\int_{Q}\chi
b(|\mathbb{T}^{n}|)F(\mathbb{T}^{n})_{ij}\frac{\partial}{\partial
x_{j}}\Big{(}\tau_{k}(|\mathbb{T}^{n}|)\Big{)}v_{i}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\quad\quad=\int_{Q}\chi
b(|\mathbb{T}^{n}|)\tau^{\prime}_{k}(|\mathbb{T}^{n}|)\frac{\partial}{\partial
x_{j}}\Big{(}|\mathbb{T}^{n}|\Big{)}F(\mathbb{T}^{n})_{ij}v_{i}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\quad\quad=\int_{Q}\chi\frac{\partial}{\partial
x_{j}}\Big{(}B_{k}(|\mathbb{T}^{n}|)\Big{)}F(\mathbb{T}^{n})_{ij}v_{i}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\quad\quad=-\int_{Q}\chi
B_{k}(|\mathbb{T}^{n}|)F(\mathbb{T}^{n})_{ij}\frac{\partial v_{i}}{\partial
x_{j}}+\chi B_{k}(|\mathbb{T}^{n}|)\frac{\partial}{\partial
x_{j}}\Big{(}F(\mathbb{T}^{n})_{ij}\Big{)}v_{i}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\quad\quad=-\int_{Q}\chi
B_{k}(|\mathbb{T}^{n}|)F(\mathbb{T}^{n})_{ij}\frac{\partial v_{i}}{\partial
x_{j}}+\chi
B_{k}(|\mathbb{T}^{n}|)\mathcal{A}_{ijpq}(\mathbb{T}^{n})\frac{\partial}{\partial
x_{j}}\Big{(}T^{n}_{pq}\Big{)}\,\mathrm{d}x\,\mathrm{d}t,$

where $\mathcal{A}(\mathbb{T})$ is the fourth-order tensor defined by

$\mathcal{A}_{ijpq}(\mathbb{T})=\frac{\partial}{\partial
T_{pq}}\left(\frac{T_{ij}}{(1+|\mathbb{T}|^{a})^{\frac{1}{a}}}\right).$

We note that we can define an inner product on $\mathbb{R}^{d\times d}$ by

$(\mathbb{S},\mathbf{U})_{\mathcal{A}(\mathbb{T})}=\sum_{i,j,p,q=1}^{d}\mathcal{A}_{ijpq}(\mathbb{T})S_{ij}U_{pq}.$

Using this definition, it follows that

$\displaystyle\Big{|}\int_{Q}\chi\mathbb{T}^{n}:\nabla\tau_{k}(|\mathbb{T}^{n}|)\otimes\boldsymbol{v}\,\mathrm{d}x\,\mathrm{d}t\Big{|}$
$\displaystyle\quad\quad\leq\int_{Q}|\chi
B_{k}(|\mathbb{T}^{n}|)||F(\mathbb{T}^{n})||\nabla\boldsymbol{v}|+\Big{|}\Big{(}\Big{(}\delta_{jl}B_{k}(|\mathbb{T}^{n}|)\frac{v_{i}}{|\boldsymbol{v}|}\Big{)}_{i,j},\chi|\boldsymbol{v}|\partial_{l}\mathbb{T}^{n}\Big{)}_{\mathcal{A}(\mathbb{T}^{n})}\Big{|}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\quad\quad\leq\Big{(}\int_{Q}\Big{(}\Big{(}\delta_{jl}B_{k}(|\mathbb{T}^{n}|)\frac{v_{i}}{|\boldsymbol{v}|}\Big{)}_{i,j},\delta_{jl}B_{k}(|\mathbb{T}^{n}|)\frac{v_{i}}{|\boldsymbol{v}|}\Big{)}_{i,j}\Big{)}_{\mathcal{A}(\mathbb{T}^{n})}\,\mathrm{d}x\,\mathrm{d}t\Big{)}^{\frac{1}{2}}$
$\displaystyle\quad\quad\quad\quad\cdot\Big{(}\int_{Q}(\chi|\boldsymbol{v}|\partial_{l}\mathbb{T}^{n},\chi|\boldsymbol{v}|\partial_{l}\mathbb{T}^{n})_{\mathcal{A}(\mathbb{T}^{n})}\,\mathrm{d}x\,\mathrm{d}t\Big{)}^{\frac{1}{2}}+\int_{Q}|\chi
B_{k}(|\mathbb{T}^{n}|)||F(\mathbb{T}^{n})||\nabla\boldsymbol{v}|\,\mathrm{d}x\,\mathrm{d}t.$

Thanks to the definition of $\mathcal{A}(\mathbb{T})$, we have

$\displaystyle\int_{Q}(\chi|\boldsymbol{v}|\partial_{l}\mathbb{T}^{n},\chi|\boldsymbol{v}|\partial_{l}\mathbb{T}^{n})_{\mathcal{A}(\mathbb{T}^{n})}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\leq\int_{Q}\chi^{2}|\boldsymbol{v}|^{2}\nabla\mathbb{T}^{n}\,\vdots\,\nabla
F(\mathbb{T}^{n})\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\leq\int_{Q}\chi^{2}|\boldsymbol{v}|^{2}\nabla\mathbb{T}^{n}\,\vdots\,\nabla
F_{n}(\mathbb{T}^{n})\,\mathrm{d}x\,\mathrm{d}t$ $\displaystyle\leq
C(\chi,\boldsymbol{v})\int_{Q}\nabla\mathbb{T}^{n}\,\vdots\,\nabla
F_{n}(\mathbb{T}^{n})\,\mathrm{d}x\,\mathrm{d}t$ $\displaystyle\leq C,$

where $C$ is a positive constant that is independent of $n$. Furthermore, we
have

$\displaystyle\int_{Q}\Big{(}\Big{(}\delta_{jl}B_{k}(|\mathbb{T}^{n}|)\frac{v_{i}}{|\boldsymbol{v}|}\Big{)}_{i,j},\delta_{jl}B_{k}(|\mathbb{T}^{n}|)\frac{v_{i}}{|\boldsymbol{v}|}\Big{)}_{i,j}\Big{)}_{\mathcal{A}(\mathbb{T}^{n})}\,\mathrm{d}x\,\mathrm{d}t$
(4.2) $\displaystyle\quad\quad\leq
C(d)\int_{Q}\frac{|B_{k}(|\mathbb{T}^{n}|)|^{2}}{(1+|\mathbb{T}^{n}|^{a})^{\frac{1}{a}}}\,\mathrm{d}x\,\mathrm{d}t,$

where $C(d)$ is a positive constant depending only on $d$. By the choice of
$\tau_{k}$, we have $B_{k}(t)=0$ if $t\leq k$. If $t\geq k$, we have

$\displaystyle\begin{aligned}
|B_{k}(t)|&=\Big{|}\int_{k}^{t}\tau^{\prime}_{k}(s)b(s)\,\mathrm{d}s\Big{|}\\\
&\leq\frac{C}{k}\int_{k}^{2k}b(s)\,\mathrm{d}s\\\ &\leq C(1+k)\\\ &\leq
C(1+t),\end{aligned}$ (4.3)

where $C$ is independent of $k$ and $t$. Furthermore, $B_{k}$ is uniformly
bounded on $[0,\infty)$. Using this and (4.3), taking the limit in (4.2) we
get

$\displaystyle\lim_{k\rightarrow\infty}\lim_{n\rightarrow\infty}\int_{Q}\Big{(}\Big{(}\delta_{jl}B_{k}(|\mathbb{T}^{n}|)\frac{v_{i}}{|\boldsymbol{v}|}\Big{)}_{i,j},\delta_{jl}B_{k}(|\mathbb{T}^{n}|)\frac{v_{i}}{|\boldsymbol{v}|}\Big{)}_{i,j}\Big{)}_{\mathcal{A}(\mathbb{T}^{n})}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\quad\quad\leq\lim_{k\rightarrow\infty}C\int_{Q}\frac{|B_{k}(|\mathbb{T}|)|^{2}}{(1+|\mathbb{T}|^{a})^{\frac{1}{a}}}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\quad\quad\leq\lim_{k\rightarrow\infty}C\int_{\\{|\mathbb{T}|>k\\}}1+|\mathbb{T}|\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\quad\quad=0,$

where the transition to the final line follows from the fact that
$\mathbb{T}\in L^{1}(Q)^{d\times d}$. In particular, we deduce that

$\lim_{k\rightarrow\infty}\lim_{n\rightarrow\infty}\int_{Q}\Big{|}\Big{(}\Big{(}\delta_{jl}B_{k}(|\mathbb{T}^{n}|)\frac{v_{i}}{|\boldsymbol{v}|}\Big{)}_{i,j},\chi|\boldsymbol{v}|\partial_{l}\mathbb{T}^{n}\Big{)}_{\mathcal{A}(\mathbb{T}^{n})}\Big{|}\,\mathrm{d}x\,\mathrm{d}t=0.$
(4.4)

By similar reasoning, we have

$\displaystyle\lim_{k\rightarrow\infty}\lim_{n\rightarrow\infty}\int_{Q}|B_{k}(|\mathbb{T}^{n}|)||F(\mathbb{T}^{n})||\nabla\boldsymbol{v}|\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle=\lim_{k\rightarrow\infty}\int_{Q}|B_{k}(|\mathbb{T}|)||F(\mathbb{T})||\nabla\boldsymbol{v}|\,\mathrm{d}x\,\mathrm{d}t$
(4.5)
$\displaystyle\leq\lim_{k\rightarrow\infty}C(\boldsymbol{v})\int_{\\{|\mathbb{T}|>k\\}}1+|\mathbb{T}|\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle=0.$

Combining (4.4) and (4.5), it follows that

$\displaystyle\lim_{k\rightarrow\infty}\lim_{n\rightarrow\infty}\int_{Q}\chi\mathbb{T}^{n}:\nabla\tau_{k}(|\mathbb{T}^{n}|)\otimes\boldsymbol{v}\,\mathrm{d}x\,\mathrm{d}t=0.$

Returning to (4.1) and noting that $\chi$ is arbitrary, we deduce that

$\int_{\Omega}\ddot{\boldsymbol{u}}(t)\cdot\boldsymbol{v}+\mathbb{T}(t):\boldsymbol{\varepsilon}(\boldsymbol{v})\,\mathrm{d}x=\int_{\Omega}\boldsymbol{f}\cdot\boldsymbol{v}\,\mathrm{d}x,$
(4.6)

for a.e. $t\in(0,T)$ and every $\boldsymbol{v}\in
C^{1}_{*}(\overline{\Omega})^{d}$. Using Lemma 2.3, it follows that (4.6)
holds for every $\boldsymbol{v}\in W^{1,2}_{*}(\Omega)^{d}$ such that
$\boldsymbol{\varepsilon}(\boldsymbol{v})\in L^{\infty}_{\\#}(\Omega)^{d}$.
Hence $(\boldsymbol{u},\mathbb{T})$ is a weak solution of the strain-limiting
problem (2.1) and the proof is complete. ∎

Although we have a full existence result from Theorem 4.1, under the condition
that $a$ is small and the dimension is $3$ we can improve the convergence
result of Theorem 4.1 for $(\mathbb{T}^{n})_{n}$.

######  Theorem 4.2.

Let $\alpha>0$, $a\in(0,\frac{2}{7})$ and $d=3$. Assume that
$\boldsymbol{u}_{0}$, $\boldsymbol{v}_{0}\in W^{1,2}_{*}(\Omega)^{d}$ are such
that $\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}\in
W^{k+1,2}_{*}(\Omega)^{d}$ for some $k>\frac{d}{2}$ with

$\displaystyle\|\boldsymbol{\varepsilon}(\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0})\|_{\infty}\leq
C_{*}<1.$

Suppose that $\boldsymbol{f}\in L^{2}(0,T;W^{1,2}_{*}(\Omega)^{d})$. Let
$(\boldsymbol{u}^{n},\mathbb{T}^{n})$ be the unique weak solution of the
regularised problem (2.5). Then there exists a couple
$(\boldsymbol{u},\mathbb{T})$, the unique weak solution of (2.1), such that

$\mathbb{T}^{n}\rightarrow\mathbb{T}\quad\text{ strongly in
}L^{1}(0,T;L^{1}_{\\#}(\Omega)^{d\times d}).$

###### Proof.

Under the conditions of the theorem, if
$(\boldsymbol{u}^{n,m},\mathbb{T}^{n,m})$ denotes the solution to the Galerkin
approximation from $V_{m}$, then we have

$\sup_{t\in[0,T]}\Big{(}\int_{\Omega}|\mathbb{T}^{n,m}(t)|^{1-a}\,\mathrm{d}x\Big{)}+\int_{Q}\frac{|\nabla\mathbb{T}^{n,m}|^{2}}{(1+|\mathbb{T}^{n,m}|)^{a+1}}\,\mathrm{d}x\,\mathrm{d}t\leq
C,$ (4.7)

for every $m\geq m_{0}$, where $C$ is independent of $n$ and $m$. We no longer
require the indicator function in the first term on the left-hand side because
$a<1$. From now on, we will assume that $m\geq m_{0}$. Using the Sobolev
embedding theorem, for every $p\in(2,6]$ we have

$\displaystyle\int_{0}^{T}\Big{(}\int_{\Omega}|\mathbb{T}^{n,m}|^{\frac{p(1-a)}{2}}\,\mathrm{d}x\Big{)}^{\frac{2}{p}}\,\mathrm{d}t\leq
C(p,a)\Big{(}\int_{Q}\frac{|\nabla\mathbb{T}^{n,m}|^{2}}{(1+|\mathbb{T}^{n,m}|)^{a+1}}\,\mathrm{d}x\,\mathrm{d}t+1\Big{)}.$
(4.8)

We note that we can choose $p$ sufficiently large from the interval $(2,6]$ so
that $p(1-a)/2$ is greater than $1$. We may take $p$ larger if needed in
subsequent calculations, but $p$ will remain bounded by $6$.

Let $q\in(1,\infty)$, to be determined later. Let $q^{\prime}=\frac{q}{q-1}$
denote the Hölder conjugate of $q$. Using Hölder’s inequality, for a.e.
$t\in(0,T)$, we have

$\displaystyle\int_{\Omega}|\mathbb{T}^{n,m}|^{1+\frac{a}{q^{\prime}}}\,\mathrm{d}x$
$\displaystyle\leq\Big{(}\int_{\Omega}|\mathbb{T}^{n,m}|^{1-a}\,\mathrm{d}x\Big{)}^{\frac{1}{q^{\prime}}}\Big{(}\int_{\Omega}|\mathbb{T}^{n,m}|^{1+2a(q-1)}\,\mathrm{d}x\Big{)}^{\frac{1}{q}}$
(4.9) $\displaystyle\leq
C^{\frac{1}{q^{\prime}}}\Big{(}\int_{\Omega}|\mathbb{T}^{n,m}|^{1+2a(q-1)}\,\mathrm{d}x\Big{)}^{\frac{1}{q}},$

where the constant $C$ comes from the first term on the left-hand side of
(4.7). To bound the other factor on the right-hand side of (4.9), we would
like to use (4.8). A pair of simultaneous restrictions on $p$ and $q$ that
enable this are

$\frac{1}{q}\leq\frac{2}{p},$ (4.10)

and

$1+2a(q-1)\leq\frac{p(1-a)}{2}.$ (4.11)

To show that we may find $p\in(2,6]$ and $q>1$ such that the above holds when
$a\in(0,\frac{2}{7}]$, we first set $p=2q$. Then (4.10) automatically holds.
We will now demand that $q\in(1,3]$ because $p\in(2,6]$.

We write $a=\frac{1}{3}-\delta$ for a $\delta\in(0,\frac{1}{3})$. Then (4.11)
reduces to

$\frac{1}{3\delta}+2\leq 3q.$

If $\delta\geq\frac{1}{21}$, then $\frac{1}{3\delta}+2\leq 9$ and $q$ can be
chosen sufficiently large from the interval $(1,3]$ so that

$\frac{1}{3\delta}+2\leq 3q\leq 9.$

Thus (4.11) holds if $\delta\geq\frac{1}{21}$, which is simply equivalent to
$a\leq\frac{2}{7}$. Thus we have that

$\displaystyle\int_{Q}|\mathbb{T}^{n,m}|^{1+\frac{a}{q^{\prime}}}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\leq
C^{\frac{1}{q^{\prime}}}\int_{0}^{T}\Big{(}\int_{\Omega}|\mathbb{T}^{n,m}|^{1+2a(q-1)}\,\mathrm{d}x\Big{)}^{\frac{1}{q}}\,\mathrm{d}t$
$\displaystyle\leq
C\Big{[}\int_{0}^{T}\Big{(}\int_{\Omega}|\mathbb{T}^{n,m}|^{\frac{p(1-a)}{2}}\,\mathrm{d}x\Big{)}^{\frac{1}{q}}\,\mathrm{d}t+1\Big{]}$
$\displaystyle\leq
C\Big{[}\int_{0}^{T}\Big{(}\int_{\Omega}|\mathbb{T}^{n,m}|^{\frac{p(1-a)}{2}}\,\mathrm{d}x\Big{)}^{\frac{2}{p}}\,\mathrm{d}t+1\Big{]}$
$\displaystyle\leq
C(a,\alpha,\boldsymbol{u}_{0},\boldsymbol{v}_{0},\boldsymbol{f},\Omega,T),$

where $C$ is a positive constant that is independent of $n$ and $m$. We note
that the choices of $p$ and $q$ were made independent of $n$ and $m$. Fix
$\delta=\frac{a}{q^{\prime}}>0$. Then, for a constant $C$ that is independent
of $n$ and $m$, we get

$\int_{Q}|\mathbb{T}^{n,m}|^{1+\delta}\,\mathrm{d}x\,\mathrm{d}t\leq C.$

Using the convergence results in Theorems 3.1, we deduce that

$\int_{Q}|\mathbb{T}^{n}|^{1+\delta}\,\mathrm{d}x\,\mathrm{d}t.$

It follows that $\mathbb{T}^{n}\rightharpoonup\mathbb{T}$ weakly in
$L^{1+\delta}(Q)^{d\times d}$. Reasoning as we did in the proof of Theorem 3.1
and noting that
$(\frac{\mathbb{T}^{n}}{n(1+|\mathbb{T}^{n}|^{1-\frac{1}{n}})})_{n}$ converges
to $\mathbf{0}$ strongly in $L^{2}(Q)^{d\times d}$ as $n\rightarrow\infty$, we
deduce that $\mathbb{T}^{n}\rightarrow\mathbb{T}$ strongly in
$L^{1}(0,T;L^{1}_{\\#}(\Omega)^{d\times d})$ as $n\rightarrow\infty$. Using
these two facts, we may repeat the reasoning from the proof of Theorem 3.1 to
deduce that $(\boldsymbol{u},\mathbb{T})$ is a weak solution of (2.1) under
these weaker conditions. ∎

######  Remark.

_We remark that the results here hold under lower regularity requirements on
the data. Indeed, rather than $\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}\in
W^{k+1,2}_{*}(\Omega)^{d}$ for some $k>\frac{d}{2}$, we need only assume that
$\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}\in W^{2,2}_{*}(\Omega)^{d}$.
However, in this case we need an extra level of approximation. This is
required to deal with the convergence of
$(\boldsymbol{\varepsilon}(P^{m}(\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0})))_{m}$.
We consider the following regularised problem:_

$\displaystyle\boldsymbol{u}_{tt}$
$\displaystyle=\mathrm{div}(\mathbb{T})+\boldsymbol{f},$ $\displaystyle\text{
in }Q,$ (4.12)
$\displaystyle\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}+\alpha\boldsymbol{u})$
$\displaystyle=\frac{\mathbb{T}}{(1+|\mathbb{T}|^{a})^{\frac{1}{a}}}+\frac{\mathbb{T}}{n}$
$\displaystyle\text{ in }Q,$ $\displaystyle\boldsymbol{u}(0,\cdot)$
$\displaystyle=\boldsymbol{u}_{0}^{k},$ $\displaystyle\text{ in }\Omega,$
$\displaystyle\boldsymbol{u}_{t}(0,\cdot)$
$\displaystyle=\boldsymbol{v}_{0}^{k},$ $\displaystyle\text{ in }\Omega.$

_The functions $\boldsymbol{u}_{0}^{k}$, $\boldsymbol{v}_{0}^{k}$ are chosen
so that they are elements of $C^{\infty}_{*}(\overline{\Omega})^{d}$ such that
if $\boldsymbol{u}_{0}$, $\boldsymbol{v}_{0}\in L^{2}_{*}(\Omega)^{d}$ with
$\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}\in W^{2,2}_{*}(\Omega)^{d}\cap
W^{1,\infty}_{*}(\Omega)^{d}$, then_

  * •

$\boldsymbol{u}_{0}^{k}\rightarrow\boldsymbol{u}_{0}$ strongly in
$L^{2}_{*}(\Omega)^{d}$;

  * •

$\boldsymbol{v}_{0}^{k}\rightarrow\boldsymbol{v}_{0}$ strongly in
$L^{2}_{*}(\Omega)^{d}$;

  * •

$\boldsymbol{v}_{0}^{k}+\alpha\boldsymbol{u}_{0}^{k}\rightarrow\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0}$
strongly in $W^{2,2}_{*}(\Omega)^{d}$;

  * •

$\boldsymbol{\varepsilon}(\boldsymbol{v}_{0}^{k}+\alpha\boldsymbol{u}_{0}^{k})\overset{\ast}{\rightharpoonup}\boldsymbol{\varepsilon}(\boldsymbol{v}_{0}+\alpha\boldsymbol{u}_{0})$
weakly-* in $L^{\infty}(\Omega)^{d\times d}$.

_We deduce that such an approximating sequence exists by applying the proof of
Lemma 4.1 in[5] to $\boldsymbol{u}_{0}$ and $\boldsymbol{v}_{0}$. We deduce
the existence of a weak solution to (4.12) by using a finite-dimensional
approximation as in Theorem 3.1. Then we take the limit as
$k\rightarrow\infty$ for each fixed $n$. Rather than considering
$f^{-1}(C_{1})$ as we did in the proof of Theorem 3.1, we use
$f_{n}^{-1}(\|\boldsymbol{\varepsilon}(\boldsymbol{v}_{0}^{k}+\alpha\boldsymbol{u}_{0}^{k})\|_{\infty})$
in the bounds on $(\boldsymbol{u}^{n,k},\mathbb{T}^{n,k})$. This is uniformly
bounded in $k$, so we may deduce appropriate convergence results for each
fixed $n$. However, since we only have weak-* convergence of
$(\boldsymbol{\varepsilon}(\boldsymbol{v}_{0}^{k}+\alpha\boldsymbol{u}_{0}^{k}))_{k}$,
we must obtain bounds of the type (3.15) and (3.21) for
$(\boldsymbol{u}^{n},\mathbb{T}^{n})$ from scratch. Since we have chosen the
regularisation $n^{-1}\mathbb{T}$, we have the following bound:_

$\displaystyle\int_{Q}\Big{(}|\dot{\mathbb{T}}^{n,k}|^{2}+|\nabla\mathbb{T}^{n,k}|^{2}+|\mathbb{T}^{n,k}|^{2}\Big{)}\,\mathrm{d}x\,\mathrm{d}t\leq
C(n),$

_where $C$ is a constant that is independent of $k$ and
$(\boldsymbol{u}^{n,k},\mathbb{T}^{n,k})$ is the weak solution of (4.12). Thus
if $(\boldsymbol{u}^{n},\mathbb{T}^{n})$ is the solution of (4.12) but with
$\boldsymbol{u}_{0}^{k}$, $\boldsymbol{v}_{0}^{k}$ replaced by
$\boldsymbol{u}_{0}$, $\boldsymbol{v}_{0}$, then we must have
$\mathbb{T}^{n}\in C([0,T];L^{2}(\Omega)^{d\times d})$. Using this fact, we
can derive estimates similarly as for the finite-dimensional solution in the
proof of Theorem 3.1. Since we are no longer using approximate initial data,
we may use $f^{-1}(C_{*})$ in the bounds that are obtained._ $\diamond$

######  Remark.

_We note that we may consider more general constitutive relations than the
function $F$. In particular, the above proofs can be adapted to a relation of
the form_

$\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}+\alpha\boldsymbol{u})=f(|\mathbb{T}|)\mathbb{T},$

_where $f:[0,\infty)\rightarrow[0,\infty)$ is a continuously differentiable
function and there exist positive constants $C_{1}$, $C_{2}$, $\kappa$ and $a$
such that_

$\frac{C_{1}s^{2}}{\kappa+s}\leq f(s)s^{2}\leq C_{2}s$ (4.13)

_and_

$\frac{\mathrm{d}}{\mathrm{d}s}\Big{(}f(s)s\Big{)}\geq\frac{C_{1}}{(\kappa+s)^{a+1}},$
(4.14)

_for every $s\geq 0$. Similarly, in the spirit of [3], we could also consider
relations of the form_

$\displaystyle\boldsymbol{\varepsilon}(\boldsymbol{u}_{t}+\alpha\boldsymbol{u})=\lambda(|\operatorname{tr}\mathbb{T}|)(\operatorname{tr}\mathbb{T})\mathbb{I}+\mu(|\mathbb{T}^{\delta}|)\mathbb{T}^{\delta},$

_where $\mu$ and $\lambda$ satisfy conditions similar to (4.13) and (4.14)._
$\diamond$

##  5\. Conclusion and open problems

We have proved the existence of a unique global-in-time large-data weak
solution to a class of initial-boundary-value problems that model the motion
of implicitly constituted strain-limiting viscoelastic solids, without any
restrictions on the parameters in the model or the number of space dimensions.
Related studies have only been pursued to date in the quasi-static case or in
one space dimension. The results presented here complement those for elastic
solids in the static case discussed in [5]. Admittedly, our focus on the
periodic setting here is a notable simplification. The extension of the
results presented in this paper to problems on general domains, with Dirichlet
and mixed Dirichlet–Neumann boundary conditions will be reported in a
forthcoming paper.

An interesting further challenge is the proof of the existence of weak
solutions to the purely elastic unsteady strain-limiting model. Although the
analysis presented in this paper suggests that such an existence result might
be within reach, it has to be borne in mind that the viscoelastic term has a
regularising effect, whose absence in the purely elastic case is likely to
lead to additional technical complications.

Finally, a related class of problems that has been discussed in the literature
are stress-rate type models. For these the constitutive relation is of the
form

$\boldsymbol{\varepsilon}(\boldsymbol{u})=F(\mathbb{T})-\gamma\mathbb{T}_{t},$
(5.1)

where $F$ is an appropriately chosen function and $\gamma>0$. In the case when
$F$ is bounded, it would be interesting to investigate the existence and
uniqueness of solutions to (2.1), where the constitutive relation is now
replaced by (5.1). Rudimentary comparisons of solutions to the stress-rate
problem and the strain-rate problem are performed in [10], but results
concerning the existence of global weak solutions to multidimensional stress-
rate type models are still lacking.

## Acknowledgements

M. Bulíček’s work is supported by the project 20-11027X financed by GAČR. M.
Bulíček is a member of the Nečas Center for Mathematical Modeling. V. Patel is
supported by the UK Engineering and Physical Sciences Research Council
[EP/L015811/1]. Y. Şengül is partially supported by the Scientific and
Technological Research Council of Turkey (TÜBITAK) under the grant 116F093.

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