# Existence and uniqueness for a viscoelastic Kelvin-Voigt model with
nonconvex stored energy

Konstantinos Koumatos
Department of Mathematics
University of Sussex
Pevensey 2 Building
Falmer, Brighton, BN1 9QH, UK<EMAIL_ADDRESS>,  Corrado Lattanzio
Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica
Università degli Studi dell’Aquila
Via Vetoio
I-67010 Coppito (L’Aquila) AQ
Italy<EMAIL_ADDRESS>,  Stefano Spirito
Department of Information Engineering, Computer Science and Mathematics
University of L’Aquila
Via Vetoio
I-67100 Coppito (L’Aquila), Italy<EMAIL_ADDRESS>and  Athanasios
E. Tzavaras
Computer, Electrical and Mathematical Science and Engineering Division
King Abdullah University of Science and Technology (KAUST)
Thuwal 23955-6900, Saudi Arabia<EMAIL_ADDRESS>
###### Abstract.

We consider nonlinear viscoelastic materials of Kelvin-Voigt type with stored
energies satisfying an Andrews-Ball condition, allowing for non convexity in a
compact set. Existence of weak solutions with deformation gradients in $H^{1}$
is established for energies of any superquadratic growth. In two space
dimensions, weak solutions notably turn out to be unique in this class.
Conservation of energy for weak solutions in two and three dimensions, as well
as global regularity for smooth initial data in two dimensions are established
under additional mild restrictions on the growth of the stored energy.

##  1\. Introduction

We consider the Cauchy problem for viscoelastic materials of the strain-rate
type in Lagrangean coordinates

$\partial_{tt}y-\mathop{\mathrm{div}}(S(\nabla y))-\Delta\partial_{t}y=0\,,$
(1.1)

where $y:(0,T)\times\mathbb{T}^{d}\to\mathbb{R}^{d}$, where $\mathbb{T}^{d}$
is the $d$-dimensional torus, $d=2,3$ and $T>0$ is arbitrary but finite, and
with initial data

$\displaystyle y|_{t=0}=y_{0},\quad$
$\displaystyle\partial_{t}y|_{t=0}=v_{0}\,.$

This second-order system describes motions of a viscoelastic material of
strain-rate type with the Piola-Kirchhoff stress tensor

$T_{R}=S(F)+\partial_{t}F\,,\quad S(F)=\frac{\partial W}{\partial F}(F)\,.$
(1.2)

It is assumed throughout that the elastic part of the stress is given as the
gradient of a strain-energy function, $S=DW$, while the viscous part of the
stress is linear, leading to (1.2). Such constitutive relations fit under the
general framework of viscoelasticity of strain-rate type, and specifically
into the class of Kelvin-Voigt type materials; we refer to Antman [2, Ch 10,
Secs 10-11] and Lakes [19, Ch 2] for general information on viscoelasticity
and the specific terminologies.

In this work, we focus on the model (1.2) with linear dependence on the
strain-rate and study the effect of nonlinear elastic behavior on various
aspects of the existence theory. Note that the constitutive relation (1.2)
violates frame-indifference and we refer the reader to the end of the
Introduction for a discussion.

The system (1.1) is expressed as a hyperbolic-parabolic system,

$\displaystyle\partial_{t}v-\mathop{\mathrm{div}}(S(F))-\Delta\,v$
$\displaystyle=0$ (1.3) $\displaystyle\partial_{t}F-\nabla\,v$
$\displaystyle=0$ $\displaystyle\mathop{\mathrm{curl}}\,F$
$\displaystyle=0\,,$

for the functions $v:(0,T)\times\mathbb{T}^{d}\to\mathbb{R}^{d}$,
$F:(0,T)\times\mathbb{T}^{d}\to M^{d\times d}$, where $v=\partial_{t}y$ stands
for the velocity and $F=\nabla y$ for the deformation gradient. System (1.3)
is supplemented with periodic boundary conditions and initial data at
$\\{t=0\\}\times\mathbb{T}^{d}$ given by

$\displaystyle v|_{t=0}=v_{0},$ (1.4) $\displaystyle F|_{t=0}=F_{0}=\nabla
y_{0}\,.$

The constraint $\mathop{\mathrm{curl}}\,F=0$ is an involution of the dynamics
and it is propagated from the initial data $F_{0}=\nabla y_{0}$ by the
kinematic compatibility equation $F_{t}=\nabla v$.

A number of studies concern the issue of existence for viscoelastic models of
strain-rate type both for linear viscosity [22, 1, 14] as well as nonlinear
strain-rate dependence [11, 9, 31, 5]; the interested reader may find a
thorough review of previous literature in [14]. Our objective is to study the
effect of nonlinear elastic response on the existence, uniqueness, and
regularity theory of (1.3). We limit attention to (1.2) in several space
dimensions, and in that sense the most relevant studies are the existence
theory of Friesecke-Dolzmann [14], and the works concerning convex stored
energies of Friedman-Nečas [11, Sec 6] and Engler [12, 13]. There are
available studies for models with nonlinear strain-rate dependence
$T_{R}=S(F,F_{t})$, under monotonicity hypotheses for $S(F,\cdot)$ and
globally Lipschitz hypotheses that restrict to linear growth in $F$, see [11,
9, 31].

In our analysis, we allow for nonconvex strain energy functions $W(F)$ with
growth conditions of polynomial type. Instead of convexity we adopt the
Andrews-Ball condition [1, 14] imposing monotonicity at infinity; namely,
$W(F)$ satisfies for some $R>0$

$(S(F_{1})-S(F_{2}),F_{1}-F_{2})\geq 0,\qquad\forall\;|F_{1}|,|F_{2}|\geq
R\,,$ (AB)

where $(F,G)={\rm tr}\,FG^{T}$ denotes the inner product on
$\mathbb{R}^{d\times d}$. On occasion a strengthened version of (AB) is
employed, requesting that there exist constants $C>0$, $R>0$ such that

$(S(F_{1})-S(F_{2}),F_{1}-F_{2})\geq
C(|F_{1}|^{p-2}+|F_{2}|^{p-2})|F_{1}-F_{2}|^{2},\qquad\forall\;|F_{1}|,|F_{2}|\geq
R\,.$ (AB′)

Condition (AB) amounts to requiring the strain energy to be a semiconvex
function while, similarly, (AB′) implies a strenghtened variant of
semiconvexity; see [14] and Lemma 2.1 below.

A-priori bounds for the system (1.3) are provided by the energy identity

$\int\tfrac{1}{2}|v|^{2}+W(F)\,dx+\int_{0}^{t}\int|\nabla
v|^{2}\,dx=\int\tfrac{1}{2}|v_{0}|^{2}+W(F_{0})\,dx\,,$

but clearly they do not suffice to yield an existence theory. Friesecke and
Dolzmann[14] in a penetrating study provide global existence of weak solutions
for (1.1) under the Andrews-Ball condition (AB). Key to their analysis is the
property of propagation of compactness for the deformation gradient, [14, Prop
3.1]; the latter is complemented with a variational time-discretization scheme
to achieve existence of weak solutions in energy space. We provide here an
example indicating that the system does not enjoy any compactification
mechanism, by showing that oscillations in the initial strain can persist in
the dynamics. The example concerns one dimensional models for phase
transitions,

$\displaystyle u_{t}$ $\displaystyle=v_{x}$ (1.5) $\displaystyle v_{t}$
$\displaystyle=\sigma(u)_{x}+v_{xx}\,,$

and combines the universal class of uniform shearing solutions with the
observation of Pego [22] and Hoff [16] that (1.5) admits solutions with
discontinuities in the strain and strain-rate. Combining these ingredients,
exact oscillatiory solutions are constructed for (1.5) with non-monotone
stress-strain laws. Our example corroborates an example of Friesecke and
Dolzmann at the level of approximating solutions for (1.1), [14, Example 2.1],
and indicates that persistent oscillations in the strain is a feature of the
viscoelasticity system.

The property of propagation of compactness can also be seen as propagation of
regularity, which in [14] is the propagation of the $L^{2}$-modulus of
continuity. In the current work, a crucial observation is that in fact
$H^{1}$-regularity of the initial deformation gradient is also propagated.
Precisely, the following energy bound holds

$\frac{d}{dt}\int|v-\tfrac{1}{2}\mathop{\mathrm{div}}F|^{2}+\tfrac{1}{2}|\nabla
F|^{2}+2W(F)dx+\int\Big{(}D^{2}\tilde{W}:(\nabla F,\nabla F)+|\nabla
v|^{2}\Big{)}dx\leq K\int|\nabla F|^{2}dx\,,$ (1.6)

for $\tilde{W}(F):=W(F)+K\frac{|F|^{2}}{2}$, which is convex because of the
condition (AB). Then, for data $(v_{0},F_{0})\in L^{2}(\mathbb{T}^{d})\times
H^{1}(\mathbb{T}^{d})$, we establish an existence theory for weak solutions
$(v,F)$ of (1.3)-(1.4) such that

$\displaystyle v\in L^{\infty}(0,T;L^{2}(\mathbb{T}^{d}))\cap
L^{2}(0,T;H^{1}(\mathbb{T}^{d}))\,,$ (1.7) $\displaystyle F\in
L^{\infty}(0,T;H^{1}(\mathbb{T}^{d})\cap L^{p}(\mathbb{T}^{d}))\,;$

see Theorem 2.3.

The propagation of $H^{1}$-regularity of the deformation gradient has several
important implications. First, we prove that solutions satisfying (1.7)
conserve the energy in $d=2$ and, if the non-linearity $S$ does not grow too
fast, in $d=3$ as well, see Theorem 2.5. Second, a main consequence of the
$H^{1}$-estimation are the uniqueness and regularity properties for solutions
in two dimensions. Uniqueness results for (1.3) or related systems are
provided in [14, 9, 31], always based on global Lipschitz assumptions for the
stress function. Instead, thanks to the propagation property of
$H^{1}$-regularity, we prove uniqueness of weak solutions $(v,F)$ of class
(1.7) for dimension $d=2$ assuming some mild restrictions on the growth of
$D^{2}W$, and thus substantially improving upon the global Lipschitz
assumptions on $S(F)$. Indeed, uniqueness is established in Theorem 2.7 under
hypothesis (AB) with the growth of $W$ restricted to $2\leq p<4$, or in
Theorem 2.8 under the strengthened condition (AB′) for any growth $p\geq 2$.
Moreover, assuming that $W(F)\sim|F|^{p}$ for $|F|$ large and using classical
energy estimates, based on the Gagliardo-Nirenberg and Brezis-Gallouet
inequalities, we establish smoothness of solutions under conditions (AB) and
$2\leq p\leq 5$ or (AB′) and $2\leq p\leq 6$; see Theorem 2.10.

These results highlight a striking analogy with the $2D$ incompressible Euler
equations, at least when the non-linearity $S(F)$ does not grow too fast.
Indeed, for $2D$ Euler, existence holds in the class of solutions with
vorticity in $L^{q}$ with $q\leq\infty$, in analogy to Theorem 2.3. Moreover,
existence and uniqueness holds for solutions with bounded vorticity, in
analogy to Theorems 2.7 and 2.8, and finally for both systems there is global
regularity given smooth initial data, in analogy to Theorem 2.10. We also
remark that besides the conceptual similarities just mentioned, the uniqueness
and global regularity proofs exploit critical (logarithmic) estimates similar
to Yudovich [33] and Beale, Kato and Majda [4].

Some related material is listed in two appendices. The key estimate (1.6),
used in the existence proof, can be understood by considering the problem of
transfer of dissipation. It was noted by DiPerna [10] that the estimate on
velocity gradients obtained by energy dissipation in viscoelasticity models
can be transferred to strain gradients; a similar observation holds for
relaxation systems [32]. Such estimates are here established for several space
dimensions, see the modulated energy estimate (A.4), and is what lies behind
the existence Theorem 2.3. Another class of approximations used for elasticity
is the so-called diffusion-dispersion approximations that occur when viscosity
is intermixed with higher-order gradient theories. An observation of Slemrod
[27] indicates that diffusion-dispersion approximations of one-dimensional
elasticity can be transformed to parabolic approximations. We show that this
is also the case in several space dimensions.

### Frame Indifference

We conclude this introduction with a short discussion on frame-indifference.
The constitutive theory of (isothermal) viscoelasticity of strain-rate type
asserts that the Piola-Kirchhoff stress $T_{R}=T_{R}(F,\dot{F})$ depends on
the deformation gradient $F$ and the strain rate $\dot{F}$. The stress is
nominally decomposed as

$T_{R}=T_{el}(F)+T_{v}(F,\dot{F})\,,$ (1.8)

where $\dot{\,}$ denotes time derivative, $T_{el}$ is the elastic part of the
stress, $T_{v}$ the viscous part, and $T_{v}(F,0)=0$. Compatibility with the
Clausius-Duhem inequality dictates,

$T_{el}(F)=\frac{\partial W}{\partial F}(F)\,,\quad
T_{v}(F,\dot{F}):\dot{F}\geq 0\,,$ (1.9)

that the elastic part is induced by a strain energy (or stored energy)
function $W(F)$ while the viscous part is dissipative. The terminology Kelvin-
Voigt model originates from the interpretation of viscoelastic behaviour
through systems of spring and dashpot mechanisms, Lakes [19, Ch 2], and
Kelvin-Voigt specifically refers to additive decomposition of the elastic and
viscous stresses.

In the continuum mechanics literature the principle of material frame
indifference is imposed on constitutive theories, which posits that two
observers moving with respect to each other with a Euclidean transformation,

$x^{*}=Q(t)x+d(t),\,\quad t^{*}=t+a\,,$

with $Q(t)$ an arbitrary proper orthogonal tensor, $Q^{T}Q=QQ^{T}=I$ and $\det
Q=1$, should observe the same constitutive relations in their respective
frames of reference [29]. When applied to the constitutive theory of
viscoelasticity of strain-rate type, material frame indifference implies that
the Piola-Kirchhoff stress tensor $T_{R}$ must satisfy
$T_{R}(F,\dot{F})=FG(C,\dot{C})$ for some symmetric tensor-valued function $G$
of $C=F^{T}F$, e.g. Antman [2, Ch 10, Secs 10-11], [24]. For (1.8)-(1.9) this
suggests

$T_{el}(F)=2F\,\frac{\partial\varphi}{\partial C}(C)\,,\quad
T_{v}(F,\dot{F})=FG_{v}(C,\dot{C})\,,$

where $\varphi(C)=W(\sqrt{C})$ and $G_{v}^{T}(C,\dot{C})=G_{v}(C,\dot{C})$.

For reasons related to analysis it is beneficial to impose a strengthened
version of (1.9) on the energy dissipation, assuming that for some $\gamma>0$

$T_{v}(F,\dot{F}):\dot{F}\geq\gamma|\dot{F}|^{2}\,.$ (1.10)

We outline an argument of Şengül [24] showing that (1.10) is incompatible with
frame indifference. Indeed, using the symmetry of $G_{v}$,

$T_{v}(F,\dot{F}):\dot{F}=FG_{v}(C,\dot{C}):\dot{F}=G_{v}(C,\dot{C}):F^{T}\dot{F}=\frac{1}{2}G_{v}(C,\dot{C}):\dot{C}$

and (1.10) becomes

$\frac{1}{2}G_{v}(C,\dot{C}):\dot{C}\geq\gamma|\dot{F}|^{2}\,.$

The latter is violated by the example $F(t)=e^{t\Omega}$ with
$\Omega^{T}=-\Omega$ a constant skew-symmetric matrix, for which $C(t)=I$,
$\dot{F}=\Omega e^{t\Omega}$ and $|\dot{F}(t)|=|\Omega|\neq 0$. This argument
indicates that the Kelvin-Voigt model in Lagrangian coordinates
$T_{R}(F,\dot{F})=S(F)+\dot{F}$ violates frame-indifference, as also do any
reasonable linear viscoelastic models as first noted by Antman [3].

The principle of material frame indifference is strictly speaking a hypothesis
imposed on the form of constitutive relations of continuum physics. It
reflects the intuition that stress results from deformations originating from
an unstrained state and it should not be affected by the superposition of an
arbitrary rigid body motion. Detractors argue that constitutive relations
reflect microscopic dynamics determined via Newton’s laws which are only
Galilean invariant. According to this viewpoint, frame indifference might be
too restrictive and should be replaced by invariance under the Galilean group
or the extended Galilean group. (The latter only requires that rotations $Q$
are time independent, and in particular this invariance admits the model
(1.2).) The reader is referred to the very interesting review by Speziale [28]
(and references therein) which tests the validity of frame indifference of
constitutive relations, in a context where fluctuations result from kinetic
modeling or from models for turbulence.

### Organization of the paper

The paper is organized as follows. Section 2 lists the hypotheses on the
stored energy, discusses their interrelations, and contains the statements of
the main results. Section 3 contains auxiliary results that are needed later
in the proofs. Section 4 contains the proof of the existence of weak
solutions, which is based on a Galerkin approximation and a compactness
argument for the constructed Galerkin iterates using the Aubin-Lions-Simon
Lemma. Section 5 contains the uniqueness proof in two space dimensions, while
Section 6 the proof of global regularity. Then, in Section 7 we provide the
construction of examples of one-dimensional oscillating solutions in the
nonlinear case with phase transitions, and then in the linear case. Appendix A
contains the energy estimate indicating the transfer of dissipation, while
Appendix B shows how the multi-dimensional diffusion-dispersion approximation
of the elasticity system can be transformed to a parabolic approximation of
conservation laws. Finally, in Appendix C we discuss the assumptions on the
stored energy, in particular the ones regarding its growth at infinity.

##  2\. Assumptions on the stored energy and main results

In this section we fix the the assumptions on the stored energy $W$ and we
present all our main results.

###  2.1. Hypotheses on the stored energy

We always assume throughout the paper that the stored energy $W$ satisfies for
some $p\geq 2$ the following hypotheses:

  * (A1)

$W\in C^{2}(\mathbb{R}^{d\times d};\mathbb{R})$.

  * (A2)

There exists $c>0$ such that

$c(|F|^{p}-1)\leq W(F).$

  * (A3)

There exists $C>0$ such that

$\displaystyle|W(F)|\leq C(1+|F|^{p});\,$ $\displaystyle|DW(F)|\leq
C(1+|F|^{p-1}).$

The following assumption characterizes the dissipative nature of the stored
energy:

  * (A4)

There exists constant $K>0$ such that

$(S(F_{1})-S(F_{2}),F_{1}-F_{2})\geq-K|F_{1}-F_{2}|^{2}.$

In some cases, as mentioned in the Introduction, we consider the following
more restrictive assumption:

  * (A4′)

There exists $C>0$ and $K>0$ such that for any $F_{1}$ and $F_{2}$ we have
that

$(S(F_{1})-S(F_{2}),F_{1}-F_{2})\geq\big{(}C(|F_{1}|^{p-2}+|F_{2}|^{p-2})-K\big{)}|F_{1}-F_{2}|^{2}.$
(2.1)

In the next lemma we prove that (A4) and (A4′) are direct consequences of (AB)
and (AB′), respectively. They imply that upon adding a quadratic function of
$F$ the stored energy becomes convex. The latter condition is called
semiconvexity, it has geometric implications on the graph of $W$, and is used
extensively in the theory of Hamilton-Jacobi equations.

######  Lemma 2.1.

Assume that $W$ satisfies (A1)-(A3). Then:

  1. (1)

If $W$ satisfies (AB), then $W$ satisfies (A4) and

$\tilde{W}(F)=W(F)+\frac{K}{2}|F|^{2}$

is convex. In particular, $D^{2}\tilde{W}\geq 0$.

  2. (2)

If $W$ satisfies (AB′), then $W$ satisfies (A4′) and

$\tilde{W}(F)=:W(F)+\frac{K}{2}|F|^{2}$

satisfies

$D^{2}\tilde{W}(F)\geq c|F|^{p-2}\mathbb{I}.$

###### Proof.

Note that (1) is proved in [14, Lemma 1.1]. Regarding (2), and following the
ideas in the proof of [14, Lemma 1.1], let $R>0$ be the radius appearing in
(AB′) and consider the balls $B_{R}(0)$ and $B_{2R}(0)$. If $F_{1}$,
$F_{2}\notin B_{R}(0)$, then the stronger assertion (AB′) holds, whereas if
$F_{1}$, $F_{2}\in B_{2R}(0)$ we find that

$\displaystyle\left|(S(F_{1})-S(F_{2}),F_{1}-F_{2})-C(|F_{1}|^{p-2}+|F_{2}|^{p-2})|F_{1}-F_{2}|^{2}\right|$
$\displaystyle\leq K_{1}|F_{1}-F_{2}|^{2}+2C(2R)^{p-2}|F_{1}-F_{2}|^{2}$
$\displaystyle\leq K|F_{1}-F_{2}|^{2},$

where $K_{1}>0$ is the Lipschitz constant of $S$ on the ball $B_{2R}(0)$.
Hence (2.1) holds and we are left to consider the case $F_{1}\notin
B_{2R}(0)$, $F_{2}\in B_{R}(0)$, since the remaining case amounts to switching
the roles of $F_{1}$ and $F_{2}$. Define

$F(\lambda)=F_{1}-\lambda\frac{F_{1}-F_{2}}{|F_{1}-F_{2}|}.$

Since $F(0)=|F_{1}|>2R$ and $|F(|F_{1}-F_{2}|)|=|F_{2}|<R$, there exists
$\lambda_{0}\in(0,|F_{1}-F_{2}|)$ such that

$|F(\lambda_{0})|=\frac{3R}{2}.$

Write $F_{0}:=F(\lambda_{0})$ and note that

$\frac{F_{1}-F_{0}}{\lambda_{0}}=\frac{F_{1}-F_{2}}{|F_{1}-F_{2}|}.$ (2.2)

We may thus compute that

$\displaystyle(S(F_{1})-S(F_{2}),F_{1}-F_{2})$
$\displaystyle=(S(F_{1})-S(F_{0}),F_{1}-F_{2})+(S(F_{0})-S(F_{2}),F_{1}-F_{2})$
$\displaystyle=(S(F_{1})-S(F_{0}),F_{1}-F_{0})\frac{|F_{1}-F_{2}|}{\lambda_{0}}+(S(F_{0})-S(F_{2}),F_{1}-F_{2})$
$\displaystyle\geq
C(|F_{1}|^{p-2}+|F_{0}|^{p-2})|F_{1}-F_{0}|^{2}\frac{|F_{1}-F_{2}|}{\lambda_{0}}$
$\displaystyle\qquad-K_{1}|F_{0}-F_{2}||F_{1}-F_{2}|,$ (2.3)

where the last inequality follows from (AB′) and the fact that $F_{1}$,
$F_{0}\notin B_{R}(0)$, whereas $F_{0}$, $F_{2}\in B_{2R}(0)$ and $K_{1}$
denotes again the Lipschitz constant of $S$ on $B_{2R}(0)$. Next, note that

$\displaystyle|F_{0}-F_{2}|$
$\displaystyle\leq\frac{5R}{2}=\frac{5R}{2}\frac{1}{|F_{1}-F_{2}|}|F_{1}-F_{2}|\leq\frac{5}{2}|F_{1}-F_{2}|,$
(2.4)

since $|F_{1}-F_{2}|>R$. Moreover, using (2.2), we find that

$\displaystyle\frac{|F_{1}-F_{0}|^{2}}{\lambda_{0}}|F_{1}-F_{2}|$
$\displaystyle=|F_{1}-F_{0}||F_{1}-F_{2}|=|F_{1}-F_{2}|^{2}\frac{|F_{1}-F_{0}|}{|F_{1}-F_{2}|}$
(2.5)

and we aim to prove that

$\frac{|F_{1}-F_{0}|}{|F_{1}-F_{2}|}\geq L>0.$ (2.6)

Indeed, as $F_{0}$, $F_{2}\in B_{2R}$,

$\lim_{|F_{1}|\to\infty}\frac{|F_{1}-F_{0}|}{|F_{1}-F_{2}|}=1$

and thus there exists $M>0$ such that whenever $|F_{1}|>M$ it holds that

$\frac{|F_{1}-F_{0}|}{|F_{1}-F_{2}|}>1-\varepsilon.$

On the other hand, when $|F_{1}|<M$, we find that

$\frac{|F_{1}-F_{0}|}{|F_{1}-F_{2}|}\geq\frac{R/2}{M+R}$

proving (2.6). Using (2.4)-(2.6), and noting that $|F_{0}|>|F_{2}|$ we find
that (2.3) implies

$(S(F_{1})-S(F_{2}),F_{1}-F_{2})\geq
LC(|F_{1}|^{p-2}+|F_{2}|^{p-2})|F_{1}-F_{2}|^{2}-\frac{5}{2}K_{1}|F_{1}-F_{2}|^{2}$

which is (2.1). To conclude the proof of Lemma 2.1, we are left to show that
(2.1) implies $D^{2}\tilde{W}(F)\geq c|F|^{p-2}\mathbb{I}$. Indeed, apply
(2.1) to $F_{1}=F$, $F_{2}=F+tH$ where $t>0$ and $H$ a matrix to obtain

$\Big{(}\frac{S(F+tH)-S(F)}{t},H\Big{)}\geq\big{[}c(|F+tH|^{p-2}+|F|^{p-2})-K\big{]}|H|^{2}.$

Taking $t\to 0$ gives

$\Big{(}D^{2}W(F)H,H\Big{)}\geq\big{(}2c|F|^{p-2}-K\big{)}|H|^{2}$

and implies the result. ∎

###  2.2. Main results

We start by providing the definition of a weak solution to the Cauchy problem
(1.3)-(1.4).

######  Definition 2.2.

Let $v_{0}\in L^{2}(\mathbb{T}^{d})$ and $F_{0}\in L^{p}(\mathbb{T}^{d})$. The
pair

$(v,F)\in L^{\infty}(0,T;L^{2}(\mathbb{T}^{d}))\cap
L^{2}(0,T;H^{1}(\mathbb{T}^{d}))\times L^{\infty}(0,T;L^{p}(\mathbb{T}^{d}))$
(2.7)

is a weak solution to the initial value problem (1.3) if the following are
satisfied:

  * •

For any $\phi,\psi\in C^{\infty}_{c}([0,T);C^{\infty}(\mathbb{T}^{d}))$

$\displaystyle\iint\,v\psi_{t}+S(F)\nabla\psi+\nabla\,v\nabla\psi\,dxdt-\int\,v_{0}(x)\psi(0,x)\,dx=0$
(2.8)
$\displaystyle\iint\,F\phi_{t}+\nabla\,v\phi\,dxdt-\int\,F_{0}(x)\phi(0,x)\,dx=0$
$\displaystyle F(t,x)=\nabla\,y(t,x)\mbox{ for a.e.
}(t,x)\in(0,T)\times\mathbb{T}^{d}$

  * •

The energy inequality holds for a.e. $t\in(0,T)$

$\int\,\frac{|v|^{2}}{2}+W(F)\,dx+\int_{0}^{t}\int\,|\nabla\,v|^{2}\,dxds\leq\int\,\frac{|v_{0}|^{2}}{2}+W(F_{0})\,dx.$
(2.9)

The first main result of the paper is the following:

######  Theorem 2.3.

Assume that $W$ satisfies (A1)-(A4) for some $p\geq 2$. Then, for any initial
data $(v_{0},F_{0})$ with $F_{0}=\nabla\,y_{0}$ a.e. in $\mathbb{T}^{d}$ and

$(v_{0},F_{0})\in L^{2}(\mathbb{T}^{d})\times L^{p}(\mathbb{T}^{d}),$ (2.10)

there exists at least one weak solution in the sense of Definition 2.2.
Moreover,

  1. (1)

if $F_{0}\in H^{1}(\mathbb{T}^{d})$ then $F$ satisfies

$F\in L^{\infty}(0,T;H^{1}(\mathbb{T}^{d})).$

  2. (2)

if $F_{0}\in H^{1}(\mathbb{T}^{d})$ and (A4′) holds, then $F$ satisfies

$\displaystyle F\in L^{\infty}(0,T;H^{1}(\mathbb{T}^{d}))$ (2.11)
$\displaystyle\nabla|F|^{\frac{p}{2}}\in L^{2}(0,T;L^{2}(\mathbb{T}^{d})).$

######  Remark 2.4.

We note that the global existence for initial data $(v_{0},F_{0})\in
L^{2}(\mathbb{T}^{d})\times L^{p}(\mathbb{T}^{d})$ was already proved in [14].
Instead and to the best of our knowledge, $\it(1)$ and $\it(2)$ are new
contributions.

The next result concerns the conservation of energy.

######  Theorem 2.5.

Assume that $W$ satisfies (A1)-(A3). For $v_{0}\in L^{2}(\mathbb{T}^{d})$ and
$F_{0}\in(H^{1}\cap L^{p})(\mathbb{T}^{d})$ let $(v,F)$ be a weak solution in
the sense of Definition 2.2. Then, if $F\in
L^{\infty}(0,T;H^{1}(\mathbb{T}^{d}))$, with $d=2$ and $p\geq 2$ or with $d=3$
and $2\leq p\leq 4$, the weak solution $(v,F)$ verifies for any $t\in[0,T]$

$\int\,\frac{|v|^{2}}{2}+W(F)\,dx+\int_{0}^{t}\int\,|\nabla\,v|^{2}\,dxds=\int\,\frac{|v_{0}|^{2}}{2}+W(F_{0})\,dx.$

######  Remark 2.6.

The regularity hypothesis $F\in L^{\infty}(0,T;H^{1}(\mathbb{T}^{d}))$
suffices to guarantee conservation of energy. On the other hand, Hypothesis
(A4) is needed in Theorem 2.3 for the existence of weak solutions in that
class. It is interesting to note that assuming (A4′) instead of (A4) does not
seem to improve the range of $p$ in the three-dimensional case.

The next results concern the uniqueness of weak solutions in the two-
dimensional case. We additionally assume that the second derivatives of $W$
have polynomial growth. Precisely, we assume that there exists $C>0$ such that

$\displaystyle|D^{2}W(F)|\leq C(1+|F|^{s}),\mbox{ with }s\geq p-2.$ (2.12)

In particular, the lower bound on $s$ in (2.12) is needed for consistency with
the coercivity assumption (A2); see Lemma C.2. The first result concerning the
uniqueness deals with assumption $(A4)$.

######  Theorem 2.7.

Assume that $d=2$ and let $\gamma\in[0,2]$. Assume that $2\leq p<2+\gamma$ and
$W$ satisfies (A1)-(A4) and (2.12) with $p-2\leq s\leq p-\gamma$. Then, for
any initial data $(v_{0},F_{0})$ with $F_{0}=\nabla\,y_{0}$ for a.e. in
$\mathbb{T}^{2}$ and

$(v_{0},F_{0})\in L^{2}(\mathbb{T}^{2})\times H^{1}(\mathbb{T}^{2}),$

there exists a unique weak solution such that

$\displaystyle F\in L^{\infty}(0,T;H^{1}(\mathbb{T}^{2})).$ (2.13)

A more satisfactory result can be obtained by invoking assumption (A4′). The
statement of uniqueness in this case follows in Theorem 2.8.

######  Theorem 2.8.

Assume that $p\geq 2$ and $d=2$ and $W$ satisfies (A1)-(A4′) and (2.12) for
some $p-2\leq s<p$. Then, for any initial data $(v_{0},F_{0})$ with
$F_{0}=\nabla\,y_{0}$ a.e. in $\mathbb{T}^{2}$ and

$(v_{0},F_{0})\in L^{2}(\mathbb{T}^{2})\times H^{1}(\mathbb{T}^{2}),$

there exists a unique weak solution such that

$\displaystyle F\in L^{\infty}(0,T;H^{1}(\mathbb{T}^{2}))$ (2.14)
$\displaystyle\nabla|F|^{\frac{p}{2}}\in L^{2}(0,T;L^{2}(\mathbb{T}^{2})).$

######  Remark 2.9.

As in Remark 2.6, the proofs of uniqueness in Theorems 2.7 and 2.8 use only
the regularity in (2.13) and (2.14). On the other hand, in view of Theorem
2.3, (A4) (resp. (A4′)) guarantee the existence of weak solutions of
regularity class (2.13) (resp. (2.14)).

The last main result in the present article concerns global regularity. We
assume that $W(F)$ behaves like $|F|^{p}$ for $|F|$ large. More precisely, we
assume that $W\in C^{4}(\mathbb{R}^{2\times 2};\mathbb{R})$ and there exists
$C>0$ such that

$\displaystyle|D^{2}W(F)|\leq C(1+|F|^{p-2}),\,\,|D^{3}W(F)|\leq
C(1+|F|^{p-3}),\,\,\,3<p,$ (2.15) $\displaystyle|D^{2}W(F)|\leq
C(1+|F|^{p-2}),\,\,|D^{3}W(F)|\leq C,\,\,\,2<p\leq 3,$
$\displaystyle|D^{2}W(F)|\leq C,\,\,|D^{3}W(F)|\leq C,\,\,\,p=2,$

and that also the fourth derivatives have a polynomial growth without a
precise order.

######  Theorem 2.10.

Assume $d=2$ that $W\in C^{4}(\mathbb{R}^{2\times 2};\mathbb{R})$ satisfies
(A2), (A3), and (2.15). Then, for any initial data $(v_{0},F_{0})$ with
$F_{0}=\nabla\,y_{0}$ a.e. in $\mathbb{T}^{2}$ and

$(v_{0},F_{0})\in H^{3}(\mathbb{T}^{2})\times H^{3}(\mathbb{T}^{2}),$

the unique weak solution of (1.3)-(1.4) constructed in Theorem 2.7 satisfies

$(v,F)\in L^{\infty}(0,T;H^{3}(\mathbb{T}^{2}))\times
L^{\infty}(0,T;H^{3}(\mathbb{T}^{2})),$

provided $W$ satisfies (A4) and $2\leq p\leq 5$ or $W$ satisfies (A4′) and
$2\leq p\leq 6$.

##  3\. Some technical lemmas

In this section we recall some classical technical lemmas which play a crucial
role in the proofs of our main results. The first lemma contains some
classical interpolation inequality. First, we recall the Gagliardo-Nirenberg-
Sobolev interpolation inequality, the critical Sobolev embedding inequality,
and we note that the precise constants in (1) and (2) are well-known and can
be deduced for example from the result in [17], see also [21, Theorem 8.5]
for(2). Moreover, in the lemma below we also include a version of the Brezis-
Gallouet inequality which can easily be deduced from the original statement in
[8]. The notation $\|\cdot\|_{p}$ , $p\in[1,\infty]$, denotes the norm of the
classical Lebesgue spaces $L^{p}(\mathbb{T}^{d})$.

######  Lemma 3.1.

Let $f\in H^{1}(\mathbb{T}^{2})$, then the following inequalities hold:

  1. (1)

For any $r>1$ and $q>1$, there exists a constant $C_{r}>0$ such that

$\|f\|_{\frac{rq}{q-1}}\leq
C_{r}\|f\|_{r}^{1-\frac{1}{q}}\|f\|_{H^{1}}^{\frac{1}{q}}.$

In particular, for $r=2$ and $q=2$ it holds that

$\|f\|_{4}\leq C\|f\|_{2}^{\frac{1}{2}}\|f\|_{H^{1}}^{\frac{1}{2}}.$

  2. (2)

There exists a constant $C>0$ such that for any $q>2$

$\|f\|_{q}\leq C\sqrt{q}\|f\|_{H^{1}}.$

  3. (3)

If, in addition, $f\in H^{2}(\mathbb{T}^{2})$, then there exists a constant
$C>0$ such that

$\|f\|_{\infty}\leq
C(1+\|f\|_{H^{1}})\left(\log\left(e+\|f\|_{H^{2}}\right)\right)^{\frac{1}{2}}.$

Next, we recall the classical result concerning the maximal $L^{p}$ regularity
for the heat equation on the torus. The result on the entire space can be
found in [18, Chapter IV, Section 3]. Here and based on this result, we
provide a short proof for the torus, which is also classical but difficult to
find in the literature.

######  Lemma 3.2.

For a smooth function $G$, let $u$ be a smooth solution of the following
initial value problem:

$\displaystyle\partial_{t}u-\Delta u$
$\displaystyle=\mathop{\mathrm{div}}G\mbox{ in }(0,T)\times\mathbb{T}^{d}$
(3.1) $\displaystyle u|_{t=0}$ $\displaystyle=0\mbox{ on
}\\{t=0\\}\times\mathbb{T}^{d}.$

Then, for any $t\in(0,T)$ and for any $1<r<\infty$

$\int_{0}^{t}\|\nabla u\|_{r}^{r}\,ds\leq C\int_{0}^{t}\|G\|_{r}^{r}\,ds,$
(3.2)

where $C=C(r,d)$.

###### Proof.

We first recall that if $\tilde{G}$ and $\tilde{f}$ are in
$C^{\infty}_{c}((0,T)\times\mathbb{R}^{d})$ and $\tilde{u}\in
C^{\infty}((0,T)\times\mathbb{R}^{d}))$ satisfy

$\displaystyle\partial_{t}\tilde{u}-\Delta\tilde{u}$
$\displaystyle=\mathop{\mathrm{div}}\tilde{G}+\tilde{f}$
$\displaystyle\tilde{u}|_{t=0}$ $\displaystyle=0$

then it follows that

$\int_{0}^{T}\|\nabla\tilde{u}\|_{L^{r}(\mathbb{R}^{d})}^{r}\,dt\leq
C(r,d)\left(\int_{0}^{T}\|\tilde{G}\|_{L^{r}(\mathbb{R}^{d})}^{r}\,dt+\int_{0}^{T}\|\tilde{f}\|_{L^{r}(\mathbb{R}^{d})}^{r}\,dt\right),$
(3.3)

see [18, Chapter IV, Section 3]. Next, given $u$, $G$ and $f$ as in (3.1), we
extend the functions periodically on the whole $\mathbb{R}^{d}$ and we denote
by $Q^{N}$ the cube parallel to the axes, centered at zero and side-length
$2N$. Let $\phi_{N}\in C^{\infty}_{c}(\mathbb{R}^{d})$ be a cut-off function
such that $\phi_{N}=1$ on $Q^{N}$ and $\mbox{supp}\phi_{N}\subset Q^{N+1}$.
Note that we can also assume $|\nabla\phi_{N}|\leq C$ and
$|\nabla^{2}\phi_{N}|\leq C$. Let $u_{N}:=u\phi_{N}\in
C^{\infty}_{c}([0,T]\times\mathbb{R}^{d})$ and note that $u_{N}$ solves

$\displaystyle\partial_{t}u_{N}-\Delta u_{N}$
$\displaystyle=\mathop{\mathrm{div}}G_{N}+f_{N}$ $\displaystyle u_{N}|_{t=0}$
$\displaystyle=0$

where $G_{N}:=G\phi_{N}$ and $f_{N}:=-2\nabla
u\cdot\nabla\phi_{N}-u\Delta\phi_{N}-\nabla\phi_{N}G$. Since
$\mbox{supp}\nabla\phi_{N}$ and $\mbox{supp}\nabla^{2}\phi_{N}$ are contained
in $Q^{N+1}\setminus Q^{N}$, we infer that

$\displaystyle\int_{0}^{T}\|f_{N}\|^{r}_{L^{r}(\mathbb{R}^{d})}\,dt$
$\displaystyle=\int_{0}^{T}\|f_{N}\|^{r}_{L^{r}(Q^{N+1})}\,dt$
$\displaystyle\leq
C(r)((2N+1)^{d}-(2N)^{d})\left(\int_{0}^{T}\|G\|_{r}^{r}+\|\nabla
u\|_{r}^{r}+\|u\|_{r}^{r}\,dt\right)$
$\displaystyle:=C(r,d,u)((2N+1)^{d}-(2N)^{d})$

since there are $((2N+1)^{d}-(2N)^{d})$ cubes with side-length one in
$Q^{N+1}\setminus Q^{N}$. Therefore, by using (3.3), the definition of
$\phi_{N}$ and the periodicity of $u$ we have

$\displaystyle(2N)^{d}\int_{0}^{T}\|\nabla u\|_{r}^{r}\,dt$
$\displaystyle=\int_{0}^{T}\|\nabla
u\|_{L^{r}(Q^{N})}^{r}\,dt\leq\int_{0}^{T}\|\nabla
u_{N}\|_{L^{r}(\mathbb{R}^{d})}^{r}\,dt$ $\displaystyle\leq
C(r,d)\int_{0}^{T}\|G_{N}\|_{L^{r}(\mathbb{R}^{d})}^{r}\,dt+C(r,d)\int_{0}^{T}\|f_{N}\|_{L^{r}(\mathbb{R}^{d})}^{r}\,dt$
$\displaystyle\leq
C(r,d)(2N+2)^{d}\int_{0}^{T}\|G\|_{r}^{r}\,dt+C(r,d,u)((2N+1)^{d}-(2N)^{d}).$

Now (3.2) follows by sending $N\to\infty$. ∎

We conclude this section by recalling the following classical generalization
of the Grönwall lemma.

######  Lemma 3.3.

Let $q>1$ and $C>0$. Suppose $F\in L^{1}(0,T)$ and that $0\leq y\in C([0,T])$
satisfies the inequality

$y(t)\leq Cq\int_{0}^{t}F(s)y(s)^{1-\frac{1}{q}}\,ds$ (3.4)

with $y(0)=0$. Then,

$y(t)\leq\left(C\int_{0}^{t}\,F(s)\,ds\right)^{q}.$ (3.5)

###### Proof.

Define

$R(t):=\int_{0}^{t}F(s)y(s)^{1-\frac{1}{q}}\,ds.$

By using (3.4) and the fact that $R$ is absolutely continuous it follows that

$\frac{d}{dt}R(t)\leq F(t)(CqR(t))^{1-\frac{1}{q}}$

and then

$\frac{d}{dt}R(t)^{\frac{1}{q}}\leq\frac{1}{(Cq)^{\frac{1}{q}}}CF(t).$

Integrating in time and using the definition of $R$ and (3.4), the inequality
(3.5) follows. ∎

##  4\. Global existence of weak solutions and conservation of energy

In this section we give the proof of Theorem 2.3. We start by defining the
approximation system for (1.3)-(1.4) which is given by a simple Fourier based
Galerkin scheme, we refer to [15] for the Galerkin scheme in a general bounded
domain $\Omega\subset\mathbb{R}^{d}$ for the system (1.1).

###  4.1. The Galerkin Approximation

Let $N\in\mathbb{N}$ and consider the following initial value problem

$\displaystyle\partial_{t}\,v^{N}-\mathop{\mathrm{div}}\,P^{N}(S(F^{N}))-\Delta\,v^{N}$
$\displaystyle=0$ (4.1) $\displaystyle\partial_{t}\,F^{N}-\nabla\,v^{N}$
$\displaystyle=0$ $\displaystyle\mathop{\mathrm{curl}}\,F^{N}$
$\displaystyle=0,$

with initial data

$\displaystyle v^{N}|_{t=0}=v^{N}_{0}$ (4.2) $\displaystyle
F^{N}|_{t=0}=F^{N}_{0}=\nabla\,y^{N}_{0}$

where the unknowns $(v^{N},F^{N})$ are defined for $k\in\mathbb{Z}^{d}$ as

$\displaystyle v^{N}=\sum_{|k|\leq N}v^{N}_{k}(t)e^{ix\cdot k}$ $\displaystyle
F^{N}=\sum_{|k|\leq N}F^{N}_{k}(t)e^{ix\cdot k}=\sum_{|k|\leq
N}i\,y^{N}_{k}(t)\otimes\,ke^{ix\cdot k}$

with $v^{N}_{k}=\bar{v}^{N}_{-k}$, $F^{N}_{k}=\bar{F}^{N}_{-k}$ and
$P^{N}:L^{2}(\mathbb{T}^{d})\to\,P^{N}(L^{2}(\mathbb{T}^{d}))$ is the
projection operator from $L^{2}(\mathbb{T}^{d})$ to the finite dimensional
subspace $P^{N}(L^{2}(\mathbb{T}^{d}))$ given by the formula

$\displaystyle P^{N}:L^{2}(\mathbb{T}^{d})$ $\displaystyle\rightarrow$
$\displaystyle P^{N}(L^{2}(\mathbb{T}^{d}))$ $\displaystyle v$
$\displaystyle\rightarrow$ $\displaystyle\sum_{|k|\leq N}v_{k}(t)e^{ix\cdot
k}$

where $v_{k}(t)=\frac{1}{|\mathbb{T}^{d}|}\int v(t,x)e^{-ix\cdot k}\,dx$ are
the Fourier coefficients of $v$ satisfying $v_{k}=\bar{v}_{-k}$.

We have the following Proposition.

######  Proposition 4.1.

Let $(v_{0},F_{0})\in L^{2}(\mathbb{T}^{d})\times L^{p}(\mathbb{T}^{d})$ and
define $(v_{0}^{N},F_{0}^{N})=(P^{N}v_{0},P^{N}F_{0})$. Then, for any
$N\in\mathbb{N}$ the system (4.1)-(4.2) admits a unique smooth solution.

###### Proof.

We first note that (4.1)-(4.2) is equivalent to the following system of
ordinary differential equations:

$\displaystyle\frac{d}{dt}v^{N}_{k}(t)-iS_{k}^{N}(t)k+|k|^{2}v^{N}_{k}(t)=0$
$\displaystyle\frac{d}{dt}F^{N}_{k}(t)-iv^{N}_{k}(t)\otimes k=0$

where

$S_{k}^{N}(t)=\int S\left(\sum_{|k|\leq N}F^{N}_{k}(t)e^{ik\cdot
x}\right)e^{ik\cdot x}\,dx.$

By Assumptions (A1), (A3) and the Cauchy-Lipschitz theorem we deduce that for
any $N\in\mathbb{N}$ there exists $T^{N}>0$ and $(v^{N},F^{N})$ solution of
(4.1)-(4.2) of the form

$\displaystyle v^{N}=\sum_{|k|\leq N}v^{N}_{k}(t)e^{ix\cdot k}\quad$
$\displaystyle F^{N}=\sum_{|k|\leq N}F^{N}_{k}(t)e^{ix\cdot k}$

and smooth on $(0,T^{N})\times\mathbb{T}^{d}$. Next, we prove that $T^{N}=T$.
By multiplying the first equation of (4.1) by $v^{N}$, after integrating in
space we get

$\frac{d}{dt}\int\frac{|v^{N}|^{2}}{2}\,dx+\int
P^{N}(S(F^{N}):\nabla\,v^{N}\,dx+\int|\nabla v^{N}|^{2}\,dx=0.$ (4.3)

Then, by multiplying the second equation of (4.1) by $P^{N}(S(F^{N}))$ and
using that $DW=S$ we have, after summing with (4.3), that

$\frac{d}{dt}\int\left(\frac{|v^{N}|^{2}}{2}+W(F^{N})\right)\,dx+\int|\nabla
v^{N}|^{2}\,dx=0.$ (4.4)

By integrating in time, using (A2), (A3), the fact that $(v_{0},F_{0})\in
L^{2}(\mathbb{T}^{d})\times L^{p}(\mathbb{T}^{d})$, and Parseval’s identity we
infer that

$\sup_{t}\left(\sum_{|k|\leq
N}|v^{N}_{k}(t)|^{2}+|F^{N}_{k}(t)|^{2}\right)\leq C$

with $C>0$ not depending on $N$. Therefore, by standard O.D.E. theory it
follows that $T^{N}=T$. ∎

###  4.2. Proof of Theorem 2.3

The following lemma contains the main estimate of the paper, resulting in the
propagation of $H^{1}$ regularity. We stress that the inequality (4.5) in
Lemma 4.2 is a mere rephrasing of the a priori  estimate (1.6). We also remark
that constants depending on fixed parameters, e.g. the domain, dimension or
fixed exponents, will be suppressed from appearing in inequalities. Instead,
we adopt the notation $\lesssim$, meaning that all the terms on the right-hand
side of $\lesssim$ are multiplied by constants depending only on the data
except the ones where the constant is explicit.

######  Lemma 4.2.

Assume that $W$ satisfies (A1)-(A4) for some $p\geq 2$ and let $(v^{N},F^{N})$
be the solution of the Galerkin approximation constructed in Proposition 4.1.
Then, for any $t\in(0,T)$

$\displaystyle\int|\nabla\,F^{N}(t,x)|^{2}\,dx$
$\displaystyle\lesssim\int_{0}^{t}\int|\nabla
v^{N}|^{2}\,dx\,ds+\sup_{t}\int|v^{N}(t,x)|^{2}\,dx$ (4.5)
$\displaystyle\qquad+\int_{0}^{t}\int|\nabla
F^{N}|^{2}\,dx\,ds+\int|v^{N}_{0}|^{2}\,dx+\int|\nabla F^{N}_{0}|^{2}\,dx.$

In addition, if $W$ satisfies (A4′), it holds that

$\displaystyle\int|\nabla\,F^{N}(t,x)|^{2}\,dx$
$\displaystyle+\int_{0}^{t}\int|\nabla|F^{N}|^{\frac{p}{2}}|^{2}\,dx\,ds\lesssim\int_{0}^{t}\int|\nabla
v^{N}|^{2}\,dx\,ds+\sup_{t}\int|v^{N}(t,x)|^{2}\,dx$ (4.6)
$\displaystyle\qquad+\int_{0}^{t}\int|\nabla
F^{N}|^{2}\,dx\,ds+\int|v^{N}_{0}|^{2}\,dx+\int|\nabla F^{N}_{0}|^{2}\,dx.$

###### Proof.

By multiplying the first equation of (4.1) by $-\mathop{\mathrm{div}}F^{N}$
and integrating by parts we get

$-\int\,v^{N}_{t}\mathop{\mathrm{div}}F^{N}\,dx+\int\mathop{\mathrm{div}}(P^{N}(S(F^{N}))\mathop{\mathrm{div}}F^{N}\,dx+\int\Delta\,v^{N}\mathop{\mathrm{div}}\,F^{N}\,dx=0.$

By using the second equation and the fact that $F^{N}$ is a gradient, after
standard manipulations, it follows that

$\frac{d}{dt}\int\left(\frac{1}{2}|\nabla\,F^{N}|^{2}-v^{N}\mathop{\mathrm{div}}F^{N}\right)\,dx+\int\mathop{\mathrm{div}}(P^{N}(S(F^{N}))\mathop{\mathrm{div}}F^{N}\,dx=\int|\nabla\,v^{N}|^{2}\,dx$

and by using the definition of $\tilde{W}$ we have

$\displaystyle\frac{d}{dt}\int\left(\frac{1}{2}|\nabla\,F^{N}|^{2}-v^{N}\mathop{\mathrm{div}}F^{N}\right)\,dx$
$\displaystyle+\int\mathop{\mathrm{div}}(P^{N}(\tilde{S}(F^{N}))\mathop{\mathrm{div}}F^{N}\,dx=\int|\nabla\,v^{N}|^{2}\,dx$
(4.7) $\displaystyle+K\int|\mathop{\mathrm{div}}\,F^{N}|^{2}\,dx.$

Regarding the third term on the left-hand side, by integrating by parts twice
and using that $\mathop{\mathrm{curl}}\,F^{N}=0$ we infer that

$\int\mathop{\mathrm{div}}(P^{N}(\tilde{S}(F^{N}))\mathop{\mathrm{div}}F^{N}\,dx=\int\frac{\partial^{2}\tilde{W}(F^{N})}{\partial
F_{i\beta}^{N}\partial
F_{j\gamma}^{N}}\partial_{\alpha}F_{i\beta}^{N}\partial_{\alpha}F_{j\gamma}^{N}\,dx\geq
0$ (4.8)

where $\tilde{S}=D\tilde{W}$ and we have used that $\tilde{W}$ is convex and
therefore $D^{2}\tilde{W}\geq 0$. Then, after integrating in time (4.7) and
using (4.8) we find that

$\displaystyle\int|\nabla\,F^{N}|^{2}\,dx$
$\displaystyle\lesssim\int_{0}^{t}\int|\nabla\,v^{N}|^{2}\,dx\,ds+K\int_{0}^{t}\int|\mathop{\mathrm{div}}\,F^{N}|^{2}\,dx\,ds$
$\displaystyle\quad+\int|v^{N}||\mathop{\mathrm{div}}F^{N}|\,dx+\int|v^{N}_{0}||\mathop{\mathrm{div}}F^{N}_{0}|\,dx+\int|\nabla\,F^{N}_{0}|^{2}\,dx.$

Therefore, (4.5) follows by a simple application of Young’s inequality.
Regarding (4.6) it is enough to note that, owing to assumption (A4′), Lemma
2.1 implies that (4.8) reads as

$\int|\nabla|F^{N}|^{\frac{p}{2}}|^{2}\,dx\lesssim\int\frac{\partial^{2}\tilde{W}(F^{N})}{\partial
F_{i\beta}^{N}\partial
F_{j\gamma}^{N}}\partial_{\alpha}F_{i\beta}^{N}\partial_{\alpha}F_{j\gamma}^{N}\,dx.$

∎

We are now in a position to prove Theorem 2.3.

###### Proof of Theorem 2.3.

We divide the proof in two steps.

Step 1: Global existence for initial data $(v_{0},F_{0})$ in
$L^{2}(\mathbb{T}^{d})\times H^{1}(\mathbb{T}^{d})$.

We first prove the existence for initial data $(v_{0},F_{0})$ in
$L^{2}(\mathbb{T}^{d})\times H^{1}(\mathbb{T}^{d})$. Note that by recalling
hypothesis (2.10), equality (4.4) and Lemma 4.2 we infer that

$\displaystyle\\{v^{N}\\}_{N}\subset L^{\infty}(0,T;L^{2}(\mathbb{T}^{d}))\cap
L^{2}(0,T;H^{1}(\mathbb{T}^{d}))$ $\displaystyle\\{F^{N}\\}_{N}\subset
L^{\infty}(0,T;H^{1}(\mathbb{T}^{d}))$

with uniform bounds. Moreover, by using (4.1) we also have that

$\displaystyle\\{\partial_{t}v^{N}\\}_{N}\subset
L^{2}(0,T;H^{-s}(\mathbb{T}^{d})),\,\,s>\frac{d}{2}+1$
$\displaystyle\\{\partial_{t}F^{N}\\}_{N}\subset
L^{2}(0,T;L^{2}(\mathbb{T}^{d})),$

also with uniform bounds. Therefore, a simple application of the Aubin-Lions-
Simon lemma [25] gives the existence of $(v,F)$ such that

$\displaystyle v\in L^{\infty}(0,T;L^{2}(\mathbb{T}^{d}))\cap
L^{2}(0,T;H^{1}(\mathbb{T}^{d}))$ $\displaystyle F\in
L^{\infty}(0,T;H^{1}(\mathbb{T}^{d}))\cap C([0,T);L^{2}(\mathbb{T}^{d}))$

and

$\displaystyle v^{N}\rightarrow v\mbox{ in }L^{2}(0,T;L^{2}(\mathbb{T}^{d}))$
(4.9) $\displaystyle F^{N}\rightarrow F\in C([0,T);L^{r}(\mathbb{T}^{d}))$

with $r<6$ if $d=3$ and $r<\infty$ if $d=2$. Next, for any $\phi\in
C^{\infty}_{c}([0,T);C^{\infty}(\mathbb{T}^{d}))$, it follows that

$P^{N}\phi\rightarrow\phi\mbox{ in }L^{q}((0,T)\times\mathbb{T}^{d}),\mbox{
for any }q<\infty,$ (4.10)

while, due to assumption (A3) and (4.9), we find that

$S(F^{N})\rightarrow S(F)\mbox{ in }L^{r}((0,T)\times\mathbb{T}^{d})\mbox{ for
any }r<\frac{p}{p-1}.$ (4.11)

Then, combining (4.9)-(4.11) it is fairly straightforward to prove that
$(v,F)$ is a weak solution of (1.3)-(1.4) in the sense of Definition 2.2.
Moreover, if $W$ satisfies assumption (A4′), by Lemma 4.2 we have that

$\\{\nabla|F^{N}|^{\frac{p}{2}}\\}_{N}\subset
L^{2}(0,T;L^{2}(\mathbb{T}^{d}))$

which together with the bounds of $\\{F^{N}\\}$ in
$L^{\infty}(0,T;H^{1}(\mathbb{T}^{d}))$ and the strong convergence in (4.9)
implies (2.11).

Step 2: Global existence for initial data $(v_{0},F_{0})\in
L^{2}\times\,L^{p}(\mathbb{T}^{d})$: Propagation of compactness.

Next, we consider the case of initial data $(v_{0},F_{0})\in
L^{2}\times\,L^{p}(\mathbb{T}^{d})$. We note that since $p\geq 2$ given
$F_{0}\in\,L^{p}(\mathbb{T}^{d})$, there always exists a sequence of
$\\{F_{0}^{n}\\}_{n}\subset\,H^{1}(\mathbb{T}^{d})$ such that

$\displaystyle F_{0}^{n}\rightarrow\,F_{0}\mbox{ in }L^{2}(\mathbb{T}^{d})$
$\displaystyle\sup_{n}\|F_{0}^{n}\|_{p}\leq C.$

From the previous part of the proof, we can claim the following: for any
$n\in\mathbb{N}$, there exists $(v^{n},F^{n})$ weak solution of (1.3) with
initial data $(v_{0},F_{0}^{n})$ satisfying the following uniform bounds

$\displaystyle\\{v^{n}\\}_{n}\subset L^{\infty}(0,T;L^{2}(\mathbb{T}^{d}))\cap
L^{2}(0,T;H^{1}(\mathbb{T}^{d}))$ (4.12) $\displaystyle\\{F^{n}\\}_{n}\subset
L^{\infty}(0,T;L^{p}(\mathbb{T}^{d})).$

Moreover, we find that

$\displaystyle\\{\partial_{t}v^{n}\\}_{n}\subset
L^{2}(0,T;W^{-1,\frac{p}{p-1}}(\mathbb{T}^{d})),$

also with uniform bounds. Therefore, proceeding as before, a simple
application of the Aubin-Lions-Simon lemma gives the existence of $(v,F)$ such
that

$\displaystyle v\in L^{\infty}(0,T;L^{2}(\mathbb{T}^{d}))\cap
L^{2}(0,T;H^{1}(\mathbb{T}^{d}))$ $\displaystyle F\in
L^{\infty}(0,T;L^{p}(\mathbb{T}^{d}))$

and

$\displaystyle v^{n}\rightarrow v\mbox{ in }L^{2}(0,T;L^{2}(\mathbb{T}^{d}))$
(4.13) $\displaystyle F^{n}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}F\in
L^{\infty}(0,T;L^{p}(\mathbb{T}^{d})).$

Next, note that

$\\{\partial_{t}F^{n}\\}_{n}\subset L^{2}(0,T;L^{2}(\mathbb{T}^{d}))$

and thus $F\in C([0,T];L^{2}(\mathbb{T}^{d}))$ and

$\frac{d}{dt}\int\,\frac{|F|^{2}}{2}\,dx=\int\,\partial_{t}\,F\,F\,dx.$ (4.14)

Indeed, this follows since $F$, $\partial_{t}F\in
L^{2}(0,T;L^{2}(\mathbb{T}^{d}))$. Then, $F\in
H^{1}(0,T;L^{2}(\mathbb{T}^{d}))$ which embeds into
$C([0,T];L^{2}(\mathbb{T}^{d}))$ as
$\|\int_{s}^{\tau}\partial_{t}F\|_{2}\leq\int_{s}^{\tau}\|\partial_{t}F\|_{2}$.
Next, we note that by (A3) and the uniform bound of $\\{F^{n}\\}_{n}$ in
$L^{\infty}(0,T;L^{p}(\mathbb{T}^{d}))$ we have that

$S(F_{n})\rightharpoonup\overline{S(F)}\mbox{ in
}L^{\frac{p}{p-1}}(\mathbb{T}^{d})$ (4.15)

where $\overline{S(F)}$ denotes the weak limit of $S(F^{n})$ which, at this
moment, may not be equal to $S(F)$. In particular, testing with
$\theta(t)\phi(t,x)$ where $\dot{\theta}$ is a function localising at time
$t$, we infer that for a.a $t\in(0,T)$, $(v,F)$ satisfies

$\iint\,v\partial_{t}\phi\,dx\,ds-\iint\,\overline{S(F)}\nabla\phi\,dx\,ds-\iint\nabla\,v\nabla\phi\,dx\,ds-\int
v(t)\phi(t)\,dx+\int\,v(0)\phi(0)\,dx=0$ (4.16)

for any $\phi\in C^{\infty}([0,T];C^{\infty}(\mathbb{T}^{d}))$. Similarly,
(4.16) holds for $(v^{n},F^{n})$. Then, subtracting the equations for
$(v^{n},F^{n})$ from (4.16), noting that $v^{n}_{0}=v_{0}$, by a simple
density argument, we may test with $\phi=y-y^{n}$ to infer that

$\displaystyle\int\,\frac{1}{2}|F-F^{n}|^{2}\,dx+$
$\displaystyle\int_{0}^{t}\int\,(\overline{S(F)}-S(F^{n}))(F-F^{n})\,dx\,ds=\int_{0}^{t}\int|v-v^{n}|^{2}\,dx\,ds$
(4.17)
$\displaystyle-\int(v-v^{n})(y-y^{n})\,dx+\int\frac{1}{2}|F_{0}-F^{n}_{0}|^{2}\,dx,$

where we have also made use of (4.14). For $\tilde{S}(F)=S(F)+KF$, (4.17)
becomes

$\displaystyle\int\,\frac{1}{2}|F-F^{n}|^{2}\,dx+\int_{0}^{t}\int\,(\tilde{S}(F)-\tilde{S}(F^{n}))(F-F^{n})\,dx\,ds=\int_{0}^{t}\int|v-v^{n}|^{2}\,dx\,ds$
$\displaystyle-\int(v-v^{n})(y-y^{n})\,dx+\int\frac{1}{2}|F_{0}-F^{n}_{0}|^{2}\,dx+\int_{0}^{t}\int(S(F)-\overline{S(F)})(F-F^{n})\,dx\,ds$
$\displaystyle+K\int_{0}^{t}\int|F-F^{n}|^{2}\,dx\,ds$

By assumption (A4) and (4.12) we find that

$\displaystyle\int\,|F-F^{n}|^{2}\,dx$
$\displaystyle\lesssim\int_{0}^{t}\int|v-v^{n}|^{2}\,dx\,ds+\left(\int|y-y^{n}|^{2}\,dx\right)^{\frac{1}{2}}$
$\displaystyle\quad+\int|F_{0}-F^{n}_{0}|^{2}\,dx+\left|\int_{0}^{t}\int\,(S(F)-\overline{S(F)})(F-F^{n})\,dx\,ds\right|$
$\displaystyle\qquad+\int_{0}^{t}\int|F-F^{n}|^{2}\,dx\,ds,$

where the second term on the right-hand side is obtained via Hölder’s
inequality and the bound $v$, $v^{n}\in L^{\infty}(0,T;L^{2}(\mathbb{T}^{d}))$
in (4.12). Next, define

$\kappa(t):=\limsup_{n\to\infty}\int|F(t)-F^{n}(t)|^{2}\,dx.$

Then, $\kappa(0)=0$, and note that

$\displaystyle v^{n}\rightarrow v\mbox{ in }L^{2}(0,T;L^{2}(\mathbb{T}^{d})),$
$\displaystyle y^{n}\rightarrow y\mbox{ in }C(0,T;L^{2}(\mathbb{T}^{d})).$

Moreover, due to (4.13) and (4.15),

$\int_{0}^{t}\int\,(S(F)-\overline{S(F)})(F-F^{n})\,dx\,ds\rightarrow 0\mbox{
as }n\to\infty$

and we obtain that

$\kappa(t)\leq\int_{0}^{t}\,\kappa(s)\,ds.$

By Grönwall’s inequality we conclude that $\kappa(t)=0$ and therefore

$F_{n}\rightarrow\,F\mbox{ in }L^{2}(0,T;L^{2}(\mathbb{T}^{d})).$ (4.18)

From (4.18) we deduce that $\overline{S(F)}=S(F)$ and thus $(v,F)$ is a weak
solution of (1.3)-(1.4). ∎

###  4.3. Proof of Theorem 2.5

We prove the theorem concerning the conservation of energy.

###### Proof of Theorem 2.5.

Let $(v,F)$ be a weak solution of the system (1.3)-(1.4) in the sense of
Definition 2.2 such that

$F\in L^{\infty}(0,T;H^{1}(\mathbb{T}^{d})).$ (4.19)

By Sobolev embedding we get

$\displaystyle F\in L^{\infty}(0,T;L^{r}(\mathbb{T}^{d})),\mbox{ for
}d=2\mbox{ and for all }r<\infty,$ $\displaystyle F\in
L^{\infty}(0,T;L^{6}(\mathbb{T}^{d}))\mbox{ for }d=3.$

Then by using the growth condition of $S$ in (A2) we have that

$S(F)\in L^{\infty}(0,T;L^{2}(\mathbb{T}^{d}))$ (4.20)

for any $p\geq 2$ in the case $d=2$ and for $2\leq p\leq 4$ in the case $d=3$.

Next, the weak formulation (2.7), (2.8) and (4.20) imply that

$\displaystyle\partial_{t}\,F=\nabla v,\mbox{ a.e. on
$(0,T)\times\mathbb{T}^{d}$}$ (4.21)
$\displaystyle\partial_{t}v=\mathop{\mathrm{div}}(S(F)+\nabla u)\mbox{ in
}L^{2}(0,T;H^{-1}(\mathbb{T}^{d})),$ (4.22)

where $H^{-1}(\mathbb{T}^{d}):=(H^{1}(\mathbb{T}^{d}))^{\prime}$. First, we
note that regarding the time chain-rule for $W(F)$, by using the bounds
(4.21), (4.19) and (2.7), and by using the weak formulation (2.8), we can
infer that $F$ can be re-defined on a set of measure zero in time so that
$F\in C([0,T];L^{2}(\mathbb{T}^{d}))$ and

$\displaystyle\lim_{t\to 0}F(t)=F_{0}\mbox{ strongly in
}L^{2}(\mathbb{T}^{d}),$ $\displaystyle\lim_{t\to 0}F(t)=F_{0}\mbox{ weakly in
}H^{1}(\mathbb{T}^{d}).$

Then, assumption (A2) on the growth of $S(F)$ and the fact that $F\in
C([0,T];L^{2}(\mathbb{T}^{d}))$ imply that $W(F(\cdot))\in
C([0,T];L^{1}(\mathbb{T}^{d}))$ and $W(F(t))\to W(F_{0})$ in
$L^{1}(\mathbb{T}^{d})$ as $t\to 0$. Finally, again by using (4.21), (2.7) and
(4.20) we have that $\partial_{t}W(F)=S(F):\partial_{t}F$ a.e. in
$(0,T)\times\mathbb{T}^{d}$ and then $\partial_{t}W(F)\in
L^{1}(0,T;L^{1}(\mathbb{T}^{d}))$. Then, by standard properties of the Bochner
integral we have that

$\int W(F(t))\,dx-\int W(F_{0})\,dx=\int\,S(F):\nabla v.$ (4.23)

Next, regarding the velocity, we have that (2.8), (2.9), (4.22), and (4.20)
imply that $\partial_{t}v\in L^{2}(0,T;H^{-1}(\mathbb{T}^{d}))$ and therefore,
by the Lions-Magenes Lemma, after a possible re-definition on a set of measure
zero in time, we have that $v\in C([0,T];L^{2}(\mathbb{T}^{d}))$ and then from
(2.8) it follows that

$\lim_{t\to 0}v(t)=v_{0}\mbox{ strongly in }L^{2}(\mathbb{T}^{d}).$

Moreover, again from the Lions-Magenes Lemma, we have that

$\displaystyle\frac{d}{dt}\int\,\frac{|v|^{2}}{2}\,dx=\langle\partial_{t}v,v\rangle_{H^{-1},H^{1}}$

and then

$\int\frac{|v(t)|^{2}}{2}\,dx-\int\frac{|v_{0}|^{2}}{2}\,dx=-\int\left(S(F)+\nabla
v\right):\nabla v\,dx.$ (4.24)

Combining (4.23) and (4.24) completes the proof. ∎

##  5\. Uniqueness in two dimensions

In this section, we restrict to two space dimensions and prove Theorem 2.7 and
Theorem 2.8.

###### Proof of Theorem 2.7.

Given the initial data $(v_{0},F_{0})$ satisfying the hypothesis of Theorem
2.7, the existence follows from the Theorem 2.3. Regarding uniqueness, let
$(v_{1},F_{1})$ and $(v_{2},F_{2})$ be two weak solutions (1.3)-(1.4) and note
that from Theorem 2.3 we have that

$\displaystyle v\in L^{\infty}(0,T;L^{2}(\mathbb{T}^{2}))\cap
L^{2}(0,T;H^{1}(\mathbb{T}^{2})),$ (5.1) $\displaystyle F\in
C(0,T;L^{r}(\mathbb{T}^{2}))\cap L^{\infty}(0,T;H^{1}(\mathbb{T}^{2})),\mbox{
for any }r<\infty.$

Since $\nabla\,v\in L^{2}(0,T;L^{2}(\mathbb{T}^{2}))$ the second equation in
(1.3) is satisfied almost everywhere on $(0,T)\times\mathbb{T}^{2}$ and,
therefore, by an approximation argument and (5.1), we deduce that for any
$1<r<\infty$ and any $t\in(0,T)$

$\int\,|F_{1}-F_{2}|^{r}\,dx\lesssim\int_{0}^{t}\int|\nabla\,v_{1}-\nabla\,v_{2}||F_{1}-F_{2}|^{r-1}\,dx\,ds$
(5.2)

Next, by setting $w=v_{1}-v_{2}$ we infer that

$\displaystyle\partial_{t}w-\Delta w$
$\displaystyle=\mathop{\mathrm{div}}(S(F_{1})-S(F_{2}))$ $\displaystyle
w|_{t=0}$ $\displaystyle=0$

weakly on $(0,T)\times\mathbb{T}^{2}$. Then, given $1<r<\infty$, by Lemma 3.2,
we find that for any $t\in(0,T)$

$\int_{0}^{t}\|\nabla\,v_{1}-\nabla\,v_{2}\|_{r}^{r}\,ds\lesssim\int_{0}^{t}\|S(F_{1})-S(F_{2})\|_{r}^{r}\,ds.$
(5.3)

In particular, by using Young’s inequality and combining (5.2) and (5.3), we
get that

$\displaystyle\int\,|F_{1}-F_{2}|^{r}\,dx$
$\displaystyle\lesssim\int_{0}^{t}\int|F_{1}-F_{2}|^{r}\,dx\,ds+\int_{0}^{t}\int|S(F_{1})-S(F_{2})|^{r}\,dx\,ds.$
(5.4)

Next, as a consequence of (2.12), we have that

$|S(F_{1})-S(F_{2})|\lesssim(1+|F_{1}|^{s}+|F_{2}|^{s})|F_{1}-F_{2}|$

and then (5.4) becomes

$\displaystyle\int\,|F_{1}-F_{2}|^{r}\,dx$
$\displaystyle\lesssim\int_{0}^{t}\int|F_{1}-F_{2}|^{r}\,dx\,ds$
$\displaystyle\quad+\int_{0}^{t}\int|F_{1}|^{rs}|F_{1}-F_{2}|^{r}\,dx\,ds$
$\displaystyle\quad+\int_{0}^{t}\int|F_{2}|^{rs}|F_{1}-F_{2}|^{r}\,dx\,ds$
$\displaystyle\qquad=I_{1}+I_{2}+I_{3}.$ (5.5)

We now estimate the term $I_{2}$. Let $q>1$ and since
$\tfrac{1}{rq}+\tfrac{q-1}{rq}+\tfrac{r-1}{r}=1$ Hölder’s inequality implies

$\displaystyle I_{2}$
$\displaystyle\leq\int_{0}^{t}\int|F_{1}|^{rs}|F_{1}-F_{2}||F_{1}-F_{2}|^{r-1}\,ds$
$\displaystyle\leq\int_{0}^{t}\|F_{1}\|_{r^{2}qs}^{rs}\|F_{1}-F_{2}\|_{\frac{rq}{q-1}}\|F_{1}-F_{2}\|_{r}^{r-1}\,ds\,.$

Note that since $0\leq\gamma\leq 2$, $2\leq p<2+\gamma$, and $p-2\leq s\leq
p-\gamma$ we have that $\frac{2}{s}>1$ and then by choosing $r=\frac{2}{s}$
and by using consecutively Lemma 3.1 (i) and (ii), we infer that

$\displaystyle I_{2}$
$\displaystyle\leq\int_{0}^{t}\|F_{1}\|_{\frac{4q}{s}}^{2}\|F_{1}-F_{2}\|_{\frac{rq}{q-1}}\|F_{1}-F_{2}\|_{r}^{r-1}\,ds$
$\displaystyle\lesssim
q\int_{0}^{t}\|\nabla\,F_{1}\|^{2}_{2}\|F_{1}-F_{2}\|_{r}^{1-\frac{1}{q}}\|\nabla\,F_{1}-\nabla\,F_{2}\|_{2}^{\frac{1}{q}}\|F_{1}-F_{2}\|_{r}^{r-1}\,ds,$

where, the suppressed constant depends on $s$, $p$ but not $q$. Then, since
$F_{1},F_{2}\in L^{\infty}(0,T;H^{1}(\mathbb{T}^{2}))$, we deduce that

$I_{2}\lesssim q\int_{0}^{t}\,\|F_{1}-F_{2}\|_{r}^{r-\frac{1}{q}}\,ds.$

Arguing in the exact same way for $I_{3}$ we get from (5.5) that

$\displaystyle\|F_{1}-F_{2}\|_{r}^{r}$
$\displaystyle\lesssim\int_{0}^{t}\|F_{1}-F_{2}\|_{r}^{r}\,ds+q\int_{0}^{t}\|F_{1}-F_{2}\|_{r}^{r-\frac{1}{q}}\,ds$
$\displaystyle\lesssim
q\,(\sup_{t}\|F_{1}-F_{2}\|^{\frac{1}{q}}_{r}+1)\int_{0}^{t}\|F_{1}-F_{2}\|_{r}^{r-\frac{1}{q}}\,ds$
$\displaystyle\lesssim q\int_{0}^{t}\|F_{1}-F_{2}\|_{r}^{r-\frac{1}{q}}\,ds$

where we have used again that $F_{1},F_{2}\in
L^{\infty}(0,T;H^{1}(\mathbb{T}^{2}))$.

Next, set $y(t)=\|F_{1}(t)-F_{2}(t)\|_{r}^{r}$ and notice that $y(t)\in
C([0,T])$; then, for any $q>1$ it holds that

$y(t)\leq Crq\int_{0}^{t}\,y(s)^{1-\frac{1}{rq}}\,ds,\,\,y(0)=0,$

where we recall that $r=\frac{2}{s}$ is fixed and $C$ does not depend on $q$.
Then, the fact that $y(t)=0$ on $(0,T)$ follows from Lemma 3.3, which implies
that $0\leq y(t)\leq\bar{y}(t)=(Ct)^{rq}$ with $C$ independent of $q$. In
particular, since $\bar{y}$ is a non-decreasing function, we may choose
$t^{*}\in(0,T)$ such that for all $q>1$

$y(t)\leq\left(\frac{1}{2}\right)^{q},\,\,\forall\,t\in(0,t^{*}).$

Taking $q\to\infty$ we deduce that $y\equiv 0$ on $(0,t^{*})$ and, by
repeating the argument, that $y\equiv 0$ on $(0,T)$.

Therefore, $F_{1}=F_{2}$ and $v_{1}-v_{2}$ solves in the weak sense

$\displaystyle\partial_{t}(v_{1}-v_{2})-\Delta(v_{1}-v_{2})$ $\displaystyle=0$
$\displaystyle(v_{1}-v_{2})|_{t=0}$ $\displaystyle=0.$

Hence, $v_{1}=v_{2}$ due to standard uniqueness results for the heat equation.
∎

We next proceed with the proof of Theorem 2.8. This follows by an argument
similar to Theorem 2.7 and we only show the few modifications required.

###### Proof of Theorem 2.8.

As in Theorem 2.7, given $(v_{0},F_{0})$ as in the statement of Theorem 2.8
the existence part follows by Theorem 2.3. Regarding uniqueness, arguing
exactly as in the proof of Theorem 2.7 we get (5.5), which we rewrite for the
reader’s convenience

$\displaystyle\int\,|F_{1}-F_{2}|^{r}\,dx$
$\displaystyle\lesssim\int_{0}^{t}\int|F_{1}-F_{2}|^{r}\,dx\,ds$
$\displaystyle\quad+\int_{0}^{t}\int|F_{1}|^{rs}|F_{1}-F_{2}|^{r}\,dx\,ds$
$\displaystyle\qquad+\int_{0}^{t}\int|F_{2}|^{rs}|F_{1}-F_{2}|^{r}\,dx\,ds$
$\displaystyle\qquad=I_{1}+I_{2}+I_{3}.$

To estimate the term $I_{2}$, set $U_{i}:=|F_{i}|^{\frac{p}{2}}$, so that

$\displaystyle I_{2}$ $\displaystyle=\int_{0}^{t}\int
U_{1}^{\frac{2rs}{p}}|F_{1}-F_{2}||F_{1}-F_{2}|^{r-1}\,dx\,ds.$

Note that $p\geq 2$ and the hypothesis $s<p$ implies that
$r:=\frac{p}{s}\in(1,\infty)$. Then, as in the proof of Theorem 2.7, by using
Lemma 3.1 we now get that

$\displaystyle I_{2}$
$\displaystyle\leq\int_{0}^{t}\|U_{1}\|_{2rq}^{2}\|F_{1}-F_{2}\|_{\frac{rq}{q-1}}\|F_{1}-F_{2}\|_{r}^{r-1}\,ds$
$\displaystyle\lesssim
q\int_{0}^{t}\|U_{1}\|^{2}_{H^{1}}\|F_{1}-F_{2}\|_{r}^{1-\frac{1}{q}}\|\nabla\,F_{1}-\nabla\,F_{2}\|_{2}^{\frac{1}{q}}\|F_{1}-F_{2}\|_{r}^{r-1}\,ds,$

where the suppressed constant depends on $r$, $p$ but not $q$. Since
$F_{1},F_{2}\in L^{\infty}(0,T;H^{1}(\mathbb{T}^{2}))$

$I_{2}\lesssim
q\int_{0}^{t}\,\|U_{1}(s)\|_{H^{1}}^{2}\|F_{1}(s)-F_{2}(s)\|_{r}^{r-\frac{1}{q}}\,ds.$

We may treat $I_{3}$ similarly to get that
$y(t)=\|F_{1}(t)-F_{2}(t)\|_{r}^{r}\in C([0,T])$ satisfies

$y(t)\leq Crq\int_{0}^{t}\,f(s)y(s)^{1-\frac{1}{rq}}\,ds,\,\,y(0)=0,$ (5.6)

where $f(s):=1+\|U_{1}(s)\|_{H^{1}}^{2}+\|U_{2}(s)\|_{H^{1}}^{2}\in
L^{1}(0,T)$, $C$ is independent of $q$ and (5.6) holds for all $q>1$. Then, by
using again Lemma 3.3 we deduce that $0\leq
y(t)\leq\bar{y}(t)=\left(C\int_{0}^{t}f(s)\,ds\right)^{q}$ and the proof can
now be concluded as in Theorem 2.7. ∎

##  6\. Global regularity

Before proving Theorem 2.10 we prove a local existence result for (1.3)-(1.4).
Of course, the local existence of smooth solutions holds under more general
hypothesis than the ones considered in the next proposition.

######  Proposition 6.1.

Under the assumptions of Theorem 2.10, there exists a time
$T^{*}=T^{*}(\|v_{0}\|_{H^{3}},\|F_{0}\|_{H^{3}})$ such that the unique weak
solution constructed in Theorem 2.7 satisfies

$(v,F)\in L^{\infty}(0,T^{*};H^{3}(\mathbb{T}^{2}))$

###### Proof.

We henceforth use the following notation: given tensors
$\\{A^{i}\\}_{i=1}^{n}$, we write

$(A^{1})(A^{2})....(A^{n})$

for linear combinations of products of the entries of the tensor inside the
brackets. Recall the Galerkin approximation

$\displaystyle\partial_{t}\,v^{N}-\mathop{\mathrm{div}}\,P^{N}(S(F^{N}))-\Delta\,v^{N}$
$\displaystyle=0$ (6.1) $\displaystyle\partial_{t}\,F^{N}-\nabla\,v^{N}$
$\displaystyle=0$ $\displaystyle\mathop{\mathrm{curl}}\,F^{N}$
$\displaystyle=0,$

and note that, in the periodic setting, the operators $P^{N}$ and $\nabla$
commute. Therefore,

$\nabla^{2}\mathop{\mathrm{div}}\,P^{N}(S(F^{N}))=P^{N}\nabla^{2}\mathop{\mathrm{div}}\,S(F^{N})$

and

$\|P^{N}\nabla^{2}\mathop{\mathrm{div}}\,S(F^{N})\|_{2}\leq
C\|\nabla^{3}\,S(F^{N})\|_{2}$

with $C$ independent of $N$. Next, note that

$\displaystyle\nabla^{3}(S(F^{N}))$
$\displaystyle=(D^{4}W(F^{N}))(\nabla\,F^{N})(\nabla\,F^{N})(\nabla\,F^{N})+(D^{3}W(F^{N}))(\nabla^{2}\,F^{N})(\nabla\,F^{N})$
$\displaystyle+(D^{2}W(F^{N}))(\nabla^{3}\,F^{N})$

and then since all the derivative of $W$ have polynomial growth, by Sobolev
embeddings it follows that there exists $m>0$ such that

$\|\nabla^{3}\,S(F^{N})\|_{2}^{2}\leq\,C\|\nabla^{3}\,F^{N}\|_{2}^{2+2m}$

again with $C$ independent of $N$. By differentiating two times the first
equation of (6.1) and multiplying the resulting equation by
$-\Delta\nabla^{2}\,v^{N}$, after integrating in space, we obtain

$\displaystyle\frac{d}{dt}\|\nabla^{3}\,v^{N}\|_{2}^{2}+\|\nabla^{4}\,v^{N}\|_{2}^{2}$
$\displaystyle\lesssim\|\nabla^{2}\mathop{\mathrm{div}}(S(F^{N}))\|_{2}^{2}$
(6.2) $\displaystyle\lesssim\|\nabla^{3}\,F^{N}\|_{2}^{2+2m}.$

On the other hand by differentiating three times the second equation and
multiplying the resulting equation by $\nabla^{3}\,F^{N}$, integrating in
space and using Young’s inequality we get

$\frac{d}{dt}\|\nabla^{3}\,F^{N}\|_{2}^{2}\lesssim\|\nabla^{3}\,F^{N}\|_{2}^{2}+\frac{1}{4}\|\nabla^{4}\,v^{N}\|_{2}^{2},$
(6.3)

where we note that no constant has been suppressed for the term
$\frac{1}{4}\|\nabla^{4}\,v^{N}\|_{2}^{2}$. Therefore, by (6.2) and (6.3),
$y(t):=(\|\nabla^{3}\,F^{N}(t)\|_{2}^{2}+\|\nabla^{3}\,v^{N}(t)\|_{2}^{2})$
satisfies

$\dot{y}(t)\lesssim y(t)+y^{1+m}(t)$

which implies that there exists $T^{*}$ independent of $N$ such that

$(v^{N},F^{N})\in L^{\infty}(0,T^{*};H^{3}(\mathbb{T}^{2}))\times
L^{\infty}(0,T^{*};H^{3}(\mathbb{T}^{2})).$

The proposition now follows by taking the limit as $N\to\infty$ to obtain

$(v,F)\in L^{\infty}(0,T^{*};H^{3}(\mathbb{T}^{2}))\times
L^{\infty}(0,T^{*};H^{3}(\mathbb{T}^{2}))$

which is the solution constructed in Theorem 2.3 whose uniqueness follows from
Theorem 2.7. ∎

We are now in a position to prove Theorem 2.10.

###### Proof of Theorem 2.10.

We only prove the _a priori_ estimates. Indeed, by using Proposition 6.1 the
result follows by a simple continuity argument. Note that we have already
proved

$\displaystyle v\in L^{\infty}(0,T;L^{2}(\mathbb{T}^{2}))\cap
L^{2}(0,T;H^{1}(\mathbb{T}^{2}))$ $\displaystyle F\in
L^{\infty}(0,T;H^{1}(\mathbb{T}^{2})).$

Next, we prove an $H^{2}$-estimate. By multiplying the first equation of (1.3)
by $-\Delta\,v$ we get

$\frac{d}{dt}\|\nabla\,v\|_{2}^{2}+\|\nabla^{2}\,v\|_{2}^{2}\lesssim\int|D^{2}\tilde{W}(F)||\nabla\,F||\nabla^{2}\,v|\,dx+K\int|\nabla\,F||\nabla^{2}\,v|\,dx$

and integrating in time on $(0,t)$, suppressing $K$, we get that

$\displaystyle\int|\nabla\,v|^{2}\,dx+\int_{0}^{t}\int|\nabla^{2}\,v|^{2}\,dx\,ds\lesssim\int|\nabla\,v_{0}|^{2}\,dx$
$\displaystyle+\int_{0}^{t}\int|D^{2}\tilde{W}(F)||\nabla\,F||\nabla^{2}\,v|\,dx\,ds$
(6.4) $\displaystyle+\int_{0}^{t}\int|\nabla\,F||\nabla^{2}\,v|\,dx\,ds.$

Next we multiply the first equation of(1.3) by
$\Delta\mathop{\mathrm{div}}\,F$ and we integrate in space to get that

$\int
v_{t}\Delta\mathop{\mathrm{div}}\,F\,dx-\int\mathop{\mathrm{div}}(S(F))\Delta\mathop{\mathrm{div}}\,F\,dx-\int\Delta\,v\Delta\mathop{\mathrm{div}}\,F\,dx=0$

and we treat every term separately. We start with the third term. By using the
second and the third equation in (1.3) we have

$\displaystyle-\int\Delta\,v\Delta\mathop{\mathrm{div}}\,F\,dx$
$\displaystyle=-\int\mathop{\mathrm{div}}\,F_{t}\Delta\mathop{\mathrm{div}}\,F\,dx=\frac{d}{dt}\int\frac{1}{2}|\nabla^{2}\,F|^{2}\,dx.$
(6.5)

Concerning the second term and using that fact that
$\mathop{\mathrm{curl}}\,F=0$ we find that

$\displaystyle-\int\partial_{\alpha}S_{i\alpha}(F)\partial_{\gamma\gamma}\partial_{\beta}F_{i\beta}\,dx$
$\displaystyle=-\int\partial_{\beta}S_{i\alpha}(F)\partial_{\gamma\gamma}\partial_{\alpha}F_{i\beta}\,dx$
$\displaystyle=-\int\frac{\partial^{2}\,W(F)}{\partial\,F_{i\alpha}\partial\,F_{j\delta}}\partial_{\beta}F_{j\delta}\partial_{\gamma\gamma}\partial_{\beta}F_{i\alpha}\,dx$
$\displaystyle=\int\frac{\partial^{2}\,W(F)}{\partial\,F_{i\alpha}\partial\,F_{j\delta}}\partial_{\beta\gamma}F_{j\delta}\partial_{\gamma\beta}F_{i\alpha}\,dx$
$\displaystyle+\int(D^{3}\,W(F))(\nabla\,F)(\nabla\,F)(\nabla^{2}\,F)\,dx$
$\displaystyle=\int\frac{\partial^{2}\,\tilde{W}(F)}{\partial\,F_{i\alpha}\partial\,F_{j\delta}}\partial_{\beta\gamma}F_{j\delta}\partial_{\gamma\beta}F_{i\alpha}\,dx$
$\displaystyle-K\int\,|\nabla^{2}\,F|^{2}\,dx$
$\displaystyle+\int(D^{3}\,W(F))(\nabla\,F)(\nabla\,F)(\nabla^{2}\,F)\,dx.$
(6.6)

Finally, concerning the first term we have

$\displaystyle\int\,v_{t}\Delta\mathop{\mathrm{div}}\,F\,dx$
$\displaystyle=\frac{d}{dt}\int
v\Delta\mathop{\mathrm{div}}\,F\,dx-\int|\nabla^{2}v|^{2}\,dx$ (6.7)

and combining (6.5)-(6.7) we infer that

$\displaystyle\frac{d}{dt}\int\left(|\nabla^{2}\,F|^{2}+v\Delta\mathop{\mathrm{div}}\,F\right)\,dx+\int\frac{\partial^{2}\,\tilde{W}(F)}{\partial\,F_{i\alpha}\partial\,F_{j\delta}}\partial_{\beta\gamma}F_{j\delta}\partial_{\gamma\beta}F_{i\alpha}\,dx$
(6.8)
$\displaystyle\lesssim\int\,|\nabla^{2}\,F|^{2}\,dx+\int|\nabla^{2}v|^{2}\,dx+\int(D^{3}\,W(F))(\nabla\,F)(\nabla\,F)(\nabla^{2}\,F)\,dx.$

Integrating (6.8) in time we get

$\displaystyle\int|\nabla^{2}\,F|^{2}\,dx+\int_{0}^{t}\int\frac{\partial^{2}\,\tilde{W}(F)}{\partial\,F_{i\alpha}\partial\,F_{j\delta}}\partial_{\beta\gamma}F_{j\delta}\partial_{\gamma\beta}F_{i\alpha}\,dx\,ds$
$\displaystyle\lesssim\int_{0}^{t}\int\,|\nabla^{2}\,F|^{2}\,dx\,ds+\int_{0}^{t}\int|\nabla^{2}v|^{2}\,dx\,ds+\int_{0}^{t}\int|D^{3}\,W(F)||\nabla\,F|^{2}|\nabla^{2}\,F|\,dx\,ds$
$\displaystyle\quad+\int\left(|\nabla^{2}\,F_{0}|^{2}+|\nabla
v_{0}|^{2}+|\Delta F_{0}|^{2}\right)\,dx+\left|\int\nabla
v\Delta\,F\,dx\right|$ (6.9)
$\displaystyle\lesssim\int_{0}^{t}\int\,|\nabla^{2}\,F|^{2}\,dx\,ds+\int_{0}^{t}\int|\nabla^{2}v|^{2}\,dx\,ds+\int_{0}^{t}\int|D^{3}\,W(F)||\nabla\,F|^{2}|\nabla^{2}\,F|\,dx\,ds$
$\displaystyle\quad+\int\left(|\nabla^{2}\,F_{0}|^{2}+|\nabla
v_{0}|^{2}+|\Delta F_{0}|^{2}\right)\,dx+\int|\nabla
v|^{2}\,dx+\frac{1}{2}\int|\nabla^{2}\,F|^{2}\,dx,$

where, by Young’s inequality, there is no suppressed constant in front of the
term $\frac{1}{2}\int|\nabla^{2}\,F|^{2}\,dx$. Also, note that by the
convexity of $\tilde{W}$, (A4), the second term on the left-hand side is
nonnegative and thus (6.9) now reads as

$\displaystyle\int|\nabla^{2}\,F|^{2}\,dx$
$\displaystyle\lesssim\int_{0}^{t}\int\,|\nabla^{2}\,F|^{2}\,dx\,ds+\int_{0}^{t}\int|\nabla^{2}v|^{2}\,dx\,ds$
(6.10)
$\displaystyle+\int_{0}^{t}\int|D^{3}\,W(F)||\nabla\,F|^{2}|\nabla^{2}\,F|\,dx\,ds$
$\displaystyle\quad+\int\left(|\nabla^{2}\,F_{0}|^{2}+|\nabla
v_{0}|^{2}\right)\,dx+\int|\nabla v|^{2}\,dx.$

We may now multiply (6.10) by a constant small enough so that, after adding up
with (6.4),

$\displaystyle\int\left(|\nabla^{2}\,F|^{2}+|\nabla\,v|^{2}\right)\,dx+\int_{0}^{t}\int|\nabla^{2}\,v|^{2}\,dx\,ds$
$\displaystyle\lesssim\int_{0}^{t}\int|\nabla^{2}\,F|^{2}\,dx\,ds$ (6.11)
$\displaystyle+\int_{0}^{t}\int|D^{3}W(F)||\nabla\,F|^{2}|\nabla^{2}\,F|\,dx\,ds$
$\displaystyle+\int_{0}^{t}\int|D^{2}\tilde{W}(F)||\nabla\,F||\nabla^{2}\,v|\,dx\,ds$
$\displaystyle+\int_{0}^{t}\int|\nabla\,F||\nabla^{2}\,v|\,dx\,ds$
$\displaystyle+\int\left(|\nabla^{2}\,F_{0}|^{2}+|\nabla\,v_{0}|^{2}\right)\,dx.$

We use now the growth conditions (2.15). We only treat the case $p>3$, which
is the case when $D^{3}\tilde{W}$ is not bounded. Thus, in the case we assume
$(A4)$ we restrict out attention to the range $3<p\leq 5$ and from (6.11) we
get

$\displaystyle\int\left(|\nabla^{2}\,F|^{2}+|\nabla\,v|^{2}\right)\,dx+\int_{0}^{t}\int|\nabla^{2}\,v|^{2}\,dx\,ds$
$\displaystyle\lesssim\int_{0}^{t}\int|\nabla^{2}\,F|^{2}\,dx\,ds$ (6.12)
$\displaystyle+\int_{0}^{t}\int(1+|F|^{p-3})|\nabla\,F|^{2}|\nabla^{2}\,F|\,dx\,ds$
$\displaystyle+\int_{0}^{t}\int(1+|F|^{p-2})|\nabla\,F||\nabla^{2}\,v|\,dx\,ds$
$\displaystyle+\int\left(|\nabla^{2}\,F_{0}|^{2}+|\nabla\,v_{0}|^{2}\right)\,dx$
$\displaystyle=I_{1}+I_{2}+I_{3}+I_{4}.$

We only have to bound $I_{2}$ and $I_{3}$. By using Young’s inequality, Lemma
3.1 (1) and the bound $F\in L^{\infty}(0,T;H^{1}(\mathbb{T}^{2}))$ we have

$\displaystyle I_{3}$
$\displaystyle\lesssim\int_{0}^{t}\left(1+\|F\|_{4(p-2)}^{p-2}\right)\|\nabla\,F\|_{4}\|\nabla^{2}\,v\|_{2}\,ds$
$\displaystyle\lesssim\int_{0}^{t}\left(1+\|F\|_{4(p-2)}^{2(p-2)}\right)\|\nabla\,F\|_{2}\|\nabla^{2}\,F\|_{2}\,ds+\frac{1}{4}\int_{0}^{t}\|\nabla^{2}\,v\|_{2}^{2}\,ds$
$\displaystyle\lesssim\int_{0}^{t}\|\nabla^{2}\,F\|_{2}\,ds+\frac{1}{4}\int_{0}^{t}\|\nabla^{2}\,v\|_{2}^{2}\,ds,$

where, as in other instances, due to Young’s inequality, there is no
suppressed constant in front of
$\frac{1}{4}\int_{0}^{t}\|\nabla^{2}\,v\|_{2}^{2}\,ds$ and can get absorbed
into the left-hand side of (6.12).

Regarding $I_{2}$, we proceed similarly and using Lemma 3.1 (1), and the
Brezis-Gallouet inequality (3) together with the bound $F\in
L^{\infty}(0,T;H^{1}(\mathbb{T}^{2}))$, we find that

$\displaystyle I_{2}$
$\displaystyle\lesssim\int_{0}^{t}\left(1+\|F\|_{\infty}^{p-3}\right)\|\nabla\,F\|_{4}^{2}\|\nabla^{2}\,F\|_{2}\,ds$
$\displaystyle\lesssim\int_{0}^{t}\left(1+(\log(e+\|\nabla^{2}\,F\|_{2}^{2}))^{\frac{p-3}{2}}\right)\|\nabla^{2}\,F\|_{2}^{2}\,ds$
$\displaystyle\lesssim\int_{0}^{t}(\log(e+\|\nabla^{2}\,F\|_{2}^{2}))^{\frac{p-3}{2}}\|\nabla^{2}\,F\|_{2}^{2}\,ds$
$\displaystyle\lesssim\int_{0}^{t}(\log(e+\|\nabla^{2}\,F\|^{2}_{2}))\|\nabla^{2}\,F\|_{2}^{2}\,ds$

where we used that $3<p\leq 5$, i.e. that $(p-3)/2\in(0,1]$. We note that for
$2\leq p\leq 3$ there is no need to use the Brezis-Gallouet inequality since
$|D^{3}W|\leq C$. Therefore
$y(t):=(\|\nabla^{2}\,F(t)\|_{2}^{2}+\|\nabla\,v(t)\|_{2}^{2})$ satisfies the
following differential inequality

$y(t)\lesssim y(0)+\int_{0}^{t}(y(s)+y^{\frac{1}{2}}(s)+y(s)\log(e+y(s)))\,ds$

which implies that

$\displaystyle v\in L^{\infty}(0,T;H^{1}(\mathbb{T}^{2})\cap
L^{2}(0,T;H^{2}(\mathbb{T}^{2})),$ (6.13) $\displaystyle F\in
L^{\infty}(0,T;H^{2}(\mathbb{T}^{2})).$

In particular, from (6.13), it follows that

$\displaystyle F\in L^{\infty}((0,T)\times\mathbb{T}^{2}),$ (6.14)
$\displaystyle\nabla\,F\in L^{\infty}(0,T;L^{q}(\mathbb{T}^{2}))\mbox{ for any
}q<\infty.$

It remains to prove the $H^{3}$-estimate. As in the proof of Proposition 6.1
note that

$\displaystyle\nabla^{3}(S(F))$
$\displaystyle=(D^{4}W(F))(\nabla\,F)(\nabla\,F)(\nabla\,F)+(D^{3}W(F))(\nabla^{2}\,F)(\nabla\,F)+(D^{2}W(F))(\nabla^{3}\,F).$

Therefore, using that $W$ and its derivative have polynomial growth, and Lemma
3.1 (1), we get from (6.13) and (6.14) that

$\|\nabla^{3}\,S(F)\|_{2}^{2}\lesssim
1+\|\nabla^{3}\,F\|_{2}+\|\nabla^{3}\,F\|_{2}^{2}.$

Arguing as in Proposition 6.1 we get the following two differential
inequalities in analogy to (6.2), (6.3):

$\displaystyle\frac{d}{dt}\|\nabla^{3}\,v\|_{2}^{2}+\|\nabla^{4}\,v\|_{2}^{2}$
$\displaystyle\lesssim\|\nabla^{2}\mathop{\mathrm{div}}(S(F))\|_{2}^{2}$
(6.15) $\displaystyle\lesssim
1+\|\nabla^{3}\,F\|_{2}+\|\nabla^{3}\,F\|_{2}^{2}$

and

$\frac{d}{dt}\|\nabla^{3}\,F\|_{2}^{2}\lesssim\|\nabla^{3}\,F\|_{2}^{2}+\frac{1}{4}\|\nabla^{4}\,v\|_{2}^{2},$
(6.16)

where there are no suppressed constants in the term
$\frac{1}{4}\|\nabla^{4}\,v\|_{2}^{2}$. Therefore from (6.15) and (6.16),
$y(t):=(\|\nabla^{3}\,F(t)\|_{2}^{2}+\|\nabla^{3}\,v(t)\|_{2}^{2})$ satisfies

$\dot{y}(t)\leq 1+y^{\frac{1}{2}}(t)+y(t)$

which implies that

$(v,F)\in L^{\infty}(0,T;H^{3}(\mathbb{T}^{2}))\times
L^{\infty}(0,T;H^{3}(\mathbb{T}^{2})).$

Then the case of $W$ satisfying (A4) and $2\leq p\leq 5$ follows by a simple
continuation argument.

Regarding the case of $W$ satisfying (A4′) and $2\leq p\leq 6$ the proof is
exactly the same except for the way we treat the term $I_{2}$ in the
inequality (6.12). We first notice that by exploiting the assumption (A4′) in
the inequality (6.9) we have that (6.12) reads as follows

$\displaystyle\int\left(|\nabla^{2}\,F|^{2}+|\nabla\,v|^{2}\right)\,dx+\int_{0}^{t}\int|\nabla^{2}\,v|^{2}+\int_{0}^{t}\int|F|^{p-2}|\nabla^{2}F|^{2}\,dx\,ds\lesssim\int_{0}^{t}\int|\nabla^{2}\,F|^{2}\,dx\,ds$
$\displaystyle+\int_{0}^{t}\int(1+|F|^{p-3})|\nabla\,F|^{2}|\nabla^{2}\,F|\,dx\,ds+\int_{0}^{t}\int(1+|F|^{p-2})|\nabla\,F||\nabla^{2}\,v|\,dx\,ds$
$\displaystyle+\int\left(|\nabla^{2}\,F_{0}|^{2}+|\nabla\,v_{0}|^{2}\right)\,dx=I_{1}+I_{2}+I_{3}+I_{4}.$

The terms $I_{1}$, $I_{3}$ and $I_{4}$ can be treated in the same as before,
while $I_{2}$ is treated as follows. First, since (A4′) implies (A4) we can
consider only the range $5<p\leq 6$. Then, by using Young’s inequality, Lemma
3.1 and the bound $\nabla F\in L^{\infty}(0,T;H^{1}(\mathbb{T}^{d}))$ we get
that

$\displaystyle I_{2}$
$\displaystyle\lesssim\int_{0}^{t}\int|\nabla\,F|^{2}|\nabla^{2}\,F|\,dx\,ds+\int_{0}^{t}\int|F|^{p-4}|\nabla\,F|^{4}\,dx\,ds+\frac{1}{2}\int_{0}^{t}\int|F|^{p-2}|\nabla^{2}\,F|^{2}\,dx\,ds$
$\displaystyle\lesssim\int_{0}^{t}\|\nabla^{2}F(s)\|_{2}^{2}\,ds+\int_{0}^{t}\left(\ln(e+\|\nabla^{2}F(s)\|_{2})\right)^{\frac{p-4}{2}}\|\nabla^{2}F(s)\|^{2}_{2}\,ds$
$\displaystyle+\frac{1}{2}\int_{0}^{t}\int|F|^{p-2}|\nabla^{2}\,F|^{2}\,ds.$

Then, by noticing that $5<p\leq 6$ implies that $\frac{p-4}{2}\in(0,1]$ we can
conclude as the previous case. ∎

##  7\. Sustained Oscillations

We next record some examples of solutions of the viscoelasticity system that
exhibit sustained oscillations.

###  7.1. A nonlinear problem with non-monotone stress-strain relation

Consider the equations of viscoelasticity in one dimension, for the motion
$y(t,x):(0,T)\times[0,1]\to\mathbb{R}$,

$y_{tt}=\sigma(y_{x})_{x}+y_{txx}$ (7.1)

which upon setting $u=y_{x}$, $v=y_{t}$ is expessed as a system

$\displaystyle u_{t}$ $\displaystyle=v_{x}$ (7.2) $\displaystyle v_{t}$
$\displaystyle=\sigma(u)_{x}+v_{xx}.$

We denote the stress with $\sigma$ instead of $S$ to comply with the standard
notation used in the one-dimensional case. The function $\sigma(u)$ is smooth
and for the purposes of this section it is typically required to be non-
monotone.

Assume there exist two positive states $0<a<b$ such that the values of the
stress function $\sigma(u)$ satisfy

$a+\sigma(ta)=b+\sigma(tb)\qquad\mbox{ $\forall t\in[1,2]$}\,.$ (7.3)

Here, the states $a,b$ are fixed and $t$ is thought of as a parameter.

The condition (7.3) restricts considerably the form of $\sigma(u)$, and we
give an example to show that it can be satisfied. Let $a,b>0$ fixed so that
$0<a<2a<b<2b$ and suppose the graph of $\sigma(u)$ is given and is strictly
increasing for $u\in(b,2b)$. Then $(a,2a)\cap(b,2b)=\emptyset$ and (7.3) fully
determines the graph of $\sigma(u)$ for $u\in(a,2a)$ from the graph in
$(b,2b)$. The emerging graph is increasing in $(a,2a)$ but the full graph will
be non-monotone, see Figure Fig. 1 where a specific example is depicted.

Figure 1. The block parts of the graph satisfy (7.3) for $1\leq t\leq 2$; the
dotted points interpolate between these parts.

The example that is constructed below is based on two properties of (7.2):

  * •

the class $u_{s}(t)=\kappa t$, $v_{s}(x)=\kappa x$ with $\kappa\in\mathbb{R}$
form a special class of universal – for any $\sigma(u)$ – uniform shear
solutions.

  * •

as pointed out by D. Hoff [16] system (7.2) admits solutions continuous in $v$
but discontinuous in $u$ and $v_{x}$. Such solutions satisfy the Rankine-
Hugoniot conditions

$\displaystyle-s[u]$ $\displaystyle=[v]$ $\displaystyle-s[v]$
$\displaystyle=[\sigma(u)+v_{x}]$

where $s$ is the shock speed and $[q]=q_{+}-q_{-}$ the jump of the quantity
$q$. As $v$ is continuous, the shocks are stationary $s=0$ and $[u]\neq 0$ has
to satisfy

$s=0\,,\quad[\sigma(u)+u_{t}]=0\,.$ (7.4)

Next, we construct a family of solutions defined on $[1,2]\times\mathbb{R}$.
To this end, fix states $a,b$ satisfying $0<a<2a<b<2b$, suppose that (7.3) is
satisfied, and denote by $S(t)$ the common value

$S(t):=a+\sigma(ta)=b+\sigma(tb)\qquad 1\leq t\leq 2\,.$

For $0<\theta<1$, define the periodic function

$F(x):=\begin{cases}a,&x\in(k,k+\theta)\\\
b,&x\in(k+\theta,k+1)\end{cases}\quad\;k\in\mathbb{Z}$

and, based on $F$, define

$U(t,x)=tF(x)\,,\quad V_{x}(t,x)=F(x)\,,\quad t\in(1,2)$ (7.5)

and $V(t,x)\,,Y(t,x)$ are defined by setting

$\displaystyle V(t,x)$
$\displaystyle=\bar{V}(x):=\int_{0}^{x}F(y)dy=\begin{cases}k\bar{V}(1)+a(x-k),&x\in(k,k+\theta)\\\
k\bar{V}(1)+a\theta+b(x-k-\theta),&x\in(k+\theta,k+1)\end{cases}$ (7.6)
$\displaystyle Y(t,x)$ $\displaystyle=\int_{0}^{x}U(t,y)dy=t\bar{V}(x)$

where $k\in\mathbb{Z}$, $t\in(1,2)$ and $\bar{V}(1)=(a\theta+b(1-\theta))$.
The function $(U(t,x),V(t,x))$ is a weak solution of (7.2) on
$[1,2]\times\mathbb{R}$ satisfying the equations in a classical sense on the
domains $(1,2)\times(k,k+\theta)$ and $(1,2)\times(k+\theta,k+1)$ and due to
(7.3) it is a weak solution with discontinuities at the interfaces $x=k$ and
$x=k+\theta$ satisfying the Rankine-Hugoniot conditions (7.4).

The functions $Y(t,x)$, $V(t,x)$, $U(t,x)$ are then rescaled and restricted to
the interval $(1,2)\times(0,1)$ as follows

$\displaystyle y_{n}(t,x)$ $\displaystyle=\frac{1}{n}Y(t,nx)\,,\qquad$
$\displaystyle v_{n}(t,x)=\frac{1}{n}(\partial_{t}Y)(t,nx)=\frac{1}{n}V(t,nx)$
(7.7) $\displaystyle u_{n}(t,x)$ $\displaystyle=U(t,nx)\,,\qquad$
$\displaystyle{v_{n}}_{x}(t,x)=V_{x}(t,nx).$

The equation (7.1) is not invariant under the scaling
$y_{n}(t,x)=\frac{1}{n}Y(t,nx)$. However, since (7.5), (7.6) is stationary, we
easily check that (7.7) are a class of exact weak solutions of (7.2) defined
for $x\in(0,1)$, $t\in[1,2]$. Moreover, one easily computes the limits

$\displaystyle v_{n}(t,x)$ $\displaystyle\to(a\theta+b(1-\theta))x\quad\mbox{
strongly in $L^{2}\big{(}(1,2)\times(0,1)\big{)}$ }$ $\displaystyle
u_{n}(t,x)$ $\displaystyle\rightharpoonup(a\theta+b(1-\theta))t\quad\mbox{
weakly-$\star$ in $L^{\infty}\big{(}(1,2)\times(0,1)\big{)}$ }$
$\displaystyle\sigma(u_{n})+\partial_{x}v_{n}=S(t)$

and

$\sigma(u_{n})\rightharpoonup\theta\sigma(at)+(1-\theta)\sigma(bt)\neq\sigma\big{(}\theta
at+(1-\theta)bt).$

The reader should note that the oscillations are induced by the oscillations
in the initial data of $u_{n}(1,x)$.

###  7.2. The linear problem

Consider next the linear one-dimensional system

$y_{tt}=\kappa y_{xx}+y_{txx}\qquad x\in(-\pi,\pi)\,,t>0$ (7.8)

where the motion $y(t,x)$ is now defined on the torus,
$y:(0,T)\times\mathbb{T}^{1}\to\mathbb{R}$. We investigate a class of
oscillatory solutions of the form

$y_{n}=e^{inx}e^{\lambda_{n}t}\,;$

these functions are periodic and will satisfy (7.8) provided $\lambda_{n}$ is
a root of

$\lambda^{2}+\lambda n^{2}+\kappa n^{2}=0.$

The two roots are

$\lambda_{\pm}=n^{2}\left(-\tfrac{1}{2}\pm\tfrac{1}{2}\sqrt{1-\tfrac{4\kappa}{n^{2}}}\right).$

Both roots are real and negative (for large $n$) and the smallest in absolute
value corresponds to the $+$ sign and has the asymptotic behaviour

$\displaystyle\lambda_{n+}$
$\displaystyle=n^{2}\left(-\kappa\frac{1}{n^{2}}-\kappa^{2}\frac{2}{n^{4}}+\dots\right)$
$\displaystyle=-\kappa-\kappa^{2}\frac{2}{n^{2}}+\dots$

Consider now the rescaled solution
$y_{n}=\tfrac{1}{n}e^{inx}e^{-\lambda_{n}t}$ and observe that the associated
$(u_{n},v_{n})$ have the behavior

$u_{n}(t,x)=ie^{inx}e^{-\lambda_{n+}t}=ie^{inx-\kappa t}g_{n}(t)$

where $g_{n}(t)\to 1$ and thus $u_{n}$ has persistent oscillations as
$n\to\infty$. Again such oscillations are induced from oscillatory initial
data. By contrast,

$v_{n}=-\frac{\lambda_{n+}}{n}e^{inx}e^{-\lambda_{n+}t}\to 0\qquad\mbox{as
$n\to\infty$}.$

##  Appendix A Transfer of dissipation

Consider the system (1.3) with small viscosity $\varepsilon>0$, written in
coordinate form,

$\displaystyle\partial_{t}\,v_{i}-\partial_{\alpha}(S_{i\alpha}(F))-\varepsilon\Delta\,v_{i}$
$\displaystyle=0$ (A.1)
$\displaystyle\partial_{t}\,F_{i\alpha}-\partial_{\alpha}\,v_{i}$
$\displaystyle=0$
$\displaystyle\partial_{\alpha}F_{i\beta}-\partial_{\beta}F_{i\alpha}$
$\displaystyle=0\,,$

with periodic boundary conditions and initial data (1.4). We assume for the
appendix that $W(F)$ is convex and as usual $S_{i\alpha}=\frac{\partial
W}{\partial F_{i\alpha}}$, and discuss the formal estimates available in the
zero-viscosity limit.

First, the energy estimate reads

$\frac{d}{dt}\int\tfrac{1}{2}|v|^{2}+W(F)\,dx+\int\varepsilon|\nabla
v|^{2}\,dx=0$ (A.2)

and provides $L^{p}$ control for the family of solutions
$(v^{\varepsilon},F^{\varepsilon})$ and the weak control $\varepsilon|\nabla
v^{\varepsilon}|^{2}\in L^{1}_{t,x}$ familiar from the theory of compensated
compactness.

Using (A.1) one obtains the identities

$\displaystyle\big{(}\partial_{\alpha}S_{i\alpha}(F)\big{)}(\partial_{\beta}F_{i\beta})-(\partial_{\alpha}v_{i})(\partial_{\alpha}v_{i})$
$\displaystyle=(\partial_{\beta}F_{i\beta})(\partial_{t}v_{i})-(\partial_{\alpha}v_{i})(\partial_{t}F_{i\alpha})-\varepsilon(\partial_{\beta}F_{i\beta})\Delta
v_{i}$
$\displaystyle=\partial_{t}\big{(}v_{i}\partial_{\beta}F_{i\beta}\Big{)}-\partial_{\alpha}\big{(}v_{i}\partial_{t}F_{i\alpha}\big{)}-\varepsilon(\partial_{\beta}F_{i\beta})\partial_{t}(\partial_{\alpha}F_{i\alpha}\big{)}.$

Integrating over the torus gives

$\frac{d}{dt}\int\varepsilon\frac{1}{2}|\mathop{\mathrm{div}}F|^{2}-v\cdot\mathop{\mathrm{div}}Fdx+\int\big{(}\partial_{\alpha}S_{i\alpha}(F)\big{)}(\partial_{\beta}F_{i\beta})-(\partial_{\alpha}v_{i})(\partial_{\alpha}v_{i})dx=0.$

Next, a use of integration by parts gives

$\displaystyle I$
$\displaystyle=\int\big{(}\partial_{\alpha}S_{i\alpha}(F)\big{)}(\partial_{\beta}F_{i\beta})-|\nabla
v|^{2}dx$ $\displaystyle=\int\frac{\partial^{2}W}{\partial F_{i\alpha}\partial
F_{k\gamma}}\partial_{\beta}F_{k\gamma}\partial_{\alpha}F_{i\beta}-|\nabla
v|^{2}dx$ $\displaystyle=\int\frac{\partial^{2}W}{\partial F_{i\alpha}\partial
F_{k\gamma}}\partial_{\beta}F_{k\gamma}\partial_{\beta}F_{i\alpha}-|\nabla
v|^{2}dx$ $\displaystyle=\int D^{2}W:(\nabla F,\nabla F)-|\nabla v|^{2}dx$

and implies

$\frac{d}{dt}\int\varepsilon^{2}\frac{1}{2}|\mathop{\mathrm{div}}F|^{2}-\varepsilon
v\cdot\mathop{\mathrm{div}}F\,dx+\int\varepsilon(D^{2}W:(\nabla F,\nabla
F)-|\nabla v|^{2})\,dx=0.$ (A.3)

Combining (A.2) with (A.3) we arrive at

$\frac{d}{dt}\int\tfrac{1}{2}|v-\tfrac{\varepsilon}{2}\mathop{\mathrm{div}}F|^{2}+\tfrac{\varepsilon^{2}}{4}|\mathop{\mathrm{div}}F|^{2}+W(F)\,dx+\frac{\varepsilon}{2}\int
D^{2}W:(\nabla F,\nabla F)+|\nabla v|^{2}\,dx=0.$ (A.4)

Identity (A.4) imples that, under conditions of uniform convexity for $W(F)$,

$D^{2}W\geq c\;\mathbb{I}\,,\quad\mbox{for some $c>0$},$

in addition to the uniform bounds $\varepsilon|\nabla
v^{\varepsilon}|^{2}\in_{b}L^{1}_{t,x}$ one also obtains control
$\varepsilon|\nabla F^{\varepsilon}|^{2}\in_{b}L^{1}_{t,x}$. The estimate
(A.4) extends to several space dimensions an observation of DiPerna [10] in
connection to the problem of zero-viscosity limits in one space dimension. A
similar property holds for relaxation approximations of the nonlinear
elasticity system in one-space dimension [32].

######  Remark A.1.

Identity (A.4) in conjunction with the energy (A.2) provides the main a-priori
estimates for system (1.3). If for example one assumes hypothesis (AB) and use
$\varepsilon=1$, Lemma 2.1 and the identity then we conclude

$\frac{d}{dt}\int|v-\tfrac{1}{2}\mathop{\mathrm{div}}F|^{2}+\tfrac{1}{2}|\nabla
F|^{2}+2W(F)\,dx+\int\Big{(}D^{2}\tilde{W}:(\nabla F,\nabla F)+|\nabla
v|^{2}\Big{)}dx\leq K\int|\nabla F|^{2}dx.$ (A.5)

Using Grönwall’s lemma, (A.5) yields control of $\int|\nabla F|^{2}dx$.

##  Appendix B Diffusion-dispersion approximations of elasticity

Diffusion-dispersion approximations arise naturally in studies of phase
transitions in elasticity. One way to introduce such systems is to consider a
nonlinear evolution $y(t,x)$ generated by an energy functional

$\frac{\partial^{2}y}{\partial t^{2}}=-\frac{\delta\mathcal{E}}{\delta
F}[y]\,,$

where $\mathcal{E}[y]$ is a functional on the motion $y$. A simple example is
provided by strain gradient theories of the form

$\mathcal{E}[y]=\int W(\nabla y)+\delta A\frac{1}{2}|\Delta y|^{2}dx\,.$

Here, $W(F)$ is assumed nonconvex to allow models with phase transitions and
the term with the higher order gradient is motivated by the Korteweg theory.
It leads to the second order nonlinear partial differential equation

$\frac{\partial^{2}y}{\partial
t^{2}}=\mathop{\mathrm{div}}\Big{(}\frac{\partial W}{\partial F}(\nabla
y)-\delta A\nabla\Delta y\Big{)}\,.$ (B.1)

It is known that for strain energies that are non-convex, viscosity is not
sufficient to select the admissible shocks and, motivated by the Korteweg
theory, Slemrod [26] and Truskinovsky [30] proposed to include the effects of
capillarity.

Adding viscosity to (B.1) produces a diffusive-dispersive approximation of the
elasticity equations, in the form of the system

$\displaystyle\partial_{t}\,v_{i}-\partial_{\alpha}S_{i\alpha}(F)$
$\displaystyle=\varepsilon\Delta\,v_{i}-\delta A\partial_{\alpha}\Delta
F_{i\alpha}$ (B.2)
$\displaystyle\partial_{t}\,F_{i\alpha}-\partial_{\alpha}\,v_{i}$
$\displaystyle=0$
$\displaystyle\partial_{\alpha}F_{i\beta}-\partial_{\beta}F_{i\alpha}$
$\displaystyle=0\,$

with $W(F)$ is nonconvex. Here $\varepsilon>0$, $\delta>0$ are positive
parameters while $A$ is a numerical constant. The involution (B.2)3 is a
constraint induced on solutions by the initial data. In one space dimension in
the limit as $\varepsilon,\delta\to 0$ diffusion dominates dispersion in (at
least) the range $\delta=O(\varepsilon^{2})$. A clever transformation of
variables discovered by Slemrod [27] indicates that (B.2) in 1-$d$ can be
transformed to a viscosity approximation for a system of conservation laws.

A generalization of this observation to multi-$d$ is provided here. Let
$\kappa$ be a parameter (to be selected) and write (B.2)1, (B.2)2,
respectively, as follows:

$\displaystyle\partial_{t}(v_{i}-\kappa\partial_{\gamma}F_{i\gamma})-\partial_{\alpha}S_{i\alpha}(F)$
$\displaystyle=(\varepsilon-\kappa)\Delta(v_{i}-\kappa\partial_{\alpha}F_{i\alpha})+[(\varepsilon-\kappa)\kappa-\delta
A]\partial_{\alpha}\Delta F_{i\alpha}$
$\displaystyle\partial_{t}F_{i\alpha}-\partial_{\alpha}\big{(}v_{i}-\kappa\partial_{\gamma}F_{i\gamma}\big{)}$
$\displaystyle=\kappa\Delta F_{i\alpha}.$

Observe that if $\kappa$ is selected by

$\displaystyle\kappa^{2}-\varepsilon\kappa+\delta A=0$ (B.3) $\displaystyle
0<\kappa<\varepsilon$

then (B.2) reduces to the hyperbolic parabolic system

$\displaystyle\partial_{t}w-\mathop{\mathrm{div}}S(F)$
$\displaystyle=(\varepsilon-\kappa)\Delta w$ (B.4)
$\displaystyle\partial_{t}F-\nabla w$ $\displaystyle=\kappa\Delta F$
$\displaystyle\mathop{\mathrm{curl}}F$ $\displaystyle=0$

which describes the evolution of the function $(w,F)$ with
$w=v-\kappa\mathop{\mathrm{div}}F$. Again $\mathop{\mathrm{curl}}F=0$ is an
involution propagated from the initial data via the equation (B.4)2.

One easily checks the solvability of (B.3). The roots of the quadratic are

$\kappa_{\pm}=\frac{\varepsilon}{2}\pm\frac{\varepsilon}{2}\sqrt{1-\frac{4\delta
A}{\varepsilon^{2}}}.$

We deduce that if $\delta=\varepsilon^{\rho}$, $\rho>2$, then for any $A$ we
can select $\kappa$ that fulfills the condition $0<\kappa<\varepsilon$; in the
borderline case $\delta=\varepsilon^{2}$ the parameter $A$ is restricted to be
$0<A\leq\frac{1}{4}$. An interesting special case occurs for
$\delta=\varepsilon^{2}$, $A=\frac{1}{4}$ and leads to a system with identity
viscosity matrix

$\displaystyle\partial_{t}w-\mathop{\mathrm{div}}S(F)$
$\displaystyle=\tfrac{\varepsilon}{2}\Delta w$
$\displaystyle\partial_{t}F-\nabla w$
$\displaystyle=\tfrac{\varepsilon}{2}\Delta F\,.$

##  Appendix C A discussion on the assumptions on the stored energy

We briefly comment on the set of assumptions (A1)-(A4) on $W$, which we
rewrite for the reader’s convenience. For $p\geq 2$ we assume $W$ satisfies

  * (A1)

$W\in C^{2}(\mathbb{R}^{d\times d};\mathbb{R})$.

  * (A2)

There exists $c>0$ such that

$c(|F|^{p}-1)\leq W(F).$

  * (A3)

There exists $C>0$ such that

$\displaystyle|W(F)|\leq C(1+|F|^{p});\,$ $\displaystyle|DW(F)|\leq
C(1+|F|^{p-1}).$

  * (A4)

There exists constant $K>0$ such that

$(DW(F_{1})-DW(F_{2}),F_{1}-F_{2})\geq-K|F_{1}-F_{2}|^{2}.$

We start by remarking that the growth condition on $DW$ in (A3) is redundant
if we assume (A4). Indeed, by Lemma 2.1, we have that $\tilde{W}$ is convex
with a $p$-growth, because of the growth of $W$. Then, it is well-known that
$D\tilde{W}$ must have a $p-1$-growth, see for example [6, Proposition 2.32],
and therefore $DW$ has a $p-1$-growth as well.

A technical, yet necessary assumption in the analysis of the uniqueness
problem is (2.12), that is the following condition on the second derivative on
$W$: there exists $C>0$ such that

$\displaystyle|D^{2}W(F)|\leq C(1+|F|^{s}),\mbox{ with }s\geq p-2.$ (C.1)

The following example shows that (C.1) is not a consequence of (A1)-(A4).

######  Example C.1.

Let $n=1$ and $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the function
$\phi(x):=e^{\frac{1}{1-|x|^{2}}}\chi_{[-1,1]}(x)$. Let $k\in\mathbb{N}$ with
$k\geq 2$ and define

$\phi_{k}(x):=e^{k}\phi(e^{3k}(x-k)).$

Note that $supp\,\phi_{k}=[k-e^{-3k},k+e^{-3k}]$. Let $g\in
C(\mathbb{R}^{+};\mathbb{R}^{+})$ given by

$g(x):=1+\sum_{k=2}^{\infty}\phi_{k}(x),\,x>0.$

Define $f:\mathbb{R}\to\mathbb{R}$ first by solving for $x>0$

$\displaystyle f^{\prime\prime}(x)=g(x),\quad$ $\displaystyle
f^{\prime}(0)=0,$ $\displaystyle f(0)=0,$ (C.2)

and then by extending the resulting $f$ in an even way for negative $x$’s.
Then, by construction $f^{\prime\prime}$ saturates the exponential growth.
Moreover, $f\in C^{2}(\mathbb{R};\mathbb{R})$ and it is convex. Therefore it
satisfies (A4). It remains to check that the growth and coercivity conditions
are satisfied. It is enough to consider the case $x>0$. Integrating (C.2) we
have that , since $\phi_{k}$ is positive for any $k$,

$\displaystyle x\leq f^{\prime}(x)$ $\displaystyle\leq
x+\sum_{k=2}^{\infty}\int_{k-e^{-3k}}^{k+e^{-3k}}e^{k}\phi(e^{3k}(y-k))\,dy$
$\displaystyle\leq x+\sum_{k=2}^{\infty}e^{-2k}\int_{-1}^{1}\phi(z)\,dz$
$\displaystyle\leq C(x+1).$

Then, after integrating again we also get that $\frac{x^{2}}{2}\leq f(x)\leq
C(x^{2}+1)$ and therefore $f$ satisfies also (A2) and (A3).

We also note that, because of the coercivity assumption (A2), the lower bound
on $s$ in (C.1) is necessary.

######  Lemma C.2.

Let $k\in\mathbb{N}$, $p>k$ and $f\in C^{k}(\mathbb{R}^{n};\mathbb{R})$
$p$-coercive. Then, if there exists $C>0$ and $s\geq 0$ such that
$|D^{k}f(x)|\leq C(1+|x|^{s})$ for any $x\in\mathbb{R}^{n}$, it must hold that
$s\geq p-k$.

###### Proof.

Without loss of generality we can assume that $n=1$, $f$ positive and $x>0$.
Moreover, it is enough to prove the lemma for $k=1$. For general $k$ it will
follow by induction. Let $p>1$, $f\in C^{1}(\mathbb{R};\mathbb{R})$, $C>0$,
and $s>0$ such that for any $x>0$

$\displaystyle C(x^{p}-1)\leq f(x);$ $\displaystyle|f^{\prime}(x)|\leq
C(1+x^{s}).$

Assume there exists $\varepsilon>0$ such that $0\leq s=p-1-\varepsilon$. Then,
for some $C>0$

$C(x^{p}-1)\leq f(x)\leq\int_{0}^{x}|f^{\prime}(y)|\,dy+C\leq C(1+x+x^{s+1}).$

Therefore, for $C>0$ we have that for any $x>0$

$x^{p}\leq C(1+x+x^{p-\varepsilon})$

which is a contradiction since $p>1$. ∎

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