# Two body hadronic $D$ decays

Yu Fusheng, Cai-Dian Lü, Xiao-Xia Wang

We analyze the decay modes of $D/D_{s}\to PP,PV$ on the basis of a hybrid
method with the generalized factorization approach for emission diagrams and
the pole dominance model for the annihilation type contributions. Our results
of PV final states are better than the previous method, while the results of
PP final states are comparable with previous diagrammatic approach.

##  1 Introduction

The CLEO-c and the two B factories already give more measurements of charmed
meson decays than ever. The BESIII and super B factories are going to give
even much more data soon. Therefore, it is a good chance to further study the
nonleptonic two-body $D$ decays. However, it is theoretically unsatisfied
since some model calculations, such as QCD sum rules or Lattice QCD, are
ultimate tools but formidable tasks. In $B$ physics, there are QCD-inspired
approaches for hadronic decays, such as the perturbative QCD approach
(pQCD),$\\!{}^{{\bf?}}$ the QCD factorization approach (QCDF),$\\!{}^{{\bf?}}$
and the soft-collinear effective theory (SCET).$\\!{}^{{\bf?}}$ But it doesn’t
make much sense to apply these approaches to charm decays, since the mass of
charm quark, of order 1.5 GeV, is neither heavy enough for a sensible
$1/m_{c}$ expansion, nor light enough for the application of chiral
perturbation theory.

After decades of studies, the factorization approach is still an effective way
to investigate the hadronic $D$ decays $\\!{}^{{\bf?}}$. However, the naive
factorization encounters well-known problems: the Wilson coefficients are
renormalization scale and $\gamma_{5}$-scheme dependent, and the color-
suppressed processes are not well predicted due to the smallness of $a_{2}$.
The generalized factorization approaches were proposed to solve these
problems, considering the significant nonfactorizable contributions in the
effective Wilson coefficients $\\!{}^{{\bf?}}$. Besides, in the naive or
generalized factorization approaches, there are no strong phases between
different amplitudes, which are demonstrated to be existing by experiments.

On the other hand, the hadronic picture description of non-leptonic weak
decays has a longer history, because of their non-perturbative feature. Based
on the idea of the vector dominance, which is discussed on strange particle
decays,$\\!{}^{{\bf?}}$ the pole-dominance model of two-body hadronic decays
was proposed.$\\!{}^{{\bf?}}$ This model has already been applied to the two-
body nonleptonic decays of charmed and bottom mesons $\\!{}^{{\bf?},{\bf?}}$.

In this work, the two-body hadronic charm decays are analyzed based on a
hybrid method with the generalized factorization approach for emission
diagrams and the pole dominance model for the annihilation type contributions
$\\!{}^{{\bf?}}$.

##  2 The hybrid method

In charm decays, we start with the weak effective Hamiltonian for the $\Delta
C=1$ transition

$\mathcal{H}_{eff}=\frac{G_{F}}{\sqrt{2}}V_{CKM}(C_{1}O_{1}+C_{2}O_{2})+h.c.,$
(1)

with the current-current operators

$\displaystyle
O_{1}=\bar{u}_{\alpha}\gamma_{\mu}(1-\gamma_{5})q_{2\beta}\cdot\bar{q}_{3\beta}\gamma^{\mu}(1-\gamma_{5})c_{\alpha},$
$\displaystyle
O_{2}=\bar{u}_{\alpha}\gamma_{\mu}(1-\gamma_{5})q_{2\alpha}\cdot\bar{q}_{3\beta}\gamma^{\mu}(1-\gamma_{5})c_{\beta}.$
(2)

In the generalized factorization method, the amplitudes are separated into two
parts

$\langle
M_{1}M_{2}|\mathcal{H}_{eff}|D\rangle=\frac{G_{F}}{\sqrt{2}}V_{CKM}a_{1,2}\langle
M_{1}|\bar{q}_{1}\gamma_{\mu}(1-\gamma_{5})q_{2}|0\rangle\langle
M_{2}|\bar{q}_{3}\gamma^{\mu}(1-\gamma_{5})c|D\rangle,$ (3)

where $a_{1}$ and $a_{2}$ correspond to the color-favored tree diagram
($\mathcal{T}$) and the color-suppressed diagram ($\mathcal{C}$) respectively.
To include the significant non-factorizable contributions, we take $a_{1,2}$
as scale- and process-independent parameters fitted from experimental data.
Besides, a large relative strong phase between $a_{1}$ and $a_{2}$ is
demonstrated by experiments. Theoretically, the existence of large phase is
reasonable for the importance of inelastic final state interactions in the
charmed meson decays, with on-shell intermediate states. Therefore, we take

$a_{1}=|a_{1}|,~{}~{}a_{2}=|a_{2}|e^{i\delta},$ (4)

where $a_{1}$ is set to be real for convenience.

On the other hand, annihilation type contributions are neglected in the
factorization approach. However, the weak annihilation ($W$-exchange and
$W$-annihilation) contributions are sizable, of order $1/m_{c}$, and have to
be considered. It is also demonstrated to be important by the difference of
life time between $D^{0}$ and $D^{+}$. The pole-dominance model is a useful
tool to calculate the considerable resonant effects of annihilation diagrams.
For simplicity, only the lowest-lying pole is considered in the single-pole
model. Taking $D^{0}\to PP,PV$ as example, the annihilation type diagram in
the pole model is shown in Fig.1(a). $D^{0}$ goes into the intermediate state
$M$ via the effective weak Hamiltonian in Eq.(1), shown by the quark line in
the Fig.1(b), and then decays into $PP(PV)$ through strong interactions.
Angular momentum should be conserved at the weak vertex, and all conservation
laws be preserved at the strong vertex. Therefore, the intermediate particles
are scalar mesons for $PP$ modes and pseudoscalar mesons for $PV$ modes. In
$D^{0}$ decays, they are $W$-exchange diagrams, but $W$-annihilation
amplitudes in the $D^{+}_{(s)}$ decay modes.

Figure 1: Annihilation diagram in the pole-dominance model

The weak matrix elements are evaluated in the vacuum insertion
approximation$\\!{}^{{\bf?}}$,

$\displaystyle\langle M|\mathcal{H}|D\rangle$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{CKM}a_{A,E}\langle
M|\bar{q}_{1}\gamma_{\mu}(1-\gamma_{5})q_{2}|0\rangle\langle
0|\bar{q}_{3}\gamma^{\mu}(1-\gamma_{5})c|D\rangle$ (5) $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{CKM}a_{A,E}f_{M}f_{D}m_{D}^{2},$

where the effective coefficients $a_{A}$ and $a_{E}$ correspond to
$W$-annihilation and $W$-exchange amplitudes respectively. Strong phases
relative to the emission diagrams are also considered in these coefficients.

For the $PV$ modes, the effective strong coupling constants are defined
through the Lagrangian

$\mathcal{L}_{VPP}=ig_{VPP}V^{\mu}(P_{1}{\partial}_{\mu}P_{2}-P_{2}{\partial}_{\mu}P_{1}),$
(6)

where $g_{VPP}$ is dimensionless and obtained from experiments. By inserting
the propagator of the intermediate state $M$, the annihilation amplitudes are

$\langle
PV|\mathcal{H}_{eff}|D\rangle=\frac{G_{F}}{\sqrt{2}}V_{CKM}a_{A,E}f_{M}f_{D}m_{D}^{2}\frac{1}{m_{D}^{2}-m_{M}^{2}}g_{VPM}2(\varepsilon^{*}\cdot
p_{D}).$ (7)

As for the $PP$ modes, the intermediate mesons are scalar particles. The
effective strong coupling constants are described by

$\mathcal{L}_{SPP}=-g_{SPP}m_{S}SPP.$ (8)

However, the decay constants of scalar mesons are very small, which is shown
in the following relation

$\displaystyle\frac{f_{S}}{\bar{f}_{S}}=\frac{m_{2}(\mu)-m_{1}(\mu)}{m_{S}},$
(9)

where $f_{S}$ is the vector decay constant used in the pole model,
$\bar{f}_{S}$ is the scale-independent scalar decay constant, $m_{1,2}$ are
the running current quark mass, and $m_{S}$ is the mass of scalar meson.
Therefore, the scalar pole contribution is very small, resulting in little
resonant effect of annihilation type contributions in the $PP$ modes. On the
contrary, large annihilation contributions are given in the $PV$ modes by
relative large decay constants of intermediate pseudoscalar mesons.

##  3 Numerical results and discussions

In this method, only the effective Wilson coefficients with relative strong
phases are free parameters, which are chosen to obtain the suitable results
consistent with experimental data. For $PP$ modes,

$\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle 0.94\pm
0.10,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{2}=(0.65\pm
0.10)e^{i(142\pm 10)^{\circ}},$ $\displaystyle a_{A}$ $\displaystyle=$
$\displaystyle(0.20\pm 0.10)e^{i(300\pm 10)^{\circ}},~{}~{}a_{E}=(1.7\pm
0.1)e^{i(90\pm 10)^{\circ}}.$ (10)

For $PV$ modes,

$\displaystyle a_{1}^{PV}$ $\displaystyle=$ $\displaystyle 1.32\pm
0.10,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{2}^{PV}=(0.75\pm
0.10)e^{i(160\pm 10)^{\circ}},$ $\displaystyle a_{A}^{PV}$ $\displaystyle=$
$\displaystyle(0.12\pm 0.10)e^{i(345\pm 10)^{\circ}},~{}~{}a_{E}^{PV}=(0.62\pm
0.10)e^{i(238\pm 10)^{\circ}}.$ (11)

All the predictions of the 100 channels are shown in the tables of
ref.$\\!{}^{{\bf?}}$. The prediction of branching ratio of the pure
annihilation process $D_{s}^{+}\to\pi^{+}\pi^{0}$ vanishes in the pole model
within the isospin symmetry. It is also zero in the diagrammatic approach in
the flavor SU(3) symmetry. Simply, two pions can form an isospin 0,1,2 state,
but 0 is ruled out because of charged final states, and isospin-2 is forbidden
for the leading order $\Delta C=1$ weak decay. The only left s-wave isospin-1
sate is forbidden by Bose-Einstein statics. In the pole model language, $G$
parity is violated in the isospin-1 case. Therefore, no annihilation amplitude
contributes to this mode.

The theoretical analysis in the $\eta-\eta^{\prime}$ sector is kind of
complicated. The predictions with $\eta^{\prime}$ in the final state are
always smaller in this hybrid method than those case of $\eta$ due to the
smaller phase space. However, it is opposite by experiments in some modes,
such as $D_{s}^{+}\to\pi^{+}\eta(\eta^{\prime})$,
$D^{0}\to\bar{K}^{0}\eta(\eta^{\prime})$. This may be the effects of SU(3)
flavor symmetry breaking for $\eta_{q}$ and $\eta_{s}$, the error mixing angle
between $\eta$ and $\eta^{\prime}$ aaaThe theoretical and phenomenological
estimates for the mixing angle $\phi$ is $42.2^{\circ}$ and $(39.3\pm
1.0)^{\circ}$, respectively.$\\!{}^{{\bf?}}$, inelastic final state
interaction, or the two gluon anomaly mostly associated to the
$\eta^{\prime}$, etc.. The mode of $D_{s}^{+}\to\rho^{+}\eta(\eta^{\prime})$
is similar with the above two cases, the opposite ratio of $\eta$ over
$\eta^{\prime}$ between theoretical prediction and the data. But this is a
puzzle by experiment measurement, which is taken more than ten years ago
$\\!{}^{{\bf?}}$. As is questioned by PDG $\\!{}^{{\bf?}}$, this branching
ratio of $(12.5\pm 2.2)\%$ considerably exceeds the recent inclusive
$\eta^{\prime}$ fraction of $(11.7\pm 1.8)\%$.

Recently, model independent diagrammatic approach is used to analyze the charm
decays $\\!{}^{{\bf?}}$. All two-body hadronic decays of $D$ mesons can be
expressed in terms of some distinct topological diagrams within the SU(3)
flavor symmetry, by extracting the topological amplitudes from the data
$\\!{}^{{\bf?}}$. Since the recent measurements of
$D_{s}^{+}\to\pi^{+}\rho^{0}$ $\\!{}^{{\bf?}}$ and $D_{s}^{+}\to\pi^{+}\omega$
$\\!{}^{{\bf?}}$ give a strong constraint on the $W-$annihilation amplitudes,
one cannot find a nice fit for $A_{P}$ and $A_{V}$ in the diagrammatic
approach to the data with $D_{s}^{+}\to\bar{K}^{*0}K^{+},\bar{K}^{0}K^{*+}$
simultaneously. Compared to the calculations in the model-independent
diagrammatic approach $\\!{}^{{\bf?}}$, our hybrid method gives more
predictions for the $PV$ modes in which the predictions are consistent with
the experimental data. It is questioned that the measurement of
$Br(D_{s}^{+}\to\bar{K}^{0}K^{*+})=(5.4\pm 1.2)\%$,$\\!{}^{{\bf?}}$ which was
taken two decades ago, was overestimated. Since $|C_{V}|<|C_{P}|$ and
$A_{V}\approx A_{P}$ as a consequence of very small rate of
$D_{s}^{+}\to\pi^{+}\rho^{0}$, it is expected that
$Br(D_{s}^{+}\to\bar{K}^{0}K^{*+})<Br(D_{s}^{+}\to\bar{K}^{*0}K^{+})=(3.90\pm
0.23)\%$. Our result in the hybrid method also agrees with this argument.

As an application of the diagrammatic approach, the mixing parameters
$x=(m_{1}-m_{2})/\Gamma$ and $y=(\Gamma_{1}-\Gamma_{2})/\Gamma$ in the
$D^{0}-\bar{D}^{0}$ mixing are evaluated from the long distance contributions
of the $PP$ and $VP$ modes $\\!{}^{{\bf?}}$. The global fit and predictions in
the diagrammatic approach are done in the SU(3) symmetry limit. However, as we
know, the nonzero values of $x$ and $y$ come from the SU(3) breaking effect.
Part of the flavor SU(3) breaking effects are considered in the factorization
method and in the pole model. Therefore, our hybrid method takes its advantage
in the analysis of $D^{0}-\bar{D}^{0}$ mixing.

## Acknowledgments

This work is partially supported by National Science Foundation of China under
the Grant No. 10735080, 11075168; and National Basic Research Program of China
(973) No. 2010CB833000.

## References

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# Solvability of the asymmetric Bingham fluid equations

###### Abstract

In this work, we investigate the asymmetric Bingham fluid equations. The
asymmetric fluid of Bingham includes symmetric and antisymmetric stresses with
such stresses appearing as an elastic response to the micro-rotational
deformations of grains in a complex fluid. We show the global-in-time
solvability of a weak solution for three dimensional boundary value problem
with Navier boundary conditions of the asymmetric Bingham fluid equations.

Anderson Luis Albuquerque de Araujo

Departamento de Matemática,

Universidade Federal de Viçosa. Viçosa, MG, Brasil

Email<EMAIL_ADDRESS>
Nikolai V. Chemetov

Universidade de Lisboa, Edificio C6, 1 Piso, Campo Grande,

1749-016 Lisboa, Portugal

Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
Marcelo M. Santos

Departamento de Matemática,

IMECC-Instituto de Matemática, Estatística e Computação Científica,

UNICAMP-Universidade Estadual de Campinas. Campinas, SP, Brazil

Email<EMAIL_ADDRESS>
Keywords: asymmetric fluids, Bingham plastic fluids, yield stress, global
existence, weak solution.

MSC (2010): 35Q35, 74A20, 76S05

##  1 Introduction

The Newtonian fluid obeys the constitutive relations that the deviation stress
tensor is a linear function of the stress rate-of-strain tensor and if the
fluid is isotropic then the stress and rate-of strain tensors are symmetric.
Many of fluids can not be described by the Newtonian constitutive relations,
such as slurries, animal blood, mud (mixtures of water, clay), viscous
polymers, polymeric suspensions. These fluids does not commence to flow till
the applied stress attains a certain optimum magnitude, called the yield
stress $\tau_{\ast}$, after of that they behave as a Newtonian fluid. An
example is toothpaste, which will not be extruded until a certain pressure is
applied to the tube. Then it is pushed out as a solid plug. The physical
reason for such behaviour is that the fluid contains particles, such as
toothpaste, paints, clay, or large molecules, such as polymers, animal blood,
which have an interaction, creating a weak rigid structure. Therefore a
certain stress is required to break this weak rigid structure. As soon as the
structure has been broken, the particles move with the fluid under viscous
forces. The particles will associate again if the stress is removed. Such
behaviour was firstly presented in an experimental study by Bingham [3], where
he proposed its mathematical model. Later on these type of fluids have been
called as Bingham plastic fluids. The Bingham plastic fluid behaves as a rigid
body at low stresses but flows as a viscous fluid at high stress. Nowadays it
is used as a mathematical model of mud flows in drilling engineering, heavy
oil, lava (being a mix with melting snow, stones), in the handling of
slurries, waxy crude oils. Recent examples concerns the propane flow within
the hydro-fracture [20]. Significant efforts of the study of the Bingham
plastic fluids have been done by Oldroyd [14], Mossolov, Miasnikov [13],
Glowinski, Wachs [11], Papanastasiou [15] and many others scientists.

The Newtonian flow in the Bingham fluid (after the load greater than the yield
stress $\tau_{\ast}$) has a drastic limitation, since it does not account the
behaviour of the fluid, that contains the particles. Most of above mentioned
fluid systems contain rigid, randomly oriented particles, irregularly shaped
particles (drops in emulsions), branched and entangled molecules in case of
polymeric systems, or loosely formed clusters of particles in suspensions,
etc. The particles may shrink and expand or change their shape, they may
rotate, independently of the rotation of the fluid. To describe accurately the
behaviour of such fluids a so-called asymmetric continuum theory [22] (or
micropolar theory [7], [9]) has been developed that ignores the deformation of
the particles but takes into account geometry, intrinsic motion of material
particles. This theory is a significant and a simple generalization of the
classical Navier-Stokes model, that describe the Newtonian fluids. Only one
new vector field, called as the angular velocity field, of rotation of
particles is introduced. As a consequence, only one equation is added, that
represents the conservation of the angular momentum. The asymmetric/micropolar
fluids belong to the class of fluids with non-symmetric stress tensor. This
class of fluids is more general than the classical Newtonian fluids.

Shelukhin, Růžička [21] suggested a mathematical model that describe the
behaviour of an asymmetric/micropolar viscous fluid of Bingham. In the paper
[19] the global solvability for the mathematical model of [21] has been
demonstrated in a special case of one dimensional flow. Later on in [18] the
authors have obtained the solvability of the stationary solution for this
model already for three dimensional case, but in a particular case when the
stress tensor does not have non-symmetric part.

The main objective of the current work is to correct the model suggested in
[21] and to show the well-posedness of a modified model.

The paper is organized as follows:

  * •

First, in the section 2 we describe the model, proposed in [21], and modify
this model.

  * •

In section 3 we explain the main idea of the modification in the Shelukhin-
Růžička model and collect some technical results that are used in our main
result, related with the proof of global-in-time solvability result for the
modified model. In particular, we introduce a potential for the viscous part
of the stress tensor and characterizes completely its sub-differential (see
Proposition 3.7).

  * •

In section 4 we formulate boundary-value problem and a global-in-time
existence theorem 4.1 for the modified model.

  * •

In section 5 we introduce an approximated problem (27), depending on a
regularization index $n\in\mathbb{N}$, and show the solvability of this
approximated problem (27) by Schauder fixed point theorem. Also we derive a
priori estimates for the solution of (27), which are independent on $n$;

  * •

Finally, in section 6 we prove Theorem 4.1, applying the Lions-Aubin
compactness theorem and a priori estimates of the section 5.

##  2 Model of asymmetric Bingham fluids

In what follows we explain the mathematical model of asymmetric Bingham fluid
that was proposed in the article [21]. For simplicity of consideration in this
article we consider a particular case when the angular velocity field is zero.

For any matrix $X\in\mathbb{R}^{3\times 3}$ we define the symmetric and
asymmetric parts

$X_{s}=\frac{1}{2}\left(X+X^{T}\right)\qquad\text{and}\qquad
X_{a}=\frac{1}{2}\left(X-X^{T}\right)$ (1)

with the adjoint matrix $X^{T}$ having the property
$\left(X^{T}\right)_{ij}=x_{ji}$. Also we denote

$X^{d}=X-\frac{\mbox{tr}\,X}{3}I\qquad\text{with}\qquad\mbox{tr}\,X=\sum_{i=1}^{3}x_{ii}.$

For any matrices $X,Y\in\mathbb{R}^{3\times 3}$ the scalar product $X:Y$ and
the modulus $|X|$ of $X$ are defined by

$X:Y=\sum_{i,j=1}^{3}x_{ij}y_{ij},\qquad|X|=\left(X:X\right)^{1/2}.$ (2)

Let $\mathbf{v}=\mathbf{v}(\mathbf{x},t)$ be the velocity of the mass center
of the material point $({\bm{\xi}},t)$ for an asymmetric Bingham fluid. We
denote the rate of strain tensor

$\mathrm{B}=\mathrm{B}(\mathbf{v})=\frac{\partial\mathbf{v}}{\partial\mathbf{x}},\qquad(\partial\mathbf{v}/\partial\mathbf{x})_{ij}=\partial
v_{i}/\partial x_{j},$ (3)

and introduce the matrix

$\mathrm{B}_{0}=\mathrm{B}_{s}+\varkappa\mathrm{B}_{a},\qquad\varkappa=\frac{2\mu_{1}}{\mu_{2}}$

where

$\qquad\mathrm{B}_{s}=\left(\frac{\partial\mathbf{v}}{\partial\mathbf{x}}\right)_{s}\qquad\text{and}\qquad\mathrm{B}_{a}=\left(\frac{\partial\mathbf{v}}{\partial\mathbf{x}}\right)_{a}$
(4)

are the symmetric and asymmetric parts of $\mathrm{B}=\mathrm{B}(\mathbf{v}),$
respectively. The positive constants $\mu_{i}$ are viscosities of the
asymmetric Bingham fluid. An instant stress state of the fluid is described by
the Cauchy stress tensor $\mathrm{T}=-p\,\mathrm{I}+\mathrm{S}$, where $p$ and
$\mathrm{S}$ are the pressure and the viscous part of the stress tensor. In
[19] the viscous part $\mathrm{S}$ of the stress tensor of the fluid was
suggested to be expressed as

$\mathrm{S}=\left\\{\begin{array}[]{ll}2\mu_{1}\mathrm{B}_{0}+\tau_{\ast}\frac{\mathrm{B}_{0}}{|\mathrm{B}_{0}|},&\mathrm{B}_{0}\neq
0\\\ \mathrm{S}_{plug},&\mathrm{B}_{0}=0\end{array}\right.$ (5)

for some tensor $\ \mathrm{S}_{plug}\in\mathbb{R}^{3\times 3},$ such
that$\quad|\mathrm{S}_{plug}|\leqslant\tau_{\ast}.$ The positive constant
$\tau_{\ast}$ is the yield stress.

Finally, we have the momentum balance law describing the motion of asymmetric
fluid of Bingham

$\rho\dot{\mathbf{v}}=\mbox{div}\,\mathrm{T}+\rho\mathbf{f},$ (6)

where $\rho$ is the density and $\mathbf{f}$ is the mass force vector.

######  Remark 2.1.

Let us introduce the potential

$V(X)=\mu_{1}|X|^{2}+\tau_{\ast}|X|,\qquad\forall\,X\in\mathbb{R}^{3\times
3}.$

Then the constitutive law (5) for the asymmetric Bingham fluid can be
formulated as $\mathrm{S}\in\partial V(\mathrm{B}_{0}).$ Let us remind that
this inclusion is equivalent to the variational inequality

$V(X)-V(\mathrm{B}_{0})\geqslant\mathrm{S}:(X-\mathrm{B}_{0}),\qquad\forall\,X\in\mathbb{R}^{3\times
3}.$ (7)

Nevertheless that the relation (5) describes a plug zone in the Bingham fluid,
there exists a significant restriction in such modelling. As we mentioned in 1
Introduction, in the articles [18], [19] the solvability of the model (5)-(6)
was shown only in the case when the asymmetric part is not present in the
tensor $\mathrm{S}$. As it is well known, one of the principal approach for
the study of problems with inclusions is the theory of monotone operators,
that was developed by Duvaut, Lions [8]. The major difficulty in the study of
Shelukhin-Růžička model [21] consists in the presence of the term
$\mathrm{S}:\mathrm{B}_{0}$ in the inequality (7). The asymmetric part
$\mathrm{B}_{a}$ of $\mathrm{B}_{0}$ does not permit to apply the theory of
monotone operators.

In the following we present our modification in the above described model
(5)-(6) in such way that permits to apply the theory of monotone operators.
Moreover, in our model the viscous part $\mathrm{S}$ will be a more general
then in the mathematical model of [21]. For vectors functions
$\mathbf{v}\in\mathbb{R}^{3}$ we consider $\mathrm{B}=\mathrm{B}(\mathbf{v})$
introduced in (3). For such defined tensor $\mathrm{B}$ we introduce the
following tensors

$\mathrm{B}_{\mu}=2\mu_{1}|\mathrm{B}_{s}|^{p-2}\mathrm{B}_{s}+\mu_{2}|\mathrm{B}_{a}|^{p-2}\mathrm{B}_{a},\qquad\mathrm{B}_{\nu}=|\mathrm{B}_{s}|^{\frac{p-2}{2}}\mathrm{B}_{s}+\nu|\mathrm{B}_{a}|^{\frac{p-2}{2}}\mathrm{B}_{a},\qquad$
$\mathrm{B}_{\nu^{2}}=|\mathrm{B}_{s}|^{p-2}\mathrm{B}_{s}+\nu^{2}|\mathrm{B}_{a}|^{p-2}\mathrm{B}_{a},$
(8)

where $p\geqslant 2$, $\mu_{1},\mu_{2}$ are the viscosities of the
viscoplastic fluid of Bingham and $\tau_{\ast}$ is a so-called plug parameter.
Let us define the viscous part $\mathrm{S}$ of asymmetric fluid of Bingham by

$\mathrm{S}=\left\\{\begin{array}[]{ll}\mathrm{B}_{\mu}+\widehat{\tau}_{\ast}\frac{\mathrm{B}_{\nu^{2}}}{|\mathrm{B}_{\nu}|^{\frac{2(p-1)}{p}}},&\mathrm{B}_{\nu}\neq
0,\\\ \mathrm{S}_{plug},&\mathrm{B}_{\nu}=0,\end{array}\right.$ (9)

for some tensor $\
\mathrm{S}_{plug}=\mathrm{S}_{plug}(\mathbf{x},t)\in\mathbb{R}^{3\times 3},$
which fulfils the restriction $|\mathrm{S}_{plug}|\leqslant\tau_{\ast}$. Here
we denote

$\widehat{\tau}_{\ast}=\frac{\tau_{\ast}}{\max(1,\nu^{2/p})}.$ (10)

The major explanation of this modification is based on Proposition 3.7 proved
in the following section.

##  3 Some useful results

In what follows the following algebraic result will be very useful.

######  Lemma 3.1.

The space of the matrices endowed with the dot product (2) is the direct sum
of the spaces of symmetric matrices and anti-symmetric matrices. More
precisely, for any matrices $X,$ $Y\in\mathbb{R}^{3\times 3},$ we have

$\displaystyle X_{s}:Y=X_{s}:Y_{s},\quad X_{a}:Y=X_{a}:Y_{a},\quad
X_{s}:Y_{a}=X_{a}:Y_{s}=0,$ $\displaystyle
X_{s}:X=X_{s}:X_{s}=\left|X_{s}\right|^{2},\qquad
X_{a}:X=\left|X_{a}\right|^{2},$

with the notation introduced in (1).

Proof. The coefficients of the matrices $X_{s}$ and $X_{a}$ are equal to
$x_{ij}^{s}=\frac{x_{ij}+x_{ji}}{2}$ and $x_{ij}^{a}=\frac{x_{ij}-x_{ji}}{2}$.
Then $x_{ij}^{s}=x_{ji}^{s}$, $x_{ij}^{a}=-x_{ji}^{a},$

$\begin{array}[]{ccl}X_{s}:Y&=&\sum_{i,j=1}^{3}x_{ij}^{s}y_{ij}=\frac{1}{2}\left[\sum_{i,j=1}^{3}x_{ij}^{s}y_{ij}+\sum_{i,j=1}^{3}x_{ji}^{s}y_{ji}\right]\\\
&=&\sum_{i,j=1}^{3}x_{ij}^{s}\frac{1}{2}\left(y_{ij}+y_{ji}\right)=\sum_{i,j=1}^{3}x_{ij}^{s}y_{ij}^{s}=X_{s}:Y_{s}\end{array}$

and

$\begin{array}[]{ccl}X_{a}:Y&=&\sum_{i,j=1}^{3}x_{ij}^{a}y_{ij}=\frac{1}{2}\left[\sum_{i,j=1}^{3}x_{ij}^{a}y_{ij}+\sum_{i,j=1}^{3}x_{ji}^{a}y_{ji}\right]\\\
&=&\frac{1}{2}\left[\sum_{i,j=1}^{3}x_{ij}^{s}y_{ij}-\sum_{i,j=1}^{3}x_{ij}^{a}y_{ji}\right]=\sum_{i,j=1}^{3}x_{ij}^{s}\frac{1}{2}\left(y_{ij}-y_{ji}\right)\\\
&=&\sum_{i,j=1}^{3}x_{ij}^{a}y_{ij}^{a}=X_{a}:Y_{a}.\end{array}$

Moreover, we have

$\begin{array}[]{rcl}\left(\frac{X\pm X^{T}}{2}\right):\left(\frac{Y\mp
Y^{T}}{2}\right)&=&\frac{1}{4}\sum_{i,j=1}^{3}(x_{ij}\pm x_{ji})(y_{ij}\mp
y_{ji})\\\
&=&\frac{1}{4}\sum_{i,j=1}^{3}(x_{ij}y_{ij}-x_{ji}y_{ji})=0.\end{array}$

$\hfill\;\blacksquare$

Before proceeding let us recall two basic theorems on convex analysis.

######  Theorem 3.2.

(see [16, Theorem 23.1] or [2, Proposition 17.2]) Let $f$ be a convex function
from $\mathbb{R}^{n}$ to $[-\infty,+\infty]$, and let $\mathbf{x}$ be a point
where $f$ is finite. Then, for each $\mathbf{y}\in\mathbb{R}^{n}$ there exists
the one-sided directional derivative of $f$ at $\mathbf{x}$ with respect to
the vector $\mathbf{y}$, i.e.

$f^{\prime}(\mathbf{x};\mathbf{y})=\lim_{\lambda\searrow
0}\frac{f(\mathbf{x}+\lambda\mathbf{y})-f(\mathbf{x})}{\lambda}.$

In fact, the difference quotient
$(f(\mathbf{x}+\lambda\mathbf{y})-f(\mathbf{x}))/\lambda$ is a non-decreasing
function of $\lambda>0$, so that

$f^{\prime}(\mathbf{x};\mathbf{y})=\inf_{\lambda>0}(f(\mathbf{x}+\lambda\mathbf{y})-f(\mathbf{x}))/\lambda.$

######  Remark 3.3.

For any function $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ positively
homogeneous of order 1, we have

$f^{\prime}(0;\mathbf{y})=f(\mathbf{y}),\quad\forall\mathbf{y}\in\mathbb{R}^{n}.$

Indeed, in this case one has

$f^{\prime}(0;\mathbf{y})=\lim_{\lambda\searrow
0}\frac{f(\lambda\mathbf{y})-f(0)}{\lambda}=\lim_{\lambda\downarrow
0}\frac{\lambda f(\mathbf{y})}{\lambda}=f(\mathbf{y}).$

Let us remember the concept of sub-differential.

######  Definition 3.4.

A vector $\mathbf{x}^{\ast}$ is said to be a sub-gradient of $f$ at
$\mathbf{x}$ if

$f(\mathbf{y})\geqslant
f(\mathbf{x})+(\mathbf{x}^{\ast},\mathbf{y}),\quad\forall\mathbf{y}\in\mathbb{R}^{n}.$

The set of all sub-gradients of $f$ at $\mathbf{x}$ is called sub-differential
of $f$ at $\mathbf{x}$ and is denoted by $\partial f(\mathbf{x}).$

######  Theorem 3.5.

(see [16, Theorem 23.2] or [2, Proposition 17.7]) Let $f$ be a convex
function, and let $\mathbf{x}$ be a point where $f$ is finite. Then
$\mathbf{x}^{\ast}$ is a sub-gradient of $f$ at $\mathbf{x}$ if and only if

$f^{\prime}(\mathbf{x};\mathbf{y})\geqslant(\mathbf{x}^{\ast},\mathbf{y}),\quad\forall\mathbf{y}\in\mathbb{R}^{n}.$

Let us also remark the following fact.

######  Remark 3.6.

Let

$\|x\|_{l^{p}(\mathbb{R}^{n})}=\sqrt[p]{|x_{1}|^{p}+...+|x_{n}|^{p}},\qquad\mathbf{x}=(x_{1},\cdots,x_{n})\in\mathbb{R}^{n},$

denote the $l^{p}$ norm in $\mathbb{R}^{n}$. Then the ${l}^{p}$ norm is
decreasing with respect to $p\in[1,\infty]$.

This is easy to prove: given $p_{1}\leqslant p_{2}$ in $[1,\infty]$ and
$\mathbf{x}=(x_{1},\cdots,x_{n})\not=0,$ let$\quad
y_{i}=|x_{i}|/\|x\|_{p_{2}}.$ Then $|y_{i}|\leqslant 1$, so

$|y_{i}|^{p_{1}}\geqslant|y_{i}|^{p_{2}},\qquad\|\mathbf{y}\|_{{l}^{p_{1}}}\geqslant
1,$

and, consequently,
$\|\mathbf{x}\|_{{l}^{p_{1}}}\geqslant\|\mathbf{x}\|_{{l}^{p_{2}}}$.

The following results are used in a crucial way in the proof of our Theorem
4.1.

######  Proposition 3.7.

For given $p\geqslant 2$ let us introduce the potential

$V(X)=\frac{2\mu_{1}}{p}|X_{s}|^{p}+\frac{\mu_{2}}{p}|X_{a}|^{p}+\widehat{\tau}_{\ast}||\mathrm{X}_{s}|^{\frac{p-2}{2}}\mathrm{X}_{s}+\nu|\mathrm{X}_{a}|^{\frac{p-2}{2}}\mathrm{X}_{a}|^{\frac{2}{p}}$
(11)

for any matrix $X\in\mathbb{R}^{3\times 3}.$ The potential $V$ is convex,
differentiable at any $X\not=0$ (equivalently, $X_{\nu}\not=0$) in
$\mathbb{R}^{3\times 3}$, with

$DV(X)=X_{\mu}+\widehat{\tau}_{\ast}\frac{X_{\nu^{2}}}{|X_{\nu}|^{\frac{2(p-1)}{p}}},$

the matrices $X_{\mu},\mathrm{\ }X_{\nu},\mathrm{\ }X_{\nu^{2}}$ are defined
in (8) (instead of $B$ we substitute $X$). Moreover:

$(a)\ \qquad B_{r_{p}}(0)\subset(\widehat{\tau}_{\ast})^{-1}\partial
V(0)\subset B_{\max\\{1,\nu^{\frac{2}{p}}\\}}(0),$ where $B_{r}(0)$ is the
closed ball of a radius $r$ at the center $0$ and

$r_{p}=\nu^{2}/(1+\left(\nu^{2}\right)^{{}^{\frac{2}{p-2}}})^{{}^{\frac{p-2}{2}}};$

$(b)\qquad$let $q=p/(p-1),$ then

$(\widehat{\tau}_{\ast})^{-1}\partial V(0)=\\{S\in\mathbb{R}^{3\times
3}:\quad|S_{s}|^{q}+\nu^{2(1-q)}|S_{a}|^{q}\leqslant 1\\}.$

Proof. We write $V(X)=U(X)+\widehat{\tau}_{\ast}W(X)$ with

$U(X)=\frac{2\mu_{1}}{p}|X_{s}|^{p}+\frac{\mu_{2}}{p}|X_{a}|^{p},\qquad
W(X)=||X_{s}|^{\frac{p-2}{2}}X_{s}+\nu|X_{a}|^{\frac{p-2}{2}}X_{a}|^{\frac{2}{p}}.$

To see that $V$ is convex, we first notice that the functions

$X\mapsto|X_{s}|^{p},\qquad X\mapsto|X_{a}|^{p}$

are convex since they are a composition of the convex function
$t\in\mathbb{R}\mapsto|t|^{\frac{p}{2}}$ with quadratic functions
$X\mapsto|X_{s}|^{2},$ $\ ~{}X\mapsto|X_{a}|^{2}$. In addition, we can check
that the function $W$ is convex using that any norm is convex and the fact
that

$W(X)=\sqrt[p]{|X_{s}|^{p}+\nu^{2}|X_{a}|^{p}}=\|(|X_{s}|,\nu^{\frac{2}{p}}|X_{a}|)\|_{{l}^{p}(\mathbb{R}^{2})}$

is also a norm. Thus, $V$ is convex because it is a linear combination of
convex functions.

The function $V$ is differentiable at any $X\in\mathbb{R}^{3\times
3}\backslash\\{0\\}$ due to the chain rule, and we can compute $DV(X)$
differentiating directly $U$ and $W$ with respect to the standard variables
$x_{ij}$ in $\mathbb{R}^{3\times 3}$, or, alternatively, using the chain rule,
one has

$\displaystyle DV(X)=$ $\displaystyle
2\mu_{1}|X_{s}|_{s}^{p-2}X+\mu_{2}|X_{a}|_{a}^{p-2}X+\widehat{\tau}_{\ast}DW(X)$
$\displaystyle=$ $\displaystyle
X_{\mu}+\widehat{\tau}_{\ast}(|X_{s}|^{p}+\nu^{2}|X_{a}|^{p})^{\frac{1}{p}-1}(|X_{s}|^{p-2}X_{s}+\nu^{2}|X_{a}|^{p-2}X_{a})$
$\displaystyle=$ $\displaystyle
X_{\mu}+\widehat{\tau}_{\ast}|X_{\nu}|^{-\frac{2(p-1)}{p}}X_{\nu}^{2}.$ (12)

Now, the function $U$ is differentiable also at $X=0$ and $DU(0)=0$. Then, the
sub-differential $\partial V(0)$ is equal to $\widehat{\tau}_{\ast}\partial
W(0)$. Let us show the items (a) and (b) in the statement of the Proposition).

By Remark 3.3,

$W^{\prime}(0;Y)=W(Y)\qquad\text{for any\quad}Y\in\mathbb{R}^{3\times 3}.$

Then, by Theorem 3.5,

$S\in\partial W(0)=\widehat{\tau}_{\ast}^{-1}\partial
V(0)\qquad\Longleftrightarrow\qquad W(Y)\geqslant
S:Y,\qquad\forall\,Y\in\mathbb{R}^{3\times 3}.$ (13)

Taking in the inequality $Y=S$ and using the Remark 3.6, we obtain

$\displaystyle|S|^{2}$
$\displaystyle\leqslant\|(|S_{s}|,\nu^{2/p}|S_{a}|)\|_{{l}^{p}(\mathbb{R}^{2})}\leqslant\max\\{1,\nu^{2/p}\\}\|(|S_{s}|,|S_{a}|)\|_{{l}^{p}(\mathbb{R}^{2})}$
$\displaystyle\leqslant\max\\{1,\nu^{2/p}\\}\|(|S_{s}|,|S_{a}|)\|_{{l}^{2}(\mathbb{R}^{2})}=\max\\{1,\nu^{2/p}\\}|S|,$

hence we have proved the claim $\partial W(\mathrm{0})\subset
B_{\max\\{1,\nu^{2/p}\\}}(0)$ of item (a) of the Proposition.

Now let us show the claim

$B_{r_{p}}(0)\subset\partial W(\mathrm{0})=\widehat{\tau}_{\ast}^{-1}\partial
V(0)$ (14)

of item (a). Let us consider arbitrary matrix $S\in\mathbb{R}^{3\times 3}$
satisfying the property

$|S||Y|\leqslant W(Y),\qquad\forall Y\in\mathbb{R}^{3\times 3}.$ (15)

Then, by the Cauchy-Schwarz inequality, we have also

$S:Y\leqslant W(Y),\qquad\forall\,Y\in\mathbb{R}^{3\times 3}.$

By Theorem 3.5 we conclude that any $S\in\mathbb{R}^{3\times 3},$ satisfying
the property (15), belongs to $\partial W(0)$. On the other hand, by the
positive homogeneity of $W$ the property (15) is equivalent to

$|S|\leqslant\min_{Y}\\{W(Y):~{}Y\in\mathbb{R}^{3\times 3}\quad\mbox{ with
}|Y|=1\\}.$

Let us demonstrate that this minimum is equal to $r_{p}$, that gives the claim
(14). the exact value of $r_{p}$ is defined in the statement of item (a) in
the Proposition. Since $1=|Y|^{2}=|Y_{s}|^{2}+|Y_{a}|^{2}$, writing
$t=|Y|_{a}^{2},$ we have $|Y_{s}|^{2}=1-t$ and

$W(Y)\equiv W(t)=\sqrt[p]{(1-t)^{\frac{p}{2}}+\nu^{2}t^{\frac{p}{2}}},\qquad
t\in[0,1].$

By a straightforward computation, we obtain that the minimum of the function

$\alpha(t)=(1-t)^{\frac{p}{2}}+\nu^{2}t^{\frac{p}{2}}\qquad\text{in the
interval \quad}[0,1]$

is $\left(r_{p}\right)^{p}$, and it is attained at
$t_{\ast}=1/(1+\nu^{\frac{4}{p-2}})$. Thus, we have proved the claim (a).

To show claim (b) we follow a similar reasoning as above, and with the help of
the Hölder inequality. Let $S\in\partial W(0)$. Then, by Theorem 3.5, we have

$S:Y\leqslant W(Y),\qquad\forall\,Y\in\mathbb{R}^{3\times 3}.$

If we take in this inequality $Y,$ having

$Y_{s}=|S_{s}|^{q-2}S_{s}\qquad\text{and}\qquad
Y_{a}=\nu^{-\frac{2q}{p}}|S_{a}|^{q-2}S_{a},$

we obtain

$|S_{s}|^{q}+\nu^{-\frac{2q}{p}}|S_{a}|^{q}\leqslant\sqrt[p]{|S_{s}|^{q}+\nu^{-2q+2}|S_{a}|^{q}}.$

This implies that

$|S_{s}|^{q}+\nu^{2(1-q)}|S_{a}|^{q}\leqslant 1,$

because $-2q/p=2(1-q)$ and $p\geqslant 2$. Reciprocally, if

$|S_{s}|^{q}+\nu^{2(1-q)}|S_{a}|^{q}\leqslant 1$

then, for all $Y\in\mathbb{R}^{3\times 3}$, using the Hölder inequality, we
have

$\displaystyle S$ $\displaystyle:$ $\displaystyle
Y=S_{s}:Y_{s}+(\nu^{-2/p}S_{a}):(\nu^{2/p}Y_{a})$ $\displaystyle\leqslant$
$\displaystyle\sqrt[q]{|S_{s}|^{q}+\nu^{-2q/p}|S_{s}|^{q}}\sqrt[p]{|Y_{s}|^{p}+\nu^{2}|Y_{a}|^{p}}\leqslant
W(Y).$

Then, again by Theorem 3.5, we obtain that $S\in\partial W(0)$.
$\hfill\;\blacksquare$

Now we show the auxiliary result that explains the definition of
$\widehat{\tau}_{\ast}$ by (10) in (9).

######  Corollary 3.8.

For any matrix $B\in\mathbb{R}^{3\times 3}\backslash\\{\mathrm{0}\\}$ one has
the estimate

$\frac{|B_{\nu}^{2}|}{\sqrt[p]{|B_{\nu}|^{2(p-1)}}}\leqslant\max\\{1,\nu^{\frac{2}{p}}\\}.$
(16)

Proof. As the derivative of the functional $W(X)$ has been calculated in (12)
and equals to

$DW(B)=\frac{B_{\nu}^{2}}{\sqrt[p]{|B_{\nu}|^{2(p-1)}}}\qquad\text{at any
\quad}B\not=0\text{.}$

Now we claim that $DW(B)\in\partial W(0)$, and thus the estimate (16) shall
follow from this fact, by (a) of Proposition 3.7, since

$\partial W(\mathrm{0})=(\widehat{\tau}_{\ast})^{-1}\partial
V(\mathrm{0})\subset B_{\max\\{1,\nu^{\frac{2}{p}}\\}}(0).$

Accounting (13) we have to show the claim

$DW(B):Y\leqslant W(Y)\qquad\text{for all}\quad Y\in\mathbb{R}^{3\times 3}.$

By Theorem 3.2 we have that the directional derivative

$DW(B):Y=\inf_{\lambda>0}\lambda^{-1}\left(W(B+~{}\lambda Y)-W(B)\right).$

In addition, $W$ is a norm by Remark 3.6, then

$W(B+\lambda Y)\leqslant W(B)+\lambda W(Y).$

Summing up, we obtain our above claim.$\hfill\;\blacksquare$

######  Remark 3.9.

By Proposition 3.7, the relation (9) is equivalent to the variational
inequality

$V(X)-V(\mathrm{B})\geqslant
S:(X-\mathrm{B}),\qquad\forall\,X\in\mathbb{R}^{3\times 3}.$

Next we present two technical results we shall use latter.

######  Lemma 3.10.

Let $W=W(X)$ be a positive convex function on $X\in\mathbb{R}^{3\times 3}.$
Then, for any natural $n$, the approximated function

$W_{n}(X)=\sqrt[p]{\left(W(X)\right)^{p}+n^{-1}}$

is also convex with respect of the parameter $X\in\mathbb{R}^{3\times 3}$.

Proof. Note that the function $\varphi(z)=\sqrt[p]{z^{p}+n^{-1}}$ is monotone
increasing and convex function for $z\geqslant 0.$ Therefore applying the
definition of convex function, we easily derive that the composition
$W_{n}(X)=\varphi(W(X))$ is also convex with respect of the parameter
$X\in\mathbb{R}^{3\times 3}$. $\hfill\;\blacksquare$

######  Lemma 3.11.

Let $n$ be an arbitrary natural number. We consider the convex potential

$V^{n}(X)=\frac{2\mu_{1}}{p}|X_{s}|^{p}+\frac{\mu_{2}}{p}|X_{a}|^{p}+\widehat{\tau}_{\ast}\sqrt[p]{||\mathrm{X}_{s}|^{\frac{p-2}{2}}\mathrm{X}_{s}+\nu|\mathrm{X}_{a}|^{\frac{p-2}{2}}\mathrm{X}_{a}|^{2}+n^{-1}},$

defined for arbitrary $X\in\mathbb{R}^{3\times 3}$. Let

$\mathrm{S}^{n}=\mathrm{B}_{\mu}+\widehat{\tau}_{\ast}\frac{\mathrm{B}_{\nu^{2}}}{\sqrt[p]{(|\mathrm{B}_{\nu}|^{2}+n^{-1})^{p-1}}}.$

Then, for any given $B\in\mathbb{R}^{3\times 3}$, we have $\frac{\partial
V^{n}}{\partial X}(\mathrm{B})=\mathrm{S}^{n}$, i.e.

$V^{n}(X)-V^{n}(\mathrm{B})\geqslant\mathrm{S}^{n}:(X-\mathrm{B}),\qquad\forall\,X\in\mathbb{R}^{3\times
3}.$

Proof. Straightforward computation. $\hfill\;\blacksquare$

##  4 Statement of the problem

Let us consider the motion of an asymmetric Bingham fluid, assuming that the
fluid is incompressible. For simplicity of considerations we admit that the
density $\rho$ is equal to $1$ and neglect the mass force vector$\
\mathbf{f}$. Then the flow equations (6) for the velocity $\mathbf{v}$ in a
bounded domain $\Omega\subset\mathbb{R}^{3}$ with the boundary $\Gamma$ are

$\mathbf{v}_{t}+\left(\mathbf{v\cdot\nabla}\right)\mathbf{v}=\mbox{div}\
\mathrm{T},\qquad\mbox{div}\,\mathbf{v}=0\qquad\text{in \ \
}\Omega_{T}=(0,T)\times\Omega,$ (17)

where $\mathrm{T}=-p\,\mathrm{I}+\mathrm{S}$ and $\mathrm{S}$ satisfies the
constitutive law (9) with the relations (3), (4), (8). We add to this system
the initial data

$\mathbf{v}|_{t=0}=\mathbf{v}_{0}\qquad\text{in}\quad\Omega.$ (18)

The system (17) is mostly supplemented with the usual Dirichlet boundary
condition. The Dirichlet condition implies the adherence of fluid particles to
the boundary. For the motion of Bingham fluids (such as the extrusion of the
toothpaste from the tube, the mud flows in drilling engineering, the propane
flow within the hydro-fracture, etc.) it is more natural to study the system
(17) with slip type boundary conditions, permitting the slippage of the fluid
against the boundary. To describe accurately physical phenomena we consider
homogeneous Navier slip boundary conditions

$\mathbf{v}\cdot\mathbf{n}=0,\;\quad\left[\mathrm{T}\,\mathbf{n}+\alpha\mathbf{v}\right]\cdot{\bm{\tau}}=0\qquad\text{on
}\ \Gamma_{T}=(0,T)\times\Gamma,$ (19)

where $\alpha$ is a positive friction coefficient. For the discussion of the
Navier slip boundary conditions we refer to the articles [4]-[6].

Let us introduce some notations to formulate our main result. We denote by
$(\cdot,\cdot)$ the dot product in $L^{2}(\Omega)$. Also we define the spaces

$\displaystyle H$ $\displaystyle=$ $\displaystyle\\{\mathbf{v}\in
L^{2}(\Omega):\,\mbox{div }\mathbf{v}=0\;\ \text{
in}\;\mathcal{D}^{\prime}(\Omega),\;\ \mathbf{v}\cdot\mathbf{n}=0\;\ \text{
in}\;H^{-1/2}(\Gamma)\\},$ $\displaystyle V$ $\displaystyle=$
$\displaystyle\\{\mathbf{v}\in H^{1}(\Omega):\,\mbox{div }\mathbf{v}=0\;\
\text{ a.e. in \ }\Omega,\;\ \mathbf{v}\cdot\mathbf{n}=0\;\ \text{
in}\;H^{1/2}(\Gamma)\\},$ $\displaystyle V_{p}$ $\displaystyle=$
$\displaystyle\\{\mathbf{v}\in V:\,\;\;|\nabla\mathbf{v}|^{p}\in
L^{1}(\Omega)\\}.$ (20)

The space $V_{p}$ is endowed with the norm
$\|\mathbf{v}\|_{V_{p}}=\|\mathbf{v}\|_{L^{2}(\Omega)}+\|\nabla\mathbf{v}\|_{L^{p}(\Omega)}.$

The main objective of our article is to show the well-posedness of the system
(17)-(19) for unknown functions $\mathbf{v}$ and $\mathrm{S}$. This result of
the well-posedness is formulated in the following theorem, in which we also
define the concept of the weak solution for the system (17)-(19). This concept
is a direct consequence of the equations (17), the boundary conditions (19)
and the integral equality

$-\int_{\Omega}\mbox{div}\,\mathrm{T}\cdot{\bm{\varphi}}\
d\mathbf{x}=-\int_{\Gamma}\left(\mathrm{T}\
\mathbf{n}\right)\cdot{\bm{\varphi}}\
d\mathbf{\gamma}+\int_{\Omega}\mathrm{T}:\frac{\partial{\bm{\varphi}}}{\partial\mathbf{x}}\,d\mathbf{x},$
(21)

which is valid for any $(3\times 3)$-matrix function $\mathrm{T}\in
H^{1}(\Omega)$ and any 3D-vector function ${\bm{\varphi}}\in H^{1}(\Omega).$

######  Theorem 4.1.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{3}$ with a $C^{1}-$smooth
boundary $\Gamma$, $p\geqslant 2$, a given real number,

$\mathbf{v}_{0}\in H\qquad\mbox{ and }\qquad\alpha\in
L^{2}(0,T;L^{\infty}(\Omega)).$ (22)

Then there exists a function $\mathbf{v}$ and a $(3\times 3)-$matrix function
$\mathrm{S}$, such that

$\mathbf{v}\in L^{\infty}(0,T;H)\cap
L^{2}(0,T;V_{p}),\quad\quad\mathbf{v}_{t}\in L^{2}(0,T;V_{p}^{\ast})$ (23)

and the pair $\left(\mathbf{v,\ }\mathrm{S}\right)$ is a weak solution of the
system (17)-(19), satisfying the integral equality

$\int\limits_{\Omega_{T}}\left[\mathbf{v}\partial_{t}{\bm{\varphi}}+\left(\mathbf{v\otimes
v}-\mathrm{S}\right):\frac{\partial{\bm{\varphi}}}{\partial\mathbf{x}}\right]\,d\mathbf{x}dt+\int\limits_{\Omega}\mathbf{v}_{0}{\bm{\varphi}}(0)\,d\mathbf{x}=\int_{\Gamma_{T}}\alpha(\mathbf{v}\cdot{\bm{\varphi}})\,d\mathbf{\gamma}\,dt$
(24)

for any test function

${\bm{\varphi}}\in H^{1}(0,T;V_{p})\quad\mbox{such that
}\quad{\bm{\varphi}}(\cdot,T)=0.$ (25)

The $(3\times 3)-$matrix function $\ \mathrm{S}\in L^{p/(p-1)}(\Omega_{T})$
fulfils the relation (9).

Moreover, if

$p\geqslant\frac{7+\sqrt{19}}{5}\approx 2.272,$

then the solution $\left(\mathbf{v,\ }\mathrm{S}\right)$ is unique.

##  5 Construction of the approximation problem

In this section we consider an approximated problem for the system (17)-(19)
and solve this approximated problem applying the Faedo-Galerkin method and the
Schauder fixed point argument (see for instance [1]).

Since the space $V_{p}$ is separable, it is the span of a countable set of
linearly independent functions
$\left\\{\mathbf{e}_{k}\right\\}_{k=1}^{\infty}$. More precisely, we can
choose this set as the eigenfunctions for the following non-linear Stokes type
equations with Navier boundary conditions:

$\left\\{\begin{array}[]{ll}-{\mbox{div}}(\mathrm{T}(\mathbf{e}_{k}))=\lambda_{k}\mathbf{e}_{k},\qquad\mathrm{div}\,\mathbf{e}_{k}=0&\quad\mbox{in}\
\Omega,\vspace{2mm}\\\ &\\\
\mathbf{e}_{k}\cdot\mathbf{n}=0,\;\quad\left[\mathrm{T}(e_{k})\,\mathbf{n}+\alpha\mathbf{e}_{k}\right]\cdot{\bm{\tau}}=0&\quad\mbox{on}\
\Gamma\end{array}\right.$

with
$\mathrm{T}(\mathbf{e}_{k})=-p_{k}\,\mathrm{I}+|\nabla\mathbf{e}_{k}|^{p-2}\nabla\mathbf{e}_{k}.$
The solvability of this problem follows from the spectral theory [10]. This
theory permits to construct this set
$\left\\{\mathbf{e}_{k}\right\\}_{k=1}^{\infty}$ as an orthogonal basis for
$V_{p}$ and an orthonormal basis for $H.$

We can consider the subspace
$V_{p}^{n}=\mathrm{span}\,\\{\mathbf{e}_{1},\ldots,\mathbf{e}_{n}\\}$ of
$V_{p}$, for any fixed natural $n.$ Let us define the vector function

$\mathbf{v}^{n}(t)=\sum_{k=1}^{n}c_{k}^{(n)}(t)\ \mathbf{e}_{k},\qquad
c_{k}^{(n)}(t)\in{\mathbb{R}},$ (26)

as the solution of the approximate system

$\left\\{\begin{array}[]{l}\int\limits_{\Omega}[\partial_{t}\mathbf{v}^{n}\mathbf{e}_{k}+\left(\mathbf{v}^{n}\mathbf{\cdot\nabla}\right)\mathbf{v}^{n}\mathbf{e}_{k}+\mathrm{S}^{n}:\frac{\partial\mathbf{e}_{k}}{\partial\mathbf{x}}]\,d\mathbf{x}+\int_{\Gamma}\alpha(\mathbf{v}^{n}\cdot{\bm{\tau}})(\mathbf{e}_{k}\cdot{\bm{\tau}})\,d\mathbf{\gamma}\,dt=0,\\\
\qquad\forall k=1,2,\dots,n,\\\
\mathbf{v}^{n}(0)=\mathbf{v}_{0}^{n}.\end{array}\right.$ (27)

The $(3\times 3)-$matrix functions $\mathrm{T}^{n},$ $\ \ \mathrm{S}^{n}$ are
prescribed by the relations

$\mathrm{T}^{n}=-p^{n}\,\mathrm{I}+\mathrm{S}^{n},\qquad\mathrm{S}^{n}=\mathrm{B}_{\mu}^{n}+\widehat{\tau}_{\ast}\frac{\mathrm{B}_{\nu^{2}}^{n}}{\sqrt[p]{(|\mathrm{B}_{\nu}^{n}|^{2}+n^{-1})^{p-1}}}$
(28)

and the matrix functions $\mathrm{B}^{n}=\mathrm{B}(\mathbf{v}^{n}),$
$\mathrm{B}_{\mu}^{n},\mathrm{\ B}_{\nu}^{n},\mathrm{\ B}_{\nu^{2}}^{n}$ are
calculated through the formulas (3), (4), (8). The function
$\mathbf{v}_{0}^{n}$ is the orthogonal projection of $\mathbf{v}_{0}\in H$
into the space $V_{p}^{n}.$ Note that the system (26)-(28) is a weak
formulation of the problem

$\left\\{\begin{array}[]{c}\mathbf{v}_{t}^{n}+\left(\mathbf{v}^{n}\mathbf{\cdot\nabla}\right)\mathbf{v}^{n}=\mbox{div}\
\mathrm{T}^{n},\qquad\mbox{div}\,\mathbf{v}^{n}=0,\qquad\text{in}\quad\Omega_{T},\\\
\mathbf{v}^{n}\cdot\mathbf{n}=0,\;\quad\left[\mathrm{T}^{n}\mathbf{n}+\alpha\mathbf{v}^{n}\right]\cdot{\bm{\tau}}=0\qquad\text{on
}\ \Gamma_{T},\\\
\mathbf{v}^{n}|_{t=0}=\mathbf{v}_{0}^{n}\qquad\text{in}\quad\Omega.\end{array}\right.$

Next we prove that the approximated problem (27) is solvable.

######  Lemma 5.1.

Let us assume that the data $\mathbf{v}_{0},$ $\alpha$ satisfy the conditions
(22). Then there exists a solution $\mathbf{v}^{n}\in L^{\infty}(0,T;V_{p})$
of the system (26)-(28), satisfying the a priori estimate

$\int\limits_{\Omega}|\mathbf{v}^{n}|^{2}d\mathbf{x}+\int\limits_{0}^{t}\left[\int\limits_{\Omega}|\frac{\partial\mathbf{v}^{n}}{\partial\mathbf{x}}|^{p}d\mathbf{x}+\int_{\Gamma}\alpha|\mathbf{v}^{n}|^{2}\,d\mathbf{\gamma}\right]dt\leqslant
A,\quad t\in[0,T],$ (29)

where $A$ is a constant independent on $n$. More precisely, $A$ depends only
on the data $\mathbf{v}_{0},\mu_{i},\nu$.

Proof. The system (27) is a system of $n$ ordinary differential equations of
the first order, which can be written in the form

$\frac{d\mathbf{c}^{(n)}}{dt}=F(\mathbf{c}^{(n)}),\qquad t\in(0,T),$

for the vector function
$\mathbf{c}^{(n)}(t)=(c_{1}^{(n)}(t),...,c_{n}^{(n)}(t))$ with $c_{k}^{(n)}$
introduced in (26). We can solve system using the Schauder fixed point
theorem.

For an arbitrary $\widehat{\mathbf{c}}^{(n)}\in C([0,T])$, we define

$\widehat{\mathbf{v}}^{n}(t)=\sum_{k=1}^{n}\widehat{c}_{k}^{(n)}(t)\
\mathbf{e}_{k}\qquad\text{and}\quad\widehat{\tau}_{\ast}(\widehat{\mathbf{c}}^{(n)})=\frac{\mathrm{1}}{\sqrt[p]{(|\mathrm{B}_{\nu}^{n}(\widehat{\mathbf{v}}^{n})|^{2}+n^{-1})^{p-1}}}.$

Let $\mathbf{c}^{(n)}$ be an unknown, such that the vector function

$\mathbf{v}^{n}(t)=\sum_{k=1}^{n}c_{k}^{(n)}(t)\ \mathbf{e}_{k}$ (30)

solves the system

$\left\\{\begin{array}[]{l}\int\limits_{\Omega}[\partial_{t}\mathbf{v}^{n}\mathbf{e}_{k}+\left(\widehat{\mathbf{v}}^{n}\mathbf{\cdot\nabla}\right)\mathbf{v}^{n}\mathbf{e}_{k}+\widehat{\mathrm{S}}^{n}:\frac{\partial\mathbf{e}_{k}}{\partial\mathbf{x}}]\,d\mathbf{x}+\int_{\Gamma}\alpha(\mathbf{v}^{n}\cdot{\bm{\tau}})(\mathbf{e}_{k}\cdot{\bm{\tau}})\,d\mathbf{\gamma}\,dt=0,\\\
\qquad\forall k=1,2,\dots,n,\vspace{2mm}\\\
\mathbf{v}^{n}(0)=\mathbf{v}_{0}^{n}\end{array}\right.$ (31)

with the $(3\times 3)-$matrix function

$\widehat{\mathrm{S}}^{n}=\mathrm{B}_{\mu}^{n}+\widehat{\tau}_{\ast}(\widehat{\mathbf{c}}^{(n)})\mathrm{B}_{\nu^{2}}^{n},$
(32)

and the matrix functions $\mathrm{B}^{n}=\mathrm{B}(\mathbf{v}^{n}),$
$\mathrm{B}_{\mu}^{n},\mathrm{\ B}_{\nu}^{n},\mathrm{\ B}_{\nu^{2}}^{n}$ given
by the formulas (3), (4) and (8). From the theory of ordinary differential
equations, it follows that the linear system (30)-(31), of $n$ ordinary linear
differential equations, has an unique solution $\mathbf{c}^{(n)}\in
C^{1}([0,T])$. Therefore, we can consider the operator $K:C([0,T])\rightarrow
C([0,T])$ defined as

$\mathbf{c}^{(n)}=K\left(\widehat{\mathbf{c}}^{(n)}\right).$

The solvability of the system (26)-(28) will be shown if we demonstrate that
this operator $K$ has a fixed point, which we shall do by the Schauder fixed
point theorem. Thus, we have to prove that this operator is compact on a
bounded convex subset $M$ of $\ C([0,T])$.

First, let us deduce a priori estimates for $\mathbf{c}^{(n)}.$ We multiply
(31)1 by $c_{k}^{(n)}$ and take the sum on the index $k=1,...,n.$ Then the
integration over the time interval $(0,t)$ gives

$\frac{1}{2}\int_{\Omega}|\mathbf{v}^{n}|^{2}d\mathbf{x}+\int_{0}^{t}\left[\int_{\Omega}\widehat{\mathrm{S}}^{n}:\frac{\partial\mathbf{v}^{n}}{\partial\mathbf{x}}d\mathbf{x}+\int_{\Gamma}\alpha|\mathbf{v}^{n}|^{2}\,d\mathbf{\gamma}\right]\,dt=\frac{1}{2}\int_{\Omega}|\mathbf{v}_{0}^{n}|^{2}d\mathbf{x}.$
(33)

Lemma 3.1 and the definition (32) of $\widehat{\mathrm{S}}^{n}$ imply

$\mathrm{B}_{\mu}^{n}:\frac{\partial\mathbf{v}^{n}}{\partial\mathbf{x}}=2\mu_{1}|\mathrm{B}_{s}^{n}|^{p}+\mu_{2}|\mathrm{B}_{a}^{n}|^{p},\qquad\mathrm{B}_{\nu^{2}}^{n}:\frac{\partial\mathbf{v}^{n}}{\partial\mathbf{x}}=|\mathrm{B}_{s}^{n}|^{p}+\nu^{2}|\mathrm{B}_{a}^{n}|^{p}$

and

$\widehat{\mathrm{S}}^{n}:\frac{\partial\mathbf{v}^{n}}{\partial\mathbf{x}}=2\mu_{1}|\mathrm{B}_{s}^{n}|^{p}+\mu_{2}|\mathrm{B}_{a}^{n}|^{p}+\widehat{\tau}_{\ast}(\widehat{\mathbf{c}}^{(n)})\left(|\mathrm{B}_{s}^{n}|^{p}+\nu^{2}|\mathrm{B}_{a}^{n}|^{p}\right).$

Therefore, we deduce the inequality

$\displaystyle\frac{1}{2}\int_{\Omega}|\mathbf{v}^{n}|^{2}d\mathbf{x}$
$\displaystyle+$
$\displaystyle\int_{0}^{t}\left[\int_{\Omega}\left(2\mu_{1}|\mathrm{B}_{s}^{n}|^{p}+\mu_{2}|\mathrm{B}_{a}^{n}|^{p}\right)d\mathbf{x}+\int_{\Gamma}\alpha|\mathbf{v}^{n}|^{2}\,d\mathbf{\gamma}\right]$
(34) $\displaystyle\leqslant$
$\displaystyle\frac{1}{2}\int_{\Omega}|\mathbf{v}_{0}^{n}|^{2}d\mathbf{x,}$

which gives the apriori estimate (29).

Also, we have

$|\mathrm{B}_{\mu}^{n}|^{2}=4\mu_{1}^{2}|\mathrm{B}_{s}^{n}|^{2(p-1)}+\mu_{2}^{2}|\mathrm{B}_{a}^{n}|^{2(p-1)},\qquad|\mathrm{B}_{\nu^{2}}^{n}|^{2}=|\mathrm{B}_{s}^{n}|^{2(p-1)}+\nu^{4}|\mathrm{B}_{a}^{n}|^{2(p-1)},$

by Lemma 3.1. Since

$\widehat{\tau}_{\ast}(\widehat{\mathbf{c}}^{(n)})\leqslant\frac{\mathrm{1}}{\sqrt[p]{n^{-(p-1)}}},\quad\text{then
\ \
}|\widehat{\mathrm{S}}^{n}|\leqslant|\mathrm{B}_{\mu}^{n}|+\frac{\mathrm{1}}{\sqrt[p]{n^{-(p-1)}}}|\mathrm{B}_{\nu^{2}}^{n}|,$

and using (34), we obtain

$||\widehat{\mathrm{S}}^{n}||_{L^{p/(p-1)}(\Omega_{T})}\leqslant C(n)$ (35)

where the constant $C(n)$ depends only on $n$.

Let us define the bounded convex set

$M=\left\\{\mathbf{c}^{(n)}\in C([0,T]):||\mathbf{c}^{(n)}||\leqslant
A\right\\},$

where the norm
$||\mathbf{c}^{(n)}||^{2}=\max_{t\in[0,T]}\sum\limits_{k=1}^{n}\left(c_{k}^{(n)}(t)\right)^{2}$
is defined on the space $C([0,T])$ and the constant $A$ is prescribed in (29).

Let us assume that $\widehat{\mathbf{c}}^{(n)}\in M$. Since
$\\{\mathbf{e}_{j}\\}_{j=1}^{\infty}$ is the orthonormal basis for the space
$H,$ then the equality (31) can be written as

$\frac{dc_{k}^{(n)}}{dt}=-\int\limits_{\Omega}\left[\left(\widehat{\mathbf{v}}^{n}\mathbf{\cdot\nabla}\right)\mathbf{v}^{n}\mathbf{e}_{k}+\widehat{\mathrm{S}}^{n}:\frac{\partial\mathbf{e}_{k}}{\partial\mathbf{x}}\right]\,d\mathbf{x}-\int_{\Gamma}\alpha(\mathbf{v}^{n}\cdot{\bm{\tau}})(\mathbf{e}_{k}\cdot{\bm{\tau}})\,d\mathbf{\gamma}\,dt$

for any $k=1,\dots,n.$ Since $V_{p}\subset H^{1}(\Omega),$ then the Sobolev
continuous embedding$\ H^{1}(\Omega)\hookrightarrow L^{6}(\Omega)\cap
L^{2}(\Gamma)$ and the Holder inequality imply

$\displaystyle|\frac{dc_{k}^{(n)}}{dt}|$ $\displaystyle\leqslant$
$\displaystyle\\{\|\widehat{\mathbf{v}}^{n}\|_{L^{4}(\Omega)}\|\nabla\mathbf{v}^{n}\|_{L^{2}(\Omega)}+\|\sqrt{\alpha}\|_{L^{\infty}(\Omega)}\|\sqrt{\alpha}\mathbf{v}^{n}\|_{L^{2}(\Gamma)}\\}\|\mathbf{e}_{k}\|_{H^{1}(\Omega)}$
(36)
$\displaystyle\quad+||\widehat{\mathrm{S}}^{n}||_{L^{p/(p-1)}(\Omega)}\|\frac{\partial\mathbf{e}_{k}}{\partial\mathbf{x}}\|_{L^{p}(\Omega)},$

that is

$\displaystyle|\frac{dc_{k}^{(n)}}{dt}|$ $\displaystyle\leqslant$
$\displaystyle
C(n)\\{\|\widehat{\mathbf{v}}^{n}\|_{L^{4}(\Omega)}\|\nabla\mathbf{v}^{n}\|_{L^{2}(\Omega)}+\|\sqrt{\alpha}\mathbf{v}^{n}\|_{L^{2}(\Gamma)}$
(37)
$\displaystyle\qquad+||\widehat{\mathrm{S}}^{n}||_{L^{p/(p-1)}(\Omega)}\\}.$

Let us recall the Gagliardo–Nirenberg-Sobolev inequality (see [23])

$\displaystyle\int_{0}^{T}\left(\|\widehat{\mathbf{v}}^{n}\|_{L^{4}(\Omega)}\|\nabla\mathbf{v}^{n}\|_{L^{2}(\Omega)}\right)^{8/7}\,dt\leqslant
C\int_{0}^{T}\left(||\widehat{\mathbf{v}}^{n}||_{L^{2}(\Omega)}^{1/4}||\nabla\mathbf{v}^{n}||_{L^{2}(\Omega)}^{7/4}\right)^{8/7}\,dt$
$\displaystyle\qquad\qquad\qquad\leqslant
C\|\widehat{\mathbf{v}}^{n}\|_{L^{\infty}\left(0,T;L^{2}(\Omega)\right)}^{2/7}\|\mathbf{v}^{n}\|_{L^{2}\left(0,T;H^{1}(\Omega)\right)}^{2}.$
(38)

Therefore, the integration of the inequality (37) over the time interval
$[t,t+h]\subset[0,T]$, the Hölder inequality and
$\widehat{\mathbf{c}}^{(n)}\in M$, plus the estimates (29), (38), imply

$|c_{k}^{(n)}(t+h)-c_{k}^{(n)}(t)|\leqslant\int_{t}^{t+h}|\frac{dc_{k}^{(n)}}{dt}|\leqslant
C(n)h^{q}\qquad\text{with\quad}q=min(\frac{1}{8},\frac{1}{2},\frac{1}{p})$

for each $k=1,\dots,n.$ Therefore the operator $K:M\rightarrow M$ is compact
by the Arzela-Ascoli theorem.

The continuity of $K$ is a direct consequence of the theorem on continuous
dependence of the solution of the Cauchy problem (30)-(31) with respect to the
coefficients $\widehat{\mathbf{c}}^{(n)}$ . Therefore, the operator $K$
fulfils the conditions of the Schauder fixed point theorem, which implies the
existence of a fixed point of $K,$ and gives the solution of the system
(26)-(28). $\hfill\;\blacksquare$

######  Lemma 5.2.

We assume that the data $\mathbf{v}_{0},$ $\alpha$ fulfil the conditions (22).
Then there exists a solution

$\mathbf{v}^{n}\in L^{\infty}(0,T;H)\cap L^{2}(0,T;V_{p})$

of the system (26)-(28) that satisfies the following estimates

$\int\limits_{\Omega}|\mathbf{v}^{n}|^{2}d\mathbf{x}+\int\limits_{0}^{t}\left[\int\limits_{\Omega}|\frac{\partial\mathbf{v}^{n}}{\partial\mathbf{x}}|^{p}d\mathbf{x}+\int_{\Gamma}\alpha|\mathbf{v}^{n}|^{2}\,d\mathbf{\gamma}\right]dt\leqslant
C,\quad t\in[0,T],$ (39)
$||\mathrm{S}^{n}||_{L^{p/(p-1)}(\Omega_{T})}\leqslant C$ (40)

and

$||\partial_{t}\mathbf{v}^{n}||_{L^{8/7}(0,T;\,\,V_{p}^{\ast})}\leqslant C.$
(41)

Here and below $C$ is a positive constant that does not depend on $n,$ but may
depend on $\mathbf{v}_{0}$ and $\alpha$.

Proof. Let us note that in Lemma 5.1 we already shown that the solution
$\mathbf{v}^{n}\in L^{\infty}(0,T;V_{p}^{n})$ of (26)-(28) satisfies (39).
Moreover, from the definition (28) of $\mathrm{S}^{n}$, we have

$|\mathrm{S}^{n}|\leqslant|\mathrm{B}_{\mu}^{n}|+\tau_{\ast}.$

This is a direct consequence of the estimate (16). Hence we obtain the a
priori estimate (40) from (39) and the equality

$|\mathrm{B}_{\mu}^{n}|^{2}=4\mu_{1}^{2}|\mathrm{B}_{s}^{n}|^{2(p-1)}+\mu_{2}^{2}|\mathrm{B}_{a}^{n}|^{2(p-1)}.$

Let us consider a subspace $V_{p}^{n}$ of of $V_{p},$ defined in Lemma 5.1.
Let $P_{n}$ be the orthogonal projection of $V_{p}$ onto $V_{p}^{n}.$ Let
${\bm{\varphi}}\in H^{1}(0,T;V_{p})$ be an arbitrary function. The first
equality of (27) is linear with respect of the functions $\mathbf{e}_{k}$,
$k=1,..n,$ then we have

$\left\\{\begin{array}[]{l}\int\limits_{\Omega}[\partial_{t}\mathbf{v}^{n}(P_{n}{\bm{\varphi}})+\left(\mathbf{v}^{n}\mathbf{\cdot}\nabla\right)\mathbf{v}^{n}\
(P_{n}{\bm{\varphi}})+\mathrm{S}^{n}:\frac{\partial(P_{n}{\bm{\varphi}})}{\partial\mathbf{x}}]\,d\mathbf{x}\\\
\qquad\qquad\qquad+\int_{\Gamma}\alpha(\mathbf{v}^{n}\cdot(P_{n}{\bm{\varphi}}))\,d\mathbf{\gamma}\,dt=0,\\\
\mathbf{v}^{n}(0)=\mathbf{v}_{0}^{n},\end{array}\right.$ (42)

since $\\{\mathbf{e}_{j}\\}_{j=1}^{\infty}$ is the orthogonal basis for the
space $V_{p}.$

As it was done in (36), we obtain

$\displaystyle|(\partial_{t}\mathbf{v}^{n},{\bm{\varphi}})_{L_{2}(\Omega)}|$
$\displaystyle\leqslant$
$\displaystyle\\{\|\mathbf{v}^{n}\|_{L^{4}(\Omega)}\|\nabla\mathbf{v}^{n}\|_{L^{2}(\Omega)}$
$\displaystyle+$
$\displaystyle\|\sqrt{\alpha}\|_{L^{\infty}(\Omega)}\|\sqrt{\alpha}\mathbf{v}^{n}\|_{L^{2}(\Gamma)}\\}\|{P_{n}\bm{\varphi}}\|_{H^{1}(\Omega)}$
$\displaystyle+$
$\displaystyle||\mathrm{S}^{n}||_{L^{p/(p-1)}(\Omega)}\|\frac{\partial({P_{n}\bm{\varphi}})}{\partial\mathbf{x}}\|_{L^{p}(\Omega)},$

then

$\displaystyle||\partial_{t}\mathbf{v}^{n}||_{V_{p}^{\ast}}=\sup_{{\bm{\varphi}}\in
V_{p}}\left\\{|(\partial_{t}\mathbf{v}^{n},{\bm{\varphi}})_{L^{2}(\Omega)}|:\quad||{\bm{\varphi}}||_{V_{p}}=1\right\\}$
$\displaystyle\leqslant
C\left\\{\|\mathbf{v}^{n}\|_{L^{4}(\Omega)}\|\nabla\mathbf{v}^{n}\|_{L^{2}(\Omega)}+\|\sqrt{\alpha}\mathbf{v}^{n}\|_{L^{2}(\Gamma)}+||\mathrm{S}^{n}||_{L^{p/(p-1)}(\Omega)}\right\\},$
(43)

since the norm
$\|{\bm{\varphi}}\|_{V_{p}}=\|{\bm{\varphi}}\|_{L^{2}(\Omega)}+\|{\bm{\varphi}}\|_{L^{p}(\Omega)}$
on the space $V_{p}$ of the continuous operator $P_{n}$ is less or equal than
$1$.

Therefore this last inequality (43), the a priori estimates (39)-(40) and the
Gagliardo–Nirenberg-Sobolev inequality (38), written for $\mathbf{v}^{n}$
instead of $\widehat{\mathbf{v}}^{n}$ imply the estimate
(41).$\hfill\;\blacksquare$

##  6 Limit transition

We have that $V_{p}\subset H^{1}(\Omega),$ then the embedding result
$H^{1}(\Omega)\hookrightarrow L^{2}(\Gamma)$ and the estimates (39)-(41) imply
the existence of sub-sequences, such that

$\displaystyle\mathbf{v}^{n}$ $\displaystyle\rightharpoonup$
$\displaystyle\mathbf{v}\quad\mbox{ *-weakly in }\quad
L^{\infty}(0,T;L^{2}(\Omega)),$ $\displaystyle\mathbf{v}^{n}$
$\displaystyle\rightharpoonup$ $\displaystyle\mathbf{v}\quad\mbox{ weakly in
}\quad L^{2}(0,T;V_{p}\cap L^{2}(\Gamma)),$ $\displaystyle\mathrm{S}^{n}$
$\displaystyle\rightharpoonup$ $\displaystyle\mathrm{S}\quad\mbox{ weakly
in}\quad L^{2}(\Omega_{T}).$ (44)

Let us remember the Aubin-Lions-Simon compactness result [17], [23].

######  Lemma 6.1.

Let $X_{0}$, $X$ and $X_{1}$ be three Banach spaces with $X_{0}\subseteq
X\subseteq X_{1}$. Suppose that

$X_{0},$ $X_{1}$ are reflexive, $X_{0}$ is compactly embedded in $X$ and that
$X$ is continuously embedded in $X_{1}$. Let

$W=\left\\{v\in L^{2}(0,T;X_{0}),\qquad\partial_{t}v\in
L^{8/7}(0,T;X_{1})\right\\}.$

Then the embedding of $W$ into $L^{2}(0,T;X)$ is compact.

Therefore the estimates (39), (41), the compact embedding
$H^{1}(\Omega)\hookrightarrow L^{2}(\Omega)$ and Lemma 6.1 give that

$\mathbf{v}^{n}\rightarrow\mathbf{v}\quad\mbox{strongly in
}~{}L^{2}(\Omega_{T})\quad\mbox{ and a.e. in }~{}\Omega_{T}.$ (45)

Hence applying the convergences (44)-(45) in (42), we deduce that the limit
functions $\mathbf{v},$ $\mathrm{S}$ fulfil the integral equality

$\int\limits_{\Omega_{T}}\left[\mathbf{v}\partial_{t}{\bm{\varphi}}+\left(\mathbf{v\otimes
v}-\mathrm{S}\right):\frac{\partial{\bm{\varphi}}}{\partial\mathbf{x}}\right]\,d\mathbf{x}dt+\int\limits_{\Omega}\mathbf{v}_{0}{\bm{\varphi}}(0)\,d\mathbf{x}=\int_{\Gamma_{T}}\alpha(\mathbf{v}\cdot{\bm{\varphi}})\,d\mathbf{\gamma}\,dt$
(46)

for any function ${\bm{\varphi}}\in
H^{1}(0,T;V_{p}),\quad\,{\bm{\varphi}}(\cdot,T)=0.$

In what follows we use the approach of the theory of variational inequalities
[8], [10] to demonstrate the relation (9), that ends the proof of Theorem 4.1.
For a fixed natural $n$ we consider the convex potential

$V^{n}=V^{n}(X),\forall\,X\in\mathbb{R}^{3\times 3},$

introduced in Lemma 3.11. By this Lemma we have

$\mathrm{S}^{n}=\frac{\partial V^{n}}{\partial
X}(\mathrm{B}^{n})\qquad\text{with~{}
}\mathrm{B}^{n}=\frac{\partial\mathbf{v}^{n}}{\partial\mathbf{x}}$ (47)

and

$V^{n}(X)-V^{n}(\mathrm{B}^{n})\geqslant\mathrm{S}^{n}:(X-\mathrm{B}^{n})\quad\text{a.e.
in }\Omega_{T},\quad\forall\,X\in\mathbb{R}^{3\times 3}.$ (48)

are equivalent.

Let us denote $\Omega_{r}=(0,r)\times\Omega$ for arbitrary $r\in(0,T),$ the
equality (33) can be written as

$\frac{1}{2}\int\limits_{\Omega}\left[|\mathbf{v}^{n}|^{2}(r)-|\mathbf{v}_{0}^{n}|^{2}\right]\,d\mathbf{x}+\int_{0}^{r}\int_{\Gamma}\alpha|\mathbf{v}^{n}|^{2}\,d\mathbf{\gamma}\,dt=-\int\limits_{\Omega_{r}}\left[\mathrm{S}^{n}:\frac{\partial\mathbf{v}^{n}}{\partial\mathbf{x}}\right]\,\,d\mathbf{x}\,dt.$
(49)

If we substitute (49) in (48), then the lower semi-continuity property of
convex functional with respect of weak convergence gives

$\displaystyle\int\limits_{\Omega_{r}}V(X)-V(\mathrm{B})\,d\mathbf{x}dt\geqslant\lim_{n\rightarrow\infty}\inf\int\limits_{\Omega_{r}}V^{n}(X)-V^{n}(\mathrm{B}^{n})\,d\mathbf{x}dt$
$\displaystyle\geqslant$
$\displaystyle\lim_{n\rightarrow\infty}\inf\\{\int\limits_{\Omega_{r}}\mathrm{S}^{n}:X\,d\mathbf{x}dt+\frac{1}{2}\int\limits_{\Omega}\left[|\mathbf{v}^{n}|^{2}(r)-|\mathbf{v}_{0}^{n}|^{2}\right]\,d\mathbf{x}+\int_{0}^{r}\int_{\Gamma}\alpha|\mathbf{v}^{n}|^{2}\,d\mathbf{\gamma}\,dt\\}$
$\displaystyle\geqslant$
$\displaystyle\int\limits_{\Omega_{r}}\mathrm{S}:X\,d\mathbf{x}dt+\frac{1}{2}\int\limits_{\Omega}\left[|\mathbf{v}|^{2}(r)-|\mathbf{v}_{0}|^{2}\right]\,d\mathbf{x}+\int_{0}^{r}\int_{\Gamma}\alpha|\mathbf{v}|^{2}\,d\mathbf{\gamma}\,dt$

by use the convergences (44). Hence for any matrix function $X\in
L^{2}(\Omega_{T})$ we have the inequality

$\displaystyle\int\limits_{\Omega_{r}}V(X)$ $\displaystyle-$ $\displaystyle
V(\mathrm{B})\,d\mathbf{x}dt\geqslant\int\limits_{\Omega_{r}}\mathrm{S}:X\,d\mathbf{x}dt$
(50) $\displaystyle+$
$\displaystyle\frac{1}{2}\int\limits_{\Omega}\left[|\mathbf{v}|^{2}(r)-|\mathbf{v}_{0}|^{2}\right]\,d\mathbf{x}+\int_{0}^{r}\int_{\Gamma}\alpha|\mathbf{v}|^{2}\,d\mathbf{\gamma}\,dt.$

Let us take $\mathbf{\varphi}=\mathbf{v}(1-sgn_{+}^{\varepsilon}(t-r))$ in the
equality (24) for a fixed $r\in(0,T),$ where

$sgn_{+}^{\varepsilon}(t)=\left\\{\begin{array}[]{cc}0,&\text{if \ }t<0;\\\
t/\varepsilon,&\text{if \ }0\leqslant t<\varepsilon;\\\ 1,&\text{if \
}\varepsilon\leqslant t.\end{array}\right.$

In the obtained equality the limit transition on $\varepsilon\rightarrow 0$
implies

$\frac{1}{2}\int\limits_{\Omega}\left[|\mathbf{v}|^{2}(r)-|\mathbf{v}_{0}|^{2}\right]\,d\mathbf{x}+\int_{0}^{r}\int_{\Gamma}\alpha|\mathbf{v}|^{2}\,d\mathbf{\gamma}\,dt=-\int\limits_{\Omega_{r}}\left[\mathrm{S}:\frac{\partial\mathbf{v}}{\partial\mathbf{x}}\right]\,\,d\mathbf{x}dt.$
(51)

Substituting (51) in (50), we derive

$\int\limits_{\Omega_{r}}V(X)-V(\mathrm{B})\,d\mathbf{x}dt\geqslant\int\limits_{\Omega_{r}}\mathrm{S}:(X-\mathrm{B})\,d\mathbf{x}dt\qquad\text{with
\ }\mathrm{B}=\frac{\partial\mathbf{v}}{\partial\mathbf{x}}.$

Since the matrix function $X\in L^{2}(\Omega_{T})$ is arbitrary, we can choose
in this inequality $X=\mathrm{B}+\varepsilon\mathrm{Z}$ for any positive
$\varepsilon$ and any matrix function $Z$$\in L^{2}(\Omega_{T}),$ which gives

$\lim_{\varepsilon\rightarrow
0^{+}}\int\limits_{\Omega_{r}}\frac{V(\mathrm{B}+\varepsilon\mathrm{Z})-V(\mathrm{B})}{\varepsilon}\,d\mathbf{x}dt\geqslant\int\limits_{\Omega_{r}}\mathrm{S}:\mathrm{Z}\,d\mathbf{x}dt.$

So, passing the limit over the sign of integration, we obtain

$\int\limits_{\Omega_{r}}V^{\prime}(\mathrm{B};\mathrm{Z})\,d\mathbf{x}dt\geqslant\int\limits_{\Omega_{r}}\mathrm{S}:\mathrm{Z}\,d\mathbf{x}dt,$

for any matrix function $\text{$Z$}\in L^{2}(\Omega_{T})$. Now, since
$V^{\prime}(\mathrm{B};\mathrm{Z})$ and $\mathrm{S}:\mathrm{Z}$ are positively
homogeneous with respect to $Z$, this implies indeed in

$\int\limits_{\Omega_{r}}V^{\prime}(\mathrm{B};\mathrm{Z})\xi\,d\mathbf{x}dt\geqslant\int\limits_{\Omega_{r}}(\mathrm{S}:\mathrm{Z})\xi\,d\mathbf{x}dt,$

for any positive function $\xi\in L^{\infty}(\Omega_{T})$ and any matrix
function $Z\in L^{2}(\Omega_{T})$. Therefore,
$V^{\prime}(\mathrm{B};\mathrm{Z})\geqslant\mathrm{S}:\mathrm{Z}$, for any
matrix function $\mathrm{Z}\in L^{2}(\Omega_{T})$, and thus,
$S(x,t)\in\partial V(B(x,t))$ by Theorem 3.5 and $S$ has the form (9) by
Proposition 3.7. In particular, if $B=0$, we have
$|S(x,t)|\leqslant\tau_{\ast}$ by (a) of Proposition 3.7.

In the sequel we will show the uniqueness result. First let us show that the
norms

$\|\mathbf{v}\|_{V_{p}}=\|\mathbf{v}\|_{L^{2}(\Omega)}+\|\nabla\mathbf{v}\|_{L^{p}(\Omega)},\qquad\|\mathbf{v}\|_{W_{p}^{1}}=\|\mathbf{v}\|_{L^{p}(\Omega)}+\|\nabla\mathbf{v}\|_{L^{p}(\Omega)}$

are equivalent. It is enough to show the following result.

######  Lemma 6.2.

There exists a constant $C$, such that

$\|\mathbf{v}\|_{W_{p}^{1}(\Omega)}\leqslant
C\|\mathbf{v}\|_{V_{p}},\quad\forall\mathbf{v}\in W_{p}^{1}(\Omega).$ (52)

Proof. Assume that the affirmation of Lemma is not true, then for any
$n\in\mathbb{N}$ there exists a vector function $\mathbf{v}_{n}\in
W_{p}^{1}(\Omega),$ such that
$\|\mathbf{v}_{n}\|_{W_{p}^{1}(\Omega)}>n\|\mathbf{v}_{n}\|_{V_{p}}.$ Let us
define
$\widetilde{\mathbf{v}}_{n}=\frac{\mathbf{v}_{n}}{\|\mathbf{v}\|_{W_{p}^{1}(\Omega)}},$
that fulfils

$\|\widetilde{\mathbf{v}}_{n}\|_{W_{p}^{1}(\Omega)}=1,\qquad\|\widetilde{\mathbf{v}}_{n}\|_{V_{p}}<\frac{1}{n}.$
(53)

Hence $\\{\widetilde{\mathbf{v}}_{n}\\}_{n=1}^{\infty}$ is compact in
$L^{p}(\Omega),$ such that there exists a strongly convergent subsequence
$\widetilde{\mathbf{v}}_{n^{\prime}}$ to some $\widetilde{\mathbf{v}}\in
L^{p}(\Omega).$ This subsequence $\widetilde{\mathbf{v}}_{n^{\prime}}$ is a
strongly convergent to $\widetilde{\mathbf{v}}$ in the space $L^{2}(\Omega)$
by the inequality

$\|\mathbf{v}\|_{L^{2}(\Omega)}\leqslant
C\|\mathbf{v}\|_{L^{p}(\Omega)},\quad\forall\mathbf{v}\in L^{p}(\Omega).$

Also from (53) we have that

$\lim_{n\rightarrow\infty}\|\nabla\widetilde{\mathbf{v}}_{n}\|_{L^{p}(\Omega)}=0,\quad\lim_{n\rightarrow\infty}\|\widetilde{\mathbf{v}}_{n}\|_{L^{p}(\Omega)}=1\qquad\text{and}\qquad\lim_{n\rightarrow\infty}\|\widetilde{\mathbf{v}}_{n}\|_{L^{2}(\Omega)}=0,$

that implies

$\|\widetilde{\mathbf{v}}\|_{L^{p}(\Omega)}=1\qquad\text{and}\qquad\|\widetilde{\mathbf{v}}\|_{L^{2}(\Omega)}=0,$

which is impossible. Hence (52) is true.$\hfill\;\blacksquare$

Since for any $\beta\in[0,1]$, and any $a,b\geqslant 0$, we have
$\left(a+b\right)^{\beta}\leqslant a^{\beta}+b^{\beta}$, then Lemma 6.2 and
the well known interpolation inequality
$\|\mathbf{v}\|_{L^{r}(\Omega)}\leqslant
C||\mathbf{v}||_{L^{2}(\Omega)}^{1-\beta}||\mathbf{v}||_{W_{p}^{1}(\Omega)}^{\beta}$,
valid for any $\mathbf{v}\in W_{p}^{1}(\Omega)$ (see e.g. of [1, Lemma 2.2]),
give the following interpolation result.

######  Lemma 6.3.

There exists a positive constant $C$, such that

$\|\mathbf{v}\|_{L^{r}(\Omega)}\leqslant
C(||\mathbf{v}||_{L^{2}(\Omega)}^{1-\beta}||\nabla\mathbf{v}||_{L^{p}(\Omega)}^{\beta}+||\mathbf{v}||_{L^{2}(\Omega)}),\quad\forall\mathbf{v}\in
V_{p},$ (54)

where $\beta=\left(\frac{1}{2}-\frac{1}{r}\right)/(\frac{5}{6}-\frac{1}{p})$.

Now we are able to prove the uniqueness result. Let us denote the difference
of two functions $\mathbf{f}_{1}$ and $\mathbf{f}_{2}$ by
$\overline{\mathbf{f}}$, i.e.
$\overline{\mathbf{f}}=\mathbf{f}_{1}-\mathbf{f}_{2}$. Let us admit the
existence of two different solutions $\mathbf{v}_{1},\mathbf{v}_{2}$ with
respective tensors $\mathrm{S}_{1},$ $\mathrm{S}_{2},$ satisfying the relation
(9). By (24) the difference $\overline{\mathbf{v}}$ fulfils the equality

$\int\limits_{\Omega_{T}}\left[\overline{\mathbf{v}}\partial_{t}{\bm{\varphi}}+\left(\overline{\mathbf{v\otimes
v}}-\overline{\mathrm{S}}\right):\frac{\partial{\bm{\varphi}}}{\partial\mathbf{x}}\right]\,d\mathbf{x}dt=\int_{\Gamma_{T}}\alpha(\overline{\mathbf{v}}\cdot{\bm{\varphi}})\,d\mathbf{\gamma}\,dt$
(55)

for any ${\bm{\varphi}}\in H^{1}(0,T;V_{p}),$ such that
${\bm{\varphi}}(\cdot,T)=0$ in $\Omega.$

It is easily to check that

$\int\limits_{\Omega}\left[\left(\overline{\mathbf{v\otimes
v}}\right):\frac{\partial\overline{\mathbf{v}}}{\partial\mathbf{x}}\right]\,d\mathbf{x}=-\int\limits_{\Omega}\left[\left(\overline{\mathbf{v}}\mathbf{\otimes}\overline{\mathbf{v}}\right):\frac{\partial\mathbf{v}_{2}}{\partial\mathbf{x}}\right]\,d\mathbf{x.}\
$

Also there exists a constant $\widetilde{C}>0,$ depending only on
$\mu_{1},\mu_{2}$ and $p,$ such that

$C(|\overline{\mathrm{B}_{s}}|^{p}+|\overline{\mathrm{B}_{a}}|^{p})\leqslant\overline{\mathrm{B}_{\mu}}:\frac{\partial\overline{\mathbf{v}}}{\partial\mathbf{x}}\leqslant\overline{\mathrm{S}}:\frac{\partial\overline{\mathbf{v}}}{\partial\mathbf{x}},$

being a consequence of the monotonicity of the second term in the relation (9)
and the inequality $(1.25)_{1}$ of Lemma 1.19, shown in [12].

Let us fix an arbitrary $r\in(0,T)$ and take
$\mathbf{\varphi}=\overline{\mathbf{v}}(1-sgn_{+}^{\varepsilon}(t-r))$ in
(55). Then the limit transition on $\varepsilon\rightarrow 0$ in the obtained
equality and the Hölder inequality give that

$\displaystyle\frac{1}{2}\int\limits_{\Omega}|\overline{\mathbf{v}}|^{2}(r)\,d\mathbf{x}+\widetilde{C}\int\limits_{\Omega_{r}}|\frac{\partial\overline{\mathbf{v}}}{\partial\mathbf{x}}|^{p}\,\,d\mathbf{x}dt$
(56) $\displaystyle\leqslant$
$\displaystyle\frac{1}{2}\int\limits_{\Omega}|\overline{\mathbf{v}}|^{2}(r)\,d\mathbf{x}+\int\limits_{\Omega_{r}}\left[\overline{\mathrm{S}}:\frac{\partial\overline{\mathbf{v}}}{\partial\mathbf{x}}\right]\,\,d\mathbf{x}dt+\int_{\Gamma_{T}}\alpha|\overline{\mathbf{v}}|^{2}\,d\mathbf{\gamma}\,dt$
$\displaystyle=$
$\displaystyle\int\limits_{\Omega_{r}}(\overline{\mathbf{v}}\mathbf{\otimes}\overline{\mathbf{v}}):\frac{\partial\mathbf{v}_{2}}{\partial\mathbf{x}}\,\,d\mathbf{x}dt.$

In the sequel we follows the ideas presented in Theorem 3.2 of [23] and
Theorem 4.29 of [12]. By Hölder’s inequality and Lemma 6.3, the right hand
side of (56) is estimated as

$\displaystyle|\int\limits_{\Omega}(\overline{\mathbf{v}}\mathbf{\otimes}\overline{\mathbf{v}})$
$\displaystyle:$
$\displaystyle\frac{\partial\mathbf{v}_{2}}{\partial\mathbf{x}}\,\,d\mathbf{x}|\leqslant
C||\nabla\mathbf{v}_{2}||_{L^{p}(\Omega)}||\overline{\mathbf{v}}||_{L^{\frac{2p}{p-1}}(\Omega)}^{2}$
(57) $\displaystyle\leqslant$ $\displaystyle
C||\nabla\mathbf{v}_{2}||_{L^{p}(\Omega)}\left(||\overline{\mathbf{v}}||_{L^{2}(\Omega)}^{\frac{5p-9}{5p-6}}||\frac{\partial\overline{\mathbf{v}}}{\partial\mathbf{x}}||_{L^{p}(\Omega)}^{\frac{3}{5p-6}}+||\overline{\mathbf{v}}||_{L^{2}(\Omega)}\right)^{2}$
$\displaystyle\leqslant$ $\displaystyle
C||\nabla\mathbf{v}_{2}||_{L^{p}(\Omega)}\left(||\overline{\mathbf{v}}||_{L^{2}(\Omega)}^{2\left(\frac{5p-9}{5p-6}\right)}||\frac{\partial\overline{\mathbf{v}}}{\partial\mathbf{x}}||_{L^{p}(\Omega)}^{\frac{6}{5p-6}}+||\overline{\mathbf{v}}||_{L^{2}(\Omega)}^{2}\right)$
$\displaystyle\leqslant$
$\displaystyle\varepsilon||\frac{\partial\overline{\mathbf{v}}}{\partial\mathbf{x}}||_{L^{p}(\Omega)}^{p}+C_{\varepsilon}||\nabla\mathbf{v}_{2}||_{L^{p}(\Omega)}^{\left(\frac{5p-6}{5p-9}\right)}||\overline{\mathbf{v}}||_{L^{2}(\Omega)}^{2}$
$\displaystyle+C||\nabla\mathbf{v}_{2}||_{L^{p}(\Omega)}||\overline{\mathbf{v}}||_{L^{2}(\Omega)}^{2},$

where the $\varepsilon-$version of Young’s inequality has been used in the
last inequality. Hence taking $\varepsilon=\widetilde{C},$ we obtain

$z(t)\leqslant C+C\int\limits_{0}^{t}f(s)z(s)\ ds\text{\qquad
with\quad}z(t)=\int\limits_{\Omega}|\overline{\mathbf{v}}|^{2}\
d\mathbf{x,}\text{\quad}f(s)=||\nabla\mathbf{v}_{2}||_{L^{p}(\Omega)}^{\left(\frac{5p-6}{5p-9}\right)}.$

Therefore, if $p\geqslant\left(\frac{5p-6}{5p-9}\right),$ that is
$p\geqslant\frac{7+\sqrt{19}}{5}\approx 2.272,$ then applying the Gronwall
inequality we obtain $z(t)=0$ a.e. in $(0,T)$ and we deduce the global-in-time
uniqueness result.$\hfill\;\blacksquare$

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