# Improved decay results for micropolar flows with nonlinear damping

Cilon F. Perusato  Departamento de Matemática. Universidade Federal de
Pernambuco, CEP 50740-560, Recife - PE. Brazil<EMAIL_ADDRESS>and
Franco D. Vega  Departamento de Matemática. Universidade Federal de
Pernambuco, CEP 50740-560, Recife - PE. Brazil<EMAIL_ADDRESS>
###### Abstract.

We examine the long-time behavior of solutions (and their derivatives) to the
micropolar equations with nonlinear velocity damping. Additionally, we get a
speed-up gain of $t^{1/2}$ for the angular velocity, consistent with
established findings for classic micropolar flows lacking nonlinear damping.
Consequently, we also obtain a sharper result regarding the asymptotic
stability of the micro-rotational velocity ${\bf w}(\cdot,t)$. Related results
of independent interest are also included.

###### Key words and phrases:

Micropolar equations, damping term, asymptotic behavior, asymptotic stability

###### 2000 Mathematics Subject Classification:

35B40 (primary), 35Q35 (secondary)

C. F. Perusato was partially supported by Cnpq through grant $\\#$ 310444/2022
- 5, bolsa PQ

##  1\. Introduction

In this work we investigate the large time behavior of solutions to the
following micropolar equations with nonlinear velocity damping in
$\mathbb{R}^{3}$.

(1.1)
$\begin{split}\mbox{\boldmath$u$}_{t}+\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$}+\nabla\>\\!{\sf{p}}=(\>\\!\mu+\chi\>\\!)\,\Delta\mbox{\boldmath$u$}\,+\,2\,\chi\,\nabla\times{\bf
w}-\eta\,|\mbox{\boldmath$u$}|^{\beta-1}\mbox{\boldmath$u$},\\\ {\bf
w}_{t}+\mbox{\boldmath$u$}\cdot\nabla\mbox{\bf w}=\gamma\;\\!\,\Delta\mbox{\bf
w}\,+\,\kappa\,\nabla\>\\!(\>\\!\nabla\cdot\;{\bf
w}\>\\!)\,+\,2\,\chi\,\nabla\times\,\mbox{\boldmath$u$}\,-\,4\,\chi\,{\bf
w},\\\ \nabla\cdot\mbox{\boldmath$u$}(\cdot,t)=0,\\\ (\mbox{\boldmath$u$},{\bf
w})(\cdot,0)=(\mbox{\boldmath$u$}_{0},{\bf
w}_{0})\in\\!\;\\!\mbox{\boldmath$L$}^{2}_{\sigma}(\mathbb{R}^{3})\times\\!\;\\!\mbox{\boldmath$L$}^{2}(\mathbb{R}^{3}),\end{split}$

where the coefficients $\mu$ (kinematic viscosity), $\gamma$ (angular
viscosity), $\chi$ (vortex or micro-rotation viscosity) and $\beta\geq
1,\,\eta$ (damping coefficients) are positive, and $\kappa$ (gyroviscosity) is
nonnegative, all assumed to be constant. The functions
$\mbox{\boldmath$u$}=\mbox{\boldmath$u$}(\mbox{\boldmath$x$},t)$, ${\bf
w}={\bf w}(\mbox{\boldmath$x$},t)$, and ${\sf p}={\sf
p}(\mbox{\boldmath$x$},t)$ are the flow velocity, micro-rotational velocity
and the total pressure, respectively, for $t>0$ and $x\in\mathbb{R}^{n}$.
Here, $\\!\;\\!\mbox{\boldmath$L$}^{2}_{\sigma}(\mathbb{R}^{n})$ denotes the
space of solenoidal fields $\>\\!\mbox{\bf
v}=(v_{1},v_{2},\ldots,v_{n})\\!\>\\!\in\mbox{\boldmath$L$}^{2}(\mathbb{R}^{n})\equiv
L^{2}(\mathbb{R}^{n})^{n}\\!\>\\!$ with $\nabla\\!\cdot\mbox{\bf v}\\!\;\\!=0$
in the distributional sense.

The so-called Leray-Hopf (or simply Leray) solutions in $\mathbb{R}^{3}\\!$
are global mappings (see [18])

$(\mbox{\boldmath$u$},\mbox{\bf w})(\cdot,t)\in C_{\sf
w}([\;\\!0,\infty),L^{2}_{\sigma}(\mathbb{R}^{3})\times
L^{2}(\mathbb{R}^{3}))\\!\;\\!\cap\hskip
2.13387ptL^{2}((0,\infty),\dot{H}^{\\!\;\\!1}(\mathbb{R}^{3})\times\dot{H}^{\\!\;\\!1}(\mathbb{R}^{3}))$

that satisfy the equations in weak sense for $t>0$ and in addition the energy
estimate

(1.2) $\,\,\|\\!\;\\!\;\\!(\mbox{\boldmath$u$},\mbox{\bf
w})(\cdot,t)\\!\;\\!\;\\!\|_{\mbox{}_{\scriptstyle\\!\>\\!L^{\\!\;\\!2}}}^{\>\\!2}\\!\;\\!+\;\\!2\\!\int_{\\!\>\\!s}^{\>\\!t}\mu\;\\!\|\\!\;\\!\;\\!D\mbox{\boldmath$u$}(\cdot,\tau)\\!\;\\!\;\\!\|_{\mbox{}_{\scriptstyle\\!\>\\!L^{\\!\;\\!2}}}^{\>\\!2}+\gamma\;\\!\|\\!\;\\!\;\\!D\mbox{\bf
w}(\cdot,\tau)\\!\;\\!\;\\!\|_{\mbox{}_{\scriptstyle\\!\>\\!L^{\\!\;\\!2}}}^{\>\\!2}d\tau\;\\!\leq\,\|\\!\;\\!\;\\!(\mbox{\boldmath$u$},\mbox{\bf
w})(\cdot,s)\\!\;\\!\;\\!\|_{\mbox{}_{\scriptstyle\\!\>\\!L^{\\!\;\\!2}}}^{\>\\!2}$

for all $t>s$, for $s=0$ and almost every $s>0$. The presence of the nonlinear
damping term $\eta\,|\mbox{\boldmath$u$}|^{\beta-1}\,\mbox{\boldmath$u$}$ is
clearly beneficial to the regularity of weak solutions as we will see. Due to
this, one has the following extra regularity property for the velocity field:
$\mbox{\boldmath$u$}(\cdot,t)\in
L^{\beta+1}(\,(0,\infty),L^{\beta+1}(\mathbb{R}^{3})\,)$, see e.g [7]. In
fact, when $\beta>3$ (or when $\beta=3$ and $4\,\eta\,(\mu+\chi)>1$) and the
initial data are also in ${\bf H}^{1}_{\sigma}\times{\bf{H}}^{1}$ there exists
a unique global strong solution of (1.1), see Z. Ye [27]. In order to have the
same result, when $1\leq\beta<3$, it is required (as usual) some smallness
condition on the initial data, as pointed out by W. Wang and Y. Long [26] .

Let us now give some physical motivation regarding (1.1). In the 1960s, C.
Eringen [10] introduced the micropolar fluids model (see the equations (1.1)
with $\eta=0$ above). It deals with a class of fluids which exhibit certain
microscopic effects arising from micro-motions and local structures of the
fluid particles that lead to a nonsymmetric stress tensor, and which are often
called polar fluids. It includes, as a particular case, the well-established
Navier-Stokes model [17]. When one also considers the nonlinear damping term
several applications can be taken into account in view of the fact that a
system with nonlinear damping arises from the resistance to the motion of the
flows. For that reason it describes various physical situations such as porous
media flow, drag or friction effects, and some dissipative mechanisms. So,
this plays an important role to balance the convection term.

The existence of Leray solutions (originally constructed for the Navier-Stokes
system in [18]) for the equations (1.1) and other similar dissipative systems
is well known, but their uniqueness and exact regularity properties are still
open,111Recent advances seem to give some indication that 3D Leray solutions
may fail to be unique (and therefore smooth), see [1, 6, 14] and references
therein.  except for small initial data in suitable spaces [23, 11, 17, 28] or
in case of nonlinear damping as mentioned before, when $\beta>3$. Recently,
interesting results regarding the regularity of solutions to the micropolar
equation without nonlinear damping (i.e., when $\eta=0$) was studied
separately for $u$ and ${\bf w}$ by using a new notion of partial suitable
solutions, see [8, 9].

###  1.1. Organization of the paper

This work is organized as follows. In subsection 1.2 below, we shall delineate
the primary contributions elucidated in this paper. In Section 2 some
preliminary results are provided to prepare the way for the derivation of the
main theorems described thereafter. Although much of this material is
essentially known, with the exception of Theorem 2.6, which is novel, several
proofs are newly presented and a few improvements are offered. In Section 3,
we present the proofs of our main results. This section is divided into two
parts: the first part is dedicated to proving Theorem 1.1, while the second
part is dedicated to providing the proof of Theorem 1.3. An appendix
supplements the discussion by providing the proof of a claim necessary to
establish Theorem 2.6.

###  1.2. Main results

We improve some previous results (originally obtained by H. Li and Y. Xiao in
[19]). Namely, we provide a faster decay for the $L^{2}$ norm of the micro-
rotational field. We also obtain a better range for the nonlinear damping
coefficient $\beta$, see c.f. Theorem 1.1 and remark 1.2 below. For the
Navier-Stokes flows with nonlinear damping, the large time decay was studied
by Y. Jia, X. Zhang and B-Q. Dong in [15]. That being mentioned, our primary
result can be stated as follows.

######  Theorem 1.1.

Let $(\mbox{\boldmath$u$}_{0},{\bf w}_{0})\in{\bf
L}^{2}_{\sigma}(\mathbb{R}^{3})\times{\bf L}^{2}(\mathbb{R}^{3})$ and
$(\mbox{\boldmath$u$},{\bf w})(\cdot,t)$ any Leray solution to (1.1). Then,

(1.3) $\|\mbox{\boldmath$u$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{n})}\to
0\quad\text{as}\,\,t\to\infty,\quad\forall\beta\geq 7/3.$

Moreover, for the micro-rotational field ${\bf w}(\cdot,t)$, one has

(1.4) $t^{1/2}\|{\bf w}(\cdot,t)\|_{L^{2}(\mathbb{R}^{n})}\to
0\quad\text{as}\,\,t\to\infty,\quad\forall\beta\geq 1.$

Now, let us delve into the results concerning the $\dot{H}^{m}$ norms of the
solutions. For convenience, we define
$\lambda_{0}(\alpha):=\limsup_{t\to\infty}\,t^{\alpha}\,\|\mbox{\boldmath$u$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{n})}$,
for some constant $\alpha\geq 0$. We assume that

(1.5) $\lambda_{0}(\alpha)<\infty,$

$\text{for some }\,\,\alpha\geq 0$.

######  Remark 1.2 (Reasonability of the assumption (1.5)).

Theorem 1.1 can be formulated in terms of the decay character, see [2, 4, 21].
Moreover, by using the so-called Fourier Splitting technique developed by M.
Schonbek in [24] , the authors in [19] show that when the initial data are in
$L^{1}\cap L^{2}$ the assumption (1.5) is valid with $\alpha=3/4$, for all
$\beta>\frac{14}{5}$. Actually, adapting the arguments presented in [19] and
here in our paper it is possible to obtain the same result for all
$\beta>8/3$, but this is out of the scope of this work. All of these facts
ensure that the assumption (1.5) above holds for some values of $\alpha\geq
0$.

We can finally state the result for all derivatives of order $m$ which is the
following asymptotic inequalities in $\dot{H}^{m}$.

######  Theorem 1.3.

Let $(\mbox{\boldmath$u$}_{0},{\bf w}_{0})\in{\bf
L}^{2}_{\sigma}(\mathbb{R}^{3})\times{\bf L}^{2}(\mathbb{R}^{3})$ and
$(\mbox{\boldmath$u$},{\bf w})(\cdot,t)$ any Leray solution to (1.1). Then,

(1.6a)
$\limsup_{t\to\infty}\,t^{\alpha+\frac{m}{2}}\|D^{m}\mbox{\boldmath$u$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}\leq
C_{\alpha,m}\;\;\lambda_{0}(\alpha)$ and, for the micro-rotational field
$\mathbf{w}$, (1.6b) $\limsup_{t\to\infty}t^{\alpha+\frac{m+1}{2}}\|D^{m}{\bf
w}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}\leq\,\frac{{C}_{\alpha,m}}{4\,\chi}\;\;\lambda_{0}(\alpha),$

for each $m\geq 2$ and $\beta\geq\frac{4\,\alpha+7}{4\,\alpha+3}$, where

$C_{\alpha,m}=2^{\alpha+m/2}\,(\mu+\chi)^{-\frac{m+1}{2}}.$

The case $m=1$ is addressed in Lemma 2.3 and in Proposition 2.4. The last
theorem allows us to obtain lower bounds for the $L^{2}$ norm of solutions,
see e.g. [12]. These estimates are closely related to the important results
provided by M. Oliver and E. Titi for the Navier-Stokes equations in [22].

As a corollary of Theorem 1.3, one can also obtain the following sharper
stability result. To obtain the following corollary, it suffices to apply the
triangle inequality and employ Theorem 1.3.

######  Corollary 1.4.

Let $(\mbox{\boldmath$u$}_{0},{\bf w}_{0})\in{\bf
L}^{2}_{\sigma}(\mathbb{R}^{3})\times{\bf L}^{2}(\mathbb{R}^{3})$ and
$(\mbox{\boldmath$u$},{\bf w})(\cdot,t)$ any Leray solution to (1.1).
$If\|(\mbox{\boldmath$u$},{\bf w})(\cdot,t)\|=O(t^{-\alpha})$, then

$\|D^{m}\mbox{\boldmath$u$}(\cdot,t)-D^{m}\tilde{\mbox{\boldmath$u$}}(\cdot,t)\|_{L^{2}}=O(t^{-\alpha-m/2})$

and

$\|D^{m}{\bf w}(\cdot,t)-D^{m}\tilde{{\bf
w}}(\cdot,t)\|_{L^{2}}=O(t^{-\alpha-m/2-1/2}),.$

for all $\beta\geq\frac{4\,\alpha+7}{4\,\alpha+3}$ and each $m\geq 2$, where
$(\tilde{\mbox{\boldmath$u$}},\tilde{{\bf w}})$ stands for a solution to the
problem (1.1) with a perturbed initial condition
$\left.(\tilde{\mbox{\boldmath$u$}},\tilde{{\bf
w}})\right|_{t=0}=\left(\mbox{\boldmath$u$}_{0}+a,{\bf w}_{0}+b\right)$ and
$a,b\in{\bf L}^{2}_{\sigma}(\mathbb{R}^{3})\times{\bf L}^{2}(\mathbb{R}^{3})$.

Notation. As usual,
${\overset{.}{H}^{1}(\mathbb{R}^{n})}=\overset{.}{H}^{1}(\mathbb{R}^{n})^{n}$
where $\overset{.}{H}^{1}(\mathbb{R}^{n})$ denotes the homogeneous Sobolev
space of order 1 , $e^{\nu\Delta t}$ denotes the heat semigroup. As shown
above, boldface letters are used for vector quantities, as in
$\mbox{\boldmath$u$}(x,t)=(u_{1}(x,t),u_{2}(x,t),u_{3}(x,t))$ denotes the
field velocity. Also $\nabla P\equiv\nabla P(\cdot,t)$ denotes the spatial
gradient of $P(\cdot,t)$ , $D_{j}=\frac{\partial}{\partial x_{j}}$ ,
$\nabla\cdot\mbox{\boldmath$u$}\>=\>D_{1}u_{1}\>+\>D_{2}u_{2}\>+\>D_{3}u_{3}$
is the (spatial) divergence of $\mbox{\boldmath$u$}(\cdot,t)$ , similarly
$\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$}=u_{1}D_{1}\mbox{\boldmath$u$}+...+u_{n}D_{n}\mbox{\boldmath$u$}$
. $\left|\cdot\right|_{2}$ denotes the Euclidean norm in $\mathbb{R}^{3}$ ,
$\left|\cdot\right|$ denotes the $\ell^{1}$ norm in $\mathbb{R}^{3}$ and
$\left\|\cdot\right\|_{L^{q}(\mathbb{R}^{3})}\>\>\>\>;\>\>\>\>1\>\leq\>q\>\leq\>\infty$
,are the standard norms of the Lebesgue spaces $L^{q}(\mathbb{R}^{3})$, with
the vector counterparts.

(1.7)
$\left\|\mbox{\boldmath$u$}(\cdot,t)\right\|_{L^{q}(\mathbb{R}^{n})}\>=\>\left\\{\sum_{i=1}^{n}\int_{\mathbb{R}^{n}}\left|u_{i}(x,t)\right|^{q}dx\right\\}^{\frac{1}{q}}$

(1.8)
$\left\|D\mbox{\boldmath$u$}(\cdot,t)\right\|_{L^{q}(\mathbb{R}^{n})}\>=\>\left\\{\sum_{i,j=1}^{n}\int_{\mathbb{R}^{n}}\left|D_{j}u_{i}(x,t)\right|^{q}dx\right\\}^{\frac{1}{q}}$

and, in general,

(1.9)
$\left\|D^{m}\mbox{\boldmath$u$}(\cdot,t)\right\|_{L^{q}(\mathbb{R}^{n})}\>=\>\left\\{\sum_{i,j_{1},...,j_{m}=1}^{n}\int_{\mathbb{R}^{n}}\left|D_{j_{1}}...D_{j_{m}}u_{i}(x,t)\right|^{q}dx\right\\}^{\frac{1}{q}}$

if $1\>\leq\>q\><\>\infty$. When, $q=\infty$,

(1.10)
$\left\|\mbox{\boldmath$u$}(\cdot,t)\right\|_{L^{\infty}(\mathbb{R}^{n})}\>=\>\max\left\\{\left\|u_{i}(\cdot,t)\right\|_{L^{\infty}(\mathbb{R}^{n})}\>\>\>:\>\>\>1\leq
i\leq n\right\\}$

and, for general $m\>\geq\>1$ :

(1.11)
$\left\|D^{m}\mbox{\boldmath$u$}(\cdot,t)\right\|_{L^{\infty}(\mathbb{R}^{n})}\>=\>\max\left\\{\left\|D_{j_{1}}...D_{j_{m}}u_{i}(\cdot,t)\right\|_{L^{\infty}(\mathbb{R}^{n})}\>\>\>:\>\>\>1\leq
i,j_{1},...,j_{m}\leq n\right\\}.$

We also defined for simplicity the following norms for $(u,w)$ as usually made
in the literature:

(1.12) $\left\|(\mbox{\boldmath$u$},{\bf
w})\right\|_{L^{q}(\mathbb{R}^{n})}^{q}\>:=\>\left\|\mbox{\boldmath$u$}\right\|_{L^{q}(\mathbb{R}^{n})}^{q}\>+\>\left\|{\bf
w}\right\|_{L^{q}(\mathbb{R}^{n})}^{q}$

and more generally, for all integers $m\>\geq\>1$,

(1.13) $\left\|(D^{m}\mbox{\boldmath$u$},D^{m}{\bf
w})\right\|_{L^{q}(\mathbb{R}^{n})}^{q}\>:=\>\left\|D^{m}\mbox{\boldmath$u$}\right\|_{L^{q}(\mathbb{R}^{n})}^{q}\>+\>\left\|D^{m}{\bf
w}\right\|_{L^{q}(\mathbb{R}^{n})}^{q}$

for all $1\>\leq q<\infty$ and when $q=\infty$,

(1.14) $\left\|(\mbox{\boldmath$u$},{\bf
w})\right\|_{L^{\infty}(\mathbb{R}^{n})}\>=\>\max\left\\{\left\|\mbox{\boldmath$u$}\right\|_{L^{\infty}(\mathbb{R}^{n})},\left\|{\bf
w}\right\|_{L^{\infty}(\mathbb{R}^{n})}\right\\}.$

##  2\. Some mathematical preliminaries

Let us recall some basic facts regarding the weak solutions of (1.1). Exactly
as in the Navier-Stokes case, it is not known that Leray solutions to the
problem (1.1) are regular and smooth for all $t>0$. However, it is known that
they do behave nicely for all, $t>0$ sufficiently large [17, 18], say
$t>t_{*}$, with

(2.1a) $(\mbox{\boldmath$u$},{\bf w})(\cdot,t)\in
C^{\infty}(\mathbb{R}^{3}\times(t_{*},\infty))$ and, for each
$m\in\mathbb{Z}_{+}$ : (2.1b) $(\mbox{\boldmath$u$},{\bf w})(\cdot,t)\in
C^{0}(\left[t_{*},\infty\right),\mbox{\boldmath$H$}^{m}(\mathbb{R}^{3}))$

and such that the elementary (strong) energy inequality

$\displaystyle\left\|(\mbox{\boldmath$u$},{\bf
w})(\cdot,t)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}\>+\>2\lambda\int_{t_{0}}^{t}\left\|(D\mbox{\boldmath$u$},D{\bf
w})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}d\tau$
$\displaystyle+\>2\kappa\int_{t_{0}}^{t}\left\|\nabla\cdot{\bf
w}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}d\tau\>+\>2\eta\int_{t_{0}}^{t}\left\|\mbox{\boldmath$u$}(\cdot,\tau)\right\|_{L^{\beta+1}(\mathbb{R}^{3})}^{\beta+1}d\tau\>\leq\>\left\|(\mbox{\boldmath$u$},{\bf
w})(\cdot,t_{0})\right\|_{L^{2}(\mathbb{R}^{3})}^{2},$

holds for all $t>t_{0}$ and for a.e. $t_{0}\geq 0$ (including $t_{0}=0$),
where $\lambda=\min\left\\{\mu,\gamma\right\\}$.

We state now some useful well-known lemmas that are necessary to proof our
main results.

######  Lemma 2.1.

For any $f\in H^{2}(\mathbb{R}^{3})$, we have

$\displaystyle\left\|f\right\|_{L^{\infty}(\mathbb{R}^{3})}$
$\displaystyle\left\|\nabla f\right\|_{L^{2}(\mathbb{R}^{3})}\leq
C\left\|f\right\|_{L^{2}(\mathbb{R}^{3})}^{\frac{1}{2}}\left\|\nabla
f\right\|_{L^{2}(\mathbb{R}^{3})}^{\frac{1}{2}}\left\|\nabla^{2}f\right\|_{L^{2}(\mathbb{R}^{3})}.$

######  Lemma 2.2.

Let $\mbox{\boldmath$u$}\in L^{r}(\mathbb{R}^{n})$ and $e^{\nu\Delta\tau}$ the
heat Kernel, then

$\displaystyle\left\|D^{\alpha}\left[e^{\nu\Delta\tau}\mbox{\boldmath$u$}\right]\right\|_{L^{2}(\mathbb{R}^{n})}\>\leq\>K(n,m)\left\|\mbox{\boldmath$u$}\right\|_{L^{r}(\mathbb{R}^{n})}(\nu\tau)^{-\frac{n}{2}\left(\frac{1}{r}-\frac{1}{2}\right)-\frac{\left|\alpha\right|}{2}}$

for all $\tau>0$ and $\alpha$ (multi-index), $1\>\leq\>r\>\leq\>2$,
$n\>\geq\>1$, and $m\>=\>\left|\alpha\right|$.

The following asymptotic result is a key property for the gradient and it is
very important to provide general time decay estimates for solutions to the
system (1.1) and to other diffusive equations (see [12] for the general
approach).

######  Lemma 2.3.

For $(\mbox{\boldmath$u$},{\bf w})$ Leray solutions of (1.1), one has

(2.2)
$\lim_{t\rightarrow+\infty}t^{\frac{1}{2}}\left\|(D\mbox{\boldmath$u$},D{\bf
w})(\cdot,t)\right\|_{L^{2}(\mathbb{R}^{3})}\>=\>0.$

###### Proof.

The next argument is adapted from [16]. Define, for simplicity,
$\mbox{\boldmath$z$}(\cdot,t):=(\mbox{\boldmath$u$},{\bf w})(\cdot,t)$ and
$D^{m}\mbox{\boldmath$z$}=(D^{m}\mbox{\boldmath$u$},D^{m}{\bf w})$ , for each
$m\geq 0$ integer. In order to show (2.2), we used (1.1) to get, after a few
computations,

(2.3) $\begin{split}&\left\|(D\mbox{\boldmath$u$},D{\bf
w})(\cdot,t)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}\>+\>2\min\left\\{\mu,\gamma\right\\}\int_{t_{0}}^{t}\left\|(D^{2}\mbox{\boldmath$u$},D^{2}{\bf
w})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}d\tau\\\
&\>\leq\>\left\|(D\mbox{\boldmath$u$},D{\bf
w})(\cdot,t_{0})\right\|_{L^{2}(\mathbb{R}^{3})}^{2}\\\
&\quad+\>C\>\int_{t_{0}}^{t}\left\|(\mbox{\boldmath$u$},{\bf
w})\right\|_{L^{2}(\mathbb{R}^{3})}^{\frac{1}{2}}\left\|(\nabla\mbox{\boldmath$u$},\nabla{\bf
w})\right\|_{L^{2}(\mathbb{R}^{3})}^{\frac{1}{2}}\left\|(\nabla^{2}\mbox{\boldmath$u$},\nabla^{2}{\bf
w})\right\|_{L^{2}(\mathbb{R}^{3})}^{2}d\tau,\end{split}$

where we have used a Lemma 2.1. By (1.2), we can choose $t_{0}\geq t_{*}$
large enough such that

$C^{2}\left\|(\mbox{\boldmath$u$}_{0},{\bf
w}_{0})\right\|_{L^{2}(\mathbb{R}^{3})}\left\|(D\mbox{\boldmath$u$},D{\bf
w})(\cdot,t_{0})\right\|_{L^{2}(\mathbb{R}^{3})}<\left(\min\left\\{\mu,\gamma\right\\}\right)^{2},$

so that (2.3) gives $\left\|(D\mbox{\boldmath$u$},D{\bf
w})(\cdot,t)\right\|_{L^{2}(\mathbb{R}^{3})}\leq\left\|(D\mbox{\boldmath$u$},D{\bf
w})(\cdot,t_{0})\right\|_{L^{2}(\mathbb{R}^{3})}$ for all t near $t_{0}$ by
continuity. Actually, with this choice, it follows from (2.3) again that

$C^{2}\left\|(\mbox{\boldmath$u$}_{0},{\bf
w}_{0})\right\|_{L^{2}(\mathbb{R}^{3})}\left\|(D\mbox{\boldmath$u$},D{\bf
w})(\cdot,s)\right\|_{L^{2}(\mathbb{R}^{3})}<\left(\min\left\\{\mu,\gamma\right\\}\right)^{2},\quad\forall
s\geq t_{0}.$

Recalling (2.3), this implies that

$\left\|(D\mbox{\boldmath$u$},D{\bf
w})(\cdot,t)\right\|_{L^{2}(\mathbb{R}^{3})}\leq\left\|(D\mbox{\boldmath$u$},D{\bf
w})(\cdot,t_{0})\right\|_{L^{2}(\mathbb{R}^{3})},$

for all $t\geq t_{0}$. Because a monotonic function $f\in C^{0}(a,\infty)\cap
L^{1}(a,\infty)$ has to satisfy $f(t)=O(1/t)$ as $t\rightarrow\infty$, we have

$\lim_{t\rightarrow\infty}t\left\|(D\mbox{\boldmath$u$},D{\bf
w})(\cdot,t)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}\>=\>0.$

∎

An important direct consequence of lemma 2.3 is the following proposition.

######  Proposition 2.4.

Let $(\mbox{\boldmath$u$},{\bf w})(\cdot,t)$ be any Leray solution of problem
(1.1) with $\left\|(\mbox{\boldmath$u$},{\bf
w})\right\|_{L^{2}(\mathbb{R}^{3})}=O(t^{-\alpha})$ , for some $\alpha\geq 0$.
Then, we have

(2.4) $\left\|(\nabla\mbox{\boldmath$u$},\nabla{\bf
w})\right\|_{L^{2}(\mathbb{R}^{3})}=O(t^{-\alpha-\frac{1}{2}}).$

###### Proof.

Define, for simplicity,
$\mbox{\boldmath$z$}(\cdot,t):=(\mbox{\boldmath$u$},{\bf w})(\cdot,t)$ and
$D^{m}\mbox{\boldmath$z$}=(D^{m}\mbox{\boldmath$u$},D^{m}{\bf w})$ , for each
$m\geq 0$ integer. In order to show (2.4), we observe, after a few
computations, that

(2.5)
$\begin{split}&(t-t_{0})^{\widetilde{\gamma}}\left\|\mbox{\boldmath$z$}(\cdot,t)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}+2\lambda\int_{t_{0}}^{t}(\tau-
t_{0})^{\widetilde{\gamma}}\left\|D\mbox{\boldmath$z$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}d\tau\\\
&\leq\widetilde{\gamma}\int_{t_{0}}^{t}(\tau-
t_{0})^{\widetilde{\gamma}-1}\left\|\mbox{\boldmath$z$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}d\tau\\\
&\leq C\int_{t_{0}}^{t}(\tau-
t_{0})^{\widetilde{\gamma}-1}\tau^{-2\alpha}d\tau\\\ &\leq
C(t-t_{0})^{-2\alpha+\widetilde{\gamma}},\end{split}$

for all $\widetilde{\gamma}>2\alpha$. By using Lemma 2.1, we get

(2.6)
$\begin{split}&(t-t_{0})^{\widetilde{\gamma}+1}\left\|D\mbox{\boldmath$z$}(\cdot,t)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}+2\lambda\int_{t_{0}}^{t}(\tau-
t_{0})^{\widetilde{\gamma}+1}\left\|D^{2}\mbox{\boldmath$z$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}d\tau\\\
&\leq(\widetilde{\gamma}+1)\int_{t_{0}}^{t}(\tau-
t_{0})^{\widetilde{\gamma}}\left\|D\mbox{\boldmath$z$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}d\tau\\\
&\;\;\;+C\int_{t_{0}}^{t}(\tau-
t_{0})^{\widetilde{\gamma}+1}\left\|\mbox{\boldmath$z$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}^{\frac{1}{2}}\left\|D\mbox{\boldmath$z$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}^{\frac{1}{2}}\left\|D^{2}\mbox{\boldmath$z$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}d\tau\\\
&\leq(\widetilde{\gamma}+1)\int_{t_{0}}^{t}(\tau-
t_{0})^{\widetilde{\gamma}}\left\|D\mbox{\boldmath$z$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}d\tau\\\
&\;\;\;+C\int_{t_{0}}^{t}(\tau-
t_{0})^{\widetilde{\gamma}+1}\left\|D\mbox{\boldmath$z$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}^{\frac{1}{2}}\left\|D^{2}\mbox{\boldmath$z$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}d\tau\end{split}$

For some $C>0$. By Lemma 2.3 , there exists a $t_{0}$ sufficiently large such
that ,

(2.7) $C^{2}\left\|(D\mbox{\boldmath$u$},D{\bf
w})(\cdot,t)\right\|_{L^{2}(\mathbb{R}^{3})}\leq\lambda^{2}\quad;\quad{\text{for
all}}\quad t\geq t_{0},$

combining (2.5), (2.6) and (2.7), we have

(2.8)
$\begin{split}(t-t_{0})^{\widetilde{\gamma}+1}\left\|D\mbox{\boldmath$z$}(\cdot,t)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}&+\lambda\int_{t_{0}}^{t}(\tau-
t_{0})^{\widetilde{\gamma}+1}\left\|D^{2}\mbox{\boldmath$z$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}d\tau\\\
&\leq(\widetilde{\gamma}+1)\int_{t_{0}}^{t}(\tau-
t_{0})^{\widetilde{\gamma}}\left\|D\mbox{\boldmath$z$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}d\tau\\\
&\leq\widetilde{C}(t-t_{0})^{-2\alpha+\widetilde{\gamma}}\end{split}$

Applying (2.8), we get

$\displaystyle\left\|(D\mbox{\boldmath$u$},D{\bf
w})(\cdot,t)\right\|_{L^{2}(\mathbb{R}^{3})}\leq\widetilde{C}t^{-\alpha-\frac{1}{2}}.$

Thus we complete the proof of (2.4).

∎

###  2.1. Linear operator related to (1.1)

In this section we described some properties regarding the linear problem
associated to (1.1). All the results presented here are useful to provide the
$\dot{H}^{m}$ estimates established in the section 4.

We first consider the linear system related to (1.1).

(2.9a)
$\mbox{\boldmath$\bar{u}$}_{t}\;=\;(\>\\!\mu+\chi\>\\!)\,\Delta\mbox{\boldmath$\bar{u}$}\,+\,2\,\chi\,\nabla\times{\bf\bar{w}},$
(2.9b) ${\bf\bar{w}}_{t}\;\\!=\,\gamma\;\\!\,\Delta{\bf{\bar{{\bf
w}}}}\,+\,\kappa\,\nabla\>\\!(\>\\!\nabla\cdot\;{\bf\bar{w}}\>\\!)\,+\,2\,\chi\,\nabla\times\,\mbox{\boldmath$\bar{u}$}\,-\,4\,\chi\,{\bf\bar{w}},$
(2.9c) $\nabla\cdot\mbox{\boldmath$\bar{u}$}(\cdot,t)=0.$

We observe that the linear system (2.9) has a solution
$\bar{\mbox{\boldmath$z$}}(\cdot,t;t_{0})\in
C^{0}([t_{0},\infty),L^{2}(\mathbb{R}^{3}))$, for each initial time $t_{0}\geq
0$, that is,
$\bar{\mbox{\boldmath$z$}}(\cdot,t;t_{0})=e^{\mathbb{A}\>\\!(t-t_{0})}\mbox{\boldmath$z$}(\cdot,t_{0})$,
where

$\mathbb{A}\mbox{\boldmath$z$}:=\left[\begin{array}[]{ccc}(\mu+\chi)\Delta&2\,\chi\;\\!\nabla\wedge\\\
2\,\chi\;\\!\nabla\wedge&\mathbb{L}-4\chi\,\mbox{Id}_{3\times 3}\\\
\end{array}\right]\\!\\!\;\\!\left[\begin{array}[]{ccc}\mbox{\boldmath$u$}\\\
{\bf w}\\\ \end{array}\right]\\!\\!\;\\!,$

and $\mathbb{L{\bf w}}:=\gamma\,\Delta{\bf w}+\kappa\nabla(\nabla\cdot{\bf
w})$ is the Lamé operator. In the frequency space, after taking the Fourier
transform of system (2.9), one obtains

$\partial_{t}\widehat{\bar{\mbox{\boldmath$z$}}}=M(\xi)\widehat{\bar{\mbox{\boldmath$z$}}},$

where $M=M(\xi)$ is the matrix of symbols

(2.10) $M=\left(\begin{array}[]{ccc}-(\mu+\chi)|\xi|^{2}Id_{3\times 3}&i\chi
R_{3}(\xi)\\\ i\chi R_{3}(\xi)&-(\gamma|\xi|^{2}+2\chi)Id_{3\times
3}-\kappa\,\xi_{i}\xi_{j}\end{array}\right).$

Here, $Id_{3\times 3}$ denotes the $3\times 3$ identity matrix and
$iR_{3}(\xi)$ denotes the rotation matrix

$iR_{3}(\xi)=i\left(\begin{array}[]{ccc}0&\xi_{3}&-\xi_{2}\\\
-\xi_{3}&0&\xi_{1}\\\ \xi_{2}&-\xi_{1}&0\end{array}\right).$

In [20] (see lemma (2.5), p. 48), the authors proved the following estimate
for the eigenvalues of $M(\xi)$:

$\lambda_{max}(M)\leq-C|\xi|^{2},\qquad C=C(\mu,\chi,\gamma)>0,$

as long as $32\chi(\mu+\chi+\gamma)>1$. As a consequence, we immediately have
the following upper bound for the semigroup
$\bigl{(}e^{\mathbb{A}t}\bigr{)}_{t\geq 0}$ associated to (2.9).

######  Lemma 2.5.

Let $\mathcal{G}\in
L^{2}_{\sigma}(\mathbb{R}^{3})\times{L}^{2}(\mathbb{R}^{3})$. If
$32\chi(\mu+\chi+\gamma)>1$, then 222Recalling that $e^{\Delta\tau}$ denotes
the heat semigroup.

(2.11) $\|e^{\mathbb{A}t}\>\\!\mathcal{G}\|_{L^{2}}\leq\left\|e^{c\,\Delta
t}\>\\!\mathcal{G}\right\|_{L^{2}},$

for all $t\geq 0$, where $e^{\mathbb{A}t}$ is the semigroup generated by the
linear operator $\mathbb{A}$ above.

###### Proof.

Using the Plancherel‘s identity, we have
$\|e^{\mathbb{A}t}\>\\!\mathcal{G}\|_{L^{2}}\leq\|e^{-C\,|\xi|^{2}}\hat{\mathcal{G}}\|=\|e^{c\,\Delta
t}\>\\!\mathcal{G}\|_{L^{2}}$ ∎

As a byproduct of our investigation, we finish this section by showing a
fundamental result regarding the $\dot{H}^{m}$ estimates for the linear system
(2.9) associated to (1.1). This should be compared to the so-called inverse
Wiegner’s theorem for solutions to the Navier-Stokes equations, see [25]. The
following estimate is needed to obtain Theorem 1.3 and also might be of
independent interest.

######  Theorem 2.6.

Let $\lambda_{0}(\alpha)<\infty$, for some $0<\alpha<3/4$. Then, one has
$\limsup_{t\to+\infty}t^{\alpha+\frac{m}{2}}\|e^{\mathbb{A}\,t}\mbox{\boldmath$z$}_{0}\|_{\dot{H}^{m}}<C_{m,\alpha}\,\lambda_{0}(\alpha).$

###### Proof.

We start with the following claim.

######  Claim 2.7.

Given any initial time $t_{0}\geq 0$, one has

(2.12)
$\limsup_{t\to\infty}\,t^{\alpha\,+\,\hat{\beta}}\,\|\mbox{\boldmath$z$}(\cdot,t)-e^{\mathbb{A}(\,t-t_{0})}\mbox{\boldmath$z$}(\cdot,t_{0})\|_{L^{2}}\leq
C_{\,\beta,\,\hat{\beta},\,\lambda_{0}(\alpha)\,},$

where

$\displaystyle
C_{\,\beta,\,\hat{\beta},\,\lambda_{0}(\alpha)\,}=\begin{cases}c_{1}\,2^{\alpha+\hat{\beta}}\,\lambda_{0}(\alpha)^{2}+\tilde{c}_{1}\,\lambda_{0}(\alpha)^{\beta},\quad
0\leq\alpha<1/4\\\
c_{2}\,2^{\alpha+\hat{\beta}}\,\lambda_{0}(\alpha)+\tilde{c}_{2}\,\lambda_{0}(\alpha)^{\beta},\quad
1/4\leq\alpha<1/2\\\
c_{3}\,2^{\alpha+\hat{\beta}}\,\lambda_{0}(\alpha)^{1/2}+\tilde{c}_{3}\,\lambda_{0}(\alpha)^{\beta},\quad
1/2\leq\alpha<3/4,\end{cases}$
$\displaystyle\hat{\beta}=\begin{cases}\alpha+1/4,\quad
0\leq\alpha<1/4\text{\,\,and\,\,}\beta\geq\frac{8\alpha+8}{4\alpha+3}\\\
1/4,\quad
1/4\leq\alpha<1/2\text{\,\,and\,\,}\beta\geq\frac{4\alpha+8}{4\alpha+3}\\\
0,\quad
1/2\leq\alpha<3/4\text{\,\,and\,\,}\beta\geq\frac{4\alpha+7}{4\alpha+3},\end{cases}$

and $c_{i}$, $\tilde{c}_{i}$ are constants.

Before proving the claim above333For the proof of claim, see the appendix
section., we obtain the Theorem 2.6 by using the following argument (adapted
from [3]). By Theorem 1.1 we have that $\lambda_{0}(0)=0$, so we can apply the
above estimate (2.12) for $\alpha=0$ which immediately leads us to
$t^{1/4}\|\mbox{\boldmath$z$}(\cdot,t)-e^{\mathbb{A}(t-t_{0})}\mbox{\boldmath$z$}(\cdot,t_{0})\|\to
0$ as $t\to\infty$. Now, we just have to notice that

(2.13)
$\begin{split}\|e^{\mathbb{A}(t-t_{0})}\mbox{\boldmath$z$}(\cdot,t_{0})\|\leq\|\mbox{\boldmath$z$}(\cdot,t)\|+\|\mbox{\boldmath$z$}(\cdot,t)-e^{\mathbb{A}(t-t_{0})}\mbox{\boldmath$z$}(\cdot,t_{0})\|\\\
=O(t^{-\alpha})+O(t^{-1/4})=O(t^{-\gamma}),\end{split}$

where $\gamma=\min\\{\alpha,1/4\\}$. Therefore, if $\alpha\leq 1/4$, then
Theorem 2.6 is already shown. The remaining cases for $\alpha$ will be
considered as follows. When $1/4<\alpha\leq 1/2$, by (2.13), we have
$\|\mbox{\boldmath$z$}(\cdot,t)\|=O(t^{-1/4})$, so that, by (2.12), we have
$\|\mbox{\boldmath$z$}(\cdot,t)-e^{\mathbb{A}(t-t_{0})}\mbox{\boldmath$z$}(\cdot,t_{0})\|=O(t^{-1/2})$.
As before, we get

$\begin{split}\|e^{\mathbb{A}(t-t_{0})}\mbox{\boldmath$z$}(\cdot,t_{0})\|_{L^{2}}\leq\|\mbox{\boldmath$z$}(\cdot,t)\|_{L^{2}}+\|\mbox{\boldmath$z$}(\cdot,t)-e^{\mathbb{A}(t-t_{0})}\mbox{\boldmath$z$}(\cdot,t_{0})\|_{L^{2}}\\\
=O(t^{-\alpha})+O(t^{-1/2})=O(t^{-\alpha})\end{split}$

and we are done in this case since $1/4<\alpha\leq 1/2$. Similarly, we can
complete the proof for the case $\alpha<3/4$. Adapting the argument presented
in [13] and further developed in [12], we just have to note that for all
$m\geq 0$,

$\|e^{\mathbb{A}(t-t_{0})}\mbox{\boldmath$z$}(\cdot,t_{0})\|_{L^{2}}=O(t^{-\alpha})\implies\|D^{m}\,e^{\mathbb{A}(t-t_{0})}\mbox{\boldmath$z$}(\cdot,t_{0})\|_{L^{2}}=O(t^{-\alpha-m/2})\,.$

For convenience, the proof of Claim 2.7 will be given in Appendix. ∎

######  Remark 2.8.

Due to this last result, the same interesting topological properties (obtained
by L. Brandolese, C. Perusato and P. Zingano in [3]) hold for solutions to
(1.1).

We are now in position to prove the main results.

##  3\. Proof of $L^{2}$ results

###  3.1. Proof of Theorem 1.1

Let $t_{0}>t_{*}$, where $t_{*}$ is the regularity time (see (2.1)). First, we
will prove the result for the micro-rotational field ${\bf w}(\cdot,t)$.
Defining $Z(\cdot,t)=e^{4\chi t}{\bf w}(\cdot,t)$ and applying Duhamel’s
Principle, for $Z(\cdot,t)$ we obtain, after rewritting for ${\bf
w}(\cdot,t)$, that

$\displaystyle{\bf w}(\cdot,t)$
$\displaystyle=\>e^{\gamma\Delta(t-t_{0})}e^{-4\chi(t-t_{0})}{\bf
w}(\cdot,t_{0})\>-\>\int_{t_{0}}^{t}e^{\gamma\Delta(t-\tau)}e^{-4\chi(t-\tau)}(\mbox{\boldmath$u$}\cdot\nabla{\bf
w})(\cdot,\tau)d\tau$
$\displaystyle\>\>\>\>\>+\>\kappa\int_{t_{0}}^{t}e^{\gamma\Delta(t-\tau)}e^{-4\chi(t-\tau)}\nabla(\nabla\cdot{\bf
w})(\cdot,\tau)d\tau$
$\displaystyle\>\>\>\>\>+\>2\chi\int_{t_{0}}^{t}e^{\gamma\Delta(t-\tau)}e^{-4\chi(t-\tau)}(\nabla\times\mbox{\boldmath$u$})(\cdot,\tau)d\tau.$

Hence,

$\displaystyle t^{\frac{1}{2}}\left\|{\bf
w}(\cdot,t)\right\|_{L^{2}(\mathbb{R}^{3})}$
$\displaystyle\leq\>t^{\frac{1}{2}}e^{-4\chi(t-t_{0})}\left\|e^{\gamma\Delta(t-t_{0})}{\bf
w}(\cdot,t_{0})\right\|_{L^{2}(\mathbb{R}^{3})}$
$\displaystyle\>\>\>+\>t^{\frac{1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|e^{\gamma\Delta(t-\tau)}(\mbox{\boldmath$u$}\cdot\nabla{\bf
w})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\>\>\>+\>\kappa
t^{\frac{1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|e^{\gamma\Delta(t-\tau)}\nabla(\nabla\cdot{\bf
w})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\>\>\>+\>2\chi
t^{\frac{1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|e^{\gamma\Delta(t-\tau)}(\nabla\times\mbox{\boldmath$u$})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle=\>I\>+\>II\>+\>III\>+\>IV$

It is clear that
$t^{\frac{1}{2}}e^{-4\chi(t-t_{0})}\left\|e^{\gamma\Delta(t-t_{0})}{\bf
w}(\cdot,t_{0})\right\|_{L^{2}(\mathbb{R}^{3})}\>\rightarrow\>0$ as
$t\to+\infty$.

For $II$ , we use Lemma 2.2 and Lemma 2.3, we have

$\displaystyle
t^{\frac{1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|e^{\gamma\Delta(t-\tau)}(\mbox{\boldmath$u$}\cdot\nabla{\bf
w})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq\>K\gamma^{-\frac{3}{4}}t^{\frac{1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|(\mbox{\boldmath$u$}\cdot\nabla{\bf
w})(\cdot,\tau)\right\|_{L^{1}(\mathbb{R}^{3})}(t-\tau)^{-\frac{3}{4}}d\tau$
$\displaystyle\leq\>K\gamma^{-\frac{3}{4}}t^{\frac{1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|\mbox{\boldmath$u$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}\left\|D{\bf
w}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}(t-\tau)^{-\frac{3}{4}}d\tau$
$\displaystyle\leq\>K\gamma^{-\frac{3}{4}}\left\|(\mbox{\boldmath$u$},{\bf
w})(\cdot,t_{0})\right\|_{L^{2}(\mathbb{R}^{3})}t^{\frac{1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|D{\bf
w}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}(t-\tau)^{-\frac{3}{4}}d\tau$
$\displaystyle\leq\>K\gamma^{-\frac{3}{4}}\left\|(\mbox{\boldmath$u$},{\bf
w})(\cdot,t_{0})\right\|_{L^{2}(\mathbb{R}^{3})}\epsilon
t^{\frac{1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\tau^{-\frac{1}{2}}(t-\tau)^{-\frac{3}{4}}d\tau$
$\displaystyle\leq\>K\gamma^{-\frac{3}{4}}\epsilon\left\|(\mbox{\boldmath$u$},{\bf
w})(\cdot,t_{0})\right\|_{L^{2}(\mathbb{R}^{3})}\left(t^{\frac{1}{4}}e^{-2\chi
t}\>+\>(4\chi)^{-\frac{1}{4}}\Gamma\left(\frac{1}{4}\right)\right).$

Therefore

(3.1)
$\lim_{t\rightarrow+\infty}t^{\frac{1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|e^{\gamma\Delta(t-\tau)}(\mbox{\boldmath$u$}\cdot\nabla{\bf
w})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau=0.$

For $III$ , we use Lemma 2.2 and Lemma 2.3, we have

$\displaystyle\kappa
t^{\frac{1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|e^{\gamma\Delta(t-\tau)}\nabla(\nabla\cdot{\bf
w})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq\>\kappa
K\gamma^{-\frac{1}{2}}t^{\frac{1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|D{\bf
w}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}(t-\tau)^{-\frac{1}{2}}d\tau$
$\displaystyle\leq\>\kappa
K\gamma^{-\frac{1}{2}}\epsilon\>t^{\frac{1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\tau^{-\frac{1}{2}}(t-\tau)^{-\frac{1}{2}}d\tau$
$\displaystyle\leq\>\kappa
K\gamma^{-\frac{1}{2}}\epsilon\left[t^{\frac{1}{2}}e^{-2\chi
t}\>+\>(4\chi)^{-\frac{1}{2}}\sqrt{\pi}\right].$

So,

(3.2) $\lim_{t\rightarrow+\infty}\kappa
t^{\frac{1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|e^{\gamma\Delta(t-\tau)}\nabla(\nabla\cdot{\bf
w})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau\>=\>0.$

For $IV$,

(3.3)
$\lim_{t\rightarrow+\infty}2\chi\;t^{\frac{1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|e^{\gamma\Delta(t-\tau)}(\nabla\times\mbox{\boldmath$u$})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau\>=\>0$

Combining (3.1), (3.2) and (3.3), we get

$\lim_{t\rightarrow+\infty}t^{\frac{1}{2}}\left\|{\bf
w}(\cdot,t)\right\|_{L^{2}(\mathbb{R}^{3})}\>=\>0.$

Thus, we complete the proof of (1.4). Now, we will show that
$\left\|\mbox{\boldmath$u$}\right\|_{L^{2}(\mathbb{R}^{3})}\rightarrow 0$. We
start rewriting the first equation in (1.1) as

$\displaystyle\mbox{\boldmath$u$}_{t}\>=\>(\mu+\chi)\Delta\mbox{\boldmath$u$}\>+\>{\bf
F}(\cdot,\tau)$

where

${\bf
F}(\cdot,\tau)\>=\>-\mbox{\boldmath$u$}\cdot\nabla{\mbox{\boldmath$u$}}\>-\>\nabla{p}\>-\>\eta\left|\mbox{\boldmath$u$}\right|^{\beta-1}\mbox{\boldmath$u$}\>+\>2\chi\nabla\times{\bf
w}.$

In other words,

$\displaystyle{\bf
F}(\cdot,\tau)\>=\>\mathbb{P}_{h}\left[-\mbox{\boldmath$u$}\cdot\nabla{\mbox{\boldmath$u$}}\>-\>\eta\left|\mbox{\boldmath$u$}\right|^{\beta-1}\mbox{\boldmath$u$}\>+\>2\chi\nabla\times{\bf
w}\right],$

where $\mathbb{P}_{h}$ denotes the Helmholtz - Leray projector. By Duhamel’s
principle, we get

$\mbox{\boldmath$u$}(\cdot,t)\>=\>e^{(\mu+\chi)\Delta(t-t_{0})}\mbox{\boldmath$u$}(\cdot,t_{0})\>+\>\int_{t_{0}}^{t}e^{(\mu+\chi)\Delta(t-\tau)}{\bf
F}(\cdot,\tau)d\tau,$

Since the Heat Kernel conmutes with the Helmholtz projector, we have

$\displaystyle\left\|\mbox{\boldmath$u$}(\cdot,t)\right\|_{L^{2}(\mathbb{R}^{3})}$
$\displaystyle\>\leq\>\left\|e^{(\mu+\chi)\Delta(t-t_{0})}\mbox{\boldmath$u$}(\cdot,t_{0})\right\|_{L^{2}(\mathbb{R}^{3})}$
$\displaystyle\;\;\;\>+\>\int_{t_{0}}^{t}\left\|e^{(\mu+\chi)\Delta(t-\tau)}(\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\;\;\;\>+\>\eta\int_{t_{0}}^{t}\left\|e^{(\mu+\chi)\Delta(t-\tau)}(\left|\mbox{\boldmath$u$}\right|^{\beta-1}\mbox{\boldmath$u$})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\;\;\;\>+\>2\chi\int_{t_{0}}^{t}\left\|e^{(\mu+\chi)\Delta(t-\tau)}(\nabla\times{\bf
w})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\;=\>I\>+\>II\>+\>III\>+\>IV.$

It is clear that
$\left\|e^{(\mu+\chi)\Delta(t-t_{0})}\mbox{\boldmath$u$}(\cdot,t_{0})\right\|_{L^{2}(\mathbb{R}^{3})}\rightarrow
0$ as $t\rightarrow+\infty$.

For $II$, we use again 2.2 and 2.3, we get

$\displaystyle\int_{t_{0}}^{t}\left\|e^{(\mu+\chi)\Delta(t-\tau)}\mbox{\boldmath$u$}(\cdot,\tau)\cdot\nabla{\mbox{\boldmath$u$}}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq
K(\mu+\chi)^{-\frac{3}{4}}\int_{t_{0}}^{t}(t-\tau)^{-\frac{3}{4}}\left\|\mbox{\boldmath$u$}(\cdot,\tau)\cdot\nabla{\mbox{\boldmath$u$}}(\cdot,\tau)\right\|_{L^{1}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq\>K(\mu+\chi)^{-\frac{3}{4}}\sqrt{3}\int_{t_{0}}^{t}(t-\tau)^{-\frac{3}{4}}\left\|\mbox{\boldmath$u$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}\left\|D\mbox{\boldmath$u$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq\>K(\mu+\chi)^{-\frac{3}{4}}\left\|(\mbox{\boldmath$u$},{\bf
w})(\cdot,t_{0})\right\|_{L^{2}(\mathbb{R}^{3})}\int_{t_{0}}^{t}(t-\tau)^{-\frac{3}{4}}\left\|(D\mbox{\boldmath$u$},D{\bf
w})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq\>K(\mu+\chi)^{-\frac{3}{4}}\left\|(\mbox{\boldmath$u$},{\bf
w})(\cdot,t_{0})\right\|_{L^{2}(\mathbb{R}^{3})}\epsilon\int_{t_{0}}^{t}(t-\tau)^{-\frac{3}{4}}\tau^{-\frac{1}{2}}d\tau$
$\displaystyle\leq\>K\>(\mu+\chi)^{-\frac{3}{4}}\left\|(\mbox{\boldmath$u$},{\bf
w})(\cdot,t_{0})\right\|_{L^{2}(\mathbb{R}^{3})}\>\epsilon\>t^{-\frac{1}{4}}.$

Therefore

(3.4)
$\lim_{t\to+\infty}\int_{t_{0}}^{t}\left\|e^{(\mu+\chi)\Delta(t-\tau)}\mbox{\boldmath$u$}(\cdot,\tau)\cdot\nabla{\mbox{\boldmath$u$}}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau\>=\>0.$

For $III$, applying lemma 2.2, Gagliardo - Sobolev - Nirenberg inequality and
lemma 2.3, we get

$\displaystyle\int_{t_{0}}^{t}\left\|e^{(\mu+\chi)\Delta(t-\tau)}\eta\left|\mbox{\boldmath$u$}\right|^{\beta-1}\mbox{\boldmath$u$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle=\>K(m)\>(\mu+\chi)^{-\frac{3}{4}}\int_{t_{0}}^{t}(t-\tau)^{-\frac{3}{4}}\left\|\eta\left|\mbox{\boldmath$u$}\right|^{\beta-1}\mbox{\boldmath$u$}(\cdot,\tau)\right\|_{L^{1}(\mathbb{R}^{3})}d\tau$
$\displaystyle\>=\>K(m)\>(\mu+\chi)^{-\frac{3}{4}}\eta\int_{t_{0}}^{t}(t-\tau)^{-\frac{3}{4}}\left\|\mbox{\boldmath$u$}\right\|_{L^{\beta}(\mathbb{R}^{3})}^{\beta}d\tau$
$\displaystyle\leq
K(m)\>(\mu+\chi)^{-\frac{3}{4}}\eta\int_{t_{0}}^{t}(t-\tau)^{-\frac{3}{4}}\left\|\mbox{\boldmath$u$}\right\|_{L^{2}(\mathbb{R}^{3})}^{\frac{6-\beta}{2}}\left\|D\mbox{\boldmath$u$}\right\|_{L^{2}(\mathbb{R}^{3})}^{\frac{3(\beta-2)}{2}}d\tau$
$\displaystyle\leq
K(m)\>(\mu+\chi)^{-\frac{3}{4}}\eta\>\left\|(\mbox{\boldmath$u$}_{0},{\bf
w}_{0})\right\|_{L^{2}(\mathbb{R}^{3})}^{\frac{6-\beta}{2}}\int_{t_{0}}^{t}(t-\tau)^{-\frac{3}{4}}\left(\epsilon\tau^{-\frac{1}{2}}\right)^{\frac{3(\beta-2)}{2}}d\tau$
$\displaystyle\leq
K(m)\>(\mu+\chi)^{-\frac{3}{4}}\eta\>C^{\frac{6-\beta}{2}}\>\epsilon^{\frac{3(\beta-2)}{2}}\int_{t_{0}}^{t}(t-\tau)^{-\frac{3}{4}}\tau^{-\frac{3(\beta-2)}{4}}d\tau.$

Now, we observe that

$\displaystyle\int_{t_{0}}^{t}(t-\tau)^{-\frac{3}{4}}\tau^{-\frac{3(\beta-2)}{4}}d\tau=\left\\{\begin{matrix}C(\beta)\>t^{\frac{7-3\beta}{4}}&;\forall\;\;\frac{7}{3}\leq\beta<\frac{10}{3}\\\
C\>t^{-\frac{3}{4}}\ln\left(\frac{t}{2}\right)\>+\>t^{\frac{7-3\beta}{4}}&;\beta=\frac{10}{3}\\\
C(\beta)\>\left[t^{-\frac{3}{4}}+t^{\frac{7-3\beta}{4}}\right]&;\forall\;\;\beta>\frac{10}{3}\end{matrix}\right.$

Therefore

(3.5)
$\lim_{t\to+\infty}\int_{t_{0}}^{t}\left\|e^{(\mu+\chi)\Delta(t-\tau)}\eta\left|\mbox{\boldmath$u$}\right|^{\beta-1}\mbox{\boldmath$u$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau\>=\>0$

for all $\beta\>\geq\>\frac{7}{3}$. Finally, for $IV$, by using Lemma 2.2 we
may assume that $t_{0}$ is large enough such that

$\displaystyle\left\|{\bf
w}(\cdot,t)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau\><\>\epsilon
t^{\frac{1}{2}},\;\;\;\;\;\;\;\;\;\forall t>t_{0}.$

So, we have

$\displaystyle
2\chi\int_{t_{0}}^{t}\left\|e^{(\mu+\chi)\Delta(t-\tau)}(\nabla\times{\bf
w})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq\>K(2\chi)(\mu+\chi)^{-\frac{1}{2}}\int_{t_{0}}^{t}(t-\tau)^{-\frac{1}{2}}\left\|{\bf
w}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq\>K(2\chi)(\mu+\chi)^{-\frac{1}{2}}\epsilon\int_{t_{0}}^{t}(t-\tau)^{-\frac{1}{2}}\tau^{-\frac{1}{2}}d\tau$
$\displaystyle\leq K(2\chi)(\mu+\chi)^{-\frac{1}{2}}\>\epsilon\>C.$

Therefore

(3.6)
$\lim_{t\rightarrow+\infty}2\chi\int_{t_{0}}^{t}\left\|e^{(\mu+\chi)\Delta(t-\tau)}(\nabla\times{\bf
w})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau\>=\>0.$

Combining (3.4), (3.5) and (3.6), we have

$\displaystyle\lim_{t\rightarrow+\infty}\left\|\mbox{\boldmath$u$}(\cdot,t)\right\|_{L^{2}(\mathbb{R}^{3})}\>=\>0.$

which completes the proof of Theorem 1.1 , for all $\beta\geq\frac{7}{3}$.

##  4\. Proof of $\dot{H}^{m}$ results

A natural consequence of Theorem 1.1 and Proposition 2.4 is the following
result.

######  Corollary 4.1.

Let $(\mbox{\boldmath$u$},{\bf w})(\cdot,t)$ be any Leray solution of (1.1)
with $\|(\mbox{\boldmath$u$},{\bf w})(\cdot,t)\|_{L^{2}}=O(t^{-\alpha})$, for
some $\alpha\geq 0$. Then, we have $\|{\bf
w}(\cdot,t)\|_{L^{2}}=O(t^{-\alpha-\frac{1}{2}})$.

###  4.1. Proof of Theorem 1.3

Let $\mbox{\boldmath$z$}(\cdot,t)=(\mbox{\boldmath$u$},{\bf w})(\cdot,t)$ any
Leray solution to (1.1). First, we note that by using the corollary 4.1 we
actually have

$\begin{split}\limsup_{t\to+\infty}t^{\alpha}\,\|\mbox{\boldmath$u$}(\cdot,t)\|_{L^{2}}<\infty\\\
\,\,\,\Rightarrow\,\,\limsup_{t\to+\infty}t^{\alpha}\,\|\mbox{\boldmath$z$}(\cdot,t)\|_{L^{2}}=\limsup_{t\to+\infty}t^{\alpha}\,\|\mbox{\boldmath$u$}(\cdot,t)\|_{L^{2}}<\infty\end{split}$

Due to this, we will show the result for
${\lambda}_{0}:=\limsup_{t\to+\infty}t^{\alpha}\|\mbox{\boldmath$z$}(\cdot,t)\|_{L^{2}}<\infty$.

Let

${\bf
Q}(\cdot,t):=\left[\begin{array}[]{ccc}\mathbb{P}_{h}\,[-\,(\mbox{\boldmath$u$}\cdot\nabla)\mbox{\boldmath$u$}-\eta|\mbox{\boldmath$u$}|^{\beta-1}\,\mbox{\boldmath$u$}\,]\\\
-\,(\mbox{\boldmath$u$}\cdot\nabla){\bf w}\end{array}\right]\\!\\!\;\\!$

We consider the following integral representation for
$\mbox{\boldmath$z$}(\cdot,t):=(\mbox{\boldmath$u$},{\bf w})(\cdot,t)$

${\mbox{\boldmath$z$}}(\cdot,t)=e^{\mathbb{A}(t-t_{0})}\,\mbox{\boldmath$z$}(\cdot,t_{0})+\int_{t_{0}}^{t}e^{\mathbb{A}(t-\tau)}{\bf
Q}(\cdot,\tau)d\tau,\quad\forall t>t_{0},$

for $t_{0}>t_{*}$ (see (2.1)) where $\mathbb{A}$ is the linear operator given
by lemma 2.5 above. Taking the operator $\nabla^{m}$, norm
$\left\|\cdot\right\|_{L^{2}(\mathbb{R}^{3})}$ and multiplying by
$t^{\alpha+\frac{m}{2}}$, one has

$\displaystyle
t^{\alpha+\frac{m}{2}}\left\|\nabla^{m}\mbox{\boldmath$u$}(\cdot,t)\right\|_{L^{2}(\mathbb{R}^{3})}$
$\displaystyle\leq\>t^{\alpha+\frac{m}{2}}\left\|\nabla^{m}e^{(\mu+\chi)\Delta(t-t_{0})}\mbox{\boldmath$u$}_{0}\right\|_{L^{2}(\mathbb{R}^{3})}$
$\displaystyle\>\>\>+\>t^{\alpha+\frac{m}{2}}\int_{t_{0}}^{t}\left\|\nabla^{m}e^{(\mu+\chi)\Delta(t-\tau)}(\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\>\>\>+\>t^{\alpha+\frac{m}{2}}\int_{t_{0}}^{t}\left\|\nabla^{m}e^{(\mu+\chi)\Delta(t-\tau)}\eta(\left|\mbox{\boldmath$u$}\right|^{\beta-1}\mbox{\boldmath$u$})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\>\>\>+\>2\chi
t^{\alpha+\frac{m}{2}}\int_{t_{0}}^{t}\left\|\nabla^{m}e^{(\mu+\chi)\Delta(t-\tau)}(\nabla\times{\bf
w})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle=\>I\>+\>II\>+\>III,$

where we have used some basic properties for the Leray projector.

To estimate term I, we invoke Theorem 2.6, which provides a inverse Wiegner-
type result. Therefore,

(4.1)
$\lim_{t\rightarrow+\infty}t^{\alpha+\frac{m}{2}}\left\|\nabla^{m}e^{(\mu+\chi)\Delta(t-t_{0})}\mbox{\boldmath$u$}(\cdot,t_{0})\right\|_{L^{2}(\mathbb{R}^{3})}\leq
C_{m,\alpha}\lambda_{0}(\alpha).$

For II,

$\displaystyle
t^{\alpha+\frac{m}{2}}\int_{t_{0}}^{\frac{t}{2}}\left\|\nabla^{m}e^{(\mu+\chi)\Delta(t-\tau)}(\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq\>(\mu+\chi)^{-\frac{m}{2}-\frac{5}{4}}\>t^{\alpha+\frac{m}{2}}\int_{t_{0}}^{\frac{t}{2}}(t-\tau)^{-\frac{m}{2}-\frac{5}{4}}\left\|\mbox{\boldmath$u$}\right\|_{L^{2}(\mathbb{R}^{3})}^{2}d\tau$
$\displaystyle\leq\>(\mu+\chi)^{-\frac{m}{2}-\frac{5}{4}}\>\left\|(\mbox{\boldmath$u$}_{0},{\bf
w}_{0})\right\|_{L^{2}(\mathbb{R}^{3})}\>t^{\alpha+\frac{m}{2}}\int_{t_{0}}^{\frac{t}{2}}(t-\tau)^{-\frac{m}{2}-\frac{5}{4}}\left\|\mbox{\boldmath$u$}\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq\>C(\mu+\chi)^{-\frac{m}{2}-\frac{5}{4}}\left\|(\mbox{\boldmath$u$}_{0},{\bf
w}_{0})\right\|_{L^{2}(\mathbb{R}^{3})}\>(\lambda_{0}(\alpha)+\epsilon)\>\underbrace{t^{\alpha+\frac{m}{2}}\int_{t_{0}}^{\frac{t}{2}}(t-\tau)^{-\frac{m}{2}-\frac{5}{4}}\tau^{-\alpha}d\tau}_{(*)}$

Note that

$\displaystyle(*)=\left\\{\begin{matrix}C_{1}(\alpha,m)\>t^{-\frac{1}{4}}&\forall\>0\leq\>\alpha<\>1\\\
C_{2}(m)\>\left[t^{-\frac{1}{4}}\>\>\ln\left(\frac{t}{2}\right)\>+\>t^{-\frac{1}{4}}\right]&\alpha=1\\\
C_{3}(\alpha,m)\>\left[t^{\alpha-\frac{5}{4}}\>+\>t^{-\frac{1}{4}}\right]&\forall\>1<\>\alpha<\>\frac{5}{4}\end{matrix}\right.$

Therefore, we have

(4.2)
$\limsup_{t\to+\infty}t^{\alpha+\frac{m}{2}}\int_{t_{0}}^{t}\left\|\nabla^{m}e^{(\mu+\chi)\Delta(t-\tau)}(\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau\>=\>0.$

For $III$, applying the Gagliardo–Nirenberg inequality, we get

$\displaystyle
t^{\alpha+\frac{m}{2}}\int_{t_{0}}^{t}\left\|\nabla^{m}e^{(\mu+\chi)\Delta(t-\tau)}(\eta\left|\mbox{\boldmath$u$}\right|^{\beta-1}\mbox{\boldmath$u$})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq\>K(m)(\mu+\chi)^{-\frac{3}{4}-\frac{m}{2}}\eta\>t^{\alpha+\frac{m}{2}}\int_{t_{0}}^{t}(t-\tau)^{-\frac{3}{4}-\frac{m}{2}}\left\|\left|\mbox{\boldmath$u$}\right|^{\beta-1}\mbox{\boldmath$u$}(\cdot,\tau)\right\|_{L^{1}(\mathbb{R}^{3})}d\tau$
$\displaystyle\>=\>K(m)(\mu+\chi)^{-\frac{3}{4}-\frac{m}{2}}\eta\>t^{\alpha+\frac{m}{2}}\int_{t_{0}}^{t}(t-\tau)^{-\frac{3}{4}-\frac{m}{2}}\left\|\mbox{\boldmath$u$}\right\|_{L^{\beta}(\mathbb{R}^{3})}^{\beta}d\tau$
$\displaystyle\leq
K(m)(\mu+\chi)^{-\frac{3}{4}-\frac{m}{2}}\eta\>t^{\alpha+\frac{m}{2}}\int_{t_{0}}^{t}(t-\tau)^{-\frac{3}{4}-\frac{m}{2}}\left\|\mbox{\boldmath$u$}\right\|_{L^{2}(\mathbb{R}^{3})}^{\frac{6-\beta}{2}}\left\|D\mbox{\boldmath$u$}\right\|_{L^{2}(\mathbb{R}^{3})}^{\frac{3(\beta-2)}{2}}d\tau$
$\displaystyle\leq
C(\mu,\chi,m,\eta)\>(\lambda_{0}(\alpha)+\epsilon)^{\frac{6-\beta}{2}}t^{\alpha+\frac{m}{2}}\int_{t_{0}}^{t}(t-\tau)^{-\frac{3}{4}-\frac{m}{2}}\tau^{-\alpha\left(\frac{6-\beta}{2}\right)}\tau^{\left(-\alpha-\frac{1}{2}\right)\left(\frac{3(\beta-2)}{2}\right)}d\tau$
$\displaystyle=\>C(\mu,\chi,m,\eta)\>(\lambda_{0}(\alpha)+\epsilon)^{\frac{6-\beta}{2}}\;\underbrace{t^{\alpha+\frac{m}{2}}\int_{t_{0}}^{t}\tau^{\frac{6-\beta(4\alpha+3)}{4}}(t-\tau)^{-\frac{3}{4}-\frac{m}{2}}d\tau}_{(*)}.$

Note that

$\displaystyle(*)=\left\\{\begin{matrix}C_{1}(\alpha,\beta,m)\>t^{\frac{4\alpha+7-\beta(4\alpha+3)}{4}}&;\;\;\;\;\;\forall\;\;\frac{4\alpha+7}{4\alpha+3}<\beta<\frac{10}{4\alpha+3}\\\
C_{2}(m)\>t^{\alpha-\frac{3}{4}}\ln\left(\frac{t}{2}\right)\>+\>C_{2}(\alpha,m)\>t^{\frac{4\alpha-3}{4}}&;\;\;\beta=\frac{10}{4\alpha+3}\\\
C_{3}(\alpha,\beta,m)\>\left[t^{\alpha-\frac{3}{4}}\>+\>t^{\frac{4\alpha+7-\beta(4\alpha+3)}{4}}\right]&;\;\;\forall\;\;\beta>\frac{10}{4\alpha+3}.\end{matrix}\right.$

Therefore, we have

(4.3)
$\limsup_{t\to+\infty}t^{\alpha+\frac{m}{2}}\>\int_{t_{0}}^{t}\left\|\nabla^{m}e^{(\mu+\chi)\Delta(t-\tau)}(\eta\left|\mbox{\boldmath$u$}\right|^{\beta-1}\mbox{\boldmath$u$})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau\>=\>0$

for all $\beta\>>\>\frac{4\alpha+7}{4\alpha+3}$ and for all
$0\>\leq\>\alpha\><\>\frac{3}{4}$ .

Therefore combining (4.1) , (4.2) and (4.3), we finally have

$\displaystyle\limsup_{t\to+\infty}t^{\alpha+\frac{m}{2}}\left\|\nabla^{m}\mbox{\boldmath$u$}\right\|_{L^{2}(\mathbb{R}^{3})}\leq
C_{\alpha,m}\lambda_{0}(\alpha).$

Thus we complete the proof of (1.6a). In order to prove assertion (1.6b) of
Theorem 1.3. As before, appliying the Duhamel’s principle, we get

$\displaystyle t^{\alpha+\frac{m+1}{2}}\left\|{\bf
w}(\cdot,t)\right\|_{L^{2}(\mathbb{R}^{3})}$
$\displaystyle\leq\>t^{\alpha+\frac{m+1}{2}}e^{-4\chi(t-t_{0})}\left\|D^{m}e^{\gamma\Delta(t-t_{0})}{\bf
w}(\cdot,t_{0})\right\|_{L^{2}(\mathbb{R}^{3})}$
$\displaystyle\>\>\>+\>t^{\alpha+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|e^{\gamma\Delta(t-\tau)}\nabla^{m}(\mbox{\boldmath$u$}\cdot\nabla{\bf
w})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\>\>\>+\>\kappa
t^{\alpha+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|e^{\gamma\Delta(t-\tau)}\nabla^{m+2}{\bf
w}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\>\>\>+\>2\chi
t^{\alpha+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|e^{\gamma\Delta(t-\tau)}\nabla^{m+1}\mbox{\boldmath$u$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle=\>I\>+\>II\>+\>III\>+\>IV.$

First, it is clear that
$t^{\alpha+\frac{m+1}{2}}e^{-4\chi(t-t_{0})}\left\|D^{m}e^{\gamma\Delta(t-t_{0})}{\bf
w}(\cdot,t_{0})\right\|_{L^{2}(\mathbb{R}^{3})}\>\rightarrow\>0$ as
$t\rightarrow+\infty$.

For II,

$\displaystyle
t^{\alpha+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|e^{\gamma\Delta(t-\tau)}D^{m}(\mbox{\boldmath$u$}\cdot\nabla{\bf
w})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq\>K(m)\gamma^{-\frac{3}{4}-\frac{m}{2}}\>t^{\alpha+\frac{m}{2}+\frac{1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|\mbox{\boldmath$u$}\cdot\nabla{\bf
w}\right\|_{L^{1}(\mathbb{R}^{3})}(t-\tau)^{-\frac{3}{4}-\frac{m}{2}}d\tau$
$\displaystyle\leq\>K(m)\gamma^{-\frac{3}{4}-\frac{m}{2}}\>t^{\alpha+\frac{m}{2}+\frac{1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|\mbox{\boldmath$u$}\right\|_{L^{2}(\mathbb{R}^{3})}\left\|\nabla{\bf
w}\right\|_{L^{2}(\mathbb{R}^{3})}(t-\tau)^{-\frac{3}{4}-\frac{m}{2}}d\tau$
$\displaystyle\leq\>K(m)\gamma^{-\frac{3}{4}-\frac{m}{2}}\>(\lambda_{0}(\alpha)+\epsilon)\>\underbrace{t^{\alpha+\frac{m}{2}+\frac{1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\tau^{-2\alpha-\frac{1}{2}}(t-\tau)^{-\frac{3}{4}-\frac{m}{2}}d\tau}_{(*)}.$

Note that

$\displaystyle(*)=\left\\{\begin{matrix}C_{1}(\alpha,m)\left(t^{\frac{1}{4}-\alpha}e^{-2\chi
t}\>+\>(4\chi)^{-1}t^{-\alpha-\frac{3}{4}}\right)&\;;\forall\;0\leq\alpha<\frac{1}{4}\\\
C_{2}(m)\>\left[e^{-2\chi
t}\ln\left(\frac{t}{2}\right)\>+\>(4\chi)^{-1}t^{-1}\right]&\alpha=\frac{1}{4}\\\
C_{3}(\alpha,m)\left(t^{\alpha-\frac{1}{4}}e^{-2\chi
t}\>+\>(4\chi)^{-1}t^{-\alpha-\frac{3}{4}}\right)&\;;\forall\;\alpha>\frac{1}{4}\\\
\end{matrix}\right.$

Now, letting $t\rightarrow+\infty$, we have

(4.4)
$\limsup_{t\to+\infty}t^{\alpha+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|e^{\gamma\Delta(t-\tau)}D^{m}(\mbox{\boldmath$u$}\cdot\nabla{\bf
w})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau\>=\>0.$

For III,

$\displaystyle\kappa\>t^{\alpha+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|e^{\gamma\Delta(t-\tau)}D^{m+2}{\bf
w}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq\>K(m)\>\kappa\>t^{\alpha+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|D^{m+2}{\bf
w}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq\>K(m)\>(\lambda_{0}(\alpha)+\epsilon)\>\kappa\>t^{\alpha+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\tau^{-\alpha-\frac{m+2}{2}}d\tau$
$\displaystyle\leq\>K(m)\>(\lambda_{0}(\alpha)+\epsilon)\>\kappa\>t^{\alpha+\frac{m+1}{2}}\left[\left(\frac{1}{\alpha+\frac{m}{2}}\right)\>t_{0}^{-\alpha-\frac{m}{2}}\>e^{-2\chi
t}\>+\>(4\chi)^{-1}\left(\frac{t}{2}\right)^{-\alpha-\frac{m+2}{2}}\right]$
$\displaystyle\leq\>K(\alpha,m)\>(\lambda_{0}(\alpha)+\epsilon)\>\kappa\>\left[t^{\alpha+\frac{m+1}{2}}\>e^{-2\chi
t}\>+\>t^{-\frac{1}{2}}(4\chi)^{-1}\right]$

so that we have

(4.5)
$\limsup_{t\to+\infty}\kappa\>t^{\alpha+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|e^{\gamma\Delta(t-\tau)}D^{m+2}{\bf
w}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau=0.$

For IV, we get

$\displaystyle 2\chi
t^{\alpha+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|e^{\gamma\Delta(t-\tau)}D^{m+1}\mbox{\boldmath$u$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq\>2\chi
K(m)t^{\alpha+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|D^{m+1}\mbox{\boldmath$u$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq\>2\chi
K(m)\>(\lambda_{0}(\alpha)+\epsilon)\>t^{\alpha+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\tau^{-\alpha-\frac{m+1}{2}}d\tau$
$\displaystyle\leq\>2\chi
K(m)\>(\lambda_{0}(\alpha)+\epsilon)\>t^{\alpha+\frac{m+1}{2}}\left[C(\alpha,m)t_{0}^{-\alpha-\frac{m}{2}+\frac{1}{2}}e^{-2\chi
t}\>+\>(4\chi)^{-1}\left(\frac{t}{2}\right)^{-\alpha-\frac{m+1}{2}}\right]$
$\displaystyle\leq\>2\chi
C(\alpha,m)\>(\lambda_{0}(\alpha)+\epsilon)\>\left[e^{-2\chi
t}\>t^{\alpha+\frac{m+1}{2}}\>+\>(4\chi)^{-1}\right].$

Letting $\epsilon\rightarrow 0$ and $t\rightarrow+\infty$ in this order, we
have

(4.6)
$\begin{split}&\limsup_{t\rightarrow+\infty}t^{\alpha+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-4\chi(t-\tau)}\left\|e^{\gamma\Delta(t-\tau)}D^{m+1}\mbox{\boldmath$u$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau\\\
&\leq\>C(\alpha,m)\lambda_{0}(\alpha).\end{split}$

Therefore, combining (4.4) , (4.5) and (4.6) , we have

$\limsup_{t\rightarrow+\infty}t^{\alpha+\frac{m+1}{2}}\left\|D^{m}{\bf
w}\right\|_{L^{2}(\mathbb{R}^{3})}\>\leq\>C(\alpha,m)\lambda_{0}(\alpha),$

which concludes the proof of (1.6b).

##  Appendix A Proof of Claim 2.7

In this section we shall prove Claim 2.7 which has an independent interest. We
start with the following lemma for the associated linear system (2.9).

######  Lemma A.1.

Let $\mbox{\boldmath$z$}(\cdot,t)$ be a weak solution for system (1.1) for all
$t>0$. If $32\chi(\mu+\chi+\gamma)>1$, then, given any pair of initial times
$\tilde{t}_{0}\geq t_{0}\geq 0$, one has

(A.1)
$\begin{split}\|D^{m}\bar{\mbox{\boldmath$u$}}(\cdot,t;{t_{0}})-D^{m}\bar{\mbox{\boldmath$u$}}(\cdot,t;\tilde{t}_{0})\|_{L^{2}(\mathbb{R}^{3})}\leq
C\,(t-\tilde{t}_{0})^{-\frac{5}{4}-\frac{m}{2}},\end{split}$ (A.2)
$\begin{split}\|D^{m}\bar{{\bf w}}(\cdot,t;{t_{0}})-D^{m}\bar{{\bf
w}}(\cdot,t;\tilde{t}_{0})\|_{L^{2}(\mathbb{R}^{3})}\leq
C\,e^{-2\,\chi\,(t-\tilde{t}_{0})}\,(t-\tilde{t}_{0})^{-\frac{5}{4}-\frac{m}{2}},\end{split}$

where $\bar{\mbox{\boldmath$u$}},\bar{{\bf w}}$ denotes the solution to the
linear system (2.9) with initial data
$\bar{\mbox{\boldmath$u$}}(\cdot,t_{0}):=\mbox{\boldmath$u$}(\cdot,t_{0})$ and
$\bar{{\bf w}}(\cdot,t_{0}):={\bf w}(\cdot,t_{0})$. To be more precise, we
denote such solution by
$\bar{\mbox{\boldmath$z$}}(\cdot,t;t_{0})=(\bar{\mbox{\boldmath$u$}}(\cdot,t;t_{0}),\bar{{\bf
w}}(\cdot,t;t_{0}))$

###### Proof.

See e.g. Theorem 2.3 in [5]. ∎

Due to lemma A.1 it is enough to prove the claim for times $t_{0}$ larger than
the regularity time $t_{*}$. In fact, if the inequality (2.12) does hold for
$t_{0}>t_{*}$, then given any $\hat{t}\in[0,t_{*}]$, Lemma A.1 gives

$\begin{split}\limsup_{t\rightarrow\infty}t^{\alpha+\hat{\beta}}\|{\mathcal{E}}_{\mbox{\boldmath$z$}}(\cdot,t;\hat{t})\|_{L^{2}}\leq\limsup_{t\rightarrow\infty}t^{\alpha+\hat{\beta}}\|{\mathcal{E}}_{\mbox{\boldmath$z$}}(\cdot,t;t_{0})\|_{L^{2}}\\\
+\limsup_{t\rightarrow\infty}t^{\alpha+\hat{\beta}}\underbrace{\|{\mathcal{E}}_{\mbox{\boldmath$z$}}(\cdot,t;t_{0})-{\mathcal{E}}_{\mbox{\boldmath$z$}}(\cdot,t;\hat{t})\|_{L^{2}}}_{=O(t^{-5/4})}\\\
=\limsup_{t\rightarrow\infty}t^{\alpha+\hat{\beta}}\|{\mathcal{E}}_{\mbox{\boldmath$z$}}(\cdot,t;t_{0})\|_{L^{2}}\leq
C_{\,\beta,\,\hat{\beta},\,\lambda_{0}(\alpha)\,},\end{split}$

where, for convenience, we define
${\mathcal{E}}_{\mbox{\boldmath$z$}}(\cdot,t;t_{0}):=\mbox{\boldmath$z$}(\cdot,t)-\bar{\mbox{\boldmath$z$}}(\cdot,t;t_{0})$.
Now, by taking $t_{0}\geq t_{\ast}$ and using the integral representations for
$\mbox{\boldmath$z$}(\cdot,t)$ and $\bar{\mbox{\boldmath$z$}}(\cdot,t;t_{0})$,
we get

$\displaystyle{\mathcal{E}}_{\mbox{\boldmath$z$}}(\cdot,t;t_{0})$
$\displaystyle\>=\>\mbox{\boldmath$z$}(\cdot,t)-\bar{\mbox{\boldmath$z$}}(\cdot,t;t_{0})$
$\displaystyle\>=\>\mbox{\boldmath$z$}(\cdot,t)-e^{\mathbb{A}(t-t_{0})}\mbox{\boldmath$z$}(\cdot,t_{0})$
$\displaystyle\>=\>\int_{t_{0}}^{t}e^{\mathbb{A}(t-\tau)}\begin{bmatrix}\mathbb{P}_{h}\left[-\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$}-\eta\left|\mbox{\boldmath$u$}\right|^{\beta-1}\mbox{\boldmath$u$}\right]\\\
-\mbox{\boldmath$u$}\cdot\nabla{\bf w}\end{bmatrix}d\tau$

and, using Lemma 2.5, this yields

(A.3)
$\begin{split}t^{\alpha+\hat{\beta}}&\left\|{\mathcal{E}}_{\mbox{\boldmath$z$}}(\cdot,t;t_{0})\right\|_{L^{2}(\mathbb{R}^{3})}\\\
&\leq\>t^{\alpha+\hat{\beta}}\,\int_{t_{0}}^{t}\left\|\,e^{c\Delta(t-\tau)}\,(\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$})\right\|_{L^{2}(\mathbb{R}^{3})}d\tau\\\
&\;\;\;+\;t^{\alpha+\hat{\beta}}\,\int_{t_{0}}^{t}\left\|\,e^{c\Delta(t-\tau)}\,(\mbox{\boldmath$u$}\cdot\nabla{\bf
w})\right\|_{L^{2}(\mathbb{R}^{3})}d\tau\\\
&\;\;\;+\;t^{\alpha+\hat{\beta}}\,\eta\int_{t_{0}}^{t}\left\|\,e^{c\Delta(t-\tau)}\,(\left|\mbox{\boldmath$u$}\right|^{\beta-1}\mbox{\boldmath$u$})(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau.\\\
\end{split}$

We just need to examine each of the integrals above by splitting the interval
$(t_{0},t)$ into $(t_{0},\mu(t))\cup(\mu(t),t)$, where
$\left.\mu(t):=\left(t+t_{0}\right)/2\right)$. So, by applying the same idea
used to obtain (4.2) and (4.3), we get

$\displaystyle
t^{\alpha+\hat{\beta}}\int_{\frac{t}{2}}^{t}\left\|e^{(\mu+\chi)\Delta(t-\tau)}(\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$}(\cdot,\tau))\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq\>C\,(\mu+\chi)^{-\frac{5}{4}}\>t^{\alpha+\hat{\beta}}\int_{\frac{t}{2}}^{t}(t-\tau)^{-\frac{5}{4}}\left\|\mbox{\boldmath$u$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}d\tau$
$\displaystyle\leq\>C\,(\mu+\chi)^{-\frac{5}{4}}(\lambda_{0}(\alpha)+\epsilon)^{2}\>t^{\alpha+\hat{\beta}}\,\int_{\frac{t}{2}}^{t}(t-\tau)^{-\frac{5}{4}}\tau^{-2\alpha}d\tau$
$\displaystyle\leq\>C\,(\mu+\chi)^{-\frac{5}{4}}(\lambda_{0}(\alpha)+\epsilon)^{2}\,2^{2\alpha-1},$

where we started by examining the integral on the interval
$\left[\frac{t}{2},t\right]$. The first half needs to be studied in three
cases depending on $\alpha$.

Case 1: $\alpha\in\left(0,\frac{1}{2}\right)$.

$\displaystyle
t^{\alpha+\hat{\beta}}\int_{t_{0}}^{\frac{t}{2}}\left\|e^{(\mu+\chi)\Delta(t-\tau)}(\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$}(\cdot,\tau))\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq\>C\,(\mu+\chi)^{-\frac{5}{4}}\>t^{\alpha+\hat{\beta}}\int_{t_{0}}^{\frac{t}{2}}(t-\tau)^{-\frac{5}{4}}\left\|\mbox{\boldmath$u$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}d\tau$
$\displaystyle\leq\>C(\mu,\chi)(\lambda_{0}(\alpha)+\epsilon)^{2}\>t^{\alpha+\hat{\beta}}\int_{t_{0}}^{\frac{t}{2}}(t-\tau)^{-\frac{5}{4}}\tau^{-2\alpha}d\tau$
$\displaystyle\leq\,C\;(\lambda_{0}(\alpha)+\epsilon)^{2}\>2^{2\alpha+\frac{1}{4}}\>t^{-\alpha+\hat{\beta}-\frac{1}{4}}=\,C\;(\lambda_{0}(\alpha)+\epsilon)^{2},$

where we take $\hat{\beta}\,=\,\alpha+\frac{1}{4}$ in the last step. The
second case is similar.

Case 2: $\alpha\in\left(\frac{1}{2},1\right)$.

$\displaystyle
t^{\alpha+\hat{\beta}}\int_{t_{0}}^{\frac{t}{2}}\left\|e^{(\mu+\chi)\Delta(t-\tau)}(\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$}(\cdot,\tau))\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq\>C\,(\mu+\chi)^{-\frac{5}{4}}\>t^{\alpha+\hat{\beta}}\int_{t_{0}}^{\frac{t}{2}}(t-\tau)^{-\frac{5}{4}}\left\|\mbox{\boldmath$u$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}^{2}d\tau$
$\displaystyle\leq\>C(\mu,\chi)\>t^{\alpha+\hat{\beta}}\int_{t_{0}}^{\frac{t}{2}}(t-\tau)^{-\frac{5}{4}}\left\|\mbox{\boldmath$u$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}\left\|\mbox{\boldmath$z$}(\cdot,t_{0})\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq\>C\;(\lambda_{0}(\alpha)+\epsilon)\>t^{\alpha+\hat{\beta}}\left(\frac{t}{2}\right)^{-\frac{5}{4}}\int_{t_{0}}^{\frac{t}{2}}\tau^{-\alpha}d\tau\leq
C\;(\lambda_{0}(\alpha)+\epsilon)\>\;t^{\hat{\beta}-\frac{1}{4}}\leq
C\;(\lambda_{0}(\alpha)+\epsilon),$

where we take $\hat{\beta}\,=\,\frac{1}{4}$ in this case.

Case 3: $\alpha\in\left(1,\frac{5}{4}\right)$.

$\displaystyle
t^{\alpha+\hat{\beta}}\int_{t_{0}}^{\frac{t}{2}}\left\|e^{(\mu+\chi)\Delta(t-\tau)}(\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$}(\cdot,\tau))\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq\>C(\mu,\chi)\>\left\|\mbox{\boldmath$z$}(\cdot,t_{0})\right\|_{L^{2}(\mathbb{R}^{3})}t^{\alpha+\hat{\beta}}\int_{t_{0}}^{\frac{t}{2}}(t-\tau)^{-\frac{5}{4}}\left\|\mbox{\boldmath$u$}(\cdot,\tau)\right\|_{L^{2}(\mathbb{R}^{3})}d\tau$
$\displaystyle\leq
C\,(\lambda_{0}(\alpha)+\epsilon)t^{\alpha+\hat{\beta}}\left({t}\right)^{-\frac{5}{4}}\underbrace{\int_{t_{0}}^{\frac{t}{2}}\tau^{-\alpha}d\tau}_{<\>\infty}\leq\>C\,(\lambda_{0}(\alpha)+\epsilon)\,t^{\alpha+\hat{\beta}-\frac{5}{4}}=\>C\,(\lambda_{0}(\alpha)+\epsilon)$

and here we finally take $\hat{\beta}\,=\,\frac{5}{4}-\alpha$. The analysis of
the remaining terms on (A.3) follows in a similar fashion.

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