# High order asymptotic inequalities for some dissipative systems

P. Braz e Silva  Departamento de Matemática. Universidade Federal de
Pernambuco, CEP 50740-560, Recife - PE. Brazil<EMAIL_ADDRESS>,  R. H.
Guterres  Departamento de Matemática. Universidade Federal de Pernambuco, CEP
50740-560, Recife - PE. Brazil<EMAIL_ADDRESS>,  C. F. Perusato
Departamento de Matemática. Universidade Federal de Pernambuco, CEP 50740-560,
Recife - PE. Brazil<EMAIL_ADDRESS>and  P. R. Zingano  Departamento
de Matemática Pura e Aplicada, Instituto de Matemática e Estatística.
Universidade Federal do Rio Grande do Sul, CEP 91509-900, Porto Alegre - RS,
Brazil<EMAIL_ADDRESS>
###### Abstract.

We obtain some important fundamental inequalities concerning the long time
behavior of high order derivatives for solutions of some dissipative systems
in terms of their $L^{2}$ algebraic decay. Some of these inequalities have not
been observed in the literature even for the fundamental Navier-Stokes
equations.To illustrate this new approach, we derive bounds for the asymmetric
incompressible fluids equations, where an improved inequalities established
for the micro-rotational field. Finally, we show how this technique can be
applied for other dissipative systems.

###### Key words and phrases:

Sharp inequalities for large time, asymptotic behaviour, decay rates,
micropolar equations, high-order norms

###### 2000 Mathematics Subject Classification:

35B40 (primary), 35Q35 (secondary)

P. Braz e Silva was partially supported by CAPES-PRINT, Brazil,
$\\#$88881.311964/2018-01, CAPES-MATHAMSUD $\\#$88881.520205/2020-0, and CNPq,
Brazil, $\\#$308758/2018-8 and $\\#$432387/2018-8

R. H Guterres acknowledges funding from Bolsa FACEPE Grant BFP-0067-1.01/19.

C. Perusato was partially supported by CAPES–PRINT - 88881.311964/2018–01 and
Propesq-UFPE - 08-2019 (Qualis A).

##  1\. Introduction

Asymptotic inequalities for dissipative systems are a topic of high interest,
specially in the context of fluid mechanics models, such as the Navier-Stokes
equations and related systems, like the asymmetric incompressible fluids
equations (micropolar Navier-Stokes), among others. Here, we derive some new
inequalities for high order derivatives of solutions for such systems. The
method used here is, by itself, interesting, since it is a direct method based
on energy inequalities. Even tough it is direct, it depends on very subtle and
technical arguments, which may eventually be used in different contexts. To
describe it, we show how to obtain bounds for solutions of the so called
asymmetric, also known as micropolar, fluids system

$\displaystyle\mbox{\boldmath$u$}_{t}+\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$}+\nabla\>\\!{\sf{p}}$
$\displaystyle=$
$\displaystyle(\>\\!\mu+\chi\>\\!)\,\Delta\mbox{\boldmath$u$}\,+\,\chi\,\nabla\times{\bf
w},$ $\displaystyle{\bf w}_{t}+\mbox{\boldmath$u$}\cdot\nabla\mbox{\bf w}$
$\displaystyle=$ $\displaystyle\gamma\;\\!\,\Delta\mbox{\bf
w}\,+\,\kappa\,\nabla\>\\!(\>\\!\nabla\cdot\;{\bf
w}\>\\!)\,+\,\chi\,\nabla\times\,\mbox{\boldmath$u$}\,-\,2\,\chi\,{\bf w},$
$\displaystyle\nabla\cdot\mbox{\boldmath$u$}(\cdot,t)$ $\displaystyle=$
$\displaystyle 0,$

with suitable initial data. We note that the results obtained here for the
linear velocity $\mathbf{u}$ in system above above also hold for solutions of
the classical Navier-Stokes equations. In particular, as far as we know, the
results on the large time dynamics given in Theorem 1.3 below are new even for
solutions of the Navier-Stokes equations.

In 1984, Kato [24] and Masuda [32] showed that solutions $\mathbf{u}$ of the
Navier-Stokes equations satisfy
$\|\mbox{\boldmath$u$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}\to 0$ as
$t\to\infty$, finally solving an open problem left by Leray [26]. Since then,
several works have made considerable progress in this direction (see e.g [35,
40, 39, 41, 42]). In particular, the well-known Fourier Splitting method,
developed by M. E. Schonbek [38, 39, 42], is a powerful tool to prove that the
$L^{2}$ norm of solutions for viscous conservation laws and for Navier-Stokes
equations decay with algebraic rate depending on the initial data. Recently,
Hagstrom et al. [21] derived an asymptotic inequality for arbitrary order
spatial derivatives of Leray solutions for the incompressible Navier-Stokes
equations in $\mathbb{R}^{n}$ with $n\leq 4$. This technique may also be
extended to dissipative systems with linear damping, such as the asymmetric
fluids system, as we show in details here (see Theorem 1.1). The bounds for
the linear velocity $\mathbf{u}$ are the same as for solutions of classical
Navier-Stokes equations obtained in [21]. The bounds for the rotational
velocity $\mathbf{w}$ are stronger, as can be seen in Theorems 1.1 and 1.3.

Firstly introduced by Eringen in [15], the asymmetric fluids model is a
generalization of the classical Navier-Stokes equations that takes into
account the microstructure of the fluid particles. It describes flows of
fluids whose particles undergo translations and rotations as well. Asymmetric
fluids flows are governed by the equations

(1.1a)
$\mbox{\boldmath$u$}_{t}\;\\!+\,\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$}\,+\;\\!\nabla\>\\!{\sf{p}}\;=\;(\>\\!\mu+\chi\>\\!)\,\Delta\mbox{\boldmath$u$}\,+\,\chi\,\nabla\times{\bf
w},$ (1.1b) ${\bf w}_{t}\;\\!+\,\mbox{\boldmath$u$}\cdot\nabla\mbox{\bf
w}\;=\;\gamma\;\\!\,\Delta\mbox{\bf
w}\,+\,\kappa\,\nabla\>\\!(\>\\!\nabla\cdot\;{\bf
w}\>\\!)\,+\,\chi\,\nabla\times\,\mbox{\boldmath$u$}\,-\,2\,\chi\,{\bf w},$
(1.1c) $\nabla\cdot\mbox{\boldmath$u$}(\cdot,t)=0,$

with initial data $\mbox{\boldmath$z$}_{0}:=(\>\\!\mbox{\boldmath$u$}_{0},{\bf
w}_{0})\in{\bf L}^{2}_{\sigma}(\mathbb{R}^{n})\\!\times\\!{\bf
L}^{2}(\mathbb{R}^{n})$ and

$\|\mbox{\boldmath$z$}(\cdot,t)-\mbox{\boldmath$z$}_{0}\|_{L^{2}(\mathbb{R}^{n})}\to
0\mbox{, as }t\to 0,$

where $\mbox{\boldmath$z$}:=(\mbox{\boldmath$u$},\mbox{$\bf w$})$. In system
(1.1), the constants $\mu,\gamma,\kappa>0$ are kinematic and spin viscosities
and $\chi>0$ is the vortex viscosity. The functions
$\mbox{\boldmath$u$}=\mbox{\boldmath$u$}(\mbox{\boldmath$x$},t)$,
$\mbox{\boldmath$w$}=\mbox{\boldmath$w$}(\mbox{\boldmath$x$},t)$, and
$\sc{p}=\sc{p}(\mbox{\boldmath$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. For a discussion of the physical meaning and the
derivation of system (1.1), see [13, 15, 31]. Observe that taking $\mbox{$\bf
w$}=0$ in system (1.1), one obtains the classical incompressible Navier-Stokes
equations.

Consider the linear system obtained simply by dropping out the nonlinear terms
from system (1.1)

(1.2a)
$\mbox{\boldmath$\bar{u}$}_{t}\;\\!+\,\nabla\>\\!{\sf{\bar{p}}}\;=\;(\>\\!\mu+\chi\>\\!)\,\Delta\mbox{\boldmath$\bar{u}$}\,+\,\,\chi\,\nabla\times{\bf\bar{w}},$
(1.2b) ${\bf\bar{w}}_{t}\;\\!+\,\gamma\;\\!\,\Delta{\bf{\bar{\mbox{$\bf
w$}}}}\,+\,\kappa\,\nabla\>\\!(\>\\!\nabla\cdot\;{\bf\bar{w}}\>\\!)\,+\,\chi\,\nabla\times\,\mbox{\boldmath$\bar{u}$}\,-\,2\,\chi\,{\bf\bar{w}},$
(1.2c) $\nabla\cdot\mbox{\boldmath$\bar{u}$}(\cdot,t)=0.$

Our main result compares the evolution of solutions
$\mbox{\boldmath$z$}=(\mathbf{u},\mathbf{w})$ for the nonlinear initial value
problem (1.1) with the evolution of solutions
$\bar{\mbox{\boldmath$z$}}=(\mathbf{\bar{u}},\mathbf{\bar{w}})$ for the linear
problem (1.2) with the same initial conditions.

For convenience, define $\nu:=\min\\{\mu,\gamma\\}$ and
$\displaystyle\lambda_{0}(\alpha):=\limsup_{t\to\infty}\,t^{\alpha}\,\|\mbox{\boldmath$u$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}$,
for $\alpha\geq 0$. We assume that

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

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

######  Remark 1.1.

In [3, 1, 34], the authors define the so called decay indicator. With this
tool, a function $\mbox{\boldmath$v$}_{0}\in L^{2}$ is classified accurately
with respect to the algebraic decay rate of the flow initiated from
$\mbox{\boldmath$v$}_{0}$. From there, the authors provide upper bounds for
solutions of the system in terms of the so called decay character. All of
these facts ensure that the assumption (1.3) above holds for some values of
$\alpha\geq 0$.

We state our results.

######  Theorem 1.1.

Let $\mbox{\boldmath$z$}_{0}\in{\bf L}^{2}_{\sigma}(\mathbb{R}^{3})\times{\bf
L}^{2}(\mathbb{R}^{3})$. If
$\mbox{\boldmath$z$}(\cdot,t)=(\mathbf{u}(\cdot,t),\mathbf{w}(\cdot,t))$ is a
weak solution of system (1.1) with initial data $\mbox{\boldmath$z$}_{0}$,
then

(1.4a)
$\limsup_{t\to\infty}\,t^{\alpha+\frac{m}{2}}\|D^{m}\mbox{\boldmath$z$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}\leq
K_{\alpha,m}\;\;\nu^{-m/2}\;\;\lambda_{0}(\alpha)$ and, for the micro-
rotational field $\mathbf{w}$, (1.4b)
$\limsup_{t\to\infty}t^{\alpha+\frac{m+1}{2}}\|D^{m}\mbox{$\bf
w$}(\cdot,t)\|_{{L^{2}(\mathbb{R}^{3})}}\leq\frac{K_{\alpha,m+1}}{2}\;\;\nu^{-\frac{m+1}{2}}\;\;\lambda_{0}(\alpha)$

for every $\alpha\geq 0$ and $m\geq 1$, where

(1.5)
$K_{\alpha,m}=\min_{\delta>0}\Bigg{\\{}\delta^{-1/2}\prod_{j=0}^{m}(\alpha+\frac{j}{2}+\delta)^{1/2}\Bigg{\\}}.$

By the Calderón-Zygmund theory, one directly obtains the following high order
inequality for the pressure:

######  Corollary 1.2.

The pressure $p(\cdot,t)$ satisfies

$\limsup_{t\to\infty}\,t^{2\alpha+\frac{m}{2}+\frac{3}{4}}\|D^{m}p(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}\leq
C_{m}\;\nu^{-(\frac{m}{2}+\frac{3}{4})}\,\lambda_{0}(\alpha)\,^{2},$

where $C_{m}$ is a positive constant depending only on $m$.

Our main result gives asymptotic inequalities for high order derivatives of
arbitrary weak solutions for systems (1.1) and (1.2), so describing their long
time dynamics. To this end, we choose $t_{0}>0$, sufficiently large to assure
that $\bar{\mbox{\boldmath$z$}}$ is smooth for $t\geq t_{0}$. Existence of
such $t_{0}$ is assured by Theorem 2.1. Then, consider the solution
$\bar{\mbox{\boldmath$z$}}$ of the linear system (1.2) with initial data
$\bar{\mbox{\boldmath$z$}}(\cdot,t_{0}):=\mbox{\boldmath$z$}(\cdot,t_{0})$. We
denote such solution by
$\bar{\mbox{\boldmath$z$}}(\cdot,t;t_{0})=(\bar{\mbox{\boldmath$u$}}(\cdot,t;t_{0}),\bar{\mbox{$\bf
w$}}(\cdot,t;t_{0}))$ and, to keep the notation simple, write
${\mathcal{E}}_{\mbox{\boldmath$z$}}(\cdot,t;t_{0}):=\mbox{\boldmath$z$}(\cdot,t)-\bar{\mbox{\boldmath$z$}}(\cdot,t;t_{0})$
and ${\mathcal{E}}_{\mbox{$\bf w$}}(\cdot,t;t_{0}):=\mbox{$\bf
w$}(\cdot,t)-\bar{\mbox{$\bf w$}}(\cdot,t;t_{0})$. Our main result is the
following:

######  Theorem 1.3.

Let $\mbox{\boldmath$z$}_{0}\in{\bf L}^{2}_{\sigma}(\mathbb{R}^{3})\times{\bf
L}^{2}(\mathbb{R}^{3})$ and $\mbox{\boldmath$z$}(\cdot,t)$ be any weak
solution to (1.1). If $0\leq\alpha\leq\frac{5}{4}$, then

(1.6a)
$\limsup_{t\to\infty}t^{\alpha+\beta+\frac{m}{2}}\|D^{m}{\mathcal{E}}_{\mbox{\boldmath$z$}}(\cdot,t;t_{0})\|_{L^{2}(\mathbb{R}^{3})}\leq
C(\nu,\chi,\alpha,m),\quad\forall t_{0}\geq 0,$ and, for the micro-rotational
field, (1.6b)
$\limsup_{t\to\infty}t^{\alpha+\beta+\frac{m+1}{2}}\|D^{m}{\mathcal{E}}_{\mbox{$\bf
w$}}(\cdot,t;t_{0})\|_{L^{2}(\mathbb{R}^{3})}\leq\frac{1}{2}C(\nu,\chi,\alpha,m+1),\quad\forall
t_{0}\geq 0,$ where $\beta=\begin{cases}\alpha+1/4,&0\leq\alpha<1/2,\\\
1/4,&1/2\leq\alpha<1,\\\ 5/4-\alpha,&1\leq\alpha<5/4,\end{cases}$
$C(\nu,\chi,\alpha,m)=\begin{cases}(\nu^{-(\frac{m}{2}+\frac{5}{4})}\,\tilde{K}_{\alpha,m}+c^{-(\frac{m}{2}+\frac{5}{4})}\,2^{\alpha+\beta+m/2})\,\lambda_{0}(\alpha)^{2},&0\leq\alpha<1/2,\\\
\nu^{-(\frac{m}{2}+\frac{5}{4})}\,\tilde{K}_{\alpha,m}\,\lambda_{0}(\alpha)^{2}+c^{-(\frac{m}{2}+\frac{5}{4})}\,\|\mbox{\boldmath$z$}_{0}\|2^{\alpha+\beta+m/2}\,\lambda_{0}(\alpha),&1/2\leq\alpha<1,\\\
\nu^{-(\frac{m}{2}+\frac{5}{4})}\,\tilde{K}_{\alpha,m}\,\lambda_{0}(\alpha)^{2}+c^{-(\frac{m}{2}+\frac{5}{4})}\,\|\mbox{\boldmath$z$}_{0}\|2^{\alpha+\beta+m/2},&1\leq\alpha<5/4,\end{cases}$
with
$\tilde{K}_{\alpha,m}=\sum_{l=0}^{m}K_{\alpha,l}^{1/4}K_{\alpha,l+1}^{3/4}K_{\alpha,m-l+1}^{1/4}K_{\alpha,m-l+2}^{3/4},$
$K_{\alpha,m}=\min_{\delta>0}\Bigg{\\{}\delta^{-1/2}\prod_{j=0}^{m}(\alpha+\frac{j}{2}+\delta)^{1/2}\Bigg{\\}}$
and $c$ is a positive constant given by Lemma 2 below.

######  Remark 1.2.

Theorem 1.3 actually holds for any initial time $t_{0}\geq 0$, due to Theorem
2.3 below. Moreover, the $\dot{H}^{m}$ decay of the related errors depend only
on $\limsup_{t\to\infty}\|\mbox{\boldmath$u$}(t)\|=:\lambda_{0}(\alpha)$. As
already mentioned, such inequality has not been observed in the literature
even for the Navier-Stokes case.

######  Remark 1.3.

It is interesting to note that $\lambda_{0}(\alpha)$ depends only on $u$. This
is due to the fact that $\mbox{$\bf w$}(\cdot,t)$ goes to zero faster than
$\mbox{\boldmath$u$}(\cdot,t)$ (see (1.7) and (1.8) below). We should also
recall that $\lambda_{0}(0)=0$ for any weak solution
$(\mbox{\boldmath$u$},\mbox{$\bf w$})$ (see [19] for instance) and that for
$(\mbox{\boldmath$u$}_{0},\mbox{$\bf w$}_{0})\in L^{1}(\mathbb{R}^{3})\cap
L^{2}_{\sigma}(\mathbb{R}^{3})\times L^{1}(\mathbb{R}^{3})\cap
L^{2}(\mathbb{R}^{3})$ one has $\lambda_{0}(3/4)<\infty$ (see [29]).

There are many results on the existence and uniqueness of solutions for
problems involving the system (1.1)$-$ see, e.g., [15, 31, 2, 12, 18, 44]. For
example, in [18] Galdi and Rionero showed existence and uniqueness of weak
solutions for the initial boundary-value problem for the micropolar system (in
this case, $\Omega\subset\mathbb{R}^{3}$ is a connected open set that replaces
the whole space $\mathbb{R}^{3}$ in (1.1) such that the solution vanishes on
$\partial\Omega\times[0,T]$). For the same problem, Lukaszewicz [30] proved
existence and uniqueness of strong solutions in 1989, and, in [31],
established global existence of weak solutions for arbitrary initial data
$(\mbox{\boldmath$u$}_{0},\mbox{$\bf w$}_{0})\in{{\bf L_{\sigma}^{2}\times
L^{2}}}$. In [37], Rojas-Medar proved local existence and uniqueness of strong
solutions. In [36], Ortega-Torres and Rojas-Medar, assuming small initial
data, proved global existence of a strong solution. In [2], Boldrini, Durán
and Rojas-Medar proved existence and uniqueness of strong solutions in
$L^{p}(\Omega)$, for $p>3$. In [5], Braz e Silva, Friz and Rojas-Medar showed
a exponential decay result for strong solutions in bounded domains on
$\mathbb{R}^{3}$. In [14], Cruz showed global existence and uniqueness of
strong solutions for sufficiently small initial conditions.

Here, we will extensively use the following properties of the linear and
micro-rotational velocities derived in Guterres et al. [19]:

(1.7) $\lim_{t\to\infty}t^{\frac{s}{2}}\|(\mbox{\boldmath$u$},\mbox{$\bf
w$})(\cdot,t)\|_{\dot{H}^{s}(\mathbb{R}^{3})}=0$

and

(1.8) $\lim_{t\to\infty}t^{\frac{s+1}{2}}\|\mbox{$\bf
w$}(\cdot,t)\|_{\dot{H}^{s}(\mathbb{R}^{3})}=0,$

for each $s\geq 0$.

It is worth to mention that the techniques in the proofs of Theorems 1.1 and
1.3 can be used to obtain similar results for other dissipative systems with
non linear terms of the type
$\nabla\cdot(\mbox{\boldmath$u$}\otimes\mbox{\boldmath$u$})$. We briefly
discuss how to carry on the argument to some other models in section 5.2.

##  2\. Auxiliary Results

In order to prove our theorems, we state some basic facts and results. Some of
them have not been observed before and might even be of independent interest
as, for example, Theorem 2.3.

###  2.1. Regularity time $t^{*}$ and monotonicity of high order derivatives

Here, we adapt for the asymmetric fluids equations some results on the
eventual smoothness of solutions for the Navier-Stokes equations recently
obtained in [6].

The classical regularity result for weak solutions due to Leray [26] is the
following: If $u$ is a weak solution of

$\displaystyle\mbox{\boldmath$u$}_{t}+\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$}+\nabla\>\\!{\sf{p}}$
$\displaystyle=$ $\displaystyle\mu\,\Delta\mbox{\boldmath$u$},$
$\displaystyle\nabla\cdot\mbox{\boldmath$u$}(\cdot,t)$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle\mbox{\boldmath$u$}(\cdot,0)$
$\displaystyle=$ $\displaystyle\mbox{\boldmath$u$}_{0}\in{\bf
L}^{2}_{\sigma}(\mathbb{R}^{3}),$

then there exists $t^{*}>0$ such that $\mbox{\boldmath$u$}(\cdot,t)\in
C^{\infty}(\mathbb{R}^{3}\times(t^{*},\infty))$ and

(2.1) $\mbox{\boldmath$u$}(\cdot,t)\in
C^{0}(\,(t^{*},\infty),H^{m}(\mathbb{R}^{n})\,),\,\,\,\forall m\geq 0.$

Moreover, the regularity time $t^{*}\geq 0$ satisfies

$t^{*}<K\,\nu^{-5}\,\|\mbox{\boldmath$z$}_{0}\|^{4}_{L^{2}(\mathbb{R}^{3})},\,\,\text{
with}\,\,\,K\leq\frac{1}{128\pi}<0.000791572.$

Using the epochs of regularity property described in [22, 26], one can adapt
the same argument from [6] to obtain the following result for the asymmetric
case.

######  Theorem 2.1.

If $\mbox{\boldmath$z$}(\cdot,t)$ is a weak solution for problem (1.1), then
there exists $t^{**}\geq 0$ such that

(2.2) $t^{*}\leq t^{**}\leq
0.000464504284\,\,\nu^{-5}\,\|\mbox{\boldmath$z$}_{0}\|^{4}_{L^{2}(\mathbb{R}^{3})}.$

This is a slight improvement of the estimate for the regularity time $t^{*}$,
which is relevant for numerical simulations.

The following Sobolev type inequality is very useful to our ends.

######  Lemma 1.

For all integers $m\geq 1$, $0\leq\ell\leq m-1$ and any $f\in
H^{m+1}(\mathbb{R}^{3})$, it holds

(2.3)
$\|D^{\ell}f\|_{L^{\infty}(\mathbb{R}^{3})}\|D^{m-\ell}f\|_{L^{2}(\mathbb{R}^{3})}\leq\|f\|^{1/2}_{L^{2}(\mathbb{R}^{3})}\|Df\|^{1/2}_{L^{2}(\mathbb{R}^{3})}\|D^{m+1}f\|_{L^{2}(\mathbb{R}^{3})}.$

###### Proof.

Through Fourier transformation, one has

(2.4)
$\|D^{\ell}f\|_{L^{2}(\mathbb{R}^{3})}\leq\|f\|^{\theta}_{L^{2}(\mathbb{R}^{3})}\|D^{k}f\|^{1-\theta}_{L^{2}(\mathbb{R}^{3})},\,\,\,\theta=\ell/k,$

which is actually valid for any dimension $n\in\mathbb{N}$. Moreover,

(2.5)
$\|f\|_{L^{\infty}(\mathbb{R}^{3})}\leq\|f\|^{1/4}_{L^{2}(\mathbb{R}^{3})}\|D^{2}f\|^{3/4}_{L^{2}(\mathbb{R}^{3})}.$

Combining the well-known inequalities (2.4) and (2.5), one directly obtains
the desired bound (2.3). ∎

One uses inequality (2.3) to easily adapt the proof of a result in [6] to the
case of asymmetric fluids.

######  Theorem 2.2.

Let $\mbox{\boldmath$z$}(\cdot,t)$ be a weak solution for problem (1.1). There
exists $t^{**}_{m}\geq t^{*}$ such that
$\|D^{m}\mbox{\boldmath$z$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}$ is
monotonically decreasing everywhere in the interval $(t^{**}_{m},\infty)$.
Morover, there exists a positive constant $K_{m}$, depending only on $m$, such
that

(2.6) $t^{**}_{m}\leq
K_{m}\,\,\nu^{-5}\,\|\mbox{\boldmath$z$}_{0}\|^{4}_{L^{2}(\mathbb{R}^{3})}.$

###  2.2. Linear operators leading to asymptotic behavior

We observe that the linear system (1.2) 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$}}(\cdot,t;t_{0})=e^{\mathbb{A}\>\\!t}\mbox{\boldmath$z$}(\cdot,t_{0})(\cdot)$,
where

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

and $\mathbb{L\mbox{$\bf w$}}:=(\mu+\chi)\Delta\mbox{$\bf
w$}+\kappa\nabla(\nabla\cdot\mbox{$\bf w$})$ is the Lamè operator. In the
frequency space, after taking the Fourier transform of system (1.2), 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

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

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 [33], 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 (1.2).

######  Lemma 2.

Let $\mathcal{G}\in
L^{2}_{\sigma}(\mathbb{R}^{3})\times{L}^{2}(\mathbb{R}^{3})$. If
$32\chi(\mu+\chi+\gamma)>1$, then 111Recalling that $e^{\Delta\tau}$ denotes
the heat semigroup.

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

for all $t\geq 0$.

It is also worth to note the following result concerning the Lamè semigroup.

######  Lemma 3.

If $\mathcal{G}\in
L^{2}_{\sigma}(\mathbb{R}^{3})\times{L}^{2}(\mathbb{R}^{3})$, then

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

for all $t\geq 0$.

######  Theorem 2.3.

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

(2.10)
$\begin{split}\|D^{m}\bar{\mbox{\boldmath$u$}}(\cdot,t;{t_{0}})-D^{m}\bar{\mbox{\boldmath$u$}}(\cdot,t;\tilde{t}_{0})\|_{L^{2}(\mathbb{R}^{3})}\leq
C\,(t-\tilde{t}_{0})^{-\frac{5}{4}-\frac{m}{2}},\end{split}$ (2.11)
$\begin{split}\|D^{m}\bar{\mbox{$\bf
w$}}(\cdot,t;{t_{0}})-D^{m}\bar{\mbox{$\bf
w$}}(\cdot,t;\tilde{t}_{0})\|_{L^{2}(\mathbb{R}^{3})}\leq
C\,e^{-2\,\chi\,(t-\tilde{t}_{0})}\,(t-\tilde{t}_{0})^{-\frac{5}{4}-\frac{m}{2}}.\end{split}$

###### Proof.

In order to prove (2.10) and (2.11), we consider a regularized system as
follows (see e.g. [7, 26]): Let $G_{\delta}$ be a standard mollifier, set
$\tilde{\mbox{\boldmath$z$}}_{0,\delta}(\cdot)\in C^{\infty}(\mathbb{R}^{6})$
as the convolution of $\mbox{\boldmath$z$}_{0}(\cdot)$ with $G_{\delta}$,
$\delta>0$ and define $\mbox{\boldmath$z$}_{\delta}$, $p_{\delta}$ as the
(unique, globally defined) classical $L^{2}$ solution for the system

$\left\\{\begin{array}[]{l}\frac{\partial{\mbox{\boldmath$u$}_{\delta}}}{\partial
t}+(\tilde{\mbox{\boldmath$u$}}_{\delta}\cdot\nabla)\mbox{\boldmath$u$}_{\delta}-(\mu+\chi)\Delta\mbox{\boldmath$u$}_{\delta}+\nabla
p_{\delta}=\chi\,\nabla\wedge\mbox{$\bf w$}_{\delta},\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt\cr\frac{\partial{\mbox{$\bf w$}_{\delta}}}{\partial
t}+(\tilde{\mbox{\boldmath$u$}}_{\delta}\cdot\nabla)\mbox{$\bf
w$}_{\delta}-\gamma\Delta\mbox{$\bf
w$}_{\delta}-\kappa\nabla(\nabla\cdot\mbox{$\bf w$}_{\delta})+2\chi\mbox{$\bf
w$}_{\delta}=\,\chi\,\nabla\wedge\mbox{\boldmath$u$}_{\delta},\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr\nabla\cdot\mbox{\boldmath$u$}_{\delta}=0,\\\ \vskip
3.0pt plus 1.0pt minus
1.0pt\cr\mbox{\boldmath$z$}_{\delta}(\cdot,0)=\tilde{\mbox{\boldmath$z$}}_{0,\delta}:=G_{\delta}\ast\mbox{\boldmath$z$}_{0}\in\bigcap_{m=1}^{\infty}{\bf
H}^{m}(\mathbb{R}^{6}),\end{array}\right.$

where
$\tilde{\mbox{\boldmath$z$}}_{\delta}(\cdot,t):=G_{\delta}\ast\mbox{\boldmath$z$}_{\delta}(\cdot,t)$.
It was shown by Leray that, for some suitable sequence $\tilde{\delta}\to 0$,
one has the $L^{2}$ weak convergence property
$\mbox{\boldmath$z$}_{\tilde{\delta}}(\cdot,t)\rightharpoonup\mbox{\boldmath$z$}(\cdot,t)$
as $\tilde{\delta}\to 0$, for all $t\geq 0$. Moreover, it is easy to obtain
the energy inequality

(2.12) $\|\mbox{\boldmath$z$}_{\delta}(\cdot,t)\|^{2}_{L^{2}}\leq
2\nu\int_{0}^{t}\|D\mbox{\boldmath$z$}_{\delta}(\cdot,\tau)\|^{2}_{L^{2}}d\tau\leq\|\mbox{\boldmath$z$}_{0}\|^{2}_{L^{2}}.$

Let

$\mbox{\boldmath$Q$}_{\delta}(\cdot,t):=\left[\begin{array}[]{ccc}(\tilde{\mbox{\boldmath$u$}}_{\delta}\cdot\nabla)\mbox{\boldmath$u$}_{\delta}-\nabla
p_{\delta}\\\ (\tilde{\mbox{\boldmath$u$}}_{\delta}\cdot\nabla)\mbox{$\bf
w$}_{\delta}\\\ \end{array}\right]\\!\\!\;\\!$

and write $\bar{\mbox{\boldmath$z$}}(\cdot,t;t_{0})$ as

$\bar{\mbox{\boldmath$z$}}(\cdot,t;t_{0})=e^{\mathbb{A}(t-t_{0})}[\mbox{\boldmath$z$}(\cdot,t_{0})-\mbox{\boldmath$z$}_{\delta}(\cdot,t_{0})]+e^{\mathbb{A}(t-t_{0})}\mbox{\boldmath$z$}_{\delta}(\cdot,t_{0}),\quad
t>t_{0}.$

Observing that

$\begin{split}\mbox{\boldmath$z$}_{\delta}(\cdot,t)=e^{\mathbb{A}t_{0}}\tilde{\mbox{\boldmath$z$}}_{0,\delta}+\int_{0}^{t_{0}}e^{\mathbb{A}(t_{0}-\tau)}\mbox{\boldmath$Q$}_{\delta}(\cdot,\tau)d\tau,\end{split}$

we obtain

$\bar{\mbox{\boldmath$z$}}(\cdot,t;t_{0})=e^{\mathbb{A}(t-t_{0})}\,[\,\mbox{\boldmath$z$}(\cdot,t_{0})-\mbox{\boldmath$z$}_{\delta}(\cdot,t_{0})\,]+e^{\mathbb{A}t}\tilde{\mbox{\boldmath$z$}}_{0,\delta}+\int_{0}^{t_{0}}e^{\mathbb{A}(t-\tau)}\mbox{\boldmath$Q$}_{\delta}(\cdot,\tau)d\tau,\quad\forall
t>t_{0}\geq 0.$

Analogously,

$\bar{\mbox{\boldmath$z$}}(\cdot,t;\tilde{t}_{0})=e^{\mathbb{A}(t-\tilde{t}_{0})}\,[\mbox{\boldmath$z$}(\cdot,\tilde{t}_{0})-\mbox{\boldmath$z$}_{\delta}(\cdot,\tilde{t}_{0})]+e^{\mathbb{A}t}\tilde{\mbox{\boldmath$z$}}_{0,\delta}+\int_{0}^{\tilde{t}_{0}}e^{\mathbb{A}(t-\tau)}\mbox{\boldmath$Q$}_{\delta}(\cdot,\tau)d\tau,\quad\forall
t>\tilde{t}_{0}\geq t_{0}.$

Therefore, given an arbitrary $\mathbb{K}\subset\mathbb{R}^{3}$ compact, it
follows, from Lemma 2,

(2.13)
$\begin{split}\|D^{m}\bar{\mbox{\boldmath$z$}}(\cdot,t;\tilde{t}_{0})-D^{m}\bar{\mbox{\boldmath$z$}}(\cdot,t;t_{0})\|_{L^{2}(\mathbb{K})}\leq
I_{m,\delta}(t)+\int_{t_{0}}^{\tilde{t}_{0}}\|e^{\mathbb{A}(t-\tau)}D^{m}\mbox{\boldmath$Q$}_{\delta}(\cdot,\tau)\|_{L^{2}(\mathbb{R}^{3})}\,d\tau\\\
\leq
I_{m,\delta}(t)+\int_{t_{0}}^{\tilde{t}_{0}}\|e^{\Delta\,(t-\tau)}D^{m}\mbox{\boldmath$Q$}_{\delta}(\cdot,\tau)\|_{L^{2}(\mathbb{R}^{3})}\,d\tau,\end{split}$

where

$\begin{split}I_{m,\delta}(t):=\|D^{m}\\{e^{\mathbb{A}(t-t_{0})}\,[\mbox{\boldmath$z$}(\cdot,t_{0})-\mbox{\boldmath$z$}_{\delta}(\cdot,t_{0})]\\}\|_{L^{2}(\mathbb{K})}\\\
+\|D^{m}\\{e^{\mathbb{A}(t-\tilde{t}_{0})}\,[\mbox{\boldmath$z$}(\cdot,\tilde{t}_{0})-\mbox{\boldmath$z$}_{\delta}(\cdot,\tilde{t}_{0})]\\}\|_{L^{2}(\mathbb{K})}.\end{split}$

Since $\|D^{m}e^{\Delta(t-\tau)}{\bf Q}_{\delta}(\cdot,\tau)\|_{L^{2}}\leq
C(t-\tau)^{-\frac{5}{4}-\frac{m}{2}}\|\mbox{\boldmath$z$}_{\delta}(\cdot,\tau)\|^{2}_{L^{2}}$,
one has

(2.14)
$\begin{split}\int_{t_{0}}^{\tilde{t}_{0}}\|e^{\Delta\,(t-\tau)}D^{m}\mbox{\boldmath$Q$}_{\delta}(\cdot,\tau)\|_{L^{2}(\mathbb{R}^{3})}\,d\tau\leq
C\,\int_{t_{0}}^{\tilde{t}_{0}}(t-\tau)^{-\frac{5}{4}-\frac{m}{2}}\|\mbox{\boldmath$z$}_{\delta}(\cdot,\tau)\|^{2}_{L^{2}(\mathbb{R}^{3})}\,d\tau\\\
\leq
C\,\int_{t_{0}}^{\tilde{t}_{0}}(t-\tau)^{-\frac{5}{4}-\frac{m}{2}}\|\mbox{\boldmath$z$}_{0}\|^{2}_{L^{2}(\mathbb{R}^{3})}\,d\tau\leq
C\,(t-\tilde{t}_{0})^{-\frac{5}{4}-\frac{m}{2}}\|\mbox{\boldmath$z$}_{0}\|^{2}_{L^{2}(\mathbb{R}^{3})}(\tilde{t}_{0}-t_{0})\\\
=C\,(t-\tilde{t}_{0})^{-\frac{5}{4}-\frac{m}{2}},\end{split}$

for some $C>0$, where we used bound (2.12).

Now, taking $\delta=\tilde{\delta}\to 0$, by the $L^{2}$ weak convergence
property mentioned above, we get $I_{m,\delta}(t)\to 0$, since, for any
$\sigma,s>0$,

(2.15) $\|D^{m}\,\\{\,e^{\Delta
s}[\,\mbox{\boldmath$z$}(\cdot,\sigma)-\mbox{\boldmath$z$}_{\delta}(\cdot,\sigma)\,]\,\\}\|_{L^{2}(\mathbb{K})}\to
0\quad\text{as}\quad\tilde{\delta}\to 0.$

Indeed, defining $F_{\delta}(\cdot,s):=D^{m}\,\\{\,e^{\Delta
s}[\,\mbox{\boldmath$z$}(\cdot,\sigma)-\mbox{\boldmath$z$}_{\delta}(\cdot,\sigma)\,]\,\\}$
one has

$F_{\delta}(\cdot,s)=H_{m}(\cdot,s)\ast[\mbox{\boldmath$z$}(\cdot,\sigma)-\mbox{\boldmath$z$}_{\delta}(\cdot,\sigma)],$

where $H_{m}\in L^{1}(\mathbb{R}^{3})\cap L^{\infty}(\mathbb{R}^{3})$. Since
$\mbox{\boldmath$z$}(\cdot,\sigma)-\mbox{\boldmath$z$}_{\delta}(\cdot,\sigma)\rightharpoonup
0$ in $L^{2}$, it follows that $F_{\tilde{\delta}}\to 0$ for each
$x\in\mathbb{R}^{3}$. On the other hand, by inequality (2.12) and the Cauchy-
Schwarz inequality, one obtains

$\begin{split}|F_{\delta}(x,s)|\leq\|H_{\delta}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}\|\mbox{\boldmath$z$}(\cdot,\sigma)-\mbox{\boldmath$z$}_{\delta}(\cdot,\sigma)\|_{L^{2}(\mathbb{R}^{3})}\\\
\leq
2\|H_{\delta}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}\|\mbox{\boldmath$z$}_{0}\|_{L^{2}(\mathbb{R}^{3})}\end{split}$

for all $x\in\mathbb{R}^{3}$. The Lebesgue‘s dominated convergence theorem
implies that $\|F_{\tilde{\delta}}(\cdot,s)\|_{L^{2}(\mathbb{K})}\to 0$ as
$\tilde{\delta}\to 0$. Hence, recalling that $\mathbb{K}$ is compact, the
proof of claim (2.15) is complete.

Letting $\tilde{\delta}\to 0$ in (2.13), (2.15) and (2.14), we obtain

$\|D^{m}\bar{\mbox{\boldmath$z$}}(\cdot,t;\tilde{t}_{0})-D^{m}\bar{\mbox{\boldmath$z$}}(\cdot,t;t_{0})\|_{L^{2}(\mathbb{K})}\leq
C(t-\tilde{t}_{0})^{-\frac{5}{4}-\frac{m}{2}},$

for each $t>\tilde{t}_{0}$ and for any compact
$\mathbb{K}\subset\mathbb{R}^{3}$. This is equivalent to the desired
inequality (2.10).

To show inequality (2.11), we work with the linearized micro-rotational
equation (1.2b) separately and take advantage of the linear damping term
$2\,\chi\bar{\mbox{$\bf w$}}$. To this end, just write $\bar{\mbox{$\bf
W$}}:=e^{2\chi(t-t_{0})}\bar{\mbox{$\bf w$}}$, apply Duhamel’s principle for
$\bar{\mbox{$\bf W$}}$ and write the result in terms of $\bar{\mbox{$\bf
w$}}(\cdot,t;t_{0})$ to get

$\begin{split}\bar{\mbox{$\bf
w$}}(\cdot,t;t_{0})=e^{-2\,\chi(t-t_{0})}\,e^{\mathbb{L}(t-t_{0})}[\mbox{$\bf
w$}(\cdot,t_{0})-\mbox{$\bf
w$}_{\delta}(\cdot,t_{0})]+\chi\int_{t_{0}}^{t}e^{-2\chi(t-s)}e^{\mathbb{L}(t-s)}\nabla\wedge\bar{\mbox{\boldmath$u$}}(\cdot,s)ds\\\
+e^{-2\chi(t-t_{0})}e^{\mathbb{L}(t-t_{0})}\mbox{$\bf
w$}_{\delta}(\cdot,t_{0}).\end{split}$

Furthermore, from system (1.2), it also holds

$\begin{split}e^{-2\chi(t-t_{0})}e^{\mathbb{L}(t-t_{0})}\mbox{$\bf
w$}_{\delta}(\cdot,t_{0})=e^{-2\chi\,t}e^{\mathbb{L}\,t}\bar{\mbox{$\bf
w$}}_{0,\delta}+\chi\,\int_{0}^{t_{0}}e^{-2\chi(t-s)}e^{\mathbb{L}(t-s)}\nabla\wedge\bar{\mbox{\boldmath$u$}}_{\delta}(\cdot,s)ds\\\
-\int_{0}^{t_{0}}e^{-2\chi(t-s)}e^{\mathbb{L}(t-s)}[(\tilde{\mbox{\boldmath$u$}}_{\delta}\cdot\nabla)\mbox{$\bf
w$}_{\delta}](\cdot,s)ds.\end{split}$

Consequently, after adding
$\pm\int_{0}^{t_{0}}e^{-2\,\chi(t-s)}e^{\mathbb{L}\,(t-s)}\nabla\wedge\bar{\mbox{\boldmath$u$}}(\cdot,t)ds$,
we get

$\begin{split}\bar{\mbox{$\bf
w$}}(\cdot,t;t_{0})=e^{-2\,\chi(t-t_{0})}\,e^{\mathbb{L}(t-t_{0})}[\mbox{$\bf
w$}(\cdot,t_{0})-\mbox{$\bf
w$}_{\delta}(\cdot,t_{0})]+e^{-2\chi\,t}e^{\mathbb{L}\,t}\bar{\mbox{$\bf
w$}}_{0,\delta}\\\
+\chi\int_{t_{0}}^{t}e^{-2\chi(t-s)}e^{\mathbb{L}(t-s)}\nabla\wedge\\{\bar{\mbox{\boldmath$u$}}-\bar{\mbox{\boldmath$u$}}_{\delta}\\}(\cdot,s)ds+\chi\int_{0}^{t}e^{-2\chi(t-s)}e^{\mathbb{L}(t-s)}\nabla\wedge\bar{\mbox{\boldmath$u$}}(\cdot,s)ds\\\
-\int_{0}^{t_{0}}e^{-2\chi(t-s)}e^{\mathbb{L}(t-s)}[(\tilde{\mbox{\boldmath$u$}}_{\delta}\cdot\nabla)\mbox{$\bf
w$}_{\delta}](\cdot,s)ds,\quad\forall t>t_{0},\end{split}$

for each $t_{0}\geq 0$. Therefore, by Lemma 3

(2.16) $\begin{split}\|D^{m}\bar{\mbox{$\bf
w$}}(\cdot,t;t_{0})-D^{m}\bar{\mbox{$\bf
w$}}(\cdot,t;\tilde{t}_{0})\|_{L^{2}(\mathbb{K})}\leq\mathcal{J}_{m,\delta}(t)\\\
+\int_{t_{0}}^{\tilde{t}_{0}}e^{-2\chi(t-s)}\|D^{m}e^{\mathbb{L}(t-s)}(\tilde{\mbox{\boldmath$u$}}_{\delta}\cdot\nabla)\mbox{$\bf
w$}_{\delta}(\cdot,s)\|_{L^{2}(\mathbb{K})}ds\\\
\leq\mathcal{J}_{m,\delta}(t)+\int_{t_{0}}^{\tilde{t}_{0}}e^{-2\chi(t-s)}\|D^{m}e^{\Delta(t-s)}(\tilde{\mbox{\boldmath$u$}}_{\delta}\cdot\nabla)\mbox{$\bf
w$}_{\delta}(\cdot,s)\|_{L^{2}(\mathbb{K})}ds,\end{split}$

where

$\begin{split}\mathcal{J}_{m,\delta}(t):=e^{-2\,\chi(t-t_{0})}\|D^{m}\\{\,e^{\Delta(t-t_{0})}[\mbox{$\bf
w$}(\cdot,t_{0})-\mbox{$\bf
w$}_{\delta}(\cdot,t_{0})]\\}\|_{L^{2}(\mathbb{K})}\\\
+\,\,e^{-2\,\chi(t-\tilde{t}_{0})}\|D^{m}\\{\,e^{\Delta(t-\tilde{t}_{0})}[\mbox{$\bf
w$}(\cdot,\tilde{t}_{0})-\mbox{$\bf
w$}_{\delta}(\cdot,\tilde{t}_{0})]\\}\|_{L^{2}(\mathbb{K})}\\\
+\,\,\chi\int_{t_{0}}^{t}e^{-2\chi(t-s)}\|D^{m+1}e^{\Delta(t-s)}\\{\bar{\mbox{\boldmath$u$}}-\bar{\mbox{\boldmath$u$}}_{\delta}\\}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}ds,\end{split}$

for an arbitrary compact $\mathbb{K}\subset\mathbb{R}^{3}$. As before, use the
energy inequality (2.12) to obtain

(2.17)
$\begin{split}\int_{t_{0}}^{\tilde{t}_{0}}e^{-2\chi(t-s)}\|D^{m}e^{\Delta(t-s)}(\tilde{\mbox{\boldmath$u$}}_{\delta}\cdot\nabla)\mbox{$\bf
w$}_{\delta}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}ds\\\ \leq
e^{-2\chi(t-\tilde{t}_{0})}\int_{t_{0}}^{\tilde{t}_{0}}(t-s)^{-\frac{5}{4}-\frac{m}{2}}\|\mbox{\boldmath$z$}_{\delta}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds\leq
C\,e^{-2\chi(t-\tilde{t}_{0})}(t-\tilde{t}_{0})^{-\frac{5}{4}-\frac{m}{2}},\,\,\,[\text{by}\eqref{eqn:weak-
inequality}]\end{split}$

with $C=C(t_{0},\tilde{t}_{0},\|\mbox{\boldmath$z$}_{0}\|)>0$. By the same
argument as before (c.f. (2.15)), one has

(2.18) $\mathcal{J}_{m,\delta}(t)\to
0\quad\text{as}\quad\delta=\tilde{\delta}\to 0.$

Therefore, by bounds (2.16), (2.17) and the convergence (2.18), it follows
that

$\|D^{m}\bar{\mbox{$\bf w$}}(\cdot,t;t_{0})-D^{m}\bar{\mbox{$\bf
w$}}(\cdot,t;\tilde{t}_{0})\|_{L^{2}(\mathbb{K})}\leq
C\,e^{-2\chi(t-\tilde{t}_{0})}(t-\tilde{t}_{0})^{-\frac{5}{4}-\frac{m}{2}},$

for each $t>\tilde{t}_{0}$ and for any compact
$\mathbb{K}\subset\mathbb{R}^{3}$. This is clearly equivalent to the desired
inequality (2.11), ∎

##  3\. Proof of Theorem 1.1

Define
$\tilde{\lambda}_{0}(\alpha):=\limsup_{t\to\infty}t^{\alpha}\|\mbox{\boldmath$z$}(\cdot,t)\|$.
We first assume $\tilde{\lambda}_{0}(\alpha)<\infty$ and prove inequalities
(1.4) with $\tilde{\lambda}_{0}(\alpha)$ replacing
${\lambda}_{0}(\alpha)=\limsup_{t\to\infty}t^{\alpha}\|\mbox{\boldmath$u$}(\cdot,t)\|$
in their right hand side. Finally, at the end of the section (see Claim 3.1
below), we finish the proof showing that
$\tilde{\lambda}_{0}(\alpha)=\lambda_{0}(\alpha)$ whenever
$\lambda_{0}(\alpha)<\infty$.

###  3.1. Proof of bound (1.4a)

For simplicity, we define the scalar function
$\zeta(\cdot,t):=\nabla\cdot\mbox{$\bf w$}(\cdot,t)$ and
$\nu:=\min\\{\mu,\gamma\\}$ as before. Let $t\geq t_{0}>t_{*}$, where $t_{*}$
is the regularity time (see (2.1)). Given $\delta>0$, taking the dot product
of equations (1.1a) and (1.1b) with
$2(s-t_{0})^{2\alpha+\delta}\mbox{\boldmath$u$}$ and
$2(s-t_{0})^{2\alpha+\delta}\mbox{$\bf w$}$ respectively, and integrating by
parts over $[t_{0},t]\times\mathbb{R}^{3}$, we get

(3.1)
$\begin{split}(t-t_{0})^{2\alpha+\delta}\|\mbox{\boldmath$z$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}^{2}+2\nu\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta}\|D\mbox{\boldmath$z$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds\\\
+2\kappa\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta}\|\zeta(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds+2\chi\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta}\|\mbox{$\bf
w$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds\\\
\leq(2\alpha+\delta)\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta-1}\|\mbox{\boldmath$z$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds.\end{split}$

Due to assumption (1.3), given $0<\epsilon<2$, we can choose $t_{0}>t_{*}$
sufficiently large such that
$t^{\alpha}\|\mbox{\boldmath$z$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}<\tilde{\lambda}_{0}(\alpha)+\epsilon$.
Combining this with bound (3.1), we get

(3.2)
$\begin{split}(t-t_{0})^{2\alpha+\delta}\|\mbox{\boldmath$z$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}^{2}+2\nu\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta}\|D\mbox{\boldmath$z$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds\\\
+2\kappa\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta}\|\zeta(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds+2\chi\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta}\|\mbox{$\bf
w$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds\\\
\leq(2\alpha+\delta)(\tilde{\lambda}_{0}(\alpha)+\epsilon)^{2}\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta-1}s^{-2\alpha}ds\\\
\leq(2\alpha+\delta)(\tilde{\lambda}_{0}(\alpha)+\epsilon)^{2}\frac{(t-t_{0})^{\delta}}{\delta}.\end{split}$

In particular,

$(t-t_{0})^{2\alpha}\|\mbox{\boldmath$z$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}^{2}\leq\frac{2\alpha+\delta}{\delta}(\tilde{\lambda}_{0}(\alpha)+\epsilon)^{2}$

and

(3.3)
$\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta}\|D\mbox{\boldmath$z$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds\leq\frac{1}{2\nu}(2\alpha+\delta)(\tilde{\lambda}_{0}(\alpha)+\epsilon)^{2}\frac{(t-t_{0})^{\delta}}{\delta},$

for all $t\geq t_{0}$. For the next step, first observe that differentiating
equations (1.1a) and (1.1b) with respect to $x_{l}$ (for example), taking the
dot product of equations (1.1a) and (1.1b) by
$2(s-t_{0})^{2\alpha+\delta+1}D_{l}\mbox{\boldmath$u$}$ and
$2(s-t_{0})^{2\alpha+\delta+1}D_{l}\mbox{$\bf w$}$, respectively, and
integrating the result over $\mathbb{R}^{3}\times[t_{0},t]$, we get, after
using Lemma 1 and summing over $1\leq l\leq 3$,

$\begin{split}(t-t_{0})^{2\alpha+\delta+1}\|D\mbox{\boldmath$z$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}^{2}\\\
+2\nu\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta+1}\|D^{2}\mbox{\boldmath$z$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds\\\
+2\kappa\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta+1}\|D\zeta(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds+2\chi\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta+1}\|D\mbox{$\bf
w$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds\\\
\leq(2\alpha+\delta+1)\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta}\|D\mbox{\boldmath$z$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds\\\
+C\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta+1}\|\mbox{\boldmath$z$}(\cdot,s)\|_{L^{\infty}(\mathbb{R}^{3})}\|D\mbox{\boldmath$z$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}\|D^{2}\mbox{\boldmath$z$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}ds\\\
\leq(2\alpha+\delta+1)\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta}\|D\mbox{\boldmath$z$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds\\\
+C\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta+1}\|\mbox{\boldmath$z$}(\cdot,s)\|_{{L^{2}(\mathbb{R}^{3})}}^{1/2}\|D\mbox{\boldmath$z$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{1/2}\|D^{2}\mbox{\boldmath$z$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds.\end{split}$

Since $\|D\mbox{\boldmath$z$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}\to 0$ as $t$
tends to infinity222Actually, for arbitrary initial data
$\mbox{\boldmath$z$}_{0}\in L^{2}$, a faster decay rate holds for the
gradient: $t^{1/2}\|D\mbox{\boldmath$z$}\|\to 0$ as $t\to\infty$. See e.g
[20], for more details., increasing $t_{0}$ if necessary, we obtain

$\begin{split}(t-t_{0})^{2\alpha+\delta+1}\|D\mbox{\boldmath$z$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}^{2}\\\
+(2-\epsilon)\,\nu\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta+1}\|D^{2}\mbox{\boldmath$z$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds\\\
+2\kappa\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta+1}\|D\zeta(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds+2\chi\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta+1}\|D\mbox{$\bf
w$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds\\\
\leq(2\alpha+\delta+1)\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta}\|D\mbox{\boldmath$z$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds.\end{split}$

This, with inequality (3.3)), immediately gives

$(t-t_{0})^{2\alpha+1}\|D\mbox{\boldmath$z$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}^{2}\leq\frac{1}{(2-\epsilon)\delta\nu}(2\alpha+\delta+1)(2\alpha+\delta)(\tilde{\lambda}_{0}(\alpha)+\epsilon)^{2}$

and

$\begin{split}\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+\delta+1}\|D^{2}\mbox{\boldmath$z$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds\\\
\leq\frac{1}{\delta((2-\epsilon)\nu)^{2}}(2\alpha+\delta+1)(2\alpha+\delta)(\tilde{\lambda}_{0}(\alpha)+\epsilon)^{2}(t-t_{0})^{\delta},\end{split}$

which proves (1.4a), when $\tilde{\lambda}_{0}(\alpha)<\infty$, for $m=1$.
Now, we proceed by an induction argument. Assume that, for some $m\geq 2$,

(3.4a)
$(t-t_{0})^{2\alpha+k}\|D^{k}\mbox{\boldmath$z$}(\cdot,t)\|^{2}_{L^{2}(\mathbb{R}^{3})}\leq\frac{1}{\delta[(2-\epsilon)\lambda]^{k}}\bigg{[}\prod_{j=0}^{k}(2\alpha+j+\delta)\bigg{]}(\tilde{\lambda}_{0}(\alpha)+\epsilon)^{2}$
and (3.4b)
$\begin{split}\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+k+\delta}\|D^{k+1}\mbox{\boldmath$z$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds\\\
\leq\bigg{(}\frac{1}{\delta[(2-\epsilon)\lambda]^{k+1}}\bigg{[}\prod_{j=0}^{k}(2\alpha+j+\delta)\bigg{]}(\tilde{\lambda}_{0}(\alpha)+\epsilon)^{2}\bigg{)}(t-t_{0})^{\delta}\end{split}$

hold for every $1\leq k\leq m-1$, and for all $t\geq t_{0}$ (for any
$t_{0}>t_{**}$, for some $t_{**}>t_{*}$ sufficiently large, see (2.1), (2.2)).
Then, applying $D_{l_{1}}D_{l_{2}}...D_{l_{m}}$ to the equation (1.1a) and
taking the dot product with
$2(s-t_{0})^{2\alpha+m+\delta}D_{l_{1}}D_{l_{2}}...D_{l_{m}}\mbox{\boldmath$u$}(\cdot,s)$
we get, (integrating over $\mathbb{R}^{3}\times[t_{0},t]$ and summing over
$1\leq l_{1},l_{2},...,l_{m}\leq 3$):

$\begin{split}(t-t_{0})^{2\alpha+m+\delta}\|D^{m}\mbox{\boldmath$u$}(\cdot,t)\|^{2}_{L^{2}(\mathbb{R}^{3})}+2(\mu+\chi)\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|D^{m+1}\mbox{\boldmath$u$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds\\\
=(2\alpha+m+\delta)\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m-1+\delta}\|D^{m}\mbox{\boldmath$u$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds\\\
+2\sum_{i,j,l_{1},l_{2},...,l_{m}=1}^{3}\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\int_{\mathbb{R}^{3}}[D_{j}D_{l_{1}}...D_{l_{m}}\mbox{\boldmath$u$}_{i}][D_{l_{1}}...D_{l_{m}}(\mbox{\boldmath$u$}_{i}\mbox{\boldmath$u$}_{j})]dxds\\\
+2\chi\sum_{l_{1},...,l_{m}=1}^{3}\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\int_{\mathbb{R}^{3}}<D_{l_{1}}...D_{l_{m}}\mbox{\boldmath$u$},\nabla\wedge
D_{l_{1}}...D_{l_{m}}\mbox{$\bf w$}>dxds.\end{split}$

Do the same for equation (1.1b). Adding up the resulting inequalities, one has

$\begin{split}(t-t_{0})^{2\alpha+m+\delta}\|D^{m}\mbox{\boldmath$z$}(\cdot,t)\|^{2}_{L^{2}(\mathbb{R}^{3})}+2\nu\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|D^{m+1}\mbox{\boldmath$z$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds\\\
+2\chi\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|D^{m+1}\mbox{\boldmath$u$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds\\\
+4\chi\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|D^{m}\mbox{$\bf
w$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds\\\
+2\kappa\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|D^{m}\zeta(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds\\\
\leq(2\alpha+m+\delta)\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m-1+\delta}\|D^{m}\mbox{\boldmath$z$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}\\\
+K_{m}\sum_{l=0}^{\big{[}\frac{m}{2}\big{]}}\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|D^{l}\mbox{\boldmath$z$}(s)\|_{L^{\infty}(\mathbb{R}^{3})}\|D^{m-l}\mbox{\boldmath$z$}(s)\|_{L^{2}(\mathbb{R}^{3})}\|D^{m+1}\mbox{\boldmath$z$}(s)\|_{L^{2}(\mathbb{R}^{3})}ds\\\
+4\chi\sum_{l_{1},...,l_{m}=1}^{3}\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\int_{\mathbb{R}^{3}}<D_{l_{1}}...D_{l_{m}}\mbox{$\bf
w$}(x,s),\nabla\wedge
D_{l_{1}}...D_{l_{m}}\mbox{\boldmath$u$}(x,s)>dxds,\end{split}$

for some constant $K_{m}>0$ (which depends only on $m$). Observing that

$\begin{split}4\chi\sum_{l_{1},...,l_{m}=1}^{3}(s-t_{0})^{2\alpha+m+\delta}\int_{\mathbb{R}^{3}}<D_{l_{1}}...D_{l_{m}}\mbox{$\bf
w$},\nabla\wedge D_{l_{1}}...D_{l_{m}}\mbox{\boldmath$u$}>dxds\\\ \leq
2\chi\|D^{m}\mbox{$\bf
w$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}+2\chi\|D^{m+1}\mbox{\boldmath$u$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})},\end{split}$

by Young‘s inequality, the bound

$\begin{split}K_{m}\sum_{l=0}^{\big{[}\frac{m}{2}\big{]}}\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|D^{l}\mbox{\boldmath$z$}(s)\|_{L^{\infty}(\mathbb{R}^{3})}\|D^{m-l}\mbox{\boldmath$z$}(s)\|_{L^{2}(\mathbb{R}^{3})}\|D^{m+1}\mbox{\boldmath$z$}(s)\|_{L^{2}(\mathbb{R}^{3})}ds\\\
\leq
K_{m}\sum_{l=1}^{m-1}\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|\mbox{\boldmath$z$}(s)\|^{1/2}_{L^{2}(\mathbb{R}^{3})}\|D\mbox{\boldmath$z$}(s)\|^{1/2}_{L^{2}(\mathbb{R}^{3})}\|D^{m+1}\mbox{\boldmath$z$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds,\end{split}$

and Lemma 1, one obtains

(3.5)
$\begin{split}(t-t_{0})^{2\alpha+m+\delta}\|D^{m}\mbox{\boldmath$z$}(\cdot,t)\|^{2}_{L^{2}(\mathbb{R}^{3})}+2\nu\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|D^{m+1}\mbox{\boldmath$z$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds\\\
+2\chi\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|D^{m}\mbox{$\bf
w$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds\\\
+2\kappa\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|D^{m}\zeta(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds\\\
\leq(2\alpha+m+\delta)\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m-1+\delta}\|D^{m}\mbox{\boldmath$z$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}\\\
+K_{m}\sum_{l=0}^{\big{[}\frac{m}{2}\big{]}}\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|\mbox{\boldmath$z$}(s)\|^{1/2}_{L^{2}(\mathbb{R}^{3})}\|D\mbox{\boldmath$z$}(s)\|^{1/2}_{L^{2}(\mathbb{R}^{3})}\|D^{m+1}\mbox{\boldmath$z$}(s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds.\end{split}$

Again, since $\|\mbox{\boldmath$z$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}$ is
bounded in $[t_{*},\infty)$ and
$\|D\mbox{\boldmath$z$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}\to 0$ as
$t\to\infty$ (see e.g [4, 20]), increasing $t_{0}>t_{*}$ if necessary,
inequality (3.5) becomes333Although it is not required here, we actually have
that $\|\mbox{\boldmath$z$}(t)\|$ is bounded in $[0,\infty)$ and tends to $0$
as $t\to\infty$. Moreover, as already mentioned,
$t^{1/2}\|D\mbox{\boldmath$z$}\|\to 0$.

(3.6)
$\begin{split}(t-t_{0})^{2\alpha+m+\delta}\|D^{m}\mbox{\boldmath$z$}(\cdot,t)\|^{2}_{L^{2}(\mathbb{R}^{3})}+(2-\epsilon)\nu\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|D^{m+1}\mbox{\boldmath$z$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds\\\
+2\chi\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|D^{m}\mbox{$\bf
w$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds\\\
+2\kappa\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|D^{m}\zeta(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds\\\
\leq(2\alpha+m+\delta)\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m-1+\delta}\|D^{m}\mbox{\boldmath$z$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})},\quad\forall
t>t_{0},\end{split}$

for each $t_{0}>t_{*}$ chosen appropriately large. Using the induction
hypothesis (3.4b) for $k=m-1$, we obtain

(3.7)
$(t-t_{0})^{2\alpha+m}\|D^{m}\mbox{\boldmath$z$}(\cdot,t)\|^{2}_{L^{2}(\mathbb{R}^{3})}\leq\frac{1}{\delta[(2-\epsilon)\nu]^{m}}\bigg{[}\prod_{j=0}^{m}(2\alpha+j+\delta)\bigg{]}(\tilde{\lambda}_{0}(\alpha)+\epsilon)^{2}.$

Moreover, from bounds (3.4b) and (3.6), we also get

(3.8)
$\begin{split}\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|D^{m+1}\mbox{\boldmath$z$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds\\\
\leq\bigg{(}\frac{1}{\delta[(2-\epsilon)\nu]^{m+1}}\bigg{[}\prod_{j=0}^{m}(2\alpha+j+\delta)\bigg{]}(\tilde{\lambda}_{0}(\alpha)+\epsilon)^{2}\bigg{)}(t-t_{0})^{\delta}.\end{split}$

Since $\epsilon>0$ is arbitrary, this shows inequality (1.4a) when
$\tilde{\lambda}_{0}(\alpha)<\infty$.

######  Remark 3.1.

Recalling inequalities (3.7) and (3.8), we have shown that for every $m\geq 0$
and each $\epsilon>0$ arbitrary, there exists
$t_{**}=t_{**}(m,\epsilon)>t_{*}$ sufficiently large so that

(3.9)
$\begin{split}(t-t_{0})^{2\alpha+m+\delta}\|D^{m}\mbox{\boldmath$z$}(\cdot,t)\|^{2}_{L^{2}(\mathbb{R}^{3})}+(2-\epsilon)\nu\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|D^{m+1}\mbox{\boldmath$z$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds\\\
\underbrace{+2\chi\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|D^{m}\mbox{$\bf
w$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds}_{(i)}\\\
+\underbrace{2\kappa\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|D^{m}\zeta(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds}_{(ii)}\\\
\leq\frac{1}{\delta[(2-\epsilon)\nu]^{m}}\bigg{\\{}\prod_{j=0}^{m}(2\alpha+j+\delta)\bigg{\\}}(\tilde{\lambda}_{0}(\alpha)+\epsilon)^{2}\,(t-t_{0})^{\delta},\end{split}$

$\forall t\geq t_{0}$, $\forall t_{0}\geq t_{**}(m,\epsilon)$. This suggests
the following: the presence of the terms (i) and (ii) on the LHS indicate that
$\|D^{m}\mbox{$\bf w$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}$ and
$\|D^{m}\zeta(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}$ may decay faster (as
$t\to\infty$) than what is given by our general result (3.7) above. Indeed, in
particular, inequality (3.7) implies that $\|D^{m}\mbox{$\bf
w$}(\cdot,t)\|_{L^{2}}=O(t^{-\alpha-\frac{m}{2}})$ and
$\|D^{m}\zeta(\cdot,t)\|_{L^{2}}=O(t^{-\alpha-\frac{m}{2}-\frac{1}{2}})$ as
$t\to\infty$ for each $m\geq 0$. However, by inequality (3.9) they actually
may decay faster when the vortex viscosity $\chi$ is present. This is exactly
the spirit of the next fundamental inequality (1.4b) to be shown below.

###  3.2. Proof of bound (1.4b)

As in the proof of inequality (2.11), taking advantage of the linear damping
term, using Lemma 3 and the Duhamel‘s principle, one has

(3.10) $\begin{split}t^{\alpha+\frac{m+1}{2}}\|D^{m}\mbox{$\bf
w$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}\leq\bigg{[}t^{\alpha+\frac{m+1}{2}}e^{-2\chi(t-t_{0})}\|D^{m}e^{\mathbb{L}(t-t_{0})}\mbox{$\bf
w$}(\cdot,t_{0})\|_{L^{2}(\mathbb{R}^{3})}\\\
\mbox{}+t^{\alpha+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-2\chi(t-s)}\|e^{\mathbb{L}(t-s)}D^{m}(\mbox{\boldmath$u$}\cdot\nabla\mbox{$\bf
w$})(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}ds\\\ \mbox{}+\chi
t^{\alpha+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-2\chi(t-s)}\|e^{\mathbb{L}(t-s)}D^{m+1}\mbox{\boldmath$u$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}ds\bigg{]}=:\mathcal{I}_{1}(t)+\mathcal{I}_{2}(t)+\mathcal{I}_{3}(t).\end{split}$

As already mentioned, in [19] the following $L^{2}$ decay for the higher-order
derivatives was proved:

$t^{k/2}\|D^{k}\mbox{\boldmath$z$}\|_{L^{2}(\mathbb{R}^{3})}\to 0\quad{\text{
as}}\quad t\to\infty,\,\,{\text{for each}}\,\,k\in\mathbb{Z}^{+}.$

This implies $\mathcal{I}_{2}(t)\to 0$ as $t\to\infty$. Moreover,
$\mathcal{I}_{1}(t)$ clearly tends to $0$ as $t\to\infty$. So, it remains to
examine the last term $\mathcal{I}_{3}(t)$. To this end, given $\epsilon>0$,
choose $t_{0}>t_{**}(m,\epsilon)$ large enough and fix $0<\sigma<1$. By
inequality (3.9), we get

$\begin{split}\mathcal{I}_{3}(t)\leq\chi
t^{\alpha+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-2\chi(t-s)}\|e^{\gamma\Delta(t-s)}D^{m+1}\mbox{\boldmath$u$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}ds\\\
\leq\chi
t^{\alpha+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-2\chi(t-s)}\|D^{m+1}\mbox{\boldmath$u$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}ds\\\
\leq\chi\,\nu^{-\frac{m+1}{2}}K_{\alpha,m+1}t^{\alpha+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-2\chi(t-s)}(\tilde{\lambda}_{0}(\alpha)+\epsilon)s^{-\alpha-\frac{m+1}{2}}ds\\\
=\chi\nu^{-\frac{m+1}{2}}K_{\alpha,m+1}(\tilde{\lambda}_{0}(\alpha)+\epsilon)t^{\alpha+\frac{m+1}{2}}\bigg{\\{}\int_{t_{0}}^{\sigma
t}e^{-2\chi(t-s)}s^{-\alpha-\frac{m+1}{2}}ds\\\ +\int_{\sigma
t}^{t}e^{-2\chi(t-s)}s^{-\alpha-\frac{m+1}{2}}ds\bigg{\\}}\\\
\leq\chi\nu^{-\frac{m+1}{2}}K_{\alpha,m+1}(\tilde{\lambda}_{0}(\alpha)+\epsilon)\bigg{\\{}t^{\alpha+\frac{m+1}{2}}e^{-2\chi
t(1-\sigma)}\bigg{(}2\frac{t_{0}^{-\alpha-\frac{m-1}{2}}-(\sigma
t)^{-\alpha-\frac{m-1}{2}}}{2\alpha+m-1}\bigg{)}\\\
+\sigma^{-\alpha-\frac{m+1}{2}}\bigg{(}\frac{1-e^{-2\chi
t(1-\sigma)}}{2\chi}\bigg{)}\bigg{\\}}.\end{split}$

Letting $\epsilon\to 0^{+}$, $t\to\infty$ and $\sigma\to 1^{-}$ (in this
order), one has

(3.11) $\limsup_{t\rightarrow\infty}\chi
t^{\alpha+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-2\chi(t-s)}\|e^{\gamma\mathbb{L}(t-s)}D^{m+1}\mbox{\boldmath$u$}\|_{L^{2}(\mathbb{R}^{3})}ds\leq\frac{K_{\alpha,m+1}}{2}\,\nu^{-\frac{m+1}{2}}\,\tilde{\lambda}_{0}(\alpha).$

This, together with inequality (3.10), leads to the desired inequality (1.4b)
in case $\tilde{\lambda}_{0}(\alpha)<\infty$, i.e.,

(3.12) $\limsup_{t\to\infty}t^{\alpha+\frac{m+1}{2}}\|D^{m}\mbox{$\bf
w$}(\cdot,t)\|_{{L^{2}(\mathbb{R}^{3})}}\leq\frac{K_{\alpha,m+1}}{2}\;\;\nu^{-\frac{m+1}{2}}\;\;\tilde{\lambda}_{0}(\alpha).$

∎

######  Claim 3.1.

The assumption $\tilde{\lambda}_{0}(\alpha)<\infty$ in the proofs above is not
necessary, since

(3.13) $\text{ If }\,\,\,{\lambda}_{0}(\alpha)<\infty,\text{ then
}\,\,t^{\alpha}\|\mbox{$\bf w$}(\cdot,t)\|_{L^{2}}\to 0\,\,\,\text{
as}\,\,\,t\to\infty,\,\,\text{i.e.},\,\,{\lambda}_{0}(\alpha)=\tilde{\lambda}_{0}(\alpha).$

###### Proof.

We divide the proof into three cases, depending on the value of $\alpha$.

If $\alpha<1/2$, then $t^{\alpha}\|\mbox{$\bf
w$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}\to 0$ as $t\to\infty$ (see convergence
(1.8) for $s=0$) and $\tilde{\lambda}_{0}(\alpha)<\infty$. Thus, if
$\alpha\leq 1/2$ , then inequality (3.12) leads to inequalities (1.4).

If $1/2\leq\alpha<1$, then $t^{1/2}\|\mbox{$\bf
w$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}\to 0$ as $t\to\infty$. Now, letting
$m=0$ in inequality (3.12), one concludes

$\limsup_{t\to\infty}t\|\mbox{$\bf
w$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}<\infty.$

This implies $\limsup_{t\to\infty}t^{\alpha}\|\mbox{$\bf
w$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}=0$, that is,
${\lambda}_{0}(\alpha)=\tilde{\lambda}_{0}(\alpha)$, leading once more to the
desired inequalities (1.4).

If $\alpha\geq 1$, just repeat step-by-step the same argument above. In at
most $2[\alpha]+1$ steps, one gets the desired result. ∎

##  4\. Proof of Theorem 1.3

First of all, note that it is enough to prove inequalities (1.6a) and (1.6b)
for times $t_{0}>t_{*}$, the regularity time. Indeed, if the inequalities
(1.6a) and (1.6b) hold for $t_{0}>t_{*}$, then given $\hat{t}\in[0,t_{*}]$,
Theorem 2.3 gives

$\begin{split}\limsup_{t\rightarrow\infty}t^{\alpha+\beta+\frac{m}{2}}\|{\mathcal{E}}_{\mbox{\boldmath$z$}}(\cdot,t;\hat{t})\|_{L^{2}}\leq\limsup_{t\rightarrow\infty}t^{\alpha+\beta+\frac{m}{2}}\|{\mathcal{E}}_{\mbox{\boldmath$z$}}(\cdot,t;t_{0})\|_{L^{2}}\\\
+\limsup_{t\rightarrow\infty}t^{\alpha+\beta+\frac{m}{2}}\|{\mathcal{E}}_{\mbox{\boldmath$z$}}(\cdot,t;t_{0})-{\mathcal{E}}_{\mbox{\boldmath$z$}}(\cdot,t;\hat{t})\|_{L^{2}}=\limsup_{t\rightarrow\infty}t^{\alpha+\beta+\frac{m}{2}}\|{\mathcal{E}}_{\mbox{\boldmath$z$}}(\cdot,t;t_{0})\|_{L^{2}}<C_{\alpha,m}.\end{split}$

The same argument works for ${\mathcal{E}}_{\mbox{$\bf w$}}$. So, from now on,
consider $t_{0}>t_{*}$.

###  4.1. Proof of inequality t(1.6a)

Using Lemma 2 and the integral representations of $z$ and
$\bar{\mbox{\boldmath$z$}}$, write

(4.1)
$\begin{split}\|D^{m}{\mathcal{E}}_{\mbox{\boldmath$z$}}(\cdot,t;t_{0})\|_{L^{2}(\mathbb{R}^{3})}=\|D^{m}\left[\mbox{\boldmath$z$}(\cdot,t)-e^{\mathbb{A}(t-t_{0})}\mbox{\boldmath$z$}(\cdot,t_{0})\right]\|_{L^{2}(\mathbb{R}^{3})}\\\
\leq\int_{t_{0}}^{t}\|e^{c\,\Delta(t-s)}D^{m}\left[\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$}\right]\|_{L^{2}(\mathbb{R}^{3})}+\|e^{c\,\Delta(t-s)}D^{m}\left[\mbox{\boldmath$u$}\cdot\nabla\mbox{$\bf
w$}\right]\|_{L^{2}(\mathbb{R}^{3})}ds.\end{split}$

We estimate in details the first term in the right hand side of inequality
(4.1), since the analysis for the second one is completely similar
444Actually, the second term decays even faster than the first one due to
inequality (1.4b). To this end, take $t>2t_{0}$ and split the domain of
integration as $[t_{0},\frac{t}{2}]\cup[\frac{t}{2},t]$. We begin by examining
the integral on the interval $[\frac{t}{2},t]$:

$\begin{split}t^{\alpha+\beta+m/2}\int_{t/2}^{t}\|e^{c\,\Delta(t-s)}D^{m}(\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$})(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}ds\\\
\leq
t^{\alpha+\beta+m/2}\int_{t/2}^{t}\sum_{l=0}^{m}\|D^{l}\mbox{\boldmath$u$}(\cdot,s)\|_{L^{4}(\mathbb{R}^{3})}\|D^{m-l+1}\mbox{\boldmath$u$}(\cdot,s)\|_{L^{4}(\mathbb{R}^{3})}ds\\\
\leq
t^{\alpha+\beta+m/2}\int_{t/2}^{t}\sum_{l=0}^{m}\|D^{l}\mbox{\boldmath$u$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{1/4}\|D^{l+1}\mbox{\boldmath$u$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{3/4}\\\
\times\|D^{m-l+1}\mbox{\boldmath$u$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{1/4}\|D^{m-l+2}\mbox{\boldmath$u$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{3/4}ds\\\
\leq
K\,\nu^{-(\frac{m}{2}+\frac{5}{4})}\,(\lambda_{0}(\alpha)+\epsilon)^{2}t^{\alpha+\beta+m/2}\int_{t/2}^{t}\sum_{l=0}^{m}s^{-\frac{\alpha}{4}-\frac{l}{8}}s^{-\frac{3\alpha}{4}-\frac{3(l+1)}{8}}s^{-\frac{\alpha}{4}-\frac{m-l+1}{8}}s^{-\frac{3\alpha}{4}-\frac{3(m-l+2)}{8}}ds\\\
\leq
K\,\nu^{-(\frac{m}{2}+\frac{5}{4})}\,(\lambda_{0}(\alpha)+\epsilon)^{2}t^{\alpha+\beta+m/2}\int_{t/2}^{t}s^{-2\alpha}s^{-\frac{2m+5}{4}}ds\leq
K(\lambda_{0}(\alpha)+\epsilon)^{2},\end{split}$

where

$K=\sum_{l=0}^{m}K_{\alpha,l}^{1/4}K_{\alpha,l+1}^{3/4}K_{\alpha,m-l+1}^{1/4}K_{\alpha,m-l+2}^{3/4}$

and

$K_{\alpha,m}=\min_{\delta>0}\Bigg{\\{}\delta^{-1/2}\prod_{j=0}^{m}(\alpha+\frac{j}{2}+\delta)^{1/2}\Bigg{\\}}.$

Note that to bound the $L^{4}$ norms in the inequality above, we have used the
classical Sobolev-Nirenberg-Gagliardo inequality
$\|f\|_{L^{4}}\leq\|f\|^{1/4}_{L^{2}}\|Df\|^{3/4}_{L^{2}}$, for any $f\in
H^{1}$.

It remains to bound the integral over $[t_{0},\frac{t}{2}]$. Since,
$\|e^{\Delta\tau}D^{m}(\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$})\|_{L^{2}}\leq\tau^{-\frac{5}{4}-\frac{m}{2}}\|\mbox{\boldmath$u$}\|^{2}_{L^{2}}$
(see [24, 25]), we have

$\begin{split}\int_{t_{0}}^{\frac{t}{2}}\|e^{c\Delta(t-s)}D^{m}\left[\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$}\right]\|_{L^{2}(\mathbb{R}^{3})}ds\leq
c^{-(\frac{m}{2}+\frac{5}{4})}\int_{t_{0}}^{\frac{t}{2}}(t-s)^{-(\frac{m}{2}+\frac{5}{4})}\|\mbox{\boldmath$u$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds.\end{split}$

We divide the analysis of the last integral into three cases depending on the
value of $\alpha$:

Case 1: $\alpha\in(0,1/2)$. Use that
$\|\mbox{\boldmath$u$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}<(\lambda_{0}(\alpha)+\epsilon)^{2}s^{-2\alpha}$
to get

$\begin{split}t^{\alpha+\beta+m/2}\int_{t_{0}}^{t/2}\|e^{c\,\Delta(t-s)}D^{m}\left[\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$}\right]\|_{L^{2}(\mathbb{R}^{3})}ds\\\
\leq
c^{-(\frac{m}{2}+\frac{5}{4})}\,(\lambda_{0}(\alpha)+\epsilon)^{2}t^{2\alpha+1/4+m/2}\int_{t_{0}}^{t/2}(t-s)^{-5/4-m/2}s^{-2\alpha}ds\\\
\leq
c^{-(\frac{m}{2}+\frac{5}{4})}\,(\lambda_{0}(\alpha)+\epsilon)^{2}2^{2\alpha+1/4+m/2}=c^{-(\frac{m}{2}+\frac{5}{4})}\,(\lambda_{0}(\alpha)+\epsilon)^{2}\,2^{\alpha+\beta+m/2}\end{split}.$

Case 2: $\alpha\in(1/2,1)$. Use that
$\|\mbox{\boldmath$u$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}<\|\mbox{\boldmath$z$}_{0}\|(\lambda_{0}(\alpha)+\epsilon)s^{-\alpha}$
to get

$\begin{split}t^{\alpha+\beta+m/2}\int_{t_{0}}^{t/2}\|e^{c\,\Delta(t-s)}D^{m}\left[\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$}\right]\|_{L^{2}(\mathbb{R}^{3})}ds\\\
\leq
c^{-(\frac{m}{2}+\frac{5}{4})}\,\|\mbox{\boldmath$z$}_{0}\|(\lambda_{0}(\alpha)+\epsilon)t^{\alpha+1/4+m/2}\int_{t_{0}}^{t/2}(t-s)^{-5/4-m/2}s^{-\alpha}ds\\\
\leq
c^{-(\frac{m}{2}+\frac{5}{4})}\,\|\mbox{\boldmath$z$}_{0}\|(\lambda_{0}(\alpha)+\epsilon)2^{\alpha+\beta+m/2}.\end{split}$

Case 3: $\alpha\in(1,5/4)$. Using the fact that $\alpha>1$ implies
$\int_{0}^{\infty}\|\mbox{\boldmath$u$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}ds<\infty$,
we get

$\begin{split}t^{\alpha+\beta+m/2}\int_{t_{0}}^{t/2}\|e^{c\,\Delta(t-s)}D^{m}\left[\mbox{\boldmath$u$}\cdot\nabla\mbox{\boldmath$u$}\right]\|_{L^{2}(\mathbb{R}^{3})}ds\\\
\leq
c^{-(\frac{m}{2}+\frac{5}{4})}\,\|\mbox{\boldmath$z$}_{0}\|t^{5/4+m/2}\int_{t_{0}}^{t/2}(t-s)^{-5/4-m/2}\|\mbox{\boldmath$u$}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}ds\\\
\leq
c^{-(\frac{m}{2}+\frac{5}{4})}\,\|\mbox{\boldmath$z$}_{0}\|2^{5/4+m/2}=c^{-(\frac{m}{2}+\frac{5}{4})}\,\|\mbox{\boldmath$z$}_{0}\|2^{\alpha+\beta+m/2}.\end{split}$

This concludes the proof of inequality (1.6a). ∎

###  4.2. Proof of inequality (1.6b)

Using Lemma 3, we get, for ${\mathcal{E}}_{\mbox{$\bf w$}}(\cdot,t)$, the
bound

(4.2)
$\begin{split}t^{\alpha+\beta+\frac{m+1}{2}}\|D^{m}{\mathcal{E}}_{\mbox{$\bf
w$}}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}\leq
t^{\alpha+\beta+\frac{m+1}{2}}e^{-2\chi(t-t_{0})}\|D^{m}e^{c\,\Delta(t-t_{0})}{\mathcal{E}}_{\mbox{$\bf
w$}}(\cdot,t_{0})\|_{L^{2}(\mathbb{R}^{3})}\\\
+t^{\alpha+\beta+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-2\chi(t-s)}\|e^{c\,\Delta(t-s)}D^{m}(\mbox{\boldmath$u$}\cdot\nabla\mbox{$\bf
w$})(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}ds\\\ +\chi
t^{\alpha+\beta+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-2\chi(t-s)}\|e^{c\,\Delta(t-s)}D^{m+1}{\mathcal{E}}_{\mbox{\boldmath$u$}}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}ds.\end{split}$

As in the proof of inequality (1.4b) (see bound (3.10)), all the terms in the
right hand side of inequality (4.2) tend to $0$ as $t$ tends to infinity
except for the last one. Hence, this is the term we will focus our attention.
By inequality (1.6a), one has

$\begin{split}\chi
t^{\alpha+\beta+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-2\chi(t-s)}\|e^{\gamma\Delta(t-s)}D^{m+1}{\mathcal{E}}_{\mbox{\boldmath$u$}}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}ds\\\
\leq\chi
t^{\alpha+\beta+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-2\chi(t-s)}\|D^{m+1}{\mathcal{E}}_{\mbox{\boldmath$u$}}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}ds\\\
\leq(C(\nu,\chi,\alpha,m+1)+\epsilon)\chi
t^{\alpha+\beta+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-2\chi(t-s)}s^{-\alpha-\beta-\frac{m+1}{2}}ds\\\
=(C(\nu,\chi,\alpha,m+1)+\epsilon)\chi
t^{\alpha+\beta+\frac{m+1}{2}}\bigg{[}\int_{t_{0}}^{\sigma
t}e^{-2\chi(t-s)}s^{-\alpha-\beta-\frac{m+1}{2}}ds\\\ +\int_{\sigma
t}^{t}e^{-2\chi(t-s)}s^{-\alpha-\beta-\frac{m+1}{2}}ds\bigg{]}\\\
\leq(C(\nu,\chi,\alpha,m+1)+\epsilon)\chi
t^{\alpha+\beta+\frac{m+1}{2}}\bigg{[}e^{-2\chi
t(1-\sigma)}\int_{t_{0}}^{{\sigma}t}s^{-\alpha-\beta-\frac{m+1}{2}}ds\\\
+({\sigma}t)^{-\alpha-\beta-\frac{m+1}{2}}\int_{{\sigma}t}^{t}e^{-2\chi(t-s)}ds\bigg{]}\\\
\leq(C(\nu,\chi,\alpha,m+1)+\epsilon)\chi\bigg{[}t^{\alpha+\beta+\frac{m+1}{2}}e^{-2\chi
t(1-\sigma)}\bigg{(}2\frac{t_{0}^{-\alpha-\beta-\frac{m-1}{2}}-({\sigma}t)^{-\alpha-\beta-\frac{m-1}{2}}}{2\alpha+2\beta+m-1}\bigg{)}\\\
+\sigma^{-\alpha-\beta-\frac{m+1}{2}}\bigg{(}\frac{1-e^{-2\chi
t(1-\sigma)}}{2\chi}\bigg{)}\bigg{]}.\end{split}$

Letting, in this order, $\epsilon\to 0^{+}$, taking the $\limsup$ as
$t\to\infty$ and letting ${\sigma}\to 1^{-}$, we get

$\limsup_{t\rightarrow\infty}t^{\alpha+\beta+\frac{m+1}{2}}\int_{t_{0}}^{t}e^{-2\chi(t-s)}\|e^{\gamma\Delta(t-s)}D^{m+1}{\mathcal{E}}_{\mbox{\boldmath$u$}}(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}ds\leq\frac{1}{2}C(\nu,\chi,\alpha,m+1).$

which leads to the desired inequality (1.6b). ∎

##  5\. Additional comments and the fundamental inequality for other related
systems

We present some direct consequences and extensions of the results previous
results. We also apply the previous arguments to quickly obtain results for
other related systems.

###  5.1. Direct consequences and extensions

From the results above, we can immediately get, through standard
interpolations, some general inequalities. Precisely,

$\limsup_{t\to\infty}t^{\alpha+\frac{s}{2}}\|\mbox{\boldmath$z$}\|_{\dot{H}^{s}(\mathbb{R}^{3})}\leq\min_{m>s}\,[\,K_{\alpha,m}\,]^{\frac{s}{m}}\nu^{-\frac{m}{s}}\lambda_{0}(\alpha),$
$\limsup_{t\to\infty}t^{\alpha+\frac{m}{2}+\frac{3}{4}}\|D^{m}\mbox{\boldmath$z$}\|_{L^{\infty}(\mathbb{R}^{3})}\leq(K_{\alpha,m}\,\nu^{-\frac{m}{2}})^{1/4}(K_{\alpha,m+2}\,\nu^{-\frac{m}{2}-1})^{3/4}\,\lambda_{0}(\alpha),$

and

$\limsup_{t\to\infty}t^{\alpha+\frac{m}{2}+\frac{3(p-2)}{4p}}\|D^{m}\mbox{\boldmath$z$}\|_{L^{p}(\mathbb{R}^{3})}\leq
C_{\nu,m,\alpha,p}\,\lambda_{0}(\alpha)\mbox{,\,\, for all }2\leq p<\infty.$

######  Remark 5.1.

Similar versions of the inequalities above can be obtained for
${\mathcal{E}}_{\mbox{\boldmath$z$}}$ and ${\mathcal{E}}_{\mbox{$\bf w$}}$. It
is also important to note that the main theorems 1.1 and 1.3 imply the
following interesting decays:

(5.1)
$\|D_{t}^{k}D_{x}^{m}\mbox{\boldmath$z$}(\cdot,t)\|_{L^{p}(\mathbb{R}^{3})}=O(t^{-\alpha-k-\frac{m}{2}-\frac{3(p-2)}{4p}}),$
$\|D_{t}^{k}D_{x}^{m}\mbox{$\bf
w$}(\cdot,t)\|_{L^{p}(\mathbb{R}^{3})}=O(t^{-\alpha-k-\frac{m+1}{2}-\frac{3(p-2)}{4p}}),$
$\|D_{t}^{k}D_{x}^{m}{\mathcal{E}}_{\mbox{\boldmath$z$}}(\cdot,t)\|_{L^{p}(\mathbb{R}^{3})}=O(t^{-\alpha-\beta-k-\frac{m}{2}-\frac{3(p-2)}{4p}}),$
$\|D_{t}^{k}D_{x}^{m}{\mathcal{E}}_{\mbox{$\bf
w$}}(\cdot,t)\|_{L^{p}(\mathbb{R}^{3})}=O(t^{-\alpha-\beta-k-\frac{m+1}{2}-\frac{3(p-2)}{4p}}),$
where $2\leq p\leq\infty$ and $\beta$ is as in Theorem 1.3.

###  5.2. Other Systems

As mentioned in the introduction, the technique presented can be used to
obtain similar estimates for other dissipative type problems. Let us now give
some examples.

####  5.2.1. Dissipative linear system

Consider

(5.2) $v_{t}=\mathbb{L}v$

where $\mathbb{L}:\mathbb{R}^{3}\to(L^{2}(\mathbb{R}^{3}))^{3}$ is a
pseudodifferential operator with symbol $M(\xi)$ such that

$M(\xi)=P^{-1}(\xi)D(\xi)P(\xi),\mbox{ \,\,\,\,}\xi-a.e.,$

where $P(\xi)\in O(3)$ and $D(\xi)_{ij}=-c_{i}|\xi|^{2\gamma}\delta_{ij}$, for
$c_{i}>c>0$ and $0<\gamma\leq 1$. For a study on the decay rates of solutions
for these types of equations involving the decay character, see [3, 1, 34]. In
fact, this approach assures that condition $\lambda_{0}(\alpha)<\infty$
(condition (1.3)) for certain $\alpha>0$ depending on the initial data.

Let $\Lambda=(-\Delta)^{1/2}$. Applying $\Lambda^{m\gamma}$ in equation (5.2)
and taking the dot product with
$(t-t_{0})^{2\alpha+m+\delta}\Lambda^{m\gamma}v$, we get

(5.3)
$\begin{split}\frac{1}{2}(t-t_{0})^{2\alpha+m+\delta}\dfrac{d}{dt}\|v(t)\|^{2}_{\dot{H}^{m\gamma}}=(t-t_{0})^{2\alpha+m+\delta}\langle|\xi|^{m\gamma}\hat{v},M|\xi|^{m\gamma}\hat{v}\rangle_{L^{2}}\\\
=-(t-t_{0})^{2\alpha+m+\delta}\langle(-D)^{1/2}P|\xi|^{m\gamma}\hat{v},(-D)^{1/2}P|\xi|^{m\gamma}\hat{v}\rangle_{L^{2}}\\\
=-(t-t_{0})^{2\alpha+m+\delta}\int_{\mathbb{R}^{n}}|(-D)^{1/2}P|\xi|^{m\gamma}\hat{v}|^{2}d\xi\\\
\leq-C(t-t_{0})^{2\alpha+m+\delta}\int_{\mathbb{R}^{n}}|\xi|^{2(m+1)\gamma}|\hat{v}|^{2}d\xi=-C(t-t_{0})^{2\alpha+m+\delta}\|v(t)\|^{2}_{\dot{H}^{(m+1)\gamma}}.\end{split}$

Therefore,

(5.4)
$\begin{split}(t-t_{0})^{2\alpha+m+\delta}\|v(t)\|^{2}_{\dot{H}^{m\gamma}}+2C\int_{t_{0}}^{t}(t-t_{0})^{2\alpha+m+\delta}\|v(s)\|^{2}_{\dot{H}^{(m+1)\gamma}}ds\\\
\leq(2\alpha+m+\delta)\int_{t_{0}}^{t}(t-t_{0})^{2\alpha+m-1+\delta}\|v(s)\|^{2}_{\dot{H}^{m\gamma}}ds.\end{split}$

Using the same induction argument as in the proof of Theorem 1.1, one has

(5.5)
$\limsup_{t\rightarrow\infty}t^{\alpha+m/2}\|v(t)\|^{2}_{\dot{H}^{m\gamma}}\leq
K_{\alpha,m}C^{-m/2}\limsup_{t\rightarrow\infty}t^{\alpha}\|v(t)\|_{L^{2}(\mathbb{R}^{3})}$

where

(5.6)
$K_{\alpha,m}=\min_{\delta>0}\Bigg{\\{}\delta^{-1/2}\prod_{j=0}^{m}(\alpha+\frac{j}{2}+\delta)^{1/2}\Bigg{\\}}.$

####  5.2.2. Tropical climate model equations

Consider the system

(5.7a)
$\mbox{\boldmath$u$}_{t}+(\mbox{\boldmath$u$}\cdot\nabla)\mbox{\boldmath$u$}-\mu\Delta\mbox{\boldmath$u$}+\nabla
p+\nabla\cdot(\mbox{\boldmath$v$}\otimes\mbox{\boldmath$v$})=0,$ (5.7b)
$\mbox{\boldmath$v$}_{t}+(\mbox{\boldmath$u$}\cdot\nabla)\mbox{\boldmath$v$}-\nu\Delta\mbox{\boldmath$v$}+\nabla\theta+(\mbox{\boldmath$v$}\cdot\nabla)\mbox{\boldmath$u$}=0,$
(5.7c)
$\theta_{t}+(\mbox{\boldmath$u$}\cdot\nabla)\theta-\eta\Delta\theta+\nabla\cdot\mbox{\boldmath$v$}=0,$
(5.7d) $\nabla\cdot\mbox{\boldmath$u$}=0,$

where $\mbox{\boldmath$u$}=(u_{1}(x,t),u_{2}(x,t),u_{3}(x,t))$ and
$\mbox{\boldmath$v$}=(v_{1}(x,t),v_{2}(x,t),v_{3}(x,t))$ are the barotropic
mode and the first baroclinic mode of the velocity, respectively;
$\theta=\theta(x,t)$ and $p=p(x,t)$ represent the scalar temperature and
scalar pressure, respectively. The coefficients $\mu$, $\nu$, and $\eta$ are
nonnegative. System (5.7) was first introduced in [16] without any dissipation
terms ($\mu=\nu=\eta=0$). For a study on its regularity and well-posedness,
see [28, 10, 11, 9, 8, 43, 27].

Following the same techniques presented in the proof of Theorem 1.1 one can
obtain the inequality

(5.8)
$\begin{split}(t-t_{0})^{2\alpha+m+\delta}\|(D^{m}\mbox{\boldmath$u$},D^{m}\mbox{\boldmath$v$},D^{m}\theta)(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}^{2}+\\\
(2-\epsilon)\min{(\mu,\nu,\eta)}\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|(D^{m+1}\mbox{\boldmath$u$},D^{m+1}\mbox{\boldmath$v$},D^{m+1}\theta)(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2}ds\\\
\leq(2\alpha+m+\delta)\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|(D^{m}\mbox{\boldmath$u$},D^{m}\mbox{\boldmath$v$},D^{m}\theta)(\cdot,s)\|_{L^{2}(\mathbb{R}^{3})}^{2},\end{split}$

for each integer $m\geq 0$. Through the same induction argument as in the
proof of Theorem 1.1, one gets

(5.9)
$\begin{split}\limsup_{t\to\infty}\,t^{\alpha+\frac{m}{2}}\|(D^{m}\mbox{\boldmath$u$},D^{m}\mbox{\boldmath$v$},D^{m}\theta)(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}\\\
\leq
K_{\alpha,m}(\min\\{\mu,\nu,\eta\\})^{-m/2}\;\;\limsup_{t\to\infty}\,t^{\alpha}\|(\mbox{\boldmath$u$},\mbox{\boldmath$v$},\theta)(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})},\end{split}$

where

(5.10)
$K_{\alpha,m}=\min_{\delta>0}\Bigg{\\{}\delta^{-1/2}\prod_{j=0}^{m}(\alpha+\frac{j}{2}+\delta)^{1/2}\Bigg{\\}}.$

####  5.2.3. Navier-stokes with rotational framework

Consider the system

(5.11)
$\begin{split}\mbox{\boldmath$u$}_{t}+(\mbox{\boldmath$u$}\cdot\nabla)\mbox{\boldmath$u$}+\omega\mathbb{J}\mbox{\boldmath$u$}+\nabla
p=&\mu\Delta\mbox{\boldmath$u$}\mbox{ \,\,\,\, in
}\mathbb{R}^{3}\times(0,\infty),\\\ \nabla\cdot\mbox{\boldmath$u$}=&0\mbox{
\,\,\,\,\,\,\,\,\,\,\,\,\,\, in }\mathbb{R}^{3}\times(0,\infty),\\\
\mbox{\boldmath$u$}(\cdot,0)=&\mbox{\boldmath$u$}_{0}\in
L^{2}_{\sigma}(\mathbb{R}^{3}),\end{split}$

where $\omega\in\mathbb{R}$ is the Coriolis parameter and
$\mathbb{J}\mbox{\boldmath$u$}:={\bf e}_{3}\wedge\mbox{\boldmath$u$}$ is the
Coriolis force acting on the fluid. Observe that
$\mathbb{J}u=(-u_{2},u_{1},0)$ and therefore this term does not affect the
energy inequality obtained for $u$. Following the same procedure as in the
proof of Theorem 1.1, we get

(5.12)
$\begin{split}(t-t_{0})^{2\alpha+m+\delta}\|D^{m}\mbox{\boldmath$u$}(\cdot,t)\|^{2}_{L^{2}(\mathbb{R}^{3})}+(2-\epsilon)\nu\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m+\delta}\|D^{m+1}\mbox{\boldmath$u$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})}ds\\\
\leq(2\alpha+m+\delta)\int_{t_{0}}^{t}(s-t_{0})^{2\alpha+m-1+\delta}\|D^{m}\mbox{\boldmath$u$}(\cdot,s)\|^{2}_{L^{2}(\mathbb{R}^{3})},\end{split}$

for each integer $m\geq 0$. Using the same induction argument as in the proof
of Theorem 1.1, one has

(5.13)
$\begin{split}\limsup_{t\to\infty}\,t^{\alpha+\frac{m}{2}}\|D^{m}\mbox{\boldmath$u$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}\leq
K_{\alpha,m}\mu^{-m/2}\;\;\limsup_{t\to\infty}\,t^{\alpha}\|\mbox{\boldmath$u$}(\cdot,t)\|_{L^{2}(\mathbb{R}^{3})}\end{split}$

where

(5.14)
$K_{\alpha,m}=\min_{\delta>0}\Bigg{\\{}\delta^{-1/2}\prod_{j=0}^{m}(\alpha+\frac{j}{2}+\delta)^{1/2}\Bigg{\\}}.$

For the existence of a global mild solution assuming small data in $H^{1/2}$,
see [23]. The equations (5.11) are also related to the Navier-Stokes flow past
rotating obstacles (see e.g. [17]).

####  5.2.4. Magneto-micropolar equations

When the magnetic field is taken into account in system (1.1), we get the so
called magneto-micropolar system

(5.15) $\left\\{\begin{aligned}
\partial_{t}\mbox{\boldmath$u$}+(\mbox{\boldmath$u$}\cdot\nabla)\mbox{\boldmath$u$}+\nabla
p&=(\mu+\chi)\Delta\mbox{\boldmath$u$}+\chi\nabla\times\mbox{$\bf
w$}+(\mbox{\boldmath$b$}\cdot\nabla)\mbox{\boldmath$b$},\\\
\partial_{t}\mbox{$\bf w$}+(\mbox{\boldmath$u$}\cdot\nabla)\mbox{$\bf
w$}&=\gamma\Delta\mbox{$\bf w$}+\nabla(\nabla\cdot\mbox{$\bf
w$})+\chi\nabla\times\mbox{\boldmath$u$}-2\chi\mbox{$\bf w$},\\\
\partial_{t}\mbox{\boldmath$b$}+(\mbox{\boldmath$u$}\cdot\nabla)\mbox{\boldmath$b$}&=\nu\Delta\mbox{\boldmath$b$}+(\mbox{\boldmath$b$}\cdot\nabla)\mbox{\boldmath$u$},\\\
\nabla\cdot\mbox{\boldmath$u$}(\cdot,t)&=\nabla\cdot\mbox{\boldmath$b$}(\cdot,t)\,=0,\end{aligned}\right.$

with initial data $\mbox{\boldmath$z$}_{0}=(\>\\!\mbox{\boldmath$u$}_{0},{\bf
w}_{0},\mbox{\boldmath$b$}_{0})\in
L^{2}_{\sigma}(\mathbb{R}^{3})\\!\times\\!L^{2}(\mathbb{R}^{3})\times
L^{2}_{\sigma}(\mathbb{R}^{3})$. The results on eventual smoothness, existence
and uniqueness for the micropolar fluids system also hold for the magneto-
micropolar system (5.15). One can derive results completely analgous to
Theorems 1.1 and 1.3, where, in this case,
$\mbox{\boldmath$z$}(\cdot,t):=(\mbox{\boldmath$u$},\mbox{$\bf
w$},\mbox{\boldmath$b$})(\cdot,t)$.

####  5.2.5. Other Parabolic Systems

Let $f_{j}:\mathbb{R}^{n}\to\mathbb{R}^{n}$, $n=2,3,4$, $1\leq j\leq N$,
denote smooth vector fields with symmetric Jacobian $A_{j}(u)=D_{u}f_{j}(u)$.
Assume $|f_{j}(u)|\leq C|u|^{2}$. Consider the parabolic system

(5.16) $u_{t}+\sum_{j=1}^{N}A_{j}(u)D_{j}u+D_{j}(A_{j}(u)u)=Pu$

where the first order terms have anti-symmetric form and the linear constant
coefficient operator $P$ satisfies

$P=\sum_{j=1}^{N}R_{j}D_{j}+\sum_{i,j=1}^{N}V_{ij}D_{i}D_{j},$
$\hat{P}(k)+\hat{P}^{*}(k)\leq-c|k|^{2}I,$

for some $c>0$. Since

$\int u^{T}D_{j}(A_{j}(u)u)dx=-\int(D_{j}u^{T})A_{j}(u)udx$

and $A_{j}=A^{T}_{j}$, an analogous reasoning as used in Theorem 1.1 can be
used to show that

$\limsup_{t\to\infty}t^{\alpha+m/2}\|D^{m}u(\cdot,t)\|_{L^{2}(\mathbb{R}^{n})}\leq\,C\lambda_{0}(\alpha),$

where
$\lambda_{0}(\alpha):=\limsup_{t\to\infty}t^{\alpha}\|u(\cdot,t)\|_{L^{2}(\mathbb{R}^{n})}$.

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