\section{Introduction}

In complex many-body systems, initially localized information quickly spreads throughout the entire system --- a process known as information scrambling. Though information is ultimately conserved, it gets encoded into global entanglement among many degrees of freedom, and hence becomes inaccessible by local measurement. Information scrambling was originally studied in the context of black hole physics \cite{Hayden2007Black,Kitaev2015,Maldacena2016Bound}, and has since emerged as a field with a wide-ranging impact across different areas in physics, e.g., quantum chaos in many-body systems \cite{Rozenbaum2017Lyapunov,Roberts2017Chaos,Cotler2017Chaos,Lin2018Out,Garciaa2018Chaos,Yan2020Information,Yan2020Recovery}, phase transition \cite{Sahu2019Scrambling,Choi2020Quantum}, and quantum machine learning \cite{Holmes2021Barren}. Considerable effort has also been made in probing this effect in various experimental systems \cite{Landsman2019Verified,Li2017Measuring,Garttner2017Measuring}.

Information scrambling is usually measured by the temporal decay of the out-of-time order correlators (OTOCs) \cite{Kitaev2015,Larkin1969,Swingle2018Unscrambling}, defined as
\begin{equation}
    \langle W^\dag(t)V^\dag W(t)V \rangle,
\end{equation}
where the average is taken over a given quantum state. $W$ and $V$ are local operators, usually considered to act on distinct subsystems. $W(t)$ is the Heisenberg evolution of $W$, which becomes a global operator as scrambling proceeds, causing decay of the OTOCs.

However, it is difficult to distinguish between scrambling and decoherence: the latter causes leakage of the system information to the environment and, in general, induces decay of the OTOC as well. Protocols that measure the OTOC often involve forward and backward evolution of the system, which in practice do not exactly match each other due to operational errors. This can also cause decay of the OTOC signals. Understanding the behavior of scrambling in presence of decoherence and operational errors is an active and ongoing research in the field \cite{Swingle2018Resilience,Knap2018Entanglement,Syzranov2018Out,Zhang2019Information,Gonzalez019Out,Yoshida2019Disentangling,Joshi2020Quantum,Touil2021Information,Dominguez2021Dynamics,Zanardi2021Information}. In this line of effect, the first experiment that provided a positive verification of scrambling was performed with trapped ions \cite{Landsman2019Verified}, using a teleportation-based protocol \cite{Yoshida2019Disentangling}. This approach uses two copies of the system and an entangled input state (Bell state) between the copies, requiring sophisticated engineering of the system and therefore hindering its practical applications. More recently, the field has witnessed an increasing number of studies of information scrambling in various experimental platforms, e.g., superconductors \cite{Zhao2021Probing}, trapped-ions \cite{Joshi2020Quantum}, and cloud-based quantum computers \cite{Braumuller2021Probing,Mi2021Information,Geller2021Quantum}. However, it remains a challenge to design a simple and robust protocol for benchmarking the true signals of scrambling from a noisy background. Thus a solution to this problem is in demand and is beneficial for the development of the field, especially for experimental studies.

In this work, we propose a solution to this task. Our approach is based on a novel quantum butterfly effect \cite{Yan2020Recovery,Sinitsyn2020Quantum} that can unambiguously distinguish scrambling and non-scrambling dynamics. Due to the global entanglement generated by scrambling, information becomes more robust against local perturbations, and hence, when scrambled and disturbed by local perturbations, can still be partially recovered through a reversed unscrambling process. In contrast, decoherence and errors only temper this effect and weaken the signal produced by true scrambling. In the following, we develop the underlying theory, introduce the benchmarking protocol, and apply it to a model problem of fast scrambling in the presence of decoherence.

\begin{figure}[b!]
    \centering
    \includegraphics[width=\columnwidth]{fig_butterfly.pdf}
    \caption{In the Lorenz picture of the butterfly effect, one compares two trajectories evolved under the same Hamiltonian, but with slightly different initial conditions. In the Bradbury picture, perturbation is applied in the past. These two pictures are equivalent in the classical world but have subtle differences in quantum dynamics.}
    \label{fig:butterfly}
\end{figure}

In chaotic classical dynamics, small perturbations in the initial conditions can trigger dramatic changes in the time evolution. This effect is well-known as the Lorenz butterfly effect. A few years before Lorenz, Bradbury in his famous story \cite{bradbury1952} introduced a variant of the butterfly effect. In this picture, one starts with an initial state in the present, evolves backward in time, applies a small perturbation, and then travels forward back to the present.

These two pictures offer equivalent descriptions of the same classical butterfly effect (Fig.~\ref{fig:butterfly}). However, they exhibit subtle differences in quantum dynamics \cite{Yan2020Recovery,Cao2021Origin}: the overlap between the two trajectories (wavefunctions) in the Lorenz picture remains a constant during time evolution, due to the unitarity of (isolated) quantum dynamics. On the other hand, in the Bradbury's picture, the overlap between the two states at time $t=0$ --- the initial input state and the final output state after the backward and forward evolution loop --- does decay as a function of the evolution time. Moreover, asymptotically, the output state always contains partial information of the initial one, with the amount of information determined by the type of perturbation. This is in sharp contrast with classical chaotic dynamics, which on average smear all the initial information over the entire accessible phase space. For these reasons, we propose to call this phenomenon the quantum anti-butterfly effect.

More precisely, the perturbation in the `past' can be described by a general quantum channel $\Lambda$:
\begin{equation}\label{eq:perturbation}
   \rho \rightarrow \Lambda(\rho) = \sum_k M^\dagger_k \rho M_k,\quad \sum_k M_kM_k^\dagger = \mathbb{I},
\end{equation}
where $\mathbb{I}$ is the identity operator, and $M_k$ are the Kraus operators. The initial state $\rho$, after the Bradbury's process aforementioned, becomes
\begin{equation}\label{eq:outstate}
    \rho(t) = U_t\Lambda(U^\dag_t \rho U_t)U^\dag_t,
\end{equation}
where $U_t$ is the evolution operator for a time $t$. This process is also recognized as the quantum twirling channel of the perturbation $\Lambda$ \cite{Dankert2009Exact}.

The fidelity of the output state of the twirling channel, with respect to the initial state, is
\begin{equation}\label{eq:fidelity}
    F(t)\equiv {\rm Tr}\left[\rho(t)\cdot\rho\right] = \sum_k {\rm Tr} \left[M^\dag_k(t)\rho M_k(t)\rho\right].
\end{equation}
This quantity is the central object for the protocol developed in the following section.

After a long time evolution with a chaotic Hamiltonian, when the evolution operator becomes sufficiently random, the asymptotic state can on average be described as
\begin{equation}\label{eq:asymstate}
    \rho_{\rm as} = p\rho + (1-p)\frac{\mathbb{I}}{d},
\end{equation}
where $d$ the dimension of the total Hilbert space and the probability $p$ is determined by the error channel, namely,
\begin{equation}\label{eq:probasym}
p = \frac{\sum_k {\rm Tr} M_k {\rm Tr}M^\dagger_k-1}{d^2 - 1}
\end{equation}
The corresponding asymptotic value of the fidelity (\ref{eq:fidelity}) has the simple expression
\begin{equation}\label{eq:asymfidelity}
    F_{\rm as} = p{\rm Tr}\left[\rho^2\right] + (1-p)/d.
\end{equation}

At this point, we would like to emphasize several remarks:\par
i) The asymptotic state (\ref{eq:asymstate}) is obtained by averaging over an assemble of random unitaries with respect to Haar measure \footnote{Technically, the same result can be obtained for an ensemble of unitaries which is as random as a unitary 2-design, since the average is performed for an expression in terms of the second moment of the unitary.}. However, the fluctuation from the mean is exponentially small in the size of the total system. Consequently, for a single unitary randomly drawn from such an ensemble, deviations from this averaged behavior are exponentially suppressed. Therefore, we can treat (\ref{eq:asymstate}) as the asymptotic state for a typical complex evolution. \par
ii) The fidelity (\ref{eq:fidelity}) is identified as a sum of special types of OTOCs between the state $\rho$ and the Kraus operators. Hence, scrambling causes decay of the fidelity in the same manner as to the OTOCs. This particular type of OTOC has been used \cite{Garttner2017Measuring} and demonstrated to have various benefits \cite{Swan2019Unifying}. As will be shown in the following, this quantity can also acquire a large universal asymptotic value (\ref{eq:asymfidelity}), which will response further to decoherence and errors. This unique feather lies in the core of our protocol for singling out information scrambling.

Inspired by the above theory of the anti-butterfly effect, we propose the following protocol to detect and benchmark information scrambling. In this protocol, the total system is sent through a Bradbury process, and only the fidelity of a subsystem is measured:

\begin{enumerate}
    \item[1.] Initialize the total system such that a small subsystem is prepared in state $\rho_S$.
    \item[2.] Evolve the system forward for a time $t$; apply a perturbation, and then evolve the system backward for the same time $t$.
    \item[3.] Measure the same subsystem, and evaluate its fidelity with respect to the initial state $\rho_S$.
\end{enumerate}

We choose to measure the fidelity of a subsystem, since evaluating that of the total system is generally infeasible for large systems. One can verify that the asymptotic value of the fidelity (\ref{eq:asymfidelity}) also applies to any subsystems.

For example, for a (large) spin-$1/2$ system with scrambling dynamics, we prepare a single target spin in any pure state and choose the perturbation channel as a projective measurement on any single spin. The fidelity of the target spin, computed with (\ref{eq:asymfidelity}) and (\ref{eq:probasym}), saturates to $\sim 0.75$. On the other hand, in presence of decoherence and errors, the asymptotic state of the target spin will become random, hence resulting in $\sim 0.5$ fidelity. If we chooses to measure the fidelity of the total system, the corresponding values of the asymptotic fidelity would be $\sim 0.5$ and zero for ideal scrambling and decoherence respectively.

We also note the similarity between our approach and randomized benchmarking (RB) protocol \cite{Emerson2005Scalable,Magesan2011Scalable} for quantum computers. Indeed, our approach inherits the main benefits of RB, i.e., it is scalable and is independent with errors in state initialization and readout.

In practice, subsystem state initialization and measurement may not be directly accessible. For these systems, the corresponding steps in the protocol can be adapted as
\begin{enumerate}
    \item[1'.] Measure a local observable with a discrete set of outcomes.
    \item[3'.] Measure the same observable, and evaluate the coincide rate, i.e., the probability that the two measurements produce the same outcome.
\end{enumerate}
It can be verified that the coincidence rate is identical to the fidelity of the same subsystem on which the measurements are performed.

\begin{figure}[t!]
    \centering
    \includegraphics[width=\columnwidth]{fig_otoc.pdf}
    \caption{Measurements of the OTOC. (a) Quantum circuit for the interference protocol (left). Structure of a single evolution step of the fast scrambling model. (b) OTOC for the cases of scrambling and decoherence (Table~\ref{table}), which produces similar signals that can hardly be distinguished.}
    \label{fig:otoc}
\end{figure}

As an illustration, we study a fast scrambling model recently proposed in Ref.~\cite{Belyansky2020Minimal}. We perform simulations of this model under decoherence and compare our protocol with a standard OTOC measurement.

The evolution unitary in the fast scrambling model consists of repetitive layers of circuit evolution. Each layer (Fig.~\ref{fig:otoc}a, right) is composed of Haar random single-qubit unitaries, immediately followed by a global entangling gate, i.e., which for $n$ qubits is given by
\begin{equation}
    \exp{\left(-i\frac{g}{2\sqrt{n}}\sum_{i<j}Z_iZ_j\right)},
\end{equation}
where $g$ is a constant parameter that controls the scrambling strength. This building block can be viewed as a trotterization of the evolution generated by a spin Hamiltonian with strong random local fields (generating single-gate random rotations) and all-to-all two-body couplings (creating the global entangling gate).

To simulate the effect of a noisy environment, we introduce errors in each layer of the single-qubit gates --- after each single-qubit Haar random gate, a Pauli $X$-gate and a Pauli $Z$-gate are applied independently with probability $p$. Note that this error channel of the system qubits can be extended to a unitary evolution in an enlarged Hilbert space including ancillary qubits. Hence, the error model describes a decoherence process as well. We now have two parameters, $g$ and $p$, that control the strength of scrambling and decoherence, respectively. Table~\ref{table} lists the parameters we considered, namely, two cases for ideal scrambling (S1 and S2), two cases for strong decoherence (D1 and D2), and the cases (I) where scrambling and decoherence become comparable and compete with each other.

\begin{table}[h]
\caption{\label{table}Cases compared for the fast scrambling model. The rate of scrambling increases with the value of $g$. The strength of the decoherence increase with the value of $p$.}
\begin{ruledtabular}
\begin{tabular}{ c c c c c c c }
 cases & S1 & S2 & D1 & D2 & I\\
 \hline
 $g$  & 1 & 2 & 0.5 & 0.5 & 1\\

 $p$  &  0 & 0 & 0.025 & 0.1 & 0.001 \\
\end{tabular}
\end{ruledtabular}
\end{table}

The OTOC measurement is achieved using the interference protocol developed in Ref.~\cite{Swingle2016Measuring}, as shown in the quantum circuit diagram in Fig.~\ref{fig:otoc}a. The intermediate $X$-gate is placed on the $i$-th qubit $q_i$. The final measurement of the average value $\langle\sigma^0_x\rangle$ (or $\langle\sigma^0_y\rangle$) on the ancillary qubit $q_0$ then gives the real (or imaginary) part of the OTOC $\langle \sigma^0_x(t)\sigma^i_x\sigma^0_x(t)\sigma^i_x\rangle$, where $t$, in analog to time, is the number of layers in the forward (and hence the backward) evolution.

\begin{figure*}[t!]
    \centering
    \includegraphics[width=\textwidth]{fig_recovery.pdf}
    \caption{Measurement of the recovery signal (fidelity) of the benchmark protocol. (a) Schematic of the quantum circuit. (b) Fidelity decay for the cases of scrambling and decoherence (Table~\ref{table}). (c,~d) Fidelity as a function of the number of layers ($t_2$) in the backward evolution, for fixed numbers of layers $t_1$ in the forward evolution. (e,~g) Density plot for the fidelity.}
    \label{fig:recovery}
\end{figure*}

Figure~\ref{fig:otoc}b compares the evolution of the OTOC for ideal scrambling (without noise) and weak scrambling with strong decoherence, which exhibit similar decay curves. Hence, these two situations are practically indistinguishable from the OTOC measurements.

We now examine the performance of our benchmark protocol for the fast scrambling model under the same conditions. The circuit diagram for this protocol is shown in Fig.~\ref{fig:recovery}a. Here, all the qubits are prepared in the computational $|0\rangle$. The recovery signal is obtained by measuring the fidelity of the first qubit $q_1$ with respect to its initial state. We also specify the intermediate perturbation between the forward and backward evolution as a projective measurement on a single qubit (other than the qubit-1) along the $Z$-axis. Note that since the evolution unitary contains Haar-random single-qubit gates in each layer, the measurement in a fixed direction is equivalent to random projective measurements. As discussed in the foregoing section, for this particular perturbation channel, the expected value of fidelity (\ref{eq:fidelity}) is $0.75$ for ideal scrambling unitaries. On the other hand, in the case of strong decoherence, when the qubits eventually lose their coherence information, the fidelity would be trivially $0.5$. Hence, the cases of scrambling with and without decoherence exhibit distinct asymptotic values of fidelity. This is clearly demonstrated in Fig.~\ref{fig:recovery}b.

To vitalize the emergence of the recovery signal produced by scrambling, we also performed simulations with different numbers of layers in the forward and backward evolution. That is, the fidelity is measured after $t_1$ and $t_2$ layers of forward and backward evolution respectively, and if $t_2>t_1$, an additional $t_2-t_1$ layers in the backward evolution are chosen as independently random. Figures~\ref{fig:recovery}(c,~d) depict the fidelity as a function of $t_2$ for fixed values of $t_1$. The density plots for the fidelity scanned through various $t_1$ and $t_2$ are shown in Fig.~\ref{fig:recovery}(e,~f). The recovery signal emerges in a finite window around the peak $t_1=t_2$. The width of the peak reflects the time scale for local dissipation. For the current fast scrambling model, due to the random single-qubit rotation in every single layer, the recovery signal disappears as soon as $t_2$ is one layer away from $t_1$.

The above simulations demonstrate that our protocol can unambiguously distinguish between scrambling and decoherence. In general, both of these two competing factors contribute to the decay of the fidelity, but they make the fidelity saturate to different values. This gives rise to a two-stage decay: In the early scrambling stage, fidelity decay is influenced by both scrambling and decoherence, until the information is fully scrambled and the fidelity reaches the saturation value (\ref{eq:asymfidelity}). In the latter decoherence stage, fidelity further decays to a smaller value, at a rate determined purely by decoherence. The appearance of the two-stage decay indicates the presence of scrambling.

\begin{figure}[h!]
    \centering
    \includegraphics[width=\columnwidth]{fig_benchmark.pdf}
    \caption{Fidelity decay for case I (Table~\ref{table}) with both scrambling and decoherence. Squares are numerical data. The red solid curve is the best fit to the ansatz (\ref{eq:ansatz}). To visualize the two stages of the fidelity decay, we also plotted (black dashed) the long time pure exponential part of the ansatz (\ref{eq:ansatz}), $a_2\exp{(-\lambda_dt)}+F^d_{\rm as}$, which clearly departs from the early decay. Insets: $F(t)-F^d_{\rm as}$ in semi-log scale, which show a single exponential decay at long times, and a sum of two exponential decays at early times. At the very beginning, fidelity decay takes a quadratic form \cite{Yan2021Decoherence}.}
    \label{fig:benchmark}
\end{figure}

Suppose $\lambda_s$ and $\lambda_d$ are the exponential decay rates corresponding to scrambling and decoherence, respectively. In the extreme cases of strong scrambling and strong decoherence, the decay of the fidelity can be described as
\begin{equation}
  F(t) =
    \begin{cases}
      \left(1-F^s_{\rm as}\right) e^{-\lambda_s t} + F^s_{\rm as}, & \lambda_s \gg \lambda_d\\
     \left(1-F^d_{\rm as}\right) e^{-\lambda_d t} + F^d_{\rm as},  & \lambda_s \ll \lambda_d
    \end{cases}
\end{equation}
where $F^s_{\rm as}$ and $F^d_{\rm as}$ are the asymptotic values of the fidelity induced by scrambling and decoherence, respectively.  In the intermediate regime, such that the scrambling and decoherence are comparable, we propose an ansatz of the fidelity decay:
\begin{equation}\label{eq:ansatz}
    F(t)=\left(a_1 e^{-\lambda_s t} + a_2 \right)e^{-\lambda_d t} + F^d_{\rm as}.
\end{equation}
With this, we can fit the observed fidelity at long times (when the system gets sufficiently scrambled and the first exponential term vanishes) to a pure exponential function and extract the decoherence rate $\lambda_d$. Then the scrambling rate $\lambda_s$ can be extracted by fitting the early time fidelity to the ansatz (\ref{eq:ansatz}). We applied this procedure to case I (Table~\ref{table}), where both scrambling and decoherence contribute substantially to the fidelity decay. Figure~\ref{fig:benchmark} shows simulations of the fidelity decay, which is described accurately by ansatz (\ref{eq:ansatz}). This allows us to separate the scrambling rate $\lambda_s = 0.216$ and the decoherence rate $\lambda_d = 0.040$, and hence positively verify the presence of scrambling. The scrambling rate is determined purely by the underlying scrambling dynamics, and should not be altered by the intermediate perturbation. We verified this with different types of perturbations (\ref{eq:perturbation}) and recovery probabilities (\ref{eq:probasym}). The extracted scrambling rates are the same with a precision to the second decimal place.

\vspace{5pt}
To summarize, we have developed a protocol to benchmark information scrambling. This approach   distinguishes between information scrambling and fake positive signals, produced by decoherence  and operational errors in experiments unambiguously. It can be also used to quantify the degree of scrambling from the noisy background. Our method  requires only a single loop of forward and backward evolution, and hence can be applied to any system with access to time-reversing the dynamics.

\begin{acknowledgements}
This work was carried out under support of the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, Condensed Matter Theory Program. BY also acknowledges support from the Center for Nonlinear Studies and the U.S DOE under the LDRD program in Los Alamos. JH was supported by the U.S. DOE through a quantum computing program sponsored by the Los Alamos National Laboratory (LANL) Information Science \& Technology Institute. This research used quantum computing resources provided by the LANL Institutional Computing Program, which is supported by the U.S. DOE National Nuclear Security Administration under Contract No. 89233218CNA000001.
\end{acknowledgements}


\section{Introduction}\label{intro}
~~~We consider the large time behavior of a
viscous incompressible flow
around a rotating rigid body in $\mathbb{R}^3$.
Assume that both a compact rigid body $\mathscr{O}$
and a viscous incompressible fluid
which occupies the outside of $\mathscr{O}$ are initially at rest; then,
the body starts to rotate with the angular velocity
which gradually increases until it reaches a small terminal one
at a certain finite time and it is fixed afterwards.
We then show that the fluid motion converges
to a steady solution obtained by Galdi \cite{galdi2003}
as time $t\rightarrow \infty$
(Theorem \ref{thm1} in Subsection \ref{maintheorem2}).
This was conjectured by Hishida \cite[Section 6]{hishida2013},
but it has remained open.
Such a question is called the starting problem and
it was originally raised by
Finn \cite{finn1965}, in which rotation was
replaced by translation of the body.
Finn's starting problem was first
studied by Heywood \cite{heywood1972}; since his paper,
a stationary solution is said to be attainable
if the fluid motion converges to it as $t\rightarrow \infty$.
Later on, by using Kato's approach \cite{kato1984}
(see also Fujita and Kato \cite{fuka1964}) together with
the $L^q$-$L^r$ estimates for the
Oseen equation established by Kobayashi and Shibata \cite{kosh1998},
Finn's starting problem was completely solved
by Galdi, Heywood and Shibata \cite{gahesh1997}.
\par Let us introduce the mathematical formulation.
Let $\mathscr{O}\subset \mathbb{R}^3$ be a compact and connected set
with non-empty interior.
The motion of $\mathscr{O}$ mentioned above is described
in terms of the angular velocity
\begin{align}
\omega(t)=\psi(t)\omega_0,\quad \omega_0=(0,0,a)^{\top}
\end{align}
with a constant $a\in \mathbb{R}$, where $\psi$ is a
function on $\mathbb{R}$ satisfying the following conditions:
\begin{align}\label{psidef}
\psi\in C^1(\mathbb{R};\mathbb{R}),\quad
|\psi(t)|\leq 1~~{\rm{for}}~t\in\mathbb{R},
~~\psi(t)=0~~{\rm{for}}~t\leq 0,~~
\psi(t)=1~~{\rm{for}} ~t\geq 1.
\end{align}
Here and hereafter, $(\cdot)^\top$ denotes the transpose.
Then the domain occupied by the fluid can be expressed as
$D(t)=\{y=O(t)x;\,x\in D\}$, where
$D=\mathbb{R}^3\setminus \mathscr{O}$ is assumed to be an exterior domain with smooth
boundary $\partial D$ and
\begin{align*}
O(t)=\left(
\begin{array}{ccc}
\cos\Psi(t)&-\sin\Psi(t)&0\\
\sin\Psi(t)&\cos\Psi(t)&0\\
0&0&1
\end{array}\right),\quad \Psi(t)=\int_0^t \psi(s)a\,ds.
\end{align*}
We consider the initial boundary value problem
for the Navier-Stokes equation
\begin{align}\label{NS1}
\left\{
\begin{array}{r@{}c@{}ll}
\partial_t w+w\cdot \nabla_y w
&{}={}&\Delta_y w-\nabla_y \pi,&y\in D(t),t>0,\\
\nabla_y\cdot w&{}={}&0,&y\in D(t),t\geq 0,\\
w|_{\partial D(t)}&{}={}&\psi(t)\omega_0\times y,&t\geq 0,\\
w(y,t)&{}\rightarrow{}&0&{\rm{as}}~|y|\rightarrow \infty,\\
w(y,0)&{}={}&0,&y\in D,
\end{array}\right.
\end{align}
where $w=(w_1(y,t),w_2(y,t),w_3(y,t))^\top$
and $\pi=\pi(y,t)$ denote  unknown velocity and
pressure of the fluid, respectively.
To reduce the problem to an equivalent one
in the fixed domain $D$,
we take the frame $x=O(t)^\top y$ attached to the body and make the
change of the unknown functions:
$u(x,t)=O(t)^\top w(y,t),~p(x,t)=\pi(y,t)$.
Then the problem (\ref{NS1}) is reduced to
\begin{align}\label{NS2}
\left\{
\begin{array}{r@{}c@{}ll}
\partial_t u+u\cdot \nabla u
&{}={}&\Delta u+(\psi(t)\omega_0\times x)\cdot \nabla u
-\psi(t)\omega_0\times u-\nabla p,
&x\in D,t>0,\\
\nabla\cdot u&{}={}&0,\hspace{2cm} x\in D,t\geq 0,\\
u|_{\partial D}&{}={}&\psi(t)\omega_0\times x,\hspace{0.3cm}t\geq 0,\\
u&{}\rightarrow{}&0\hspace{2.2cm}{\rm{as}}~|x|\rightarrow \infty,\\
u(x,0)&{}={}&0,\hspace{2cm} x\in D.
\end{array}\right.
\end{align}
\par The purpose of this paper
is to show that (\ref{NS2})
admits a global solution which tends to a solution $u_s$
for the stationary problem
\begin{align}\label{sta}
\left\{
\begin{array}{r@{}c@{}ll}
u_s\cdot \nabla u_s&{}={}&\Delta u_s+(\omega_0\times x)\cdot \nabla u_s
-\omega_0\times u_s-\nabla p_s,&x\in D,\\
\nabla\cdot u_s&{}={}&0,\hspace{1.3cm}x\in D,\\
u_s|_{\partial D}&{}={}&\omega_0\times x,\\
u_s&{}\rightarrow{}&0\hspace{1.5cm}{\rm{as}}~|x|\rightarrow \infty.
\end{array}\right.
\end{align}
The rate of convergence in $L^r$ with
$r\in (3,\infty]$ is also studied.
In \cite{galdi2003}, Galdi successfully proved that
if $|\omega_0|$ is sufficiently small,
problem (\ref{sta}) has a
unique smooth solution $(u_s,p_s)$ with pointwise estimates
\begin{align}\label{pointwise}
|u_s(x)|\leq \frac{C|\omega_0|}{|x|},\quad |\nabla u_s(x)|+|p_s(x)|
\leq \frac{C|\omega_0|}{|x|^2}.
\end{align}
We note that the decay rate (\ref{pointwise})
is scale-critical, which is also captured in terms of
the Lorentz space (weak-Lebesgue space) $L^{3,\infty}$.
This was in fact done by Farwig and Hishida \cite{fahi2007}
even for the external force being in
a Lorentz-Sobolev space of order $(-1).$
\par Let us mention some difficulties of our problem
and how to overcome them in this paper.
In \cite{gahesh1997},
the $L^q$-$L^r$ estimates for the Oseen semigroup play an important role.
In the rotational case with constant angular velocity,
Hishida and Shibata \cite{hish2009}
also established the $L^q$-$L^r$ estimates of
the semigroup generated by the Stokes operator
with the additional term $(\omega_0\times x)\cdot \nabla-\omega_0\times$.
If we use this semigroup as in \cite{gahesh1997},
we have to treat the term
$(\psi(t)-1)(\omega_0\times x)\cdot\nabla v$, which is however
no longer perturbation from the semigroup on account of
the unbounded coefficient $\omega_0\times x$, where $v=u-\psi(t)u_s$.
In this paper,
we make use of the evolution operator $\{T(t,s)\}_{t\geq s\geq 0}$ on
the solenoidal space
$L^q_{\sigma}(D)~(1<q<\infty$),
which is a solution operator to the linear problem
\begin{align}\label{EV1}
\left\{
\begin{array}{r@{}c@{}ll}
\partial_t u&{}={}&\Delta u+(\psi(t)\omega_0\times x)\cdot
\nabla u-\psi(t)\omega_0\times u-\nabla p,&x\in D,t>s,\\
\nabla\cdot u&{}={}&0,\hspace{0.3cm} x\in D,t\geq s,\\
u|_{\partial D}&{}={}&0,\hspace{0.3cm} t> s,\\
u&{}\rightarrow{}&0\hspace{0.45cm}{\rm{as}}~|x|\rightarrow \infty,\\
u(x,s)&{}={}&f,\hspace{0.3cm} x\in D.
\end{array}\right.
\end{align}
Hansel and Rhandi \cite{harh2014} succeeded in the proof of generation of
this evolution operator with the $L^q$-$L^r$ smoothing rate.
They constructed the evolution operator in their own way since the
corresponding semigroup is not analytic
(Hishida \cite{hishida1999}, Farwig and Neustupa \cite{fane2007}).
Recently, Hishida \cite{hishida2018,hishidapre}
developed the $L^q$-$L^r$ decay estimates of the
evolution operator, see Section 3. With those estimates,
we solve the integral equation which perturbation
from the stationary solution $u_s$ obeys.
However, it is difficult to perform analysis
with the standard Lebesgue space
on account of the scale-critical pointwise estimates (\ref{pointwise}).
Thus, we first construct a solution for the weak formulation
in the framework of Lorentz
space by the strategy due to Yamazaki \cite{yamazaki2000}.
We next identify this solution with a local solution
possessing better regularity in a neighborhood of each time.
The later procedure is actually adopted
by Kozono and Yamazaki \cite{koya1998}.
Furthermore, we derive the $L^{\infty}$ decay which is not observed
in \cite{gahesh1997}.
When the stationary solution possesses
the scale-critical rate $O(1/|x|)$,
Koba \cite{koba2017} first derived the $L^{\infty}$ decay
of perturbation with less sharp rate in
the context of stability analysis, see also Remark \ref{rmk3}.
Although he used both the $L^1$-$L^r$ estimates of
the Oseen semigroup $T(t)$ and
the $L^q$-$L^r$ estimates (yielding the $L^q$-$L^\infty$ estimates)
of the composite
operator $T(t)P{\rm{div}}\,$, where $P$ denotes the Fujita-Kato projection
(see Subsection \ref{notation}),
it turns out that either of them is enough to accomplish the proof.
In this paper, we employ merely the $L^1$-$L^r$ estimates of the
adjoint evolution operator $T(t,s)^*$ to simplify the argument.
\par The paper is organized as follows.
In Section 2 we introduce the notation and give the main theorems.
Section 3 is devoted to some preliminary results
on the stationary problem and the evolution operator.
In Section 4 we give the proof of the main theorems.

\section{Main theorems} \label{maintheorem}
~~~In this section, we first introduce some notation and
after that, we give our main theorems.
\subsection{Notation}\label{notation}
~~~We introduce some function spaces.
Let $D\subset \mathbb{R}^3$ be an exterior domain with smooth boundary.
By $C^{\infty}_0(D)$, we
denote the set of all $C^{\infty}$ functions
with compact support in $D$. For $1\leq q\leq \infty$
and nonnegative integer $m$, $L^q(D)$ and
$W^{m,q}(D)$ denote the standard Lebesgue and Sobolev spaces, respectively.
We write the $L^q$ norm as $\|\cdot\|_q$.
The completion of $C_0^\infty(D)$ in $W^{m,q}(D)$ is
denoted by $W_0^{m,q}(D)$.
Let $1<q<\infty$ and $1\leq r\leq \infty$.
Then the Lorentz spaces $L^{q,r}(D)$ are defined by
\begin{align*}
L^{q,r}(D)=\{f:{\rm{Lebesgue ~measurable~ function}}
\mid\|f\|^*_{q,r}<\infty\},
\end{align*}
where
\begin{align*}
\|f\|^*_{q,r}=
\begin{cases}
\left(\displaystyle\int_0^{\infty}\big(t
\mu(\{x\in D\mid|f(x)|>t\})^{\frac{1}{q}}\big)^r\frac{dt}{t}\right)
^{\frac{1}{r}}
&1\leq r<\infty,\\
\displaystyle\sup_{t>0}
t\mu(\{x\in D\mid|f(x)|>t\})^{\frac{1}{q}}&r=\infty
\end{cases}
\end{align*}
and $\mu(\cdot)$ denotes the Lebesgue measure on $\mathbb{R}^3$.
The space $L^{q,r}(D)$ is a
quasi-normed space and it is even
a Banach space equipped with
norm $\|\cdot\|_{q,r}$ equivalent to $\|\cdot\|^*_{q,r}$.
The real interpolation functor
is denoted by $(\cdot,\cdot)_{\theta,r}$,
then we have
\begin{align*}
L^{q,r}(D)=\left(L^{q_0}(D),L^{q_1}(D)\right)_{\theta,r},
\end{align*}
where $1\leq q_0<q<q_1\leq \infty$ and $0<\theta<1$
satisfy $1/q=(1-\theta)/q_0+\theta/q_1$, while
$1\leq r\leq \infty$, see Bergh-L\"{o}fstr\"{o}m \cite{belobook1976}.
We note that if $1\leq r<\infty$, the dual of the space
$L^{q,r}(D)$ is $L^{q/(q-1),r/(r-1)}(D)$.
It is well known that if $1\leq r<\infty$,
the space $C^{\infty}_{0}(D)$ is dense in $L^{q,r}(D)$, while
the space $C^{\infty}_{0}(D)$ is not dense in $L^{q,\infty}(D)$.
\par We next introduce some solenoidal function spaces.
Let $C_{0,\sigma}^\infty(D)$ be the
set of all $C_0^{\infty}$-vector fields $f$ which satisfy ${\rm{div}}\, f=0$
in $D$. For $1<q<\infty$,
$L^{q}_{\sigma}(D)$ denote the
completion of $C_{0,\sigma}^\infty(D)$ in $L^q(D)$.
For every $1<q<\infty$, we have the following Helmholtz decomposition:
\begin{align*}
L^q(D)=L^q_{\sigma}(D)\oplus\{\nabla p\in L^q(D)\mid
p\in L^q_{\rm{loc}}(\overline{D})\},
\end{align*}
see Fujiwara and Morimoto \cite{fumo1977},
Miyakawa \cite{miyakawa1982}, and Simader and Sohr \cite{siso1992}.
Let $P_q$ denote the Fujita-Kato projection
from $L^q(D)$ onto $L^q_{\sigma}(D)$
associated with the decomposition. We remark that the adjoint operator
of $P_q$ coincides
with $P_{q/(q-1)}$. We simply write $P=P_q$\,. By real interpolation,
it is possible to extend $P$ to a bounded operator on $L^{q,r}(D)$.
We then define the solenoidal Lorentz spaces
$L^{q,r}_{\sigma}(D)$ by
\begin{align*}
L^{q,r}_{\sigma}(D)=PL^{q,r}(D)=
\left(L^{q_0}_\sigma(D),L^{q_1}_\sigma(D)\right)_{\theta,r},
\end{align*}
where $1<q_0<q<q_1<\infty$ and $0<\theta<1$
satisfy $1/q=(1-\theta)/q_0+\theta/q_1$,
while $1\leq r\leq \infty$, see Borchers and Miyakawa \cite{bomi1995}.
We then have the duality relation
$L^{q,r}_{\sigma}(D)^*=L_{\sigma}^{q/(q-1),r/(r-1)}(D)$
for $1<q<\infty$ and $1\leq r<\infty$.
We denote various constants by $C$
and they may change from line to line.
The constant dependent on $A,B,\cdots$ is denoted by $C(A,B,\cdots)$.
Finally, if there is no confusion,
we use the same symbols for denoting spaces of scalar-valued functions and
those of vector-valued ones.

\subsection{Main theorems}\label{maintheorem2}
~~~It is reasonable to look for a solution to (\ref{NS2}) of the form
\begin{align*}
u(x,t)=v(x,t)+\psi(t)u_s,\quad p(x,t)=\phi(x,t)+\psi(t)p_s.
\end{align*}
Then the perturbation $(v,\phi)$
satisfies the following initial boundary value problem
\begin{align}
\left\{
\begin{array}{r@{}c@{}ll}
\partial_t v&{}={}&\Delta v+(\psi(t)\omega_0\times x)\cdot\nabla v
-\psi(t)\omega_0\times v\\
&&\hspace{4.3cm}+(Gv)(x,t)+H(x,t)-\nabla \phi,
&x\in D,t>0,\\
\nabla\cdot v&{}={}&0,\hspace{0.3cm} x\in D,t\geq 0,\\
v|_{\partial D}&{}={}&0,\hspace{0.3cm}t> 0,\label{a}\\
v&{}\rightarrow{}&0\hspace{0.47cm}{\rm{as}}~|x|\rightarrow \infty,\\
v(x,0)&{}={}&0,\hspace{0.3cm} x\in D,
\end{array}\right.
\end{align}
where
\begin{align}
&(Gv)(x,t)=-v\cdot\nabla v-\psi(t)v\cdot\nabla u_s-\psi(t)u_s\cdot \nabla v,
\label{g}\\
&H(x,t)=\psi(t)(\psi(t)-1)\{-u_s\cdot \nabla u_s-\omega_0
\times u_s+(\omega_0\times x)\cdot\nabla u_s\}
-\psi'(t)u_s.\label{h}
\end{align}
In what follows, we concentrate ourselves
on the problem (\ref{a}) instead of (\ref{NS2}).
In fact, if we obtain the solution $v$ of (\ref{a})
which converges to $0$ as $t\rightarrow \infty$,
the solution $u$ of (\ref{NS2}) converges to $u_s$ as
$t\rightarrow \infty$.
By using the evolution operator $\{T(t,s)\}_{t\geq s\geq 0}$
on $L^q_{\sigma}(D)~(1<q<\infty$) associated with (\ref{EV1}),
problem (\ref{a}) is converted into
\begin{align}\label{integraleq}
v(t)=&\int_0^t T(t,\tau)P[(Gv)(\tau)+H(\tau)]\,d\tau.
\end{align}
\par We are now in a position to give our attainability theorem.
\begin{thm}\label{thm1}
Let $\psi$ be a function on $\mathbb{R}$ satisfying (\ref{psidef}) and
put $\alpha:=\displaystyle\max_{t\in\mathbb{R}}|\psi'(t)|.$
For $q\in (6,\infty)$, there exists a constant $\delta(q)>0$
such that if $(\alpha+1)|a|\leq \delta,$
problem (\ref{integraleq}) admits a
solution $v$ which possesses the following properties:
\begin{align*}
({\rm{i}})\,
v\in BC_{w^*}\big((0,\infty);L^{3,\infty}_{\sigma}(D)\big),~\,
\|v(t)\|_{3,\infty}\rightarrow 0~\, {\rm{as}}~ t\rightarrow 0,~\,
\sup_{0<t<\infty}\|v(t)\|_{3,\infty}\leq C(\alpha+1)|a|,
\end{align*}
where $BC_{w^*}(I;X)$
is the set of bounded and weak-$\ast$ continuous
functions on the interval $I$ with values in $X$, the constant $C$
is independent of $a$ and $\psi$;
\begin{align*}
&({\rm{ii}})\,v\in C\big((0,\infty);L^r_{\sigma}(D)\big)
\cap C_{w^*}\big((0,\infty);L^\infty(D)\big),~
\nabla v\in C_w\big((0,\infty);L^r(D)\big)~{\rm for}~r\in (3,\infty);\\
&({\rm{iii}})\,({\rm{Decay}})\quad
\|v(t)\|_{r}=O(t^{-\frac{1}{2}+\frac{3}{2r}})\quad
{\rm{as}}~t\rightarrow \infty
\quad {\rm{for~all~}} r\in(3,q),\\
&\hspace{2.585cm}\|v(t)\|_{q,\infty}
=O(t^{-\frac{1}{2}+\frac{3}{2q}})\quad
{\rm{as}}~t\rightarrow \infty,\\
&\hspace{2.585cm}\|v(t)\|_{r}=O(t^{-\frac{1}{2}+\frac{3}{2q}})
\quad {\rm{as}}~t\rightarrow \infty
\quad {\rm{for~all~}} r\in(q,\infty].
\end{align*}
\end{thm}

\begin{rmk}
We can obtain the $L^q$ decay of $v(t)$ like
$O(t^{-1/2+3/(2q)}\log t)$ as $t\rightarrow\infty$,
but it is not clear whether $\|v(t)\|_q=O(t^{-1/2+3/(2q)})$ holds.
\end{rmk}

To prove Theorem \ref{thm1},
the key step is to construct a solution of the weak formulation
\begin{align}\label{NS6}
(v(t),\varphi)=\int_0^t
\big(v(\tau)\otimes v(\tau)
&+\psi(\tau)\{v(\tau)\otimes u_s+u_s\otimes v(\tau)\},
\nabla T(t,\tau)^*\varphi\big)\,d\tau\nonumber\\&
+\int_0^t(H(\tau),T(t,\tau)^*\varphi)\,d\tau,
\quad \forall \varphi\in C^{\infty}_{0,\sigma}(D)
\end{align}
as in Yamazaki \cite{yamazaki2000}, where $T(t,\tau)^*$ denotes the
adjoint of
$T(t,\tau)$ and, here and in what follows,
$(\cdot,\cdot)$ stands for various duality pairings.
In this paper, a function $v$ is called a solution of (\ref{NS6}) if
$v\in L^{\infty}_{\rm{loc}}\big([0,\infty);L^{3,\infty}_{\sigma}(D)\big)$
satisfies (\ref{NS6}) for a.e. $t$.
By following Yamazaki's approach, we can easily see that
the solution obtained in Theorem \ref{thm1} is unique in the small, see
Proposition \ref{thm2}. In the following theorem, we give another result
on the uniqueness without assuming the smallness of solutions.
\begin{thm}\label{thm3}
Let $q\in (3,\infty).$
Then there exists a constant $\widetilde{\delta}>0$
independent of $q$ and $\psi$
such that if $|a|\leq \widetilde{\delta}$,
problem (\ref{NS6}) admits at most one solution
within the class
\begin{align*}
\big\{v\in L^{\infty}_{\rm{loc}}
\big([0,\infty);L^{3,\infty}_{\sigma}(D)\big)\cap
L^{\infty}_{\rm{loc}}
\big(0,\infty;L^q_\sigma(D)\big)\,\big|\,
\lim_{t\rightarrow 0}\|v(t)\|_{3,\infty}=0\big\}.
\end{align*}
\end{thm}

\begin{rmk}
Theorem \ref{thm3} asserts that if the angular velocity is small enough
and if $\widetilde{v}$ is a solution
within the class above which is not necessarily small,
then it coincides with the solution obtained in Theorem \ref{thm1}.
\end{rmk}

\section{Preliminary results}\label{preliminary}
~~~~In this section, we prepare some results on the
stationary solutions and the evolution operator.
For the stationary problem (\ref{sta}), Galdi \cite{galdi2003} proved the following result.
\begin{prop}[\cite{galdi2003}]\label{galdi2003}
There exists a constant $\eta\in(0,1]$ such that if
$|\omega_0|=|a|\leq \eta$, the stationary problem (\ref{sta}) admits a unique solution $(u_s,p_s)$ with
the estimate
\begin{align*}
\sup_{x\in D}\big\{(1+|x|)|u_s(x)|\big\}+\sup_{x\in D}\big\{(1+|x|^2)(|\nabla u_s(x)|+|p_s(x)|)\big\}\leq C|a|,
\end{align*}
where the constant $C$ is independent of $a$.
\end{prop}
From now on, we assume that the angular velocity $\omega_0=(0,0,a)^\top$ always satisfies
$|\omega_0|=|a|\leq \eta$.
Proposition \ref{galdi2003} then yields
\begin{align*}
u_s\in L^{3,\infty}(D)\cap L^{\infty}(D),\quad \nabla u_s\in L^{\frac{3}{2},\infty}(D)\cap L^{\infty}(D),
\quad |x|\nabla u_s\in L^{3,\infty}(D)\cap L^\infty(D)
\end{align*}
and
\begin{align}\label{hstaest2}
H(t)\in L^{3,\infty}(D),\quad
\|H(t)\|_{3,\infty}\leq C(a^2+\alpha|a|)
\end{align}
for all $t>0$. Here, $H(t)$ is defined by (\ref{h}) and
$\alpha=\displaystyle \sup_{t\in\mathbb{R}}|\psi'(t)|.$

\par We next collect some results on the evolution operator associated
with (\ref{EV1}).
We define the linear operator by
\begin{align*}
\mathscr{D}_q(L(t))&=\{
u\in L^q_{\sigma}(D)\cap W_0^{1,q}(D)\cap W^{2,q}(D)\mid
(\omega_0\times x)\cdot \nabla u\in L^q(D)\},\\
L(t)u&=-P[\Delta u+ (\psi(t)\omega_0\times x)\cdot \nabla u- \psi(t)\omega_0 \times u].
\end{align*}
Then the problem (\ref{EV1}) is formulated as
\begin{align}\label{EV2}
\partial_t u+L(t)u=0,\quad t\in (s,\infty);\quad u(s)=f
\end{align}
in $L^q_{\sigma}(D)$.
We can see that (\ref{psidef}) implies
\begin{align}\label{globalholder}
\psi(t)\omega_0\in C^\theta ([0,\infty);\mathbb{R}^3)\cap L^{\infty}(0,\infty;\mathbb{R}^3)
\end{align}
for all $\theta\in (0,1)$. In fact, we have
\begin{align}\label{supholder}
\sup_{0\leq t<\infty}|\psi(t)\omega_0|=|a|,\quad
\sup_{0\leq s<t<\infty}\frac{|\psi(t)\omega_0-\psi(s)\omega_0|}{(t-s)^{\theta}}\leq |a|
\max_{t\in \mathbb{R}}|\psi'(t)|
\end{align}
for all $\theta\in (0,1)$.
We fix, for instance, $\theta=1/2.$
Under merely the local H\"{o}lder continuity of the angular velocity,
Hansel and Rhandi \cite{harh2014} proved the following proposition
(see also Hishida \cite{hishidapre} concerning the assertion 1).
Indeed they did not derive the assertion 4,
but it directly follows from the real interpolation.
For completeness, we give its proof.
\begin{prop}[\cite{harh2014}]\label{harhprop}
Let $1<q<\infty$. Suppose (\ref{psidef}).
The operator family $\{L(t)\}_{t\geq 0}$ generates
a strongly continuous evolution operator
$\{T(t,s)\}_{t\geq s\geq 0}$ on $L^q_{\sigma}(D)$ with the following properties:
\begin{enumerate}
\item Let $q\in(3/2,\infty)$ and $s\geq 0$.
For every $f\in Z_q(D)$ and $t\in (s,\infty),$ we have $T(t,s)f\in Y_q(D)$ and
$T(\cdot,s)f\in C^1((s,\infty); L^q_{\sigma}(D))$ with
\begin{align*}
\partial_t T(t,s)f+L(t)T(t,s)f=0,\quad t\in(s,\infty)
\end{align*}
in $L^q_{\sigma}(D)$, where
\begin{align*}
&Y_q(D)=\{u\in L^q_{\sigma}(D)\cap W_0^{1,q}(D)
\cap W^{2,q}(D)\mid|x|\nabla u\in L^q(D)\},\\
&Z_q(D)=\{u\in L^q_\sigma(D)\cap W^{1,q}(D)\mid|x|\nabla u\in L^q(D)\}.
\end{align*}
\item For every $f\in Y_q(D)$ and $t>0$,
we have $T(t,\cdot)f\in C^1\big([0,t]; L^q_{\sigma}(D)\big)$ with
\begin{align*}
\partial_s T(t,s)f=T(t,s)L(s)f\quad s\in [0,t]
\end{align*}
in $L^q_{\sigma}(D)$.
\item Let $1<q\leq r<\infty,$ and
$m,\mathcal{T}\in (0,\infty)$. There is a constant $C=C(q,r,m,\mathcal{T},D)$ such that
\begin{align}\label{gradT}
\|\nabla T(t,s)f\|_{r}
\leq C(t-s)^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{r})-\frac{1}{2}}
\|f\|_{q}
\end{align}
holds for all $0\leq s<t\leq \mathcal{T}$ and $f\in L^{q}_{\sigma}(D)$ whenever
\begin{align}\label{condition m}
(1+\max_{t\in \mathbb{R}}|\psi'(t)|\,)|a|\leq m
\end{align}
is satisfied.
\item Let $1<q<r<\infty, 1\leq\rho_1,\rho_2\leq\infty
~{\rm{and}}~ m,\mathcal{T}\in (0,\infty)$.
There is a constant $C=C(q,r,\rho_1,\rho_2,m,\mathcal{T},D)$ such that
\begin{align}\label{gradT2}
\|\nabla T(t,s)f\|_{r,\rho_2}
\leq C(t-s)^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{r})-\frac{1}{2}}
\|f\|_{q,\rho_1}
\end{align}
holds for all $0\leq s<t\leq \mathcal{T}$ and $f\in L^{q,\rho_1}_{\sigma}(D)$
whenever (\ref{condition m}) is satisfied.
\end{enumerate}
\end{prop}
\hspace{-0.6cm}{\bf{Proof of the assertion 4.}}~
We choose $r_0,r_1$ such that $1<q<r_0<r<r_1<\infty.$
From the assertion 3 and the real interpolation, we have
\begin{align}
&\|\nabla T(t,s)f\|_{r_0,\rho_1}\leq C(t-s)^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{r_0})-\frac{1}{2}}
\|f\|_{q,\rho_1},\label{lqlr00}\\
&\|\nabla T(t,s)f\|_{r_1,\rho_1}\leq C(t-s)^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{r_1})-\frac{1}{2}}
\|f\|_{q,\rho_1}.\label{lqlr1}
\end{align}
By the reiteration theorem
for real interpolation
(see for instance \cite[Theorem 3.5.3]{belobook1976}), we obtain
\begin{align}\label{realinterpolation}
L_\sigma^{r,\rho_2}(D)=
(L_{\sigma}^{r_0,\rho_1}(D),L_{\sigma}^{r_1,\rho_1}(D))_{\beta,\rho_2},\quad
\|u\|_{r,\rho_2}\leq C\|u\|_{r_0,\rho_1}^{1-\beta}\|u\|^{\beta}_{r_1,\rho_1},\quad
\frac{1}{r}=\frac{1-\beta}{r_0}+\frac{\beta}{r_1}
\end{align}
which combined with (\ref{lqlr00}) and
(\ref{lqlr1}) concludes (\ref{gradT2}).\qed
\vspace{0.2cm}
\par We know that the adjoint operator $T(t,s)^*$ is also a strongly
continuous evolution operator and
satisfies the backward semigroup property
\begin{align*}
T(\tau,s)^*T(t,\tau)^*=T(t,s)^*\quad (t\geq \tau\geq s\geq 0),\quad T(t,t)^*=I,
\end{align*}
see Hishida \cite[Subsection 2.3]{hishida2018}.
Under the assumption (\ref{globalholder}) with some $\theta\in(0,1)$,
Hishida \cite{hishida2018,hishidapre} established
the following
$L^q$-$L^r$ decay estimates. The assertion 3 is not found there
but can be proved in the same way as above.
We note that the idea of deduction of (\ref{intesti})
below is due to Yamazaki \cite{yamazaki2000} once we have the assertion 5.
\begin{prop}[\cite{hishida2018,hishidapre}]\label{hishidaprop}
\quad Let $m\in (0,\infty)$ and suppose (\ref{psidef}).
\begin{enumerate}
\item Let $1<q\leq r\leq\infty~(q\ne\infty)$.
Then there exists a constant $C=C(m,q,r,D)$ such that
\begin{align}
&\|T(t,s)f\|_{r}\leq
C(t-s)^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{r})}\|f\|_{q}\label{lqlr0}\\
&\|T(t,s)^*g\|_{r}\leq
C(t-s)^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{r})}\|g\|_{q}\label{ad0}
\end{align}
hold for all $t> s\geq 0$ and $f,g\in L^q_{\sigma}(D)$
whenever (\ref{condition m}) is satisfied.
\item Let $1<q\leq r<\infty, ~1\leq \rho\leq \infty$.
Then there exists a constant $C=C(m,q,r,\rho,D)$
such that
\begin{align}
&\|T(t,s)f\|_{r,\rho}\leq
C(t-s)^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{r})}
\|f\|_{q,\rho}\label{lqlr}\\
&\|T(t,s)^*g\|_{r,\rho}
\leq
C(t-s)^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{r})}\|g\|_{q,\rho}\label{ad}
\end{align}
hold for all $t> s\geq 0$ and $f,g\in L^{q,\rho}_{\sigma}(D)$
whenever (\ref{condition m}) is satisfied.
\item Let $1<q<r<\infty, 1\leq \rho_1,\rho_2\leq \infty$.
Then there exists a constant
$C=C(m,q,r,\rho_1,\rho_2,D)$ such that
\begin{align}
&\|T(t,s)f\|_{r,\rho_2}\leq C(t-s)^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{r})}\|f\|_{q,\rho_1}\label{lqlr2}\\
&\|T(t,s)^*g\|_{r,\rho_2}\leq C(t-s)^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{r})}\|g\|_{q,\rho_1}\label{ad2}
\end{align}
hold for all $t> s\geq 0$ and $f,g\in L^{q,\rho_1}_{\sigma}(D)$
whenever (\ref{condition m}) is satisfied.
\item Let $1<q\leq r\leq 3$.
Then there exists a constant $C=C(m,q,r,D)$
such that
\begin{align}
&\|\nabla T(t,s)f\|_{r}\leq C(t-s)^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{r})-\frac{1}{2}}
\|f\|_{q}\label{grad0}\\
&\|\nabla T(t,s)^*g\|_{r}\leq C(t-s)^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{r})-\frac{1}{2}}
\|g\|_{q}\label{adgrad0}
\end{align}
hold for all $t> s\geq 0$ and $f,g\in L^{q}_{\sigma}(D)$
whenever (\ref{condition m}) is satisfied.
\item Let $1<q\leq r\leq 3,~1\leq \rho<\infty$.
Then there exists a constant $C=C(m,q,r,\rho,D)$ such that
\begin{align}
&\|\nabla T(t,s)f\|_{r,\rho}\leq C(t-s)^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{r})-\frac{1}{2}}
\|f\|_{q,\rho}\label{grad}\\
&\|\nabla T(t,s)^*g\|_{r,\rho}\leq C(t-s)^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{r})-\frac{1}{2}}
\|g\|_{q,\rho}\label{adgrad}
\end{align}
hold for all $t> s\geq 0$ and $f,g\in L^{q,\rho}_{\sigma}(D)$
whenever (\ref{condition m}) is satisfied.
\item Let $1<q\leq r\leq 3$ with $1/q-1/r=1/3$.
Then there exists a constant $C=C(m,q,D)$ such that
\begin{align}\label{intesti}
\int_0^t\|\nabla T(t,s)^*g\|_{r,1}\,ds\leq C\|g\|_{q,1}
\end{align}
holds for all $t>0$ and $g\in L^{q,1}_{\sigma}(D)$
whenever (\ref{condition m}) is satisfied.
\end{enumerate}
\end{prop}

To prove the $L^{\infty}$ decay estimate in Theorem \ref{thm1},
we also prepare the following $L^1$-$L^r$ estimates.
The following estimates for data being in $C_0^\infty(D)^3$ are enough
for later use, but it is clear that, for instance,
the composite operator $T(t,s)P$ extends to
a bounded operator from $L^1(D)$ to $L^r_\sigma(D)$
with the same estimate.

\begin{lem}\label{lem,infty,1}
Let $m\in (0,\infty)$ and suppose (\ref{psidef}).
\begin{enumerate}
\item Let $1<r<\infty$.
Then there is a constant $C=C(m,r,D)>0$ such that
\begin{align}
&\|T(t,s)Pf\|_r\leq C(t-s)^{-\frac{3}{2}(1-\frac{1}{r})}\|f\|_1,\label{l1lr}\\
&\|T(t,s)^*Pg\|_r\leq C(t-s)^{-\frac{3}{2}(1-\frac{1}{r})}\|g\|_1\label{l1lrad}
\end{align}
for all $t>s\geq 0$ and $f,g\in C^\infty_0(D)^3$ whenever
(\ref{condition m}) is satisfied.
\item Let $1<r<\infty$ and $1\leq\rho\leq \infty$.
Then there is a constant
$C=C(m,r,\rho,D)>0$ such that
\begin{align}
&\|T(t,s)Pf\|_{r,\rho}\leq C(t-s)^{-\frac{3}{2}(1-\frac{1}{r})}\|f\|_1,\label{l1lr2}\\
&\|T(t,s)^*Pg\|_{r,\rho}\leq C(t-s)^{-\frac{3}{2}(1-\frac{1}{r})}\|g\|_1\label{l1lr2ad}
\end{align}
for all $t>s\geq 0$ and $f,g\in C^\infty_0(D)^3$
whenever (\ref{condition m}) is satisfied.
\item Let $1<r\leq 3,\,1\leq\rho<\infty$.
Then there is a constant
$C=C(m,r,\rho,D)>0$ such that
\begin{align}
\|\nabla T(t,s)Pf\|_{r,\rho}\leq C(t-s)^{-\frac{3}{2}(1-\frac{1}{r})-\frac{1}{2}}\|f\|_1,
\label{gradl1lr}\\
\|\nabla T(t,s)^*Pg\|_{r,\rho}\leq C(t-s)^{-\frac{3}{2}(1-\frac{1}{r})-\frac{1}{2}}\|g\|_1\label{gradl1lrad}
\end{align}
for all $t>s\geq 0$ and $f,g\in C^{\infty}_0(D)^3$ whenever
(\ref{condition m}) is satisfied.
\end{enumerate}
\end{lem}
\begin{proof}
The proof is simply based on duality argument
(see Koba \cite[Lemma 2.15]{koba2017}), however,
we give it for completeness.
Let $1<q\leq r<\infty$ and $1/r+1/r'=1$.
By using (\ref{ad0}),
we see that
\begin{align}\label{argue}
|(T(t,s)Pf,\varphi)|=|(f,T(t,s)^*\varphi)|\leq \|f\|_1\|T(t,s)^*\varphi\|_\infty
\leq C(t-s)^{-\frac{3}{2}(1-\frac{1}{r})}\|f\|_1\|\varphi\|_{r'}
\end{align}
for all $\varphi \in L^{r'}_\sigma(D)$,
which implies (\ref{l1lr}). We next show (\ref{l1lr2}).
We fix $q$ such that $1<q<r$.
Combining the estimate (\ref{lqlr2}) with (\ref{l1lr}),
we have
\begin{align*}
\|T(t,s)Pf\|_{r,\rho}\leq
C(t-s)^{-\frac{3}{2}(\frac{1}{q}-\frac{1}{r})}
\left\|T\left(\frac{t+s}{2},s\right)Pf\right\|_q
\leq C(t-s)^{-\frac{3}{2}(1-\frac{1}{r})}\|f\|_1.
\end{align*}
Finally, in view of (\ref{grad}) and (\ref{l1lr2}), we have
\begin{align*}
\|\nabla T(t,s)Pf\|_{r,\rho}\leq
C(t-s)^{-\frac{1}{2}}
\left\|T\left(\frac{t+s}{2},s\right)Pf\right\|_{r,\rho}
\leq C(t-s)^{-\frac{3}{2}(1-\frac{1}{r})-\frac{1}{2}}\|f\|_1
\end{align*}
which implies (\ref{gradl1lr}).
The proof for the adjoint $T(t,s)^*$ is accomplished in the same way.
\end{proof}

\section{Proof of the main theorems}\label{proof}
~~~In this section we prove the main theorems
(Theorem~\ref{thm1} and Theorem~\ref{thm3}).
We first give some key estimates and then show Theorem~\ref{thm3}.
After that, following Yamazaki \cite{yamazaki2000},
we construct a solution with some decay properties
for (\ref{NS6})
and then derive the $L^\infty$ decay of the solution.
We finally identify
the solution above with a local solution
possessing better regularity
for the integral equation (\ref{integraleq}) in a
neighborhood of each time $t>0$.
\par Let us define the function spaces
\begin{align*}
&X=\big\{v\in BC_{w^*}\big((0,\infty);L^{3,\infty}_{\sigma}(D)\big)
\,\big|\,\lim_{t\rightarrow 0}
\|v(t)\|_{3,\infty}=0\big\},\\
&X_q=\big\{v\in X\,\big|\,t^{\frac{1}{2}-\frac{3}{2q}}v
\in BC_{w^*}\big((0,\infty);L_{\sigma}^{q,\infty}(D)
\big)\big\},~~3<q<\infty.
\end{align*}
Both are Banach spaces endowed with
norms $\|\cdot\|_X=\|\cdot\|_{X,\infty}$ and
$\|\cdot\|_{X_q}=\|\cdot\|_{X_q,\infty}$, respectively, where
\begin{align*}
&\|v\|_{X,t}=\sup_{0<\tau<t}\|v(\tau)\|_{3,\infty},\quad
\|v\|_{X_q,t}=\|v\|_{X,t}
+[v]_{q,t},\quad
[v]_{q,t}=\sup_{0<\tau<t}
\tau^{\frac{1}{2}-\frac{3}{2q}}\|v(\tau)\|_{q,\infty}
\end{align*}
for $t\in (0,\infty].$
\begin{lem}\label{key}
\begin{enumerate}
\item ~Let $v,w\in X$ and set
\begin{align*}
&\braket{\mathcal{I}(v,w)(t),\varphi}:=
\int_0^t (v(\tau)\otimes w(\tau),
\nabla T(t,\tau)^*\varphi)\,d\tau,\\
&\braket{\mathcal{J}(v)(t),\varphi}:=
\int_0^t (\psi(\tau)\{v(\tau)\otimes u_s+u_s\otimes v(\tau)\},
\nabla T(t,\tau)^*\varphi)\,d\tau
\end{align*}
for all $\varphi\in C_{0,\sigma}^{\infty}(D)$.
Then $\mathcal{I}(v,w),\mathcal{J}(v)\in X$ and there exists a positive
constant $C$ such that
\begin{align}\label{e1}
\|\mathcal{I}(v,w)\|_{X,t}\leq C\|v\|_{X,t}\|w\|_{X,t},\quad\quad
\|\mathcal{J}(v)\|_{X,t}\leq C\|u_s\|_{3,\infty}\|v\|_{X,t}
\end{align}
hold for any $v,w\in X$ and $t\in(0,\infty]$.
\item ~Let $q\in(3,\infty)$. If $v\in X_q$, $w\in X$,
then $\mathcal{I}(v,w),\mathcal{J}(v)\in X_q$ and
there exists a positive constant $C=C(q)$ such that
\begin{align}\label{e2}
\|\mathcal{I}(v,w)\|_{X_q,t}\leq C\|v\|_{X_q,t}\|w\|_{X,t},
\quad \|\mathcal{J}(v)\|_{X_q,t}\leq C\|u_s\|_{3,\infty}\|v\|_{X_q,t}
\end{align}
hold for every $v\in X_q$, $w\in X$ and $t\in(0,\infty]$.
\item ~We set
\begin{align*}
\braket{\mathcal{K}(t),\varphi}:=\int_0^t (H(\tau),T(t,\tau)^*\varphi)\,d\tau
\end{align*}
for $\varphi\in C_{0,\sigma}^\infty(D)$.
Let $q\in(3,\infty)$. Then $\mathcal{K}\in X_q$ and
there exist positive constants $C$ independent of $q$
and $C'=C'(q)$ such that
\begin{align}\label{kest}
\|\mathcal{K}\|_{X,t}\leq C(a^2+\alpha|a|),\quad
\|\mathcal{K}\|_{X_q,t}\leq C'(a^2+\alpha|a|)
\end{align}
hold for every $t\in (0,\infty]$.
\end{enumerate}
\end{lem}
\begin{proof}
Estimates (\ref{e1}) and (\ref{e2}) can be proved in the same way as done by
Yamazaki \cite[Lemma 6.1.]{yamazaki2000}, see also
Hishida and Shibata \cite[Section 8]{hish2009}, however, we
briefly give the proof of (\ref{e1})$_1$ and (\ref{e2})$_1$.
By (\ref{intesti}), we have
\begin{align*}
|\braket{\mathcal{I}(v,w)(t),\varphi}|\leq \|v\|_{X,t}\|w\|_{X,t}
\int_0^t\|\nabla T(t,\tau)^*\varphi\|_{3,1}
\leq C\|v\|_{X,t}\|w\|_{X,t}\|\varphi\|_{\frac{3}{2},1},
\end{align*}
which yields (\ref{e1})$_1$.
We choose $r$ such that $1/3+1/q+1/r=1$ to find
\begin{align*}
|\braket{\mathcal{I}(v,w)(t),\varphi}|\leq [v]_{X_q,t}\|w\|_{X,t}
\int_0^t\tau^{-\frac{1}{2}+\frac{3}{2q}}
\|\nabla T(t,\tau)^*\varphi\|_{r,1}\,d\tau= [v]_{q,t}\|w\|_{X,t}
\left(\int_0^{\frac{t}{2}}+\int_{\frac{t}{2}}^t\right).
\end{align*}
In view of (\ref{adgrad}), we have
\begin{align*}
\int_0^{\frac{t}{2}}\leq C\int_0^{\frac{t}{2}}\tau^{-\frac{1}{2}
+\frac{3}{2q}}(t-\tau)^{-1}\,d\tau\|\varphi\|_{q',1}
\leq Ct^{-\frac{1}{2}+\frac{3}{2q}}\|\varphi\|_{q',1},
\end{align*}
where $1/q+1/q'=1$,
whreas, (\ref{intesti}) implies
\begin{align*}
\int_{\frac{t}{2}}^t
\leq \left(\frac{t}{2}\right)^{-\frac{1}{2}+\frac{3}{2q}}
\int_0^t\|\nabla T(t,\tau)^*\varphi\|_{r,1}\,d\tau
\leq Ct^{-\frac{1}{2}+\frac{3}{2q}}\|\varphi\|_{q',1}
\end{align*}
from which together with (\ref{e1})$_1$, we obtain (\ref{e2})$_1$.
The estimate (\ref{e1}) leads us to
\begin{align*}
\lim_{t\rightarrow 0}\|\mathcal{I}(v,w)(t)\|_{3,\infty}=0,\quad
\lim_{t\rightarrow 0}\|\mathcal{J}(v)(t)\|_{3,\infty}=0.
\end{align*}
\par Let us consider the weak-$\ast$ continuity
of $\mathcal{I}(v,w)$ with values in $L^{3,\infty}_{\sigma}(D)$
(resp. $L^{q,\infty}_{\sigma}(D)$) when $v\in X$ (resp. $v\in X_q$),
$w\in X$. Here, we need a different argument from
\cite{yamazaki2000} because of the non-autonomous character
as well as the non-analyticity of the corresponding semigroup.
Since $C_{0,\sigma}^{\infty}(D)$ is dense in
$L_{\sigma}^{\kappa,1}$ ($\kappa=3/2,\,q'$) and since
we know (\ref{e1}) and (\ref{e2}), it suffices to show that
\begin{align}\label{w*contiI}
|\braket{\mathcal{I}(v,w)(t)-\mathcal{I}(v,w)(\sigma),\varphi}|
\rightarrow 0~~ {\rm{as}} ~~\sigma\rightarrow t
\end{align}
for all $0<t<\infty$ and $\varphi\in C^{\infty}_{0,\sigma}(D)$.
Let $0<\sigma<t$. By using the backward semigroup property, we have
\begin{align*}
|\braket{\mathcal{I}(v,w)(t)-\mathcal{I}(v,w)(\sigma),\varphi}|&\leq
\int_0^\sigma|(v(\tau)\otimes w(\tau),
\nabla T(\sigma,\tau)^*(T(t,\sigma)^*\varphi-\varphi)|\,d\tau
\\&\hspace{1.5cm}
+\int_\sigma^t
|(v(\tau)\otimes w(\tau),\nabla T(t,\tau)^*\varphi)|\,d\tau
=:I_1+I_2.
\end{align*}
The estimate (\ref{intesti}) yields
\begin{align*}
I_1&\leq \|v\|_{X,t}\|w\|_{X,t}\int_0^\sigma\|
\nabla T(\sigma,\tau)^*(T(t,\sigma)^*\varphi-\varphi)\|_{3,1}\,d\tau\\
&\leq C\|v\|_{X,t}\|w\|_{X,t}\,
\|T(t,\sigma)^*\varphi-\varphi\|_{\frac{3}{2},1}\rightarrow 0
\quad{\rm{as}}~ \sigma \rightarrow t.
\end{align*}
Furthermore, (\ref{adgrad}) yields
\begin{align*}
I_2\leq \|v\|_{X,t}\|w\|_{X,t}\int_\sigma^t
\|\nabla T(t,\tau)^*\varphi\|_{3,1}\,d\tau
\leq C\|v\|_{X,t}\|w\|_{X,t}(t-\sigma)^{\frac{1}{2}}\|\varphi\|_{3,1}
\rightarrow 0 \quad {\rm{as}}~\sigma \rightarrow t.
\end{align*}
We can discuss the other case $0<t<\sigma$ similarly and
thus we obtain (\ref{w*contiI}).
By the same manner, we can obtain the desired weak-$\ast$ continuity
of $\mathcal{J}$. We thus conclude the assertion 1 and 2.
\par We next consider $\mathcal{K}(t)$.
We use (\ref{ad}) as well as (\ref{hstaest2})
to obtain
\begin{align*}
|\braket{\mathcal{K}(t),\varphi}|\leq C(a^2+\alpha|a|)
\int_0^{\min\{1,t\}}|\|T(t,\tau)^*\varphi\|_{\frac{3}{2},1}\,d\tau
\leq C(a^2+\alpha|a|)\min\{1,t\}\|\varphi\|_{\frac{3}{2},1}
\end{align*}
for $\varphi\in C_{0,\sigma}^\infty(D)$ and $t>0$
which yields $\mathcal{K}(t)\in L^{3,\infty}_\sigma(D)$ with
\begin{align*}
\|\mathcal{K}\|_{X,t}\leq C(a^2+\alpha|a|)
\quad {\rm{for}} ~t\in(0,\infty],\quad
\lim_{t\rightarrow 0} \|\mathcal{K}(t)\|_{3,\infty}=0.
\end{align*}
To derive the estimate
$[\mathcal{K}]_{q,t}\leq C(a^2+\alpha|a|)$,
we consider two cases: $0<t\leq 2$ and $t\geq 2$.
For $0<t\leq2$, (\ref{ad}) yields
\begin{align*}
|\braket{\mathcal{K}(t),\varphi}|
\leq C(a^2+\alpha|a|)
\int_0^t\|T(t,\tau)^*\varphi\|_{\frac{3}{2},1}\,d\tau
\leq C(a^2+\alpha|a|)\|\varphi\|_{q',1}.
\end{align*}
For $t\geq 2$, we have
\begin{align*}
|\braket{\mathcal{K}(t),\varphi}|
\leq C(a^2+\alpha|a|)
\int_0^1\|T(t,\tau)^*\varphi\|_{\frac{3}{2},1}\,d\tau
\leq C(a^2+\alpha|a|)\,
t^{-\frac{1}{2}+\frac{3}{2q}}\|\varphi\|_{q',1}.
\end{align*}
We thus obtain (\ref{kest}).
It remains to show the weak-$\ast$ continuity. To this end, it is
sufficient to show that
\begin{align}\label{w*contiJ}
|\braket{\mathcal{K}(t)-\mathcal{K}(\sigma),\varphi}|\rightarrow 0~~ {\rm{as}}
~\sigma\rightarrow t
\end{align}
for all $t\in (0,\infty)$
due to (\ref{kest}). To prove (\ref{w*contiJ}),
we suppose  $0<\sigma<t$.
We use the backward semigroup property to observe
\begin{align*}
\braket{\mathcal{K}(t)-\mathcal{K}(\sigma),\varphi}
=\int_0^\sigma\big(H(\tau),T(\sigma,\tau)^*(T(t,\sigma)^*
\varphi-\varphi)\big)\,d\tau
+\int_\sigma^t (H(\tau),T(t,\tau)^*\varphi)\,d\tau.
\end{align*}
By applying (\ref{ad}), we find that
\begin{align*}
&\left|
\int_0^\sigma\big(H(\tau),T(\sigma,\tau)^*(T(t,\sigma)^*\varphi-\varphi)
\big)\,d\tau
\right|
\leq C(a^2+\alpha|a|)\,
\sigma\,\|T(t,\sigma)^*\varphi-\varphi\|_{\frac{3}{2},1}
\rightarrow 0
\quad {\rm{as}} ~\sigma\rightarrow t,\\
&\left|
\int_\sigma^t (H(\tau),T(t,\tau)^*\varphi)\,d\tau
\right|
\leq C(a^2+\alpha|a|)(t-\sigma)\|\varphi\|_{\frac{3}{2},1}
\rightarrow 0\quad {\rm{as}} ~\sigma\rightarrow t.
\end{align*}
The other case $t<\sigma$ is discussed similarly. Hence we have
(\ref{w*contiJ}). The proof is complete.
\end{proof}

\begin{comment}
\begin{lem}\label{uniquenessresult}
Let $r\in (3,\infty).$ Then there exists a constant
$\widetilde{\delta}>0$ independent of $r$ and $\psi$ such that if
the angular velocity $|\omega_0|=|a|\leq \widetilde{\delta}$,
(\ref{NS6}) admits at most one solution within the class
\begin{align*}
\{v\in BC_{w^*}((0,\infty);L_\sigma^{3,\infty}(D))
\cap L^\infty_{\rm{loc}}(0,\infty;L^r_\sigma(D))\,|\,
\lim_{t\rightarrow 0}\|v(t)\|_{3,\infty}=0\}.
\end{align*}
\end{lem}
\end{comment}
\hspace{-0.6cm}{\bf{Proof of Theorem \ref{thm3}.}}~
The idea of the proof is traced back to Fujita and Kato
\cite[Theorem 3.1.]{fuka1964}.
Let $v_1$ and $v_2$ be the solutions of (\ref{NS6}). Then we have
\begin{align}
(v_1(t)-v_2(t),\varphi)=&\int_0^t
\big(v_1(\tau)\otimes(v_1(\tau)-v_2(\tau))+
(v_1(\tau)-v_2(\tau))
\otimes v_2(\tau)\nonumber\\&+\psi(\tau)(v_1(\tau)-v_2(\tau))\otimes u_s+
\psi(\tau)u_s\otimes(v_1(\tau)-v_2(\tau)),
\nabla T(t,\tau)^*\varphi\big)\,d\tau\label{v1v2}
\end{align}
for $\varphi\in C_{0,\sigma}^\infty(D).$
By applying (\ref{e1}) to (\ref{v1v2}) and
by Proposition \ref{galdi2003},
we have
\begin{align*}
\|v_1-v_2\|_{X,t}&
\leq C(\|v_1\|_{X,t}+\|v_2\|_{X,t}+\|u_s\|_{3,\infty})
\|v_1-v_2\|_{X,t}\\
&\leq C(\|v_1\|_{X,t}+\|v_2\|_{X,t}+|a|)\|v_1-v_2\|_{X,t}.
\end{align*}
Suppose
\begin{align*}
|a|<\frac{1}{2C}=:\widetilde{\delta}.
\end{align*}
Since $\|v_j(t)\|_{3,\infty}\rightarrow 0$ as $t\rightarrow 0$ ($j=1,2$),
one can choose $t_0>0$ such that
$C(\|v_1\|_{X,t_0}+\|v_2\|_{X,t_0})<1/2$,
which implies $v_1=v_2$ on $(0,t_0]$.
Hence, (\ref{v1v2}) is written as
\begin{align}\label{v1v22}
(v_1(t)-v_2(t),\varphi)=
&\int_{t_0}^t\big(v_1(\tau)\otimes(v_1(\tau)-v_2(\tau))+
(v_1(\tau)-v_2(\tau))
\otimes v_2(\tau)\nonumber\\&
+\psi(\tau)(v_1(\tau)-v_2(\tau))\otimes u_s+
\psi(\tau)u_s\otimes(v_1(\tau)-v_2(\tau)),
\nabla T(t,\tau)^*\varphi\big)\,d\tau.
\end{align}
We fix $\mathcal{T}\in (t_0,\infty)$ and set
$[v]_{q,t_0,t}=\displaystyle\sup_{t_0\leq\tau\leq t}\|v(\tau)\|_q$
for $t\in (t_0,\mathcal{T})$.
It follows from
(\ref{v1v22}) that
\begin{align}\label{v1v2r}
[v_1-v_2]_{q,t_0,t}\leq C_*(t-t_0)^{\frac{1}{2}-\frac{3}{2q}}
[v_1-v_2]_{q,t_0,t}\,,\quad
t\in (t_0,\mathcal{T}),
\end{align}
where
$C_*=C_*(t_0,\mathcal{T})=C([v_1]_{q,t_0,\mathcal{T}}+[v_2]_{q,t_0,\mathcal{T}}+2\|u_s\|_q\big).$
In fact, the estimate (\ref{adgrad0}) yields
\begin{align*}
\int_{t_0}^t\big|&\big(v_1(\tau)\otimes(v_1(\tau)-v_2(\tau)),
\nabla T(t,\tau)^*\varphi\big)\big|\,d\tau\\
&\leq C[v_1(\tau)]_{q,t_0,\mathcal{T}}[v_1-v_2]_{q,t_0,t}
\int_{t_0}^t\|\nabla
T(t,\tau)^*\varphi\|_{(1-\frac{2}{q})^{-1}}\,d\tau\\
&\leq C[v_1(\tau)]_{q,t_0,\mathcal{T}}
[v_1-v_2]_{q,t_0,t}(t-t_0)^{\frac{1}{2}-\frac{3}{2q}}
\|\varphi\|_{(1-\frac{1}{q})^{-1}}
\end{align*}
for all $\varphi\in C_{0,\sigma}^{\infty}(D)$ and $t\in (t_0,\mathcal{T})$.
Since the other terms in (\ref{v1v22}) are treated similarly,
we obtain (\ref{v1v2r}). We take
\begin{align*}
\xi=\min\left\{\left(\frac{1}{2C_*}\right)^
{(\frac{1}{2}-\frac{3}{2q})^{-1}},\mathcal{T}-t_0\right\}
\end{align*}
which leads us to $v_1=v_2$ on $(0,t_0+\xi).$
Even though we replace $t_0$ by $t_0+\xi,t_0+2\xi,\cdots$,
we can discuss similarly. Hence, $v_1=v_2$ on $(0,\mathcal{T})$.
Since $\mathcal{T}$ is arbitrary, we conclude $v_1=v_2$. \qed
\vspace{0.2cm}
\par To prove Theorem \ref{thm1},
we begin to construct a solution of (\ref{NS6})
by applying Lemma \ref{key}.
\begin{prop}\label{thm2}
Let $\psi$ be a function on $\mathbb{R}$ satisfying (\ref{psidef}).
We put $\alpha=\displaystyle\max_{t\in\mathbb{R}}|\psi'(t)|$.
\begin{enumerate}
\item There exists $\delta_1>0$ such that
if $(\alpha+1)|a| \leq \delta_1$, problem (\ref{NS6})
admits a unique solution
within the class
\begin{align*}
\big\{v\in BC_{w^*}\big((0,\infty);L^{3,\infty}_{\sigma}(D)\big)\,\big|
\,&\lim_{t\rightarrow 0}\|v(t)\|_{3,\infty}=0,\\&
\sup_{0<\tau<\infty}\|v(\tau)\|_{3,\infty}
\leq C(\alpha+1)|a|\big\},
\end{align*}
where $C>0$
is independent of $a$ and $\psi$.
\item Let $3<q<\infty$. Then there exists $\delta_2(q)\in(0,\delta_1]$
such that if $(\alpha+1)|a|\leq \delta_2$,
\begin{align*}
t^{\frac{1}{2}-\frac{3}{2q}}v
\in BC_{w^*}\big((0,\infty);L_{\sigma}^{q,\infty}(D)\big),
\end{align*}
where $v(t)$ is the solution obtained above.
\end{enumerate}
\end{prop}
\begin{proof}
We first show the assertion 1 by the contraction mapping principle.
Given $v\in X$, we define
\begin{align*}
\braket{(\Phi v)(t),\varphi}=\rm{the~RHS~of}~(\ref{NS6}),\quad
\varphi\in C_{0,\sigma}^\infty(D).
\end{align*}
Lemma \ref{key} implies that $\Phi v\in X$ with
\begin{align}
&\|\Phi v\|_{X}\leq C_1\|v\|_{X}^2+C_2|a|\|v\|_{X}
+C_3(a^2+\alpha |a|),\label{thm1pf1}\\
&\|\Phi v-\Phi w\|_{X}
\leq (C_1\|v\|_{X}+C_1\|w\|_{X}+C_2|a|)\|v-w\|_{X}\label{thm1pf2}
\end{align}
for every $v,w\in X$. Here,
$C_1,C_2,C_3,C_4$ are constants
independent of $v,w,a$ and $\psi$.
Hence, if we take $a$ satisfying
\begin{align*}
(\alpha+1)|a|<\min
\left\{\frac{1}{2C_2},\frac{1}{16C_1C_3},\eta\right\}=:\delta_1,
\end{align*}
where $\eta\in (0,1]$ is a constant given in
Proposition \ref{galdi2003},
then we obtain a unique solution $v$ within the class
\begin{align*}
\{v\in X\mid\|v\|_X\leq 4C_3(\alpha+1)|a|\}
\end{align*}
which completes the proof of the assertion 1.
\par We next show the assertion 2.
By applying Lemma \ref{key},
we see that $\Phi v\in X_q$ together with
(\ref{thm1pf1})--(\ref{thm1pf2}) in which
$X$ norm was replaced by $X_q$ norm and
the constants $C_i~(i=1,2,3)$ are also replaced by some others
$\widetilde{C}_i(q)(\geq C_i)$.
If we assume
\begin{align}\label{acondiq}
(\alpha+1)|a|<\min\left\{\frac{1}{2\widetilde{C}_2},
\frac{1}{16\widetilde{C}_1\widetilde{C}_3}
,\eta\right\}=:\delta_2\,(\leq \delta_1),
\end{align}
we can obtain a unique solution $\hat{v}$ within the class
\begin{align*}
\{v\in X_q\mid\|v\|_{X_q}
\leq 4\widetilde{C}_3(\alpha+1)|a|\}.
\end{align*}
Under the condition (\ref{acondiq}),
let $v$ be the solution obtained in the assertion 1. Then we have
(\ref{v1v22}) in which $v_1$, $v_2$ are replaced by $v$ and $\hat{v}$.
By applying (\ref{e1}), we see that
\begin{align*}
\|v-\hat{v}\|_X\leq
\big\{C_1(\|v\|_X+\|\hat{v}\|_X)+C_2|a|\big\}\|v-\hat{v}\|_X
\leq
\big\{8\,\widetilde{C}_1\widetilde{C}_3(1+\alpha)|a|
+\widetilde{C}_2|a|\big\}\|v-\hat{v}\|_X.
\end{align*}
Furthermore, the condition (\ref{acondiq}) yields
\begin{align*}
8\,\widetilde{C}_1\widetilde{C}_3(1+\alpha)|a|+\widetilde{C}_2|a|<1
\end{align*}
which leads us to $v=\hat{v}$. The proof is complete.
\end{proof}
We note that Proposition \ref{thm2} implies
\begin{align}\label{cw*r}
t^{\frac{1}{2}-\frac{3}{2r}}v\in
BC_{w}\big((0,\infty);L^r_{\sigma}(D)\big)
\end{align}
for all $r\in (3,q)$ by the interpolation inequality
\begin{align*}
\|f\|_r\leq C\|f\|_{3,\infty}^{1-\beta}\|f\|_{q,\infty}^\beta,
\quad
\frac{1}{r}=\frac{1-\beta}{3}+\frac{\beta}{q},
\end{align*}
see (\ref{realinterpolation}).
\par Let $q\in (6,\infty),$ then
the solution obtained in Proposition \ref{thm2}
also fulfills the following decay properties.
\begin{prop}\label{linftydecay}
\par Let $\psi$ be a function on $\mathbb{R}$ satisfying (\ref{psidef}) and
we put $\alpha:=\displaystyle\max_{t\in\mathbb{R}}|\psi'(t)|$.
Suppose that $6<q<\infty$. Then, under the same condition
as in the latter part of
Proposition \ref{thm2}, the solution $v$ obtained in
Proposition \ref{thm2} satisfies $v(t)\in L^r(D)~(t>0)$ with
\begin{align}\label{q<decay}
\|v(t)\|_{r}=O(t^{-\frac{1}{2}+\frac{3}{2q}})
\quad {\rm{as}}~t\rightarrow \infty
\end{align}
for $r\in (q,\infty]$.
\end{prop}

\begin{proof}
We first show (\ref{q<decay}) with $r=\infty$, that is,
$v(t)\in L^\infty(D)$ for $t>0$ with
\begin{align}\label{inftydecay}
\|v(t)\|_{\infty}=O(t^{-\frac{1}{2}+\frac{3}{2q}})
\quad {\rm{as}}~t\rightarrow \infty.
\end{align}
We note by continuity that $C^\infty_{0,\sigma}(D)$ can be
replaced by $PC_0^{\infty}(D)$
as the class of test functions in (\ref{NS6}).
Hence, it follows that
\begin{align}\label{thm1pf6}
\sup_{\varphi\in C_0^{\infty}(D),\|\varphi\|_1\leq 1}|(v(t),\varphi)|
\leq N_1+N_2+N_3+N_4,
\end{align}
where
\begin{align*}
&N_1=\sup_{\varphi\in C_0^{\infty}(D),\|\varphi\|_1\leq 1}
\int_0^t |(v(\tau)\otimes v(\tau),\nabla T(t,\tau)^*P\varphi)|\,d\tau,\\
&N_2=\sup_{\varphi\in C_0^{\infty}(D),\|\varphi\|_1\leq 1}
\int_0^t |(v(\tau)\otimes u_s,\nabla T(t,\tau)^*P\varphi)|\,d\tau,\\
&N_3=\sup_{\varphi\in C_0^{\infty}(D),\|\varphi\|_1\leq 1}
\int_0^t |u_s\otimes v(\tau),\nabla T(t,\tau)^*P\varphi)|\,d\tau,\\
&N_4=
\sup_{\varphi\in C_0^{\infty}(D),\|\varphi\|_1\leq 1}
\int_0^t |(H(\tau),T(t,\tau)^*P\varphi)|\,d\tau.
\end{align*}
We begin by considering $N_1$.
In view of (\ref{gradl1lrad}), we have
\begin{align*}
\int_0^t|(v(\tau)\otimes v(\tau),\nabla T(t,\tau)^*P\varphi)|\,d\tau
&\leq C[v]_{q,\infty}^2
\int_0^t\tau^{-1+\frac{3}{q}}
\left\|
\nabla T(t,\tau)^*P\varphi\right\|_{(1-\frac{2}{q})^{-1},1}\,d\tau\\
&\leq C[v]_{q,\infty}^2\int_0^t
\tau^{-1+\frac{3}{q}}(t-\tau)^{-\frac{3}{q}-\frac{1}{2}}\,d\tau
\,\|\varphi\|_1\\&
\leq C[v]_{q,\infty}^2t^{-\frac{1}{2}}\|\varphi\|_1
\end{align*}
for all $\varphi\in C_0^{\infty}(D)$ and $t>0$.
Here, the integrability is ensured because of $q\in (6,\infty)$.
Hence we obtain
\begin{align}\label{thm1pf7}
N_1\leq Ct^{-\frac{1}{2}}\quad {\rm{for}} ~t>0.
\end{align}
We next consider $N_2$.
By applying (\ref{gradl1lrad}), it follows that
\begin{align*}
\int_0^t|(v(\tau)\otimes u_s,\nabla T(t,\tau)^*P\varphi)|\,d\tau&\leq
[v]_{q,\infty}\|u_s\|_{q,\infty}\int_0^t\tau^{-\frac{1}{2}+\frac{3}{2q}}
\|\nabla T(t,\tau)^*P\varphi\|_{(1-\frac{2}{q})^{-1},1}\,d\tau
\nonumber\\
&\leq C [v]_{q,\infty}\|u_s\|_{q,\infty}t^{-\frac{3}{2q}}\|\varphi\|_1
\end{align*}
for $t>0$. We thus have
\begin{align}
N_2\leq Ct^{-\frac{3}{2q}}\quad {\rm{for}} ~t>0.
\end{align}
We next intend to derive the rate of decay $N_2$
as fast as possible. To this
end, we split the integral into
\begin{align}\label{split}
\int_0^t |(v(\tau)\otimes u_s,\nabla T(t,\tau)^*P\varphi)|\,d\tau
=\int_0^{\frac{t}{2}}+\int^{t-1}_{\frac{t}{2}}+\int_{t-1}^t
\end{align}
for $t>2$. We apply (\ref{gradl1lrad}) again to find
\begin{align*}
\int_0^{\frac{t}{2}}&
\leq \|u_s\|_{3,\infty}\|v\|_X
\int_0^{\frac{t}{2}}\|\nabla T(t,\tau)^*P\varphi\|_{3,1}\,d\tau
\leq Ct^{-\frac{1}{2}}\|\varphi\|_1,\\
\int_{\frac{t}{2}}^{t-1}&\leq\|u_s\|_{3,\infty}[v]_{q,\infty}
\int_{\frac{t}{2}}^{t-1}\tau^{-\frac{1}{2}+\frac{3}{2q}}
\|\nabla T(t,\tau)^*P\varphi\|_{(1-\frac{1}{3}-\frac{1}{q})^{-1},1}\,d\tau
\leq Ct^{-\frac{1}{2}+\frac{3}{2q}}\|\varphi\|_1
\end{align*}
and
\begin{align*}
\int_{t-1}^{t}\leq \|u_s\|_{q,\infty}[v]_{q,\infty}
\int_{t-1}^t\tau^{-\frac{1}{2}+\frac{3}{2q}}\|
\nabla T(t,\tau)^*P\varphi\|_{(1-\frac{2}{q})^{-1},1}\,d\tau
\leq Ct^{-\frac{1}{2}+\frac{3}{2q}}\|\varphi\|_1
\end{align*}
for all $\varphi\in C_0^{\infty}(D)$ and $t>2$.
Summing up the estimates above, we are led to
\begin{align}\label{thm1pf8}
N_2\leq Ct^{-\frac{1}{2}+\frac{3}{2q}}\quad {\rm{for}} ~t>2.
\end{align}
Similarly, we have
\begin{align}
&N_3\leq Ct^{-\frac{3}{2q}}\quad {\rm{for}} ~t>0,\\
&N_3\leq Ct^{-\frac{1}{2}+\frac{3}{2q}}\quad {\rm{for}} ~t>2.
\end{align}
It is easily seen from (\ref{psidef}),
(\ref{hstaest2}) and (\ref{l1lr2ad}) that
\begin{align*}
\int_0^t |(H(\tau),T(t,\tau)^*P\varphi)|\,d\tau
&\leq
C(a^2+\alpha|a|)\int_0^{\min\{1,t\}}
\|T(t,\tau)^*P\varphi\|_{\frac{3}{2},1}\,d\tau\\
&\leq C(a^2+\alpha|a|)\int_0^{\min\{1,t\}}
(t-\tau)^{-\frac{1}{2}}\,d\tau\,\|\varphi\|_1
\end{align*}
for all $\varphi\in C_0^{\infty}(D)$ and $t>0$, which yields
\begin{align}
&N_4\leq Ct^{\frac{1}{2}}\quad {\rm{for}} ~t>0,\\
&N_4\leq Ct^{-\frac{1}{2}}\quad {\rm{for}} ~t>2.\label{thm1pf9}
\end{align}
Combining (\ref{thm1pf6})--(\ref{thm1pf9})
implies $v(t)\in L^\infty(D)$ ($t>0$) and
(\ref{inftydecay}). In view of the interpolation relation
\begin{align*}
(L^{q,\infty}(D),L^{\infty}(D))_{1-\frac{q}{r},r}=L^r(D),\quad
\|f\|_r\leq C\|f\|^{\frac{q}{r}}_{q,\infty}
\|f\|^{1-\frac{q}{r}}_\infty,\quad q<r<\infty,
\end{align*}
we obtain (\ref{q<decay}) for $r\in (q,\infty]$ as well.
This completes the proof.
\end{proof}

\begin{rmk}\label{rmk3}
When the stationary solution possesses the scale-critical rate
$O(1/|x|)$, the $L^\infty$ decay of perturbation with less sharp
rate $O(t^{-\frac{1}{2}+\varepsilon})$ was derived first by
Koba \cite{koba2017} in the context of stability analysis, where
$\varepsilon>0$ is arbitrary. If we have a look only at the
$L^\infty$ decay rate, our rate is comparable with his result
since $q\in (6,\infty)$ is arbitrary. However, we are not able to
prove Proposition \ref{linftydecay} by his method.
This is because he doesn't split the integrals in $N_2$ and $N_3$,
so that the rate of $L^\infty$ decay is slower than the one of
$L^{q,\infty}$ decay. From this point of view, Proposition
\ref{linftydecay} is regarded as a slight improvement of his result.
\end{rmk}

We next show that the solution $v$ obtained in Proposition \ref{thm2}
actually satisfies the integral equation
(\ref{integraleq}) by
identifying $v$ with a local solution $\widetilde{v}$ of (\ref{integraleq})
in a neighborhood of each time $t>0$.
To this end, we need the following lemma on the uniqueness.
The proof is similar to the argument in the second half
(after (\ref{v1v22})) of the proof of Theorem \ref{thm3} and thus we may
omit it.

\begin{lem}\label{uniqueness}
Let $3<r<\infty,~0\leq t_0<t_1<\infty$
and $v_0\in L^r_{\sigma}(D)$. Then the problem
\begin{align}\label{NS7}
(v(t),\varphi)=(v_0,T(t,t_0)^*\varphi)
&+\int_{t_0}^t\big(v(\tau)\otimes v(\tau)
+\psi(\tau)\{v(\tau)\otimes u_s+u_s\otimes v(\tau)\},
\nabla T(t,\tau)^*\varphi\big)\,d\tau\nonumber\\&
+\int_{t_0}^t(H(\tau),T(t,\tau)^*\varphi)\,d\tau,\quad
\forall \varphi\in C^{\infty}_{0,\sigma}(D)
\end{align}
on $(t_0,t_1)$ admits at most one solution within the class
$L^{\infty}(t_0,t_1;L_{\sigma}^r(D))$.
Here, $H$ is given by (\ref{h}).
\end{lem}

Given $v_0\in L^r_{\sigma}(D)$ with $r\in (3,\infty)$, let us
construct a local solution of the integral equation
\begin{align}\label{intt0}
v(t)=T(t,t_0)v_0+\int_{t_0}^t &T(t,\tau)P[(Gv)(\tau)+H(\tau)]\,d\tau,
\end{align}
where $G$ and $H$ are defined by (\ref{g}) and (\ref{h}),
respectively. For $0\leq t_0<t_1<\infty$ and $r\in (3,\infty)$,
we define the function space
\begin{align}
&Y_r(t_0,t_1)=\big\{v\in C\big([t_0,t_1];L^{r}_{\sigma}(D)\big)
\,\big|\,
(\cdot-t_0)^{\frac{1}{2}}\nabla v(\cdot)\in
BC_w\big((t_0,t_1];L^r(D)\big)\big\}
\end{align}
which is a Banach space equipped with norm
\begin{align}
&\|v\|_{Y_r(t_0,t_1)}=\sup_{t_0\leq\tau\leq t_1}\|v(\tau)\|_r
+\sup_{t_0<\tau\leq t_1}
(\tau-t_0)^{\frac{1}{2}}\|\nabla v(\tau)\|_r\label{Yr}
\end{align}
and set
\begin{align}
&U_1(v,w)(t)=\int_{t_0}^t
T(t,\tau)P[v(\tau)\cdot \nabla w(\tau)]\,d\tau,\quad
U_2(v)(t)=\int_{t_0}^t T(t,\tau)P[\psi(\tau)v(\tau)\cdot \nabla u_s]\,d\tau,\nonumber\\
&U_3(v)(t)=\int_{t_0}^t T(t,\tau)
P[\psi(\tau)u_s\cdot \nabla v(\tau)]\,d\tau,\quad
U_4(t)=\int_{t_0}^t T(t,\tau)PH(\tau)\,d\tau.\label{Udef}
\end{align}

\begin{lem}\label{localkey}
Let $3<r<\infty$ and $0\leq t_0<t_1\leq t_0+1$.
Suppose that $v,w\in Y_r(t_0,t_1)$.
Then $U_1(v,w),U_2(v),U_3(v),U_4\in Y_r(t_0,t_1)$.
Furthermore, there exists a constant $C=C(r,t_0)$ such that
\begin{align}
&\|U_1(v,w)\|_{Y_r(t_0,t)}\leq C(t-t_0)^{\frac{1}{2}-\frac{3}{2r}}
\|v\|_{Y_r(t_0,t)}\|w\|_{Y_r(t_0,t)\,,}\label{Y1}\\
&\|U_2(v)\|_{Y_r(t_0,t)}\leq
C(t-t_0)^{1-\frac{3}{2r}}\|\nabla u_s\|_r
\|v\|_{Y_r(t_0,t)\,,}\label{Y2}\\
&\|U_3(v)\|_{Y_r(t_0,t)}\leq C(t-t_0)^{\frac{1}{2}-\frac{3}{2r}}
\|u_s\|_r\|v\|_{Y_r(t_0,t)\,,}\label{Y3}\\
&\|U_4\|_{Y_r(t_0,t)}
\leq C(t-t_0)^{\frac{1}{2}+\frac{3}{2r}}(a^2+\alpha|a|)\label{Y4}
\end{align}
for all $t\in (t_0,t_1].$
\end{lem}

\begin{proof}
In view of (\ref{lqlr0}), we have
\begin{align}
\|U_1(t)\|_r\leq C\int_{t_0}^t (t-\tau)^{-\frac{3}{2r}}
\|v\|_r\|\nabla w\|_r\,d\tau
\leq C(t-t_0)^{-\frac{3}{2r}+\frac{1}{2}}
\|v\|_{Y_r(t_0,t)}\|w\|_{Y_r(t_0,t)}.\label{Y11}
\end{align}
Furthermore, (\ref{gradT}) with $\mathcal{T}=t_0+1$ yields
\begin{align}\label{Y12}
\|\nabla U_1(t)\|_r
\leq C(t-t_0)^{-\frac{3}{2r}}\|v\|_{Y_r(t_0,t)}
\|w\|_{Y_r(t_0,t)}.
\end{align}
By (\ref{Y11}) and (\ref{Y12}), we obtain (\ref{Y1}).
Similarly, we can show (\ref{Y2})--(\ref{Y4}).
We note that the estimate (\ref{Y4})
follows from (\ref{gradT2}) with $\mathcal{T}=t_0+1$
together with (\ref{hstaest2}).
\par We next show the continuity of $U_1$ with respect to $t$.
Let $t_2\in [t_0,t_1]$. If $t_2<t,$ we have
\begin{align*}
U_1(t)-U_1(t_2)&=\int_{t_0}^{t_2}(T(t,t_2)-1)
T(t_2,\tau)P[v(\tau)\cdot\nabla w(\tau)]\,d\tau
+\int_{t_2}^t T(t,\tau)P[v(\tau)\cdot
\nabla w(\tau)]\,d\tau\\&=:U_{11}(t)+U_{12}(t).
\end{align*}
Lebesgue's convergence theorem yields that
$\|U_{11}(t)\|_r\rightarrow 0$ as $t\rightarrow t_2$, while
\begin{align*}
\|U_{12}(t)\|_r\leq C(t-t_2)^{\frac{1}{2}-\frac{3}{2r}}
\|v\|_{Y_r(t_0,t_1)}\|w\|_{Y_r(t_0,t_1)}
\rightarrow 0 \quad {\rm{as}}~t\rightarrow t_2.
\end{align*}
To discuss the case $t<t_2$, we need the following device.
Let $(t_0+t_2)/2\leq \tilde{t}<t<t_2$,
where $\tilde{t}$ will be determined later, then
\begin{align*}
U_1(t)-U_1(t_2)=\left(\int_{t_0}^{\tilde{t}}+\int_{\tilde{t}}^t\right)
T(t,\tau)&P[v(\tau)\cdot \nabla w(\tau)]\,d\tau\\
&-\left(\int_{t_0}^{\tilde{t}}+\int_{\tilde{t}}^{t_2}\right)
T(t_2,\tau)P[v(\tau)\cdot \nabla w(\tau)]\,d\tau.
\end{align*}
We observe that
\begin{align*}
\int_{\tilde{t}}^t&\|T(t,\tau)P[v(\tau)\cdot\nabla w(\tau)]\|_r\,d\tau
+\int_{\tilde{t}}^{t_2}
\|T(t_2,\tau)P[v(\tau)\cdot\nabla w(\tau)]\|_r\,d\tau\\&
\leq C\|v\|_{Y_r(t_0,t_1)}\|w\|_{Y_r(t_0,t_1)}\left(
\int_{\tilde{t}}^t(t-\tau)^{-\frac{3}{2r}}
(\tau-t_0)^{-\frac{1}{2}}\,d\tau+
\int_{\tilde{t}}^{t_2}(t_2-\tau)^{-\frac{3}{2r}}
(\tau-t_0)^{-\frac{1}{2}}\,d\tau\right)\\
&\leq\frac{2C}{1-\frac{3}{2r}}\|v\|_{Y_r(t_0,t_1)}\|w\|_{Y_r(t_0,t_1)}
\left(\frac{t_0+t_2}{2}-t_0\right)^{-\frac{1}{2}}
(t_2-\tilde{t}\,)^{1-\frac{3}{2r}}.
\end{align*}
For any $\varepsilon>0$, we choose $\tilde{t}$ such that
\begin{align*}
\frac{2C}{1-\frac{3}{2r}}\|v\|_{Y_r(t_0,t_1)}\|w\|_{Y_r(t_0,t_1)}
\left(\frac{t_0+t_2}{2}-t_0\right)^{-\frac{1}{2}}
(t_2-\tilde{t}\,)^{1-\frac{3}{2r}}<\varepsilon
\end{align*}
which yields
\begin{align*}
\|U_1(t)-U_1(t_2)\|_r&\leq
\int_{t_0}^{\tilde{t}}
\left\|\big(T(t,\tau)-T(t_2,\tau)\big)
P[v(\tau)\cdot\nabla w(\tau)]\right\|_r\,d\tau+\varepsilon
\quad {\rm{for~}}\tilde{t}<t<t_{2}
\end{align*}
and therefore,
\begin{align}\label{111}
\limsup_{t\rightarrow t_2}\|U_1(t)-U_1(t_2)\|_r
\leq \limsup_{t\rightarrow t_2}\int_{t_0}^{\tilde{t}}
\left\|\big(T(t,\tau)-T(t_2,\tau)\big)
P[v(\tau)\cdot\nabla w(\tau)]\right\|_r\,d\tau+\varepsilon.
\end{align}
Since
$\big\|\big(T(t,\tau)-T(t_2,\tau)\big)
P[v(\tau)\cdot\nabla w(\tau)]\big\|_r=
\big\|\big(T(t,\tilde{t}\,)-T(t_2,\tilde{t}\,)\big)
T(\tilde{t},\tau)P[v(\tau)\cdot\nabla w(\tau)]\big\|_r$
tends to $0$ as
$t\rightarrow t_2$ for $t_0<\tau<\tilde{t}$,
it follows from Lebesgue's convergence theorem
that the integral term in (\ref{111}) tends to $0$ as $t\rightarrow t_{2}$.
Since $\varepsilon>0$ is arbitrary, we have
\begin{align}\label{Y1conti}
U_1\in C\big([t_0,t_1];L^r_{\sigma}(D)\big).
\end{align}
Furthermore, we find $\nabla U_1\in C_w\big((t_0,t_1];L^r(D)\big)$
on account of (\ref{Y12}) and (\ref{Y1conti})
together with the relation
\begin{align*}
(\nabla U_1(t)-\nabla U_1(t_2),\varphi)
=-(U_1(t)-U_1(t_2),\nabla \cdot\varphi)
\end{align*}
for all $t_2\in (t_0,t_1]$ and
$\varphi\in C_0^{\infty}(D)^{3\times 3}$.
Since $U_2,U_3$ and $U_4$ are discussed similarly,
the proof is complete.
\end{proof}

The following proposition provides a local solution of (\ref{intt0}).
\begin{prop}\label{localexistence}
Let $3<r<\infty$, $t_0\geq 0$ and $v_0\in L^r_{\sigma}(D)$.
There exists $t_1\in (t_0,t_0+1]$ such that (\ref{intt0})
admits a unique solution $v\in Y_r(t_0,t_1)$.
Moreover, the length of the existence interval can be estimated
from below by
\begin{align*}
t_1-t_0\geq \zeta(\|v_0\|_r),
\end{align*}
where $\zeta(\cdot):[0,\infty)\rightarrow (0,1)$
is a non-increasing function defined by (\ref{zeta}) below.
\end{prop}

\begin{proof}
We put
\begin{align*}
(\Psi v)(t)={\rm{the~RHS~of~(\ref{intt0})}}.
\end{align*}
By applying Lemma \ref{localkey}, we have
\begin{align*}
&\|\Psi v\|_{Y_r(t_0,t)}\leq
(C_1\|v\|_{Y_r(t_0,t)}^2
+C_2\|v\|_{Y_r(t_0,t)}
+C_3)(t-t_0)^{\frac{1}{2}-\frac{3}{2r}}+C_4\|v_0\|_r,\\
&\|\Psi v-\Psi w\|_{Y_r(t_0,t)}\leq
\{C_1(\|v\|_{Y_r(t_0,t)}
+\|w\|_{Y_r(t_0,t)})+C_2\}(t-t_0)^{\frac{1}{2}-\frac{3}{2r}}
\|v-w\|_{Y_r(t_0,t)}
\end{align*}
for all $t\in(t_0,t_0+1]$ and $v,w\in Y_r(t_0,t)$.
We note that the constants $C_i$ may be
dependent on $\|u_s\|_r$, $\|\nabla u_s\|_r$, $\alpha$ and $a$.
We choose $t_1\in(t_0,t_0+1]$ such that
\begin{align}
&C_2(t_1-t_0)^{\frac{1}{2}-\frac{3}{2r}}<\frac{1}{2},\label{t12}\\
&8C_1(t_1-t_0)^{\frac{1}{2}-\frac{3}{2r}}
\{C_3(t_1-t_0)^{\frac{1}{2}-\frac{3}{2r}}+C_4\|v_0\|_r\}
<\frac{1}{2}\label{t13}
\end{align}
which imply
\begin{align}\label{t11}
&\lambda:=\big\{1-C_2(t_1-t_0)^{\frac{1}{2}-\frac{3}{2r}}\big\}^2
-4C_1(t_1-t_0)^{\frac{1}{2}-\frac{3}{2r}}
\big\{C_3(t_1-t_0)^{\frac{1}{2}-\frac{3}{2r}}+C_4\|v_0\|_r\big\}>0.
\end{align}
We set
\begin{align*}
&\Lambda:=\frac{1-C_2(t_1-t_0)^{\frac{1}{2}-\frac{3}{2r}}
-\sqrt{\lambda}}{2C_1(t_1-t_0)^{\frac{1}{2}-\frac{3}{2r}}}
<4(C_3+C_4\|v_0\|_r),\\
&Y_{r,\Lambda}(t_0,t_1):=\{v\in Y_r(t_0,t_1)\mid\|v\|_{Y_r(t_0,t_1)}
\leq \Lambda\}.
\end{align*}
Then we find that the map $\Psi:Y_{r,\Lambda}(t_0,t_1)\rightarrow
Y_{r,\Lambda}(t_0,t_1)$ is well-defined and also contractive.
Hence we obtain a local solution.
Indeed, the conditions (\ref{t12}) and (\ref{t13}) are accomplished by
\begin{align*}
t_1-t_0<\min\left\{1,\left(\frac{1}{2C_2}\right)
^{(\frac{1}{2}-\frac{3}{2r})^{-1}},
\left(\frac{1}{16C_1(C_3+C_4\|v_0\|_r)}\right)
^{(\frac{1}{2}-\frac{3}{2r})^{-1}}\right\}.
\end{align*}
Thus, it is possible to take $t_1$ such that
\begin{align}\label{zeta}
t_1-t_0\geq
\frac{1}{2}\min\left\{1,\left(\frac{1}{2C_2}\right)
^{(\frac{1}{2}-\frac{3}{2r})^{-1}},
\left(\frac{1}{16C_1(C_3+C_4\|v_0\|_r)}\right)
^{(\frac{1}{2}-\frac{3}{2r})^{-1}}\right\}=:\zeta(\|v_0\|_r).
\end{align}
The proof is complete.
\end{proof}

\begin{lem}\label{betterregular}
Let $3<r<\infty$, $t_0\geq 0$ and $v_0\in L^r_\sigma(D).$
The local solution $v$ obtained in Proposition \ref{localexistence}
also possesses the following properties:
\begin{align}\label{regular}
v\in C\big((t_0,t_1];L^\kappa_\sigma(D)\big)
\cap C_{w^*}\big((t_0,t_1];L^\infty(D)\big)
\end{align}
for every $\kappa\in (r,\infty)$ and
\begin{align}
\nabla v\in C_w\big((t_0,t_1];L^\gamma(D)\big)
\end{align}
for every $\gamma\in(r,\infty)$ satisfying
\begin{align}\label{rlrelation}
\frac{2}{r}-\frac{1}{\gamma}<\frac{1}{3}.
\end{align}
\end{lem}

\begin{proof}
By using (\ref{lqlr0}) and (\ref{lqlr2}) and the semigroup property,
we find $v(t)\in L^{\infty}(D)$ with
\begin{align}\label{inftybdd}
\|v(t)\|_\infty\leq C(t-t_0)^{-\frac{3}{2r}}\big\{\|v_0\|_r+
\|v\|^2_{Y_r(t_0,t_1)}+
\|v\|_{Y_r(t_0,t_1)}(\|u_s\|_r+\|\nabla u_s\|_r)+(a^2+\alpha|a|)
\big\}
\end{align}
for all $t\in (t_0,t_1]$. Moreover, for each $t_2\in(t_0,t_1]$,
we know from $v\in C([t_0,t_1];L^r_\sigma(D))$ that
\begin{align}
\big(v(t),\varphi\big)-\big(v(t_2),\varphi\big)
\rightarrow 0\quad {\rm as}~t\rightarrow t_2
\end{align}
for all $\varphi\in C_0^{\infty}(D)$, which combined with (\ref{inftybdd})
yields $v\in C_{w^*}\big((t_0,t_1];L^\infty(D)\big)$. Since
\begin{align*}
\|v(t)-v(t_2)\|_\kappa\leq \|v(t)-v(t_2)\|^{\frac{r}{\kappa}}_r
\|v(t)-v(t_2)\|_\infty^{1-\frac{r}{\kappa}}
\end{align*}
for $\kappa\in(r,\infty)$ and $t_2\in(t_0,t_1]$,
it follows from (\ref{inftybdd}) that
\begin{align}\label{contikappa1}
v\in C\big((t_0,t_1];L_\sigma^\kappa(D)\big)
\quad{\rm for}~\kappa\in (r,\infty).
\end{align}
\par The estimates (\ref{gradT}) and (\ref{gradT2}) with $\mathcal{T}=t_0+1$
imply that if we assume (\ref{rlrelation}),
we have $\nabla v(t)\in L^\gamma (D)$ with
\begin{align}
\|\nabla v(t)\|_\gamma\leq
C(t-t_0)^{-\frac{3}{2}(\frac{1}{r}-\frac{1}{\gamma})-\frac{1}{2}}\big\{
\|v_0\|_r+\|v\|^2_{Y_r(t_0,t_1)}
&+\|v\|_{Y_r(t_0,t_1)}(\|u_s\|_r+\|\nabla u_s\|_r)\nonumber\\&
+(a^2+\alpha|a|)\big\}\label{nablabdd}
\end{align}
for all $t\in (t_0,t_1]$.
Here, we note that (\ref{rlrelation}) is needed for estimates of
$\nabla U_1$ and $\nabla U_3$ given in (\ref{Udef}).
On account of (\ref{contikappa1}), (\ref{nablabdd}) and
\begin{align*}
(\nabla v(t)-\nabla v(t_2),\varphi)
=-(v(t)-v(t_2),\nabla\cdot\varphi)
\end{align*}
for all $t_2\in (t_0,t_1]$ and $\varphi\in C_0^\infty(D)^{3\times 3},$
we find the weak continuity of $\nabla v$ with values in $L^{\gamma}(D)$.
The proof is complete.
\end{proof}

We close the paper with completion of the proof of Theorem \ref{thm1}.
\vspace{0.2cm}\\
\noindent{\bf Proof of Theorem \ref{thm1}}\quad It remains to show that
the solution $v$ obtained in Proposition \ref{thm2}
also satisfies (\ref{integraleq}) with
\begin{align}\label{contikappa}
v\in C\big((0,\infty);L^\kappa_\sigma(D)\big)\cap
C_{w^*}\big((0,\infty);L^\infty(D)\big),\quad
\nabla v\in C_w\big((0,\infty);L^\kappa(D)\big)
\end{align}
for all $3<\kappa<\infty$.
Let $t_*\in (0,\infty).$
By applying Proposition \ref{localexistence}
and Lemma \ref{betterregular} with $r=6$,
we can see that for each $t_0\in [t_*/2,t_*)$,
there exists $\widetilde{v}\in Y_6(t_0,t_1)$
which satisfies (\ref{intt0}) and therefore, (\ref{NS7}) with $v_0=v(t_0)$
such that
\begin{align*}
\widetilde{v}\in C\big((t_0,t_1];L^\kappa_\sigma(D)\big)\cap
C_{w^*}\big((t_0,t_1];L^\infty(D)\big),\quad
\nabla\, \widetilde{v}\in C_w\big((t_0,t_1];L^\kappa(D)\big)
\end{align*}
for all $\kappa\in [6,\infty)$.
Moreover, the length of the existence interval can be estimated by
\begin{align*}
t_1-t_0\geq \zeta(\|v(t_0)\|_6)\geq
\zeta\left(C_5\left(\frac{t_*}{2}\right)^{-\frac{1}{4}}\right)
=:\varepsilon,
\end{align*}
where $\zeta(\cdot)$ is the non-increasing function
in Proposition \ref{localexistence} because of
\begin{align*}
\|v(t)\|_6\leq C_5\left(\frac{t_*}{2}\right)^{-\frac{1}{4}}
\end{align*}
for all $t\geq t_*/2$, see (\ref{cw*r}).
We note that the solution $v$ obtained in Proposition \ref{thm2}
also satisfies (\ref{NS7})  with $v_0=v(t_0)$
since $C^\infty_{0,\sigma}(D)$ can be replaced
by $L^{6/5}_\sigma(D)$ as the class of test functions in (\ref{NS6}).
Let us take $t_0:=\max\{t_*/2,t_*-\varepsilon/2\}$
so that $t_*\in (t_0,t_1)$,
in which $v=\widetilde{v}$ on account of Lemma \ref{uniqueness}.
Since $t_*$ is arbitrary,
we conclude (\ref{contikappa}) for $\kappa\in [6,\infty)$.
It is also proved by applying
Proposition \ref{localexistence} with $r\in (3,6)$
that the solution belongs
to the class (\ref{contikappa}) for $\kappa\in (3,6)$ as well.
The proof is complete.\qed
\vspace{0.3cm}\\
\noindent{\bf{Acknowledgment.}}~~The author would like to
thank Professor Toshiaki Hishida for valuable
discussions to improve this paper.