\section{Introduction}\label{introsec}

We study the long-time behavior of the three-dimensional incompressible viscous fluids rotated about the vertical axis with a constant angular speed.
The Navier--Stokes equations (NSE) written in the rotating frame are used to describe the fluid dynamics in this case, see, e.g., \cite{VanyoBook2001}.
We denote by $\bvec x\in\ensuremath{\mathbb R}^3$ the spatial variables, $t\in\ensuremath{\mathbb R}_+=[0,\infty)$ the time variable, and  $\{\bvec e_1,\bvec e_2,\bvec e_3\}$ the standard canonical basis of $\ensuremath{\mathbb R}^3$.
The NSE for the rotating fluids  are
\begin{align}
\label{NPDE}
\frac{\partial \bvec u}{\partial t} - \nu \Delta \bvec u + (\bvec u\cdot\nabla) \bvec u + \nabla p +
\Omega \bvec e_3\times \bvec u &=0,\\
\label{divfree}
{\rm div}\, \bvec u&=0,
\end{align}
where $\bvec u(\bvec x,t)$ is the velocity field, $p$ is the pressure adjusted by the fluid's constant density, gravity and centrifugal force,
 $\nu>0$ is the kinematic viscosity, and $\frac{1}{2}\Omega \bvec e_3$ is the angular velocity of the rotation.

Above, $\Omega \bvec e_3\times \bvec u$ represents the Coriolis force exerted on the fluid. We will write
\begin{equation*}
\bvec e_3\times \bvec u=J \bvec u,\text{ where }
J=\begin{pmatrix}
  0&-1&0\\1&0&0\\0&0&0
 \end{pmatrix}.
\end{equation*}

Equations \eqref{NPDE} and \eqref{divfree} comprise a system of nonlinear partial differential equations with the unknowns $\bvec u$ and $p$, while the constants $\nu>0$ and $\Omega$ are given. We will study this system within the context of spatially periodic functions.

Let $L_1,L_2,L_3>0$ be the spatial periods and denote $\bvec L=(L_1,L_2,L_3)$.  A function $f:\ensuremath{\mathbb R}^3\to\ensuremath{\mathbb R}^m$, for some $m\in\ensuremath{\mathbb N}$, is $\bvec L$-periodic if
\begin{equation*}
f(\bvec x+L_j \bvec e_j)=f(\bvec x)\text{ for all } \bvec x\in\ensuremath{\mathbb R}^3,\ j=1,2,3.
\end{equation*}

Consider the $\bvec L$-periodic solutions $(\bvec u,p)$, that is, $\bvec u(\cdot,t)$ and $p(\cdot,t)$ are $\bvec L$-periodic for all $t> 0$.

Let $L_*=\max\{L_1,L_2,L_3\}$ and $\lambda_1=(2\pi/L_*)^2$.
Under the transformation
\begin{equation*} \bvec u(\bvec x,t)=\lambda_1^{1/2}\nu\, \bvec v(\lambda_1^{1/2}\bvec x,\lambda_1\nu
 t),
 \quad p(\bvec x,t)=\lambda_1\nu^2 q(\lambda_1^{1/2}\bvec x,\lambda_1\nu t),
 \quad \Omega=\lambda_1\nu\omega,
 \end{equation*}
where $\bvec v(\bvec y,\tau)$ and $q(\bvec y,\tau)$ are $\lambda_1^{1/2}\bvec L$-periodic adimensional functions,
system  \eqref{NPDE} and \eqref{divfree} becomes
\begin{equation*} \lambda_1^{3/2}\nu^2\Big(\frac{\partial \bvec v}{\partial \tau} - \Delta_{\bvec y} \bvec v + (\bvec v\cdot\nabla_{\bvec y}) \bvec v + \nabla_{\bvec y} q + \omega J \bvec v  \Big)=0\text{ and }
 \lambda_1\nu\, {\rm div}_{\bvec y} \bvec v=0,
\end{equation*}
thus,
\begin{equation}\label{vyeq}
\frac{\partial \bvec v}{\partial \tau} - \Delta_{\bvec y} \bvec v + (\bvec v\cdot\nabla_{\bvec y}) \bvec v + \nabla_{\bvec y} q + \omega J \bvec v =0
\text{ and }
{\rm div}_{\bvec y} \bvec v=0.
\end{equation}

Thanks to \eqref{vyeq},  we can assume hereafter, without loss of generality, that the Navier--Stokes system \eqref{NPDE} and \eqref{divfree} has
\begin{equation}\label{fixpar} \nu=1,\quad  L_*=2\pi,\quad  \lambda_1=1.
\end{equation}

In dealing with $\bvec L$-periodic functions, it is convenient to formulate the equations and functional spaces using the domain (the three-dimensional flat torus)
$${\mathbb T_\bvec L}\stackrel{\rm def}{=} (\ensuremath{\mathbb R}/L_1\ensuremath{\mathbb Z})\times (\ensuremath{\mathbb R}/L_2\ensuremath{\mathbb Z})\times (\ensuremath{\mathbb R}/L_3\ensuremath{\mathbb Z}).$$

Regarding the notation in this paper, a vector in $\ensuremath{\mathbb C}^3$ is viewed as a column vector, and we denote the dot product between $\bvec x,\bvec y\in\ensuremath{\mathbb C}^3$ by
\begin{equation*}
\bvec x\cdot \bvec y =\bvec y^{\rm T}\bvec x=\bvec x^{\rm T}\bvec y.
\end{equation*}
Hence, the standard inner product in $\ensuremath{\mathbb C}^3$ is $\bvec x\cdot\bar\bvec y$.

For $\bvec k=(k_1,k_2,k_3)\in\ensuremath{\mathbb Z}^3$, denote
\begin{equation}\label{kcheck}
\check \bvec k=(\check k_1,\check k_2,\check k_3)\stackrel{\rm def}{=} 2\pi (k_1/L_1,k_2/L_2,k_3/L_3),
\end{equation}
and, in case $\bvec k\ne \mathbf 0$,
\begin{equation}\label{ktil}
\widetilde \bvec k=(\widetilde k_1,\widetilde k_2,\widetilde k_3)\stackrel{\rm def}{=} \check\bvec k/|\check\bvec k|,
\end{equation}
\begin{equation}\label{Xk}
  X_\bvec k\stackrel{\rm def}{=} \{ \bvec z\in \ensuremath{\mathbb C}^3: \bvec z\cdot\check\bvec k=0 \}=\{ \bvec z\in \ensuremath{\mathbb C}^3: \bvec z\cdot\widetilde\bvec k=0 \}.
\end{equation}

In the following, we present the functional setting and functional formulations for the NSE. We refer the reader to the books \cite{LadyFlowbook69,CFbook,TemamAMSbook,TemamSIAMbook,FMRTbook} for more details.

Denote the inner product and norm in $L^2({\mathbb T_\bvec L})^3$ by $\inprod{\cdot,\cdot}$ and $|\cdot|$, respectively.
The latter notation is also used for the modulus of a complex number and the length of a vector in $\ensuremath{\mathbb C}^n$, but its meaning will be clear from the context.

Each $\bvec u\in L^2({\mathbb T_\bvec L})^3$ has the Fourier series
\begin{equation}\label{uF}
\bvec u(\bvec x)=\sum_{\bvec k\in\ensuremath{\mathbb Z}^3} \widehat \bvec u_\bvec k e^{i\check\bvec k\cdot\bvec x},
\end{equation}
where $i=\sqrt{-1}$, $\widehat\bvec u_\bvec k\in \ensuremath{\mathbb C}^3$ are the Fourier coefficients with the reality condition $\widehat \bvec u_{-\bvec k}= \overline{\widehat \bvec u_\bvec k}$.
If $\bvec u$ has zero spatial average over ${\mathbb T_\bvec L}$ then $\widehat\bvec u_{\mathbf 0}=0$.

We now focus on the case of solutions $\bvec u(\bvec x,t)$ with zero spatial average over ${\mathbb T_\bvec L}$, for any $t\ge 0$. The general case will be studied in section \ref{nonzerosec}. For brevity, denote
\begin{equation*}
\sideset{}{'}\sum=\sum_{\bvec k\in\ensuremath{\mathbb Z}^3\setminus\{\mathbf 0\}}.
\end{equation*}

Let $\mathcal V$ be the space of zero-average, divergence-free $\bvec L$-periodic trigonometric polynomial vector fields, that is, it consists  of
functions
$$u=\sideset{}{'}\sum  \widehat\bvec u_\bvec k e^{i \check \bvec k\cdot \bvec x}$$
where  $\widehat\bvec u_\bvec k\in X_\bvec k$,
 $\widehat\bvec u_{-\bvec k}=\overline{\widehat\bvec u_\bvec k}$ for all $\bvec k\in\ensuremath{\mathbb Z}^3\setminus\{\mathbf 0\}$,
and $\widehat\bvec u_\bvec k\ne \mathbf 0$ for only finitely many $\bvec k$'s.

Let $H$, respectively (resp.) $V$, be the closure of $\mathcal V$ in $L^2({\mathbb T_\bvec L})^3$, resp. $H^1({\mathbb T_\bvec L})^3$.

We use the following embeddings and identification
$$V\subset H=H'\subset V',$$
where each space is dense in the next one, and the embeddings are compact.

Let $\mathcal P$ denote the orthogonal (Leray) projection from $L^2({\mathbb T_\bvec L})^3$ onto $H$.
More precisely,
\begin{equation*}
\mathcal P \Big(\sum_{\bvec k\in\ensuremath{\mathbb Z}^3}  \widehat \bvec u_\bvec k e^{i\check\bvec k\cdot\bvec x}\Big)=\sideset{}{'}\sum [\widehat \bvec u_\bvec k-(\widehat \bvec u_\bvec k\cdot \widetilde\bvec k)\widetilde\bvec k] e^{i\check\bvec k\cdot\bvec x}
=\sum_{\bvec k\in\ensuremath{\mathbb Z}^3} \widehat P_\bvec k \widehat \bvec u_\bvec k e^{i\check\bvec k\cdot\bvec x}
=\sideset{}{'}\sum \widehat P_\bvec k \widehat \bvec u_\bvec k e^{i\check\bvec k\cdot\bvec x},
\end{equation*}
where $\widehat P_\bvec k$'s are symmetric $3\times 3$ matrices given by
\begin{equation}\label{Pk}
\widehat P_{\mathbf 0}=0,\quad
\widehat P_\bvec k=I_3-\widetilde \bvec k \widetilde\bvec k^{\rm T}\text{ for }\bvec k\in\ensuremath{\mathbb Z}^3\setminus\{\mathbf 0\}.
\end{equation}

The Stokes operator $A$ is a bounded linear mapping from $V$ to its dual space $V'$ defined by
\begin{equation*}
\inprod{A\bvec u,\bvec v}_{V',V}=
\doubleinprod{\bvec u,\bvec v}
\stackrel{\rm def}{=} \sum_{j=1}^3 \inprod{ \frac{\partial \bvec u}{\partial x_j} , \frac{\partial \bvec v}{\partial x_j} }, \text{ for all } \bvec u,\bvec v\in V.
\end{equation*}

As an unbounded operator on $H$, the operator $A$ has the domain $\mathcal D(A)=V\cap H^2({\mathbb T_\bvec L})^3$, and, under  the current consideration of periodicity conditions,
\begin{equation*} A\bvec u = - \mathcal P \Delta \bvec u=-\Delta \bvec u\in H, \text{ for all }\bvec u\in\mathcal D(A).
\end{equation*}
With the Fourier series, this reads as
\begin{equation*}
Au=\sideset{}{'}\sum |\check \bvec k|^2 \widehat\bvec u_\bvec k e^{i \check \bvec k\cdot \bvec x},
\text{ for }
u=\sideset{}{'}\sum \widehat\bvec u_\bvec k e^{i \check \bvec k\cdot \bvec x}\in\mathcal D(A).
\end{equation*}

The spectrum  of $A$ is known to be
\begin{equation}\label{spA}
\sigma(A)=\big\{|\check\bvec k|^2: \ \bvec k\in\ensuremath{\mathbb Z}^3, \bvec k\ne \mathbf 0\big\},
\end{equation}
and each $\lambda\in \sigma(A)$ is an eigenvalue with finite multiplicity.
We order
\begin{equation}\label{Biglam}
\sigma(A)=\{\Lambda_n:n\in\ensuremath{\mathbb N}\},\text{ where the sequence $(\Lambda_n)_{n=1}^\infty$ is strictly increasing}.
\end{equation}

The additive semigroup generated by $\sigma(A)$ is
\begin{equation}\label{semispecA}
\langle \sigma(A)\rangle \stackrel{\rm def}{=} \Big\{\sum_{j=1}^n \alpha_j \,:\, n\in\ensuremath{\mathbb N}, \alpha_j\in\sigma(A)\text{ for } 1\le j\le n\Big\}.
\end{equation}

The set $\langle \sigma(A)\rangle$ is ordered as a strictly increasing sequence $(\mu_n)_{n=1}^\infty$, i.e.,
\begin{equation}\label{mudef}
\langle \sigma(A)\rangle=\{\mu_n:n\in\ensuremath{\mathbb N}\},\quad \mu_{n+1}>\mu_n \quad\forall n\in\ensuremath{\mathbb N}.
\end{equation}

Note, by \eqref{fixpar}, \eqref{spA}, \eqref{Biglam} and \eqref{mudef}, that $\mu_1=\Lambda_1=\lambda_1=1$.
Also,
\begin{equation}\label{musum}
\mu_m+\mu_n\in \langle \sigma(A)\rangle \quad\forall m,n\in\ensuremath{\mathbb N},
\end{equation}
\begin{equation}\label{speclim}
\Lambda_n\to\infty \text{ and } \mu_n\to\infty \text{ as }n\to\infty.
\end{equation}

For $\Lambda\in\sigma (A)$, we denote by  $R_\Lambda$ the orthogonal projection from $H$ onto the eigenspace of $A$ corresponding to $\Lambda$,
and set $$P_\Lambda=\sum_{\lambda\in \sigma(A),\lambda\le \Lambda}R_\lambda.$$
Note that each vector space $P_\Lambda H$ is finite dimensional.

\medskip
For $\alpha,\sigma \in \ensuremath{\mathbb R}$ and  $u=\sideset{}{'}\sum \widehat \bvec u_\bvec k e^{i\check\bvec k\cdot \bvec x}\in H$, define
$$A^\alpha u=\sideset{}{'}\sum |\check\bvec k|^{2\alpha} \widehat \bvec u_\bvec k e^{i\check\bvec k\cdot
\bvec x},\quad e^{\sigma A^{1/2}} u=\sideset{}{'}\sum e^{\sigma
|\check\bvec k|} \widehat \bvec u_\bvec k e^{i\check\bvec k\cdot
\bvec x},$$
and
$$A^\alpha e^{\sigma A^{1/2}} u=\sideset{}{'}\sum |\check\bvec k|^{2\alpha}e^{\sigma
|\check\bvec k|} \widehat \bvec u_\bvec k e^{i\check\bvec k\cdot
\bvec x}.$$

For $\alpha,\sigma\ge 0$, the  Gevrey spaces are defined by
\begin{equation*}
G_{\alpha,\sigma}=\mathcal D(A^\alpha e^{\sigma A^{1/2}} )\stackrel{\rm def}{=} \{ u\in H: |u|_{\alpha,\sigma}\stackrel{\rm def}{=} |A^\alpha
e^{\sigma A^{1/2}}u|<\infty\}.
\end{equation*}
In particular, when $\sigma=0$  the domain of the fractional operator $A^\alpha$ is
\begin{equation*}
\mathcal D(A^\alpha)=G_{\alpha,0}=\{ u\in H: |A^\alpha u|=|u|_{\alpha,0}<\infty\}.
\end{equation*}

Thanks to the zero-average condition, the norm $|A^{m/2}u|$ is equivalent to $\|u\|_{H^m(\Omega)^3}$ on the space $\mathcal D(A^{m/2})$, for $m=0,1,2,\ldots$

Note that $\mathcal D(A^0)=H$,  $\mathcal D(A^{1/2})=V$, and  $\|u\|\stackrel{\rm def}{=} |\nabla u|$ is equal to $|A^{1/2}u|$, for all $u\in V$.
Also, the norms $|\cdot|_{\alpha,\sigma}$ are increasing in $\alpha$, $\sigma$, hence, the spaces $G_{\alpha,\sigma}$ are decreasing in $\alpha$, $\sigma$.

Observe, for $\sigma>0$, that  $G_{\alpha,\sigma}$ consists of real analytic divergence-free vector fields and its norm is stronger than all the Sobolev norms. Any asymptotic expansions in such Gevrey spaces induce estimates of the remainders in their very strong norms, see \eqref{vremain} and \eqref{uremain} below. Those estimates can be beneficial to numericals, and have potential applications in studying practical aspects of the physics of fluids. See, e.g., \cite{DoeringTiti} for a consequence of the estimates in Gevrey norms connected with the Kolmogorov theory of turbulence.
From the technical point of view, the use of the Gevrey norms enables an immediate regularity bootstrapping in all Sobolev spaces via the method by Foias and Temam \cite{FT-Gevrey}.
It avoids the lengthy, gradual bootstrapping of step-by-step increment in Sobolev regularity. For example, the original paper by Foias and Saut \cite{FS87} has to establish that the time derivatives of all orders of the solution and remainders belong to all Sobolev spaces.
See also the contrast between the two bootstrapping mechanisms in the proof of Proposition 3.4 of \cite{HM2}.

\medskip
Regarding the nonlinear terms in the NSE, a bounded linear map $B:V\times V\to V'$ is defined by
\begin{equation*}
\inprod{B(\bvec u,\bvec v),\bvec w}_{V',V}=b(\bvec u,\bvec v,\bvec w)\stackrel{\rm def}{=} \int_{\mathbb T_\bvec L} ((\bvec u\cdot \nabla) \bvec v)\cdot \bvec w\, d\bvec x, \text{ for all } \bvec u,\bvec v,\bvec w\in V.
\end{equation*}
In particular,
\begin{equation}\label{defBuv}
B(\bvec u,\bvec v)=\mathcal P ((\bvec u\cdot \nabla) \bvec v), \text{ for all }\bvec u,\bvec v\in\mathcal D(A).
\end{equation}
In fact, if $u=\sideset{}{'}\sum \widehat\bvec u_\bvec k  e^{i \check \bvec k\cdot \bvec x}\in \mathcal D(A)$ and
$v=\sideset{}{'}\sum \widehat\bvec v_\bvec k  e^{i \check \bvec k\cdot \bvec x}\in \mathcal D(A)$, then $(u\cdot \nabla )v$ has zero spatial average and
\begin{align}\label{unabv}
(u\cdot\nabla)v&=\sideset{}{'}\sum \widehat\bvec b_\bvec k  e^{i \check \bvec k\cdot \bvec x},\text{ where }
\widehat\bvec b_\bvec k = \sum_{\check\bvec m+\check j=\check\bvec k} i (\widehat\bvec u_\bvec m\cdot \check \bvec k) \widehat\bvec v_\bvec j, \\
\intertext{and, consequently, }
\label{BF}
B(u,v)&=\sideset{}{'}\sum \widehat P_\bvec k \widehat\bvec b_\bvec k  e^{i \check \bvec k\cdot \bvec x}.
\end{align}

Applying the projection $\mathcal P$ of the equation \eqref{NPDE}, we obtain
\begin{equation}\label{ueq1}
\frac{du}{dt} +Au + B(u,u) +\Omega \mathcal P J u =0,
\end{equation}
with solution $u\in H$.
In the case of non-rotation, i.e. $\Omega=0$, equation \eqref{ueq1} is the standard NSE
\begin{equation}\label{ueqzero}
\frac{du}{dt} +Au + B(u,u) =0.
\end{equation}

Since $\mathcal P u=u$, for $u\in H$, equation \eqref{ueq1} is equivalent to
\begin{equation}\label{NSO}
\frac{du}{dt} +Au + B(u,u) +\Omega S u =0
\end{equation}
with $u\in H$, where $S=\mathcal P J \mathcal P$. Equation \eqref{NSO} will be the focus of our study.

Note that if $u$ is as in \eqref{uF}, then
\begin{equation}\label{SF}
Su=\sideset{}{'}\sum \widehat P_\bvec k  J\widehat P_\bvec k \widehat \bvec u_\bvec k e^{i\check\bvec k\cdot\bvec x}.
\end{equation}

We have the following elementary properties:
\begin{equation}\label{Sper}
\inprod{Su,A^\alpha e^{\sigma A^{1/2}} u}=0, \text{ for all } \alpha,\sigma\ge 0\text{ and } u\in \mathcal D(A^\alpha e^{\sigma A^{1/2}}),
\end{equation}
\begin{equation}\label{borth}
b(u,v,w)=-b(u,w,v) \text{ and } b(u,v,v)=0, \text{ for all } u,v,w\in V.
\end{equation}

Because of relation \eqref{Sper} with $\alpha=\sigma=0$, the energy balance/inequality for \eqref{NSO} is the same as for \eqref{ueqzero}.
Hence, the following definitions of  weak and regular solutions for \eqref{NSO} are quite similar to those for \eqref{ueqzero}, see, e.g., \cite{CFbook,FMRTbook,TemamAMSbook}.

\begin{definition}\label{lhdef}
{\rm (a)} A Leray-Hopf weak solution $u(t)$ of \eqref{NSO} is a mapping from $[0,\infty)$ to $H$ such that
\begin{equation*}
u\in C([0,\infty),H_{\rm w})\cap L^2_{\rm loc}([0,\infty),V),\
u'\in L^{4/3}_{\rm loc}([0,\infty),V'),
\end{equation*}
and satisfies
\begin{equation*}
\frac{d}{dt} \inprod{u(t),w}+\doubleinprod{u(t),w}+b(u(t),u(t),w)+\Omega \inprod{Su,w}=0
\end{equation*}
in the distribution sense in $(0,\infty)$ (in fact in $L^{4/3}_{\rm loc}([0,\infty))$), for all $w\in V$, and the energy inequality
\begin{equation*}
\frac12|u(t)|^2+\int_{t_0}^t \|u(\tau)\|^2d\tau\le \frac12|u(t_0)|^2
\end{equation*}
holds for $t_0=0$ and almost all $t_0\in(0,\infty)$, and for all $t\ge t_0$.
Here, $H_{\rm w}$ denotes the topological vector space $H$ with the weak topology.

{\rm (b)}  We say a function $u(t)$ is a Leray-Hopf weak solution on $[T,\infty)$ if $u(T+\cdot)$ is a Leray-Hopf weak solution.

A Leray-Hopf weak solution $u(t)$ on $[T_0,\infty)$, for some $T_0\in\ensuremath{\mathbb R}_+$, is a \emph{regular solution} on $[T_0,T_0+T)$, for some $0<T\le \infty$, if
\begin{equation*}
u\in C([T_0,T_0+T),V)\cap L^2_{\rm loc}([T_0,T_0+T),\mathcal D(A)), \text{ and }u'\in L^2_{\rm loc}([T_0,T_0+T),H).
\end{equation*}

A \emph{global regular solution} is a regular solution on $[0,\infty)$.
\end{definition}

Same as for equation \eqref{ueqzero}, the basic questions, for equation \eqref{NSO}, about the uniqueness of weak solutions, and the global existence of regular solutions are still open, see  Theorem \ref{typicalthm} below.
Regarding the second question, it is proved in Babin-Mahalov-Nicolaenko \cite{BMN99b} that for any initial data $u_0\in V$, there is $\Omega_0>0$ depending on $u_0$ such that global regular solutions exist for all $|\Omega|>\Omega_0$.
Moreover, it is also showed in \cite{BMN99b} that the long-time dynamics of \eqref{NSO}, with an additional non-potential body force, is close to that of a so-called ``$2\frac12$-dimensional NSE'' when $|\Omega|$ is sufficiently large.
However, these basic questions will be bypassed and the mentioned results of \cite{BMN99b} will not be needed in this study.
It is due to the eventual regularity and decay (to zero) of the solutions, see Proposition \ref{decay2} below.
We, instead, will focus on the refined analysis of that decay.

When $\Omega=0$, the long-time dynamics of equation \eqref{ueqzero}, which covers the case of potential forces, is studied in details early in \cite{FS83,FS84a,FS84b,FS87,FS91}, and later in \cite{FHOZ1,FHOZ2,FHN1,FHN2,FHS1,HM1}. (The case of non-potential forces is treated in \cite{HM2,CaH1,CaH2}. See also the survey paper \cite{FHS2} for more information.) Briefly speaking, it is proved in \cite{FS87} that any solution $u(t)$ of \eqref{ueqzero} admits an asymptotic expansion, as $t\to\infty$,
\begin{equation}\label{exreal}
u(t)\sim \sum_{n=1}^\infty q_n(t)e^{-\mu_n t},
\end{equation}
where $q_n(t)$ is an $\mathcal V$-valued polynomial in $t$. See Definition \ref{Sexpand} below for more information.

When $\Omega\ne 0$, one can initially view equation \eqref{NSO} the same as \eqref{ueqzero} with the linear operator $\tilde A=A+\Omega S$ replacing $A$, and follow \cite{FS87} to obtain the asymptotic expansions. However, the spectrum of $\tilde A$, and its eigenspaces will be more complicated. The expansions will be of the form
\begin{equation}\label{excomplex}
u(t)\sim \sum_{\mu\in\langle \sigma(\tilde A)\rangle} q_\mu(t)e^{-\mu t},
\end{equation}
where $\langle \sigma(\tilde A)\rangle$ is defined similarly to \eqref{semispecA}.

One can see that the additive semigroup $\langle \sigma(\tilde A)\rangle$ is a set of complex numbers, much more complicated than $\langle \sigma(A)\rangle$, and consequently expansion \eqref{excomplex} is considerably more complicated than \eqref{exreal}. The construction of $q_\mu(t)$ must consider various scenarios including resonance and non-resonance cases, which are harder to track for complex values $\mu$'s.
To avoid all these technicalities, we propose another approach that converts \eqref{NSO} to \eqref{ueqzero} with a time-dependent bilinear form.
By maintaining the operator $A$, the expansions will be similar to \eqref{exreal}, and the proof will be direct and clean in the spirit of \cite{FS87}, taking advantage of recent improvements in \cite{HM1,HM2}. This approach was, in fact, successfully used in \cite{BMN99b} in studying the global well-posedness for regular solutions of \eqref{NSO}.

\medskip
\noindent\textbf{Rewriting the variational NSE using the Poincar\'e waves.}
In dealing with term $\Omega S u$ in \eqref{NSO}, we will make a change of variables using the exponential operator $e^{t S}$.
Clearly, $S$ is a bounded linear operator on $L^2({\mathbb T_\bvec L})^3$ with norm $\|S\|_{\mathcal L(L^2({\mathbb T_\bvec L})^3)}=1$.
For our problem, we restrict $e^{tS}$ to $H$ only, and have the isometry group $e^{tS}:H\to H$, $t\in\ensuremath{\mathbb R}$, is analytic in $t$.

It is well-known, see e.g. \cite{BMN99b,CheminBook2006},  that one has, for $u=\sideset{}{'}\sum  \widehat\bvec u_\bvec k e^{i\check\bvec k\cdot\bvec x}\in H$,
\begin{equation}\label{etS}
e^{tS}u=\sideset{}{'}\sum  E_\bvec k(\widetilde k_3 t)\widehat\bvec u_\bvec k e^{i\check\bvec k\cdot\bvec x},
\quad \text{where }
E_\bvec k(t)= \cos(t)I_3+\sin(t)J_\bvec k,
\end{equation}
with $J_\bvec k$ being the $3\times 3$ matrix for which $J_\bvec k\bvec z=\widetilde\bvec k \times \bvec z$, for all $\bvec z\in \ensuremath{\mathbb C}^3$.
Explicitly, the matrix $J_\bvec k$, for $\bvec k\in\ensuremath{\mathbb Z}^3\setminus\{\mathbf 0\}$, is
\begin{equation}\label{Jk}
J_\bvec k=\begin{pmatrix}
  0 & -\widetilde k_3 & \widetilde k_2\\
  \widetilde k_3   & 0 & - \widetilde k_1 \\
  - \widetilde k_2 &  \widetilde k_1 &0
 \end{pmatrix}.
\end{equation}

For the reader's convenience, we include  a simple proof of \eqref{etS} in Appendix \ref{apd}.

One sees that
\begin{align*}
(E_\bvec k(t))^*&=E_\bvec k(-t),\\
|E_\bvec k(t)\bvec z|&=|\bvec z|,\text{ for all } \bvec z\in X_\bvec k.
\end{align*}

From these properties, we deduce
\begin{align}
\label{eaj}
(e^{tS})^*&=e^{-tS} \quad(\text{on } H),\\
\label{ESiso}
|e^{tS}u|_{\alpha,\sigma}&=|u|_{\alpha,\sigma},\text{ for all }\alpha,\sigma\ge 0 \text{ and } u\in\mathcal D(A^\alpha e^{\sigma A^{1/2}}).
\end{align}

Also, some relations between $S$ and $A$ are
\begin{equation}\label{SAcom}
ASu=SAu,\quad e^{tS}Au=Ae^{tS}u,\text{ for all }u\in\mathcal D(A)\text{ and } t\in\ensuremath{\mathbb R}.
\end{equation}

The bi-linear form in \eqref{NSO} will be transformed to a similar, but time-dependent one that we describe below.

Let $t,\Omega\in\ensuremath{\mathbb R}$. Define $b(t,\cdot,\cdot,\cdot):V^3\to\ensuremath{\mathbb R}$ and $b_\Omega(t,\cdot,\cdot,\cdot):V^3\to\ensuremath{\mathbb R}$ by
\begin{equation*}
b(t,u,v,w)=b(e^{-tS}u,e^{-tS}v,e^{-tS}w),\quad b_\Omega(t,u,v,w)=b(\Omega t,u,v,w),
\end{equation*}
for all $u,v,w\in V$. We then  define $B(t,\cdot,\cdot):V\times V\to V'$ by
\begin{equation}\label{defBt}
\inprod{B(t,u,v),w}_{V',V}=b(t,u,v,w), \text{ for all } u,v,w\in V.
\end{equation}
In particular, thanks to \eqref{defBuv} and \eqref{eaj},
\begin{equation}\label{defBtuv}
B(t,u,v)=e^{ t S}B(e^{- t S}u,e^{- t S}v), \text{ for all } u,v\in\mathcal D(A).
\end{equation}

Define $B_\Omega(t,u,v)=B(\Omega t,u,v)$.

\medskip
We now rewrite equation \eqref{NSO} using the Poincar\'e waves $e^{-\Omega t S}w$, for $w\in H$ and $t\in\ensuremath{\mathbb R}$.

Let $u(t)$ be a solution of \eqref{NSO}. Set $v(t)=e^{\Omega t S}u(t)$, or equivalently, $u(t)=e^{-\Omega t S} v(t)$.

If $u\in C^1((0,\infty),H)\cap C((0,\infty),\mathcal D(A))$ then, with \eqref{SAcom} taken into account, $v$  solves
\begin{equation}\label{veq}
\frac{dv}{dt} +Av+B_\Omega(t,v,v)=0,\quad t>0.
\end{equation}

With this formulation, equation \eqref{veq} resembles more with \eqref{ueqzero} than \eqref{NSO}.
The difference between \eqref{veq} and \eqref{ueqzero} is the time-dependent bi-linear form $B_\Omega(t,\cdot,\cdot)$.
However, this bi-linear form turns out to possess many features similar to $B(\cdot,\cdot)$ itself.

For example, from \eqref{defBt} and \eqref{borth}, we have, for all $\Omega,t\in\ensuremath{\mathbb R}$ and $u,v,w\in V$, that
\begin{equation}\label{BS1}
\inprod{ B_\Omega(t,u,v),w}_{V',V}=-\inprod{ B_\Omega(t,u,w),v}_{V',V},
\end{equation}
and, consequently,
\begin{equation}\label{BS2}
\inprod{B_\Omega(t,u,v),v}_{V',V}=0.
\end{equation}

This prompts the following definition of weak solutions of \eqref{veq}.

\begin{definition}\label{lhv}
A Leray-Hopf weak solution $v(t)$ of \eqref{veq} is a mapping from $[0,\infty)$ to $H$ such that
\begin{equation*}
v\in C([0,\infty),H_{\rm w})\cap L^2_{\rm loc}([0,\infty),V),\quad v'\in L^{4/3}_{\rm loc}([0,\infty),V'),
\end{equation*}
and satisfies
\begin{equation*}
\frac{d}{dt} \inprod{v(t),w}+\doubleinprod{v(t),w}+b_\Omega(t,v(t),v(t),w)=0
\end{equation*}
in the distribution sense in $(0,\infty)$ (in fact in $L^{4/3}_{\rm loc}([0,\infty))$), for all $w\in V$, and the energy inequality
\begin{equation*}
\frac12|v(t)|^2+\int_{t_0}^t \|v(\tau)\|^2d\tau\le \frac12|v(t_0)|^2
\end{equation*}
holds for $t_0=0$ and almost all $t_0\in(0,\infty)$, and all $t\ge t_0$.

Other definitions in {\rm (b)} of Definition \ref{lhdef} are extended to the solution $v(t)$.
\end{definition}

Similar to the case $\Omega=0$, we have the following existence, uniqueness and regularity results for equations \eqref{NSO} and \eqref{veq}.

\begin{theorem}\label{typicalthm}
Let $\Omega\ne 0$ be a given number.

\begin{enumerate}[label=\rm(\roman*)]
\item For any $u_0\in H$, there exists a Leray-Hopf weak solution $u(t)$ of \eqref{NSO}, resp. $v(t)$ of \eqref{veq}, with initial data $u_0$.
Moreover, there is $T_0=T_0(u_0)\ge 0$ such that $u$, resp. $v$, is a regular solution on $[T_0,\infty)$.

\item  For any $u_0\in V$, there exists a unique regular solution $u(t)$ of \eqref{NSO}, resp. $v(t)$ of \eqref{veq}, with initial data $u_0$,  on an interval $[0,T)$ for some $T>0$. If $\|u_0\|$ is sufficiently small, then $T=\infty$.

\item For any Leray-Hopf weak solution $u(t)$ of \eqref{NSO}, resp. $v(t)$ of \eqref{veq}, and any number $\sigma>0$, there exists $T_*>0$ such that $u(t)$, resp. $v(t)$, belongs to $G_{1/2,\sigma}$, for all $t\ge T_*$, and the equation \eqref{NSO}, resp. \eqref{veq}, holds in  $\mathcal D(A)$ on $(T_*,\infty)$ with classical time derivative.
\end{enumerate}
\end{theorem}

Parts (i) and (ii) of Theorem \ref{typicalthm} are standard. Part (iii) can be proved by using the same technique of Foias-Temam \cite{FT-Gevrey}, noticing that we have the orthogonality \eqref{Sper}. See details of similar calculations in \cite{HM1}, and also more specific statements in Proposition \ref{decay2} below.

The current paper is focused on a different question, namely, the precise long-time dynamics of the solutions of \eqref{NSO} for all $\Omega\ne 0$. Even though they eventually, as $t\to\infty$, go to zero, our goal is to provide a detailed description of such a decay. In the case with the zero spatial average of the solutions, we will show that each solution possesses an asymptotic expansion which is similar to \eqref{exreal}, but contains some oscillating terms that are from the rotation.
The oscillating parts, in this case, are written in terms of sinusoidal functions of time. (See a similar result by Shi \cite{Shi2000} for dissipative wave equations.)
In the general case of non-zero spatial average, similar asymptotic expansions are obtained with the oscillating terms being expressed by the ``double sinusoidal'' functions.

This paper is organized as follows.
In section \ref{expsec}, we introduce our abstract asymptotic expansions, for large time, in terms of oscillating-decaying functions.
Our basic classes of functions are the S-polynomials and SS-polynomials, see Definition \ref{polyS}, below. Fundamental properties of these two classes are studied. In particular, when the forcing term of a linear ordinary differential equation (ODE) is a perturbation of an S-polynomial, then the  ODE's solution can be approximated by S-polynomials, see Lemma \ref{polylem}, below. This lemma turns out to be a building block in our constructions of polynomials in the asymptotic expansions.
In section \ref{zerosec}, we establish the asymptotic expansions for solutions of \eqref{veq} and \eqref{NSO}, below, in Theorems \ref{vthm} and \ref{uthm}, respectively. The expansions are in terms of S-polynomials and exponential functions. It is worth mentioning that these results are established for \emph{all} $\Omega\ne 0$ and for all Leray-Hopf weak solutions.
In particular, it does not rely on Babin-Mahalov-Nicolaenko's global well-posedness result \cite{BMN99b}, which, as mentioned after Definition \ref{lhdef}, requires $\Omega$  to be \emph{sufficiently large} depending on the initial data.
In section \ref{nonzerosec}, we derive the asymptotic expansions for solutions without the zero spatial averages.
This is done by using the specific Galilean-type transformation \eqref{Gtrans}. Unlike the previous section, this yields the expansions in terms of SS-polynomials.
In section \ref{specialsec}, we present some special solutions that form infinite dimensional linear manifolds that are invariant under the flows generated by the solutions of \eqref{NSO}. Especially, Remark \ref{heli} contains a subclass of solutions for which the helicity, a meaningful physical quantity in fluid dynamics \cite{Moffatt69,Moffatt92}, vanishes. The case of non-zero spatial average of the solutions is treated in Theorem \ref{specthm2}, below.

\section{Abstract asymptotic expansions and their properties}
\label{expsec}

We introduce here the classes of functions which will appear in our asymptotic expansions.

\begin{definition}\label{polyS}
Let $X$ be a vector space over the scalar field $\ensuremath{\mathbb K}=\ensuremath{\mathbb R}$ or $\ensuremath{\mathbb K}=\ensuremath{\mathbb C}$.
\begin{enumerate}[label=\rm(\alph*)]
 \item A function $g:\ensuremath{\mathbb R}\to X$ is an $X$-valued S-polynomial if it is a finite sum of the functions in the set
\begin{equation*}
\Big \{ t^m \cos(\omega t)Z,\ t^m \sin(\omega t) Z: m\in\ensuremath{\mathbb N}\cup\{0\},\ \omega\in\ensuremath{\mathbb R}, \ Z\in X\Big\}.
\end{equation*}

\item A function $g:\ensuremath{\mathbb R}\to X$ is an $X$-valued SS-polynomial if it is a finite sum of the functions in the set
\begin{align*}
&\Big\{t^m \cos\big( a\cos(\omega t)+b\sin(\omega t)+ct+d\big)Z,\\
&\quad t^m \sin\big( a\cos(\omega t)+b\sin(\omega t)+ct+d\big)Z:\\
&\quad m\in\ensuremath{\mathbb N}\cup\{0\},\ a,b,c,d,\omega\in \ensuremath{\mathbb R},\  Z\in X\Big\}.
\end{align*}

\item Denote by $\mathcal{F}_0(X)$, resp., $\mathcal{F}_1(X)$ and $\mathcal{F}_2(X)$ the set of all $X$-valued polynomials, resp., S-polynomials and SS-polynomials.
\end{enumerate}
\end{definition}

Clearly, $\mathcal{F}_0(X)$, $\mathcal{F}_1(X)$ and $\mathcal{F}_2(X)$ are vector spaces over $\ensuremath{\mathbb K}$, and $\mathcal{F}_0(X)\subset \mathcal{F}_1(X)\subset \mathcal{F}_2(X)$.

If $f\in \mathcal{F}_1(X)$, then we can write $f$ as
\begin{equation}\label{fpow}
f(t)=\sum_{n=0}^N t^n f_n(t),
\end{equation}
where
\begin{equation}\label{fcos}
f_n(t)=\sum_{j=1}^{N_n} [a_{n,j}\cos(\omega_{n,j} t)+b_{n,j}\sin(\omega_{n,j}t)],
\end{equation}
with $N_n\in \ensuremath{\mathbb N}$, $a_{n,j},b_{n,j}\in X$, and $\omega_{n,j}\ge 0$ with the mapping $j\mapsto \omega_{n,j}$ being strictly increasing for each fixed $n$.

\begin{definition}\label{Sexpand}
Let $(X,\|\cdot\|_X)$ be a normed space and  $(\alpha_n)_{n=1}^\infty$ be a  sequence of strictly increasing non-negative real numbers.
Let $\mathcal{F}=\mathcal{F}_0$, $\mathcal{F}_1$, or $\mathcal{F}_2$.
A function $f:[T,\infty)\to X$, for some $T\in\ensuremath{\mathbb R}_+$, is said to have an asymptotic expansion
\begin{equation}\label{ex1}
f(t)\sim\sum_{n=1}^\infty f_n(t)e^{-\alpha_n t} \quad\text{in } X,
\end{equation}
where each $f_n$ belongs to $\mathcal{F}(X)$, if one has, for any $N\ge 1$, that
\begin{equation}\label{ex2}
\Big \|f(t)-\sum_{n=1}^N f_n(t)e^{-\alpha_n t}\Big \|_X=\mathcal O(e^{-(\alpha_N+\varepsilon_N)t}),\text{ as }t\to\infty,
\end{equation}
for some $\varepsilon_N>0$.
\end{definition}

With this definition, the precise statement of \eqref{exreal} is that it holds, with $\mathcal{F}(X)=\mathcal{F}_0(X)$, in the space $X=G_{\alpha,\sigma}$, for all $\alpha,\sigma\ge 0$.

The expansion \eqref{ex1} with $\mathcal{F}=\mathcal{F}_1$ is equivalent to the one used in \cite{Shi2000}, for the dissipative wave equations, while with the largest class $\mathcal{F}=\mathcal{F}_2$ is new.
For other related asymptotic expansions for solutions of NSE, see \cite{CaH1,CaH2}.

One can observe, same as in \cite[Remark 2.3]{HM2}, that if \eqref{ex2} holds for all $N$ then
\begin{equation}\label{ex3}
\Big \|f(t)-\sum_{n=1}^N f_n(t)e^{-\alpha_n t}\Big \|_X=\mathcal O(e^{-\alpha t}),\text{ as }t\to\infty,
\end{equation}
for all $N$ and all $\alpha\in (\alpha_N,\alpha_{N+1})$.

\begin{lemma}\label{unilem}
 Given $N\in\ensuremath{\mathbb N}$, if $f_1,f_2,\ldots,f_N\in \mathcal{F}_1(X)$ satisfy \eqref{ex2}, then such  S-polynomials $f_n$'s are unique.
\end{lemma}
\begin{proof}
Let $g_1,g_2,\ldots,g_N$ be functions in $\mathcal{F}_1(X)$ that satisfy
\begin{equation*}
\Big \|f(t)-\sum_{n=1}^N g_n(t)e^{-\alpha_n t}\Big \|_X=\mathcal O(e^{-(\alpha_N+\varepsilon_N')t}),\text{ as }t\to\infty\text{ for some }\varepsilon_N'>0.
\end{equation*}

Let $h_n=f_n-g_n$. By the triangle inequality, we have
\begin{equation*}
\Big \|\sum_{n=1}^N h_n(t)e^{-\alpha_n t}\Big \|_X
\le \Big \|f(t)-\sum_{n=1}^N f_n(t)e^{-\alpha_n t}\Big \|_X+\Big \|f(t)-\sum_{n=1}^N g_n(t)e^{-\alpha_n t}\Big \|_X,
\end{equation*}
hence
\begin{equation}\label{exh}
\Big \|\sum_{n=1}^N h_n(t)e^{-\alpha_n t}\Big \|_X=\mathcal O(e^{-(\alpha_N+\varepsilon)t}),\text{ as }t\to\infty,\text{ for }\varepsilon=\min\{\varepsilon_N,\varepsilon_N'\}>0.
\end{equation}

Suppose that not all $h_n$'s are zero functions. Let $n_0$ be the smallest number such that $h_{n_0}\ne 0$.
By multiplying \eqref{exh} with $e^{\alpha_{n_0} t}$, we deduce
\begin{equation} \label{hnot}
\|h_{n_0}(t)\|_X=\mathcal O(e^{-\varepsilon' t}), \text{ for some }\varepsilon'>0.
\end{equation}

We write $h_{n_0}(t)$ in the form of \eqref{fpow}.
Suppose the highest order term (with respect to the power of $t$) of $h_{n_0}(t)$ is $t^d z(t)$,
where $d$ is a non-negative integer, and $z$ is a non-zero function of the form as in the RHS of \eqref{fcos}.
Then dividing \eqref{hnot} by $t^d$ yields
\begin{equation} \label{zX}
\lim_{t\to\infty}\|z(t)\|_X=0.
\end{equation}

Suppose
\begin{equation}\label{zo}
z(t)=\sum_{n=1}^{N_d} [a_n\cos(\omega_n t)+b_n\sin(\omega_n t)],
\end{equation}
where $N_d\in\ensuremath{\mathbb N}$, $a_n,b_n\in X$ and $\omega_n\in\ensuremath{\mathbb R}$, for $1\le n\le N_d$.

Let $Y$ be the vector space spanned by $\{a_n,b_n\in X: 1\le n\le N_d \}$.
Let $\mathcal Y=\{Y_j: 1\le j\le m\}$ be a basis of $Y$.
By representing the vectors $a_n$'s and $b_n$'s in basis $\mathcal Y$, we can rewrite $z(t)$ as
\begin{equation*}
z(t)=\sum_{j=1}^m z_j(t) Y_j,
\end{equation*}
where $m\in\ensuremath{\mathbb N}$, each $z_j:\ensuremath{\mathbb R}\to \ensuremath{\mathbb K}$, for $1\le j\le m$, is a linear combination of the functions $\cos(\omega_n t)$ and $\sin(\omega_n t)$, for $1\le n\le N_d$, in \eqref{zo}.

For $y=\sum_{j=1}^m y_j Y_j\in Y$, with $y_j\in\ensuremath{\mathbb K}$ for $1\le j\le m$, define the norm
\begin{equation*}
\|y\|_Y=\Big(\sum_{j=1}^m |y_j|^2\Big)^{1/2}.
\end{equation*}

On the finite dimensional space $Y$, the norms $\|\cdot\|_X$ and $\|\cdot\|_Y$ are equivalent. Therefore, \eqref{zX} gives
\begin{equation*}
\lim_{t\to\infty}\|z(t)\|_Y=0, \text{ which implies }
\lim_{t\to\infty} z_j(t)=0, \text{ for } 1\le j\le m.
\end{equation*}

By Lemma \ref{uniR}, we obtain $z_j=0$ for all $j$. Hence $z=0$, which leads to a contradiction.
Thus, $h_n=0$, for all $n=1,2,\ldots,N$.  We conclude $f_n=g_n$, for $1\le n \le N$. The proof is complete.
\end{proof}

\begin{remark}\label{uniG}
We discuss a consequence of Lemma \ref{unilem}. Suppose \eqref{ex2} is satisfied with $X=G_{\beta_i,\sigma_i}$  and $f_n=g_n^{(i)}\in \mathcal{F}_1(X)$, for $i=1,2$ and $1\le n\le N$.
Let $\bar\beta=\min\{\beta_1,\beta_2\}$ and $\bar\sigma=\min\{\sigma_1,\sigma_2\}$. Then \eqref{ex2} is satisfied with both $f_n=g_n^{(1)}$ and $f_n=g_n^{(2)}$ on the same space $X=G_{\bar\beta,\bar\sigma}$. By the virtue of Lemma \ref{unilem}, we have $g_n^{(1)}=g_n^{(2)}$, for $1\le n\le N$.
\end{remark}

\subsection{Properties of S-polynomials and SS-polynomials}
\label{ss21}

We start this subsection with some elementary properties of the functions introduced in Definition \ref{polyS} above.

\begin{lemma}\label{Spolyinvar}
Let $X$ and $\ensuremath{\mathbb K}$ be as in Definition \ref{polyS}, and $\mathcal{F}(X)=\mathcal{F}_1(X)$ or $\mathcal{F}(X)=\mathcal{F}_2(X)$.
Let $f$ be any function in $\mathcal{F}(X)$.
\begin{enumerate}[label={\rm(\roman*)}]
\item  The functions $t\mapsto f(T+t)$ and $t\mapsto f(k t)$ belong to $\mathcal{F}(X)$ for any numbers $T,k\in\ensuremath{\mathbb R}$.

\item If $g\in  \mathcal{F}_1(\ensuremath{\mathbb K})$, then $gf\in \mathcal{F}(X)$.

\item In case $X$ is a subspace of $\ensuremath{\mathbb C}^n$, with  $\ensuremath{\mathbb K}=\ensuremath{\mathbb C}$, then $e^{i\omega t}f\in \mathcal{F}(X).$

\item Suppose $Y$ is another vector space over $\ensuremath{\mathbb K}$ and $L$ is a linear mapping from $X$ to $Y$. Then
$Lf\in \mathcal{F}(Y)$.
\end{enumerate}
\end{lemma}
\begin{proof}
Parts (i) and (ii) can be easily verified by using elementary trigonometric identities such as the sine and cosine of a sum, and the product to sum formulas.
Part (iii) is obtained by applying part (ii) to the function $g(t)=e^{i\omega t}$ which, in fact, belongs to $\mathcal{F}_1(\ensuremath{\mathbb C})$.
Part (iv) is obvious.
\end{proof}

\begin{lemma}\label{SPlem1}
Let $\mathcal{F}=\mathcal{F}_1$ or $\mathcal{F}=\mathcal{F}_2$.
Then, a function $f$ belongs to $\mathcal{F}(\mathcal V)$ if and only if
\begin{equation} \label{fsum}
f(t)=\sideset{}{'}\sum_{\text{\rm finitely many $\bvec k$}} \bvec f_\bvec k(t)e^{i\check\bvec k\cdot\bvec x}, \text{ with each}\quad \bvec f_\bvec k\in \mathcal{F}(X_\bvec k), \text{ and }
\bvec f_{-\bvec k}=\overline{\bvec f_\bvec k},
\end{equation}
where $X_\bvec k$ is as in \eqref{Xk}.
\end{lemma}
\begin{proof}
We prove for the case $\mathcal{F}=\mathcal{F}_1$. The arguments for the other case $\mathcal{F}=\mathcal{F}_2$ are similar and omitted.

Suppose $f\in \mathcal{F}_1(\mathcal V)$. We write
\begin{equation*}
f(t)=\sum_{j=1}^N t^{m_j}[a_j \cos(\omega_j t)+b_j\sin(\omega_j t)]u_j, \text{ each } a_j,b_j\in\ensuremath{\mathbb R}, \ u_j\in\mathcal V.
\end{equation*}
By writing the finite Fourier series of each $u_j$ and combining the coefficients for $e^{i\check\bvec k\cdot\bvec x}$,
we find that the Fourier series of $f$ is of the form as in \eqref{fsum} with each $\bvec f_\bvec k(t)$ being a finite sum of
$t^{m_j}[a_j \cos(\omega_j t)+b_j\sin(\omega_j t)] \bvec z$ for some $\bvec z\in X_\bvec k$. Thus, $\bvec f_\bvec k\in \mathcal{F}_1(X_\bvec k)$.
The last relation in \eqref{fsum} is the standard condition for $f$ to be real-valued.

Now, suppose $f$ is as in \eqref{fsum}. For each $\bvec k$, consider the function
$$F_\bvec k(t)=\bvec f_\bvec k(t)e^{i\check\bvec k\cdot\bvec x}+\bvec f_{-\bvec k}(t)e^{-i\check\bvec k\cdot\bvec x}
=\bvec f_\bvec k(t)e^{i\check\bvec k\cdot\bvec x}+\overline{\bvec f_\bvec k(t)}e^{-i\check\bvec k\cdot\bvec x}.$$

If  $\bvec f_\bvec k(t)$ contains $t^m\cos(\omega t)\bvec z$ for some $\bvec z\in X_\bvec k$, then
$F_\bvec k(t)$ contains
 \begin{equation} \label{aterm}
 t^m\cos(\omega t) (\bvec z e^{i\check\bvec k\cdot\bvec x} + \bar \bvec z e^{-i\check\bvec k\cdot\bvec x}).
 \end{equation}
Since $\bvec z e^{i\check\bvec k\cdot\bvec x} + \bar \bvec z e^{-i\check\bvec k\cdot\bvec x}\in\mathcal V$, the function in \eqref{aterm} belongs to $\mathcal{F}_1(\mathcal V)$.
Similar property holds for $\sin(\omega t)$ replacing $\cos(\omega t)$, and we obtain $F_\bvec k\in \mathcal{F}_1(\mathcal V)$.
Then $f$ being a finite sum of such $F_\bvec k$'s yields $f\in \mathcal{F}_1(\mathcal V)$.
\end{proof}

The following are important properties relating the S- and SS- polynomials with the rotation and nonlinear terms in the rotational NSE.

\begin{lemma}\label{SPVlem}
Let  $\Lambda\in \sigma(A)$, two functions $f,g\in \mathcal{F}_1(P_\Lambda H)$, and $\Omega\in\ensuremath{\mathbb R}$.
Then
\begin{equation}\label{Sf1}
e^{\Omega tS}f(t)\in \mathcal{F}_1(P_\Lambda H),\end{equation}
\begin{equation}\label{Sf2}
B(f(t),g(t))\in \mathcal{F}_1(P_{4\Lambda} H),
\end{equation}
\begin{equation}\label{Sf3}
B_\Omega(t,f(t),g(t))\in \mathcal{F}_1(P_{4\Lambda}H).
\end{equation}
\end{lemma}
\begin{proof}
(a) By Lemma \ref{SPlem1}, we can write $f(t)$ as
\begin{equation} \label{ft}
f(t)=\sideset{}{'}\sum_{|\check \bvec k|^2\le \Lambda} \bvec f_\bvec k(t)e^{i\check\bvec k\cdot\bvec x}, \text{ with each } \bvec f_\bvec k\in \mathcal{F}_1(X_\bvec k), \text{ and }
\bvec f_{-\bvec k}=\overline{\bvec f_\bvec k}.
\end{equation}

Applying \eqref{etS} yields
\begin{equation}\label{eOf}
e^{\Omega t S} f(t)=\sideset{}{'}\sum_{|\check k|^2\le \Lambda} E_\bvec k(\Omega \widetilde k_3 t)\bvec f_\bvec k(t) e^{i\check\bvec k\cdot\bvec x}.
\end{equation}
Applying Lemma \ref{Spolyinvar} to $\mathcal{F}=\mathcal{F}_1$, $g(t):=\cos(\Omega \widetilde k_3 t)$ and then $g(t):=\sin(\Omega \widetilde k_3 t)$, one has that each $E_\bvec k(\Omega \widetilde k_3 t)\bvec f_\bvec k(t)$ belongs to $\mathcal{F}_1(X_\bvec k)$. Then by the virtue of the sufficient condition in Lemma \ref{SPlem1},  we have $e^{\Omega t S} f(t)\in \mathcal{F}_1(\mathcal V)$. This and the restriction $|\check\bvec k|^2\le \Lambda$ in \eqref{eOf} give \eqref{Sf1}.

(b) We prove \eqref{Sf2}. By Lemma \ref{SPlem1} again, we can assume, in addition to \eqref{ft}, that
\begin{equation*}
g(t)=\sideset{}{'}\sum_{|\check \bvec k|^2\le \Lambda} \bvec g_\bvec k(t)e^{i\check\bvec k\cdot\bvec x}, \text{ with each } \bvec g_\bvec k\in \mathcal{F}_1(X_\bvec k), \text{ and }
\bvec g_{-\bvec k}=\overline{\bvec g_\bvec k}.
\end{equation*}

By \eqref{BF}, we have
\begin{equation} \label{Btsum}
B(f(t),g(t))=\sideset{}{'}\sum \mathbf B_\bvec k(t) e^{i \check \bvec k\cdot\bvec x},
\end{equation}
where
\begin{equation}\label{Bk}
\mathbf B_\bvec k (t)
= \sum_{\stackrel{0<|\check\bvec m|^2,|\check\bvec j|^2\le \Lambda,}{\check \bvec m+\check \bvec j=\check \bvec k}}
 (\bvec f_\bvec m(t)\cdot i\, \check\bvec k)  \widehat P_\bvec k \bvec g_\bvec j(t).
\end{equation}

Thanks to formula \eqref{Bk} for $\mathbf B_\bvec k$, we only need to sum over $\bvec k$ in \eqref{Btsum} with
\begin{equation*}
|\check \bvec k|^2=|\check \bvec m+\check \bvec j|^2\le 2(|\check \bvec m|^2+|\check \bvec j|^2)\le 4\Lambda.
\end{equation*}
Thus, $B(f(t),g(t))\in P_{4\Lambda}H$ for all $t\in\ensuremath{\mathbb R}$.

Note that $\widehat P_\bvec k =\widehat P_{-\bvec k} $.
In the sum \eqref{Bk}, we will pair $\check \bvec m+\check \bvec j=\check \bvec k$ for $\mathbf B_\bvec k(t)$ with $(-\check \bvec m)+(-\check \bvec j)=(-\check \bvec k)$ for $\mathbf B_{-\bvec k}(t)$.
Because $\big(\bvec f_\bvec m\cdot i\, \check\bvec k\big) \widehat P_\bvec k\bvec g_\bvec j$, for $\mathbf B_\bvec k(t)$,
and
$ \big(\bvec f_{-\bvec m}\cdot  (-i\,\check\bvec k)\big) \widehat P_{-\bvec k}\bvec g_{-\bvec j}$, for $\mathbf B_{-\bvec k}(t)$,
are conjugates of each other,  so are  $\mathbf B_\bvec k(t)$ and $\mathbf B_{-\bvec k}(t)$.

Using Lemma \ref{Spolyinvar} (ii) and (iv), one can verify that
$\big(\bvec f_\bvec m\cdot i\, \check\bvec k\big) \widehat P_\bvec k \bvec g_\bvec j$ belongs to $\mathcal{F}_1(X_\bvec k)$,
hence  $\mathbf B_\bvec k(t)\in \mathcal{F}_1(X_\bvec k)$.

 By combining the above facts with Lemma \ref{SPlem1}, we conclude \eqref{Sf2}.

(c) Next, we prove \eqref{Sf3}. Because $f(t),g(t)\in \mathcal{F}_1(P_\Lambda H)$, we apply \eqref{Sf1} to have $e^{-\Omega tS}f(t)$ and $e^{-\Omega tS}g(t)$ belong to $\mathcal{F}_1(P_\Lambda H)$,
 which, by \eqref{Sf2}, imply
 $$B(e^{-\Omega tS}f(t),e^{-\Omega tS}g(t))\in \mathcal{F}_1(P_{4\Lambda} H),$$
 which, in turn, thanks to \eqref{Sf1} again, further implies
 $$e^{\Omega tS} B(e^{-\Omega tS}f(t),e^{-\Omega tS}g(t))\in \mathcal{F}_1(P_{4\Lambda} H).$$
Therefore, thanks also to \eqref{defBtuv}, we conclude \eqref{Sf3}.
 \end{proof}

\subsection{Approximating solutions of certain linear ODEs with S-polynomials}
\label{ss22}

In our proofs, we often need the following integrals.

\begin{lemma}\label{intlem}
Let $\alpha,\omega\in\ensuremath{\mathbb R}$ with $\alpha^2+\omega^2>0$, and $m$ be a non-negative integer.
Then each integral
 \begin{equation}\label{tei}
 \int t^m e^{\alpha t}\cos(\omega t) dt,\quad  \int t^m e^{\alpha t}\sin(\omega t) dt
 \end{equation}
is of the form $$p(t)e^{\alpha t}\cos(\omega t)+q(t) e^{\alpha t}\sin(\omega t)+const.,$$
where $p(t)$ and $q(t)$ are polynomials of degrees at most $m$.
\end{lemma}

Although this lemma is elementary, we give a proof in Appendix \ref{apd} that yields simple and explicit formulas for the integrals in \eqref{tei}, see \eqref{tmI} below.

The next lemma essentially originates from Foias-Saut \cite{FS87}, but is stated and proved in the same convenient form as \cite[Lemma 4.2]{HM2}.

\begin{lemma}\label{polylem}
Let $(X,\|\cdot\|_X)$ be a Banach space. Suppose $y$ is a function in $C([0,\infty),X)$, with distribution derivative $y'\in L^1_{\rm loc}([0,\infty),X)$, that solves the following ODE
 \begin{equation*}
 y'(t)+ \beta y(t) =p(t)+g(t)
 \end{equation*}
 in the $X$-valued distribution sense on $(0,\infty)$, where  $\beta\in \ensuremath{\mathbb R}$ is a constant, $p(t)$ is an $X$-valued S-polynomial, and $g\in L^1([0,\infty),X)$ satisfies
 \begin{equation}\label{gM}
 \|g(t)\|_X\le Me^{-\delta t}, \text{ for all } t\ge 0,  \text{ and some } M,\delta>0.
 \end{equation}

 Define $q(t)$, for $t\in  \ensuremath{\mathbb R}$, by
\begin{equation}\label{qdef}
 q(t)=
\begin{cases}
e^{-\beta t}\int_{-\infty}^t e^{\beta\tau }p(\tau) d\tau&\text{if }\beta >0,\\
y(0) +\int_0^\infty  g(\tau)d\tau + \int_0^t p(\tau)d\tau &\text{if }\beta =0,\\
-e^{-\beta t}\int_t^\infty e^{\beta\tau }p(\tau) d\tau&\text{if }\beta <0.
\end{cases}
\end{equation}

Then $q(t)$ is an $X$-valued S-polynomial  that  satisfies
\begin{equation}\label{pode2}
q'(t)+\beta q(t) = p(t),\text{ for all } t\in \ensuremath{\mathbb R},
\end{equation}
and the following estimates hold:

\begin{enumerate}[label={\rm (\roman*)}]
  \item
  If $\beta>0$ then
  \begin{equation}\label{zsquare}
\|y(t)-q(t)\|_X^2 \le 2e^{-2\beta t}\|y(0)-q(0)\|_X^2 + 2t \int_0^t e^{-2\beta(t-\tau)} \|g(\tau)\|_X^2 d\tau,
\text{ for all } t\ge 0.
\end{equation}

  \item If either

  {\rm (a)} $\beta=0$, or

  {\rm (b)}  $\beta<0$ and
  \begin{equation}\label{limycond} \lim_{t\to\infty} (e^{\beta t}\|y(t)\|_X)=0,\end{equation}
then
  \begin{equation}\label{g1b3}
  \|y(t)-q(t)\|_X^2\le \Big(\frac{M}{\delta-\beta}\Big)^2e^{-2\delta t}, \quad \text{ for all } t\ge 0.
  \end{equation}
  \end{enumerate}
\end{lemma}
\begin{proof}
Thanks to Lemma \ref{intlem}, $q(t)$ is an $X$-valued S-polynomial. The rest of this lemma is the same as \cite[Lemma 4.2]{HM2}, except for the relaxed estimate \eqref{zsquare} which we verify now.
Consider $\beta>0$. Let $z(t)=y(t)-q(t)$. Recall the inequality after (4.13) in \cite[Lemma 4.2]{HM2}, for all $t\ge 0$,
\begin{equation*}
\|z(t)\|_X\le e^{-\beta t} \|z(0)\|+ \int_{t_0}^t e^{-\beta(t-\tau)}\|g(\tau)\|_X d\tau.
\end{equation*}

Using Cauchy-Schwarz's and H\"older's inequalities, we estimate
\begin{align*}
\|z(t)\|_X^2
&\le 2e^{-2\beta t}\|z(0)\|_X^2+ 2\Big( \int_0^t e^{-\beta (t-\tau)}\|g(\tau)\|_X d\tau\Big)^2\\
& \le 2e^{-2\beta t}\|z(0)\|_X^2 + 2t \int_0^t e^{-2\beta(t-\tau)} \|g(\tau)\|_X^2 d\tau.
\end{align*}
Therefore, we obtain \eqref{zsquare}.
\end{proof}

\section{The case of zero spatial average solutions}\label{zerosec}

We obtain two main asymptotic expansion results, one for equation \eqref{veq}, and the other for equation \eqref{NSO}.
We recall that the numbers $\mu_n$'s are defined in \eqref{mudef} with basic properties \eqref{musum} and \eqref{speclim}.

\begin{theorem}
\label{vthm}
For any Leray-Hopf weak solution $v(t)$ of \eqref{veq}, there exist unique $\mathcal V$-valued S-polynomials $q_n$'s, for all $n\in\ensuremath{\mathbb N}$, such that it holds, for any $\alpha,\sigma >0$ and $N\ge 1$, that
\begin{equation}\label{vremain}
\Big|v(t)-\sum_{n=1}^N q_n(t)e^{-\mu_nt}\Big|_{\alpha,\sigma}= \mathcal O
\big(e^{-\mu t}\big),\text{ as } t\to\infty,\text{ for all }\mu\in(\mu_N,\mu_{N+1}).
\end{equation}
That is,
\begin{equation*}
v(t)\sim \sum_{n=1}^\infty q_n(t)e^{-\mu_nt}\quad \text{ in } G_{\alpha,\sigma}, \text{ for all } \alpha,\sigma>0.
\end{equation*}
\end{theorem}

Theorem \ref{vthm} is our key technical result. With this, we immediately obtain the asymptotic expansions for solutions of \eqref{NSO}.

\begin{theorem}
\label{uthm}
Let $u(t)$ be any Leray-Hopf weak solution of \eqref{NSO}. Then there exist unique $\mathcal V$-valued S-polynomials $Q_n$'s, for all $n\in\ensuremath{\mathbb N}$,  such that
it holds, for any $\alpha,\sigma >0$ and $N\ge 1$, that
\begin{equation}\label{uremain}
\Big|u(t)-\sum_{n=1}^N Q_n(t)e^{-\mu_n t} \Big|_{\alpha,\sigma}= \mathcal O
\big(e^{-\mu t}\big),\text{ as } t\to\infty,\text{ for all }\mu\in(\mu_N,\mu_{N+1}).
\end{equation}
That is,
\begin{equation*}
u(t)\sim \sum_{n=1}^\infty Q_n(t)e^{-\mu_nt}\quad \text{ in } G_{\alpha,\sigma}, \text{ for all } \alpha,\sigma>0.
\end{equation*}

Moreover, each $Q_n$ is related to $q_n$ in Theorem \ref{vthm} via relation \eqref{e1}, below.
\end{theorem}
\begin{proof}
Let $T_*>0$ be as in Theorem \ref{typicalthm}(iii).
Set $v(t)=e^{\Omega t S} u(t)$. Then $v(t)$ is a regular solution of \eqref{veq} on $[T_*,\infty)$.
Applying Theorem \ref{vthm} to solution $v(T_*+t)$, we have, for all $\alpha,\sigma>0$ and $N\ge 1$, that
\begin{equation}\label{vremT}
\Big|v(T_*+t)-\sum_{n=1}^N q_n(t)e^{-\mu_nt}\Big|_{\alpha,\sigma}= \mathcal O
\big(e^{-\mu t}\big),\text{ as }  t\to\infty,\text{ for all }\mu\in(\mu_N,\mu_{N+1}),
\end{equation}
where all $q_n$'s belong to $\mathcal{F}_1(\mathcal V)$.
By shifting the time variable, we obtain from \eqref{vremT} that
\begin{equation}\label{vrem2}
\Big|v(t)-\sum_{n=1}^N q_n(t-T_*)e^{-\mu_n (t-T_*)}\Big|_{\alpha,\sigma}= \mathcal O
\big(e^{-\mu t}\big), \text{ as } t\to\infty.
\end{equation}

Let
\begin{equation}\label{e1} Q_n(t)=e^{\mu_n T_*} e^{-\Omega t S} q_n(t-T_*).
\end{equation}
Rewrite the left-hand side of \eqref{vrem2} as
\begin{equation*}
\Big|e^{\Omega t S} \Big(u(t)-\sum_{n=1}^N Q_n(t) e^{-\mu_nt}\Big)\Big|_{\alpha,\sigma},
\text{ which equals }\Big|u(t)-\sum_{n=1}^N Q_n(t)e^{-\mu_n t}\Big|_{\alpha,\sigma}
\end{equation*}
thanks to the isometry \eqref{ESiso}. Thus, we obtain \eqref{uremain}.
Thanks to Lemma \ref{Spolyinvar}(i), each $q_n(t-T_*)$ is a  $\mathcal V$-valued S-polynomial, and hence, by \eqref{Sf1} of Lemma \ref{SPVlem}, so is each $Q_n(t)$.
The uniqueness of the S-polynomials $Q_n$'s follows from Lemma \ref{unilem}.
\end{proof}

Our proof of Theorem \ref{vthm} uses the Gevrey norm technique.
We recall a convenient estimate in \cite[Lemma 2.1]{HM1} for the Gevrey norms of the the bi-linear form $B(\cdot,\cdot)$, which is a generalization of the original inequality in \cite[Lemma 2.1]{FT-Gevrey}, and also the Sobolev estimates in \cite{FHOZ1}.

\textit{
There exists a constant $K\ge 1$ such that for any numbers $\alpha\ge 1/2$, $\sigma\ge 0$, and any functions $v,w\in G_{\alpha+1/2,\sigma}$,  one has
\begin{equation}\label{Buv}
|B(v,w)|_{\alpha,\sigma}
\le K^\alpha |v|_{\alpha+1/2,\sigma}  |w|_{\alpha+1/2,\sigma}.
\end{equation}
}

The same estimate as \eqref{Buv} can be obtained for $B(t,v,w)$.

\begin{lemma}\label{Btuv}
For any numbers $\alpha\ge 1/2$, $\sigma\ge 0$, any functions $v,w\in G_{\alpha+1/2,\sigma}$  and any $t\in \ensuremath{\mathbb R}$, one has
\begin{equation}\label{Bas}
|B(t,v,w)|_{\alpha,\sigma}
\le K^\alpha |v|_{\alpha+1/2,\sigma}  |w|_{\alpha+1/2,\sigma},
\end{equation}
where $K$ is the constant in \eqref{Buv}.
\end{lemma}
\begin{proof}
 By the isometry \eqref{ESiso} and inequality \eqref{Buv}, we have
 \begin{align*}
  |B(t,v,w)|_{\alpha,\sigma}&=|B(e^{-tS}v,e^{-tS}w)|_{\alpha,\sigma}\\
&  \le K^\alpha |e^{-tS} v|_{\alpha+1/2,\sigma}  |e^{-tS} w|_{\alpha+1/2,\sigma}
  =  K^\alpha |v|_{\alpha+1/2,\sigma}  |w|_{\alpha+1/2,\sigma},
\end{align*}
which proves \eqref{Bas}.
\end{proof}

As another preparation for the proof of Theorem \ref{vthm}, we establish the relevant estimates for the Gevrey norms of $v(t)$, when $t$ is large.

\begin{proposition}
\label{decay2}
Let $v^0\in H$ and $v(t)$ be a Leray-Hopf weak solution of \eqref{veq}.
For any $\sigma>0$,   there exist $T,D_\sigma>0$  such that
	\begin{equation*}
|v(t)|_{1/2,\sigma+1}\le D_\sigma e^{-t},\quad \text{ for all } t\ge T.
	\end{equation*}
Moreover, for any $\alpha\ge 0$ there exists $D_{\alpha,\sigma}>0$ such that
\begin{equation}\label{cl10}
|v(t)|_{\alpha+1/2,\sigma}\le D_{\alpha,\sigma} e^{-t},\quad \text{ for all }\ t\ge T.
\end{equation}
\end{proposition}
\begin{proof}
Thanks to the isometry \eqref{ESiso}, properties \eqref{BS1}, \eqref{BS2}, and inequality \eqref{Bas}, the proof, with $\mu_1=1$ under the current setting, is exactly the same as in \cite[Theorem 2.4]{HM1} and is omitted.
\end{proof}

We now are ready to prove Theorem \ref{vthm}.

\begin{proof}[Proof of Theorem \ref{vthm}]
This proof follows \cite{FS87} and \cite{HM1,HM2} with necessary modifications.

Firstly, we note, by part (iii) of Theorem \ref{typicalthm},  that there exists $T_*>0$ such that
equation \eqref{veq} holds in the classical sense in $\mathcal D(A)$ on $(T_*,\infty)$.

Let $\sigma>0$ be fixed. For each $N\in\ensuremath{\mathbb N}$, our main statement is

\textbf{(H$_N$)}
\textit{There exist $\mathcal V$-valued S-polynomials $q_n$'s for $n=1,2,\ldots,N$,
such that
\begin{equation}\label{vnrem}
\Big|v(t)-\sum_{n=1}^N q_n(t)e^{-\mu_n t}\Big|_{\alpha,\sigma}=\mathcal O(e^{-(\mu_N+\varepsilon)t})\text{ as }t\to\infty,
\end{equation}
for all $\alpha>0$, and some $\varepsilon=\varepsilon_{N,\alpha}>0$. Moreover, each $v_n(t)\stackrel{\rm def}{=} q_n(t)e^{-\mu_n t}$, for $n=1,2,\ldots,N$, solves the equation
\begin{equation}\label{vneq}
v_n'(t)+Av_n(t) +\sum_{\stackrel{1\le m,j \le n-1}{\mu_m+\mu_j=\mu_n}} B_{\Omega}(t,v_m(t),v_j(t))=0, \text{ for all } t\in\ensuremath{\mathbb R}.
\end{equation}
}

\noindent\textbf{Claim:} \textbf{(H$_N$)} holds true for any $N\in\ensuremath{\mathbb N}$.

We prove this Claim by induction in $N$.

\medskip
\noindent\textbf{First step $N=1$.}
Given $\alpha\ge 1/2$.
By estimate \eqref{cl10}, there exist $T_0>T_*$ and $d_0>0$ such that
\begin{equation}\label{vdec}
|v(t)|_{\alpha+1/2,\sigma}\le d_0 e^{-\mu_1 t},\quad\text{for all }t\ge T_0.
\end{equation}

Set $w_0=e^{\mu_1 t}v(t)$. We have, for $t\in(T_0,\infty)$, that
\begin{equation*}
w_0'=e^{\mu_1 t}( v' +\mu_1 v)
=e^{\mu_1 t} (-Av -B_{\Omega}(t,v,v)) +\mu_1 v),
\end{equation*}
hence
\begin{equation} \label{w0eq}
w_0' +(A-\mu_1)w_0 =H_0(t)\stackrel{\rm def}{=} - e^{\mu_1 t}B_{\Omega}(t,v(t),v(t)).
\end{equation}

Estimate \eqref{vdec} and inequality \eqref{Bas} imply
\begin{equation}\label{HOT}
|H_0(T_0+t)|_{\alpha,\sigma}
\le e^{\mu_1 (T_0+t)} K^\alpha |v(T_0+t)|_{\alpha+1/2,\sigma}^2
\le M_0 e^{-\mu_1 t},\quad\text{for all } t\ge 0,
\end{equation}
where $M_0=K^\alpha d_0^2 e^{-\mu_1 T_0}$.

For $k\in\ensuremath{\mathbb N}$, applying the projection $R_{\Lambda_k}$ to equation \eqref{w0eq} gives
\begin{equation} \label{Rkw0}
(R_{\Lambda_k} w_0)' +(\Lambda_k-\mu_1)R_{\Lambda_k} w_0 =R_{\Lambda_k} H_0(t).
\end{equation}

We apply Lemma \ref{polylem} to equation \eqref{Rkw0} in the space $X=R_{\Lambda_k}H$ with norm $\|\cdot\|_X=|\cdot|_{\alpha,\sigma}$,  solution $y(t)=R_{\Lambda_k} w_0(T_0+t)$, S-polynomial  $p(t)\equiv 0$, constant  $\beta=\Lambda_k-\mu_1\ge 0$, function $g(t)=R_{\Lambda_k}H_0(T_0+t)$ and numbers $M=M_0$, $\delta=\mu_1$ in \eqref{gM}.

When $k=1$, we have $\beta=0$, then by Lemma \ref{polylem}(ii), it follows that
\begin{equation}\label{R1w0}
|R_{\Lambda_1} w_0(T_0+t)-\xi_1|_{\alpha,\sigma}=\mathcal O(e^{-\mu_1 t}),
\end{equation}
where
\begin{equation*}
\xi_1=R_{\Lambda_1}w_0(T_0)+\int_0^\infty e^{\mu_1 \tau} R_{\Lambda_1} H_0(T_0+\tau)d\tau,
\end{equation*}
which exists and belongs to $R_{\Lambda_1}H$.

When $k\ge 2$, we have $\beta\ge \mu_2-\mu_1>0$, and it follows Lemma \ref{polylem}(i) that $q(t)$ defined by \eqref{qdef} is $0$, and, by \eqref{zsquare}, one has
\begin{align*}
&|R_{\Lambda_k} w_0(T_0+t)|_{\alpha,\sigma}^2
\le 2e^{-2(\Lambda_k-\mu_1) t}|R_{\Lambda_k} w_0(T_0)|_{\alpha,\sigma}^2 + 2t \int_0^t e^{-2(\Lambda_k-\mu_1)(t-\tau)} |R_{\Lambda_k} H_0(T_0+\tau)|_{\alpha,\sigma}^2 d\tau  \\
&\le 2e^{-2(\mu_2-\mu_1) t}|R_{\Lambda_k} w_0(T_0)|_{\alpha,\sigma}^2 + 2t \int_0^t e^{-2(\mu_2-\mu_1)(t-\tau)} |R_{\Lambda_k} H_0(T_0+\tau)|_{\alpha,\sigma}^2 d\tau.
\end{align*}

Summing up this inequality in $k$ gives
\begin{align*}
&|({\rm Id}-R_{\Lambda_1})w_0(T_0+t)|_{\alpha,\sigma}^2
=\sum_{k=2}^\infty |R_{\Lambda_k} w_0(T_0+t)|_{\alpha,\sigma}^2\\
&\le 2e^{-2(\mu_2-\mu_1) t}\sum_{k=2}^\infty |R_{\Lambda_k} w_0(T_0)|_{\alpha,\sigma}^2 + 2t \int_0^t e^{-2(\mu_2-\mu_1)(t-\tau)}\sum_{k=2}^\infty |R_{\Lambda_k} H_0(T_0+\tau)|_{\alpha,\sigma}^2 d\tau\\
&\le 2e^{-2(\mu_2-\mu_1) t}|w_0(T_0)|_{\alpha,\sigma}^2 + 2te^{-2(\mu_2-\mu_1)t} \int_0^t e^{2(\mu_2-\mu_1)\tau} | H_0(T_0+\tau)|_{\alpha,\sigma}^2 d\tau.
\end{align*}

Using \eqref{HOT}, we obtain
\begin{align*}
|({\rm Id}-R_{\Lambda_1})w_0(T_0+t)|_{\alpha,\sigma}^2
\le 2e^{-2(\mu_2-\mu_1) t}\Big(|w_0(T_0)|_{\alpha,\sigma}^2 +  2 M_0 t \int_0^t e^{2(\mu_2-2\mu_1)\tau} d\tau\Big).
\end{align*}

Since $2\mu_1>\mu_1$ and, by  \eqref{musum}, $2\mu_1\in \langle \sigma(A)\rangle$, then, thanks to \eqref{mudef}, $2\mu_1\ge \mu_2$.
Using this simple fact to estimate the last integral yields
\begin{align}\label{IR1w0}
|({\rm Id}-R_{\Lambda_1})w_0(T_0+t)|_{\alpha,\sigma}^2
&\le 2e^{-2(\mu_2-\mu_1) t}\big(|w_0(T_0)|_{\alpha,\sigma}^2 +  2 M_0 t^2\big).
\end{align}

Combining \eqref{R1w0}, \eqref{IR1w0} and the fact $\mu_2-\mu_1\le \mu_1$, gives
\begin{equation}\label{w0x1}
|w_0(t)-\xi_1|_{\alpha,\sigma}
\le |R_{\Lambda_1}w_0(t)-\xi_1|_{\alpha,\sigma}  + |({\rm Id}-R_{\Lambda_1})w_0(t)|_{\alpha,\sigma}=\mathcal O(e^{-\varepsilon t}),
\end{equation}
for any number $\varepsilon$ such that $0<\varepsilon<\mu_2-\mu_1$.

Define
\begin{equation}\label{q1} q_1(t)\equiv \xi_1.\end{equation}

Multiplying \eqref{w0x1} by $e^{-\mu_1 t}$ yields \eqref{vnrem} for $N=1$.
Also, since $\xi_1\in R_{\Lambda_1}H$, it is clear that $v_1(t)=\xi_1 e^{-\Lambda_1 t}$ satisfies  $v_1'(t)+Av_1(t)=0$ on $\ensuremath{\mathbb R}$.
Hence, $v_1$ satisfies \eqref{vneq} with $n=1$, because the sum of the bi-linear terms in \eqref{vneq} is void.
Note that $q_1$ does not depend on $\alpha$. Therefore, the statement \textbf{(H$_N$)} holds true for $N=1$.

\medskip
\noindent\textbf{Induction step.} Let $N\ge 1$ and assume the statement \textbf{(H$_N$)} holds true.
Let $q_n$, for $n=1,2,\ldots, N,$ be the $\mathcal V$-valued  S-polynomials in \textbf{(H$_N$)}.
There exists $\Lambda\in\sigma(A)$ such that
\begin{equation}\label{qnL}
q_n\in\mathcal{F}_1(P_\Lambda H), \text{ for all } 1\le n\le N.
\end{equation}

Let $v_n(t)=q_n(t)e^{-\mu_n t}$, denote $s_N(t)=\sum_{n=1}^N v_n(t)$
and $r_N(t)=v(t)-s_N(t)$. Let $\alpha\ge 1/2$.

(a) By the definition of $v_n$, we have, for $n\ge 2$,
\begin{equation}\label{h3}
|v_n(t)|_{\alpha+1/2,\sigma}= \mathcal O(e^{-(\mu_n-\delta) t})\quad\forall \delta>0,
\end{equation}
and, thanks to \eqref{q1},
\begin{equation*}
|v_1(t)|_{\alpha+1/2,\sigma}= \mathcal O(e^{-\mu_1 t}).
\end{equation*}
The last two properties imply
\begin{equation}\label{h2}
|s_N(T+t)|_{\alpha+1/2,\sigma} =\mathcal O(e^{-\mu_1 t}).
\end{equation}

By the induction hypothesis \textbf{(H$_N$)} applied to $\alpha+1/2$, there exists $\varepsilon>0$ such that
\begin{equation}\label{h1}
|r_N(t)|_{\alpha+1/2,\sigma}=\mathcal O(e^{-(\mu_N+\varepsilon)t}).
\end{equation}

We derive a differential equation for $r_N(t)$, for $t>T_*$. We calculate from \eqref{veq} and \eqref{vneq} for $n=1,2,\ldots,N$ that
\begin{align*}
 r_N'
 &=v'-\sum_{n=1}^N v_n' =-Av-B_\Omega(t,v,v)-\sum_{n=1}^N \Big\{ -Av_n - \sum_{\stackrel{1\le m,j \le n-1}{\mu_m+\mu_j=\mu_n}} B_{\Omega}(t,v_m,v_j)\Big\}\\
 &=-Ar_N -B_\Omega(t,r_N(t),v(t))-B_\Omega(t,s_N(t),r_N(t))-B_\Omega(t,s_N(t),s_N(t))\\
 &\quad + \sum_{\mu_m+\mu_j\le \mu_N} B_{\Omega}(t,v_m,v_j).
\end{align*}

We manipulate the last two terms as
\begin{align*}
&-B_\Omega(t,s_N(t),s_N(t))+ \sum_{\mu_m+\mu_j\le \mu_N} B_{\Omega}(t,v_m,v_j)
=-\sum_{\stackrel{1\le m,j\le N}{\mu_m+\mu_j\ge\mu_{N+1}}} B_\Omega(t,v_m(t),v_j(t))\\
&  =-\sum_{\stackrel{1\le m,j\le N}{\mu_m+\mu_j=\mu_{N+1}}} B_\Omega(t,v_m,v_j)-\sum_{\stackrel{1\le m,j\le N}{\mu_m+\mu_j\ge\mu_{N+2}}} B_\Omega(t,v_m(t),v_j(t)).
\end{align*}

 Thus, we obtain
\begin{equation}\label{vNeq}
r_N'(t)+Ar_N(t) +\sum_{\stackrel{1\le m,j\le N}{\mu_m+\mu_j=\mu_{N+1}}} B_\Omega(t,v_m(t),v_j(t))=h_N,
\text{ for } t>T_*,
\end{equation}
where
\begin{equation}\label{hdef}
h_N(t)=-B_\Omega(t,r_N(t),v(t))-B_\Omega(t,s_N(t),r_N(t))-\sum_{\stackrel{1\le m,j\le N}{\mu_m+\mu_j\ge  \mu_{N+2}}} B_\Omega(t,v_m(t),v_j(t)).
\end{equation}

(b) We estimate each term on the right-hand side of \eqref{hdef}. Note from \eqref{vdec}, \eqref{h1}, \eqref{h3}, \eqref{h2} and \eqref{Bas} that
\begin{align}
\label{BO1}
  |B_\Omega(t,r_N(t),v(t))|_{\alpha,\sigma} &=\mathcal O(e^{-(\mu_N+\mu_1+\varepsilon)t}),  \\
\label{BO2}
  |B_\Omega(t,s_N(t),r_N(t))|_{\alpha,\sigma} &= \mathcal O(e^{-(\mu_N+\mu_1+\varepsilon)t}),
\end{align}
and for $1\le m,j\le N$ with $\mu_m+\mu_j\ge  \mu_{N+2}$,
\begin{equation}
|B_\Omega(t,v_m(t),v_j(t))|_{\alpha,\sigma}=\mathcal O(e^{-(\mu_m+\mu_j-2\delta)t})=\mathcal O(e^{-(\mu_{N+2}-2\delta)t}),\forall \delta>0.\label{Bsum}
\end{equation}

Observe that $\mu_N+\mu_1>\mu_N$, and, by \eqref{musum}, $\mu_N+\mu_1\in\langle \sigma(A)\rangle$. This and \eqref{mudef} imply $\mu_N+\mu_1\ge \mu_{N+1}$. Obviously, $\mu_{N+2}>\mu_{N+1}$.
Then, by taking $\delta$ sufficiently small in \eqref{Bsum}, we have from \eqref{hdef}, \eqref{BO1}, \eqref{BO2} and \eqref{Bsum} that
\begin{equation}\label{hNO}
|h_N(t)|_{\alpha,\sigma}=\mathcal O(e^{-(\mu_{N+1}+\delta_N)t}),\quad  \text{for some }\delta_N\in(0,\mu_{N+2}-\mu_{N+1}).
\end{equation}

(c) Define $w_{N}(t)=e^{\mu_{N+1}t}r_N(t)$, and  $w_{N,k}(t)=R_{\Lambda_k}w_{N}(t)$, for $k\in\ensuremath{\mathbb N}$.
We have from \eqref{vNeq} that
\begin{equation}\label{wNk}
\frac{d}{dt} w_{N,k} +(\Lambda_k-\mu_{N+1})w_{N,k}=-\sum_{\stackrel{1\le m,j\le N}{\mu_m+\mu_j=\mu_{N+1}}} R_{\Lambda_k}B_\Omega(t,q_m,q_j) +R_{\Lambda_k}H_N(t),
\end{equation}
where $H_N(t)=e^{\mu_{N+1}t}h_N(t)$.
By \eqref{qnL} and \eqref{Sf3},
\begin{equation} \label{BLam}
B_\Omega(t,q_m(t),q_j(t))\in \mathcal{F}_1(P_{4\Lambda}H),
\end{equation}
which implies that the finite sum
$$\sum_{\stackrel{1\le m,j\le N}{\mu_m+\mu_j=\mu_{N+1}}} B_\Omega(t,q_m(t),q_j(t))\in \mathcal{F}_1(P_{4\Lambda}H).$$
Consequently,
\begin{equation}\label{B4L}
\sum_{\stackrel{1\le m,j\le N}{\mu_m+\mu_j=\mu_{N+1}}} R_{\Lambda_k}B_\Omega(t,q_m(t),q_j(t))
\in \mathcal{F}_1(R_{\Lambda_k}H).
\end{equation}

By the first property in Lemma \ref{Spolyinvar}(i),
\begin{equation*}
\sum_{\stackrel{1\le m,j\le N}{\mu_m+\mu_j=\mu_{N+1}}} R_{\Lambda_k}B_\Omega(T+t,q_m(T+t),q_j(T+t))\in \mathcal{F}_1(R_{\Lambda_k}H), \quad \text{for all } T\in \ensuremath{\mathbb R}.
\end{equation*}

By \eqref{hNO},
$|H_N(t)|_{\alpha,\sigma}=\mathcal O(e^{-\delta_N t})$.
Then there exist $T_N>T_*$ and $M_N>0$ such that
\begin{equation}\label{HNrate}
|H_N(T_N+t)|_{\alpha,\sigma}\le M_N e^{-\delta_N t}, \quad\text{for all }t\ge 0.
\end{equation}

We will apply Lemma \ref{polylem} again to equation \eqref{wNk} in the space $X=R_{\Lambda_k}H$ with norm $\|\cdot\|_X=|\cdot|_{\alpha,\sigma}$, solution $y(t)=w_{N,k}(T_N+t)$, constant $\beta=\Lambda_k-\mu_{N+1}$, S-polynomial
$$p(t)=-\sum_{\stackrel{1\le m,j\le N}{\mu_m+\mu_j=\mu_{N+1}}} R_{\Lambda_k}B_\Omega(T_N+t,q_m(T_N+t),q_j(T_N+t)),$$
function $g(t)=R_{\Lambda_k}H_N(T_N+t)$, numbers $M=M_N$ and $\delta=\delta_N$ in \eqref{HNrate}.

We consider three cases.

\medskip
\textbf{Case $\Lambda_k= \mu_{N+1}$.}  Then $\beta=0$ in \eqref{wNk}.
Let
\begin{equation*}
\xi_{N+1}\stackrel{\rm def}{=} R_{\mu_{N+1}}r_N(T_N)+\int_0^\infty e^{\mu_{N+1}\tau} R_{\mu_{N+1}}H_N(T_N+\tau)\ d\tau,
\end{equation*}
which exists and belongs to $R_{\mu_{N+1}}H$.
Define
\begin{equation} \label{p1}
p_{N+1,k}(t)=\xi_{N+1}-\int_0^t \sum_{\stackrel{1\le m,j\le N}{\mu_m+\mu_j=\mu_{N+1}}} R_{\mu_{N+1}} B_\Omega(T_N+ \tau,q_m(T_N+\tau),q_j(T_N+\tau))d\tau.
\end{equation}

\medskip
\textbf{Case $\Lambda_k\le \mu_N$.} Then $\beta<0$ in \eqref{wNk}.
Note, by \eqref{h1}, that
\begin{equation*}
e^{\beta t}|y(t)|_{\alpha,\sigma}=e^{\Lambda_k t}|R_{\Lambda_k}r_N(T_N+t)|_{\alpha,\sigma}
\le e^{\mu_N t}|R_{\Lambda_k}r_N(T_N+t)|_{\alpha,\sigma}=\mathcal O(e^{-\varepsilon t}).
\end{equation*}
Hence condition \eqref{limycond} is met. Define
\begin{equation}\label{i1}
\begin{aligned}
p_{N+1,k}(t)&=e^{-(\Lambda_k-\mu_{N+1})t}\int_t^\infty e^{(\Lambda_k-\mu_{N+1})\tau}\\
&\quad \cdot \Big(\sum_{\stackrel{1\le m,j\le N}{\mu_m+\mu_j=\mu_{N+1}}}R_{\Lambda_k}B_\Omega(T_N+ \tau,q_m(T_N+\tau),q_j(T_N+\tau)) \Big) d\tau.
\end{aligned}
\end{equation}

In the above two cases of $\Lambda_k$,  by applying Lemma \ref{polylem}(ii), we obtain $p_{N+1,k}\in \mathcal{F}_1(R_{\Lambda_k}H)$
that, by \eqref{g1b3}, satisfies
\begin{equation}\label{smallk}
|w_{N,k}(T_N+t)-p_{N+1,k}(t)|_{\alpha,\sigma}^2=\mathcal O(e^{-2\delta_N t}).
\end{equation}

\medskip
\textbf{Case $\Lambda_k\ge \mu_{N+2}$.} Then $\beta>0$ in \eqref{wNk}. Define
\begin{equation}\label{i2}
\begin{aligned}
p_{N+1,k}(t)
&=-e^{-(\Lambda_k-\mu_{N+1})t}\int_{-\infty}^t e^{(\Lambda_k-\mu_{N+1})\tau} \\
&\quad \cdot \Big(\sum_{\stackrel{1\le m,j\le N}{\mu_m+\mu_j=\mu_{N+1}}}R_{\Lambda_k}B_\Omega(T_N+ \tau,q_m(T_N+\tau),q_j(T_N+\tau)) \Big) d\tau.
\end{aligned}
\end{equation}

Applying Lemma \ref{polylem}(i), we obtain $p_{N+1,k}\in \mathcal{F}_1(R_{\Lambda_k}H)$, and, by \eqref{zsquare},
\begin{equation}  \label{tilwNk}
\begin{aligned}
|w_{N,k}(T_N+t)-p_{N+1,k}(t)|_{\alpha,\sigma}^2
&\le 2 e^{-2(\Lambda_k-\mu_{N+1})t} |w_{N,k}(T_N)- p_{N+1,k}(0)|_{\alpha,\sigma}^2 \\
&\quad + 2t\int_0^t e^{-2(\mu_{N+2}-\mu_{N+1})(t-\tau)} |R_{\Lambda_k} H_N(T_N+\tau)|_{\alpha,\sigma}^2 d\tau.
\end{aligned}
\end{equation}

\medskip
Denote $z_{N,k}(t)= w_{N,k}(T_N+t)-p_{N+1,k}(t)$, which is the remainder on the left-hand side of \eqref{smallk} and \eqref{tilwNk}.

On the one hand, because there are only finitely many $k$'s with $\Lambda_k\le \mu_{N+1}$, it follows from \eqref{smallk} that
\begin{equation}\label{sumsmall}
\sum_{\Lambda_k\le \mu_{N+1}} |z_{N,k}(t)|_{\alpha,\sigma}^2=\mathcal O(e^{-2\delta_N t}).
\end{equation}

On the other hand, summing inequality \eqref{tilwNk} in $k$ for which $\Lambda_k\ge \mu_{N+2}$, we obtain
\begin{align}
\sum_{\Lambda_k\ge \mu_{N+2}} |z_{N,k}(t)|_{\alpha,\sigma}^2
&\le  4 e^{-2(\mu_{N+2}-\mu_{N+1})t} \sum_{\Lambda_k\ge \mu_{N+2}}  |w_{N,k}(T_N)- p_{N+1,k}(0)|_{\alpha,\sigma}^2 \notag \\
&\quad + 2t\int_0^t e^{-2(\mu_{N+2}-\mu_{N+1})(t-\tau)}\sum_{\Lambda_k\ge \mu_{N+2}} |R_{\Lambda_k} H_N(T_N+\tau)|_{\alpha,\sigma}^2 d\tau.\label{zz}
\end{align}

For the terms on the right-hand side of \eqref{zz}, we have
\begin{align*}
\sum_{\Lambda_k\ge \mu_{N+2}}  |w_{N,k}(T_N)- p_{N+1,k}(0)|_{\alpha,\sigma}^2
&\le 2\sum_{\Lambda_k\ge \mu_{N+2}}  |R_{\Lambda_k} w_N(T_N)|_{\alpha,\sigma}^2+2\sum_{\Lambda_k\ge \mu_{N+2}}|p_{N+1,k}(0)|_{\alpha,\sigma}^2\\
&\le 2|w_N(T_N)|_{\alpha,\sigma}^2+2\sum_{\Lambda_k\ge \mu_{N+2}}|p_{N+1,k}(0)|_{\alpha,\sigma}^2.
\end{align*}
It follows from \eqref{B4L} and \eqref{i2} that $p_{N+1,k}\equiv 0$, for $\Lambda_k>\max\{4\Lambda,\mu_{N+1}\}$.
Thus, $$\sum_{\Lambda_k\ge \mu_{N+2}}|p_{N+1,k}(0)|_{\alpha,\sigma}^2\text{ is a finite sum, and hence is finite.}
$$
Also, the last integral in \eqref{zz} has
\begin{equation*}
\sum_{\Lambda_k\ge \mu_{N+2}} |R_{\Lambda_k} H_N(T_N+\tau)|_{\alpha,\sigma}^2\le |H_N(T_N+\tau)|_{\alpha,\sigma}^2\le M_N^2 e^{-2\delta_N \tau}.
\end{equation*}

Therefore, there exists $C_0>0$ such that
\begin{align*}
\sum_{\Lambda_k\ge \mu_{N+2}} |z_{N,k}(t)|_{\alpha,\sigma}^2
&\le  4C_0 e^{-2(\mu_{N+2}-\mu_{N+1})t} + 2M_N^2t e^{-2(\mu_{N+2}-\mu_{N+1})t} \int_0^t e^{2(\mu_{N+2}-\mu_{N+1}-\delta_N)\tau} d\tau.
\end{align*}

Recall that $\delta_N<\mu_{N+2}-\mu_{N+1}$ in \eqref{HNrate}. Calculating the last integral explicitly, we easily find
\begin{equation}\label{sumlarge}
\sum_{\Lambda_k\ge \mu_{N+2}} |z_{N,k}(t)|_{\alpha,\sigma}^2
\le  4C_0 e^{-2(\mu_{N+2}-\mu_{N+1})t} +  \frac{M_N^2 t  e^{-2\delta_N t}}{\mu_{N+2}-\mu_{N+1}-\delta_N}\le C_N e^{-\delta_N t},
\end{equation}
for some constant $C_N>0$.

Combining \eqref{sumsmall} and \eqref{sumlarge} gives
\begin{equation}\label{sumall}
\sum_{k=1}^\infty |z_{N,k}(t)|_{\alpha,\sigma}^2=\mathcal O(e^{-\delta_N t}).
\end{equation}

(d) By the first limit in \eqref{speclim}, we can select a number $\Lambda_*\in\sigma(A)$ to be the smallest $\Lambda_n$ such that $\Lambda_n\ge\max\{4\Lambda,\mu_{N+1}\}$.
By \eqref{BLam} and formulas \eqref{p1}, \eqref{i1}, \eqref{i2}, we have
\begin{equation} \label{pprop}
p_{N+1,k}\in \mathcal{F}_1(P_{\Lambda_k}H)\subset \mathcal{F}_1(P_{\Lambda_*}H),\text{ for }\Lambda_k\le \Lambda_*,
 \text{ and }
p_{N+1,k}=0, \text{ for }\Lambda_k>\Lambda_*.
\end{equation}

Define, for $t\in\ensuremath{\mathbb R}$,
\begin{equation}\label{qq}
q_{N+1}(t)=\sum_{k=1}^\infty p_{N+1,k}(t-T_N).
\end{equation}
Thanks to \eqref{pprop}, the sum in \eqref{qq} is only a finite sum, and
$q_{N+1}\in \mathcal{F}_1(P_{\Lambda_*}H)$.

Obviously,
\begin{equation} \label{Rqp}
R_{\Lambda_k}q_{N+1}(t+T_N)=p_{N+1,k}(t),
\end{equation}
hence, $z_{N,k}(t)=R_{\Lambda_k}\big(w_N(t+T_N)-q_{N+1}(t+T_N)\big)$.
It follows from \eqref{sumall} that
\begin{equation*}
|w_{N}(t+T_N)-q_{N+1}(t+T_N)|_{\alpha,\sigma}^2=\sum_{k=1}^\infty |z_{N,k}(t)|_{\alpha,\sigma}^2
=\mathcal O(e^{-\delta_N t}),
\end{equation*}
which implies that
\begin{equation*}
|w_{N}(t)-q_{N+1}(t)|_{\alpha,\sigma}=\mathcal O(e^{-\delta_N t/2}).
\end{equation*}
Multiplying this equation by $e^{-\mu_{N+1}t}$, we obtain the desired statement \eqref{vnrem} for $N+1$.

(e) It remains to prove the ODE \eqref{vneq}, for $n=N+1$. By \eqref{pode2} of Lemma \ref{polylem}, we have, for each $k$, that
$$p_{N+1,k}'(t)+(\Lambda_k-\mu_{N+1})p_{N+1,k}(t)+ \sum_{\stackrel{1\le m,j\le N}{\mu_m+\mu_j=\mu_{N+1}}} R_{\Lambda_k}B_{\Omega}(T_N+t,q_m(T_N+t),q_j(T_N+t)) =0.$$

From this and \eqref{Rqp}, we deduce
$$(R_{\Lambda_k} q_{N+1}(t) )'+(\Lambda_k-\mu_{N+1})R_{\Lambda_k}q_{N+1}(t) + R_{\Lambda_k}\sum_{\stackrel{1\le m,j\le N}{\mu_m+\mu_j=\mu_{N+1}}} B_{\Omega}(t,q_m(t),q_j(t)) =0.$$

Multiplying this equation by $e^{-\mu_{N+1}t}$ yields
$$(R_{\Lambda_k} v_{N+1}(t))'+A(R_{\Lambda_k}v_{N+1}(t)) + R_{\Lambda_k}\sum_{\stackrel{1\le m,j\le N}{\mu_m+\mu_j=\mu_{N+1}}} B_{\Omega}(t,v_m(t),v_j(t)) =0,$$
where $v_{N+1}(t)= q_{N+1}(t)e^{-\mu_{N+1}t}$.
Note in this equation that $v_{N+1}(t)$ and  $B_{\Omega}(t,v_m(t),v_j(t))$ all belong to $P_{\Lambda_*}H$.
Then summing up the equation in $k$, for which $\Lambda_k\le \Lambda_*$, yields that  the ODE system \eqref{vneq} holds for $n=N+1$.
According to Remark \ref{uniG}, the S-polynomial $q_{N+1}$ is independent of $\alpha$.
Therefore, the statement \textbf{(H$_{N+1}$)} is proved.

By the induction principle, the statement \textbf{(H$_N$)} holds true for all $N$, i.e., the Claim is proved.

Now, we note that at step $n=N+1$, the $q_n$'s, for $n=1,2,\ldots,N$, are those from the step $n=N$, and are used in the construction of $q_{N+1}$. Therefore, for each $\sigma>0$, such recursive construction gives the existence of  the S-polynomials $q_n$'s, for all $n\in\ensuremath{\mathbb N}$. By Remark \ref{uniG} again, all these $q_n$'s are, in fact,  independent of $\sigma$. Therefore, \eqref{vnrem} holds true for all $N$, $\alpha$, $\sigma$. Then estimate \eqref{vremain} follows thanks to \eqref{ex3}.
Finally, by Lemma \ref{unilem}, the S-polynomials $q_n$'s are unique.
The proof is complete.
\end{proof}

\begin{remark}
 The statement and proof of Theorem \ref{vthm} can be presented simply with $G_{0,\sigma}$ for all $\sigma\ge 0$. Nonetheless, general calculations in the above proof for $G_{\alpha,\sigma}$ are flexible and can be applied to cover the case when a non-potential force is included in the NSE and has limited regularity in $G_{\alpha,\sigma}$ for a fixed $\sigma$, see \cite{HM2,CaH1,CaH2}.
\end{remark}

\begin{remark}\label{limrmk}
 It is not known whether the polynomials $Q_n$'s in Theorem \ref{uthm} have any limits as $|\Omega|\to\infty$. However, such a question can be answered for some special solutions,
 see Remark \ref{lim2rmk} below.
\end{remark}

\section{The case of non-zero spatial average solutions}
\label{nonzerosec}

In this section, we establish the asymptotic expansions for the solutions with non-zero spatial averages.
Below, $C_{\bvec x,t}^{2,1}$ denotes the class of continuous functions $f(\bvec x,t)$ with continuous partial derivatives $\partial f/\partial x_k,\partial^2 f/\partial x_k\partial x_j$ (for $k,j=1,2,3$,) and $\partial f/\partial t$.
Notation $C_{\bvec x}^{1}$ is meant similarly.

\begin{assumption}
 Throughout this section, $\bvec u(\bvec x,t)\in C_{\bvec x,t}^{2,1}(\ensuremath{\mathbb R}^3\times (0,\infty))\cap C(\ensuremath{\mathbb R}^3\times[0,\infty))$ and $p(\bvec x,t)\in C_{\bvec x}^{1}(\ensuremath{\mathbb R}^3\times (0,\infty))$
 are $\bvec L$-periodic  functions that form a solution $(\bvec u,p)$ of the NSE \eqref{NPDE} and \eqref{divfree}.
\end{assumption}

\medskip
Note that any $\bvec L$-periodic function on $\ensuremath{\mathbb R}^3$ can be considered as a function on the flat torus ${\mathbb T_\bvec L}$.
In particular, $u(t)=\bvec u(\cdot,t)$ and $p(t)=p(\cdot,t)$ can be considered as functions on ${\mathbb T_\bvec L}$.

Suppose $\bvec f$ is a $\bvec L$-periodic vector field on $\ensuremath{\mathbb R}^3$ and $\bvec f\in \mathcal D(A^\alpha e^{\sigma A^{1/2}})$, for some $\alpha,\sigma\ge 0$.
Let $\bvec g(\bvec x)=\bvec f(\bvec x+\bvec X_0)$, for some fixed $\bvec X_0\in\ensuremath{\mathbb R}^3$, then $\bvec g\in H$.
Let $\bvec f(\bvec x)=\sideset{}{'}\sum  \widehat\bvec f_\bvec k e^{i \check \bvec k\cdot \bvec x}$, then
\begin{equation*}
\bvec g(\bvec x)=\sideset{}{'}\sum  \widehat\bvec g_\bvec k e^{i \check \bvec k\cdot \bvec x}, \text{ where } \widehat\bvec g_\bvec k=\widehat\bvec f_\bvec k e^{i\check\bvec k\cdot \bvec X_0}.
\end{equation*}
It follows that
$|\widehat\bvec g_\bvec k|=|\widehat\bvec f_\bvec k|$, and consequently,
 \begin{equation}\label{normshift}
| \bvec f(\cdot+\bvec X_0) |_{\alpha,\sigma}=| \bvec g(\cdot) |_{\alpha,\sigma}=| \bvec f(\cdot)|_{\alpha,\sigma}.
 \end{equation}

\medskip
Returning to the solution $(\bvec u,p)$, denote, for $t\ge 0$,
\begin{equation*}
\bvec U(t)=\frac1{L_1L_2L_3}\int_{\mathbb T_\bvec L} \bvec u(\bvec x,t)d\bvec x.
\end{equation*}

Integrating equation \eqref{NPDE} over the domain ${\mathbb T_\bvec L}$ gives
\begin{equation*}
\bvec U'(t)+\Omega J \bvec U(t)=0,\quad t>0.
\end{equation*}
Hence,
\begin{equation} \label{Ut}
\bvec U(t)=e^{-\Omega t J} \bvec U(0)=\begin{pmatrix}
                           \cos(\Omega t)& \sin(\Omega t)&0\\
                           -\sin(\Omega t)&\cos(\Omega t)&0\\
                           0&0&1
                          \end{pmatrix} \bvec U(0).
\end{equation}

\begin{theorem}\label{SSthm}
There exist $\mathcal V$-valued SS-polynomials $\mathcal Q_n(t)$'s, for all $n\in \ensuremath{\mathbb N}$, such that
 \begin{equation}\label{uU}
 u(t)-\bvec U(t)\sim \sum_{n=1}^\infty \mathcal Q_n(t) e^{-\mu_n t} \text{ in } G_{\alpha,\sigma},\text { for all }\alpha,\sigma>0.
 \end{equation}
\end{theorem}
\begin{proof}
For $t\ge 0$, define
$\bvec V(t)=\int_0^t \bvec U(\tau)d\tau$,
which, by \eqref{Ut}, is
\begin{equation} \label{Vt}
\bvec V(t)= \frac1\Omega\begin{pmatrix}
                           \sin(\Omega t)& 1-\cos(\Omega t)&0\\
                           \cos(\Omega t)-1&\sin(\Omega t)&0\\
                           0&0& \Omega t
                          \end{pmatrix} \bvec U(0).
\end{equation}

We use the following hyper-Galilean transformation \cite{Foias-TAMU2002}:
\begin{equation}\label{Gtrans}
\bvec w(\bvec x,t)=\bvec u(\bvec x+\bvec V(t),t)-\bvec U(t), \quad \vartheta (\bvec x,t)= p(\bvec x+\bvec V(t),t).
\end{equation}

By simple calculations, one can verify
\begin{equation*}
\bvec w_t-\Delta \bvec w +(\bvec w\cdot \nabla) \bvec w+\Omega J\bvec w=-\nabla \vartheta
\text{ and } {\rm div }\, \bvec w=0,
\end{equation*}
that is, $(\bvec w,\vartheta)$ is also a classical solution of the system \eqref{NPDE} and \eqref{divfree} on $\ensuremath{\mathbb R}^3\times(0,\infty)$.

Note that  $(\bvec w,\vartheta)$ is $\bvec L$-periodic, and  $\bvec w(\cdot,t)$ has zero average for each $t\ge 0$.
Applying Theorem \ref{uthm} to the solution $w(t)=\bvec w(\cdot,t)$ we obtain
 \begin{equation}\label{wexp}
 w(t)\sim \sum_{n=1}^\infty Q_n(t) e^{-\mu_n t} \text{ in } G_{\alpha,\sigma },\text{ for all }\alpha,\sigma> 0,
 \end{equation}
 where  $Q_n(t)$'s are $\mathcal V$-valued S-polynomials.

Assume each $Q_n(t)$ is a mapping $\bvec x\mapsto \bvec Q_n(\bvec x,t)$.
Note that each $Q_n(t)$ belongs to $\mathcal V$, hence $\bvec Q_n(\bvec x,t)$, as a function of $\bvec x\in\ensuremath{\mathbb R}^3$, is $\bvec L$-periodic.
Let $N\in\ensuremath{\mathbb N}$, $\alpha,\sigma>0$ and $\mu\in (\mu_N,\mu_{N+1})$. We have from \eqref{wexp} and \eqref{ex3} that
 \begin{equation*}
\Big| \bvec u(\bvec x+\bvec V(t),t)-\bvec U(t)-\sum_{n=1}^N \bvec Q_n(\bvec x,t)e^{-\mu_n t} \Big|_{\alpha,\sigma}=\mathcal O(e^{-\mu t}).
 \end{equation*}
This and \eqref{normshift} imply
  \begin{equation*}
\Big| \bvec u(\bvec x,t)-\bvec U(t)-\sum_{n=1}^N \bvec Q_n(\bvec x-\bvec V(t),t)e^{-\mu_n t} \Big|_{\alpha,\sigma}=\mathcal O(e^{-\mu t}),
 \end{equation*}
which means
 \begin{equation}\label{uUQ}
\Big| u(t)-\bvec U(t)-\sum_{n=1}^N \mathcal Q_n(t) e^{-\mu_n t} \Big|_{\alpha,\sigma}=\mathcal O(e^{-\mu t}),
 \end{equation}
where  $\mathcal Q_n(t)=\bvec Q_n(\cdot-\bvec V(t),t)$ for $n\in\ensuremath{\mathbb N}$.
It remains to prove that $\mathcal Q_n\in \mathcal{F}_2(\mathcal V)$, for all $n\in \ensuremath{\mathbb N}$.
Suppose
$$ \bvec Q_n(\bvec x,t)=\sideset{}{'}\sum_{\rm{finitely\ many\ }\bvec k} \widehat \bvec Q_{n,\bvec k}(t) e^{i\check\bvec k\cdot\bvec x},\text{ with }\widehat \bvec Q_{n,\bvec k}(t)\in \mathcal{F}_1(X_\bvec k).$$
 Then
 \begin{equation*}
\mathcal Q_n(t)
=\sideset{}{'}\sum_{\rm{finitely\ many\ }\bvec k} \widehat \bvec Q_{n,\bvec k}(t)e^{-i\check\bvec k\cdot \bvec V(t)}e^{i \check\bvec k\cdot\bvec x}
=\sideset{}{'}\sum_{\rm{finitely\ many\ }\bvec k} \widehat {\mathcal Q}_{n,\bvec k}(t) e^{i\check\bvec k\cdot\bvec x},
 \end{equation*}
 where
 $\widehat {\mathcal Q}_{n,\bvec k}(t)=\widehat \bvec Q_{n,\bvec k}(t)\big[\cos(\check\bvec k\cdot \bvec V(t))-i \sin(\check\bvec k\cdot \bvec V(t))\big]$.

Using the formula of $\bvec V(t)$ in \eqref{Vt}, one can see that
\begin{equation*}
\check\bvec k\cdot \bvec V(t)=r_1\cos(\Omega t)+r_2\sin(\Omega t)+r_3 t + r_4,
\end{equation*}
for some numbers $r_2,r_2,r_3,r_4\in \ensuremath{\mathbb R}$, that depend on $\check\bvec k$.

By using the product to sum formulas between trigonometric functions, we have
$$
\widehat \bvec Q_{n,\bvec k}(t) \cos(\check\bvec k\cdot \bvec V(t)), \widehat \bvec Q_{n,\bvec k}(t)  \sin(\check\bvec k\cdot \bvec V(t))
\in \mathcal{F}_2(X_\bvec k).
$$
Therefore, $\widehat {\mathcal Q}_{n,\bvec k}(t)\in \mathcal{F}_2(X_\bvec k)$, and, by Lemma \ref{SPlem1}, $\mathcal Q_n(t)\in \mathcal{F}_2(\mathcal V)$.
With this fact, we obtain the expansion \eqref{uU} from estimate \eqref{uUQ}.
\end{proof}

\begin{remark}
We can roughly rewrite \eqref{uU} as an asymptotic expansion of $u(t)$ as
 \begin{equation*}
 u(t)\sim  \bvec U(t) + \sum_{n=1}^\infty \mathcal Q_n(t) e^{-\mu_n t} .
 \end{equation*}
This shows that $\bvec U(t)$ is the leading order term in the asymptotic approximation, as $t\to\infty$,  i.e., the  $\mathcal O(1)$ term, is explicitly determined by \eqref{Ut}, and is responsible for the non-zero average of $u(t)$. The following order terms in the approximation are all exponentially decaying and are with zero spatial averages.
\end{remark}

\section{Some special solutions}\label{specialsec}

We present next some special solutions of \eqref{NPDE} and \eqref{divfree}. They are inspired by examples in \cite{FS84b,FHN1,FHN2} for the case without rotation.

Let $\bvec k=(k_1,k_2,k_3)\in \ensuremath{\mathbb Z}^3\setminus\{\mathbf 0\}$ with $\widetilde\bvec k=(\widetilde k_1,\widetilde k_2,\widetilde k_3)$ be fixed.
For any $m\in\ensuremath{\mathbb Z}_*=\ensuremath{\mathbb Z}\setminus\{0\}$, we note from \eqref{kcheck}, \eqref{ktil} and \eqref{Jk} that
\begin{equation} \label{mkprop}
(m\bvec k)\check{}=m\check\bvec k,\quad
\widetilde{m\bvec k}={\rm sgn}(m)\widetilde\bvec k,\text{ and }
J_{m\bvec k}=\rm{sgn }(m)J_\bvec k.
\end{equation}

We define $V_\bvec k$ to be the space of all $u\in V$ such that
\begin{equation} \label{uset}
u=\sum _{m\in\ensuremath{\mathbb Z}_*} \widehat \bvec u_{m\bvec k} e^{i m \check\bvec k\cdot\bvec x}.
\end{equation}

By \eqref{unabv},  one immediately sees that
\begin{equation}\label{specialB}
(u\cdot \nabla)v=0, \text{ for all } u,v\in V_\bvec k.
\end{equation}

Let $u\in V_\bvec k$ as in \eqref{uset} and $t\in\ensuremath{\mathbb R}$.  We have, by \eqref{etS}, \eqref{uset} and \eqref{mkprop},
\begin{equation*}
e^{tS}u=\sum_{m\in\ensuremath{\mathbb Z}_*} E_\bvec k(\widetilde k_3 t)\widehat \bvec u_{m\bvec k} e^{i m \check\bvec k\cdot\bvec x}.
\end{equation*}
Hence
\begin{equation}\label{euV}
e^{tS}u\in V_\bvec k.
\end{equation}

Combining \eqref{specialB} and \eqref{euV} yields
\begin{equation}\label{specialBS}
[(e^{tS}u)\cdot\nabla ](e^{tS}v)=0,\text{ for all }u,v\in V_\bvec k \text{ and } t\in \ensuremath{\mathbb R}.
\end{equation}

\begin{theorem}\label{specthm1}
Let $\bvec k\in \ensuremath{\mathbb Z}^3\setminus\{\mathbf 0\}$ be fixed and $u_0\in V_\bvec k$, then problem \eqref{NSO}, with initial condition $u(0)=u_0$, has a unique global regular solution
 \begin{equation}\label{specialu}
 u(t)=e^{-tA}e^{-\Omega t S}u_0,\text{ for } t\ge 0.
 \end{equation}
Moreover, $u(t)$ solves the linear equation
\begin{equation}\label{linearrot}
u_t + Au + \Omega S u =0, \text{ for } t>0.
\end{equation}
 \end{theorem}
\begin{proof}
 Let $u_0=\sum _{m\in\ensuremath{\mathbb Z}_*} \widehat \bvec u_{0,m\bvec k} e^{i m \check\bvec k\cdot\bvec x}\in V_\bvec k$.
Set $v(t)=e^{-tA}u_0$. First, we have
\begin{equation}\label{stokesv}
v_t+Av=0,\text{ for all }t>0.
\end{equation}

Note that $u(t)$ defined by \eqref{specialu} equals $e^{-\Omega t S} v(t)$. Then $u(0)=u_0$ and, by \eqref{stokesv},  $u(t)$ solves
\eqref{linearrot}.

Clearly, $v(t)\in V_\bvec k$, for all $t\ge 0$, hence,   by \eqref{specialBS},
\begin{equation}\label{strongb}
(u(t)\cdot\nabla) u(t) =0,
\end{equation}
which gives $B(u(t),u(t))=0$. The last fact and \eqref{linearrot} imply that $u(t)$ also solves \eqref{NSO}.

Since $u(t)$ is a global, regular solution, it is unique.
\end{proof}

\begin{remark}\label{heli}
We find some special solutions for which the helicity $\mathcal H(t)\stackrel{\rm def}{=} \inprod{\nabla\times u(t),u(t)}$ vanishes for all $t\ge 0$.
(See \cite{Moffatt69,Moffatt92} for the physics of helicity, and \cite{FHN1,FHN2} for its analysis in the case of non-rotating fluids.)
We consider the solution in Theorem \ref{specthm1}, which can be written explicitly as
\begin{equation}\label{helisol}
u(t)=\sum_{m\in\ensuremath{\mathbb Z}_*} e^{-m^2|\check\bvec k|^2 t}\Big\{\cos(\widetilde k_3\Omega t)\widehat \bvec u_{0,m\bvec k} -\sin(\widetilde k_3\Omega t)\widetilde\bvec k\times \widehat \bvec u_{0,m\bvec k}\Big\} e^{i m \check\bvec k\cdot\bvec x}.
\end{equation}

Then the vorticity is
\begin{equation*}
\nabla \times u(t)=\sum_{m\in\ensuremath{\mathbb Z}_*} e^{-m^2|\check\bvec k|^2 t}im|\check\bvec k| \Big\{\cos(\widetilde k_3\Omega t)\widetilde\bvec k\times\widehat \bvec u_{0,m\bvec k} + \sin(\widetilde k_3\Omega t) \widehat \bvec u_{0,m\bvec k}\Big\} e^{i m \check\bvec k\cdot\bvec x},
\end{equation*}
and the helicity is
\begin{align*}
\mathcal H(t)
&=L_1L_2L_3\sum_{m\in\ensuremath{\mathbb Z}_*} e^{-2m^2|\check\bvec k|^2 t}im|\check\bvec k|
\Big\{\cos^2(\widetilde k_3\Omega t)J_\bvec k\widehat \bvec u_{0,m\bvec k}\cdot \overline{\widehat \bvec u}_{0,m\bvec k}
- \sin^2(\widetilde k_3\Omega t) \widehat \bvec u_{0,m\bvec k}\cdot J_\bvec k\overline{\widehat \bvec u}_{0,m\bvec k}\\
&\quad
+ \cos(\widetilde k_3\Omega t)\sin(\widetilde k_3\Omega t)
\Big[\widehat \bvec u_{0,m\bvec k}\cdot \overline{\widehat \bvec u}_{0,m\bvec k}
-  J_\bvec k\widehat \bvec u_{0,m\bvec k}\cdot J_\bvec k\overline{\widehat \bvec u}_{0,m\bvec k}\Big]
\Big\}\\
&=L_1L_2L_3\sum_{m\in\ensuremath{\mathbb Z}_*} e^{-2m^2|\check\bvec k|^2 t}im|\check\bvec k|
(J_\bvec k\widehat \bvec u_{0,m\bvec k}\cdot \overline{\widehat \bvec u}_{0,m\bvec k}).
\end{align*}
Hence,
\begin{align*}
\mathcal H(t)
&= L_1L_2L_3\sum_{m\in\ensuremath{\mathbb Z}_*} e^{-2m^2|\check\bvec k|^2 t} 2 m|\check\bvec k|
\Big[ \big ({\rm Re}(\widehat \bvec u_{0,m\bvec k})\times {\rm Im}(\widehat \bvec u_{0,m\bvec k} )\big )\cdot \widetilde\bvec k\Big].
\end{align*}

Thus,  $\mathcal H(t)=0$ for all $t\ge 0$, provided that
\begin{equation} \label{helicond1}
\big ({\rm Re}(\widehat \bvec u_{0,m\bvec k})\times {\rm Im}(\widehat \bvec u_{0,m\bvec k} )\big )\cdot \widetilde\bvec k =0, \text{ for all }m\in \ensuremath{\mathbb Z}_*.
\end{equation}
However, since $\widehat \bvec u_{0,m\bvec k}$ is orthogonal to $\widetilde\bvec k$ then ${\rm Re}(\widehat \bvec u_{0,m\bvec k})\times {\rm Im}(\widehat \bvec u_{0,m\bvec k})$ is co-linear with $\widetilde\bvec k$, and as result \eqref{helicond1} is equivalent to
\begin{equation} \label{helicond}
{\rm Re}(\widehat \bvec u_{0,m\bvec k})\times {\rm Im}(\widehat \bvec u_{0,m\bvec k})=0, \text{ for all }m\in \ensuremath{\mathbb Z}_*.
\end{equation}

This class of solutions \eqref{helisol}, \eqref{helicond} with vanishing helicity  for the NSE of rotating fluids has more restrictive wave vectors, i.e., the $m\bvec k$'s in \eqref{helisol}, than those studied in \cite[Proposition 6.4]{FHN1} for the NSE without the rotation.
\end{remark}

\begin{corollary}\label{cor53}
 There exist infinitely many vector spaces of infinite dimensions that are invariant under the NSE \eqref{NSO},
 and each space is not a subspace of any of the others.
\end{corollary}
\begin{proof}
According to Theorem \ref{specthm1}, each vector space $V_\bvec k$ is invariant under the NSE \eqref{NSO}, and each has infinite dimension.
 Moreover, there are infinite many $\bvec k$'s which are pairwise not co-linear. For those $\bvec k$'s, the corresponding $V_\bvec k$'s are the vector spaces for which the statement holds true.
\end{proof}

Next, we consider the case of non-zero spatial average solutions.

\begin{theorem}\label{specthm2}
 Suppose $u_0=\bvec U_0+w_0$, where $\bvec U_0$ is a constant vector in $\ensuremath{\mathbb R}^3$, and
\begin{equation*}
w_0=\sum _{m\in\ensuremath{\mathbb Z}_*} \widehat \bvec u_{0,m\bvec k} e^{i m \check\bvec k\cdot\bvec x}\in V_\bvec k.
\end{equation*}

Let $\bvec U(t)=e^{-\Omega t J}\bvec U_0$ and $\bvec V(t)=\int_0^t\bvec U(\tau)d\tau$.
Define, for $\bvec x\in\ensuremath{\mathbb R}^3$ and $t> 0$,
 \begin{align}
 \label{ulast}
\bvec u(\bvec x,t)&=\bvec U(t)+\sum_{m\in\ensuremath{\mathbb Z}_*} e^{-m^2|\check\bvec k|^2 t}e^{-i m \check\bvec k\cdot \bvec V(t)} E_\bvec k(-\widetilde k_3\Omega t)\widehat \bvec u_{0,m\bvec k} e^{i m \check\bvec k\cdot\bvec x},\\
\label{plast}
p(\bvec x,t)&=p_*(t)-\Omega \sum_{m\in\ensuremath{\mathbb Z}_*}\frac{i}{m |\check\bvec k|} e^{-m^2|\check\bvec k|^2 t}e^{-i m \check\bvec k\cdot \bvec V(t)} \notag \\
&\quad \cdot \big[ \cos(\widetilde k_3\Omega t) J \widetilde\bvec k  + \sin(\widetilde k_3\Omega t) \bvec e_3\big ]\cdot \widehat\bvec u_{0,m\bvec k} e^{i m \check\bvec k\cdot\bvec x},
\end{align}
where $p_*(t)$ is any scalar function.
Then $(\bvec u,p)$ is a solution of \eqref{NPDE} and \eqref{divfree} on $\ensuremath{\mathbb R}^3\times (0,\infty)$.
\end{theorem}
\begin{proof}
Let $w(t)=e^{-tA}e^{-t \Omega S} w_0$, which we write as $w(t)=\bvec w(\cdot,t)$.
By Theorem \ref{specthm1}, particularly, \eqref{linearrot} and \eqref{strongb}, we have
\begin{equation*}
(\bvec w\cdot\nabla) \bvec w=0\text{ and } \bvec w_t-\Delta \bvec w + \Omega J\bvec w=-\nabla q,
\end{equation*}
where the scalar function $q(\bvec x,t)$ satisfies the geostrophic balance
\begin{equation*}
\Omega {\rm div }(J \bvec w)=-\Delta q.
\end{equation*}

We solve this equation by
\begin{equation}\label{qone}
q(\bvec x,t)=p_*(t)+\Omega (-\Delta)^{-1}{\rm div }(J \bvec w(\bvec x,t)),
\end{equation}
where the inverse operator $(-\Delta)^{-1}$ is meant to apply to functions having zero spatial average over ${\mathbb T_\bvec L}$.

We note that $\bvec w(\bvec x,t)$ is Gevrey-regular, for each $t>0$. Therefore, the following calculations are valid in the classical sense.
Define
\begin{equation} \label{up}
\bvec u(\bvec x,t)=\bvec w(\bvec x-\bvec V(t),t)+\bvec U(t)\text{ and }p(\bvec x,t)=q(\bvec x-\bvec V(t),t).
\end{equation}
(The functions $\bvec u$ and $p$ in \eqref{up} will be proved to agree with those in \eqref{ulast} and \eqref{plast} later.)

Since $\bvec w$ is divergence-free, then, clearly, so is $\bvec u$.
We calculate, with the shorthand notation $\bvec u=\bvec u(\bvec x,t)$ and $\bvec w=\bvec w(\bvec x-\bvec V(t),t)$,
\begin{equation*}
\bvec u_t = \bvec w_t - (\bvec U\cdot\nabla )\bvec w + \bvec U'=\bvec w_t - (\bvec U\cdot\nabla )\bvec w-\Omega J \bvec U,
\end{equation*}
\begin{equation*}
(\bvec u\cdot\nabla)\bvec u=[(\bvec w+\bvec U)\cdot\nabla] \bvec w=[\bvec U\cdot\nabla] \bvec w.
\end{equation*}
Therefore,
\begin{align*}
\Big[\bvec u_t-\Delta\bvec u+(\bvec u\cdot\nabla)\bvec u+\Omega J\bvec u\Big]_{(\bvec x,t)}
&=\Big[\bvec w_t-\Delta \bvec w  + \Omega J\bvec w\Big]_{(\bvec x-\bvec V(t),t)} \\
&=-\Big[\nabla q\Big]_{(\bvec x-\bvec V(t),t)} =-\Big[\nabla p\Big]_{(\bvec x,t)} .
\end{align*}

We conclude that $(\bvec u,p)$ defined by \eqref{up} is a solution of \eqref{NPDE} and \eqref{divfree} on $\ensuremath{\mathbb R}^3\times (0,\infty)$.
It remains to calculate them explicitly.
First, we have
\begin{equation}\label{wlast}
\bvec w(\bvec x,t)=\sum_{m\in\ensuremath{\mathbb Z}_*} e^{-m^2|\check\bvec k|^2 t}E_\bvec k(-\widetilde k_3\Omega t)\widehat \bvec u_{0,m\bvec k} e^{i m \check\bvec k\cdot\bvec x}.
\end{equation}
Then, thanks to \eqref{up} and \eqref{wlast}, we obtain formula \eqref{ulast} for $\bvec u(\bvec x,t)$.
Next, we calculate ${\rm div}(J\bvec w)$ by
\begin{equation}\label{divJ}
{\rm div}( J\bvec w(\bvec x,t))
 =\sum_{m\in\ensuremath{\mathbb Z}_*} e^{-m^2|\check\bvec k|^2 t}i\,m \check\bvec k^{\rm T} JE_\bvec k(-\widetilde k_3\Omega t)\widehat \bvec u_{0,m\bvec k} e^{i m \check\bvec k\cdot\bvec x}.
\end{equation}
Denote $\bvec z=\widehat \bvec u_{0,m\bvec k} \in X_\bvec k$, then
\begin{align*}
\check\bvec k^{\rm T} JE_\bvec k(t)\bvec z
&=\cos(t)  |\check\bvec k| \widetilde \bvec k \cdot  J\bvec z +\sin(t) |\check\bvec k| \widetilde \bvec k \cdot (\bvec e_3\times J_\bvec k \bvec z)\\
&=|\check\bvec k| \big[ \cos(t) (J^{\rm T}\widetilde \bvec k) \cdot \bvec z -\sin(t) \bvec e_3\cdot J_\bvec k^2 \bvec z\big]\\
&= |\check\bvec k| [-\cos(t) J\widetilde\bvec k +\sin(t) \bvec e_3]\cdot \bvec z.
\end{align*}
(We used the fact $J$ is anti-symmetric and relation \eqref{Jksq} below.)
Thus, together with  \eqref{qone} and \eqref{divJ},
\begin{equation}\label{qlast}
q(\bvec x,t)=p_*(t)-\Omega\sum_{m\in\ensuremath{\mathbb Z}_*}\frac{i\,  e^{-m^2|\check\bvec k|^2 t}}{m |\check\bvec k|}
\big[ \cos(\widetilde k_3\Omega t) J \widetilde\bvec k  + \sin(\widetilde k_3\Omega t) \bvec e_3]\cdot \widehat\bvec u_{0,m\bvec k} e^{i m \check\bvec k\cdot\bvec x}.
\end{equation}
From \eqref{up} and \eqref{qlast}, we obtain formula \eqref{plast} for $p(\bvec x,t)$. The proof is complete.
\end{proof}

\begin{remark}\label{lim2rmk}
Let $\bvec k$ and $u_0$ be as in Theorem \ref{specthm1} and its proof.
We rewrite \eqref{specialu} as
\begin{equation}\label{uLam}
u(t)=\sum_{n=1}^\infty Q_{n,\Omega}(t)e^{-\Lambda_n t},
\end{equation}
where $Q_{n,\Omega}(t)=e^{-\Omega t S}R_{\Lambda_n}u_0$.
By \eqref{Sf1}, $Q_{n,\Omega}\in \mathcal{F}_1(\mathcal V)$. Therefore, \eqref{uLam} is the asymptotic expansion of $u(t)$.

 By \eqref{etS} and \eqref{uset}, we have either $Q_{n,\Omega}=0$, or there is a unique number $m\in\ensuremath{\mathbb N}$ with $m^2 |\check\bvec k|^2=\Lambda_n$ and
\begin{equation}\label{Qome}
Q_{n,\Omega}(t)=(\cos(\tilde k_3\Omega  t)I_3-\sin(\tilde k_3\Omega  t)J_\bvec k)
[ \widehat \bvec u_{m\bvec k} e^{i m \check\bvec k\cdot\bvec x} + \widehat \bvec u_{-m\bvec k} e^{-i m \check\bvec k\cdot\bvec x}] .
\end{equation}

In case $k_3=0$, we then have $Q_{n,\Omega}(t)=R_{\Lambda_n}u_0$ which is independent of $\Omega$.

Consider the case $k_3\ne 0$. Given $T>0$, define the time averaging function
 \begin{equation}\label{barQ}
 \bar Q_{n,\Omega}(t)=\frac1T\int_t^{t+T} Q_{n,\Omega}(\tau) d\tau.
 \end{equation}
One can see from \eqref{Qome} and \eqref{barQ} that
 \begin{equation*}
 \lim_{\Omega\to\pm\infty}\bar Q_{n,\Omega}(t)=0,\quad\text{ for any } t\in \ensuremath{\mathbb R}.
 \end{equation*}
\end{remark}