text:
\section{Introduction}\label{s:intro}

The two-dimensional Navier--Stokes equations (2d NSE) have been the
subject of many studies and its basic mathematical properties (existence,
uniqueness, regularity, etc.) are now well understood;
see, e.g., \cite{doering-gibbon:aanse,temam:nsenfa} for reviews.
As a tool to understand various geophysical flows, it is often desirable
to include the effect of planetary rotation, but a constant rotation rate
(the so-called $f$-plane approximation) has no effect on the dynamics
when periodic boundary conditions are used.
To feel the effect of rotation, we need to go to the so-called
$\beta$-plane approximation, in which the rotation is given by
$f_0+\beta y$.

Simple physical arguments and numerical studies
\cite{rhines:75,maltrud-vallis:91} suggest that
a rotation rate that varies as $\beta y$ tends to force
the solution to become more zonal (a zonal flow is one that does not
depend on $x$).
In this article, we prove that this is indeed the case, by obtaining
a bound $|\tilde\omega(t)|_{H^s}^2\le \beta^{-1} M_s(f;\cdots)$, valid for large time $t$,
on the non-zonal part $\tilde\omega$ of the flow in terms of the forcing $f$.

With the further assumption that the forcing is independent of time,
it has been shown that the Navier--Stokes equations possess a global
attractor $\mathcal{A}$ of finite Hausdorff dimension.
The long-known and nearly optimal bound on this dimension
\cite{constantin-foias-temam:88} also applies to our rotating case,
but it does not take into account the effect of the rotation.
Using our bounds on $\tilde\omega$, we show that the dimension of $\mathcal{A}$
is zero for $\varepsilon$ sufficiently small, reducing the long-time
dynamics to a single steady (and stable) flow determined completely
by the forcing.
This is to be contrasted with the situation for larger (but still small)
$\varepsilon$, where the solution, although nearly zonal, evolves in time even
though $\partial_tf=0$.

Among the works similar in spirit to the present article, we mention
\cite{gallagher-straymond:07} where weak convergence to zonal flow is
proved for the (more difficult) $\beta$-plane shallow-water equations.
A related result for the inviscid Euler equation can be found in
\cite{tgs:87}.
The technique of using rapid oscillations to obtain better bounds have
been used in different contexts in, e.g., \cite{babin-al:aa:97,schochet:94}.

In the rest of this section, we describe the problem and set up the notation.
In Section~\ref{s:pre}, we review basic results on 2d NSE which will be
needed later.
The heart of this article is Section~\ref{s:bd0}, where $L^2$ and $H^s$
bounds are obtained for the non-zonal component of the flow.
An application of these bounds to the dimension of the global attractor
follows in Section~\ref{s:attr}.
The proof of an $L^\infty$ Agmon inequality is presented in the Appendix.

\medskip
In dimensional form, the two-dimensional Navier--Stokes equations read
\begin{equation}\label{q:fbeta}
   \partial_t{\boldsymbol{v}} + {\boldsymbol{v}}\cdot\nabla{\boldsymbol{v}} + \beta y{\boldsymbol{v}}^\perp + \nabla p
	= \mu\Delta{\boldsymbol{v}} + f_{\vb}^{}
\end{equation}
where the constant rotation $f_0$ has been dropped since it has
no effect (i.e.\ in 2d NSE, there is no difference between equatorial
and mid-latitude $\beta$-planes).
Here ${\boldsymbol{v}}=(u,v)$ is the velocity with ${\boldsymbol{v}}^\perp:=(-v,u)$
and $p$ is the pressure obtained by enforcing the incompressibility
constraint $\gb\!\cdot\!\vb=0$.
In what follows, we will work with the dimensionless form
\begin{equation}\label{q:uv}\begin{aligned}
   &\partial_t{\boldsymbol{v}} + {\boldsymbol{v}}\cdot\nabla{\boldsymbol{v}} + \frac{Y}\varepsilon {\boldsymbol{v}}^\perp + \nabla p
	= \mu\Delta{\boldsymbol{v}} + f_{\vb}^{},\\
   &\gb\!\cdot\!\vb = 0.
\end{aligned}\end{equation}

We work with ${\boldsymbol{x}}=(x,y)\in\mathscr{M}:=[0,L_1]\times[-L_2/2,L_2/2]$, with periodic
boundary conditions in both directions.
Note that we have replaced $\beta y$ in \eqref{q:fbeta} by $Y(y)/\varepsilon$,
where $Y(-L_2/2)=L_2/2$ and $Y(y)=y$ for $y\in(-L_2/2,L_2/2]$.
Furthermore, we assume the following symmetry on the velocity
\begin{equation}\label{q:sym}
   u(x,-y,t) = u(x,y,t)
   \quad\textrm{and}\quad
   v(x,-y,t) = -v(x,y,t).
\end{equation}
It is readily verified that if the initial data ${\boldsymbol{v}}(0)$ and the forcing
$f_{\vb}^{}(t)$ also satisfy this symmetry, which we henceforth assume,
it persists for all $t\ge0$.
Note also that periodicity and \eqref{q:sym} imply that
\begin{equation}\label{q:BCv}
   v(x,-L_2/2,t) = v(x,L_2/2,t) = 0.
\end{equation}
With no loss of generality, we require that the integral over $\mathscr{M}$
of ${\boldsymbol{v}}$ vanishes.

In two dimensions, it is convenient to work with the vorticity
$\omega:=\nabla^\perp\!\cdot\!{\boldsymbol{v}}=\partial_xv-\partial_yu$, whose evolution equation is
\begin{equation}\label{q:t00}
   \partial_t\omega + {\boldsymbol{v}}\cdot\nabla\omega + \frac{Y'}\varepsilon\, v
	= \mu\Delta\omega + f.
\end{equation}
Here $f:=\nabla^\perp\!\cdot\!f_{\vb}^{}$ and we can recover the velocity
using ${\boldsymbol{v}}=\nabla^\perp\Delta^{-1}\omega$.
By our assumption on ${\boldsymbol{v}}$, the integral of $\omega$ over $\mathscr{M}$ is zero;
similarly, $\Delta^{-1}$ is defined uniquely by the zero-integral condition.
The symmetry \eqref{q:sym} implies that $\omega(x,-y,t)=-\omega(x,y,t)$ and
\begin{equation}\label{q:BCw}
    \omega(x,-L_2/2,t) = \omega(x,L_2/2,t) = 0.
\end{equation}
Now $Y'(y)=1-L_2\,\delta(y-L_2/2)$, where $\delta$ is the Dirac distribution.
Using the fact that $v(x,\pm L_2/2,t)=0$, we replace $vY'$ by $v$
in \eqref{q:t00} and write
\begin{equation}\label{q:dwdt}
   \partial_t\omega + {\boldsymbol{v}}\cdot\nabla\omega + \frac1\varepsilon\, v
	= \mu\Delta\omega + f.
\end{equation}
This is the form that we will be mostly working with.

It is also convenient to write \eqref{q:dwdt} in the usual functional form
\begin{equation}\label{q:abl}
   \partial_t\omega + B(\omega,\omega) + \frac1\varepsilon L\omega + \mu A\omega = f,
\end{equation}
where $B(\omega,\omega^\sharp):=(\nabla^\perp\Delta^{-1}\omega)\cdot\nabla\omega^\sharp$,
$L\omega:=\partial_x\Delta^{-1}\omega$ and $A:=-\Delta$.
The following properties, valid whenever the expressions make sense,
are readily verified by integration by parts and the boundary conditions
\eqref{q:BCv}--\eqref{q:BCw}
\begin{equation}\label{q:bla}\begin{aligned}
   &(B(\omega,\omega^\sharp),\omega^\sharp)_{L^2}^{} = 0,\\
   &(L\omega,\omega)_{L^2}^{} = 0,\\
   &(A\omega,\omega)_{L^2}^{} = |\nabla\omega|_{L^2}^2.
\end{aligned}\end{equation}

\section{Preliminary Estimates}\label{s:pre}

The estimates derived in this section are standard from the theory of 2d NSE
(see, e.g., \cite{temam:iddsmp,doering-gibbon:aanse,robinson:idds}),
with very minor modifications to handle the Coriolis term.
We gather them here for later use.

We start by noting that the vanishing of spatial integrals of ${\boldsymbol{v}}$
and $\omega$ implies the equivalence of the norms $|\omega|_{H^s}^{}$ and
$|\nabla^s\omega|_{L^2}^{}:=|(-\Delta)^{s/2}\omega|_{L^2}^{}$,
which will thus be used interchangeably below.
We denote by $c_0$ the constant in Poincar{\'e} inequality
\begin{equation}\label{q:poidef}
   c_0\,|\nabla^s\omega|_{L^2}^{} \le |\nabla^{s+1}\omega|_{L^2}^{}.
\end{equation}
Besides the usual Sobolev and interpolation inequalities for two and
one dimensions (for functions depending on $y$ only), we note the
one-dimensional Agmon inequality
\begin{equation}
   |\bar w|_{L^\infty}^{} \le c\,|\bar w|_{L^2}^{1/2}|\nabla\bar w|_{L^2}^{1/2}.
\end{equation}
A version we use for the two-dimensional case is in Appendix~\ref{s:linfty}.

The $L^2$ estimate for the velocity is obtained by multiplying \eqref{q:uv}
by ${\boldsymbol{v}}$ and using Cauchy--Schwarz,
\begin{equation}\label{q:bdv}
   \ddt{}|{\boldsymbol{v}}|_{L^2}^2 + \mu\,|\nabla{\boldsymbol{v}}|_{L^2}^2 \le \frac{c}\mu\,|f_{\vb}^{}|_{L^2}^2\,.
\end{equation}
Assuming that $f_{\vb}^{}\in L_t^\infty L_{{\boldsymbol{x}}}^2$, we thus have
${\boldsymbol{v}}\in L_t^\infty L_{{\boldsymbol{x}}}^2 \cap L_t^2 H_{{\boldsymbol{x}}}^1$.
Here and henceforth, $L_t^pL_{{\boldsymbol{x}}\vphantom{t}}^q:=L^p((0,\infty);L^q(\mathscr{M}))$, and
$H_{{\boldsymbol{x}}}^s$ and $|\nabla^s\omega|$ below are defined in the usual way.
We denote
\begin{equation}\label{q:kmdef}
   \supt{w} := \sup_{t>0}\,|w(t)|_{L^2}^{}.
\end{equation}
Here and elsewhere in this article, $c$ denotes a generic constant depending
only on $\mathscr{M}$ whose value may not be the same each time it appears,
while numbered constants such as $c_1^{}$ have fixed values.

Now let $\varphi\in L^\infty$ be such that $\varphi'(t)=1$ for $t\in[0,\sfrac12]$
and $\varphi(t)=1$ for $t>2$, so $\varphi(t)\simeq\tanh t$.
Multiplying \eqref{q:dwdt} by $\varphi\omega$ in $L^2$, or equivalently,
multiplying (\ref{q:uv}a) by $-\varphi\Delta{\boldsymbol{v}}$ in $L^2$, we obtain
\begin{equation}\label{q:bd0}
   \ddt{}\bigl(\varphi|\omega|_{L^2}^2\bigr) + \mu\,\varphi\,|\nabla\omega|_{L^2}^2
	\le \varphi'\,|\omega|^2 + \frac{c}\mu\,\varphi\,|f_{\vb}^{}|_{L^2}^2\,.
\end{equation}
Assuming henceforth that
$\supt{f_{\vb}^{}}<\infty$, we have
$\omega\in L_t^\infty L_{{\boldsymbol{x}}}^2 \cap L_t^2 H_{{\boldsymbol{x}}}^1$ and,
for $t\ge T_0(|{\boldsymbol{v}}(0)|_{L^2}^{},\supt{f_{\vb}^{}};\mu)$,
\begin{equation}\label{q:bd0u}
   |\omega(t)|_{L^2}^{} \le \frac{c}{\mu}\,\supt{f_{\vb}^{}}.
\end{equation}
Note that $T_0$ does not depend on $\omega(0)$ and that
the requirement $f_{\vb}^{}\in L_t^\infty L_{{\boldsymbol{x}}}^2$ can be
weakened in $t$, but we shall not do so here.

A bound in $H^m$ is obtained as follows.
Fix a multi-index $\alpha=(\alpha_1,\alpha_2)$ with
$|\alpha|:=\alpha_1+\alpha_2=m$, and multiply \eqref{q:dwdt} by
$\mathsf{D}^{2\alpha}\omega:=\partial_x^{2\alpha_1}\partial_y^{2\alpha_2}\omega$ in $L^2$,
\begin{equation}
   \frac12\ddt{\;}|\mathsf{D}^\alpha\omega|^2 + ({\boldsymbol{v}}\cdot\nabla\omega,\mathsf{D}^{2\alpha}\omega) + \frac1\varepsilon (v,\mathsf{D}^{2\alpha}\omega) + \mu\,|\nabla\mathsf{D}^\alpha\omega|^2 = (f,\mathsf{D}^{2\alpha}\omega)
\end{equation}
where here and henceforth $|\cdot|$ and $(\cdot,\cdot)$ denote
$L^2$ norm and inner product.
The linear term involving $1/\varepsilon$ vanishes, and one then proceeds as usual:
Using the fact that $({\boldsymbol{v}}\cdot\nabla\mathsf{D}^\alpha\omega,\mathsf{D}^\alpha\omega)=0$, the
nonlinear term is bounded as
\begin{equation}\label{q:whs}\begin{aligned}
   \bigl|({\boldsymbol{v}}\cdot\nabla\omega,\mathsf{D}^{2\alpha}\omega)\bigr|
	&\le \sum_{1\le|\beta|\le|\alpha|} \bigl|((\mathsf{D}^\beta{\boldsymbol{v}})\cdot\nabla \mathsf{D}^{\alpha-\beta}\omega,\mathsf{D}^\alpha\omega)\bigr|\\
	&\le c\,{\textstyle\sum}_\beta\,\bigl|\mathsf{D}^{\beta-1}\omega\bigr|_{L^4}^{} \bigl|\mathsf{D}^{\alpha-\beta+1}\omega\bigr|_{L^4}^{} \bigl|\mathsf{D}^\alpha\omega\bigr|_{L^2}^{}\\
	&\le c\,{\textstyle\sum}_\beta\,\bigl|\mathsf{D}^{\beta-1}\omega\bigr|_{H^{1/2}}^{} \bigl|\mathsf{D}^{\alpha-\beta+1}\omega\bigr|_{H^{1/2}}^{} \bigl|\mathsf{D}^\alpha\omega\bigr|_{L^2}^{}\\
	&\le c(m)\,{\textstyle\sum}_{l=1}^m\,|\omega|_{H^{l-1/2}}^{}|\omega|_{H^{m-l+3/2}}^{}|\omega|_{H^m}^{}
\end{aligned}\end{equation}
where we have used Sobolev inequalities for the second and
third line, and where $l:=|\beta|$ in the last line.
Using the interpolation inequalities
\begin{equation}\label{q:interp0}\begin{aligned}
   &|\omega|_{H^{l-1/2}}^{} \le c\,|\omega|_{L^2}^{(2m-2l+3)/(2m+2)} |\omega|_{H^{m+1}}^{(2l-1)/(2m+2)}\\
   &|\omega|_{H^{m-l+3/2}}^{} \le c\,|\omega|_{L^2}^{(2l-1)/(2m+2)} |\omega|_{H^{m+1}}^{(2m-2l+3)/(2m+2)},
\end{aligned}\end{equation}
followed by Cauchy--Schwarz and summing over $\alpha$, we obtain
\begin{equation}\label{q:bdwm0}
   \ddt{\;}|\omega|_{H^m}^2 + \frac{3\mu}{2}\,|\omega|_{H^{m+1}}^2
	\le \frac{c(m)}\mu\,|\omega|_{L^2}^2|\omega|_{H^m}^2 + \frac{c'(m)}\mu\,|f|_{H^{m-1}}^2
\end{equation}
for $m=1,2,\cdots$.
Proceeding by Gronwall and induction on \eqref{q:bdwm0},
we have the following uniform bounds independent of the initial data
\begin{equation}\label{q:bdwm}\begin{aligned}
   &|\omega(t)|_{H^m}^2 \le \Bigl(\frac{c(m)}\mu\,\supt{\nabla^{m-1}f}^2 + 1\Bigr)^{m+1}\\
   &\mathrm{e}^{-\nu t'} \int_t^{t+t'} \mathrm{e}^{\nu\tau} |\omega(\tau)|_{H^{m+1}}^2 \>\mathrm{d}\tau \le \frac{c}\mu\,\Bigl(\frac{c(m)}\mu\,\supt{\nabla^{m-1}f}^2 + 1\Bigr)^{m+1}
\end{aligned}\end{equation}
valid for all $t\ge T_m(|{\boldsymbol{v}}(0)|_{L^2}^{},\supt{\nabla^{m-1}f}^{};\mu)$.
Here $\nu=c_0^2\mu$ with $c_0^{}$ the constant in Poincar{\'e} inequality
\eqref{q:poidef}.
Note that for $T_m$ to depend only on $|{\boldsymbol{v}}(0)|_{L^2}^{}$, and for the validity
of \eqref{q:bdwm} for ${\boldsymbol{v}}(0)\not\in H^1$, we need to multiply
by $\varphi$ as in \eqref{q:bd0}, but this was not done explicitly for conciseness.

\section{Bounds on the Non-zonal Component}\label{s:bd0}

Assuming sufficient regularity for $f$, which implies that for $\omega$
for any $t>0$, we expand them in Fourier series
\begin{equation}\label{q:wkdef}\begin{aligned}
   &\omega({\boldsymbol{x}},t) = {\textstyle\sum}_{\boldsymbol{k}}\,\omega_{\boldsymbol{k}}(t)\, \mathrm{e}^{\mathrm{i}{\boldsymbol{k}}\cdot{\boldsymbol{x}}-\mathrm{i}\Omega_{\boldsymbol{k}} t/\varepsilon}\\
   &f({\boldsymbol{x}},t) = {\textstyle\sum}_{\boldsymbol{k}}\,f_{\boldsymbol{k}}(t)\, \mathrm{e}^{\mathrm{i}{\boldsymbol{k}}\cdot{\boldsymbol{x}}}
\end{aligned}\end{equation}
where ${\boldsymbol{k}}=(k_1,k_2)\in\Zahl_L:=\{(2\pi l_1/L_1,2\pi l_2/L_2): (l_1,l_2)\in\mathbb{Z}^2\}$
and $\Omega_{\boldsymbol{k}}:=-k_1/|{\boldsymbol{k}}|^2$ is i times the eigenvalue of
the linear operator $L$ for wavenumber ${\boldsymbol{k}}$.
Since $\omega$ and $f$ have vanishing integrals over $\mathscr{M}$,
$\omega_{\boldsymbol{0}}=0$ and $f_{\boldsymbol{0}}=0$.
Here and in what follows, sums over wavenumbers are understood to be taken
over $\Zahl_L$.
In terms of Fourier components, \eqref{q:dwdt} reads
\begin{equation}\label{q:dwkdt}
   \ddt{\omega_{\boldsymbol{l}}} + {\textstyle\sum}_{{\boldsymbol{j}}{\boldsymbol{k}}}\,B_{{\boldsymbol{j}}{\boldsymbol{k}}{\boldsymbol{l}}}\omega_{\boldsymbol{j}}\omega_{\boldsymbol{k}}\,\mathrm{e}^{\mathrm{i}(\Omega_{\boldsymbol{l}}-\Omega_{\boldsymbol{j}}-\Omega_{\boldsymbol{k}})t/\varepsilon}
	+ \mu\,|{\boldsymbol{l}}|^2\omega_{\boldsymbol{l}} = f_{\boldsymbol{l}}\,\mathrm{e}^{\mathrm{i}\Omega_{\boldsymbol{l}} t/\varepsilon}
\end{equation}
where the coefficient of the nonlinear term is
\begin{equation}\label{q:bjkldef}
   B_{{\boldsymbol{j}}{\boldsymbol{k}}{\boldsymbol{l}}} = (B(\mathrm{e}^{\mathrm{i}{\boldsymbol{j}}\cdot{\boldsymbol{x}}},\mathrm{e}^{\mathrm{i}{\boldsymbol{k}}\cdot{\boldsymbol{x}}}),\mathrm{e}^{\mathrm{i}{\boldsymbol{l}}\cdot{\boldsymbol{x}}})
	= |\mathscr{M}|\,\frac{{\boldsymbol{j}}\wedge{\boldsymbol{k}}}{|{\boldsymbol{j}}|^2}\,\delta_{{\boldsymbol{j}}+{\boldsymbol{k}}-{\boldsymbol{l}}}
\end{equation}
with ${\boldsymbol{j}}\wedge{\boldsymbol{k}}:=j_1k_2-j_2k_1$.
We note that the linear term $\varepsilon^{-1}L\omega$ has been removed
from \eqref{q:dwkdt} by including $\exp(-\mathrm{i}\Omega_{\boldsymbol{k}} t/\varepsilon)$
in (\ref{q:wkdef}a).

Let us split $\omega$ into a slow part $\bar\omega$, for which $\Omega_{\boldsymbol{k}}=0$,
and the remaining fast part $\tilde\omega:=\omega-\bar\omega$, viz.,
\begin{equation}\begin{aligned}
   &\bar\omega({\boldsymbol{x}},t) = {\textstyle\sum}_{k_1=0}\,\omega_{\boldsymbol{k}}(t)\, \mathrm{e}^{\mathrm{i}{\boldsymbol{k}}\cdot{\boldsymbol{x}}}\\
   &\tilde\omega({\boldsymbol{x}},t) = {\textstyle\sum}_{k_1\ne0}\,\omega_{\boldsymbol{k}}(t)\, \mathrm{e}^{\mathrm{i}{\boldsymbol{k}}\cdot{\boldsymbol{x}}-\mathrm{i}\Omega_{\boldsymbol{k}} t/\varepsilon}.
\end{aligned}\end{equation}
We note that, also having zero integrals over $\mathscr{M}$,
$\tilde\omega$ and $\bar\omega$ are orthogonal in $H^m$ for $m=0,1,\cdots$.
For convenience, we also define
\begin{equation}
   \bar\omega_{\boldsymbol{k}} := \begin{cases} \omega_{\boldsymbol{k}} &\textrm{if } k_1=0\\ 0 &\textrm{otherwise,}\end{cases}
   \qquad\textrm{and}\qquad
   \tilde\omega_{\boldsymbol{k}} := \begin{cases} \omega_{\boldsymbol{k}} &\textrm{if } k_1\ne0\\ 0 &\textrm{otherwise.}\end{cases}
\end{equation}
Our objective in this section is to obtain long-time bounds for $\tilde\omega$
that tend to zero as $\varepsilon\to0$.

\subsection{Bound in $L^2$}\label{s:l2}

The development in this subsection largely follows that in
\cite{temam-dw:lbal} for the primitive equations, the main difference
being the absence of a spectral gap (that is, the eigenvalues of the
antisymmetric operator $L$ accumulate at zero in the present case).

We start by multiplying \eqref{q:abl} by $\tilde\omega$ in $L^2$,
\begin{equation}\label{q:dwtdt0}
   (\partial_t\omega,\tilde\omega) + (B(\omega,\omega),\tilde\omega) + \frac1\varepsilon (L\omega,\tilde\omega)
	+ \mu\,(A\omega,\tilde\omega) = (f,\tilde\omega).
\end{equation}
Now, using (\ref{q:bla}a) twice and the fact that $B(\bar\omega,\bar\omega)=0$,
\begin{equation}\begin{aligned}
   (B(\omega,\omega),\tilde\omega) &= (B(\omega,\tilde\omega),\tilde\omega) + (B(\omega,\bar\omega),\tilde\omega)\\
	&= (B(\bar\omega,\bar\omega),\tilde\omega) + (B(\tilde\omega,\bar\omega),\tilde\omega)\\
	&= - (B(\tilde\omega,\tilde\omega),\bar\omega).
\end{aligned}\end{equation}
Thus \eqref{q:dwtdt0} becomes
\begin{equation}\label{q:dwtdt1}
   \frac12\ddt{\;}|\tilde\omega|^2 + \mu\,|\nabla\tilde\omega|^2 = (B(\tilde\omega,\tilde\omega),\bar\omega) + (f,\tilde\omega).
\end{equation}
Dropping the nonlinear term for the moment, the fact that $\tilde\omega$ is
rapidly varying while $f$ is slow implies that the effective forcing
from the rhs becomes weaker for smaller $\varepsilon$.
This essentially is the mechanism for the attenuation of the fast part $\tilde\omega$;
the nonlinear term will be handled in the proof below.
Recalling the definition \eqref{q:kmdef},
we state the result of this subsection.

\begin{theorem}\label{t:o1}
Assume that the initial data ${\boldsymbol{v}}(0)\in L^2(\mathscr{M})$
and that the forcing is bounded as $\supt{\nabla^2f}+\supt{\partial_tf}<\infty$.
Then there exist $\mathcal{T}_0(|{\boldsymbol{v}}(0)|_{L^2}^{},\supt{\nabla^2f},\supt{\partial_tf};\mu)$ and
$M_0(\supt{\nabla^2f},\supt{\partial_tf};\mu)$ such that, for $t\ge \mathcal{T}_0$,
\begin{equation}\label{q:bdwt}\begin{aligned}
   &|\tilde\omega(t)|_{L^2}^2 \le \varepsilon\,M_0,\\
   &\mu\,\mathrm{e}^{-\nu (t+t')}\int_t^{t+t'} \mathrm{e}^{\nu\tau} |\nabla\tilde\omega(\tau)|^2 \>\mathrm{d}\tau
	\le \varepsilon\,M_0.
\end{aligned}\end{equation}
\end{theorem}

\noindent{\sc Proof.}
Recalling that $\nu=\muc_0^2$, we obtain from \eqref{q:dwtdt1}
\begin{equation}\label{q:dwtdt}
   \ddt{\;}\bigl(\mathrm{e}^{\nu t}|\tilde\omega|^2\bigr) + \mu\mathrm{e}^{\nu t}|\nabla\tilde\omega|^2
	\le 2\mathrm{e}^{\nu t}(B(\tilde\omega,\tilde\omega),\bar\omega) + 2\mathrm{e}^{\nu t}(f,\tilde\omega).
\end{equation}

We integrate the last term from $0$ to $t$ by parts,
\begin{equation}\begin{aligned}
   \int_0^t \mathrm{e}^{\nu\tau} (f,\tilde\omega)\>\mathrm{d}\tau
	&= |\mathscr{M}|\, {\textstyle\sum}_{\boldsymbol{k}}' \int_0^t f_{\boldsymbol{k}}(\tau)\overline{\tilde\omega_{\boldsymbol{k}}(\tau)}\,
		\mathrm{e}^{\mathrm{i}\Omega_{\boldsymbol{k}}\tau/\varepsilon+\nu\tau} \>\mathrm{d}\tau\\
	&= -\mathrm{i}\varepsilon|\mathscr{M}|\,\sump{{\boldsymbol{k}}}\, \frac1{\Omega_{\boldsymbol{k}}}\Bigl[ f_{\boldsymbol{k}}\overline{\tilde\omega_{\boldsymbol{k}}}\,\mathrm{e}^{\mathrm{i}\Omega_{\boldsymbol{k}}\tau/\varepsilon+\nu\tau}\Bigr]_0^t\\
	&\qquad {}+ \mathrm{i}\varepsilon|\mathscr{M}|\, \sump{{\boldsymbol{k}}}\, \frac1{\Omega_{\boldsymbol{k}}} \int_0^t \ddtau{\;}\bigl[ f_{\boldsymbol{k}}\overline{\tilde\omega_{\boldsymbol{k}}}\,\mathrm{e}^{\nu\tau}\bigr] \mathrm{e}^{\mathrm{i}\Omega_{\boldsymbol{k}}\tau/\varepsilon} \>\mathrm{d}\tau
\end{aligned}\end{equation}
where the prime on the sums indicates that the resonant terms (i.e.\ those
with $\Omega_{\boldsymbol{k}}=0$) are excluded.
Defining the operator $\partial_t^*$ by, for any $w$ for which it makes sense,
\begin{equation}\label{q:dstaudef}\begin{aligned}
   &\partial_t^* w := \mathrm{e}^{-tL/\varepsilon}\partial_t\bigl(\mathrm{e}^{tL/\varepsilon}w\bigr)\\
   \Rightarrow\qquad
   &\partial_t^*\tilde\omega := \partial_t\tilde\omega + \frac1\varepsilon L\tilde\omega
	= -\tilde B(\omega,\omega) - \mu A\tilde\omega + \tilde f,
\end{aligned}\end{equation}
and defining the operator $\mathsf{I}_\Omega$ by
\begin{equation}\label{q:IOdef}
   \mathsf{I}_\Omega\tilde f({\boldsymbol{x}},t)
	:= \sump{{\boldsymbol{k}}} \frac1{\mathrm{i}\Omega_{\boldsymbol{k}}}\, f_{\boldsymbol{k}}(t)\,\mathrm{e}^{\mathrm{i}{\boldsymbol{k}}\cdot{\boldsymbol{x}}}
	= \mathrm{i} \sump{{\boldsymbol{k}}} \frac{|{\boldsymbol{k}}|^2}{k_1}\, f_{\boldsymbol{k}}(t)\,\mathrm{e}^{\mathrm{i}{\boldsymbol{k}}\cdot{\boldsymbol{x}}},
\end{equation}
which being the restricted inverse of $L$ is also antisymmetric, we can write
\begin{equation}\label{q:i00}\begin{aligned}
   \int_0^t \mathrm{e}^{\nu\tau} (f,\tilde\omega)\>\mathrm{d}\tau
	&= \varepsilon\,(\mathsf{I}_\Omega\tilde f,\tilde\omega)(t)\mathrm{e}^{\nu t} - \varepsilon\,(\mathsf{I}_\Omega\tilde f,\tilde\omega)(0)\\
	&- \varepsilon \int_0^t \bigl[ \nu(\mathsf{I}_\Omega\tilde f,\tilde\omega) + (\partial_\tau\mathsf{I}_\Omega\tilde f,\tilde\omega)
		+ (\mathsf{I}_\Omega\tilde f,\dy_\tau^*\tilde\omega)\bigr] \mathrm{e}^{\nu\tau} \>\mathrm{d}\tau.
\end{aligned}\end{equation}
Using \eqref{q:IOdef}, the endpoint terms can be bounded as
\begin{equation}\label{q:j04}
   |(\mathsf{I}_\Omega\tilde f,\tilde\omega)| \le c\,|\nabla\tilde f|\,|\nabla\tilde\omega|.
\end{equation}
We now bound the terms in the integrand.
First,
\begin{equation}
   \bigl|(\partial_\tau\mathsf{I}_\Omega\tilde f,\tilde\omega)\bigr|
	= \bigl|(\partial_\tau\tilde f,\mathsf{I}_\Omega\tilde\omega)\bigr|
	\le c\,|\partial_\tau\tilde f|\,|\Delta\tilde\omega|.
\end{equation}
Next, using (\ref{q:dstaudef}b) and noting
the fact that $(\mathsf{I}_\Omega\tilde f,\tilde f)=0$, we bound the last term in \eqref{q:i00} by
\begin{equation}
   \bigl|(\mathsf{I}_\Omega\tilde f,\mu\Delta\tilde\omega)\bigr|
	\le \mu c\,|\Delta\tilde f|\,|\Delta\tilde\omega|;
\end{equation}
and, using Sobolev and interpolation inequalities,
\begin{equation}\begin{aligned}
   \bigl|(\mathsf{I}_\Omega\tilde f,B(\omega,\omega))\bigr|
	&\le c\,|\nabla\tilde f|_{L^2}^{}|\nabla B(\tilde\omega,\tilde\omega)|_{L^2}^{}\\
   &\le c\,|\nabla\tilde f|_{L^2}^{} |\omega|_{L^4}^{}|\nabla\omega|_{L^4}^{}
	+ c\,|\nabla\tilde f|_{L^2}^{} |\nabla^{-1}\omega|_{L^\infty}^{}|\Delta\omega|_{L^2}^{}\\
	&\le c\,|\nabla\tilde f|\,|\nabla\omega|\,|\Delta\omega|.
\end{aligned}\end{equation}
Thus the integral in \eqref{q:i00} is bounded as
\begin{equation}\label{q:j02}\begin{aligned}
   \biggl|\int_0^t &[\nu(\mathsf{I}_\Omega\tilde f,\tilde\omega) + (\partial_\tau\mathsf{I}_\Omega\tilde f,\tilde\omega)
		+ (\mathsf{I}_\Omega\tilde f,\dy_\tau^*\tilde\omega)]\,\mathrm{e}^{\nu\tau} \>\mathrm{d}\tau\biggr|\\
	&\le c \int_0^t \bigl[ \mu\,|\Delta\tilde f|\,|\Delta\tilde\omega|
		+ |\partial_\tau\tilde f|\,|\Delta\tilde\omega| + |\nabla\tilde f|\,|\nabla\omega|\,|\Delta\omega| \bigr] \mathrm{e}^{\nu\tau}\>\mathrm{d}\tau\\
	&\le c\int_0^t \bigl\{ (1+\mu)\,|\Delta\tilde\omega|^2 + \mu\,|\Delta\tilde f|^2 + |\partial_\tau\tilde f|^2 + |\Delta\omega|\,|\nabla\omega|\,|\nabla\tilde f| \bigr\}\,\mathrm{e}^{\nu\tau}\>\mathrm{d}\tau.
\end{aligned}\end{equation}

We now treat the penultimate term in \eqref{q:dwtdt}.
First, we write
\begin{equation}\label{q:Bsym}\begin{aligned}
   (B(\tilde\omega,\tilde\omega),\bar\omega) &= {\textstyle\sum}_{{\boldsymbol{j}}{\boldsymbol{k}}{\boldsymbol{l}}}\,B_{{\boldsymbol{j}}{\boldsymbol{k}}{\boldsymbol{l}}}\tilde\omega_{\boldsymbol{j}}\tilde\omega_{\boldsymbol{k}}\overline{\bar\omega_{\boldsymbol{l}}}\,\mathrm{e}^{-\mathrm{i}(\Omega_{\boldsymbol{j}}+\Omega_{\boldsymbol{k}})t/\varepsilon}\\
	 &= \frac12 {\textstyle\sum}_{{\boldsymbol{j}}{\boldsymbol{k}}{\boldsymbol{l}}}\,(B_{{\boldsymbol{j}}{\boldsymbol{k}}{\boldsymbol{l}}}+B_{{\boldsymbol{k}}{\boldsymbol{j}}{\boldsymbol{l}}})\,\tilde\omega_{\boldsymbol{j}}\tilde\omega_{\boldsymbol{k}}\overline{\bar\omega_{\boldsymbol{l}}}\,\mathrm{e}^{-\mathrm{i}(\Omega_{\boldsymbol{j}}+\Omega_{\boldsymbol{k}})t/\varepsilon}
\end{aligned}\end{equation}
and then note that $B_{{\boldsymbol{j}}{\boldsymbol{k}}{\boldsymbol{l}}}+B_{{\boldsymbol{k}}{\boldsymbol{j}}{\boldsymbol{l}}}=0$ in the resonant case,
i.e.\ when $\Omega_{\boldsymbol{j}}+\Omega_{\boldsymbol{k}}=0$ and $l_1=0$.
Furthermore, we have
\begin{equation}\label{q:nores}\begin{aligned}
   B_{{\boldsymbol{j}}{\boldsymbol{k}}{\boldsymbol{l}}} + B_{{\boldsymbol{k}}{\boldsymbol{j}}{\boldsymbol{l}}}
	&= \biggl(\frac{{\boldsymbol{j}}\wedge{\boldsymbol{k}}}{|{\boldsymbol{j}}|^2} + \frac{{\boldsymbol{k}}\wedge{\boldsymbol{j}}}{|{\boldsymbol{k}}|^2}\biggr)\,|\mathscr{M}|
	& &= ({\boldsymbol{j}}\wedge{\boldsymbol{k}})\,\biggl(\frac1{|{\boldsymbol{j}}|^2} - \frac1{|{\boldsymbol{k}}|^2}\biggr)\,|\mathscr{M}|\\
	&= j_1l_2\,\biggl(\frac1{|{\boldsymbol{j}}|^2} - \frac1{|{\boldsymbol{k}}|^2}\biggr)\,|\mathscr{M}|
	& &= -l_2\,(\Omega_{\boldsymbol{j}}+\Omega_{\boldsymbol{k}})\,|\mathscr{M}|
\end{aligned}\end{equation}
whenever ${\boldsymbol{j}}+{\boldsymbol{k}}={\boldsymbol{l}}$ and $l_1=0$.
Motivated by \eqref{q:Bsym}, we introduce the bilinear symmetric
operator $B_\Omega$ by
\begin{equation}\label{q:BOdef}\begin{aligned}
   (B_\Omega(\tilde\omega^\sharp,\tilde\omega^\flat),\bar\omega) &:=
	\frac{\mathrm{i}}{2} \sump{{\boldsymbol{j}}{\boldsymbol{k}}{\boldsymbol{l}}}\, \frac{B_{{\boldsymbol{j}}{\boldsymbol{k}}{\boldsymbol{l}}}+B_{{\boldsymbol{k}}{\boldsymbol{j}}{\boldsymbol{l}}}}{\Omega_{\boldsymbol{j}}+\Omega_{\boldsymbol{k}}}\,\tilde\omega_{\boldsymbol{j}}^\sharp\tilde\omega_{\boldsymbol{k}}^\flat\overline{\bar\omega_{\boldsymbol{l}}}\,\mathrm{e}^{-\mathrm{i}(\Omega_{\boldsymbol{j}}+\Omega_{\boldsymbol{k}})t/\varepsilon}\\
	&= \frac{|\mathscr{M}|}{2\mathrm{i}} \sump{{\boldsymbol{j}}{\boldsymbol{k}}{\boldsymbol{l}}}\, l_2 \,\tilde\omega_{\boldsymbol{j}}^\sharp\tilde\omega_{\boldsymbol{k}}^\flat\overline{\bar\omega_{\boldsymbol{l}}}\,\mathrm{e}^{-\mathrm{i}(\Omega_{\boldsymbol{j}}+\Omega_{\boldsymbol{k}})t/\varepsilon}
\end{aligned}\end{equation}
for any $\tilde\omega^\sharp$, $\tilde\omega^\flat$ and $\bar\omega$, where the prime on the sum
again indicates that resonant terms (for which $\Omega_{\boldsymbol{j}}+\Omega_{\boldsymbol{k}}=0$)
are omitted.
We note that, thanks to \eqref{q:nores}, the resonant terms are also absent
in $(B(\tilde\omega,\tilde\omega),\bar\omega)$.
Integrating by parts, we have
\begin{equation}\label{q:j01}\begin{aligned}
   \int_0^t \mathrm{e}^{\nu\tau}(B(\tilde\omega,\tilde\omega),\bar\omega) \>\mathrm{d}\tau
	&= \varepsilon\,\mathrm{e}^{\nu t} (B_\Omega(\tilde\omega,\tilde\omega),\bar\omega)(t) - \varepsilon\,(B_\Omega(\tilde\omega,\tilde\omega),\bar\omega)(0)\\
	&+ \varepsilon \int_0^t \mathrm{e}^{\nu\tau} \bigl[ \nu (B_\Omega(\tilde\omega,\tilde\omega),\bar\omega)
		+ 2\,(B_\Omega(\dy_\tau^*\tilde\omega,\tilde\omega),\bar\omega)\\
		&\hbox to116pt{}+ (B_\Omega(\tilde\omega,\tilde\omega),\partial_\tau\bar\omega) \bigr] \>\mathrm{d}\tau.
\end{aligned}\end{equation}
For the last term in the integrand, we use the fact that
$\bar B(\tilde\omega,\bar\omega) = \bar B(\bar\omega,\tilde\omega) = \bar B(\bar\omega,\bar\omega) = 0$ to write
\begin{equation}
   \partial_\tau\bar\omega = -\bar B(\tilde\omega,\tilde\omega) - \mu A\bar\omega + \bar f
\end{equation}
and estimate, using $H^1\subset L^\infty$ for $\bar f$ and \eqref{q:agmon}
for the $L^\infty$ estimates,
\begin{equation}\begin{aligned}
   \bigl|&(B_\Omega(\tilde\omega,\tilde\omega),\partial_\tau\bar\omega)\bigr|
	\le c\,|\tilde\omega|_{L^2}^{}|\partial_y\tilde\omega|_{L^2}^{}|\bar f|_{L^\infty}^{}
	+ \mu c\,|\tilde\omega|_{L^4}^{}|\partial_y\tilde\omega|_{L^4}^{}|\Delta\bar\omega|_{L^2}^{}\\
	&\hbox to160pt{} {}+ c\,|\tilde\omega|_{L^\infty}^{}|\partial_y\tilde\omega|_{L^2}^{}|\nabla^{-1}\tilde\omega|_{L^\infty}^{}|\nabla\tilde\omega|_{L^2}^{}\\
	&\le c\,|\tilde\omega|\,|\nabla\tilde\omega|\,|\bar f'|
	+ \mu c\,|\tilde\omega|^{1/2}|\nabla\tilde\omega|\,|\Delta\omega|^{3/2}+ c\,|\nabla\omega|^3|\tilde\omega|\,\Bigl(\log\frac{|\Delta\omega|}{c_0|\nabla\omega|}+c'\Bigr).
\end{aligned}\end{equation}
For the term involving $(B_\Omega(\dy_\tau^*\tilde\omega,\tilde\omega),\bar\omega)$, we bound,
using $\dy_\tau^*\tilde\omega+\tilde B(\omega,\omega) + \mu A\tilde\omega = \tilde f$
and the inequality $|\bar\omega|_{L^\infty}^{}\le c\,|\bar\omega|^{1/2}|\bar\omega'|^{1/2}$,
\begin{equation}\begin{aligned}
   \bigl|(B_\Omega(\dy_\tau^*\tilde\omega,\tilde\omega),\bar\omega)\bigr|
	&\le c\,|\partial_y\tilde f|\,|\tilde\omega|\,|\bar\omega|_{L^\infty}^{} + c\,|\tilde f|\,|\partial_y\tilde\omega|\,|\bar\omega|_{L^\infty}^{}\\
	&\qquad {}+ \mu c\,|\Delta\omega|\,|\tilde\omega|\,|\bar\omega'|_{L^\infty}^{}
	+ c\,|\nabla^{-1}\omega|_{L^\infty}^{}|\nabla\omega|\,|\tilde\omega|\,|\bar\omega'|_{L^\infty}^{}\\
	&\le c\,|\nabla\tilde f|\,|\tilde\omega|\,|\bar\omega'| + c\,|\tilde f|\,|\nabla\tilde\omega|\,|\bar\omega'|
	+ \mu c\,|\tilde\omega|\,|\bar\omega'|^{1/2}|\Delta\omega|^{3/2}\\
	&\qquad {}+ c\,|\omega|^2|\nabla\omega|^{3/2}|\bar\omega''|^{1/2}\Bigl(\log\frac{|\nabla\omega|}{c_0|\omega|}+1\Bigr)^{1/2}
\end{aligned}\end{equation}
where all unadorned norms are $L^2$.
Finally, we bound
\begin{equation}\begin{aligned}
   \bigl|(B_\Omega(\tilde\omega,\tilde\omega),\bar\omega)\bigr|
	&\le c\,|\tilde\omega|\,|\partial_y\tilde\omega|\,|\bar\omega|_{L^\infty}^{}\\
	&\le c\,|\tilde\omega|\,|\nabla\tilde\omega|\,|\bar\omega|^{1/2}|\bar\omega'|^{1/2}.
\end{aligned}\end{equation}
Using these also to bound the endpoint terms,
the integral in \eqref{q:j01} is bounded as
\begin{equation}\label{q:j03}\begin{aligned}
   \biggl|\int_0^t \mathrm{e}^{\nu\tau}&(B(\tilde\omega,\tilde\omega),\bar\omega) \>\mathrm{d}\tau\biggr|\\
	&\le \varepsilon c\,\bigl[|\tilde\omega|\,|\nabla\tilde\omega|\,|\bar\omega|^{1/2}|\bar\omega'|^{1/2}\bigr](t)\,\mathrm{e}^{\nu t}
	+ \varepsilon c\,\bigl[|\tilde\omega|\,|\nabla\tilde\omega|\,|\bar\omega|^{1/2}|\bar\omega'|^{1/2}\bigr](0)\\
	& {}+ \varepsilon \int_0^t \Bigl\{
		c\,|\nabla f|\,|\tilde\omega|\,|\nabla\omega| + c\,|\tilde f|\,|\nabla\omega|^2
		+ \mu c\,|\omega|^{1/2}\,|\nabla\omega|\,|\Delta\omega|^{3/2}\\
	&\hbox to75pt{} {}+ c\,|\omega|\,|\nabla\omega|^{5/2}|\Delta\omega|^{1/2}\Bigl(\log\frac{|\Delta\omega|}{c_0|\nabla\omega|}+c'\Bigr) \Bigl\}\,\mathrm{e}^{\nu\tau} \>\mathrm{d}\tau.
\end{aligned}\end{equation}

Putting together \eqref{q:j04}, \eqref{q:j02} and \eqref{q:j03}, we have
\begin{equation}\label{q:bd1z}\begin{aligned}
   |\tilde\omega(&t)|^2 + \mu \int_0^t |\nabla\tilde\omega|^2 \mathrm{e}^{\nu(\tau-t)} \>\mathrm{d}\tau
	\le \mathrm{e}^{-\nu t}|\tilde\omega(0)|^2\\
	&+ \varepsilon \cnst2\,(1+\mathrm{e}^{-\nu t})\,\sup_{0\le t'\le t}\,\bigl\{
		|\nabla\tilde f|\,|\nabla\tilde\omega| + |\omega|^{3/2}|\nabla\omega|^{3/2} \bigr\}\\
	&+ \varepsilon \cnst3(\mu) \int_0^t \Bigl\{ |\Delta\tilde f|^2 + |\partial_\tau\tilde f|^2
		+ |\nabla\tilde f|\,|\nabla\omega|\,|\Delta\omega| + |\Delta\omega|^2\bigl(1+|\nabla\omega|\bigr)\\
	&\hbox to80pt{} {}+ |\omega|\,|\nabla\omega|^{5/2}|\Delta\omega|^{1/2}\Bigl(\log\frac{|\Delta\omega|}{c_0|\nabla\omega|}+c'\Bigr) \Bigl\}\,\mathrm{e}^{\nu(\tau-t)} \>\mathrm{d}\tau.
\end{aligned}\end{equation}
We now shift the origin of time such that $t=0$ corresponds to $T_2$
in \eqref{q:bdwm}.
The hypothesis that $\supt{\nabla^2f}+\supt{\partial_tf}<\infty$ then implies that
both the endpoints and the integral in \eqref{q:bd1z} are bounded uniformly
for all $t>0$, independently of the initial data provided that
${\boldsymbol{v}}\in L^2$ initially.
Rewriting the bound in \eqref{q:bd1z} as
\begin{equation}\label{q:bdl2}
   |\tilde\omega(t)|^2 + \mu \int_0^t |\nabla\tilde\omega|^2 \mathrm{e}^{\nu(\tau-t)} \>\mathrm{d}\tau
	\le \mathrm{e}^{-\nu t}|\tilde\omega(0)|^2
	+ \frac{\varepsilon}2\,M_0(\supt{\nabla^2f},\supt{\partial_tf};\mu),
\end{equation}
the proof is complete.

We also note from \eqref{q:bd1z} that the hypothesis
$f\in L_t^\infty H_{{\boldsymbol{x}}}^2$ and $\partial_t f\in L_t^\infty L_{{\boldsymbol{x}}}^2$
can be weakened to
$f\in L_t^2 H_{{\boldsymbol{x}}}^2\cap L_t^\infty H_{{\boldsymbol{x}}}^1$ and $\partial_t f\in L_t^2 L_{{\boldsymbol{x}}}^2$.

\subsection{Bounds in $H^s$}\label{s:hs}
With a little extra work, $H^s$ bounds for $\tilde\omega$ that scales as $\sqrt\varepsilon$
can also be obtained.
We do this explicitly for $|\nabla\tilde\omega|$ and sketch the computation
for $s=2,3,\cdots$.

\medskip
For the $H^1$ bound, we multiply \eqref{q:abl} by $A\tilde\omega$ in $L^2$ to get
\begin{equation}
   \frac12\ddt{\;}|\nabla\tilde\omega|^2 + \mu\,|\Delta\tilde\omega|^2
	+ (B(\omega,\omega),A\tilde\omega) = (\tilde f,A\tilde\omega),
\end{equation}
which implies [cf.\ \eqref{q:dwtdt}]
\begin{equation}
   \ddt{\;}\bigl(\mathrm{e}^{\nu t}|\nabla\tilde\omega|^2\bigr) + \mu\,\mathrm{e}^{\nu t}|\Delta\tilde\omega|^2
	\le 2\mathrm{e}^{\nu t}\,(B(\omega,\omega),\Delta\tilde\omega)
	- 2\mathrm{e}^{\nu t}(\tilde f,\Delta\tilde\omega).
\end{equation}
As in the $L^2$ case, we integrate from 0 to $t$,
\begin{equation}\begin{aligned}
   \mathrm{e}^{\nu t}|\nabla\tilde\omega(t)|^2 - |\nabla\tilde\omega(0)|^2
	&+ \mu\! \int_0^t |\Delta\tilde\omega|^2 \,\mathrm{e}^{\nu\tau}\>\mathrm{d}\tau\\
	&\le 2 \int_0^t \bigl\{ (B(\omega,\omega),\Delta\tilde\omega) - (\tilde f,\Delta\tilde\omega) \bigr\}\, \mathrm{e}^{\nu\tau} \>\mathrm{d}\tau.
\end{aligned}\end{equation}
The forcing term gives
\begin{equation}\label{q:j00}\begin{aligned}
   \int_0^t \mathrm{e}^{\nu\tau} (f,\Delta\tilde\omega)\>\mathrm{d}\tau
	&= \varepsilon\,(\mathsf{I}_\Omega\tilde f,\Delta\tilde\omega)(t)\mathrm{e}^{\nu t} - \varepsilon\,(\mathsf{I}_\Omega\tilde f,\Delta\tilde\omega)(0)\\
	&+ \varepsilon \int_0^t \bigl[ \nu(\mathsf{I}_\Omega\tilde f,\Delta\tilde\omega) + (\partial_\tau\mathsf{I}_\Omega\tilde f,\Delta\tilde\omega)
		+ (\mathsf{I}_\Omega\tilde f,\Delta\dy_\tau^*\tilde\omega)\bigr] \mathrm{e}^{\nu\tau} \>\mathrm{d}\tau,
\end{aligned}\end{equation}
which can be bounded as in the $L^2$ case, giving
\begin{equation}\begin{aligned}
    -&2\int_0^t (\tilde f,\Delta\tilde\omega)\,\mathrm{e}^{\nu\tau}\>\mathrm{d}\tau
        \le \varepsilon c\,\bigl[|\Delta\tilde f|\,|\Delta\tilde\omega|\bigr](t)\,\mathrm{e}^{\nu t}
	+ \varepsilon c\,\bigl[|\Delta\tilde f|\,|\Delta\tilde\omega|\bigr](0)\\
	&+ \varepsilon c \int_0^t \bigl\{ (1+\mu)\,|\nabla^3\omega|^2 + \mu\,|\nabla^3\tilde f|^2 + |\nabla\partial_\tau\tilde f|^2 + |\nabla^3\omega|\,|\nabla\omega|\,|\Delta\tilde f| \bigr\}\,\mathrm{e}^{\nu\tau}\>\mathrm{d}\tau.
\end{aligned}\end{equation}

For the nonlinear term, we use the fact that $B(\bar\omega,\bar\omega)=0$ to write
\begin{equation}
   (B(\omega,\omega),A\tilde\omega) = (B(\bar\omega,\tilde\omega),A\tilde\omega) + (B(\tilde\omega,\bar\omega),A\tilde\omega) + (B(\tilde\omega,\tilde\omega),A\tilde\omega),
\end{equation}
and, using $(B(\omega^\sharp,\tilde\omega),A\tilde\omega) = (B(\nabla\omega^\sharp,\tilde\omega),\nabla\tilde\omega)$,
previously used in \eqref{q:whs}, we bound
\begin{equation}\label{q:bdawt}\begin{aligned}
   \bigl|(B(\tilde\omega,\tilde\omega),A\tilde\omega)\bigr|
	&= \bigl|(B(\nabla\tilde\omega,\tilde\omega),\nabla\tilde\omega)\bigr|
	\le c\,|\tilde\omega|_{L^2}^{}|\nabla\tilde\omega|_{L^4}^2
	\le c\,|\tilde\omega|\,|\nabla\tilde\omega|\,|\Delta\tilde\omega|\\
	&\le \frac\mu4\,|\Delta\tilde\omega|^2 + \frac{c}{\mu}\,|\tilde\omega|^2|\nabla\tilde\omega|^2\\
   \bigl|(B(\bar\omega,\tilde\omega),A\tilde\omega)\bigr|
	&= \bigl|(B(\bar\omega',\tilde\omega),\partial_y\tilde\omega)\bigr|
	\le c\,|\bar\omega|_{L^\infty}^{}|\nabla\tilde\omega|_{L^2}^2\\
   \bigl|(B(\tilde\omega,\bar\omega),A\tilde\omega)\bigr|
	&\le |\Delta\tilde\omega|_{L^2}^{}|\nabla^{-1}\tilde\omega|_{L^\infty}^{}|\bar\omega'|_{L^2}^{}
	\le |\Delta\tilde\omega|_{L^2}^{}|\nabla\tilde\omega|_{L^2}^{}|\bar\omega'|_{L^2}^{}\\
	&\le \frac\mu4\,|\Delta\tilde\omega|^2
		+ \frac{c}\mu\,|\nabla\tilde\omega|^2|\bar\omega'|^2.
\end{aligned}\end{equation}
Using Poincar{\'e} inequality on the last term in (\ref{q:bdawt}c), we obtain
\begin{equation}\begin{aligned}
   2\int_0^t (B(\omega,\omega),\Delta\tilde\omega)\, &\mathrm{e}^{\nu\tau}\>\mathrm{d}\tau\\
	&\le c_2(\mu) \int_0^t \Bigl\{\bigl[|\bar\omega|_{L^\infty}^{} + |\nabla\omega|^2\bigr]\,|\nabla\tilde\omega|^2 + \frac\mu2\,|\Delta\tilde\omega|^2 \Bigr\}\,\mathrm{e}^{\nu\tau}\>\mathrm{d}\tau.
\end{aligned}\end{equation}
After moving the $|\Delta\tilde\omega|^2$ to the left-hand side,
a factor of $\varepsilon$ can be obtained by pulling the square bracket outside the
integral and using (\ref{q:bdwt}b).
Collecting, we have
\begin{equation}\label{q:bdwt1}\begin{aligned}
   &|\nabla\tilde\omega(t)|^2 + \frac\mu2 \int_0^t \mathrm{e}^{\nu(\tau-t)}|\Delta\tilde\omega|^2 \>\mathrm{d}\tau
	\le \mathrm{e}^{-\nu t}|\nabla\tilde\omega(0)|^2
	+ c\varepsilon\,\sup_{t'>0}\,|\Delta\tilde f(t')|\,|\Delta\omega(t')|\\
	&\quad {}+ c\varepsilon \int_0^t \bigl\{ (1+\mu)\bigl(|\nabla^3\omega|^2 + |\nabla^3\tilde f|^2\bigr) + |\nabla\partial_\tau\tilde f|^2  + |\nabla\omega|^2|\Delta\tilde f|^2 \bigr\}\, \mathrm{e}^{\nu(\tau-t)}\>\mathrm{d}\tau\\
	&\quad {}+ \varepsilon\,\cnst3(\mu)\,M_0\,\sup_{t'>0}\,\bigl\{|\bar\omega(t')|_{L^\infty}^{} + |\nabla\omega(t')|^2\bigr\}.\\
\end{aligned}\end{equation}
Arguing as in the $L^2$ case,
$f\in L_t^\infty H_{{\boldsymbol{x}}}^3$ and $\partial_t\tilde f\in L_t^\infty H_{{\boldsymbol{x}}}^1$
gives us an $\mathsf{O}(\sqrt\varepsilon)$ bound for $\tilde\omega(t)$ in $L_t^\infty H_{{\boldsymbol{x}}}^1$
uniform for large $t$.

\medskip
Bounds in $H^s$ can now be obtained inductively.
Assuming that Theorem~\ref{t:os} below holds for $s-1$ (we just showed
that it does for $s=2$), we multiply \eqref{q:abl} by $A^s\tilde\omega$
and integrate the resulting equation in time as above.
We bound the nonlinear term
$(B(\omega,\omega),A^s\tilde\omega)=(B(\tilde\omega,\tilde\omega),A^s\tilde\omega)+(B(\bar\omega,\tilde\omega),A^s\tilde\omega)+(B(\tilde\omega,\bar\omega),A^s\tilde\omega)$
as follows.
The first term is bounded exactly as in \eqref{q:whs}--\eqref{q:interp0},
\begin{equation}\label{q:k01}
   \bigl|(B(\tilde\omega,\tilde\omega),A^s\tilde\omega)\bigr|
	\le \frac\mu4\,|\nabla^{s+1}\tilde\omega|^2 + \frac{c(s)}\mu\,|\tilde\omega|^2|\nabla^s\tilde\omega|^2.
\end{equation}
We bound the next term by [cf.\ \eqref{q:whs}], with $|\alpha|=s$
and $1\le|\beta|=r\le s$,
\begin{equation}\label{q:k02}\begin{aligned}
   \bigl|(B(\bar\omega,\tilde\omega),A^s\tilde\omega)\bigr|
	&\le c\,{\textstyle\sum}_{\alpha\beta}\,|\mathsf{D}^\beta\bar{\boldsymbol{v}}|_{L^\infty}^{}|\mathsf{D}^{\alpha-\beta}\nabla\tilde\omega|_{L^2}^{}|\mathsf{D}^\alpha\tilde\omega|_{L^2}^{}\\
	&\le c(s)\,{\textstyle\sum}_{r=1}^s\,|\nabla^r\bar\omega|\,|\nabla^{s-r+1}\tilde\omega|\,|\nabla^s\tilde\omega|\\
	&\le c(s)\,|\nabla^s\bar\omega|\,|\nabla^s\tilde\omega|^2.
\end{aligned}\end{equation}
Finally, we bound the last term as, where now $0\le|\beta|=r\le s=|\alpha|$,
\begin{equation}\label{q:k03}\begin{aligned}
   \bigl|(B&(\tilde\omega,\bar\omega),A^s\tilde\omega)\bigr|
	\le c\,{\textstyle\sum}_{\alpha\beta}\,|\mathsf{D}^\beta\tilde{\boldsymbol{v}}|_{L^4}^{}|\mathsf{D}^{\alpha-\beta}\nabla\bar\omega|_{L^2}^{}|\mathsf{D}^\alpha\tilde\omega|_{L^4}^{}\\
	&\le c(s)\,{\textstyle\sum}_{r=0}^s\,|\tilde\omega|_{H^{r-1/2}}^{}\,|\bar\omega|_{H^{s-r+1}}^{}\,|\tilde\omega|_{H^{s+1/2}}^{}\\
	&\le \frac\mu4\,|\nabla^{s+1}\tilde\omega|^2 + c(s,\mu)\,{\textstyle\sum}_{r=0}^s\,|\nabla^{r-1}\tilde\omega|^{2/3}|\nabla^r\tilde\omega|^{2/3}|\nabla^s\tilde\omega|^{2/3}|\nabla^{s-r+1}\bar\omega|^{4/3}\\
	&\le \frac\mu4\,|\nabla^{s+1}\tilde\omega|^2 + c(s,\mu)\,|\nabla^s\tilde\omega|^2\,|\nabla^{s+1}\bar\omega|^{4/3}.
\end{aligned}\end{equation}
Moving the $|\nabla^{s+1}\tilde\omega|^2$ in \eqref{q:k01} and \eqref{q:k03} to the
left-hand side of the main inequality, the right-hand side depends
at most on $|\nabla^s\tilde\omega|^2$,
which is of $\mathsf{O}(\varepsilon)$ in $L_t^2$ from step $s-1$, and on $|\nabla^{s+1}\bar\omega|^2$.
As before, the worst term (i.e.\ that requires the highest derivative on
$f$) in fact comes from bounding $(f,A^s\tilde\omega)$.

\vfill\eject

We summarise our results as:

\begin{theorem}\label{t:os}
Let the initial data ${\boldsymbol{v}}(0)\in L^2(\mathscr{M})$ and the forcing be bounded as
\begin{equation}
   K_s(f):= \supt{\nabla^{s+2}f} + \supt{\nabla^s\partial_tf} < \infty.
\end{equation}
Then there exist $\mathcal{T}_s(|{\boldsymbol{v}}(0)|_{L^2}^{},K_s;\mu)$ and
$M_s(K_s;\mu)$ such that
\begin{equation}\label{q:bdws}\begin{aligned}
   &|\nabla^s\tilde\omega(t)|_{L^2}^2 \le \varepsilon\,M_s,\\
   &\mu\,\mathrm{e}^{-\nu (t+t')}\int_t^{t+t'} \mathrm{e}^{\nu\tau} |\nabla^s\tilde\omega(\tau)|^2 \>\mathrm{d}\tau
	\le \varepsilon\,M_s
\end{aligned}\end{equation}
for all $t\ge \mathcal{T}_s$.
\end{theorem}

\subsection{Higher-order Bounds}\label{s:ho}

As in \cite{temam-dw:lbal}, one can obtain bounds that scale as $\varepsilon^{n/2}$
for $\tilde\omega$ when the force $f$ is independent of time;
see \cite{mah:th}.

\section{Stability and the Global Attractor}\label{s:attr}

When the forcing $f$ is independent of time, the existence of the global attractor
$\mathcal{A}$ follows, just as for the non-rotating 2d Navier--Stokes equations,
from the uniform long-time bounds in Section~\ref{s:pre},
where the planetary rotation does not appear at all.
In the non-rotating case, the Hausdorff dimension of $\mathcal{A}$ is bounded by
\begin{equation}\label{q:dim0}
   \textrm{dim}_\textrm{H}^{}\,\Attr \le c\,G^{2/3}(1 + \log G)^{1/3}
\end{equation}
where in our notation the Grashof number is
\begin{equation}
   G := |\nabla^{-1} f|_{L^2}^{}/\mu^2.
\end{equation}
The rotation not posing any extra essential difficulty,
the usual analysis, e.g.\ \cite[\S9.2]{doering-gibbon:aanse},
carries over essentially line-by-line to our case,
giving the bound \eqref{q:dim0} also for the rotating case \eqref{q:dwdt}.

As discussed in the introduction, and following our results that the
flow becomes more zonal (``ordered'') as $\varepsilon\to0$, we expect the
dimension of the attractor to decrease as $\varepsilon\to0$.
In this section, we use a simple computation similar to that used
for Theorem~\ref{t:o1} to show that $\textrm{dim}_\textrm{H}^{}\,\Attr=0$ for $\varepsilon$
sufficiently small.

\begin{theorem}\label{t:attr}
Let the forcing $f$ be time independent, $\partial_tf=0$, and assume the
hypotheses of Theorem~\ref{t:o1}, i.e.\ ${\boldsymbol{v}}(0)\in L^2(\mathscr{M})$ and
\begin{equation}
   |\nabla^2f|_{L^2}^{} < \infty.
\end{equation}
Then there exists an $\varepsilon_*(|\nabla^2f|;\mu)$ such that,
for all $\varepsilon<\varepsilon_*$,
\begin{equation}\label{q:dimaz}
   \textrm{dim}_\textrm{H}^{}\,\Attr = 0.
\end{equation}
\end{theorem}

Since $\mathcal{A}$ is connected, \eqref{q:dimaz} implies that $\mathcal{A}$
consists of a single point, that is, a steady flow $\omega_*$ to
which all bounded solutions converge.
Following Theorems \ref{t:o1} and~\ref{t:os}, this steady flow is nearly,
but not exactly, zonal (except in the non-generic case when $\tilde f=0$).
Heuristically, an approximation to $\omega_*$ is the steady flow
\begin{equation}
   \omega_*^{(1)} = -\mu^{-1}\Delta^{-1}\bar f + \varepsilon L^{-1}\tilde f,
\end{equation}
which satisfies
\begin{equation}
   \frac1\varepsilon L\omega_*^{(1)} + B(\omega_*^{(1)},\omega_*^{(1)}) + \mu A\omega_*^{(1)} = f
\end{equation}
up to $\mathsf{O}(\varepsilon)$.
More careful work would be needed to determine $\omega_*$ exactly.

In turbulence parlance, the smallness of $\varepsilon$ demanded by
Theorem~\ref{t:attr} implies that the Rhines scale
\cite{vallis:aofd} is so large that it overwhelms the entire
spectral range, rendering the dynamics trivial.

A general result related to ours is described in
\cite[ch.~18]{chepyzhov-vishik:aemp}, where the trajectory attractor
$\mathcal{A}_\epsilon$ of a dynamical system depending on $t/\epsilon$
(formally, in our case $\mathcal{A}_\epsilon$ would simply be the attractor
$\mathcal{A}$ for $\varepsilon>0$) converges weakly to the attractor $\mathcal{A}_0$ of
the corresponding averaged system.
Formally averaging our equations following this construction (which
does not apply directly to our case, in which the oscillations have
an infinite number of frequencies which accumulate at zero),
we obtain purely zonal NSE, whose dynamics is trivial and whose attractor
thus has dimension zero.
This is of course consistent with our results: strong convergence
at finite $\varepsilon$ of $\mathcal{A}$ to a point (which becomes zonal as
$\varepsilon\to0$).

\medskip\noindent{\sc Proof.}
Fix a solution $\omega(t)$ of \eqref{q:abl} that lives on $\mathcal{A}$,
so the bounds \eqref{q:bdws} hold for all $t$.
We consider a nearby solution $\omega(t)+\phi(t)$.
The linearised evolution equation for $\phi$ is then
\begin{equation}\label{q:dftdt}\begin{aligned}
   \partial_t\phi &= -(\nabla^\perp\Delta^{-1}\omega)\cdot\nabla\phi - (\nabla^\perp\Delta^{-1}\phi)\cdot\nabla\omega(t)
	- \frac1\varepsilon \partial_x\Delta^{-1}\phi + \mu\Delta\phi\\
	&= -B(\omega,\phi) - B(\phi,\omega) - \frac1\varepsilon L\phi - \mu A\phi
	=: \mathcal{L}(t)\phi.
\end{aligned}\end{equation}
Multiplying this by $\phi$ in $L^2$ and noting that $(B(\omega,\phi),\phi)=0$,
we obtain
\begin{equation}\label{q:dfdt}\begin{aligned}
   \frac12\ddt{\;}|\phi|^2 + \mu\,|\nabla\phi|^2 &= (B(\phi,\phi),\omega)\\
	&= (B(\phi,\phi),\bar\omega) + (B(\phi,\phi),\tilde\omega).
\end{aligned}\end{equation}

For the first term, we split $\phi=\bar\phi+\tilde\phi$ in analogy with $\omega=\bar\omega+\tilde\omega$
to get
\begin{equation}
   (B(\phi,\phi),\bar\omega) = (B(\tilde\phi,\tilde\phi),\bar\omega)
\end{equation}
using the (now familiar) facts that $B(\bar\phi,\bar\phi)=0$ and all
tilde-bar-bar terms vanish.
Using Poincar{\'e} inequality in \eqref{q:dfdt} gives us [cf.\ \eqref{q:dwtdt}]
\begin{equation}
   \ddt{\;}\bigl(\mathrm{e}^{\nu t}|\phi|^2\bigr) + \mu\mathrm{e}^{\nu t}|\nabla\phi|^2
	\le 2\,\mathrm{e}^{\nu t}(B(\tilde\phi,\tilde\phi),\bar\omega) + 2\,\mathrm{e}^{\nu t}(B(\phi,\phi),\tilde\omega),
\end{equation}
which integrates to
\begin{equation}\label{q:l01}\begin{aligned}
   |\phi(t)|^2\,\mathrm{e}^{\nu t} &+ \mu \int_0^t |\nabla\phi|^2\,\mathrm{e}^{\nu\tau} \>\mathrm{d}\tau\\
	&\le |\phi(0)|^2
	+ 2 \int_0^t \bigl\{ (B(\tilde\phi,\tilde\phi),\bar\omega) + (B(\phi,\phi),\tilde\omega) \bigr\} \,\mathrm{e}^{\nu\tau} \>\mathrm{d}\tau.
\end{aligned}\end{equation}
We bound the last term of the integrand using
\begin{equation}\begin{aligned}
   (B(\phi,\phi),\tilde\omega)
	&\le c\,|\nabla^{-1}\phi|_{L^\infty}^{}|\nabla\phi|_{L^2}^{}|\tilde\omega|_{L^2}^{}\\
	&\le \cnst4\,|\nabla\phi|^2|\tilde\omega|_{L^2}.
\end{aligned}\end{equation}
The other term needs to be integrated by parts,
\begin{equation}\label{q:l02}\begin{aligned}
   &\int_0^t (B(\tilde\phi,\tilde\phi),\bar\omega)\,\mathrm{e}^{\nu\tau}\>\mathrm{d}\tau
	= \varepsilon\,(B_\Omega(\tilde\phi,\tilde\phi),\bar\omega)(t)\,\mathrm{e}^{\nu t}
	- \varepsilon\,(B_\Omega(\tilde\phi,\tilde\phi),\bar\omega)(0)\\
	&\hbox to15pt{} {}- \varepsilon \int_0^t \bigl\{ \nu\,(B_\Omega(\tilde\phi,\tilde\phi),\bar\omega)
		+ (B_\Omega(\tilde\phi,\tilde\phi),\partial_\tau\bar\omega)
		+ 2\,(B_\Omega(\dy_\tau^*\tilde\phi,\tilde\phi),\bar\omega)
		\bigr\}\,\mathrm{e}^{\nu\tau}\>\mathrm{d}\tau
\end{aligned}\end{equation}
where $\partial_t^*\phi=-B(\omega,\phi)-B(\phi,\omega)-\mu A\phi$.
We bound the endpoint terms using
\begin{equation}
   2\,\bigl|(B_\Omega(\tilde\phi,\tilde\phi),\bar\omega)\bigr| \le \cnst5 |\tilde\phi|^2|\bar\omega'|_{L^\infty}^{}.
\end{equation}
It remains to bound the integrand in \eqref{q:l02}:
\begin{equation}\begin{aligned}
   \bigl|(B_\Omega(\tilde\phi,\tilde\phi),\bar\omega)\bigr|
	&\le c\,|\partial_y\tilde\phi|_{L^2}^{}|\tilde\phi|_{L^4}^{}|\bar\omega|_{L^4}^{}\\
	&\le c\,|\nabla\tilde\phi|^2|\bar\omega|_{L^4}^{}
\end{aligned}\end{equation}
\begin{equation}\begin{aligned}
   \bigl|(B_\Omega(\tilde\phi,\tilde\phi),\partial_t\bar\omega)\bigr|
	&\le c\,|\partial_y\tilde\phi|_{L^2}^{}|\tilde\phi|_{L^{10}}^{}|\partial_t\bar\omega|_{L^{5/2}}^{}\\
	&\le c\,|\nabla\tilde\phi|^2|\partial_t\bar\omega|_{L^{5/2}}^{}
\end{aligned}\end{equation}
Recalling \eqref{q:dftdt} for the last term in \eqref{q:l02}, we bound
\begin{align*}
   \bigl|(B_\Omega(\tilde B(\phi,\omega),\tilde\phi),\bar\omega)\bigr|
	&\le c\,|\nabla^{-1}\phi|_{L^\infty}^{}|\nabla\omega|_{L^2}^{}|\tilde\phi|_{L^{10}}^{}|\bar\omega'|_{L^{5/2}}^{}\\
	&\le c\,|\nabla\phi|^2|\bar\omega'|_{L^{5/2}}^{}|\nabla\omega|_{L^2}^{}
\end{align*}
\begin{align}
   \bigl|(B_\Omega(\tilde B(\omega,\phi),\tilde\phi),\bar\omega)\bigr|
	&\le c\,|\nabla^{-1}\omega|_{L^\infty}^{}|\nabla\phi|_{L^2}^{}|\tilde\phi|_{L^{10}}^{}|\bar\omega'|_{L^{5/2}} \notag\\
	&\le c\,|\nabla\phi|^2|\bar\omega'|_{L^{5/2}}^{}|\nabla\omega|_{L^2}^{}
\end{align}
\begin{align*}
   \bigl|(B_\Omega(\Delta\tilde\phi,\tilde\phi),\bar\omega)\bigr|
	&= \bigl|(B_\Omega(\nabla\tilde\phi,\nabla\tilde\phi),\bar\omega)\bigr| + \bigl|(B_\Omega(\partial_y\tilde\phi,\tilde\phi),\bar\omega')\bigr|\\
	&\le c\,|\nabla\tilde\phi|^2|\bar\omega'|_{L^\infty}^{} + c\,|\nabla\tilde\phi|_{L^2}^{}|\tilde\phi|_{L^{10}}^{}|\bar\omega''|_{L^{5/2}}^{}\\
	&\le c\,|\nabla\tilde\phi|^2|\bar\omega''|_{L^{5/2}}^{}.
\end{align*}
Collecting, \eqref{q:l01} now implies
\begin{equation}\label{q:a0}\begin{aligned}
   |\phi(t)|^2\bigl(1 - \varepsilon\cnst5|\bar\omega'(t)|_{L^\infty}^{}\bigr)
	+ \int_0^t \bigl\{ \mu - \varepsilon\,N(\tau) &- \cnst4|\tilde\omega(\tau)|_{L^2}^{} \bigr\}\,|\nabla\tilde\phi|^2\,\mathrm{e}^{\nu(\tau-t)}\>\mathrm{d}\tau\\
	&\le \mathrm{e}^{-\nu t}|\phi(0)|^2\bigl(1 + \varepsilon\cnst5|\bar\omega'(0)|_{L^\infty}^{}\bigr)
\end{aligned}\end{equation}
where
\begin{equation}
   N(t) := \cnst6 \bigl\{ \mu\,|\bar\omega''|_{L^{5/2}}^{}
		+ |\bar\omega'|_{L^{5/2}}^{}|\nabla\omega|_{L^2}^{}
		+ |\partial_t\bar\omega|_{L^{5/2}}^{} + |\bar\omega|_{L^4}^{} \bigr\}(t).
\end{equation}
By \eqref{q:bdwm}, $f\in H^2$ implies that $\omega\in H^3$ with
uniform bound in $t$ since we are already on the attractor,
and by Theorem~\ref{t:o1} we can find an $\varepsilon_*$ so small that,
for $\varepsilon<\varepsilon_*$,
\begin{equation}
   \sup_{t>0}\, \bigl\{ \varepsilon N(t) + \cnst4|\tilde\omega(t)|_{L^2}^{} \bigr\} < \mu.
\end{equation}
Requiring furthermore that $\varepsilon_*$ also implies, for all $\varepsilon<\varepsilon_*$,
\begin{equation}
  \varepsilon\cnst5\,\sup_{t>0}\,|\bar\omega'(t)|_{L^\infty}^{} < 1.
\end{equation}
These and \eqref{q:a0} then imply that
\begin{equation}
   |\phi(t)|^2 \le C(\cdots)\,\mathrm{e}^{-\nu t}|\phi(0)|^2,
\end{equation}
in other words, all phase space volumes contract and thus
the global attractor has dimension zero.

It is clear from the above proof that our solution $\omega(t)$
is linearly stable.
Since \eqref{q:dftdt} only differs by $B(\phi,\phi)$ from the
nonlinear system,
the fact that $(B(\phi,\phi),\phi)=0$ implies that stability
also holds under the same hypotheses for the full nonlinear system.

redpajama_set_name: RedPajamaArXiv


Will the Darling River Survive?
The health of the Darling River system is at a tipping point. Can the system survive the next round of negotiations over how it should be managed? Terry Korn, president of the Australian Floodplain Association discusses a major issue of concern which could seriously impact on recovery of water for the environment, floodplain graziers, Aboriginal culture and small communities downstream of Bourke.
The Australian government has committed almost $15 billion to the largest rural restructure program in Australia and expects to effect significant changes to water management in the Murray-Darling Basin without affecting the reliability of water supply to the irrigation industry. This is an admirable but unreal aspiration.


Canada supports international efforts to ensure judicial independence and access to justice.
This week, the Honourable Jody Wilson-Raybould, Minister of Justice and Attorney General of Canada, attended the inaugural Marrakech International Justice Conference. She spoke about Canada's perspective on three important issues: judicial independence, judicial diversity and the importance of adhering to the rule of law.
Other ministers of justice, public prosecutors, and members of the judiciary from North America, Africa, Europe, the Middle East, and Asia came together to discuss the independence of the judiciary and the rights of litigants within the rules of justice.
While in Morocco, Canada signed a common declaration between the Department of Justice and the Ministry of Justice for the Kingdom of Morocco aimed at fostering cooperation through the exchange of knowledge and experience related to their respective legal systems.
More than 40 ministers of justice attended the inaugural conference being hosted by the Kingdom of Morocco.
In 2016, bilateral trade between Canada and Morocco totaled over $767.0 million in sectors such as agriculture, education, aerospace, tourism, and mining.


text:
Valerie Stimac
Author HomepageRSS feed for Author
Valerie Stimac Bailey is a California-based freelance writer and travel blogger. She runs several websites on travel, space tourism, and even Harry Potter — she's basically been lucky to turn all of her favorite things into her professional career. When not busy writing, you can find her out exploring as near (or far) as she can, stargazing, or going to the movies.
That Faint Light in the Night Sky Could Be the Gegenschein
The gegenschein, "faint light" in German, occurs under very specific astronomical conditions when the sun reaches the exact opposite of Earth from wherever you're stargazing.
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How to Find Orion's Belt in the Night Sky
Winter is the perfect time to look for Orion's Belt in the Northern Hemisphere. If you're new to stargazing, we'll show you how to find it.
Get Ready for the Quadrantids, the First Meteor Shower of the New Year
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Walking the Pilgrim's Path of El Camino de Santiago
Whether to deepen faith or just to have some time for personal reflection, thousands of people walk, ride or bike el Camino de Santiago, an ancient pilgrim's route, every year. Want to join them?
Researchers Find First (Potential) Planet Outside the Milky Way
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NASA Wants YOU! to Help Prepare Humankind for Mars
Before humans can go to Mars, NASA needs to know how to prepare for the challenges. So, they're looking for a few brave men and women to live in isolation for a year at the Johnson Space Center. Want to join them?
Is Space Tourism Worth It?
Today, Blue Origin flew its billionaire founder Jeff Bezos and three others into space, a week after Virgin Galactic flew its billionaire founder Richard Branson on the same mission. Will space tourism be more than a millionaire's hobby?
Turns Out Earth's Pulse 'Beats' Every 27.5 Million Years, But Why?
Researchers have been asking this question for almost a century and now we're a little closer to the answer. Something else to ponder: Every 27.5 million years there is usually a mass extinction.
4 Ways to Experience Outer Space Right Here on Earth
Amazon CEO Jeff Bezos and his brother plan to blast into space next month. But even if you're not related to a billionaire, you can still experience the fun of space at a 'down-to-Earth' price.
How Many Moons Does Saturn Have?
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What's the Order of the Planets in the Solar System?
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How Long Does It Take to Get to Mars?
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Scientists Detected a Space Hurricane for the First Time. What's That?
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How Long Does It Take to Get to the Moon?
Humans have made it to the moon a number of times so we know how long the journey should take. But sometimes it may be much faster than that.
19 Amazing Facts About the Canadian Provinces
O Canada ... the country where you can order a cocktail with a human toe in it (and other quirky things about those Canucks up North).
Is the Sodium Lauryl Sulfate in My Shampoo Safe?
Is the same substance that makes your shampoo so pleasantly sudsy really going to give you cancer? Here's the real dirt on whether sodium lauryl sulfate is bad for you.
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Slow Blinking at Your Cat Really Bonds You Both — Try It!
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7 Eye-catching Facts About the Bright Star Vega
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redpajama_set_name: RedPajamaCommonCrawl


The Laneside Grill at Kingpin's Alley is the perfect complement to satisfy your hunger while you're out there on the lanes throwing strikes. Our menu, with its wide array of tempting selections, is sure to have something for everyone. We have thick and juicy burgers, some great french fries, chicken tenders with a variety of sauces and our pizza is second to none. Our wraps are a top selling selection and the chicken wings are big and made to your specs.
You're going to need a drink to wash it all down and for that we have several Pepsi product selections on the fountain.


Pelvic floor muscle training is effective and recommended as first-line therapy for female patients with stress urinary incontinence. However, standard pelvic floor physiotherapy concentrates on voluntary contractions even though the situations provoking stress urinary incontinence (for example, sneezing, coughing, running) require involuntary fast reflexive pelvic floor muscle contractions. Training procedures for involuntary reflexive muscle contractions are widely implemented in rehabilitation and sports but not yet in pelvic floor rehabilitation. Therefore, the research group developed a training protocol including standard physiotherapy and in addition focused on involuntary reflexive pelvic floor muscle contractions.
The aim of the planned study is to compare this newly developed physiotherapy program (experimental group) and the standard physiotherapy program (control group) regarding their effect on stress urinary incontinence. The working hypothesis is that the experimental group focusing on involuntary reflexive muscle contractions will have a higher improvement of continence measured by the International Consultation on Incontinence Modular Questionnaire Urinary Incontinence (short form), and — regarding secondary and tertiary outcomes — higher pelvic floor muscle activity during stress urinary incontinence provoking activities, better pad-test results, higher quality of life scores (International Consultation on Incontinence Modular Questionnaire) and higher intravaginal muscle strength (digitally tested) from before to after the intervention phase. This study is designed as a prospective, triple-blinded (participant, investigator, outcome assessor), randomized controlled trial with two physiotherapy intervention groups with a 6-month follow-up including 48 stress urinary incontinent women per group. For both groups the intervention will last 16 weeks and will include 9 personal physiotherapy consultations and 78 short home training sessions (weeks 1–5 3x/week, 3x/day; weeks 6–16 3x/week, 1x/day). Thereafter both groups will continue with home training sessions (3x/week, 1x/day) until the 6-month follow-up.
To compare the primary outcome, International Consultation on Incontinence Modular Questionnaire (short form) between and within the two groups at ten time points (before intervention, physiotherapy sessions 2–9, after intervention) ANOVA models for longitudinal data will be applied.
This study closes a gap, as involuntary reflexive pelvic floor muscle training has not yet been included in stress urinary incontinence physiotherapy, and if shown successful could be implemented in clinical practice immediately.
Stress urinary incontinence (SUI), the most common urinary incontinence subtype in women with a prevalence of 24.8 % , is defined as involuntary loss of urine during effort or physical exertion (for example, during sporting activities) or upon sneezing or coughing . Activities typically provoking SUI raise the intra-abdominal pressure and impact loading on the pelvic floor muscles (PFM) and require strong, rapid, and reflexive PFM contractions to maintain continence [3–6]. Fast and strong PFM contractions result in the generation of an adequate squeeze pressure in the proximal urethra, which maintains a pressure higher than that in the bladder, thus preventing leakage . PFM function regarding power (rate of force development) is impaired in incontinent women compared to continent women [4, 6].
PFM training — defined as a program of repeated voluntary PFM contractions taught and supervised by a health care professional — is the most commonly used physiotherapy treatment for women with SUI, is effective with all types of female incontinence, and, therefore, is recommended as a first-line therapy [8, 9]. As endorsed by the International Consultation on Incontinence, PFM training should generally be the first step of treatment before surgery . However, standard SUI physiotherapy concentrates on voluntary contractions even though the situations provoking SUI such as sneezing, coughing, jumping, and running  require involuntary fast reflexive PFM contractions . Although training procedures following the concepts of training science and sports medicine are generally well known and widely implemented in rehabilitation and sports [11, 12], an optimal and well-standardized training protocol for involuntary, fast, and reflexive PFM contractions still remains unknown.
Consequently, the research group of the present study developed a standardized therapy program, which includes the standard physiotherapy and additionally focuses on involuntary fast reflexive PFM contractions. The additional exercises are well known and applied in physiotherapy, however not yet regarding SUI.
Therefore, the aim of the present study is to compare two different physiotherapy programs: standard training plus involuntary reflexive PFM training versus standard training alone, regarding their effect on SUI and the impact of incontinence symptoms on quality of life. Both programs include standard physiotherapy. Both follow the concepts of training science, that is, periodization/exercise sequence and training of specific muscle strength components [11, 12]. One program focuses on voluntary fast contractions (standard physiotherapy; control group); the other one additionally focuses on involuntary fast reflexive PFM contractions (experimental group). The secondary objective, based on secondary and tertiary outcomes, is a deeper insight into the functioning of the PFM (PFM activity characteristics and muscle action forms) by evaluating their activity measured by electromyography (EMG) in the quality of life of patients with SUI and in patients' therapy adherence.
The present study is a single-centered, prospective, triple-blinded (participant, investigator, outcome assessor), parallel group, non-inferiority randomized controlled trial with two physiotherapy intervention groups with a 6-month follow-up.
Participants will be blinded against the type of received physiotherapy (control group, experimental group). The participant information document will not provide any information regarding the differences in the experimental and control therapy protocols in a way that women could find out about their group allocation. All investigators involved in data acquisition, data reduction, data analyses, and statistics will also be blinded against group allocation. The physiotherapists in charge of the therapy cannot be blinded against group allocation and therefore will not be involved in data acquisition, data reduction, data analyses, or statistics.
With ethics committee approval (Ethics Committee of the Canton of Bern reference number 249/14 on 12 November 2014; Chairperson Prof. Dr. Ch. Seiler), which is in accordance with the Declaration of Helsinki and the Swiss Human Research Act, and written informed consent, patients will be included on the condition that they are female adults aged 18–70 years, suffer from SUI (based on the patient's history) or mixed urinary incontinence (with dominant SUI), are one year post-partal, parous, nulliparous, pre- or post-menopausal, have a BMI of 18–30 kg/m2, are medically and physically fit for the measurement and therapeutic exercises (running, jumps), and, in case of systemic or local estrogen treatment, stable for the past 3 months prior to inclusion.
Exclusion criteria are: urge incontinence or predominant urgency in incontinence; prolapse > grade 1 POP-Q  (uterus, cystocele, rectocele during Valsalva maneuver); pregnancy (urine test to accomplish); current urinary tract or vaginal infection, menstruation on the day of examination; lactation period not yet finished; contraindications for measurements or interventions, for example, acute inflammatory or infectious disease, tumor, fracture; de novo systemic or local estrogen treatment (<3 months); de novo drug treatment with anticholinergics or other bladder active substances (tricyclic antidepressants, selective serotonin reuptake inhibitors). Interventions and measurements would be individually modified in case of temporary loss of physical fitness for certain exercises.
The primary outcome measure will be the International Consultation on Incontinence Modular Questionnaire Urinary Incontinence short form (ICIQ-UIsf), which provides a brief and robust assessment of the impact of symptoms of urinary incontinence on quality of life and outcome of treatment and is scored on a scale from 0 (not) – 21 (severely affected). Previous research has examined the ICIQ-UIsf questionnaire and found it to have good reliability and constructive validity  and to correlate well with urodynamic findings . This questionnaire is validated in the German language . For measurement time points see Table 1.
The secondary outcome measure will be the EMG activity of the PFM during rest, maximal voluntary contractions (MVCs), voluntary fast contractions (VFCs), and during involuntary contractions during squat jumps (SJs), counter movement jumps (CMJs), drop jumps (DJs), and treadmill running at 7, 9, and 11 km/h . As there is a relation between high impact activities and SUI prevalence , those different and comprehensive test exercises will be chosen to represent typical situations of involuntary urine loss and different muscle action forms (isometric-concentric voluntary muscle actions: MVC, VFC; involuntary eccentric-concentric muscle actions: SJ, CMJ, DJ, running).
Additionally, the 20-min pad test [19, 20] and the International Consultation on Incontinence Modular Questionnaire Urinary Incontinence Symptoms Quality of Life (ICIQ-LUTSqol)  will be implemented as secondary outcomes. The pad test will provide information regarding the efficacy of the therapy protocols to the UI symptoms (such as changes in the leakage volume) and the ICIQ-LUTSqol evaluates the impact on the participant's quality of life with reference to social effects. It consists of 20 questions, which lead to a summary score between 19 and 76 points (greater values indicate higher impact on quality of life) and is available in German .
As tertiary outcomes PFM strength will be digitally assessed according to the Oxford Grading Scale , and adherence to the home exercise program will be assessed, that is, how many of the total therapy sessions will be completed individually. The home exercise adherence data will be collected with a simple training diary questionnaire.
For measurement time points, see Table 1.
A vaginal surface EMG probe (3-pol-STIMPON® electrode (Innocept Biobedded Medizintechnik GmbH, Gladbeck, Germany) in a differential configuration will be used to measure PFM activity. The STIMPON electrode is a certified device (patent number EP 0 963 217 B1, CE 0482) which is widely applied in pelvic floor rehabilitation. The probe is made of polypropylene and with this soft surface therefore optimally designed to adapt its form to individual vaginal cavities and not to slip out of position during impact loads. The single reference adhesive surface electrode (Ambu Blue Sensor N, Ballerup, Denmark) will be fixed on the right iliac crest.
Running will be conducted on a Kettler Marathon TX1 treadmill device (Ense-Parsit, Germany) at speeds of 7, 9, and 11 km/h and 1° inclination. For the DJ a 20-cm step will be used.
To detect events necessary for the parameterization of the time-dependent signals such as heel strike (= initial contact = time point zero (T0)) during running, two load cells (Type KMB52 K 10KN, Megatron, Putzbrunn, Germany) are attached under the treadmill. T0 of SJ, CMJ, and DJ will be identified by using a force plate (Type 9286BA, Kistler Winterthur, Switzerland). Electrodes will be connected to the transmitter by short wire and the signals will be sent wirelessly to the receiver (TeleMyo 2400 G2, Noraxon European Service Center, Cologne, Germany).
The subjects will be instructed in the vaginal insertion of the surface EMG probe using ultrasound lubrication and will perform the insertion themselves after emptying their bladder. They will wear a loose running suit and running shoes of the same type but individual size. PFM EMG will be measured for 30 seconds (s) without any voluntary contraction and twice for 5 s during MVC (contraction maximal as possible), which will be instructed and controlled by intravaginal manual muscle testing before the EMG measurements, and ten times for VFC in a standing position, with a 15-s break between the single measurements. After that they will perform five SJs, five CMJs, and five DJs, again with 15 s between the single measurements, each jump being demonstrated and practiced before EMG recordings. Thereafter, the subjects will perform a warm-up of treadmill walking (5 km/h) for 30 s, then running at 7, 9, and 11 km/h consecutively until they reach a steady state. As soon as they reach the steady state at the respective speed, the data acquisition will be started: all data will be measured continuously for 10 s and the first ten step cycles of the right leg will be analyzed. Between the measurements of the different speeds the treadmill will be stopped, followed by a 1-minute break until the same procedure is restarted with the next speed .
All signals will be sampled at a rate of 3 kHz (sampling interval (dt) equals 0.333 ms) using a measurement amplifier and 1-bit analog-to-digital converter (ME-2600i, SisNova Engineering, Zug, Switzerland) and the software package "Analoge und digitale Signalverarbeitung" (ADS) version 1.12 (uk-labs, Kempen, Germany).
The EMG signals will firstly be first order high-pass filtered with a cut-off frequency of 10 Hz by EMG preamplifier leads to reject or eliminate artifacts and later digitally low-pass filtered by Noraxon Receiver Hardware with a cut-off frequency of 500 Hz. Secondly, to identify amplitude peaks during MVC, EMG will be calculated as RMS (200 ms moving window). 100 % of EMG equals the average of the two peak amplitude values during the two 5-s sessions of MVC. Thirdly, EMG variables of pre-activity and reflexivity will be calculated as RMS values within 30-ms intervals [24–26], averaged over 10 steps, and normalized to peak MVC (% EMG). The variables of pre-activity and reflexivity (30-ms intervals) will follow former study protocol intervals [17, 24, 25]. All variables will be analyzed using the software package MATLAB (MathWorks, Natick, MA, USA). Outcomes will be averaged over the repetitions of each exercise.
Alternative hypothesis for primary outcome: It is hypothezised that the experimental group focusing on involuntary pelvic floor muscle contractions will have a statistically higher improvement of continence measured by ICIQ-UIsf questionnaire from before to after the intervention phase.
Sample size calculation was performed with G*Power software , using the statistical model of an ANOVA approach (repeated measures, within-between interactions). To this day there is no comparable training study (RCT) to estimate sample size based on available data from ICIQ-UIsf. Consequently, sample size was estimated theoretically and an effect size of = 0.1, indicating a small effect  will be accepted. The sample size was calculated for the primary outcome ICIQ-UIsf with the following assumptions: α = 0.05, power (1–β error probability) = 0.8, number of groups = 2, number of measurements = 10; correlations among repeated measures were estimated conservatively low with 0.5. Based on these assumptions, a total sample size of N = 80 was estimated. In anticipation of dropouts (10 %: n = 8) or a violation of normality assumption (+10 %: +n = 8), a final sample size of N = 96 (48 participants per group) results.
Analysis of the patients will follow the CONSORT flow diagram (see Fig. 1) through the phases of the study (enrollment (assessed, excluded, randomized), allocation (control group, experimental group with received intervention or not received intervention), follow-up (lost to follow-up, discontinued intervention), and analysis (excluded, included)) .
The outcomes of the control and experimental group will be analyzed by the repeated measures within-between interactions ANOVA approach and an intention-to-treat analysis (last observation carried forward method). No subgroup analyses are planned.
Primary analysis: For the descriptive analyses means, standard deviations, 95 % confidence intervals, and median, quartiles, minima, maxima will be used.
To compare the primary outcome ICIQ-UIsf between and within the two groups (control group, experimental group) at ten time points (before intervention, physiotherapy sessions 2–9, after intervention), ANOVA models for longitudinal data according to Brunner et al.  will be used. All statistical analyses will be realized after the final measurement of the last patient at the time point after intervention. The statistics will be calculated by SPSS and R software packages in the current versions. The repeated measure design with ten time points allows us to monitor how patients change over time, in both short-term (before/during intervention) and long-term situations (before/after intervention).
Secondary and tertiary analyses: All secondary and tertiary outcomes will be analyzed following the same approach and with the same statistical procedures as for primary outcomes. Generally, rational data will be checked for normality using Q-Q-plots. If the normality assumption is not violated, parametric ANOVA models for longitudinal data will be used . As for intra-session retest procedures (EMG amplitude and frequency), data will also be described and checked for their reliability (ICC, SEM, MD, ANOVA).
Patients will be recruited from the urogynecology consultation of the Women's Hospital, Bern University Hospital and University of Bern, Switzerland. All participants for the study will be provided a participant information sheet and a consent form describing the study and providing sufficient information for them to make an informed decision about their participation in the study.
Each participant will be specifically informed that the participation in the study is voluntary and that she may withdraw from the study at any time and that withdrawal of consent will not affect her subsequent medical assistance and treatment.
Patients with written informed consent are randomly allocated to one of the two therapy groups (control group, experimental group). The allocation sequence will be generated by the independent urogynecology secretariat using online randomization software (http://randomization.com); allocation ratio = 1:1 (control group:experimental group). The allocation will be concealed in sealed, opaque, sequentially numbered envelopes, which will be stored at the secretariat and can be opened one at a time for each eligible patient. Physiotherapists treating patients will be informed about enrollment and group allocation of their patient by the independent secretariat.
Baseline (before intervention phase), after intervention phase, and 6-month follow-up measurements (of primary, secondary, and tertiary outcomes) will take place at the Bern University of Applied Sciences movement laboratory by two experienced pelvic floor physical therapists, who are blinded regarding group allocation of participants. Measurements during the 16-week intervention phase (ICIQ-UIsf; primary outcome) will be taken by the treating physiotherapists. However, to guarantee blinding of the outcome, the participant will fill in the questionnaire in the absence of the treating physiotherapist and seal it in an envelope, which will be given to the outcome assessor.
Both therapy programs (control group, experimental group) are based on the latest position stand paper of the American College of Sports Medicine [11, 32], PFM motor learning concepts [33, 34], and strength training concepts [11, 12, 32, 35, 36]. The planned progression of training for strength, power, and hypertrophy [11, 32, 33, 35, 36] is shown in Table 2.
Training procedures for motor learning, strength, hypertrophy, and power training phases follow the training principles of variation/periodization, muscle action and velocity of muscle action, loading, volume, exercise selection, rest period, and frequency [11, 12] for both groups. The training program will last 16 weeks and will contain 9 personal physiotherapy consultations and 78 home training sessions of approximately 15 minutes (weeks 1–5 3x/week, 3x/day; weeks 6–16 3x/week, 1x/day) (Tables 1 and 2). Motor learning and strength and hypertrophy phases are comparable for both groups. However, the main difference between the programs is the applied type of muscle action (control group: isometric, concentric; experimental group: isometric, concentric, eccentric und eccentric-concentric) and speed of movement (control group: voluntary slow to moderate (to explosive) [33, 35]; experimental group: explosive, reactive, reflexive). After completion of the 16-week training program, the participants will participate in no further personal physiotherapy consultations; however, they will continue home training sessions 3x/week until the 6-month follow-up. During the intervention period participants may continue with their individual activities of daily life.
The personal physiotherapy consultations will take place in the physiotherapy center of the Women's Hospital, Bern University Hospital and University of Bern, Switzerland. The treating physiotherapists are all specialized and experienced in PFM rehabilitation.
An additional document file presents the intervention (therapy plan of intervention group and control group) in detail [see Additional file 1].
In the current study there are no anticipated risks or inconveniences, as the applied intervention and examinations are well known and widely applied in pelvic floor standard physiotherapy. The additional exercises of the experimental group are also well known and applied in physiotherapy; however, they have not yet been used with SUI patients.
All the women are asked during every personal physiotherapy and measurement consultation whether they experience any adverse effects.
A summary flow chart is provided in Fig. 1 (CONSORT study flow diagram).
SUI is increasingly recognized as a health and economic problem which not only troubles the affected women, but also implies a substantial economic burden on health and social services . Physiotherapy, the first-line basic SUI therapy [8, 9], represents only a mere 2 % of total costs  and therefore seems to be good value for the money. As activities or efforts typically provoking SUI occur within milliseconds [39, 40], the focus of physiotherapy PFM training protocols on fast and reflexive contractions seems crucial for SUI patients. To the best of the authors' knowledge, the present study is the first one investigating a PFM protocol focusing on PFM reactivity, that is, investigating involuntary PFM contractions rather than concentrating exclusively on PFM voluntary contractions (standard physiotherapy). Should this newly developed PFM therapy protocol be shown successful, the clinical impact would be high and the application in clinical practice and therefore the benefit for SUI patients immediate. Additionally, the new therapy protocol promises to gain high acceptance by patients as it offers a higher suitability and practicability for daily use, because it has more dynamic whole body movement elements integrated than does standard SUI physiotherapy.
Trial protocol and updates: see clinicaltrials.gov.
Funding: This study is funded by the Swiss National Science Foundation (SNSF; (320030_153424/1). The granting organization Swiss National Science Foundation is not involved in the protocol and conduct of the present study.
Many thanks also to Jacqueline Buerki for proofreading the article.
LR and AK are responsible for this study. LR, HL, JPB, and AK conceived and developed the study design and drafted and revised the protocol. CL contributed to the study, especially regarding the development of the therapy protocols and is responsible for the physiotherapy interventions. LR and HL are responsible for data acquisition and analyses. HL is responsible for the coordination between the assessing gynecologists and their secretary, the physiotherapists, and the investigators. AK is responsible for patient recruitment. HL prepared the manuscript and is the corresponding author. All authors read and approved the final manuscript.
International Consultation on Incontinence Modular Questionnaire (ICIQ). ICIQ Structure Short Form. 2014. http://www.iciq.net/ICIQ-UIshortform.html. Accessed 14 Oct 2015.
International Consultation on Incontinence Modular Questionnaire (ICIQ). ICIQ LUTSquol Module. 2014. http://www.iciq.net/ICIQ.LUTSqolmodule.html. Accessed 14 Oct 2015.


When you are trying to learn how to play better baseball, you must be open to consistently learning new strategies and fundamentals. Each time you progress in the ranks, there are always expectations set and new tactics to learn. So, in essence you can always learn something. This article is here to teach you a few helpful tips to better your game.
Batting helmets need to be worn when you hit the ball. These helmets provide protection against injuries to the head. The best helmets have shields that guard the face too.
If a ball is headed straight towards you and you're being blinded by the sun, use your glove to block it. This will block out the glare from the sun and allow you to see what you are trying to catch.
To catch players off base, tag them using the ball. Speed is key to tagging people out. Improving your take-off and recoveries is key to success here.
To make sure that you do not lose your baseballs when you go to the park to practice hitting, write an identifying mark on each of them. If you use a Sharpie or other permanent marker, the wear and tear of playing ball will take longer to erase your name from the ball's surface.
To keep hitters from sitting on your fastball, it is important to develop one or two other solid pitches as well. One should be a curve ball or slider, and the other should be a change-up. To fool the batter, your arm movement should be the same for all of your pitches.
Avoid sliding into base head first. Sliding head first might seem like a good choice when it is a close play. However, these slides are dangerous. The other player is much more likely to stomp on your hands and fingers with his cleats. That can stop you from playing the rest of the season.
Are you now ready to play some baseball with your team? Taking the tips you've read about out there with you on the field will help you play a much better game. Your teammates are counting on you as well as your coach, and you just might make that play that everyone talks about after the game.


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Michel'le of 'Marriage Boot Camp' Survived an Abusive Affair with Dr Dre – Meet Her New Man
By Kagweni Micheni
https://news.amomama.com/218308-michelle-marriage-boot-camp-survived-abu.html
"Marriage Boot Camp: Hip Hop Edition" alum Michel'le has survived abusive affairs, but it seems she has found a new man. Here's what we know about him.
After surviving an abusive relationship, Michel'le Toussant of "Mariage Boot Camp: Hip Hop Edition" seems to have found herself a new man.
The singer-songwriter revealed she was dating younger man Robert "Stew" Stewart when they appeared on the reality show.
The couple met at a party where Stewart, who is a chef, was cooking for the guests. He revealed that as soon as he saw Toussant, he was captivated. However, he was unsure who she was until he heard her instantly recognizable voice.
After staring at each other throughout the night, he finally got the courage to approach her with the songstress adding,
"So to make a long story short, he get my number and he calls me about two months later, and we did this for a year."
During an interview, Toussant shared that they had chosen to make their relationship public by going on the show as they wanted to show people who they really are.
She added that she had not made great choices in the men that she had previously dated so she wanted to use the platform to see if their connection was real and strengthen their relationship.
Stewart revealed that he had struggled with her fame.
Michel'le Toussaint and Chef Robert Stewart attend the premiere of WE TV's "Marriage Boot Camp: Hip Hop Edition"| Photo: Getty Images
Earlier this year, the "Cut Throat Kitchen" star opened up about their relationship as they tried to work through their issues.
Stewart revealed that he had struggled with her fame, particularly when it comes to Toussant's fans obsessing over her past. He said,
"To no fault of my own I am not in the music industry and I mean she is older than me so a lot of that stuff that was happening I was still learning how to walk so…I have no idea what is going on with that."
Before meeting the chef, the "Marriage Boot Camp" star was linked to Dr. Dre and Death Row Records co-founder Marion "Surge" Knight. She had a son with the former and a daughter with Knight.
In an interview, she revealed that she was the victim of abuse, telling her that she was physically abused by Dre and King, who is currently incarcerated and facing murder charges.
Instagram/realjudgelynn
Judge Lynn Toler's Three-Decade Marriage Was Saved by 'Divorce Court' — inside Her Experience
Styles P Asked If Adjua Blames Him for Their Daughter's Death in a Preview of 'Marriage Boot Camp'
Michel'le Is Suge Knight's Former Wife of 6 Years and the Mother of His Daughter — Meet the Singer
While Toussant and Stewart may be in a new relationship, fans may not entirely be convinced that all is great between the pair. A lie detector test taken during the show indicated that they had lied about their feelings for each other.
While they both left the show together, viewers thought that they might be happier separately. However, their relationship continues and we wish them the very best.


US History is another popular subject taken by student as one of their SAT II subject test. US history as much popular as World History subject. Thousands of students opt for US History subject test. Normally it's a 60 minutes long exam containing 90 multiple choice questions. If you have thing for History then you must choose US history as one of your SAT II Subject.
The average score for US history is around 650+. You must check out the latest average score from college boards website. It's quite a high score compared to other subject and it simply signifies that students taking the exam are doing good. People keep asking what is a good SAT score. I will say that anything close to 700 or above is quite good to get admission to good college of your choice. However to reach close to 700, you need to study hard and from standard text books.
Barron's updated edition provides: a diagnostic test, five full-length practice tests with all questions answered and explained, and an in-depth review of U.S. history, starting with European colonization of North America and covering major topics. Subject review quizzes following every five chapters.
Continuous practice of previous papers is very much necessary to score high marks. You must make some sort of time table to study each and every subject that you are going to take in SAT II. Most of you will sure be taking more than one subject test and hence it's very important to divide time and read every subject.
We will be coming up study material PDF ebooks which will help you with extra preparation. Till then buy any one of the books and start your preparation for SAT II US history.
This entry was posted in SAT II US History Books, SAT Subject Test Books, Uncategorized and tagged How to score high makrs in US History SAT II Test, Prep Books for SAT II US History, Preparation Books for SAT Ii US History, SAT II US History Prep Books, Study Guides for SAT II World History, Study Material for SAT II US History Subject Test, Top Books for SAT II US History Test on June 19, 2014 by satyenhacks.


A Short History of the Boston Celtics Banners that symbolize basketball excellence. This article describes the evolution of the famous flags – and sets the record straight about the chronology of the franchise's retired numbers.
Note: This article was written in 2015. Since then, the Celtics retired Paul Pierce's #34 on February 11, 2018.


http://www.jenneth.info/archives/000326.html

"In my opinion, mobile phones are much easier to drop than PDAs. This is coming from first-hand experience, having dropped many, many mobile phones over the years and not that many PDAs (thank goodness!). When holding a PDA, you tend to rest it on the palm of your hand while gripping either side with your fingers - a mobile phone, on the other hand, is usually held only with your fingers, with partial support at the base of your palm, depending on the size of the phone. Recently I was unlucky enough to drop my Qtek 7070 while walking along a concrete pavement. This was when I was using a holster-style case (also made by Piel Frama) and I had just finished talking on the phone. As I was walking, I attempted to replace the phone back in the holster and as I though it was safely in, I let go of the phone. SMASH! I actually witnessed my pride and joy (at this point it was only a few days old) crash to the cement, actually bounce, with the SD card flying out underneath a car. Needless to say I was absolutely beside myself and had to be comforted by random passers-by. At that point I knew that I needed a solid leather case for my phone that provided all-over protection and that allowed me to talk on the phone while using it. That day or not long after, I contact Piel Frama about their new Qtek 7070 case."

Check out the article for photos and her thoughts on this case.
I bought the same case from Peil Frama. Its very pretty and I get a lot of ohs and ahs from friends.
However if you use the belt clip and snap the case to it, the case and phone can spin around (it uses the standard clips that most US cases use where the case snaps into the clip). Now if you have the leather flap unbuttoned and the case twirls 180s. the top of the case ends up pointing down and the phone may slide out and hurl itself to the ground, :oops: like the one did to the reviewer. They shouod have made the flap use some kind of velcro attachement so it always clings to the case, even after you have taken the phone out. I plan to modify my case by addng a thin velcro stip around the border to do just that.


we will highlight some of the challenges in creating social interaction in VR.
Both in the format of human avatar interaction and human agent interaction.
the avatar and agent we're interacting with.
the representation of the user themselves, the self-avatar.
We'll explain how to implement it and why it is important.
>> We'll also talk you through how to test your social VR interaction experience.
What kind of questions to ask and how to ask them.
we need to learn a bit more about how it works in real life.


This is the first and only shop we have been to since California became a recreational state. We come out to our property in Joshua Tree for 4-6 months yearly and HOTN is always our first stop upon arrival from Maryland. Love this place, the staff is always super pleasant and knowledgeable. We keep coming back as we have no reason to go elsewhere!
Love this place! Been going here regularly for months. They are awesome, I can't go anywhere else now!
Not many of my preferred brands here. Service was fine.
I love it. Helpful staff and great product.
Everyone is super helpful, and informative. Anything you need to know they can answer..they have great sales, and i love their rewards points. Very happy to have found!


Excited to be asked to participate in an amazing show hosted by Dominic Bonuccelli!
Back in the studio, working hard. Taking my inspiration from the music legends who have come before.
Background image on this website is by Dominic Arizona Bonuccelli! Click the picture below to be directed to his website!


The exhibition Apocalypse, A Cycle of Bela Csikos Sessia is opened in the Studio Josip Račić. It will be on view until December 20, 2009.
The Apocalypse or Revelations is the last book of the Bible; according to tradition, the author is considered to be St John. The text is highly complicated in its symbolism, for this is John's vision of the temptations that the Church and the faithful must go through, as well as the last battle of Christ against Evil (Satan, Antichrist), which will, naturally, conclude with the victory of God before the last day.
In a literary and symbolic sense, the motifs from Apocalypse are close to the poetics of Bela Csikos Sessia, and so this segment of his work should be seen along the lines of Symbolism rather than within the context of religious painting, which did not interest Csikos all that much, particularly not in the sense of the immediate transfer of religious messages.
The Modern Gallery holds part of the Csikos Apocalypse cycle; these are drawings done in pastels, Indian ink, white chalk and charcoal in 1903. With complementary colours, with intermittent contour lines and the dynamics of his composition, Csikos with expressive artistic resources shows all the drama of the literary model. For this cycle, he made many preparations: photographs, sketches in his sketchbooks, studies in pencil and variants in pastel, charcoal, Indian ink and white chalk.
The cycle is also interesting because of Csikos's iconography that does not literally take over the models in which the Apocalypse is presented in Western iconography as a whole. The selection of scenes, for example, is in line with his conception of the symbolic messages of the story called Apocalypse.
The author of the exhibition and the catalogue is Dajana Vlaisavljević, senior curator of the Modern Gallery.


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Силва () — населённый пункт в Смилтенском крае Латвии. Входит в состав Лаункалнской волости. Нахоидится у юго-восточной окраины города Смилтене на региональной автодороге  (Смилтене — Велена — Гулбене). По данным на 2015 год, в населённом пункте проживал 401 человек.

История
Населённый пункт образовался как центр деревообработки (в переводе с латыни силва — дерево).

В советское время населённый пункт входил в состав Лаункалнского сельсовета Валкского района. В селе располагался Смилтенский леспромхоз.

В настоящее время в селе действует несколько деревообрабатывающих компаний.

Примечания

Населённые пункты Смилтенского края
Лаункалнская волость

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