\subsection{Variation during the slow decay}
\label{sec:small-section-slow-variation}

The interval of slow decay is long enough for the logarithmic
wall derivatives to grow with time. Their effect on the shear
derivative is nevertheless weighted by the small wall
amplitudes. Two time moments of those amplitudes control this
interaction: the zeroth moment bounds direct growth, and the
first moment bounds the contribution of the growing wall
derivatives.

We consider the family in
Theorem~\ref{thm:small-section-slow-decay}, now with
$\zeta,\nu>0$. Write
\[
 z(t,\rho)=\phi_{t_0+t}\Xi_h(\rho,\zeta,\nu)
          =(p,s,n,x,y)(t,\rho),\qquad 0\leq t\leq T.
\]
Derivatives in $\rho$ keep $h,a,q,D,\zeta,\nu$ and $t$ fixed;
in particular, $t_0$ and $T$ are fixed. We use $\Dt$ for
differentiation in normalized time. The walls remain positive,
so their logarithmic derivatives are defined. Set
\begin{equation}
 S=\partial_\rho s,\qquad
 \ell=(\partial_\rho\log n,\partial_\rho\log x,
                         \partial_\rho\log y),\qquad
 L=\|\ell\|_\infty.
 \label{eq:slow-variation-variables}
\end{equation}
Set
\begin{equation}
 b=n^2+x^2+y^2+n(x+y),\qquad
 J_0=\int_0^T b(t)\dd t,\qquad
 J_1=\int_0^T tb(t)\dd t.
 \label{eq:slow-variation-moments}
\end{equation}
We suppress $\rho$ when discussing a fixed orbit.

Let $\delta=\partial_\rho$, $\chi=-p$, and
$H=-x^2+y^2+nx-ny$. Differentiating the constraint and the
two curvature polynomials gives
\begin{align}
 \delta p&=\frac{sS+\delta Q/2}{\chi},
       \label{eq:slow-variation-constraint}\\
 \delta Q&=2\bigl((n-x-y)\delta n+(x-n-y)\delta x
                               +(y-n-x)\delta y\bigr),\notag\\
 \delta H&=(x-y)\delta n+(n-2x)\delta x
                               +(2y-n)\delta y.\notag
\end{align}
The constraint identity holds for nearby chart parameters
and flow times, so its differential is the tangent
constraint for the actual flow.
Differentiating \eqref{eq:normalized-flow}, and using
\eqref{eq:slow-variation-constraint} to eliminate
$\delta p$, yields
\begin{equation}
 \begin{aligned}
 \Dt S&=QS+s\delta Q+\sqrt3\,\delta H,\\
 \Dt\ell_n&=(1+\chi^{-1})\delta Q+2s\chi^{-1}S,\\
 \Dt\ell_x&=(1-(2\chi)^{-1})\delta Q
                            +(-s\chi^{-1}+\sqrt3)S,\\
 \Dt\ell_y&=(1-(2\chi)^{-1})\delta Q
                            +(-s\chi^{-1}-\sqrt3)S.
 \end{aligned}
 \label{eq:slow-variation-equations}
\end{equation}
In the last three equations the wall rate cancels when
the quotient $\delta N_i/N_i$ is differentiated in time.

By definition, $|\delta n|\leq nL$, $|\delta x|\leq xL$,
and $|\delta y|\leq yL$. Nonnegativity of the walls and
$(n+x+y)^2\leq2b$ therefore imply
\begin{equation}
 |Q|\leq2b,\qquad |\delta Q|\leq4bL,
                       \qquad |\delta H|\leq2bL.
 \label{eq:slow-variation-polynomial-bounds}
\end{equation}
For example, each coefficient inside the parentheses in
$\delta Q$ has absolute value at most $n+x+y$.
The bound for $\delta H$ follows by bounding its three
terms by $(x+y)nL$, $(n+2x)xL$, and $(2y+n)yL$.

On the region \eqref{eq:slow-decay-region}, the constraint
gives $p\leq-15/16$ and $|s|\leq1/8$.
Consequently $\chi^{-1}\leq5/4$ and
$|s|\chi^{-1}\leq1/4$.
The coefficients of $\delta Q$ in the last three rows of
\eqref{eq:slow-variation-equations} have absolute value
at most $9/4$, and the coefficients of $S$ at most $2$.
Together with $\sqrt3\leq7/4$, this proves
\begin{equation}
 |\Dt S|\leq2b|S|+4bL,\qquad
 \|\Dt\ell\|_\infty\leq2|S|+9bL.
 \label{eq:slow-variation-inequalities}
\end{equation}

\begin{lemma}
\label{lem:slow-variation-moments}
Suppose $T\geq0$ and an actual orbit remains in
\eqref{eq:slow-decay-region} for $0\leq t\leq T$, with
$0<a\leq1/256$. Write
$f_0=n(0)^2+n(0)(x(0)+y(0))$. Then
\begin{align}
 J_0&\leq\frac23f_0+\frac{y(0)^2}{a}
                    +\frac{x(0)^2}{12a}e^{12aT},
       \label{eq:slow-variation-moment-zero}\\
 J_1&\leq\frac49f_0+\frac{y(0)^2}{a^2}
                    +\frac{Tx(0)^2}{12a}e^{12aT}.
       \label{eq:slow-variation-moment-one}
\end{align}
If also $T=(D/a)\log(1/a)$, $D\geq0$, and
$x(0)e^{6aT}\leq a^2$, then
\begin{equation}
 J_0\leq4a,\qquad J_1\leq\frac1{256}+(2+D)a.
 \label{eq:slow-variation-linear-moments}
\end{equation}
\end{lemma}

\begin{proof}
The wall rates \eqref{eq:slow-decay-wall-rates} show that
\[
 f=n^2+n(x+y)\quad\hbox{satisfies}\quad
 \Dt f\leq-\frac32 f.
\]
Indeed, the rates of $n^2$, $nx$, and $ny$ are respectively
$2\lambda_1$, $\lambda_1+\lambda_2$, and
$\lambda_1+\lambda_3$, all at most $-3/2$.
Also $\frac{d}{dt}y^2\leq-ay^2$.

For any nonnegative differentiable $g$ with
$\Dt g\leq-cg$ and $c>0$, integration gives
\[
 \int_0^T g\dd t\leq\frac{g(0)}c,
 \qquad
 \int_0^T tg\dd t\leq\frac{g(0)}{c^2}.
\]
For the second inequality, differentiate
$(1+ct)g(t)$ to obtain an upper bound $-c^2tg(t)$,
and use its nonnegative endpoint value.
Apply these inequalities to $f$ and $y^2$.
The remaining square satisfies
$x(t)^2\leq x(0)^2e^{12at}$, so
\[
 \int_0^T x(t)^2\dd t
       \leq\frac{x(0)^2}{12a}e^{12aT},\qquad
 \int_0^T tx(t)^2\dd t
       \leq T\frac{x(0)^2}{12a}e^{12aT}.
\]
Since $b=f+y^2+x^2$, these are
\eqref{eq:slow-variation-moment-zero} and
\eqref{eq:slow-variation-moment-one}.

For the simpler bounds, the initial region gives
\[
 f_0\leq3a^3,\qquad
 \frac{y(0)^2}{a}\leq\frac a{256},\qquad
 \frac{y(0)^2}{a^2}\leq\frac1{256}.
\]
The additional growing-wall assumption gives
\[
 \frac{x(0)^2}{12a}e^{12aT}\leq\frac{a^3}{12}.
\]
Moreover, $a\log(1/a)\leq1$, and hence
\[
 T\frac{a^3}{12}
       =\frac{Da}{12}\,a\log(1/a)\leq Da.
\]
Insert these bounds in the two moment estimates and use
$a^3\leq a$. This proves
\eqref{eq:slow-variation-linear-moments}.
\end{proof}

The following estimate explains why the first time moment
is sufficient. It applies directly to
\eqref{eq:slow-variation-inequalities}.

\begin{lemma}
\label{lem:slow-variation-feedback}
Let $T\geq0$, let $S:[0,T]\to\R$ and
$\ell:[0,T]\to\R^3$ be differentiable, and let
$b\geq0$ be continuous.
Suppose \eqref{eq:slow-variation-inequalities} holds,
with $L=\|\ell\|_\infty$. Put $L_0=L(0)$ and define
$J_0,J_1$ by \eqref{eq:slow-variation-moments}. Then,
with
\begin{equation}
 c=e^{9J_0},\qquad
 B=(|S(0)|+4cL_0J_0)e^{2J_0+8cJ_1},
 \label{eq:slow-variation-general-bound}
\end{equation}
one has
\begin{equation}
 |S(t)|\leq B,\qquad L(t)\leq c(L_0+2tB)
       \quad(0\leq t\leq T).
 \label{eq:slow-variation-general-conclusion}
\end{equation}
In particular, if $|S(0)|\leq2$ and
\begin{equation}
 9J_0\leq\log2,\qquad
 2J_0+16J_1\leq\frac12,\qquad L_0J_0\leq\frac1{16},
 \label{eq:slow-variation-small-moments}
\end{equation}
then $|S(t)|\leq5$ and $L(t)\leq2L_0+20t$.
\end{lemma}

\begin{proof}
We use the integral norm comparison
\begin{equation}
 \|u(t)\|\leq
 \left(\|u(0)\|+\int_0^t e(v)\dd v\right)
             \exp\left(\int_0^t k(v)\dd v\right)
 \label{eq:slow-variation-integral-comparison}
\end{equation}
when $\|\Dt u\|\leq k\|u\|+e$ and $k,e$ are
continuous and nonnegative. To verify it also at zeros
of $u$, compare its norm with
\[
 \left(\|u(0)\|+\delta+\int_0^t e(v)\dd v\right)
       \exp\left(\int_0^t k(v)\dd v+\delta t\right),
       \qquad \delta>0.
\]
At any contact this function has strictly larger derivative
than the permitted growth of the norm. Norm comparison
followed by $\delta\downarrow0$ proves
\eqref{eq:slow-variation-integral-comparison}.

Define a nondecreasing envelope
\[
 M(t)=|S(0)|+\int_0^t
             \bigl(2b(v)|S(v)|+4b(v)L(v)\bigr)\dd v.
\]
Then $|S(t)|\leq M(t)$. Apply
\eqref{eq:slow-variation-integral-comparison} to $\ell$.
For $0\leq v\leq t$, $|S(v)|\leq M(t)$, so
\[
 L(t)\leq c\left(L_0+2\int_0^t|S(v)|\dd v\right)
           \leq c(L_0+2tM(t)).
\]
It follows that
\[
 \Dt M(t)\leq(2+8ct)b(t)M(t)+4cL_0b(t).
\]
A second application of
\eqref{eq:slow-variation-integral-comparison}, with
$\int_0^T(2+8ct)b(t)\dd t=2J_0+8cJ_1$, gives
$M(t)\leq B$. Substituting this into the preceding
bound for $L$ proves
\eqref{eq:slow-variation-general-conclusion}.

Under \eqref{eq:slow-variation-small-moments},
$c\leq2$ and $e^{2J_0+8cJ_1}\leq e^{1/2}<2$.
Also $|S(0)|+4cL_0J_0\leq5/2$. Thus $B\leq5$,
which gives the last assertion.
\end{proof}

\begin{theorem}
\label{thm:small-section-slow-variation}
There are constants $C\geq32$ and $h_*>0$ with the
following property. For every $0<h\leq h_*$, $q>2$,
$D\geq0$, and $P>6D+2$, there is $a_*>0$ such that
for $0<a\leq a_*$, uniformly over
\begin{equation}
 \begin{gathered}
 0\leq\varepsilon\leq\frac1{128},\qquad
 0<\zeta\leq\min\{a^P,2\varepsilon a\},\\
 0<\nu\leq2\varepsilon a,
                  \qquad a\leq\rho\leq2a,
 \end{gathered}
 \label{eq:slow-variation-label-domain}
\end{equation}
the actual flow and the derivatives in
\eqref{eq:slow-variation-variables} satisfy
\begin{equation}
 \begin{split}
 L_0&\leq2+4Cq|\log a|,\\
 |\partial_\rho s(t,\rho)|&\leq5,\\
 L(t)&\leq2L_0+20t
             \leq4+8Cq|\log a|+20t
             \quad(0\leq t\leq T).
 \end{split}
 \label{eq:slow-variation-actual-bounds}
\end{equation}
In particular, for each physical wall,
\begin{equation}
 |\partial_\rho N_i(t,\rho)|
       \leq N_i(t,\rho)(2L_0+20t).
 \label{eq:slow-variation-relative-walls}
\end{equation}
The moments satisfy both
\eqref{eq:slow-variation-linear-moments} and
\eqref{eq:slow-variation-small-moments}.
\end{theorem}

\begin{proof}
Take the constant $C$ in the initial transport estimates
large enough that $C\geq32$, and decrease $h_*$ to meet
the hypotheses of
Theorem~\ref{thm:small-section-slow-decay} and
Proposition~\ref{prop:small-section-core-coordinates}.
At $t=0$ in the present coordinates,
\eqref{eq:projection-cutoff-bounds} gives
\[
 |S(0)|\leq2,\qquad
 |\ell_n(0)|\leq2,\qquad
 |\ell_x(0)|,|\ell_y(0)|\leq8Ct_0
                                  =4Cq|\log a|.
\]
Here $a\leq1$ and $t_0=(q/2)\log(1/a)$.
This proves the claimed initial bound for $L_0$.

Theorem~\ref{thm:small-section-slow-decay} keeps the actual orbit in
\eqref{eq:slow-decay-region} throughout $[0,T]$ and
gives $x(0)e^{6aT}<a^2/40$.
Lemma~\ref{lem:slow-variation-moments} therefore yields
\eqref{eq:slow-variation-linear-moments}.
Shrink $a_*$ so that, in addition to the preceding scale
conditions,
\begin{equation*}
 \begin{split}
 36a&\leq\log2,\qquad
       (40+16D)a\leq\frac14,\\
 8a+16Cq|a\log a|&\leq\frac1{16}
       \qquad(0<a\leq a_*).
 \end{split}
\end{equation*}
Such a choice is possible because $a|\log a|\to0$.
The moment bounds give
\[
 2J_0+16J_1\leq\frac1{16}+(40+16D)a\leq\frac12,
\]
and
\[
 L_0J_0\leq(2+4Cq|\log a|)4a
              =8a+16Cq|a\log a|\leq\frac1{16}.
\]
The remaining inequality $9J_0\leq\log2$ follows from
$J_0\leq4a$.

The ordinary flow derivatives satisfy
\eqref{eq:slow-variation-inequalities}, so
Lemma~\ref{lem:slow-variation-feedback} proves
\eqref{eq:slow-variation-actual-bounds}.
Multiplying the logarithmic bounds by the positive walls
proves \eqref{eq:slow-variation-relative-walls}.
All scale choices precede $\varepsilon,\zeta,\nu,\rho$,
which gives the stated uniformity.
\end{proof}

The shear response remains bounded while the logarithmic
wall response grows at most linearly with elapsed time.
For the radial phase, we next retain the cancellation
between the shear and weak-wall equations before
integrating their response.