\section{A moving comparison family on a fixed Cartesian grid} \label{sec:moving-approximation} \subsection{The retained coarse space} For $h=1/M$, let $\mathcal T_h$ be the uniform square grid on $\Omega=(0,1)^2$. Let $S_h\subset H^2_0(\Omega)$ be its clamped bicubic Hermite space. On each cell the polynomials belong to $\mathbb Q_3$, and the nodal degrees of freedom are the value, $\partial_x$, $\partial_y$, and $\partial_{xy}$. Shared degrees of freedom give $C^1$ conformity; the boundary degrees of freedom vanish. This is the Cartesian Bogner--Fox--Schmit construction \citep{bfs2020}. If a coarse cell $K$ is divided into four equal squares, let $B_h(K)$ be the space of $C^1$ piecewise bicubics clamped on $\partial K$, extended by zero outside $K$. Its four degrees of freedom are the Hermite data at the center of $K$. For an arbitrary set of enriched cells $\mathcal A$, define \begin{equation}\label{eq:moving-cartesian-space} S_{h,\mathcal A}=S_h+\sum_{K\in\mathcal A}B_h(K), \qquad X_{h,\mathcal A}=\curl S_{h,\mathcal A}, \qquad \curl\psi=(\partial_y\psi,-\partial_x\psi). \end{equation} Thus $X_{h,\mathcal A}\subset U$, with exact distributional incompressibility. Every configuration contains the full coarse space $\curl S_h$. The construction specifies a conforming local enrichment; it is not identified with all hanging-node Cartesian implementations. Write $I_h$ for the coarse Hermite interpolant. The estimates needed below are the standard fixed-grid bounds \begin{equation}\label{eq:moving-hermite-estimates} \norm{I_h\varphi-\varphi}_{H^j(\Omega)} \le C h^{4-j}\norm\varphi_{H^4(\Omega)}, \qquad j=1,2, \qquad \varphi\in H^4(\Omega)\cap H^2_0(\Omega). \end{equation} The mixed-derivative nodal values are well defined at this regularity. No $H^3$ interpolation assertion involving those point values is used. For each time $t$, let $Z_{h,\mathcal A,\eps,\mathcal P}(t)$ denote the subspace of $X_{h,\mathcal A}$ whose regularized moments on $\Gamma(t)$ vanish for every multiplier in the panel space $\Lambda_{\mathcal P}(t)$. All moments in this definition are exact integrals. Let $\bar\eps$ bound every regularization width in a run, and define \begin{align} D_h(t)&=\bigcup\{K\in\mathcal T_h: K\cap\{x:\dist(x,\Gamma(t))\le\bar\eps\}\ne\varnothing\}, \label{eq:moving-collar-cells}\\ K_h(t)&=\{v_h\in\curl S_h:v_h=0\text{ on }D_h(t)\}. \label{eq:moving-common-core} \end{align} Compact support of the regularizing kernel implies \begin{equation}\label{eq:moving-common-inclusion} K_h(t)\subset V(t)\cap Z_{h,\mathcal A,\eps,\mathcal P}(t) \quad\text{for every }\mathcal A,\mathcal P, \quad 0<\eps\le\bar\eps. \end{equation} Only the velocity is zero on these cells. The streamfunction may take a nonzero constant on the immersed collar, retaining the circulation mode of the multiply connected fluid region. The spaces $K_h(t)$ depend on time. They are not asserted to have a useful fixed intersection over the entire trajectory. \subsection{Regularity and the moving streamfunction trace} For a piecewise smooth field, define \[ \mathcal M_t(v)=\norm v_{W^{3,\infty}(B(t))} +\norm v_{W^{3,\infty}(F(t))}. \] The quantitative analysis assumes \begin{equation}\label{eq:moving-recovery-regularity} \begin{gathered} w\in H^1(0,T;L^2(\Omega)^2),\qquad w(t)\in V(t),\\ M_0=\esssup_{t\in(0,T)}\mathcal M_t(w(t))<\infty, \qquad m_1(t)=\mathcal M_t(\partial_t w(t))\in L^2(0,T). \end{gathered} \end{equation} Here $\partial_t$ is the Eulerian derivative at a fixed spatial point. Its restrictions to the two phases are meant in the Sobolev sense. It is not assumed to belong to $V(t)$ or even to global $H^1(\Omega)^2$. A moving interface generally produces a jump in this derivative. In particular, piecewise spatial regularity of a material derivative must not be substituted for the Eulerian assumption in \eqref{eq:moving-recovery-regularity} without accounting for the additional spatial derivative in the transport term. There is a unique normalized clamped streamfunction $\psi(t)$ with $\curl\psi(t)=w(t)$. On the square it can be written as \begin{equation}\label{eq:moving-streamfunction} \psi(t,x,y)=\int_0^y w_1(t,x,s)\,ds. \end{equation} The divergence constraint and outer zero trace give $\partial_x\psi=-w_2$, $\partial_y\psi=w_1$, and $\psi\in H^2_0(\Omega)$. The same identities after weak time differentiation show $\psi\in H^1(0,T;H^1(\Omega))$ and $\nabla\psi_t=(-w_{2,t},w_{1,t})$. The streamfunction and its time derivative are separately $W^{4,\infty}$ in the two phases, with norms bounded by $C\mathcal M_t(w)$ and $C m_1(t)$, respectively. Because $\nabla\psi=0$ on the connected interface, there is a scalar $c_\psi(t)$ such that $\psi|_{\Gamma(t)}=c_\psi(t)$. This scalar has an $H^1(0,T)$ representative and satisfies \begin{equation}\label{eq:moving-constant-derivative} c_\psi'(t)=\psi_t|_{\Gamma(t)}. \end{equation} To justify the moving trace derivative, one may pull back by a smooth fixed-box flow equal to the prescribed rigid motion near the body. Such a flow is generated by the compactly supported solenoidal lift constructed in Lemma~\ref{lem:moving-lift}. Composition of $\psi\in H^1_tH^1_x\cap L^\infty_tH^2_x$ with this flow has weak derivative $(\psi_t+g\cdot\nabla\psi)$ composed with the flow, in $L^2_tH^1_x$. This chain rule follows first for smooth functions and then by approximation in $H^1_tH^1_x\cap L^2_tH^2_x$; the $L^\infty_tH^2_x$ bound is used only for uniform estimates. The continuous $H^1$ trace map on the fixed interface commutes with that weak derivative. Since $\nabla\psi|_{\Gamma(t)}=0$, the transport term has zero trace, which proves \eqref{eq:moving-constant-derivative}. Let $d(t,x)$ be signed distance to $\Gamma(t)$ in a uniform tubular neighborhood. The smooth collision-free rigid trajectory gives a time-independent tubular radius and uniformly bounded spatial derivatives through the order used below. Near the interface, \begin{equation}\label{eq:moving-distance-transport} d_t+b(t,x)\cdot\nabla d=0. \end{equation} Indeed, rigid transformations preserve signed distance. In particular $\norm{d_t}_{W^{4,\infty}}$ is uniformly bounded. With $q=\psi-c_\psi(t)$, normal integration on either side of the interface gives \begin{equation}\label{eq:moving-quadratic-vanishing} \begin{aligned} |q|&\le C|d|^2\mathcal M_t(w),& |\nabla q|&\le C|d|\mathcal M_t(w),\\ |q_t|&\le C|d|m_1(t),& |\nabla q_t|&\le C m_1(t). \end{aligned} \end{equation} The second line follows from \eqref{eq:moving-constant-derivative}; it does not require $w_t$ to vanish on the moving interface. \subsection{A constructive approximation with an Eulerian derivative bound} \begin{lemma}[Moving common recovery]\label{lem:moving-recovery}\label{lem:collar} Assume \eqref{eq:moving-recovery-regularity}. For $h+\bar\eps$ sufficiently small, put $\delta=4(\bar\eps+\sqrt2h)$. There is an explicitly constructed trajectory $z_h(t)=Q_h(t)w(t)\in K_h(t)$ such that \begin{align} \esssup_t\norm{z_h(t)-w(t)}_U &\le C\sqrt\delta\,M_0,\label{eq:collar-rate}\\ \esssup_t\norm{z_h(t)-w(t)}_0 &\le C\delta^{3/2}M_0,\label{eq:moving-recovery-L2}\\ \norm{\partial_t(z_h-w)}_{L^2(0,T;L^2)} &\le C\sqrt\delta\, \bigl(\norm{m_1}_{L^2(0,T)}+\sqrt T M_0\bigr). \label{eq:collar-time} \end{align} The constant is independent of all enrichment sets, panel partitions, individual regularization widths, and their numbers of changes. It may depend on the prescribed trajectory, its clearance from the outer boundary, and the fixed geometry. The trajectory $z_h$ belongs to $H^1(0,T;\curl S_h)$. \end{lemma} \begin{proof} Choose $\chi\in C_c^\infty(\mathbb R)$ equal to one for $|s|\le1$ and zero for $|s|\ge2$. For $2\delta$ smaller than the uniform tubular radius and the outer clearance, define \begin{equation}\label{eq:moving-flattening} \psi_\delta(t,x)=\psi(t,x)-\chi(d(t,x)/\delta)q(t,x) \end{equation} in the tube, and set $\psi_\delta=\psi$ elsewhere. It equals $c_\psi(t)$ throughout $|d|\le\delta$, preserves the outer clamped condition, and removes all original interface derivative jumps. Thus $\psi_\delta(t)\in H^4(\Omega)\cap H^2_0(\Omega)$. The support of $\psi_\delta-\psi$ has area $O(\delta)$. By \eqref{eq:moving-quadratic-vanishing}, its derivatives of orders zero, one, and two are bounded by $C\delta^2\mathcal M_t(w)$, $C\delta\mathcal M_t(w)$, and $C\mathcal M_t(w)$. Fourth derivatives on the transition strip are bounded by $C\delta^{-2}\mathcal M_t(w)$. Consequently, \begin{equation}\label{eq:moving-flattening-spatial-bounds} \begin{aligned} \norm{\psi_\delta-\psi}_{H^1}&\le C\delta^{3/2}\mathcal M_t(w),\\ \norm{\psi_\delta-\psi}_{H^2}&\le C\sqrt\delta\,\mathcal M_t(w),\\ \norm{\psi_\delta}_{H^4}&\le C(1+\delta^{-3/2})\mathcal M_t(w). \end{aligned} \end{equation} For example, four cutoff derivatives multiply $q=O(d^2)$, and three cutoff derivatives multiply $\nabla q=O(d)$; both contributions have size at most $C\delta^{-2}\mathcal M_t(w)$ in the strip. The Eulerian time derivative must include the moving-cutoff term: \begin{equation}\label{eq:moving-flattening-time-formula} \partial_t(\psi_\delta-\psi) =-\chi(d/\delta)q_t -\delta^{-1}\chi'(d/\delta)d_tq. \end{equation} The first spatial derivatives of this expression are bounded by $C(m_1(t)+M_0)$ on a strip of area $O(\delta)$, using \eqref{eq:moving-quadratic-vanishing} and the uniform distance bounds. Its zeroth derivative is bounded by $C\delta(m_1(t)+M_0)$. Moreover, the fourth spatial derivatives of $\psi_{\delta,t}$ are bounded by $C\delta^{-3}(m_1(t)+M_0)$ on the transition strip. The worst terms are four spatial cutoff derivatives times $q_t=O(\delta)$, and the fourth derivative of $\delta^{-1}\chi'(d/\delta)d_tq$, whose most singular factor has size $\delta^{-5}q=O(\delta^{-3})$. Away from the strip, $\psi_{\delta,t}$ is either $\psi_t$ or the spatial constant $c_\psi'$. Therefore \begin{equation}\label{eq:moving-flattening-time-bounds} \begin{aligned} \norm{\partial_t(\psi_\delta-\psi)}_{H^1} &\le C\sqrt\delta\,(m_1(t)+M_0),\\ \norm{\psi_{\delta,t}}_{H^4} &\le C(1+\delta^{-5/2})(m_1(t)+M_0). \end{aligned} \end{equation} These identities hold as weak time derivatives as well: the original moving-interface jumps have been removed in an open tube, and the right-hand side of \eqref{eq:moving-flattening-time-formula} is in $L^2_tH^4_x$ after adding $\psi_t$. Define \begin{equation}\label{eq:moving-Q-definition} z_h(t)=Q_h(t)w(t)=\curl I_h\psi_\delta(t). \end{equation} Every coarse cell meeting the $\bar\eps$ tube is contained in the $\bar\eps+\sqrt2h$ tube. The latter lies strictly inside $|d|<\delta$, where $\psi_\delta$ is constant. Exact constant reproduction by $I_h$ gives $z_h(t)\in K_h(t)$. Apply \eqref{eq:moving-hermite-estimates} and \eqref{eq:moving-flattening-spatial-bounds}. The velocity $H^1$ and $L^2$ errors are bounded, respectively, by \[ C\bigl(\sqrt\delta+h^2(1+\delta^{-3/2})\bigr)M_0, \qquad C\bigl(\delta^{3/2}+h^3(1+\delta^{-3/2})\bigr)M_0. \] Since $\delta\ge4\sqrt2h$, these imply \eqref{eq:collar-rate}--\eqref{eq:moving-recovery-L2}. Only the fixed interpolant $I_h$ commutes with time differentiation. Using its $H^1$ estimate on $\psi_{\delta,t}$ gives \[ \norm{\partial_t(z_h-w)}_0 \le C\bigl(\sqrt\delta+h^3(1+\delta^{-5/2})\bigr) (m_1(t)+M_0) \le C\sqrt\delta\,(m_1(t)+M_0). \] Integration proves \eqref{eq:collar-time}. The bounds also give $\psi_\delta\in H^1_tH^4_x$ for fixed $h,\bar\eps$, hence the asserted finite-dimensional $H^1$ trajectory. Changes in the set of cells in \eqref{eq:moving-collar-cells} do not introduce jumps in this trajectory. \end{proof} In particular, the estimate is for $\partial_t(Q_h(t)w(t)-w(t))$. The generally false identity $\partial_t(Q_h(t)w(t))=Q_h(t)\partial_t w(t)$ is not used. Nor is convergence of this derivative in the $U$ norm asserted. \begin{lemma}[Time modulus across a moving interface] \label{lem:moving-time-modulus} Under \eqref{eq:moving-recovery-regularity}, $w$ has a continuous $U$-valued representative and, for $0\le s