Effective finite computation detected by a fixed particle

Fix a computable viscosity $\nu>0$.  From a deterministic one-tape
machine, with head moves in $\{-1,0,1\}$, and a finite input, one can
effectively construct a smooth force on $\mathbb T_1^3$ such that:
\begin{enumerate}[label=(\roman*),itemsep=2pt]
\item The force is divergence free, has zero spatial mean, and is
one-periodic from time zero.  Equation~\eqref{sv:NS} has a global
smooth solution with pressure zero, unique in the stated classical class.
\item The velocity is one-periodic, vanishes near every integer time,
has zero instantaneous self-advection, and has uniformly bounded kinetic
energy.  Every mixed derivative of force and velocity is globally bounded.
\item For the fixed label $a_*=(1/4,1/2,1/4)$ and fixed open strip
$O=\{1/2<x_1<7/8\}$, the material trajectory enters $O$ at a finite
real time exactly when the machine halts, including an initially halted
machine.
\end{enumerate}
The construction may use one Cartesian velocity component at a time.
For arbitrary fixed real $\nu>0$ the same formulas hold and are
effective relative to that viscosity.

Let $P_i,Q_i\subset[2,3]^2$, $1\le i\le N$, be closed rectangles
with rational vertices and positive side lengths.  Suppose the sources are pairwise
positively separated and the targets are pairwise positively separated.
Let $F_i:P_i\to Q_i$ be the center-to-center affine map with any
positive rational diagonal linear part, with $F_i(P_i)=Q_i$.  For each computable $\nu>0$
there is an effective smooth mean-zero solenoidal force on
$\mathbb T_{10}^3$, one-periodic from time zero, whose unique solution
in the classical class has pressure zero and period map
\[
 (y,2)\longmapsto(F_i(y),2)\qquad(y\in P_i).
\]
Its velocity is zero near integer times and has zero self-advection;
all mixed derivatives of force and velocity are bounded.  No condition
is imposed on source--target intersections.  For $N=0$ use zero.

Fix a positive computable viscosity $\nu$. From a deterministic Turing machine $M$ and a
finite input $w$ one can effectively construct a smooth force on
$[0,\infty)\times\R^3$, supported for all times in the fixed compact set
\[
 K=[-2,1]\times[-1,2]\times[-1,1],
\]
such that, for every pair of nonnegative integers $J,l$ and spatial
multi-index $\alpha$,
\begin{equation}
 \sup_{t\geq0,\,x\in\R^3}
 (1+t)^J\bigl|\partial_t^l\partial_x^\alpha f(t,x)\bigr|<\infty.
 
\end{equation}
The Navier--Stokes equation with this force and zero initial velocity has
an explicit global smooth solution, unique in the finite-interval class \eqref{eq:energy-class}.  Its material trajectory from
the origin satisfies
\begin{equation}
 M\text{ halts on }w
 \quad\Longleftrightarrow\quad
 X_1(t,0)<-1\text{ for some }t\geq0.
 
\end{equation}
The velocity has the same fixed support and rapid mixed-derivative bounds.
The pressure is zero. The force is a general body force. Because $K$ is fixed,
the same bounds hold with $(1+t+|x|)^J$ in place of $(1+t)^J$.

Fix a positive computable viscosity. From a deterministic Turing machine
and a finite input one can effectively
construct a smooth force $f$ on $[0,\infty)\times\mathbb R^3$, with
spatial support contained in a fixed compact set independent of time,
such that for every $J,l\geq0$ and spatial multi-index $\alpha$,
\begin{equation}
 \sup_{t\geq0,\,X\in\mathbb R^3}
 (1+t+|X|)^J
 |\partial_t^l\partial_X^\alpha f(t,X)|<\infty.
 
\end{equation}
The solution with zero initial velocity is explicit, global, and unique
in the class \eqref{eq:energy-class}, with pressure zero.  Its velocity satisfies
the same support and derivative estimates.  The material particle
initially at $(-1,0,0)$ reaches $X_1>0$ if and only if the machine halts.

Fix a computable viscosity $\nu>0$.  A finite deterministic machine and
a finite input effectively determine a smooth force $g$ on the unit
flat torus $\T^3=(\R/\mathbb Z)^3$ with zero spatial mean.  The
Navier--Stokes equations with force $g$ and zero initial velocity have
a smooth solution $(U,0)$ with zero spatial mean.  Its particle from
\[
 a_*=(1/4,1/2,1/2)
\]
enters the fixed open set
\[
 O=\{x\in\T^3:1/2<x_1<1\pmod 1\}
\]
at some finite time if and only if the machine halts on that input.
For every pair of nonnegative integers $r,J$ and every spatial
multi-index $\alpha$, there is an effective constant $C_{r,\alpha,J}$
such that
\begin{equation}
 \|\partial_t^r\partial_x^\alpha U(t)\|_\infty
 +\|\partial_t^r\partial_x^\alpha g(t)\|_\infty
 \le C_{r,\alpha,J}(1+t)^{-J},\qquad t\ge0.
\end{equation}
There is also an effective smooth mean-zero solenoidal force $f$
with the same bounds and exactly the same velocity and material event.
It is $f=g-\nabla\phi$, with pressure $-\phi$, where $\phi$ is the
mean-zero periodic solution of $\Delta\phi=\operatorname{div}g$.

For either force, uniqueness holds among periodic classical solutions
for which the velocity, its time derivative and its spatial derivatives
through order two, and the pressure and its spatial gradient, are
continuous on every $[0,T]\times\T^3$.  Pressure is normalized to have
zero spatial mean.

For each positive computable $\nu$, a machine and a finite input
effectively determine each of the following two forces.
\begin{enumerate}
\item On $\T^3$, a smooth force has a unique global smooth solution
in the torus class above, with pressure zero, for which
\eqref{eq:torus-event} holds exactly when the machine halts.
The force is supported in $K\times\T$ for a compact
$K\subset(0,1)^2$ independent of time. Its derivatives are bounded on every finite time
interval; their bounds may grow with that interval.
\item On $\R^2\times\T$, a smooth force and all its mixed derivatives
are globally bounded. On each finite time interval its horizontal
support is compact. It has a unique global smooth solution in
\eqref{eq:comparison-class}, with pressure zero and integrable
nonnegative third component. Event~\eqref{eq:plane-event} holds exactly
when the machine halts.
\end{enumerate}
The sets and thresholds in both observations are independent of the
machine and input. The compact set $K$ in the first construction may
depend on that instance, but is independent of time. Both force prescriptions install all the required
local instructions, without executing the selected computation.