Time localization of the actual presingular fields #
The velocity and pressure are multiplied by the constructed smooth switch from
SmoothCutoffs. This gives zero initial velocity without an initial-value
assumption on the original field. The divergence, residual and blowup claims
below use the concrete operators and quantified blowup definition from
ProblemStatement.
This transformation does not construct the incoming singular fields or a smooth global force extension. It supplies the time-switch portion of Proposition 11.4.
Activated velocity, defined pointwise by timeSwitch z.1 • u z.
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Activated pressure, defined pointwise by timeSwitch z.1 * p z.
Equations
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No hypothesis on u (0,x) is required.
The entire spacetime field agrees locally, so all its existing local jets
agree at every time strictly later than 3/4.
Restricting to a spatial slice is smooth also at time zero, because the slice maps entirely into the relative presingular domain.
Exact residual of the constructed activation: χ R + χ' u + (χ²-χ)(u·∇)u.
Local agreement proves residual agreement without any smoothness assumption on the original fields. The same concrete derivative operators are used.
Activating the field does not remove speed blowup, because the switch is identically one on the final quarter of the time interval.
A constructed pair with zero initial velocity, retaining the incoming presingular smoothness, periods, divergence constraint and terminal blowup. The input field is not assumed to vanish initially.