PAPER / ARXIV:2609.14292
Weishuo Liu
RESUMO
We introduce a compatibility--realization framework for constructing singular solutions to the forced three-dimensional incompressible Navier--Stokes equations. The framework separates exact balance and matching constraints, expressed in a chosen geometric representation, from the sufficient conditions of a particular completion scheme. Completed realizations are organized by observable fibers, distinguishing prescribed core kinematics from the surrounding force and transport mechanisms. Conditional on the source-dependent completion input specified in this paper, we obtain families with zero initial velocity, smooth compactly supported forcing, bounded kinetic energy and finite-time unbounded velocity. These include a reversal of leading viscous work at fixed axis motion and first velocity gradient, a signed transition through a stationary singular center under one fixed canonical pressure trace, and a connected stretching family crossing the threshold of zero leading direct viscous vorticity supply. Near a critical profile, the realizable trace data contain a neighborhood in an infinite-dimensional space of analytic axial-velocity and pressure perturbations, while the central first velocity gradient and the critical vorticity balance remain fixed. We also identify the full kernel and an explicit right inverse of the joint angular-transport operator in the chosen realization class. The results establish non-rigidity within observable fibers and show that axis kinematics alone do not determine the local pressure--viscosity partition. Direct profile selection and dynamical stress generation are treated as coupled components of construction, rather than as definitions of the entire design space.
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