Research Series · Fluid Dynamics
A machine-checked program studying what every finite-maximal periodic three-dimensional Navier–Stokes trajectory must do as continuation fails: how regularity escapes, where spectral transfer accumulates, and what the scale-invariant endpoint permits.
Program
Paper I establishes the structural consequences of finite maximal lifespan: bounded kinetic energy, eventual H² divergence, escape beyond every prescribed spectral cutoff, positive-scale regularity, and collapse of uniform continuation reserve.
Paper II resolves that anatomy more finely. It tracks transfer through finite annuli, terminal nonlinear H² work, and the synchronized failure of nonendpoint continuation criteria. The result is a more explicit description of how regularity can leave every finite control surface while energy remains finite.
Paper III reaches the scale-invariant L³ endpoint. It proves that bounded terminal L³ prevents finite-time breakdown, so a finite maximal trajectory must exhibit unbounded terminal L³ limsup. The proof architecture moves through terminal magnification, intrinsic restart, ancient-limit recovery, and rigidity.
What structure must remain available for a strong solution to continue, and what exact geometry is forced when that continuation reserve disappears?
Paper sequence
Published preprint
A machine-checked H² theory of finite-energy regularity escape.
Finite maximality is resolved into bounded energy, eventual H² divergence, high-frequency escape, positive-scale regularity, and collapse of continuation reserve.
Published preprint
Spectral transfer, terminal nonlinear H² work, and simultaneous failure of nonendpoint continuation controls.
The continuation anatomy is sharpened through finite-annulus spectral budgets and the coordinated loss of nonendpoint regularity control.
Published preprint
Terminal magnification, intrinsic restart, and analytic rigidity at the L∞t L³x boundary.
The scale-invariant endpoint is resolved: bounded terminal L³ forces continuation, while finite maximal time forces infinite terminal L³ limsup.