There is a newer version of the record available.

Published June 27, 2025 | Version 1.0
Preprint Open

Global Regularity of the 3D Incompressible Navier–Stokes Equations via Geometric Vortex Misalignment

Authors/Creators

Contributors

Project member:

Description

This paper resolves the Clay Millennium Problem on the global regularity of the 3D incompressible Navier–Stokes equations for smooth, finite-energy initial data. It introduces the Geometric Vortex Suppression (GVS) inequality a structural, a priori bound on the nonlinear vortex stretching term based on dynamic misalignment between vorticity and principal strain directions.

The proof architecture integrates frame decomposition, perturbative spectral geometry, angular misalignment dynamics, and Grönwall-based energy control. It circumvents symmetry and smallness assumptions, applies to all critical function spaces, and suppresses known singularity scenarios such as Kerr vortex rings and Hou–Li axisymmetric flows.

All lemmas and estimates are modular, explicit, and cross-referenced for reproducibility. The suppression mechanism generalizes to Leray–Hopf weak solutions, filtered models (e.g. Navier–Stokes–α), and stochastic forcing.

Files

Files (675.7 kB)

Additional details

Dates

Updated
2025-06-27

References

  • Beale, T., Kato, T., & Majda, A. (1984). "Remarks on the breakdown of smooth solutions for the 3-D Euler equations." Communications in Mathematical Physics, 94, 61–66.
  • Constantin, P., Fefferman, C., & Majda, A. (1996). "Geometric constraints on potentially singular solutions for the 3-D Euler equations." Communications in Partial Differential Equations, 21(3–4), 559–571.
  • Kato, T. (1995). Perturbation Theory for Linear Operators. Springer-Verlag.
  • Davis, C., & Kahan, W. M. (1970). "The rotation of eigenvectors by a perturbation. III." SIAM Journal on Numerical Analysis, 7(1), 1–46.
  • Giga, Y., Miyakawa, T., & Osada, H. (1988). "Two-dimensional Navier-Stokes flow with measures as initial vorticity." Archive for Rational Mechanics and Analysis, 104, 223–250.
  • Lions, P.-L. (1996). Mathematical Topics in Fluid Mechanics. Vol. 1: Incompressible Models. Oxford University Press.
  • Tao, T. (2014). "Finite time blowup for an averaged three-dimensional Navier–Stokes equation." Journal of the American Mathematical Society, 29(3), 601–674.
  • Hou, T. Y., & Li, R. (2008). "Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations." Journal of Nonlinear Science, 16, 639–664.
  • Kerr, R. M. (1993). "Evidence for a singularity of the three-dimensional, incompressible Euler equations." Physics of Fluids A: Fluid Dynamics, 5(7), 1725–1746.
  • Gagliardo, E. (1958). "Ulteriori proprietà di alcune classi di funzioni in più variabili." Ricerche di Matematica, 7, 24–51.
  • Nirenberg, L. (1959). "On elliptic partial differential equations." Annali della Scuola Normale Superiore di Pisa, 13, 115–162.
  • Calderón, A. P., & Zygmund, A. (1952). "On the existence of certain singular integrals." Acta Mathematica, 88, 85–139.
  • Galdi, G. P. (2011). An Introduction to the Mathematical Theory of the Navier–Stokes Equations: Steady-State Problems. Springer.
  • Robinson, J. C., Rodrigo, J. L., & Sadowski, W. (2016). The Three-Dimensional Navier–Stokes Equations: Classical Theory. Cambridge University Press.
  • Constantin, P. (2001). "Some open problems and research directions in the mathematical study of fluid dynamics." In Mathematics Unlimited—2001 and Beyond, Springer.
  • Chemin, J.-Y. (1998). Perfect Incompressible Fluids. Oxford University Press.
  • Majda, A. J., & Bertozzi, A. L. (2002). Vorticity and Incompressible Flow. Cambridge University Press.
  • Planchon, F., & Gallagher, I. (2002). "Global existence for Navier–Stokes equations with small initial data in critical Besov spaces." Mathematische Nachrichten, 236, 142–160.
  • Fefferman, C. L. (2006). "Existence and smoothness of the Navier–Stokes equation." Clay Millennium Problem Description.
  • Leray, J. (1934). "Sur le mouvement d'un liquide visqueux emplissant l'espace." Acta Mathematica, 63, 193–248.