MathLabs

Open problem, Differential equations and dynamical systems, posed 2000

Navier–Stokes existence and smoothness

OpenMillenniumSmale #15

For the incompressible Navier–Stokes equations in R3\mathbb{R}^3, ∂tui+∑j=13uj∂jui=νΔui−∂ip,div⁡u=0,\partial_t u_i + \sum_{j=1}^3 u_j \partial_j u_i = \nu \Delta u_i - \partial_i p, \qquad \operatorname{div} u = 0, with smooth, divergence-free initial velocity u0u_0 that decays rapidly at infinity, must a smooth solution (u,p)(u,p) with globally bounded kinetic energy exist for all time t≥0t \ge 0? Or can a smooth solution instead develop a singularity in finite time?

Research frontier as of 2026

As of 2026 the problem remains fully open in all four cases posed by Clay. Weak (Leray–Hopf) solutions are known to exist globally, and partial regularity theory bounds how large a potential singular set could be, but neither global smoothness nor a genuine blow-up example has been established. Conditional regularity criteria (e.g. Prodi–Serrin, Beale–Kato–Majda) show that smoothness would follow if certain norms of the solution stayed bounded, but no one can currently guarantee that they do.

Best known results

  • Leray–Hopf weak solutions exist globally in time for any finite-energy initial data, but their uniqueness and smoothness remain unknown.
  • Caffarelli–Kohn–Nirenberg (1982): the set of possible singularities has parabolic Hausdorff dimension at most 1 in space-time.
  • Conditional regularity criteria (Prodi–Serrin, Beale–Kato–Majda) guarantee smoothness if certain velocity or vorticity norms stay bounded, but these bounds are not known to hold a priori.

Tools and where they stop

ToolAchievedWhere it stops
Energy methods / Leray–Hopf weak solutionsGlobal-in-time existence of weak solutions with bounded energyUniqueness and smoothness of these weak solutions are not known
Partial regularity theory (Caffarelli–Kohn–Nirenberg)Bounds the size of any possible singular setDoes not rule out singularities altogether, or show global smoothness

Open questions

  • Do smooth solutions exist for all time from every smooth initial condition, or can finite-time blow-up occur?
  • Are Leray–Hopf weak solutions unique?

References

  1. Charles L. Fefferman (Clay Mathematics Institute) (2000). Existence and smoothness of the Navier–Stokes equation
  2. Jean Leray (1934). Sur le mouvement d'un liquide visqueux emplissant l'espace
  3. Luis Caffarelli, Robert Kohn, Louis Nirenberg (1982). Partial regularity of suitable weak solutions of the Navier-Stokes equations