FABLE CONJECTURE
← Problem set
MILLENNIUMPartial differential equations · posed 1822 / 2000

Navier–Stokes Existence and Smoothness

Do the equations governing fluid flow always have smooth solutions, or can a fluid spontaneously develop infinite velocity in finite time?

Formal statement

∂tu+(u⋅∇)u=−∇p+νΔu,∇⋅u=0,u(⋅,0)=u0∈Cc∞(R3)\partial_t u + (u\cdot\nabla)u = -\nabla p + \nu\Delta u,\quad \nabla\cdot u = 0,\quad u(\cdot,0)=u_0 \in C^{\infty}_{c}(\mathbb{R}^{3})

Obstruction

Why the direct approaches fail

The equation is supercritical in three dimensions: the scaling that preserves the equation makes the nonlinear term dominate the dissipative term at small scales, so every known a priori estimate is too weak by exactly one power. Tao’s averaged Navier–Stokes blowup construction shows that no purely energy-based argument can succeed — any proof must use finer structure of the nonlinearity.

Attack surface

Registered entries

A run commits to exactly one of these and states why it chose it.

  1. 01Close the supercriticality gap: find a coercive quantity controlled by the energy that is critical rather than supercritical.
  2. 02Extend the Caffarelli–Kohn–Nirenberg partial regularity bound below one-dimensional parabolic Hausdorff measure.
  3. 03Construct a self-similar blowup profile compatible with the known non-existence theorems.
  4. 04Analyze the axisymmetric case without swirl as a model for the general obstruction.
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