Navier-Stokes Existence and Smoothness: Clay Millennium Problem Statement

In 2000, the Clay Mathematics Institute picked seven of the hardest unsolved problems in mathematics and put $1 million on each one. The Navier-Stokes existence and smoothness problem made the list.

This page explains the Clay problem statement. It is not the official Clay Mathematics Institute page; it is a source-linked guide to what Clay is asking, what data are allowed, and what would count as a proof or counterexample. If you only want the current status, see Is the Navier-Stokes problem solved?

The question, stripped down: do the equations of fluid motion always produce smooth, well-behaved solutions, or can they blow up?

Nobody has claimed the prize. There has been real progress in understanding what a solution or a breakdown would look like, but the Clay Millennium Prize problem itself remains wide open.

The Clay Millennium Prize for Navier-Stokes is stated in the official problem description by Charles Fefferman. Two formulations are given, one on $\mathbb{R}^3$ and one on $\mathbb{T}^3$ with periodic boundary conditions. A valid solution must resolve one of the accepted alternatives in the Clay statement.

The prize requires either:

• (A) Existence and smoothness: prove that the appropriate smooth divergence-free initial data generate a smooth solution for all $t \geq 0$ with the stated energy control.
• (B) Breakdown: exhibit admissible smooth data, with the allowed forcing in the relevant formulation, for which no globally smooth solution exists.

This page paraphrases and explains the official statement; for the authoritative text, use the Clay source linked below.

The Clay problem is not asking whether fluid simulations work, whether engineers can solve pipe-flow examples, or whether weak solutions exist. Those are separate questions. The prize question is about global smoothness or finite-time breakdown for the three-dimensional incompressible equations.

To win the prize, a proof has to land in one of two buckets:

1. Global smoothness: show that every admissible smooth initial flow stays smooth for all future time.
2. Breakdown: give an admissible smooth setup where a globally smooth solution cannot exist.

The core system is the 3D incompressible Navier-Stokes equation

$$\partial_t u + (u \cdot \nabla)u = -\nabla p + \nu \Delta u + f, \qquad \nabla \cdot u = 0.$$

Clay's formulation separates whole-space and periodic cases. In the whole-space existence direction, the initial velocity is smooth, divergence-free, and rapidly decaying; the target is a globally smooth solution with finite energy for all time. In the breakdown direction, the task is to construct admissible data, under one of the official alternatives, for which the required globally smooth solution fails to exist.

Here's what the problem actually asks, in plain language:

Setup: Take any initial fluid velocity that's perfectly smooth (no sharp edges, no discontinuities) and dies off at infinity. Far from the action, the fluid sits still.

Question: Does the velocity remain smooth and finite for all future time? Or can it blow up?

Two answers. Only two.

1. Yes, always smooth. Prove that no matter what smooth initial state you pick, the solution stays smooth forever. Every initial condition, every time.
2. No, blowup happens. Find one specific smooth starting configuration, possibly together with a smooth external force, where the solution breaks down. Just one is enough.

Following Fefferman's formulation on $\mathbb{R}^3$ with $f \equiv 0$:

Hypotheses: Let $u_0 \in C^\infty(\mathbb{R}^3)$ be divergence-free. Assume for every $\alpha$ and $K$ there exist constants $C_{\alpha,K}$ such that

$$|\partial^\alpha u_0(x)| \leq \frac{C_{\alpha,K}}{(1 + |x|)^K} \quad \text{on } \mathbb{R}^3.$$

Conclusion (to prove): There exist $p \in C^\infty(\mathbb{R}^3 \times [0,\infty))$ and $u \in C^\infty(\mathbb{R}^3 \times [0,\infty))$ satisfying the Navier-Stokes equations, $u(x,0) = u_0(x)$, and the energy bound

$$\int_{\mathbb{R}^3} |u(x,t)|^2 \, dx < C \quad \text{for all } t \geq 0.$$

Three things put Navier-Stokes on that shortlist:

• Practical importance. These equations run most of fluid dynamics: aircraft design, climate models, blood flow, ocean currents. Even without a complete proof, engineers use these equations successfully in many regimes; the open problem is about whether the 3D equations can always be justified mathematically.
• Mathematical depth. It draws on analysis, geometry, topology, and physics simultaneously.
• Sheer stubbornness (explore why). Over 180 years of effort by some of the greatest mathematicians who ever lived, and we still don't know the answer.

A bright undergrad can state the question in five minutes. No one has found an answer. That gap between a simple statement and an unreachable proof is what defines a Millennium Problem.

The problem's difficulty is rooted in the supercritical nature of the 3D equations. The natural energy estimate

$$\frac{1}{2}\|u(t)\|_{L^2}^2 + \nu \int_0^t \|\nabla u(s)\|_{L^2}^2 \, ds \leq \frac{1}{2}\|u_0\|_{L^2}^2$$

places $u$ in $L^\infty_t L^2_x \cap L^2_t \dot{H}^1_x$, which is below the critical scaling. The Navier-Stokes equations are invariant under

$$u(x,t) \mapsto \lambda u(\lambda x, \lambda^2 t), \quad p(x,t) \mapsto \lambda^2 p(\lambda x, \lambda^2 t)$$

and the critical space is $L^3(\mathbb{R}^3)$ (or $\dot{H}^{1/2}$). The energy class $L^2$ is supercritical. It lies below the critical scaling threshold and does not by itself control the small-scale nonlinear cascade, leaving a gap that all existing techniques struggle to bridge.

The essential milestones:

• 1822: Navier derives the equations from molecular considerations.
• 1845: Stokes gives the modern derivation from continuum mechanics.
• 1934: Leray proves that "weak" solutions always exist. A massive result, but these solutions might not be smooth.
• 1982: Caffarelli, Kohn, and Nirenberg prove that singularities (more on partial regularity), if they exist, are extremely small: in the parabolic geometry natural to these equations, the singular set has zero one-dimensional parabolic Hausdorff measure.
• 1984: Beale, Kato, and Majda prove (originally for Euler, with Navier-Stokes analogues) that blowup can only happen if the vorticity becomes infinite.
• 2000: Clay names it a Millennium Problem.
• Today: Still open. Active work on critical-space approaches, Type-I/II blowup classification, and computer-assisted proof.

Foundational results, selectively:

• Leray (1934): Global weak solutions $u \in L^\infty_t L^2_x \cap L^2_t \dot{H}^1_x$ exist, proved via compactness. He introduced the Leray projector and the concept of turbulent solutions. The starting gun for everything that followed.
• Hopf (1951): Extended Leray's construction to bounded domains.
• Ladyzhenskaya, Prodi, Serrin (1960s): Regularity criteria. If $u \in L^p_t L^q_x$ with $2/p + 3/q \leq 1$, $q > 3$, then the solution is smooth. Escauriaza, Seregin, and Šverák settled the endpoint $L^\infty_t L^3_x$ case in 2003.
• Caffarelli, Kohn, Nirenberg (1982): $\mathcal{P}^1(\Sigma) = 0$. The singular set has zero one-dimensional parabolic Hausdorff measure.
• Beale, Kato, Majda (1984): Originally proved for incompressible Euler: blowup happens iff $\int_0^{T^*} \|\omega(\cdot,t)\|_{L^\infty} \, dt = \infty$. Analogous criteria hold for Navier-Stokes.
• Koch, Tataru (2001): Local well-posedness for small data in $\mathrm{BMO}^{-1}$. This is the largest critical space where well-posedness is known.
• Seregin (2012): At a blowup time $T^*$, the $L^3$ norm must diverge: $\|u(t)\|_{L^3} \to \infty$ as $t \to T^*$. Strictly stronger than ESS (2003), which only showed failure of uniform boundedness.

This article is part of The Problem.

If you came here wondering whether someone already solved it, start with Is the Navier-Stokes Problem Solved?.

Then explore why it's so hard, or see how mathematicians have broken it into subproblems. For the structural reasons the 2D problem is tractable while 3D remains open, see Why 2D Is Easier Than 3D.

This article is part of The Problem.

Want the short answer on whether it's been solved? See Is the Navier-Stokes Problem Solved? That page also clarifies the gap between weak existence and global smooth regularity.

The mathematical obstacles are laid out in Why It's Hard. For a decomposition into tractable pieces (weak solutions, partial regularity, blowup classification), see Subproblems. And for why the 2D case is settled while 3D isn't, see Why 2D Is Easier Than 3D.