The Millennium Problem

In the year 2000, the Clay Mathematics Institute selected seven of the most important unsolved problems in mathematics and offered a $1 million prize for each. The Navier-Stokes existence and smoothness problem is one of them.

The question, informally: do the equations of fluid motion always have smooth, well-behaved solutions, or can they break down?

To date, no one has claimed the prize. The problem remains wide open — though there has been remarkable progress in understanding what a solution (or a breakdown) would look like.

The Clay Millennium Prize for Navier-Stokes is stated in the official problem description by Charles Fefferman (2000). Two formulations are given — one on $\mathbb{R}^3$ and one on $\mathbb{T}^3$ (periodic boundary conditions). A valid solution must address one of them.

The prize requires either:

• (A) Existence and smoothness: Prove that for any $u_0 \in C^\infty(\mathbb{R}^3)$ with $\nabla \cdot u_0 = 0$ and suitable decay, there exists a smooth solution $(u, p)$ for all $t \geq 0$ with controlled growth.
• (B) Breakdown: Exhibit smooth, divergence-free initial data and a smooth external force for which no smooth solution exists for all $t > 0$.

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

Setup: Take any initial fluid velocity that is perfectly smooth (no sharp edges or discontinuities) and dies off at infinity (the fluid is essentially still far away from the action).

Question: Will the fluid velocity remain smooth and finite for all future time? Or is it possible that the velocity could become infinite at some point — a "blowup"?

The answer is one of two possibilities:

1. Yes, always smooth — prove that no matter what smooth initial state you start from, the solution stays smooth forever.
2. No, blowup is possible — find a specific smooth starting configuration where the solution eventually breaks down.

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

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

The Navier-Stokes problem earned its place among the seven Millennium Problems because it sits at the intersection of:

• Practical importance — these equations underpin most of fluid dynamics, from aircraft design to climate modeling
• Mathematical depth — the problem touches analysis, geometry, topology, and physics simultaneously
• Resistance to known techniques (explore why) — despite 180+ years of work by some of the greatest mathematicians, neither global regularity nor finite-time blowup has been established

It's one of the few Millennium Problems where the statement can be explained to a non-mathematician, yet the solution has eluded the world's best minds for nearly two centuries.

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 subcritical — it does not control the nonlinearity at small scales, leaving a gap that all existing techniques struggle to bridge.

Key milestones in the story:

• 1823 — 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 huge breakthrough, but these solutions might not be smooth)
• 1982 — Caffarelli, Kohn, and Nirenberg prove that singularities (more on partial regularity), if they exist, must be very rare (the set of singular times has measure zero)
• 1984 — Beale, Kato, and Majda show that blowup can only happen if the vorticity becomes infinite
• 2000 — Clay names it a Millennium Problem
• Today — the problem remains open, with active work on critical-space approaches, Type-I/II blowup classification, and computer-assisted proof

A selective timeline of foundational results:

• Leray (1934): Existence of global weak solutions $u \in L^\infty_t L^2_x \cap L^2_t \dot{H}^1_x$ via compactness. Introduced the Leray projector and the concept of turbulent solutions.
• Hopf (1951): Extended Leray's construction to bounded domains.
• Ladyzhenskaya–Prodi–Serrin (1960s): Regularity criteria — $u \in L^p_t L^q_x$ with $2/p + 3/q \leq 1$, $q \geq 3$ implies smoothness. The critical case $L^\infty_t L^3_x$ resolved by Escauriaza–Seregin–Šverák (2003).
• Caffarelli–Kohn–Nirenberg (1982): The one-dimensional parabolic Hausdorff measure of the singular set is zero: $\mathcal{P}^1(\Sigma) = 0$.
• Beale–Kato–Majda (1984): Blowup iff $\int_0^{T^*} \|\omega(\cdot,t)\|_{L^\infty} \, dt = \infty$.
• Koch–Tataru (2001): Local well-posedness for small data in $\mathrm{BMO}^{-1}$, the largest critical space where this is known.
• Seregin (2012): If $u \in L^\infty_t L^3_x$ near a hypothetical blowup time, the solution is regular — ruling out Type-I blowup with bounded $L^3$ norm.

If you came here asking whether the problem has already been solved, start with Is the Navier-Stokes Problem Solved?.

Then explore why it's so hard to solve, or see how mathematicians have broken it into subproblems.

For the short status answer and the distinction between weak existence and global smooth regularity, see Is the Navier-Stokes Problem Solved?.

For the mathematical obstacles underlying the regularity problem, see Why It's Hard. For a decomposition into tractable components — weak solutions, partial regularity, blowup classification — see Subproblems.