Is the Navier-Stokes Problem Solved?

No. As of 2026, the Navier-Stokes existence and smoothness problem remains unsolved. Nobody has proved that smooth solutions always exist in three dimensions, and nobody has shown they can break down. The Clay Millennium Prize ($1 million) sits there unclaimed, waiting for someone to crack a problem tied to equations formulated in the 19th century and still open today.

The equations themselves aren't in question. Engineers and scientists use Navier-Stokes every day to design aircraft, predict weather, and model blood flow. Simulations work. But here's what's unresolved: a purely mathematical question about whether the equations always produce well-behaved solutions, or whether they might eventually predict something impossible, like infinite velocity concentrating at a single point in space.

As of 2026, the Clay Millennium Prize for Navier-Stokes existence and smoothness remains open. Neither global regularity nor finite-time blowup has been established for the 3D incompressible Navier-Stokes equations.

Precisely: given smooth, divergence-free initial data $u_0 \in C^\infty(\mathbb{R}^3)$ with suitable decay (or on $\mathbb{T}^3$), it's unknown whether a unique smooth solution $(u, p)$ exists for all $t \geq 0$ with $u \in C^\infty(\mathbb{R}^3 \times [0, \infty))$. No counterexample has been constructed.

It's not completely dark. Mathematicians have chipped away at this for over a century, and they've built up a surprisingly detailed picture of what's known and what isn't:

• Weak solutions exist (Leray, 1934). If you weaken the notion of solution, global solutions exist. But whether they stay smooth and unique is still open.
• 2D is solved (Ladyzhenskaya, 1969). Two dimensions? Done. Smooth solutions exist for all time, and the difficulty is entirely, stubbornly specific to 3D.
• Singularities are rare (Caffarelli-Kohn-Nirenberg, 1982). Even if singularities exist in 3D, they're confined to a set with zero one-dimensional measure, meaning a set so thin it has no length at all.
• Short-time solutions exist. Smooth? Yes, at least briefly. The question: can they always be continued forever?

So the gap is narrow but deep. We know solutions start smooth and we know weak solutions persist globally, yet nobody can prove that smoothness survives for all time in three dimensions.

Key established results:

• Leray (1934): Global existence of weak (distributional) solutions satisfying the energy inequality $\|u(t)\|_{L^2}^2 + 2\nu \int_0^t \|\nabla u(s)\|_{L^2}^2 \, ds \leq \|u_0\|_{L^2}^2$. Uniqueness? Open.
• Ladyzhenskaya (1969): For the 2D incompressible problem, smooth divergence-free data yield a unique global smooth solution.
• Caffarelli-Kohn-Nirenberg (1982): Partial regularity for suitable weak solutions: the singular set has zero one-dimensional parabolic Hausdorff measure, $\mathcal{P}^1(S) = 0$.
• Local well-posedness: For $u_0 \in H^s(\mathbb{R}^3)$ with $s > 3/2$, a unique smooth solution exists on $[0, T^*)$; this also holds in the scaling-critical space $\dot{H}^{1/2}(\mathbb{R}^3)$ (Fujita-Kato, 1964), extending to strictly larger critical spaces including $BMO^{-1}$ (Koch-Tataru, 2001). Does $T^* = \infty$?

Here's the gap: we know weak solutions exist globally, and we know strong solutions exist locally with full uniqueness. Whether the strong solution can always be extended to all time is the question that remains wide open.

Every year or two, a preprint drops claiming to solve the Navier-Stokes problem. The cycle is predictable: excitement, expert scrutiny, then someone finds the gap. None has been accepted by the expert community as a correct resolution.

Part of the confusion comes from mixing up what "solved" actually means:

• "We can simulate fluids on computers." Sure. But numerical simulation isn't a mathematical proof; simulations chop space and time into finite pieces, and the question is about what happens in the continuous equations before you do any chopping at all.
• "Engineers use these equations successfully." They do. But practical success doesn't tell us whether the equations are internally consistent in every possible scenario a mathematician can dream up.
• "The 2D problem is solved." Correct. But the 3D problem is fundamentally different because the mechanism that makes 2D work (no vortex stretching, which keeps vorticity bounded) simply doesn't apply in three dimensions.

Claimed proofs appear regularly. They fail for predictable reasons:

• Incorrect a priori estimates: assuming control of a critical or supercritical norm that hasn't actually been established.
• Conflation of weak and strong solutions: proving properties of Leray-Hopf solutions that would require the very regularity being claimed.
• Dimensional analysis errors: arguments that close in 2D (where enstrophy gives $H^1$ control and subcritical Sobolev embeddings suffice) but fail completely in 3D, where those same embeddings no longer control the nonlinearity.
• Circular bootstraps: the hypothesis implicitly assumes what's being proved.

Why does everything fail? The 3D problem is supercritical with respect to the natural energy norm, so standard techniques like energy estimates and Gronwall-type arguments simply don't provide enough control. That's the wall every claimed proof runs into.

To claim the Clay prize, you'd need to do one of two things:

1. Prove global regularity: show that for any smooth initial conditions, the solution stays smooth forever. No infinite velocities. No breakdowns. The equations always behave.
2. Construct a blowup: find smooth initial conditions where the classical mathematical solution breaks down in finite time, or otherwise satisfy one of the official Clay breakdown formulations.

Either result would be massive. Global regularity would resolve the Clay problem and establish that the incompressible model is mathematically well-posed for all smooth data. A blowup? That would force us to rethink what happens at extreme scales and might point toward entirely new physics we haven't imagined yet.

Per Fefferman's Clay formulation, a valid resolution requires one of:

1. (A) Existence and smoothness: For every $u_0 \in C^\infty(\mathbb{R}^3)$ with $\nabla \cdot u_0 = 0$ and $|\partial_x^\alpha u_0(x)| \leq C_{\alpha K} (1 + |x|)^{-K}$ for all $\alpha, K$, prove existence of $(u, p) \in C^\infty(\mathbb{R}^3 \times [0,\infty))$ satisfying the equations, with $\int_{\mathbb{R}^3} |u(x,t)|^2 \, dx < C$ for all $t \geq 0$.
2. (B) Breakdown: Exhibit $u_0 \in C^\infty(\mathbb{R}^3)$ (divergence-free, with suitable decay) and $f \in C^\infty(\mathbb{R}^3 \times [0,\infty))$ such that no $(u,p) \in C^\infty(\mathbb{R}^3 \times [0,\infty))$ satisfies the equations.

Analogous formulations on $\mathbb{T}^3$ are also accepted. Fefferman's full statement includes separate cases with and without external forcing ($f = 0$ and $f \neq 0$); the above distills the essential alternatives.

• 1822: Navier derives the equations from molecular considerations.
• 1845: Stokes gives them their modern form.
• 1934: Leray proves weak solutions exist globally. Huge.
• 1969: Ladyzhenskaya solves 2D.
• 1982: Caffarelli, Kohn, and Nirenberg prove partial regularity, establishing that any singularities must be extraordinarily rare, confined to a set of zero one-dimensional measure.
• 1984: Beale, Kato, and Majda prove for the 3D Euler equations that breakdown of a smooth solution forces divergence of the vorticity time integral. Related continuation criteria also apply to Navier-Stokes.
• 2000: Clay names it a Millennium Problem. One million dollars.
• 2014: Tao constructs blowup for an averaged version of the equations (preprint; published 2016), showing there's no purely structural obstruction to singularity formation.
• 2026: Open.

• 1822: Navier. Molecular stress.
• 1845: Stokes. Continuum.
• 1934: Leray. The foundational result: global weak solutions in $L^2$, the Leray projector, and the energy inequality that would shape an entire century of mathematical fluid analysis and define every approach that followed.
• 1951: Hopf extends to bounded domains.
• 1962: Serrin establishes conditional regularity: smooth if $u \in L^p_t L^q_x$ with $2/p + 3/q < 1$ (endpoint by Fabes-Jones-Rivière).
• 1969: Ladyzhenskaya. 2D done.
• 1982: CKN. $\mathcal{P}^1(S) = 0$.
• 1984: Beale-Kato-Majda (for 3D Euler). If $T^* < \infty$, then $\int_0^{T^*} \|\omega(s)\|_{L^\infty} \, ds = \infty$. Analogous continuation criteria hold for Navier-Stokes.
• 2000: Clay.
• 2014: Tao. Averaged Navier-Stokes blowup (JAMS 2016), showing that any proof of regularity for the true equations must exploit finer features of the Navier-Stokes nonlinearity than those preserved by the averaged model.
• 2026: Open.

Part of The Problem.

Go deeper: why is the problem so hard?, what subproblems are mathematicians working on, and what approaches have they tried?

The formal Clay statement lives on the Millennium Problem page, and if you want to understand which version of the equations this problem actually targets, see Incompressible vs. Compressible Navier-Stokes.

Part of The Problem.

Details: Clay formulation. Obstacles: Why It's Hard.

For the full picture of partial results, open subquestions, and every strategy that's been attempted over the past century of work on this problem, see Subproblems and Approaches. Which formulation the Clay problem studies: Incompressible vs. Compressible.