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UnsolvedFluid dynamics, PDEMillennium Prize Problem

Navier–Stokes existence and smoothness

Do smooth solutions of the 3D incompressible Navier–Stokes equations exist for all time?

Exact statement

∂u∂t+(u⋅∇)u=−1ρ∇p+ν∇2u+f,∇⋅u=0\frac{\partial \mathbf u}{\partial t}+(\mathbf u\cdot\nabla)\mathbf u=-\frac{1}{\rho}\nabla p+\nu\nabla^{2}\mathbf u+\mathbf f,\qquad \nabla\cdot\mathbf u=0
u(x,0)=u0(x) smooth, divergence-free, x∈R3 (or T3)\mathbf u(x,0)=\mathbf u_0(x)\ \text{smooth, divergence-free},\ x\in\mathbb R^{3}\ (\text{or }\mathbb T^{3})

For smooth, divergence-free initial velocity in three dimensions (whole space with fast decay, or periodic), prove that a smooth solution exists for all time with bounded energy, or give an example where it breaks down in finite time.

Status

Open in three dimensions. Solved (globally smooth) in two dimensions; no 3D proof or blow-up example is known.

Posed: Equations: Navier (1822) and Stokes (1845); Millennium formulation by Charles Fefferman (2000).

Prize of US $1,000,000 offered by the Clay Mathematics Institute (Millennium Prize Problems, announced 24 May 2000).

In plain words

These equations are Newton's second law for a viscous, incompressible fluid: the velocity u\mathbf u changes because of transport by the flow itself, pressure pp, viscosity ν\nu and outside forces. They model water, air at low speeds, blood flow and weather, and engineers solve them numerically every day.

The open question is mathematical: could a smooth flow in 3D develop a singularity, where velocity becomes infinite at a point in finite time? The nonlinear term (u⋅∇)u(\mathbf u\cdot\nabla)\mathbf u can concentrate energy at small scales faster than viscosity is known to smooth it out.

Key results so far

  1. 1934: Leray proves global weak solutions exist in 3D (not known to be smooth or unique).
  2. 1930s–1960s: In 2D, global existence and smoothness is a theorem (Leray; Ladyzhenskaya).
  3. 1964: Fujita and Kato: smooth solutions exist for a short time, and for all time when the initial data are small.
  4. 1982: Caffarelli, Kohn and Nirenberg: the possible singular set of a suitable weak solution has one-dimensional parabolic Hausdorff measure zero.
  5. 2016: Tao constructs finite-time blow-up for an 'averaged' Navier–Stokes equation, showing energy methods alone cannot settle the problem.
  6. 2019: Buckmaster and Vicol prove non-uniqueness of very weak solutions.

Exact special solutions and solved cases

Plane Couette flow

u=(Uyh,0,0)\mathbf u=\Big(\frac{U y}{h},0,0\Big)

Fluid between a fixed plate and a plate moving at speed U.

Poiseuille flow in a pipe

uz(r)=G4μ (R2−r2)u_z(r)=\frac{G}{4\mu}\,(R^{2}-r^{2})

Steady flow driven by a constant pressure gradient G = −dp/dz in a pipe of radius R (μ = ρν).

Taylor–Green vortex (2D)

u=sin⁡xcos⁡y e−2νt,v=−cos⁡xsin⁡y e−2νtu=\sin x\cos y\,e^{-2\nu t},\quad v=-\cos x\sin y\,e^{-2\nu t}

An exact decaying solution of the 2D equations.

What our solver can and cannot do

No computer program or AI can solve the Navier–Stokes existence and smoothness by being asked. If you paste it into the MathGPT solver, you get this reference information, labelled as such, never an invented proof. The solver can still compute related special cases and examples. Open the solver.

Frequently asked questions

What does the Navier–Stokes existence and smoothness say?
For smooth, divergence-free initial velocity in three dimensions (whole space with fast decay, or periodic), prove that a smooth solution exists for all time with bounded energy, or give an example where it breaks down in finite time.
Has the Navier–Stokes existence and smoothness been solved?
Open in three dimensions. Solved (globally smooth) in two dimensions; no 3D proof or blow-up example is known.
Is there a prize?
Prize of US $1,000,000 offered by the Clay Mathematics Institute (Millennium Prize Problems, announced 24 May 2000).
Can MathGPT solve the Navier–Stokes existence and smoothness?
No. No one has a proof, so no solver or AI can give one. MathGPT recognises the problem and shows the exact statement, its status and the established facts, labelled as reference information.

References

  • C. Fefferman, 'Existence and smoothness of the Navier–Stokes equation', Clay Mathematics Institute (2000).
  • J. Leray, Acta Mathematica 63 (1934).
  • L. Caffarelli, R. Kohn, L. Nirenberg, Comm. Pure Appl. Math. 35 (1982).
  • T. Tao, 'Finite time blowup for an averaged three-dimensional Navier–Stokes equation', J. Amer. Math. Soc. 29 (2016).

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