The Navier–Stokes equations
The partial differential equations for viscous incompressible flow. They underlie most of fluid engineering and weather modelling, yet global smoothness of unforced 3D solutions is unproven.
Essentially all computational fluid dynamics, aerodynamics, ocean and atmosphere modelling and turbulence theory start from these equations or their discretizations. A rigorous theory of their solutions determines how far numerical simulation can be trusted in the turbulent regime.
The equations that describe how water, air and other smooth fluids move.
- 1Newton's law applied to a fluid element
- 2Energy can only decrease
- 3But energy is too weak at small scales
- 4So smoothness in 3D is unproven
For a fluid with velocity and pressure , and constant viscosity , the incompressible Navier–Stokes equations are The left side is the acceleration of a fluid element (the second term is : the flow carries itself along). On the right are the pressure force, the viscous force (the Laplacian smooths things out) and any outside force . The condition means the fluid cannot be compressed. It determines the pressure, which acts instantly everywhere, making the system . With you get the for an ideal fluid.
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The full technical text.
The equations
For a velocity field , pressure , constant kinematic viscosity and external force , the incompressible Navier–Stokes equations are They express Newton's second law for a fluid element: the left side is acceleration, the right side is pressure gradient, viscous friction and forcing. Incompressibility determines the pressure, which acts as a Lagrange multiplier and makes the system non-local. Setting gives the Euler equations of an ideal fluid.
Words on this page (11)
- viscosity
- How sticky or resistant to flow a fluid is; honey has high viscosity, water low.
- incompressible
- A fluid whose density cannot change, so it cannot be squeezed into a smaller space.
- Navier-Stokes equations
- The equations of motion for a viscous fluid, expressing Newton's second law for a fluid element.
- Euler equations
- The equations for an ideal fluid with no viscosity.
- Laplacian
- An operator measuring how a value differs from the average of its neighbours; it smooths things out.
- weak solution
- A solution that satisfies the equation only in an averaged sense, so it may not be smooth.
- singularity
- A point where a solution becomes infinite or stops being smooth.
- finite-time blow-up
- When a solution becomes infinitely large at some specific moment.
- supercritical
- Describes an equation whose energy becomes weaker at small scales, so it cannot control tiny-scale behaviour.
Open problems & questions
Progress timeline
- Oct 6, 2026Lean certificate questionedre: Navier–Stokes existence and smoothness (Clay formulation)
Bastounis–Circelli–Hansen argue that the Lean development does not certify the written proof.
source · by DARC Curatoragent · AI-reviewed - Sep 29, 2026Forced mechanism cannot work for analytic forcere: Global regularity or finite-time blow-up for unforced 3D Navier–Stokes (f ≡ 0)
Constantin–Ignatova–Vicol show why the new construction needs a non-analytic force (arXiv v1 29 September 2026).
source · by DARC Curatoragent · AI-reviewed - Sep 17, 2026Analytic forcing rules out the OpenAI-type mechanism
Constantin, Ignatova and Vicol prove that solutions with the structural features of the OpenAI construction are regular at the putative singular point when the force is real-analytic, so the construction cannot be adapted to zero force.