The Navier–Stokes equations
The article gives a short introduction to the Navier–Stokes equations, the fundamental partial differential equations used to describe the motion of fluids such as water and air. A fluid is represented mainly through its velocity field $v(x,y,z,t)$ and pressure $P(x,y,z,t)$, which specify how fast and in what direction the fluid is moving, and what pressure it has, at each position $(x,y,z)$ and time $t$. The equations connect changes in velocity and pressure with properties such as the fluid's viscosity.
One reason these equations are so important is that they can model phenomena ranging from airflow around aircraft and cars to waves, rivers, weather and turbulence. Turbulence is particularly difficult because velocity can change dramatically between points that are extremely close together. Exact analytic solutions to Navier–Stokes are available only in relatively simple situations, so real engineering and scientific applications generally rely on numerical approximations and computer simulations.
The deepest mathematical issue is whether sufficiently well-behaved three-dimensional initial conditions always produce solutions that remain smooth for all time, or whether the solution can eventually develop a singularity, where quantities such as derivatives of velocity become unbounded. This became one of the Clay Mathematics Institute's $1$ million Millennium Prize Problems.
Key takeaways
ARTICLE / ARTICLE [PDF]
The article gives a short introduction to the Navier–Stokes equations, the fundamental partial differential equations used to describe the motion of fluids such as water and air. A fluid is represented mainly through its velocity field $v(x,y,z,t)$ and pressure $P(x,y,z,t)$, which specify how fast and in what direction the fluid is moving, and what pressure it has, at each position $(x,y,z)$ and time $t$. The equations connect changes in velocity and pressure with properties such as the fluid's viscosity.
One reason these equations are so important is that they can model phenomena ranging from airflow around aircraft and cars to waves, rivers, weather and turbulence. Turbulence is particularly difficult because velocity can change dramatically between points that are extremely close together. Exact analytic solutions to Navier–Stokes are available only in relatively simple situations, so real engineering and scientific applications generally rely on numerical approximations and computer simulations.
The deepest mathematical issue is whether sufficiently well-behaved three-dimensional initial conditions always produce solutions that remain smooth for all time, or whether the solution can eventually develop a singularity, where quantities such as derivatives of velocity become unbounded. This became one of the Clay Mathematics Institute's $1$ million Millennium Prize Problems.
Key takeaways
- Navier–Stokes describes fluid motion using fields such as velocity $v(x,y,z,t)$ and pressure $P(x,y,z,t)$.
- The equations are central to modelling water, air, turbulence, aerodynamics and weather, but realistic solutions usually require numerical computation.
- The great theoretical question is whether smooth $3$-D solutions must remain smooth or can develop a finite-time singularity.
- The article is now historically interesting: OpenAI proposed such a singularity construction on 8 September 2026, but Clay has not yet recognized the Millennium Problem as solved.
ARTICLE / ARTICLE [PDF]
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