Navier–Stokes Existence & Smoothness

Do smooth solutions exist for all incompressible flows?

Last updated: 2025-10-25

The 3D incompressible Navier–Stokes equations govern fluids from airflow to blood flow. We do not know whether smooth solutions exist for all time for smooth initial data.

A resolution would reshape turbulence theory, forecasting, and engineering.

The equations

On a periodic domain Ω with viscosity ν>0, the velocity u(x,t) and pressure p satisfy:

Existence & smoothness asks: for smooth divergence-free u₀, does a unique smooth u(x,t) exist for all t≥0?
Beale–Kato–Majda: blow-up (if any) requires \(\int_0^T \|\omega\|_\infty dt = \infty\), where \(\omega=\nabla\times u\).

\partial_t u + (u\cdot\nabla)u + \nabla p = \nu\,\Delta u,\qquad \nabla\cdot u = 0\quad \text{on}\ \Omega

Vorticity filaments in 3D turbulence (illustrative render).

Why it matters

If smoothness holds, turbulence has a rigorous foundation and numerical schemes can be justified more broadly. If blow-up occurs, we learn where classical fluid models break and what new physics (or regularizations) are required.

The equations

Why it matters

Further reading