Fluid DynamicsUnsolved
Navier–Stokes Existence and Smoothness
Do smooth, globally defined solutions exist for the 3D incompressible Navier–Stokes equations?
Overview
The problem asks whether smooth, physically reasonable initial conditions for the 3D incompressible Navier–Stokes equations always produce global, smooth solutions—or whether singularities can form in finite time.
Why It Matters
Navier–Stokes equations model fluids from turbulence to weather. Understanding existence and smoothness is central to mathematical physics.
Current Status
Global existence and smoothness in 3D remains open. Many partial results exist for special cases and weaker solution concepts.
What a Solution Would Require
A proof of global regularity for all appropriate initial data, or a construction of a finite-time blow-up with rigorous justification.