Navier-Stokes Existence and Smoothness
Asks whether smooth 3D incompressible fluid flows can develop singularities.
The Navier-Stokes existence and smoothness problem asks whether, in three dimensions, smooth and globally defined solutions to the incompressible Navier-Stokes equations exist for all time given any smooth initial velocity field of finite energy. The equations have governed our understanding of fluid dynamics since the 19th century, yet mathematicians cannot prove that solutions remain smooth or show they develop singularities. Jean Leray proved the existence of weak solutions in 1934, and Caffarelli, Kohn, and Nirenberg established partial regularity in 1982, but the full regularity question in 3D remains wide open. Terence Tao's 2016 work on averaged Navier-Stokes showed that any resolution must exploit the specific nonlinear structure of the equations.
Formula
The fundamental PDE governing viscous incompressible fluid flow: momentum balance coupled with divergence-free constraint.
Do smooth initial data always produce globally smooth solutions in 3D, or can singularities form in finite time?
Leray proved global weak solutions always exist in 3D — but uniqueness and regularity of these solutions remain open.
Summary
The Navier-Stokes existence and smoothness problem asks whether, in three dimensions, smooth and globally defined solutions to the incompressible Navier-Stokes equations exist for all time given any smooth initial velocity field of finite energy. The equations have governed our understanding of fluid dynamics since the 19th century, yet mathematicians cannot prove that solutions remain smooth or show they develop singularities. Jean Leray proved the existence of weak solutions in 1934, and Caffarelli, Kohn, and Nirenberg established partial regularity in 1982, but the full regularity question in 3D remains wide open. Terence Tao's 2016 work on averaged Navier-Stokes showed that any resolution must exploit the specific nonlinear structure of the equations.
