NavierStokesGleichungen

Navier–Stokes global regularity refers to the mathematical question of whether solutions to the three-dimensional incompressible Navier–Stokes equations with smooth initial data remain smooth for all time. The incompressible Navier–Stokes equations describe the motion of a viscous, incompressible fluid and, in three dimensions, are typically written as ∂t u + (u · ∇)u = -∇p + νΔu + f, with ∇ · u = 0, where u(x,t) is the velocity field, p(x,t) is the pressure, ν>0 is the kinematic viscosity, and f represents external forces. Solutions are often sought in a domain Ω with appropriate boundary conditions, such as the no-slip condition u = 0 on ∂Ω.

In three dimensions, it is not yet known whether smooth initial data u0 in a suitable function space generate a global-in-time smooth solution. By contrast, in two dimensions global regularity is established. In three dimensions, global weak solutions exist for all time (Leray-Hopf solutions) but uniqueness and smoothness of these weak solutions are open issues, and singularities may form in finite time if regularity fails.

Several partial results illuminate the problem. Local well-posedness holds for smooth or sufficiently regular data. Global existence is known for small initial data in various critical spaces. Regularity criteria have been developed, such as the Beale–Kato–Majda criterion linking blow-up to the growth of vorticity, and Prodi–Serrin–type criteria connecting space–time integrability of the velocity to smoothness.

The Navier–Stokes global regularity problem is one of the seven Millennium Prize Problems. A resolution—either a proof of global regularity or a counterexample—would be a major milestone in mathematical fluid dynamics and analysis.