Global Smooth Solutions for the Incompressible Navier–Stokes Equations with Small Viscosity

Establish whether global-in-time smooth solutions exist for the incompressible Navier–Stokes equations with small positive viscosity and divergence-free initial data, i.e., for the system of equations describing the time evolution of an incompressible velocity field driven by nonlinear advection, pressure, and forcing, determine if smooth solutions persist for all time when viscosity is small.

Background

The paper formulates the incompressible Navier–Stokes equations and emphasizes that, despite extensive paper, a general mathematical solution theory is not known. In particular, the existence of global smooth solutions under small viscosity remains an open foundational problem in the mathematical theory of turbulence.

This question is closely related to the broader issue of well-posedness and regularity for the Navier–Stokes equations and is highlighted as a key uncertainty impacting field-theoretic approaches to turbulence.