Abstract

The incompressible Navier−Stokes equations govern the motion of viscous fluids and remain a challenging open problem in modern mathematics. While global regularity in three dimensions is still unresolved, a partial result, derived by Caffarelli, Kohn and Nirenberg (1982), provides a quantitative description of where singularities may occur. In particular, a dimensional bound on the singular set was obtained.

In this talk, we introduce the framework of suitable weak solutions, a localized refinement of the classical Leray−Hopf theory, and present the idea of deriving the statement by Caffarelli, Kohn and Nirenberg: the singular set 𝒮 of any suitable weak solution satisfies dim_H(𝒮) ≤ 1 and even 𝒫¹(𝒮) = 0. The proof combines parabolic scaling, local energy inequalities and a scale−invariant ε−regularity mechanism. We further briefly address the sharpness of this bound.