Navier–Stokes finite-time blow-up with smooth initial conditions

Determine whether solutions to the Navier–Stokes equations of fluid mechanics with smooth initial conditions develop finite-time blow-up (form singularities in finite time) or remain smooth for all time. Establish the conditions under which global regularity fails or holds for these smooth initial data, to clarify the long-time behavior of such flows.

Background

In discussing the Cauchy–Kovalevskaya theorem, the paper notes that while analytic data yield local existence and uniqueness, solutions can blow up in finite time, limiting global existence. The authors highlight the Navier–Stokes equations as a paradigmatic case where the question of blow-up versus global regularity for smooth initial conditions remains unresolved.

This open problem serves as a key example of how even classical PDEs central to physics may lack global solutions, emphasizing that local well-posedness does not guarantee long-time existence, and that such phenomena impact the empirical and conceptual adequacy of modeling assumptions.