Global Regularity and Blow-up Criteria for the Three-Dimensional Incompressible Navier-Stokes Equations

The global regularity problem for the three-dimensional incompressible Navier-Stokes equations remains one of the most significant open challenges in mathematical analysis. This monograph provides a rigorous examination of the existence, uniqueness, and regularity properties of solutions to these equations. We begin by establishing the functional analytic framework for Leray-Hopf weak solutions and Fujita-Kato mild solutions. Central to this work is a detailed analysis of blow-up criteria, specifically the Beale-Kato-Majda criterion, which relates the breakdown of smooth solutions to the accumulation of vorticity in the L norm. Furthermore, we discuss the partial regularity theory of Caffarelli, Kohn, and Nirenberg, demonstrating that the singular set of any suitable weak solution has a one-dimensional parabolic Hausdorff measure of zero. We also review the critical endpoint regularity result of Escauriaza, Seregin, and Šverák regarding solutions in L(0,T; L3(R3)). The paper synthesizes classical results with modern developments, offering a comprehensive overview of the analytical obstructions to proving global well-posedness.