An Overview of the Smoothness and Existence Problem in the Navier–Stokes Equations

This paper provides a structural overview of the Navier–Stokes equations, one of the central unresolved problems in fluid dynamics. Focus is placed on the ex- istence and smoothness of solutions for three-dimensional viscous, incompressible flows. Theoretical implications for turbulence, numerical simulations, and applied mathematics are discussed, highlighting the critical role of nonlinear terms in the potential formation of singularities.