The Navier-Stokes Problem from a Physical Perspective.pdf

<p dir="ltr">The Navier-Stokes problem, posed by the Clay Mathematics Institute, asks whether solutions to the partial differential equations (PDEs) for 3D incompressible flow exist globally and remain smooth. Rather than offering a direct solution, this paper investigates the problem's origins and proposes an alternative framework grounded in physical first principles. We begin with the physical axiom that the universe is a discrete lattice structure with minimum spatiotemporal units (ℓₚ, tₚ). Upon this foundation, we replace standard differential operators with finite difference operators. This reframes the Navier-Stokes PDE into a deterministic finite difference algorithm. Within this discrete system, this paper (1) proves the global boundedness of solutions based on a discrete energy law, thereby precluding the formation of singularities, and (2) rigorously establishes the global existence of the solution. Furthermore, we show that in the limit as the grid spacing approaches zero, the discrete solution converges to a Leray-Hopf weak solution of the standard Navier-Stokes equations. While this result does not directly prove the regularity of the continuum problem, it provides fundamental insight into why singularities are unlikely to occur in physical systems and establishes a stable numerical foundation for exploring the properties of the solution.</p><p><br></p>