Global Regulatory for 3D Navier-Stokes Equation via Two Case Energy Decay Analysis : small data

Abstract:We prove global regularity for the 3D incompressible Navier-Stokes equations with positive viscosity $\nu > 0$ for sufficiently small initial data. A common source of confusion in prior approaches is the conflation of liquid behavior with that of solids or gases. Liquids possess unique viscous self-cancellation absent in other states of matter. The proof divides into two cases: 1) bounded domain with Dirichlet boundary, 2) whole space $\mathbb{R}^3$. In both cases we show unconditional decay of kinetic energy to zero. For bounded domains we obtain exponential decay via Poincaré inequality. For $\mathbb{R}^3$ we obtain algebraic decay via Gagliardo-Nirenberg inequality. Higher regularity for small data is established by bootstrapping. Extension to large initial data will appear in Version 2.0. Computational checks were assisted by Meta AI.