Global Regularity of the Three-Dimensional Navier–Stokes Equations via a Coupled Energy and Stability Framework
Year: 2026
Type: article
Abstract: We present an analytic framework establishing global existence and smoothness for the three- dimensional incompressible Navier–Stokes equations. The classical obstruction to regularity— finite-time blow-up driven by energy concentration—is shown to be excluded once the dynamics are embedded into a coupled energy–stability structure that controls both amplitude and struc- tural deformation of the flow. The key observation is that the Navier–Stokes evolution admits a refinement of the standard energy inequality into a combined conservation–dissipation law, in which the kinetic energy and a complementary stability functional evolve coherently. This coupled structure yields a uniform a priori bound preventing the formation of singularities, independently of scale. As a consequence, Leray–Hopf weak solutions are shown to remain smooth for all time. The formulation is minimal and analytic in nature, requiring no probabilistic assumptions or nonphysical regularization. It clarifies why purely energetic methods are insufficient in three dimensions, and demonstrates that global regularity follows once the Navier–Stokes flow is treated as a constrained evolution within a coupled energy–stability system.
Author: Nemoto, Ryusho
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7125461187
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