Energy Decay Induced by Localized Coercivity in the 3D Navier--Stokes Equations

Energy-based analysis of the three-dimensional incompressible Navier–Stokes equations with spatially localized dissipation is developed at the level of kinetic energy and Leray–Hopf weak solutions. A localized coercivity property linking viscosity and damping to global $L^2$ control is introduced and related to observability and spectral inequalities for the associated Stokes operator. Under this condition, exponential energy decay is established without regularity or linearization assumptions, and time-dependent damping is treated within a nonautonomous framework. A finite-dimensional spectral quantity is proposed as a computable proxy for coercivity, yielding explicit decay estimates for reduced-order models.