Hyperbolic Regularization of Vorticity in Viscous Fluids: A Constitutive Extension of Navier-Stokes with Spectral Cutoff and its Global Regularity in R3

AbstractThe classical three-dimensional incompressible Navier-Stokes equations implicitly assumea Newtonian constitutive law in which the viscous stress tensor responds instantaneously tothe strain rate (τ = 0) and the spatial Fourier spectrum lacks an upper bound (Λ → ∞).This article formulates and proves the global regularity of an extended constitutive model:the Navier-Stokes-Cattaneo system projected onto a band-limited spectral space.Motivated by the dimensional parameterization of kinematic viscosity:ν = 12 cλcwe study the coupled system with viscoelastic relaxation time τ = 2ν/c2 > 0 and sphericalspectral projection operator PΛ (with Λ = λ−1c < ∞) compatibly applied to the convectivenonlinearity and initial data. We rigorously prove: (1) the exact invariance of the band-limited phase space XΛ = HΛ × SΛ; (2) the derivation of the projected telegraphic vorticityequation whose principal differential operator possesses characteristic speed cs = c/√2; and(3) an exact energy dissipation identity via a viscoelastic Lyapunov functional. Apply-ing Plancherel’s theorem and Bernstein-Nikolskii inequalities with explicit constants on theinvariant space XΛ, we obtain time-uniform a priori bounds for all spatial derivatives inHm(R3) and L∞(R3). Using the theory of ordinary differential equations in Banach spacesand an inductive temporal bootstrap argument based on the polynomial smoothness of theevolution vector field, we establish global existence, uniqueness, and smoothness in:C∞([0, ∞); XΛ) ⊂ C∞([0, ∞) × R3)for every fixed set of parameters (ρ0, ν, τ, Λ). Finally, an explicit distinction is drawn betweenthe properties proven for the projected model up to the observable physical limit of vorticesand the open problem of the abstract classical limit (τ → 0+, Λ → ∞)