Hyperbolic Regularization of Vorticity in Viscous Fluids: A Constitutive Extension of Navier-Stokes with Spectral Cutoff and its Global Regularity in R3 V3

AbstractThe classical three-dimensional incompressible Navier-Stokes equations implicitly assumeaNewtonian constitutive law in which the viscous stress tensor responds instantaneously to the strain rate (τ = 0) and the spatial Fourier spectrum lacks an upperbound(Λ →∞). Thispaperformulatesandprovestheglobalregularityofanextendedconstitutive model: the Navier-Stokes-Cattaneo system projected onto a band-limitedspectral space. Motivated by the dimensional parameterization of kinematic viscosity:ν = 12 cλcwe study the coupled system with viscoelastic relaxation time τ = 2ν/c2 > 0andspherical spectral projection operator PΛ (with Λ = λ−1c <∞)compatiblyapplied to the convective nonlinearity and initial data. Werigorouslyprove: (1)theexactinvarianceoftheband-limited phase space XΛ = HΛ ×SΛ; (2) the derivation of the projected telegraphequation for vorticity whose principal differential operator has characteristic speedcs = c/√2;and(3)anexactenergydissipation identity via a viscoelastic Lyapunov functional. Applying the Plancherel and Bernstein-Nikolskii inequalities with explicit constants on the invariant space XΛ, uniform-in-time a priori bounds are obtained for allspatial derivatives in Hm(R3) and L∞(R3). Using ordinary differential equation theoryin Banach spaces and an inductive temporal bootstrap argument based on the polynomial smoothness of the evolution field, global existence, uniqueness, and smoothnessare proven in:C∞([0,∞);XΛ) ⊂ C∞([0,∞)×R3)for any fixed set of parameters (ρ0,ν,τ,Λ). Finally, an explicit distinction is made between the properties proven for the projected model up to the observable physicallimit of vortices and the open problem of the abstract classical limit (τ → 0+, Λ → ∞).