Nonlinear Regularization of Viscous Incompressible and Compressible Hydrodynamics: A Comparative Analysis of Approaches

Nonlinear Regularization of Viscous Incompressible and Compressible Hydrodynamics: A Comparative Analysis of Approaches Description This work presents a comprehensive review of regularization methods for the Navier–Stokes equations, addressing both theoretical well-posedness and numerical stability for flows with extreme velocity gradients. The study systematically examines classical approaches - including von Neumann-Richtmyer artificial viscosity, TVD/WENO schemes, LES/RANS turbulence models, and entropy-stable schemes - while providing detailed mathematical analysis of nonlinear damping terms of the form λ|u|^α u. The central contribution is the Moss Solution, a physically-motivated regularization with α = 2 where the coupling parameter λ = G/c³ is uniquely determined by fundamental constants (gravitational constant G, speed of light c) rather than empirical fitting. Global existence and regularity for the modified system is established through a careful Gagliardo–Nirenberg interpolation argument, extending known results beyond the α ≥ 3 threshold. Key features of this work: Complete mathematical framework: Energy estimates, Gagliardo–Nirenberg interpolation chain, proof of global regularity for α = 2 Systematic comparison: Detailed analysis of Moss Solution versus Leray regularization, Navier–Stokes–Voigt, hyperviscosity, and stochastic forcing Historical analogy: Parallel drawn between Navier–Stokes blow-up and Planck's resolution of the ultraviolet catastrophe - both represent physical completion of incomplete classical theories Cosmological connection: Vacuum response mechanism derived within the Theory of Temporal Spheres (TTS), linking hydrodynamic regularization to dodecahedral spacetime topology Planck scale bounds: Natural UV cutoff emerging from the same constants (G, c, ℏ) that define the damping parameter Experimental predictions: Testable consequences for quark-gluon plasma, neutron star mergers, relativistic jets, and high-Reynolds-number CFD simulations Practical CFD applications: One-line code modification providing physically-grounded stabilization for aerospace and combustion simulations The work proposes a paradigm shift: rather than treating numerical blow-up as a computational nuisance requiring ad hoc patches, it recognizes blow-up as a signal of physical incompleteness—classical equations neglect vacuum response. When this response is included, global smooth solutions exist, and classical hydrodynamics emerges as an approximation valid when vacuum effects are negligible. References: See bibliography in manuscript (50+ citations including foundational works by Leray, Nirenberg, Gagliardo, Serrin, Ladyzhenskaya, and contemporary results on damped Navier–Stokes systems). "The vacuum is not nothing - it is the fabric from which everything emerges." --- Ξυα Mσςς[email protected]