Topological–Disturbance-Field Fluid Mechanics: An Axiomatic Reconstruction from Global Realism, Corrected Kinetic Theory, and Memory-Extended Continuum Dynamics

Abstract

We present a closed topological–disturbance-field reconstruction of fluid mechanics from three physical axioms: the reality of a spacetime-vacuum substrate, the existence of finite topological material carriers, and persistent causal coupling between carrier motion and retarded disturbance. These axioms generate a noncircular chain from carrier–substrate Hamiltonian dynamics through Liouville, BBGKY, history-dependent Boltzmann, and enlarged Chapman–Enskog equations to objective continuum memory laws. The physical continuum state therefore contains tensorial, scalar, thermal, and compositional memories in addition to density, velocity, temperature, and species variables. Classical Navier–Stokes–Fourier theory is recovered as a controlled low-memory, short-range, weak-nonlocality projection rather than assumed as a complete microscopic foundation. The theory derives exact mass, momentum, energy, species, and entropy balances; separates kinetic, core, topological, historical, and field-momentum stresses; and determines the conversion of relaxed mechanical memory into internal heat without double counting. A common microscopic collision spectrum generates viscosity, thermal conductivity, diffusion, relaxation times, memory lengths, and reciprocal cross effects. Exact energy identities establish stability across shear, thermoacoustic, rotating, stratified, multicomponent, electromagnetic, radiative, electrokinetic, reactive, and multiphase sectors. Porous flow, granular media, cavitation, nonlinear rheology, superfluids, plasmas, combustion, thin films, shallow water, dynamic wetting, active fluids, and fluid–structure interaction are formulated as conservative extensions of the same three-axiom system. For the complete periodic three-dimensional incompressible velocity–memory equations, finite-carrier exclusion and causal concentration response generate high-order coercivity dominating nonlinear transport. Every finite periodic \(H^s\) datum, \(s>5/2\), consequently produces a unique global solution that depends continuously on its initial data, remains smooth for all positive time, and is spatially analytic for every \(t>0\), without a smallness restriction. Sixth-order burst dissipation provides one explicit sufficient realization; the invariant requirement is finite-resolution nonconcentration, high-order coercivity, and exact dissipation balance. The classical memoryless equations remain a degenerate boundary projection and are not substituted for the complete physical state. All constitutive operators are traceable to the three axioms and computable through topology-adapted spectral approximation with explicit residual certificates. Within the declared axiomatic and realization class, global existence, uniqueness, stability, smoothness, and positive-time analyticity of the complete memory Navier–Stokes evolution are thereby closed. The remaining program consists of calculating material-specific operators, conducting cross-regime experiments, and reconstructing fluid measurement and prediction from a common first-principles foundation. **Keywords** Global realism; topological fluid mechanics; disturbance field; memory Navier–Stokes equations; corrected Boltzmann equation; causal continuum mechanics; finite topological carriers; nonlocal transport; turbulence; global regularity.