We develop a geometric framework for fluid dynamics with kappa-distributed velocity distributions, extending the classical Navier-Stokes equations to systems where the molecular velocity distribution deviates from Maxwellian equilibrium. The theory rests on three pillars: (1) the Fokker-Planck collision operator as the Wasserstein gradient flow of free energy, yielding transport coefficients without collision integrals; (2) a moment method that derives viscosity using only second moments of the distribution, avoiding singularities associated with fourth moments; and (3) an evolution equation for the non-Maxwellianity parameter η derived directly from second-moment kinetics. The central result is an aligned singularity structure: the viscosity μ(η) = ρθ²τc/2(1−3η/2) diverges as η → 2/3, while the η production rate vanishes proportionally to (1−3η/2). These opposing behaviors, controlled by the same factor, provide natural geometric regularization—the system self-limits, preventing finite-time blowup without imposed constraints. We present complete coupled systems for both incompressible and compressible flows, including derivations of all transport coefficients (shear viscosity, bulk viscosity, thermal conductivity) from first principles. The framework applies to turbulent flows at small scales, weakly collisional plasmas, and rarefied gases where power-law tails invalidate the Maxwellian assumption underlying classical fluid mechanics. Keywords: kappa distribution, Navier-Stokes equations, kinetic theory, Boltzmann equation, non-Maxwellian, Fokker-Planck, Wasserstein gradient flow, viscosity
Shlomo Ta'asan (Tue,) studied this question.