The Kappa-Navier-Stokes Equations: Fluids Deviating From Maxwellian Velocity Distribution

We derive the κ-Navier-Stokes (κ-NS) equations governing compressible and incompressible-
pressible flows whose molecular velocity distribution follows a kappa (power-law) rather than
a Maxwellian distribution. The derivation rests on three ingredients: the Fokker-Planck
collision operator as the Wasserstein gradient flow of free energy; an SO(3)-symmetric dis-
sipation metric that fixes the viscous stress up to two parameters, and the moment method
applied to the kappa distribution using only second moments. The resulting system couples
the standard fluid equations to a transport equation for the non-Maxwellianity parameter
η = 1/κ ∈ [0, 23 ). The shear viscosity µ(η) = µ0 /(1 − 3/2 η) diverges at η = 2/3 , while the pro-
duction of η vanishes there with the same factor: an aligned-singularity mechanism
that geometrically prevents velocity-gradient blow-up. Full derivations are collected in the
appendices for the interested reader; the body of the paper presents the equations, the key
lemmas, and their physical interpretation.

Keywords:   kappa distribution, Navier-Stokes equations, kinetic theory, Boltzmann equation, non-Maxwellian, Fokker-Planck, Wasserstein gradient flow, viscosity