Well-posedness regime for the macroscopic density–direction system

Determine the parameter regime for which the macroscopic system for the particle density ρ(t,x) and the mean-nematic direction Ω(t,x), given by equations ∂tρ = DxΔxρ + μ ∇x·(K(η(ρ)) ρ ∇xρ) and ∂tΩ + μ Π2(ρ) (∇xρ·∇x)Ω + μ(σ−ν) PΩ⊥ ΔxΩ = 1_{a=2} λ Π3(Ω,ρ), is well-posed (existence, uniqueness, and appropriate regularity), as functions of the parameters Dx, Du, μ, λ, χ and the scaling a.

Background

The paper derives macroscopic equations for the density ρ and mean-nematic direction Ω from a kinetic model featuring an anisotropic Gaussian-type repulsive potential. The resulting PDE system consists of a nonlinear diffusion equation for ρ and a non-conservative evolution equation for Ω that includes transport, diffusion, and, when a=2, additional terms arising from higher-order corrections of the interaction potential.

The authors emphasize that the macroscopic equations depend on multiple parameters (Dx, Du, μ, λ, χ, and the scaling parameter a) and involve nonlinear coefficients such as K(η(ρ)) and Π2(ρ). They note that constraints like ν>σ (i.e., Dx/μ > Du/λ) appear necessary for well-posedness in Ω, but a complete characterization of the well-posedness regime remains unresolved.