Explicit lower bound for the nonlinear diffusion coefficient K(η(ρ))

Derive an explicit lower bound (in particular, establish non-negativity) for the nonlinear diffusion coefficient K(η(ρ)) = 1 − χ^2/n − σ (n−1)/n S2(η(ρ)) η′(ρ) appearing in the density equation, as a function of the anisotropy χ, the ratio σ = Du/λ, and the order parameter S2(η(ρ)).

Background

The macroscopic density equation contains a nonlinear diffusion term with coefficient K(η(ρ)). The authors provide numerical evidence that K is non-negative and may admit a lower bound of 1−χ^2, but they lack an analytical proof. Ensuring K ≥ 0 is important for the parabolic character and stability of the density equation.

An explicit analytic lower bound would clarify the role of anisotropy and noise parameters and would strengthen the well-posedness theory for the macroscopic density equation.