Positivity of the ν coefficient in the homogenized Euler model

Prove that the coefficient ν, defined by ν ≡ ζ / (⟨K^{-1}⟩^3 − μ^2) in the homogenized effective system (equations (homog-xxt)) for the one-dimensional compressible Euler equations with spatially periodic entropy K(x) = e^{-s(x)/γ}, is strictly positive for all admissible periodic profiles K(x). Equivalently, establish that the linearized dispersion relation ω^2 = c^2 k^2 / (1 + μ δ^2 k^2 + ν δ^4 k^4) has ν > 0 for general periodic K(x), guaranteeing linear dispersivity for all wavenumbers k ∈ ℝ.

Background

The paper derives a homogenized, constant-coefficient dispersive model (system (homog-xxt)) that approximates long-wavelength behavior of the 1D compressible Euler equations in Lagrangian coordinates when the entropy field s(x) is periodic. Linearizing around an equilibrium state (u=0, p=p_*), the authors obtain a dispersion relation ω^2(1 + μ δ^2 k^2 + ν δ^4 k^4) = c^2 k^2 with μ ≥ 0. They empirically observe ν > 0 for all tested periodic K(x) profiles, which would imply linear dispersivity at all wavenumbers if ν were proven positive in general.

The coefficient ν is defined in terms of averages over one period of K(x) and its inverse: μ = ⟨K^{-1}⟩⟨⟨K^{-1}⟩⟩ / ⟨K^{-1}⟩^2 and ζ = ⟨K^{-1}⟦(K^{-1})^2⟧⟩, with ν ≡ ζ / (⟨K^{-1}⟩^3 − μ^2). Proving ν > 0 for all admissible periodic K(x) would provide a rigorous foundation for the linear stability and dispersive character of the effective model across all wavelengths.