THE β-COHERENT QUARTIC METAMATERIALS: VISCOUS REGULARIZATION, FROBENIUS FILTERING, AND EFFECTIVE METRIC

Abstract

We prove global well-posedness, uniform hyperbolicity τcoh≈0.061 s, ℱ≥0.999669, Q=49.92 and H-convergence for the viscosity-regularized quartic system ρ(∂tv+v·∇v)−μΔv+∇p=βcut∇·(|∇Φ|²∇Φ)χδ, ∂tΦ+v·∇Φ−βcutΔ̂β(Φ)+γradΦ=−S′p(Φ)+J, Δ̂β(Φ)=−β∇·(|∇Φ|²∇Φ). Geometry Ωshape=Ωcav∪Ωthroat∪Ωtip, |Ωcav|=4πR³/3, Sthroat=πrth², 𝒜=|Ωcav|/(SthroatLthroat)=4R³/(3rth²Lthroat), δ=√(2ν/ωc), C₁=2πδ/rth, C₂=4δ/Lthroat·cot(α/2), ℱvisc=1+C₁𝒜³+C₂𝒜², β₀=β₀^{chem}·𝒜³·𝒢form/ℱvisc, β0,sat=β₀^{chem}𝒢form/C₁. Lemma (F(a)−F(b))·(a−b)≥¼|a−b|⁴ with F(a)=|a|²a, DF(z)=|z|²I+2z⊗z. Cutoff q=5 gives N²=1−2ε≥¾ with ε=β0,cut𝒢⁴/(ρc²). Filter S¹_{log p}=ℝ/(2π/log p)ℤ, p≥2 prime, E(ω)=|1−p^{iω}|²=2−2cos(ω log p), Sp[Φ]=∫E|Φ̂|², S′p=2ℱ⁻¹[EΦ̂] 8-Lipschitz L⁴→L^{4/3}. Scaling Φλ(t,x)=Φ(λ⁴t,λx) ⇒ E(λ⁴ω)=E(ω) ∀ω ⇒ λ=1 in ℝ₊, hence external.