Quantitative vanishing-regularization limits for degenerate cross-diffusion with changing boundary strata

Abstract

We study a vanishing-regularization limit for a degenerate cross-diffusion system on the simplex when the active boundary stratum of a smooth reference solution may change in space and time. The mobility $M(u)=\operatorname{diag}u-u\otimes u$ loses rank when a component vanishes, so a fixed relative entropy or an unweighted two-point gradient estimate cannot remain uniformly coercive across moving faces. We introduce a geometric-relative framework that separates the single emerging normal direction from the active tangential subsystem. Its ingredients are a global resolvent geodesic distance, a boundary-oriented rank-one perspective correction, an intrinsic Schur-normal De Giorgi alternative, a mean-free tangential iteration with temporal-flux control, and a terminal coupled Schur compactness argument. These mechanisms yield a non-circular first-exit theorem for smooth approximations and then pass to admissibly approximable entropy weak solutions. If the strong solution stays a fixed distance from the codimension-two skeleton and its vanishing components satisfy $|\nabla v_i|\lesssim v_i^α$ and $|\{0<v_i<r\}|\lesssim r^σ$, then the augmented relative energy is bounded by $ρ_λ=λ^{2α+σ}$ if $2α+σ<2$, $ρ_λ=λ^2|\logλ|$ if $2α+σ=2$, and $ρ_λ=λ^2$ if $2α+σ>2$. For a hypersurface contact $v_i\asymp d^p$, $p\ge2$, this gives $ρ_λ=O(λ^{2-1/p})$ and $\|u^λ-v\|_{L^\infty(0,T;L^2)}=O(λ^{1-1/(2p)})$.