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)=diagu−u⊗ 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 |∇ vi|≲ viα and |{0<vi<r}|≲ rσ, then the augmented relative energy is bounded by ρλ=λ2α+σ if 2α+σ<2, ρλ=λ2|logλ| if 2α+σ=2, and ρλ=λ2 if 2α+σ>2. For a hypersurface contact viasymp dp, p≥2, this gives ρλ=O(λ2−1/p) and |uλ−v|L∞(0,T;L2)=O(λ1−1/(2p)).
Related
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BibTeX
@misc{shchetinin2026quantitative,
title = {{Quantitative vanishing-regularization limits for degenerate cross-diffusion with changing boundary strata}},
author = {E. Yu. Shchetinin and A. A. Shevchuk and S. I. Salpagarov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22079},
eprint = {2610.22079},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22079},
}
RIS
TY - UNPB
TI - Quantitative vanishing-regularization limits for degenerate cross-diffusion with changing boundary strata
AU - E. Yu. Shchetinin
AU - A. A. Shevchuk
AU - S. I. Salpagarov
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22079
UR - https://arxiv.org/abs/2610.22079
AB - 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)=diagu−u⊗ 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 |∇ v_i|≲ v_i^α and |{0<v_i<r}|≲ 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_iasymp d^p, p≥2, this gives ρ_λ=O(λ^2−1/p) and |u^λ−v|_L^∞(0,T;L^2)=O(λ^1−1/(2p)).
ER -