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Improved convergence rates in the fast-reaction approximation of the triangular Shigesada-Kawasaki-Teramoto system

Hector Bouton

TL;DR

This work quantifies the fast-reaction (triangular) limit of the Shigesada–Kawasaki–Teramoto cross-diffusion system on bounded domains with $d\le 3$. It introduces energy-based regularity tools and a new pair of functionals to establish uniform-in-$\varepsilon$ estimates and to control the convergence toward the limit SKT system, obtaining explicit rates on the whole domain in $L^{\infty}(0,T;L^{2}(\Omega))$ and $L^{2}(0,T;H^{1}(\Omega))$ as $O(\varepsilon^{1/2})$, plus an improved rate in $L^{2}(0,T;L^{2}(\Omega))$ that depends on the initial layer via $\int_{\Omega}|Q^{\varepsilon}(0)|^{2}\varepsilon$. In interior subdomains, the paper proves higher-regularity convergence in $L^{\infty}(0,T;H^{l}(\tilde{\Omega}))$ for all $l>0$ with rates $O(\varepsilon^{1/2})$ plus data-dependent terms $O(\varepsilon)$, under appropriate smoothness and affine hypotheses on the diffusion-reaction coefficients. An initial-layer analysis shows the rapid relaxation of the fast variable on the time scale $t_{\varepsilon}=\varepsilon|\ln\varepsilon|/(2S)$, with refined estimates at $t_{\varepsilon}$ and $t'_{\varepsilon}$, clarifying the transient behavior before the asymptotic regime is reached. The approach combines entropy-energy bounds, maximal-regularity arguments, and high-order interior regularity to deliver explicit convergence rates in the fast-reaction setting for a biologically meaningful cross-diffusion system.

Abstract

We consider the fast-reaction approximation to the triangular Shigesada-Kawasaki-Teramoto model on a bounded domain in the physical dimension $d\le 3$. We provide explicit convergence rates on the whole domain in $\textnormal{L}^\infty\textnormal{L}^2\cap\textnormal{L}^2\textnormal{H}^1$ and in the interior we prove convergence with an explicit rate in any $\textnormal{L}^\infty\textnormal{H}^l$ for all $l > 0$.

Improved convergence rates in the fast-reaction approximation of the triangular Shigesada-Kawasaki-Teramoto system

TL;DR

This work quantifies the fast-reaction (triangular) limit of the Shigesada–Kawasaki–Teramoto cross-diffusion system on bounded domains with . It introduces energy-based regularity tools and a new pair of functionals to establish uniform-in- estimates and to control the convergence toward the limit SKT system, obtaining explicit rates on the whole domain in and as , plus an improved rate in that depends on the initial layer via . In interior subdomains, the paper proves higher-regularity convergence in for all with rates plus data-dependent terms , under appropriate smoothness and affine hypotheses on the diffusion-reaction coefficients. An initial-layer analysis shows the rapid relaxation of the fast variable on the time scale , with refined estimates at and , clarifying the transient behavior before the asymptotic regime is reached. The approach combines entropy-energy bounds, maximal-regularity arguments, and high-order interior regularity to deliver explicit convergence rates in the fast-reaction setting for a biologically meaningful cross-diffusion system.

Abstract

We consider the fast-reaction approximation to the triangular Shigesada-Kawasaki-Teramoto model on a bounded domain in the physical dimension . We provide explicit convergence rates on the whole domain in and in the interior we prove convergence with an explicit rate in any for all .
Paper Structure (14 sections, 14 theorems, 233 equations)

This paper contains 14 sections, 14 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Main results and key ideas
  3. Review of the literature
  4. Notations
  5. Organization of the paper
  6. Regularity estimates : proof of Theorem 2
  7. Initial regularity
  8. Part I: estimate in LinftyH1 and L2H2
  9. Part 2: Higher regularity estimates
  10. Rate of convergence : proof of Theorem 1
  11. Regularity of the limit problem
  12. Part I: convergence in LinftyL2 and L2H1
  13. Part 2: convergence in a subdomain in LinftyHl for all l
  14. Initial layer

Key Result

Proposition 1

Let $d \leqslant 4$ and $\Omega \subset \mathbb{R}^d$ be a smooth bounded domain, and let $d_u,d_v, \sigma >0$, $r_u,r_v\ge 0$, $d_{ij} > 0$ for $i,j=1,2$. We suppose that $u_\mathrm{in}, v_\mathrm{in} \ge 0$ are initial data which lie in $\mathcal{C}^2(\overline{\Omega})$ and are compatible with th

Theorems & Definitions (28)

  • Proposition 1: Theorem 1 of boutondesvillettes2025
  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Proposition 2
  • Remark 6
  • Corollary 1
  • ...and 18 more