# Parabolic-elliptic and indirect-direct simplifications in chemotaxis systems driven by indirect signalling

## TL;DR

This paper analyzes mathematical models of cell movement influenced by chemical signals, focusing on simplifications in different dimensions and their convergence rates.

## Contribution

The paper introduces new methods for analyzing singular limits in chemotaxis systems up to critical dimensions.

## Key findings

- Parabolic-elliptic simplification is studied up to the critical dimension N=4.
- Indirect-direct simplification is analyzed up to the critical dimension N=2.
- Convergence rates and initial layer effects are revealed for both scenarios.

## Abstract

Singular limits for the following indirect signalling chemotaxis system \documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{array}{lllllll} \partial _t n = \Delta n - \nabla \cdot (n \nabla c ) & \text {in } \Omega \times (0,\infty ) , \\ \varepsilon \partial _t c = \Delta c - c + w & \text {in } \Omega \times (0,\infty ), \\ \varepsilon \partial _t w = \tau \Delta w - w + n & \text {in } \Omega \times (0,\infty ), \\ \partial _\nu n = \partial _\nu c = \partial _\nu w = 0, & \text {on } \partial \Omega \times (0,\infty ) \end{array} \right. \end{aligned}$$\end{document}∂tn=Δn-∇·(n∇c)inΩ×(0,∞),ε∂tc=Δc-c+winΩ×(0,∞),ε∂tw=τΔw-w+ninΩ×(0,∞),∂νn=∂νc=∂νw=0,on∂Ω×(0,∞)are investigated. More precisely, we study parabolic-elliptic simplification, or PES, \documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon \rightarrow 0^+$$\end{document}ε→0+ with fixed \documentclass[12pt]{minimal}
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\begin{document}$$\tau >0$$\end{document}τ>0 up to the critical dimension \documentclass[12pt]{minimal}
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\begin{document}$$N=4$$\end{document}N=4, and indirect-direct simplification, or IDS, \documentclass[12pt]{minimal}
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\begin{document}$$(\varepsilon ,\tau )\rightarrow (0^+,0^+)$$\end{document}(ε,τ)→(0+,0+) up to the critical dimension \documentclass[12pt]{minimal}
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\begin{document}$$N=2$$\end{document}N=2. These are relevant in biological situations where the signalling process is on a much faster time scale compared to the species diffusion and all interactions. Showing singular limits in critical dimensions is challenging. To deal with the PES, we carefully combine the entropy function, an Adam-type inequality, the regularisation of slow evolution, and an energy equation method to obtain strong convergence in representative spaces. For the IDS, a bootstrap argument concerning the \documentclass[12pt]{minimal}
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\begin{document}$$L^p$$\end{document}Lp-energy function is devised, which allows us to obtain suitable uniform bounds for the singular limits. Moreover, in both scenarios, we also present the convergence rates, where the effect of the initial layer and the convergence to the critical manifold are also revealed.