Title: Parabolic-elliptic and indirect-direct simplifications in chemotaxis systems driven by indirect signalling Show Chinese title

Title: 抛物-椭圆型和间接-直接简化在由间接信号驱动的趋化系统中的应用

Abstract: Singular limits for the following indirect signalling chemotaxis system \begin{align*} \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) %(n,c,w)_{t=0} = (n_0,c_0,w_0) & \text{on } \Omega, \end{array} \right. \end{align*} are investigated. More precisely, we study parabolic-elliptic simplification, or PES, $\varepsilon\to 0^+$ with fixed $\tau>0$ up to the critical dimension $N=4$, and indirect-direct simplification, or IDS, $(\varepsilon,\tau)\to (0^+,0^+)$ up to the critical dimension $N=2$. 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 $L^p$-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. Show Chinese abstract

Abstract: 对于以下间接信号传递趋化系统\begin{align*} \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) %(n,c,w)_{t=0} = (n_0,c_0,w_0) & \text{on } \Omega, \end{array} \right. \end{align*}的奇异极限进行了研究。 更准确地说，我们研究了抛物-椭圆简化，或 PES，$\varepsilon\to 0^+$，在固定$\tau>0$直到临界维数$N=4$的情况下，以及间接-直接简化，或 IDS，$(\varepsilon,\tau)\to (0^+,0^+)$，直到临界维数$N=2$。 这些情况在生物情境中是相关的，其中信号传递过程的时间尺度比物种扩散和所有相互作用要快得多。 在临界维数中显示奇异极限是具有挑战性的。 为了处理 PES，我们仔细结合熵函数、一种 Adam 类不等式、慢演化正则化和能量方程方法，以在代表性空间中获得强收敛。 对于 IDS，设计了一个关于$L^p$-能量函数的归纳论证，这使我们能够为奇异极限获得合适的统一界限。 此外，在这两种情况下，我们也展示了收敛速率，其中初始层的影响和收敛到临界流形的情况也被揭示出来。