Quasi-Steady State Approximation for Partial Differential Equations

具有多个时间尺度的系统出现在科学和工程的各个领域。在这个项目中，我们将研究这些系统的准稳态近似（QSSA），重点是反应扩散方程。QSSA是一种从动力学中快速去除平衡自由度的缩减方法。这种方法起源于化学，但有广泛的应用。在这个项目中，我们将超越常微分方程（ODE）的情况，专注于偏微分方程（PDE）的QSSA，这是一个具有挑战性的主题，因为它的复杂性和丰富的数学结构。主要目标是通过合并两种主要方法，即基于对偶或熵方法的功能分析技术和基于几何慢流形约简的技术，严格研究PDE的QSSA。特别是，我们将从考虑空间扩散的酶反应的Michaelis-Menten动力学的严格推导开始，我们的目标是在功能分析和几何设置中扩展现有技术。此外，我们将对同一模型的结果进行对比、比较和组合，同时研究QSSA对其他一般模型有效性的结构性假设。我们的项目还包括通过能量/熵估计以及分岔理论处理完整和简化模型的长期动力学的组件。我们期望通过这个项目的潜在成功，PDE的QSSA理论将得到显著的发展。

Systems with multiple time scales arise in all areas of science and engineering. In this project, we are going to study the quasi-steady state approximation (QSSA) for these systems with a focus on reaction-diffusion equations. QSSA is a reduction method to remove rapidly equilibrating degrees of freedom from the dynamics. The method originated in chemistry but has found broadrange applications. In this project, we are going beyond the case of ordinary differential equations (ODE) and focus on QSSA for partial differential equations (PDE), which is a challenging topic due to its complexity and rich mathematical structures. A main goal is to rigorously study QSSA for PDE by merging two main approaches, namely functional-analytic techniques based on duality or entropy methods, and techniques based upon geometric slow manifold reductions. In particular, we are going to start with rigorous derivation of Michaelis-Menten kinetics for enzyme reaction taking into account spatial diffusion, where we aim to extend existing techniques in the functional-analytic and geometric settings. In addition, we are going to contrast, compare, and combine the results for the same model, while investigating structural assumptions for validity of QSSA for other general models. Our project also includes components dealing with long-term dynamics of the full and reduced models via energy/entropy estimates as well as via bifurcation theory. It is our expectation that the theory of QSSA for PDE will be significantly advanced through the potential success of this project.