Quasi-Steady State Approximation for Partial Differential Equations

多时间尺度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简化方法。该方法起源于化学，但已被广泛应用。在这个项目中，我们超越了常微分方程（ODE）的情况，专注于偏微分方程（PDE）的QSSA，由于其复杂性和丰富的数学结构，这是一个具有挑战性的话题。一个主要目标是严格研究QSSA PDE合并两个主要的方法，即基于对偶或熵方法的功能分析技术，和基于几何慢流形减少的技术。特别是，我们将开始严格推导酶反应的Michaelis-Menten动力学，同时考虑空间扩散，我们的目标是扩展现有的技术在功能分析和几何设置。此外，我们将对比，比较，并结合联合收割机的结果为同一个模型，同时调查的结构假设的有效性QSSA的其他一般模型。我们的项目还包括通过能量/熵估计以及通过分叉理论处理完整和简化模型的长期动态的组件。我们期望通过这个项目的潜在成功，偏微分方程的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.