Quasi-Steady State Approximation for Partial Differential Equations

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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.