During the course of my postdoctoral research, my first result "Rigorous derivation of Michaelis-Menten kinetics in the presence of slow diffusion", joint with Bao Quoc Tang, was published in SIAM Journal on Mathematical Analysis. Although the Michaelis-Menten kinetics was proposed more than one hundred years ago in the ODE setting, our work is the first rigorous proof of the Michaelis-Menten kinetics in the PDE setting. We then develop our mathematical methods and ideas to successfully deal with a fast reaction limit in a model for plant-growth dynamics with autotoxicity, joint with Jeffrey Morgan, Cinzia Soresina and Bao Quoc Tang, where we first proposed the projection method to obtain an unconditional result on the convergence rate of this singular limit - a highly challenging question in the topic of fast reaction limits. On the other hand, the reductions from parabolic-parabolic chemotaxis systems to their parabolic-elliptic relatives, called fast signal diffusion limits, are well-known and have been widely used formally for about three decades but have been rigorously studied only for the last several years. My third result proposed rigorous proof for a fast signal diffusion limit in a competitive chemotaxis system of prey and predator, joint with Cordula Reisch and Juan Yang, where the initial layer's effect has been first analysed for this kind of limit. In another mindset, the connection between abstract geometrical theory of dynamical systems and fast reaction limits in cross-diffusion systems was first established in collaboration with Laurent Desvillettes, Christian Kuehn, Jan-Eric Sulzbach and Bao Quoc Tang. More results are obtained in preparation, where we answer essential questions from the above topics.
My current interests generally fall within the analysis of evolutionary partial differential equations in chemistry, biology, and physics, mainly including. In more detail,
Topic 1: PDE theory for chemical reaction network, focused on the Michaelis-Menten kinetics and links to applications in biomedicine;
Topic 2: Fast reaction limits in reaction-diffusion systems, their convergence rates and geometrical theory of dynamical systems; and
Topic 3: Chemotaxis systems: Global existence and blowing up, large-time behaviours, and fast signal diffusion limits.
I am also interested in kinetic equations of the Boltzmann type in the derivation of macroscopic equations, as well as high friction limits. The following explanation will describe my main research subjects, including brief state-of-the-art.