Mathematics of Active Gels: Stability & Traveling Waves
Leonid Berlyand, Penn State University
Sala conferenze IMATI-CNR, Pavia – Martedì 14 Maggio 2019 h.16:00
Abstract. In this review talk we demonstrate that mathematical analysis can be used in the study of active gels.
We first observe that out-of-equilibrium state of active matter such (e.g., active gels) leads to mathematical challenges and requires developments of novel mathematical tools. Next we discuss three models of active gels that capture key biophysical features (such as persistent & turning motions and symmetry breaking), while having a minimal set of parameters and variables.
Our goal is to provide theoretical understanding of cell polarity phenomenon via mathematical analysis of stability/instability and bifurcation from steady states to traveling waves. This is done by identification of key mathematical structures behind the models such as gradient coupling in Phase-Field model, Liouville equation, Keller-Segel cross-diffusion, and nonlinearity due to free boundary conditions, e.g. Hele-Show type. We
employ mathematical techniques of (i) sharp interface limit via asymptotic analysis, (ii) construction of steady states and traveling waves via Crandall-Rabinowitz bifurcation theory and (iii) topological methods such as Lerey-Schauder degree theory.
These are joint works with V. Rybalko (ILTPE, Kharkiv, Ukraine), J. Fuhrman (PSU & Mainz, Germany), M. Potomkin (PSU, USA).