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Tuesday, May 02, 2023
4:00 PM - 5:00 PM
Annenberg 104

Special ACM Seminar

Asymptotic Stability in a Free Boundary PDE Model of Active Matter
Leonid Berlyand, Professor of Mathematics, Department of Mathematics, Pennsylvania State University,
Speaker's Bio:
Leonid Berlyand received his Ph. D. in 1985 from Kharkiv University (Ukraine). He joined the Pennsylvania State University (PSU) in 1991 and he is currently a Professor of Mathematics and a member of the Materials Research Institute at PSU. He is a founding co-director of PSU Centers for Interdisciplinary Mathematics and for Mathematics of Living and Mimetic Matter. He is known for his works at the interface between mathematics and other disciplines such as physics, materials sciences, life sciences, and most recently computer science. He co-authored three books and more than 100 publications. His interdisciplinary works received research awards from leading research agencies in the USA, such as NSF, the US Department of Energy, and the National Institute of Health as well as inter-nationally (Bi-National Science Foundation and NATO). Most recently his work was recognized with the Humboldt Research Award of 2021. His teaching excellence was recognized by C.I. Noll Award for Excellence in Teaching by Eberly College of Science at Penn State.

We begin with a brief overview of the rapidly developing research area of active matter (a.k.a. active materials). These materials are intrinsically out of equilibrium resulting in novel physical properties whose modeling requires development of new mathematical tools. We next focus on studying the onset of motion of a living cell (e.g., a keratocyte) driven by myosin contraction with focus on a transition from unstable radial stationary states to stable asymmetric moving states. We introduce a two-dimensional free-boundary PDE model that generalizes a previous one-dimensional model by combining a Keller-Segel model, Hele-Shaw kinematic boundary condition, and the Young-Laplace law with a novel nonlocal regularizing term. This nonlocal term precludes blowup or collapse of the cell by ensuring that membrane-cortex interaction is sufficiently strong. We found a family of asymmetric traveling wave solutions bifurcating from stationary solutions. Our main result is the nonlinear asymptotic stability of traveling wave solutions that model observable steady cell motion. We derived and rigorously justified an explicit asymptotic formula for the stability determining eigenvalue via asymptotic expansions in a small speed of cell. Our spectral analysis reveals the physical mechanisms of stability/instability. It also leads to a novel spectral properties due to the non-self-adjointness of the linearized problem which is a signature of active matter out-of-equilibrium systems. This results in striking math features such as collapse of eigenspaces and presence of generalized eigenvalues and we determine their physical origins.

This is joint work with V. Rybalko and C. Safsten published in Transactions of AMS (2023) and Phys. Rev.B, 2022.

If time permits, we will discuss work in progress on fingering instability in multicellular tissue spreading.

For more information, please contact Diana Bohler by phone at 626-395-1768 or by email at [email protected].