ABSTRACT OF THE TALK
The Poisson-Nernst-Planck (PNP) system is a well-known model of ion transport, which belongs to Keller-Segel type systems and plays a crucial role in the study of many physical and biological phenomena. With a small parameter$\varepsilon $, PNP systems over annular domains have steady state boundary layer solutions with radial symmetry, which profiles form boundary layers near boundary curves and become flat in the interior domain as $\varepsilon $ approaches zero. For the stability of boundary layer solutions to the radial PNP systems, we estimate the solution of the perturbed problem of the radial PNP system (with respect to the boundary layer solution) with global electroneutrality. We prove that the $H_{r}^{-1}$ norm of the solution of the perturbed problem decays exponentially (in time) with exponent independent of $\varepsilon $ if the coefficient of the Robin boundary condition of electrostatic potential has a suitable positive lower bound. The main difficulty is that the gradients of boundary layer solutions on boundaries may blow up as $\varepsilon $ tends to zero. The main idea of our argument is to transform the perturbed problem into another parabolic system with a new and useful energy law for the proof of the exponential decay estimate.
BIOGRAPHY OF THE SPEAKER
Professor Tai-Chia Lin got his Ph.D. degree of Mathematics from Courant Institute of Mathematical Sciences (NYU) in the year 1996, supervised by Professor Fang-Hua Lin. He is a full professor (since 2004) of Mathematics at Taiwan University. He was a long term researcher of IMA (Institute of Mathematics and its Application, University of Minnesota, USA) during 8/1/2008-7/31/2009 and Shapiro visiting professor at Penn State University (USA) during 8/2-9/9 (2012) and 7/8-8/22 (2013). Besides, he was a researcher of Texas A&M University at Qatar during 2012-2014. His research interest includes nonlinear Schrödinger equation, Poisson-Boltzmann equations and Posson-Nernst-Plank equations for physical and biological systems.