The 2-D Peskin problem describes coupled motion of a 1-D closed elastic string and the ambient Stokes flow in the plane. In existing works, its global well-posedness has been proved when the initial string configuration is close to an equilibrium, which is an evenly-stretched circular configuration. In other words, initial shape of the string needs to be almost circular, and the string is almost evenly-stretched. In this talk, we present some recent progress on pursuing global solutions for a wider class of initial datum. We start from a 1-D simplified model of the 2-D Peskin problem, and show global solutions exist for rather arbitrary initial datum in the energy class. Then we return to the original Peskin problem, and show that certain geometric quantities of the string satisfy extremum principles and decay estimates. As a result, we can prove global well-posedness when the initial data satisfies a medium-size geometric condition on the string shape, while no assumption on the size of stretching is needed. This talk is based on a joint work with Dongyi Wei.
Jiajun Tong is an assistant professor in mathematics at Beijing International Center for Mathematical Research, Peking University. Before joining BICMR in 2021, he obtained Ph.D. from NYU in 2018, and he was a Hedrick Assistant Adjunct Professor at UCLA during 2018-2021. His research interests lie in partial differential equations and applied analysis, especially evolution free boundary problems, PDEs in fluid dynamics, and calculus of variations.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai