Speaker: Po-Chun Kuo (Purdue University)
Title: Dynamics of immersed interface problems in Stokes flow
Abstract: Immersed interface problems in Stokes flow are a fluid structure interaction problem. One of the simplest of such problems is the 2D Peskin problem, in which a 1D closed elastic structure is immersed in a 2D Stokes fluid. This has been studied computationally and analytically. We extend the 2D Peskin problem into two different cases:
(1) 2D inextensible interface problem.
(2) 3D Peskin problem.
In the 2D inextensible interface problem, we assume that the interface is inextensible. Through the boundary integral method, we reformulate the problem into two contour equations, an evolution equation and a tension determination equation. We first study the well-posedness and the regularity of the generalized tension determination problem in Hölder spaces. Next, we use a suitable time-weighted Hölder space to study the well-posedness and the regularity of the dynamic problem.
We also study the Peskin problem in the 3D case. With the boundary integral method, the 3D Peskin may be reformulated to an evolution equation on a unit sphere 𝕊2 for the elastic interface. We use more than one local chart to prove that the problem is well-posed in low-regularity Hölder spaces. Moreover, we prove that the elastic membrane becomes smooth instantly in time.