April 21, 2023
Contact: Junping Shi
Summary
{{https://xinyue-zhao.github.io, Xinyue Zhao}} (Vanderbilt)
Full Description
Title: Theoretical and Numerical Analysis of Non-Radially Symmetric Bifurcation Solutions in a Modified Hele-Shaw Problem
Abstract: Free boundary problems (the time-dependent versions are also known as moving boundary problems) deal with systems of partial differential equations (PDEs) where the domain boundary is apriori unknown. Due to this special characteristic, it is challenging to study free boundary problems both theoretically and numerically. In this talk, I will present a study of a modified Hele-Shaw problem, which is a classical example of free boundary problems. By utilizing the Crandall-Rabinowitz bifurcation theorem, I will demonstrate the existence of a series of non-radially symmetric bifurcation solutions.
I will also introduce a novel neural-network-based method for solving the problem numerically. In the simulations, the method is first verified by computing the bifurcation solutions that are guided by the theoretical bifurcation analysis. Furthermore, the new method's capabilities are validated by computing some non-radially symmetric solutions that cannot be characterized by any theorems.
