Hele-Shaw Flow Near Cusp Singularities

Abstract

This thesis discusses the radial version of the Hele-Shaw problem. Different from the channel version, traveling-wave solutions do not exist in this version. Under algebraic potentials, in the case that the droplets expand, in finite time, cusps will appear on the boundary and classical solutions may not exist afterwards. Physicists have suggested that for (2p+1,2)-cusps, that near cusp singularities of Hele-Shaw flow, after scaling X, Y by some powers of time t respectively, the main part of Y(X, t) is a one-parameter family and does not depend on time t. They have also suggested that the solutions of the Hele-Shaw problem are connected with dispersionless KdV (dKdV) hierarchy. In this study, we rigorously proved that this is the case for (3,2)-cusps when the droplets are simply connected and the external potentials are algebraic. We gave exact solutions and showed that the main parts of the exact solutions are some special solutions of the dispersionless string equation. More over, borrowed from the physical paper$\cite{Teo}$ with a little more details, we showed the arguments of how these special solutions are related to dKdV hierarchy.