Shocks and finite-time singularities in Hele-Shaw flow

TL;DR

This paper introduces a dispersive weak solution framework for Hele-Shaw flow with cusp singularities, using shock graphs and algebro-geometrical methods, revealing self-similar solutions involving elliptic functions.

Contribution

It develops a novel dispersive solution approach for Hele-Shaw flow singularities, linking physical shock phenomena with complex algebraic geometry.

Findings

Dispersive solutions resolve ill-defined flow at singularities.

Shock graphs grow and branch, modeling fluid decompression.

Self-similar solutions involve elliptic functions for cusp singularities.

Abstract

Hele-Shaw flow at vanishing surface tension is ill-defined. In finite time, the flow develops cusp-like singularities. We show that the ill-defined problem admits a weak {\it dispersive} solution when singularities give rise to a graph of shock waves propagating in the viscous fluid. The graph of shocks grows and branches. Velocity and pressure jump across the shock. We formulate a few simple physical principles which single out the dispersive solution and interpret shocks as lines of decompressed fluid. We also formulate the dispersive weak solution in algebro-geometrical terms as an evolution of the Krichever-Boutroux complex curve. We study in detail the most generic (2,3) cusp singularity, which gives rise to an elementary branching event. This solution is self-similar and expressed in terms of elliptic functions.