Unstable fingering patterns of Hele-Shaw flows as a dispersionless limit of the Kortweg-de vries hierarchy

We show that unstable fingering patterns of two-dimensional flows of viscous fluids with open boundary are described by a dispersionless limit of the Korteweg-de Vries hierarchy. In this framework, the fingering instability is linked to a known instability leading to regularized shock solutions for nonlinear waves, in dispersive media. The integrable structure of the flow suggests a dispersive regularization of the finite-time singularities.