Unstable fingering patterns of Hele-Shaw flows as a dispersionless limit   of the KdV 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 KdV 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. 
R. Teodorescu, A. Zabrodin, P. Wiegmann