Regularization of moving boundaries in a Laplacian field by a mixed   Dirichlet-Neumann boundary condition: exact results
  The dynamics of ionization fronts that generate a conducting body, are in simplest approximation equivalent to viscous fingering without regularization. Going beyond this approximation, we suggest that ionization fronts can be modeled by a mixed Dirichlet-Neumann boundary condition. We derive exact uniformly propagating solutions of this problem in 2D and construct a single partial differential equation governing small perturbations of these solutions. For some parameter value, this equation can be solved analytically which shows that the uniformly propagating solution is linearly convectively stable. 
B. Meulenbroek, U. Ebert, L. Schaefer