In this talk we will discuss regularization of one-phase Hele-Shaw flow, starting from initially Lipschitz surface. We show that if the Lipschitz constant of the initial free boundary is small, then for small positive time the solution of (HS) is smooth and solves (HS) in the classical sense. Global and local version of the result will be presented with estimates on the free boundary speed in t...

We discuss the asymptotic behaviour of weak solutions to the Hele-Shaw and one-phase Stefan problems in exterior domains. We prove that, if the space dimension is greater than one, the asymptotic behaviour is given in both cases by the solution of the Dirichlet exterior problem for the Laplacian in the interior of the positivity set and by a singular, radial and self-similar solution of the Hel...

Existence and uniqueness of solutions for Hele-shaw moving boundary problem for power-law fluid is established in the framework of viscosity solutions.

We show, both theoretically and experimentally, that the interface between two viscous fluids in a Hele-Shaw cell can be nonlinearly unstable before the Saffman-Taylor linear instability point is reached. We identify the family of exact elastica solutions [Nye et al., Eur. J. Phys. 5, 73 (1984)]] as the unstable branch of the corresponding subcritical bifurcation which ends up at a topological ...

We analyze a recent experiment of Sharon et al. (2003) on the coarsening, due to surface tension, of fractal viscous fingering patterns (FVFPs) grown in a radial Hele-Shaw cell. We argue that an unforced Hele-Shaw model, a natural model for that experiment, belongs to the same universality class as model B of phase ordering. Two series of numerical simulations with model B are performed, with t...

Gravitactic bioconvective patterns created by Tetrahymena pyriformis in a Hele-Shaw apparatus were realized and compared with theoretical results. There were found to be two thresholds for bio-convection development: the first indicates the transition from the diffusion to the steady convection state; the second corresponds to the transition from the steady to the unsteady convection state. The...

With the Lattice Boltzmann method (using the BGK approximation) we investigate the dynamics of Hele-Shaw flow under conditions corresponding to various experimental systems. We discuss the onset of the instability (dispersion relation), the static properties (characterization of the interface) and the dynamic properties (growth of the mixing zone) of simulated Hele-Shaw systems. We examine the ...