Breakdown of smoothness for the Muskat problem
  In this paper we show that there exist analytic initial data in the stable regime for the Muskat problem such that the solution turns to the unstable regime and later breaks down i.e. no longer belongs to $C^4$. 
Angel Castro, Diego Cordoba, Charles Fefferman and Francisco Gancedo
math.AP math-ph math.MP physics.flu-dyn