Weak solutions to lubrication systems describing the evolution of   bilayer thin films
  The existence of global nonnegative weak solutions is proved for coupled one-dimensional lubrication systems that describe the evolution of nanoscopic bilayer thin polymer films that take account of Navier-slip or no-slip conditions at both liquid-liquid and liquid-solid interfaces. In addition, in the presence of attractive van der Waals and repulsive Born intermolecular interactions existence of positive smooth solutions is shown. 
Sebastian Jachalski, Georgy Kitavtsev, Roman Taranets
math.AP