J.-L. Lions' Problem Concerning Maximal Regularity of Equations Governed   by Non-Autonomous Forms
  An old problem due to J.-L. Lions going back to the 1960s asks whether the abstract Cauchy problem associated to non-autonomous forms has maximal regularity if the time dependence is merely assumed to be continuous or even measurable. We give a negative answer to this question and discuss the minimal regularity needed for positive results. 
Stephan Fackler
math.FA math.AP