When solving an initial value problem for a PDE, with initial data in some function space class, one popular way to proceed is to apply the contraction mapping (or Picard iteration) method in a suitable function space (usually one related to the function space that the data is in). When this works, this usually gives local or global existence of the solution, as well as uniqueness (if one restricts the class of solutions to an appropriate function space), and continuous (in fact, Lipschitz and analytic) dependence on the initial data (in certain metrics).

However, for some equations it is known that (Lipschitz or analytic) continuous dependence or uniqueness fails at certain regularities. As such, this precludes the possibility of using Picard iteration to build solutions at that regularity, even if existence of the solution can be obtained by other means.

This is particularly the case for "supercritical" equations in which the regularity provided by the various a priori controlled quantities is too weak. Unfortunately, this class of equations includes important examples such as three-dimensional Navier-Stokes, which is a large reason as to why the global regularity problem for this equation is considered hard (see my blog post http://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/ for more discussion).