  We study the mean curvature flow in 3-dimensional null hypersurfaces. In a
spacetime a hypersurface is called null, if its induced metric is degenerate.
The speed of the mean curvature flow of spacelike surfaces in a null
hypersurface is the projection of the codimension-two mean curvature vector
onto the null hypersurface. We impose fairly mild conditions on the null
hypersurface. Then for an outer un-trapped initial surface, a condition which
resembles the mean-convexity of a surface in Euclidean space, we prove that the
mean curvature flow exists for all times and converges smoothly to a marginally
outer trapped surface (MOTS). As an application we obtain the existence of a
global foliation of the past of an outermost MOTS, provided the null
hypersurface admits an un-trapped foliation asymptotically.
