Thierry Gallay, Alexander Mielke
Convergence results for a coarsening model using global linearization
(641K, Postscript)

ABSTRACT.  We study a coarsening model describing the dynamics of interfaces 
in the one-dimensional Allen-Cahn equation. Given a partition of 
the real line into intervals of length greater than one, the model 
consists in constantly eliminating the shortest interval of 
the partition by merging it with its two neighbors. We show that 
the mean-field equation for the time-dependent distribution of 
interval lengths can be explicitly solved using a global linearization 
transformation. This allows us to derive rigorous results on the 
long-time asymptotics of the solutions. If the average length of 
the intervals is finite, we prove that all distributions approach 
a uniquely determined self-similar solution. We also obtain global 
stability results for the family of self-similar profiles which 
correspond to distributions with infinite expectation.