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.