The LSW model with encounters has been suggested by Lifshitz and
Slyozov as a regularization of their classical mean-field model for
domain coarsening to obtain universal self-similar long-time
behavior. We rigorously establish that an exponentially decaying
self-similar solution to this model exist, and show that this
solutions is isolated in a certain function space. Our proof relies
on setting up a suitable fixed-point problem in an appropriate
function space and careful asymptotic estimates of the solution to a
corresponding homogeneous problem.
Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challenging problem to identify reduced macroscopic models that describe some effective dynamics on large spatial and temporal scales. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can by derived from the microscopic system. In the first part we develop a general approach to this problem by considering non-canonical Hamiltonian structures on the tangent bundle. This approach can be applied to all Hamiltonian lattices (or Hamiltonian PDEs) and involves three building blocks: (i) the embedding of the microscopic system, (ii) an invertible two-scale transformation that encodes the underlying scaling of space and time, (iii) an elementary model reduction that is based on a Principle of Consistent Expansions. In the second part we exemplify the reduction approach and derive various reduced PDE models for the atomic chain. The reduced equations are either related to long wave-length motion or describe the macroscopic modulation of an oscillatory microstructure.
Recently, Dreyer and Duderstadt have proposed a modification of the
Becker-Doering cluster equations which now have a nonconvex
Lyapunov function. We start with existence and uniqueness results
for the modified equations. Next we derive an explicit criterion for
the existence of equilibrium states and solve the minimization
problem for the Lyapunov function. Finally, we discuss the long time
behavior in the case that equilibrium solutions do exist.