  We prove the existence of self-similar solutions to the Fradkov model for
two-dimensional grain growth, which consists of an infinite number of
nonlocally coupled transport equations for the number densities of grains with
given area and number of neighbours (topological class). For the proof we
introduce a finite maximal topological class and study an appropriate
upwind-discretization of the time dependent problem in self-similar variables.
We first show that the resulting finite dimensional differential system has
nontrivial steady states. Afterwards we let the discretization parameter tend
to zero and prove that the steady states converge to a compactly supported
self-similar solution for a Fradkov model with finitely many equations. In a
third step we let the maximal topology class tend to infinity and obtain
self-similar solutions to the original system that decay exponentially.
Finally, we use the upwind discretization to compute self-similar solutions
numerically.
