Existence of stationary solutions and local minima for 2D models of fine structure dynamics

Abstract

In this paper we study two-dimensional models for the motion of a viscoelastic material with a non-monotone stress-strain relationship. We prove existence of infinitely many stationary solutions to two model problems. This is achieved by constructing sequences of increasingly oscillatory functions, whose limit is a stationary solution. These equilibria may have arbitrarily small energy. We also prove that it is always possible to construct paths in phase space that strictly decrease the energy. This result negates the existence of local minima for the energy and asymptotically stable equilibria. These results are important first steps towards understanding the dynamics of fine structure in more than one dimension.