58E35, 58E50, 34A36, 91B60.

This paper presents novel results about the structure of solutions for certain evolutionary variational inequality problems. We show that existence of piecewise solutions is dependant upon the form of the constraint set underlying the evolutionary variational inequality problem considered. We discuss our results in the context of double-layered dynamics theory and we apply them to the modelling of traffic network equilibrium problems, in particular to the study of the evolution of such problems in a neighbourhood of a steady state.;