In this project the principal investigator will analyze rigorously several mathematical models of nonlocal phenomena in continuum mechanics. In particular, she will study the question of the existence of solutions of integral models for heat flow with finite wave speeds and with singular memory functions. This work is related to parallel work on similar equations arising in viscoelasticity. The principal investigator will also study nonconvex variational problems with spatially nonlocal terms of the type that occur in ferromagnetism and superconductivity, in collaboration with colleagues at her institution. Many phenomena in the physical world involve what applied mathematicians and scientists call "nonlocal" processes. An example of a nonlocal process is the response characteristic of an elastic material that has been subjected to stretching or some other deformation. The behavior of the material depends on both the past deformations (a nonlocal effect) and the deformation you are applying now (a local effect). As an illustration, pick up your telephone. The cord will deform in some way. All of those twists and turns in the cord are the result of all the past twisting it has received, along with the twist you are applying now. In this project the principal investigator will study rigorously mathematical models of nonlocal processes of the type that occur in heat flow, ferromagnetism and superconductivity.
