Over the time-scales of seconds to years that are associated with processes such as seismic wave propagation, tidal deformation, and rotational variations, the Earth's crust, mantle and inner core are viscoelastic solids, while the outer core and oceans are compressible fluids whose viscosity's are sufficiently low as to be commonly neglected. Were it not for the presence of the fluid regions, an essentially complete mathematical description of the dynamics could be developed, and the numerical solution of the resulting equations would present no essential difficulties. In this talk I will discuss some of the remaining challenges, both theoretical and computational, that arise within these applications due to the presence of fluid regions and point to some possible methods for their resolution.
