Existence results in the linear dynamics of quasicrystals with phason diffusion and non-linear gyroscopic effects

Quasicrystals are characterized by quasi-periodic arrangements of atoms. The description of their mechanics involves deformation and a (so-called emph{phason}) vector field accounting at macroscopic scale of local phase changes, due to atomic flips necessary to match quasi-periodicity under the action of the external environment. Here we discuss the mechanics of quasicrystals, commenting a shift from its initial formulation, as standard elasticity in a space with dimension twice the ambient one, to a more elaborated setting, which seems to account more deeply for the physics in hands. In the new setting we tackle two problems. First we discuss the linear dynamics of quasicrystals including a phason diffusion. We prove existence of weak solutions and their uniqueness under rather general boundary and initial conditions. We then consider phason rotational inertia, non-linearly coupled with the $mathrm{curl}$ of the macroscopic velocity, and prove once again existence of weak solutions to the pertinent balance equations.