Local in time solution to an integro-differential system for motion with large deformations and defects

In this paper we consider and generalize a model, recently proposed and analytically investigated in its quasi-stationary approximation by the authors and a co-author, for the motion of a medium with large deformations and conditional compatibility, with occurrence of defects when the magnitude of an internal force is above a given threshold. The model takes the form of a system of integro-differential coupled equations, expressed in terms of the stretch and the rotation tensors variables. Here, its derivation is generalized to consider mixed boundary conditions, which may represent a wider range of physical applications then the case with Dirichlet boundary conditions considered in our previous contribution. This also introduces nontrivial technical difficulties in the theoretical framework, related to the definition and the regularity of the solutions of elliptic operators with mixed boundary conditions. As a novel contribution, we develop the analysis of the fully non-stationary version of the system where we consider inertia. In this context, we prove the existence of a local in time weak solution in three space dimensions, employing techniques from PDEs and convex analysis.