On an equation from the theory of field dislocation mechanics

Luc Tartar and Amit Acharya

Bulletin of the Italian Mathematical Union, (9)IV, 409-444, 2011

Global existence and uniqueness results for a quasilinear system of partial differential equations in one space dimension and time representing the transport of dislocation density are obtained. Stationary solutions of the system are also studied, and an infinite dimensional class of equilibria is derived. These time (in)dependent solutions include both periodic and aperiodic spatial distributions of smooth fronts of plastic distortion representing dislocation twist boundary microstructure. Dominated by hyperbolic transport-like features and at the same time containing a large class of equilibria, our system differs qualitatively from regularized systems of hyperbolic conservation laws and neither does it fit into a gradient flow structure.