Dislocation dynamics

Dislocations are microscopic defects in crystals that change over time (due for instance to shear stresses on the crystal).

One dimensional case

If we have a finite number of parallel (horizontal) lines on a 2D crystal each given by the equation $y=y_i$ ($y_i \in \mathbb{R}$) then a simplified model for the evolution of these lines says that the positions of these lines evolve according to the system of ODEs

\[ \dot{y}_i=F-V\;'_0(y_i) - \sum \limits_{j \neq i} V\;'(y_i-y_j) \;\;\;\text{ for } i=1,...,N, \]

One can consider the case in which $N \to +\infty$ and consider the evolution of a density of dislocation lines. If $u(x,t)$ denotes the limiting density, then the it solves the integro-differential equation

\[ u_t +|u_x|\Lambda^s u = 0 \;\;\;\text{ for all } (x,t) \in \mathbb{R}\times\mathbb{R}_+ \]

in the case where the interaction potential $V\;\;$ satisfies $V\;'(y)=-\frac{1}{y^s}$. We note that $\Lambda$ above denotes the Zygmund operator, also known was $(-\Delta)^{1/2}$. For this one dimensional model (which enjoys a maximum principle) a complete theory in terms of viscosity solutions, including existence, uniqueness and regularity was recently developed . Further, one can go back between this model and the nonlocal porous medium equation in 1-d by integrating the solution $u$ with respect to $x$.

Higher dimensions: dislocation lines

If we drop the assumption that the dislocations occur along parallel lines we can study a different regime of the problem, instead of considering a density of parallel dislocation lines we can focus on the evolution of the shape of a single dislocation line. If $\Gamma_t$ is a dislocation line and the boundary of an open set $\Omega_t$, and if we let

\[ \rho(x,t) = 1_{\Omega_t}(x):= \left \{ \begin{array}{rl} 1 & \mbox{ if } x \in \Omega_t \\ 0 & \mbox{ if } x \not \in \Omega_t \end{array} \right.\]

Then this characteristic function is expected to solve an Eikonal equation with a nonlocal velocity

\[ \left \{ \begin{array}{rll} \rho_t & = (k\star \rho) \left |\nabla \rho \right | & \text{ in } \mathbb{R}^2\times (0,T)\\ \rho(.,0) & = 1_{\Omega_0} & \text{ in } \mathbb{R}^2 \end{array}\right. \]