Dislocation dynamics

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

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 $V\;'(y)=-\frac{1}{y^s}$. 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 [1]. Further, one can go back between this model and the non-local porous medium equation in 1-d by integrating the solution $u$ with respect to $x$.


  1. Biler, Piotr; Monneau, Régis; Karch, Grzegorz (2009), "Nonlinear Diffusion of Dislocation Density and Self-Similar Solutions", Communications in Mathematical Physics 294 (1): 145–168, doi:10.1007/s00220-009-0855-8, ISSN 0010-3616 
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