# Dislocation dynamics

(Difference between revisions)
 Revision as of 02:36, 24 January 2012 (view source)Nestor (Talk | contribs) (Created page with "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) ...")← Older edit Revision as of 02:37, 24 January 2012 (view source)Nestor (Talk | contribs) Newer edit → Line 9: Line 9: $u_t +|u_x|\Lambda^s u = 0 \;\;\;\text{ for all } (x,t) \in \mathbb{R}\times\mathbb{R}_+$ $u_t +|u_x|\Lambda^s u = 0 \;\;\;\text{ for all } (x,t) \in \mathbb{R}\times\mathbb{R}_+$ - 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 . + 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 . 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 $x$. == References == == References ==

## Revision as of 02:37, 24 January 2012

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 $x$.

## References

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