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imported>Luis |
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| Dislocations are microscopic defects in crystals that change over time (due for instance to shear stresses on the crystal).
| | Can you explain how this page works? Are you suggesting that every time someone writes an article we should update this page with the bibliography? ([[User:Luis|Luis]] 09:54, 28 January 2012 (CST)) |
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| == One dimensional case: dislocation densities ==
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| 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
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| \[ \dot{y}_i=F-V\;'_0(y_i) - \sum \limits_{j \neq i} V\;'(y_i-y_j) \;\;\;\text{ for } i=1,...,N, \]
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| 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
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| \[ u_t +|u_x|\Lambda^s u = 0 \;\;\;\text{ for all } (x,t) \in \mathbb{R}\times\mathbb{R}_+ \]
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| 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 <ref name="BMK" />. 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 $x$.
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| == Higher dimensions: dislocation lines ==
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| 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
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| \[ \rho(x,t) = 1_{\Omega_t}(x):= \left \{ \begin{array}{rl}
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| 1 & \mbox{ if } x \in \Omega_t \\
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| 0 & \mbox{ if } x \not \in \Omega_t
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| \end{array} \right.\]
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| Then this characteristic function is expected to solve an Eikonal equation with a nonlocal velocity< ref name="FLCM"/>
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| \[ \left \{ \begin{array}{rll}
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| \rho_t & = (k\star \rho) \left |\nabla \rho \right | & \text{ in } \mathbb{R}^2\times (0,T)\\
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| \rho(.,0) & = 1_{\Omega_0} & \text{ in } \mathbb{R}^2
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| \end{array}\right.
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| \]
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| == References ==
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| {{reflist|refs=
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| <ref name="BMK">{{Citation | last1=Biler | first1=Piotr | last2=Monneau | first2=Régis | last3=Karch | first3=Grzegorz | title=Nonlinear Diffusion of Dislocation Density and Self-Similar Solutions | doi=10.1007/s00220-009-0855-8 | year=2009 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=294 | issue=1 | pages=145–168}}</ref>
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| < ref name="FLCM">{{Citation | last1=Forcadel | first1=N. | last2=Lio | first2=F. | last3=Cardaliaguet | first3=P. | last4=Monneau | first4=Régis | title=Dislocation dynamics: a non-local moving boundary | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=2007 | journal=Free boundary problems | pages=125–135}}</ref>
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| }}
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