Dislocation dynamics - Revision history

imported>Nestor at 03:28, 24 January 2012

2012-01-24T03:28:09Z

← Older revision		Revision as of 22:28, 23 January 2012
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	\end{array} \right.\]		\end{array} \right.\]

	Then this characteristic function is expected to solve an Eikonal equation with a nonlocal velocity< ref name="FLCM"/>		Then this characteristic function is expected to solve an Eikonal equation with a nonlocal velocity<ref name="FLCM" />

	\[ \left \{ \begin{array}{rll}		\[ \left \{ \begin{array}{rll}
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	{{reflist\|refs=		{{reflist\|refs=
	<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>		<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>
	< 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>		<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>
	}}		}}

imported>Nestor at 03:27, 24 January 2012

2012-01-24T03:27:34Z

← Older revision		Revision as of 22:27, 23 January 2012
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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>		<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>
	< 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>		< 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>


	}}		}}

imported>Nestor at 03:27, 24 January 2012

2012-01-24T03:27:07Z

← Older revision		Revision as of 22:27, 23 January 2012
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	\end{array} \right.\]		\end{array} \right.\]

	Then this characteristic function is expected to solve an Eikonal equation with a nonlocal velocity		Then this characteristic function is expected to solve an Eikonal equation with a nonlocal velocity< ref name="FLCM"/>

	\[ \left \{ \begin{array}{rll}		\[ \left \{ \begin{array}{rll}
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	{{reflist\|refs=		{{reflist\|refs=
	<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>		<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>
			< 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>


	}}		}}

imported>Nestor: /* One dimensional case */

2012-01-24T03:17:23Z

One dimensional case

← Older revision		Revision as of 22:17, 23 January 2012
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	Dislocations are microscopic defects in crystals that change over time (due for instance to shear stresses on the crystal).		Dislocations are microscopic defects in crystals that change over time (due for instance to shear stresses on the crystal).

	== One dimensional case ==		== One dimensional case: dislocation densities ==

	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		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

imported>Nestor: /* One dimensional case */

2012-01-24T03:16:58Z

One dimensional case

← Older revision		Revision as of 22:16, 23 January 2012
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	\[ u_t +\|u_x\|\Lambda^s u = 0 \;\;\;\text{ for all } (x,t) \in \mathbb{R}\times\mathbb{R}_+ \]		\[ u_t +\|u_x\|\Lambda^s u = 0 \;\;\;\text{ for all } (x,t) \in \mathbb{R}\times\mathbb{R}_+ \]

	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 <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 $x$.		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 $x$.

	== Higher dimensions: dislocation lines ==		== Higher dimensions: dislocation lines ==

imported>Nestor at 03:12, 24 January 2012

2012-01-24T03:12:25Z

@@ Line 1: / Line 1: @@
 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
@@ Line 10: / Line 12: @@
 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 <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 $x$.
 == References ==

imported>Nestor at 02:38, 24 January 2012

2012-01-24T02:38:11Z

← Older revision		Revision as of 21:38, 23 January 2012
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	\[ u_t +\|u_x\|\Lambda^s u = 0 \;\;\;\text{ for all } (x,t) \in \mathbb{R}\times\mathbb{R}_+ \]		\[ u_t +\|u_x\|\Lambda^s u = 0 \;\;\;\text{ for all } (x,t) \in \mathbb{R}\times\mathbb{R}_+ \]

	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 <ref name="BMK" />. 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 $x$.		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 <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 $x$.

	== References ==		== References ==

imported>Nestor at 02:37, 24 January 2012

2012-01-24T02:37:45Z

← Older revision		Revision as of 21:37, 23 January 2012
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	\[ u_t +\|u_x\|\Lambda^s u = 0 \;\;\;\text{ for all } (x,t) \in \mathbb{R}\times\mathbb{R}_+ \]		\[ u_t +\|u_x\|\Lambda^s u = 0 \;\;\;\text{ for all } (x,t) \in \mathbb{R}\times\mathbb{R}_+ \]

	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 <ref name="BMK" />.		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 <ref name="BMK" />. 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 ==

imported>Nestor: 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) ..."

2012-01-24T02:36:30Z

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) ..."

New page

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 <ref name="BMK" />.

== References ==
{{reflist|refs=
<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>
}}