Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one

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April 2012, 32(4): 1255-1286. doi: 10.3934/dcds.2012.32.1255

Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one

María del Mar González ^1, and Regis Monneau ^2,

1.	Universitat Politècnica de Catalunya, ETSEIB - Departament de MA1, Av. Diagonal, 647, 08028 Barcelona, Spain
2.	Université Paris-Est, Cermics, Ecole des Ponts ParisTech, 6 et 8 avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France

Received July 2010 Revised August 2011 Published October 2011

Abstract
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We consider a reaction-diffusion equation with a half-Laplacian. In the case where the solution is independent on time, the model reduces to the Peierls-Nabarro model describing dislocations as transition layers in a phase field setting. We introduce a suitable rescaling of the evolution equation, using a small parameter $\varepsilon$. As $\varepsilon$ goes to zero, we show that the limit dynamics is characterized by a system of ODEs describing the motion of particles with two-body interactions. The interaction forces are in $1/x$ and correspond to the well-known interaction between dislocations.

Keywords: singular limit, Non-local Allen-Cahn equation, particle systems., Peierls-Nabarro model, transition layer, reaction-diffusion equation.

Mathematics Subject Classification: 35Q99, 35B40, 35J25, 35D30, 35G25, 70F9.

Citation: María del Mar González, Regis Monneau. Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1255-1286. doi: 10.3934/dcds.2012.32.1255

References:

[1]	G. Alberti, G. Bouchitté and P. Seppecher, Un résultat de perturbations singulières avec la norme $H^{1/2}$, C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 333-338.
[2]	G. Alberti, G. Bouchitté and P. Seppecher, Phase transition with the line-tension effect, Arch. Rational Mech. Anal., 144 (1998), 1-46. doi: 10.1007/s002050050111.
[3]	O. Alvarez, P. Hoch, Y. Le Bouar and R. Monneau, Dislocation dynamics: Short-time existence and uniqueness of the solution, Arch. Ration. Mech. Anal., 181 (2006), 449-504. doi: 10.1007/s00205-006-0418-5.
[4]	G. Barles and C. Imbert, Second-order elliptic integro-differential equations: Viscosity solutions' theory revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 567-585.
[5]	H. Brezis, "Analyse Fonctionnelle. Théorie et Applications," Collection Mathématiques Appliquées pour la Maîtrise [Collection of Applied Mathematics for the Master's Degree], Masson, Paris, 1983.
[6]	L. Bronsard and D. Hilhorst, On the slow dynamics for the Cahn-Hilliard equation in one space dimension, Proc. Roy. Soc. London Ser. A, 439 (1992), 669-682. doi: 10.1098/rspa.1992.0176.
[7]	L. Bronsard and R. V. Kohn, On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math., 43 (1990), 983-997. doi: 10.1002/cpa.3160430804.
[8]	X. Cabré and Y. Sire, Non-linear equations for fractional Laplacians I: Regularity, maximum principles and Hamiltoniam estimates,, preprint, ().
[9]	X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732. doi: 10.1002/cpa.20093.
[10]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
[11]	J. Carr and R. L. Pego, Metastable patterns in solutions of $u_t=\epsilon^2u_{mathcalxmathcalx}-f(u)$, Comm. Pure Appl. Math., 42 (1989), 523-576. doi: 10.1002/cpa.3160420502.
[12]	X. Chen, Generation, propagation, and annihilation of metastable patterns, J. Differential Equ., 206 (2004), 399-437. doi: 10.1016/j.jde.2004.05.017.
[13]	F. Da Lio, N. Forcadel and R. Monneau, Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocation dynamics, J. Eur. Math. Soc. (JEMS), 10 (2008), 1061-1104. doi: 10.4171/JEMS/140.
[14]	J. Droniou and C. Imbert, Fractal first-order partial differential equations, Arch. Ration. Mech. Anal., 182 (2006), 299-331. doi: 10.1007/s00205-006-0429-2.
[15]	C. Denoual, Dynamic dislocation modeling by combining Peierls Nabarro and Galerkin methods, Phys. Rev. B, 70 (2004), 024106. doi: 10.1103/PhysRevB.70.024106.
[16]	A. El Hajj, H. Ibrahim and R. Monneau, Dislocation dynamics: From microscopic models to macroscopic crystal plasticity, Continuum Mechanics and Thermodynamics, 21 (2009), 109-123. doi: 10.1007/s00161-009-0103-7.
[17]	S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J. Dynamics Differential Equ., 14 (2002), 85-137. doi: 10.1023/A:1012980128575.
[18]	L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.
[19]	E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116.
[20]	A. Fino, H. Ibrahim and R. Monneau, The Peierls-Nabarro model as a limit of a Frenkel-Kontorova model,, preprint, ().
[21]	N. Forcadel, C. Imbert and R. Monneau, Homogenization of some particle systems with two-body interactions and of the dislocation dynamics, Discrete Contin. Dyn. Syst., 23 (2009), 785-826. doi: 10.3934/dcds.2009.23.785.
[22]	G. Fusco and J. K. Hale, Slow-motion manifolds, dormant instability, and singular perturbations, J. Dynam. Differential Equations, 1 (1989), 75-94. doi: 10.1007/BF01048791.
[23]	A. Garroni and S. Müller, $\Gamma$-limit of a phase-field model of dislocations, SIAM J. Math. Anal., 36 (2005), 1943-1964.
[24]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224, Springer-Verlag, Berlin, 1983.
[25]	M.d.M. González, Gamma convergence of an energy functional related to the fractional Laplacian, Calc. Var. Partial Differential Equations, 36 (2009), 173-210.
[26]	C. P. Grant, Slow motion in one-dimensional Cahn-Morral systems, SIAM J. Math. Anal., 26 (1995), 21-34.
[27]	J. R. Hirth and L. Lothe, "Theory of Dislocations," Second edition, Malabar, Florida: Krieger, 1992.
[28]	C. Imbert and P. E. Souganidis, Phasefield theory for fractional diffusion-reaction equations and applications,, preprint, ().
[29]	W. D. Kalies, R. C. A. M. Van der Vorst and T. Wanner, Slow motion in higher-order systems and $\Gamma$-convergence in one space dimension, Nonlinear Anal., 44 (2001), 33-57. doi: 10.1016/S0362-546X(99)00245-X.
[30]	M. Kurzke, The gradient flow motion of boundary vortices, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), no. 1, 91-112.
[31]	P. Lévy, Sur les intégrales dont les éléments sont des variables aléatoires indépendantes, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2), 3 (1934), 337-366.
[32]	E. H. Lieb and M. Loss, "Analysis," Second edition, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001.
[33]	V. G. Maz'ja, "Sobolev Spaces," Translated from the Russian by T. O. Shaposhnikova, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985.
[34]	R. Monneau and S. Patrizi, Homogenization of the Peierls-Nabarro model for dislocation dynamics and the Orowan's law,, preprint, ().
[35]	A. B. Movchan, R. Bullough and J. R. Willis, Stability of a dislocation: Discrete model, Eur. J. Appl. Math., 9 (1998), 373-396. doi: 10.1017/S0956792598003489.
[36]	F. R. N. Nabarro, Fifty-year study of the Peierls-Nabarro stress, Material Science and Engineering A, 234-236 (1997), 67-76. doi: 10.1016/S0921-5093(97)00184-6.
[37]	G. Palatucci, O. Savin and A. Valdinoci, Local and global minimizers for a variational energy involving a fractional form,, preprint, ().
[38]	L. Silvestre, "Regularity of the Obstacle Problem for a Fractional Power of the Laplace Operator," Ph.D thesis, The University of Texas at Austin, 2005.
[39]	E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces," Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971.
[40]	J. F. Toland, The Peierls-Nabarro and Benjamin-Ono equations, J. Funct. Anal., 145 (1997), 136-150. doi: 10.1006/jfan.1996.3016.
[41]	H. Wei, Y. Xiang and P. Ming, A generalized Peierls-Nabarro model for curved dislocations using discrete Fourier transform, Communications in Computational Physics, 4 (2008), 275-293.

show all references

References:

[1]	G. Alberti, G. Bouchitté and P. Seppecher, Un résultat de perturbations singulières avec la norme $H^{1/2}$, C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 333-338.
[2]	G. Alberti, G. Bouchitté and P. Seppecher, Phase transition with the line-tension effect, Arch. Rational Mech. Anal., 144 (1998), 1-46. doi: 10.1007/s002050050111.
[3]	O. Alvarez, P. Hoch, Y. Le Bouar and R. Monneau, Dislocation dynamics: Short-time existence and uniqueness of the solution, Arch. Ration. Mech. Anal., 181 (2006), 449-504. doi: 10.1007/s00205-006-0418-5.
[4]	G. Barles and C. Imbert, Second-order elliptic integro-differential equations: Viscosity solutions' theory revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 567-585.
[5]	H. Brezis, "Analyse Fonctionnelle. Théorie et Applications," Collection Mathématiques Appliquées pour la Maîtrise [Collection of Applied Mathematics for the Master's Degree], Masson, Paris, 1983.
[6]	L. Bronsard and D. Hilhorst, On the slow dynamics for the Cahn-Hilliard equation in one space dimension, Proc. Roy. Soc. London Ser. A, 439 (1992), 669-682. doi: 10.1098/rspa.1992.0176.
[7]	L. Bronsard and R. V. Kohn, On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math., 43 (1990), 983-997. doi: 10.1002/cpa.3160430804.
[8]	X. Cabré and Y. Sire, Non-linear equations for fractional Laplacians I: Regularity, maximum principles and Hamiltoniam estimates,, preprint, ().
[9]	X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732. doi: 10.1002/cpa.20093.
[10]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
[11]	J. Carr and R. L. Pego, Metastable patterns in solutions of $u_t=\epsilon^2u_{mathcalxmathcalx}-f(u)$, Comm. Pure Appl. Math., 42 (1989), 523-576. doi: 10.1002/cpa.3160420502.
[12]	X. Chen, Generation, propagation, and annihilation of metastable patterns, J. Differential Equ., 206 (2004), 399-437. doi: 10.1016/j.jde.2004.05.017.
[13]	F. Da Lio, N. Forcadel and R. Monneau, Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocation dynamics, J. Eur. Math. Soc. (JEMS), 10 (2008), 1061-1104. doi: 10.4171/JEMS/140.
[14]	J. Droniou and C. Imbert, Fractal first-order partial differential equations, Arch. Ration. Mech. Anal., 182 (2006), 299-331. doi: 10.1007/s00205-006-0429-2.
[15]	C. Denoual, Dynamic dislocation modeling by combining Peierls Nabarro and Galerkin methods, Phys. Rev. B, 70 (2004), 024106. doi: 10.1103/PhysRevB.70.024106.
[16]	A. El Hajj, H. Ibrahim and R. Monneau, Dislocation dynamics: From microscopic models to macroscopic crystal plasticity, Continuum Mechanics and Thermodynamics, 21 (2009), 109-123. doi: 10.1007/s00161-009-0103-7.
[17]	S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J. Dynamics Differential Equ., 14 (2002), 85-137. doi: 10.1023/A:1012980128575.
[18]	L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.
[19]	E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116.
[20]	A. Fino, H. Ibrahim and R. Monneau, The Peierls-Nabarro model as a limit of a Frenkel-Kontorova model,, preprint, ().
[21]	N. Forcadel, C. Imbert and R. Monneau, Homogenization of some particle systems with two-body interactions and of the dislocation dynamics, Discrete Contin. Dyn. Syst., 23 (2009), 785-826. doi: 10.3934/dcds.2009.23.785.
[22]	G. Fusco and J. K. Hale, Slow-motion manifolds, dormant instability, and singular perturbations, J. Dynam. Differential Equations, 1 (1989), 75-94. doi: 10.1007/BF01048791.
[23]	A. Garroni and S. Müller, $\Gamma$-limit of a phase-field model of dislocations, SIAM J. Math. Anal., 36 (2005), 1943-1964.
[24]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224, Springer-Verlag, Berlin, 1983.
[25]	M.d.M. González, Gamma convergence of an energy functional related to the fractional Laplacian, Calc. Var. Partial Differential Equations, 36 (2009), 173-210.
[26]	C. P. Grant, Slow motion in one-dimensional Cahn-Morral systems, SIAM J. Math. Anal., 26 (1995), 21-34.
[27]	J. R. Hirth and L. Lothe, "Theory of Dislocations," Second edition, Malabar, Florida: Krieger, 1992.
[28]	C. Imbert and P. E. Souganidis, Phasefield theory for fractional diffusion-reaction equations and applications,, preprint, ().
[29]	W. D. Kalies, R. C. A. M. Van der Vorst and T. Wanner, Slow motion in higher-order systems and $\Gamma$-convergence in one space dimension, Nonlinear Anal., 44 (2001), 33-57. doi: 10.1016/S0362-546X(99)00245-X.
[30]	M. Kurzke, The gradient flow motion of boundary vortices, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), no. 1, 91-112.
[31]	P. Lévy, Sur les intégrales dont les éléments sont des variables aléatoires indépendantes, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2), 3 (1934), 337-366.
[32]	E. H. Lieb and M. Loss, "Analysis," Second edition, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001.
[33]	V. G. Maz'ja, "Sobolev Spaces," Translated from the Russian by T. O. Shaposhnikova, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985.
[34]	R. Monneau and S. Patrizi, Homogenization of the Peierls-Nabarro model for dislocation dynamics and the Orowan's law,, preprint, ().
[35]	A. B. Movchan, R. Bullough and J. R. Willis, Stability of a dislocation: Discrete model, Eur. J. Appl. Math., 9 (1998), 373-396. doi: 10.1017/S0956792598003489.
[36]	F. R. N. Nabarro, Fifty-year study of the Peierls-Nabarro stress, Material Science and Engineering A, 234-236 (1997), 67-76. doi: 10.1016/S0921-5093(97)00184-6.
[37]	G. Palatucci, O. Savin and A. Valdinoci, Local and global minimizers for a variational energy involving a fractional form,, preprint, ().
[38]	L. Silvestre, "Regularity of the Obstacle Problem for a Fractional Power of the Laplace Operator," Ph.D thesis, The University of Texas at Austin, 2005.
[39]	E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces," Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971.
[40]	J. F. Toland, The Peierls-Nabarro and Benjamin-Ono equations, J. Funct. Anal., 145 (1997), 136-150. doi: 10.1006/jfan.1996.3016.
[41]	H. Wei, Y. Xiang and P. Ming, A generalized Peierls-Nabarro model for curved dislocations using discrete Fourier transform, Communications in Computational Physics, 4 (2008), 275-293.

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