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

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

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.
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 & Continuous Dynamical Systems - A, 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. Google Scholar

[2]

G. Alberti, G. Bouchitté and P. Seppecher, Phase transition with the line-tension effect,, Arch. Rational Mech. Anal., 144 (1998), 1. doi: 10.1007/s002050050111. Google Scholar

[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. doi: 10.1007/s00205-006-0418-5. Google Scholar

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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. Google Scholar

[5]

H. Brezis, "Analyse Fonctionnelle. Théorie et Applications,", Collection Mathématiques Appliquées pour la Maîtrise [Collection of Applied Mathematics for the Master's Degree], (1983). Google Scholar

[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. doi: 10.1098/rspa.1992.0176. Google Scholar

[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. doi: 10.1002/cpa.3160430804. Google Scholar

[8]

X. Cabré and Y. Sire, Non-linear equations for fractional Laplacians I: Regularity, maximum principles and Hamiltoniam estimates,, preprint, (). Google Scholar

[9]

X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions,, Comm. Pure Appl. Math., 58 (2005), 1678. doi: 10.1002/cpa.20093. Google Scholar

[10]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. Google Scholar

[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. doi: 10.1002/cpa.3160420502. Google Scholar

[12]

X. Chen, Generation, propagation, and annihilation of metastable patterns,, J. Differential Equ., 206 (2004), 399. doi: 10.1016/j.jde.2004.05.017. Google Scholar

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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. doi: 10.4171/JEMS/140. Google Scholar

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J. Droniou and C. Imbert, Fractal first-order partial differential equations,, Arch. Ration. Mech. Anal., 182 (2006), 299. doi: 10.1007/s00205-006-0429-2. Google Scholar

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C. Denoual, Dynamic dislocation modeling by combining Peierls Nabarro and Galerkin methods,, Phys. Rev. B, 70 (2004). doi: 10.1103/PhysRevB.70.024106. Google Scholar

[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. doi: 10.1007/s00161-009-0103-7. Google Scholar

[17]

S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems,, J. Dynamics Differential Equ., 14 (2002), 85. doi: 10.1023/A:1012980128575. Google Scholar

[18]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19,, American Mathematical Society, (1998). Google Scholar

[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. Google Scholar

[20]

A. Fino, H. Ibrahim and R. Monneau, The Peierls-Nabarro model as a limit of a Frenkel-Kontorova model,, preprint, (). Google Scholar

[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. doi: 10.3934/dcds.2009.23.785. Google Scholar

[22]

G. Fusco and J. K. Hale, Slow-motion manifolds, dormant instability, and singular perturbations,, J. Dynam. Differential Equations, 1 (1989), 75. doi: 10.1007/BF01048791. Google Scholar

[23]

A. Garroni and S. Müller, $\Gamma$-limit of a phase-field model of dislocations,, SIAM J. Math. Anal., 36 (2005), 1943. Google Scholar

[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, (1983). Google Scholar

[25]

M.d.M. González, Gamma convergence of an energy functional related to the fractional Laplacian,, Calc. Var. Partial Differential Equations, 36 (2009), 173. Google Scholar

[26]

C. P. Grant, Slow motion in one-dimensional Cahn-Morral systems,, SIAM J. Math. Anal., 26 (1995), 21. Google Scholar

[27]

J. R. Hirth and L. Lothe, "Theory of Dislocations,", Second edition, (1992). Google Scholar

[28]

C. Imbert and P. E. Souganidis, Phasefield theory for fractional diffusion-reaction equations and applications,, preprint, (). Google Scholar

[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. doi: 10.1016/S0362-546X(99)00245-X. Google Scholar

[30]

M. Kurzke, The gradient flow motion of boundary vortices,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 91. Google Scholar

[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. Google Scholar

[32]

E. H. Lieb and M. Loss, "Analysis," Second edition, Graduate Studies in Mathematics, 14,, American Mathematical Society, (2001). Google Scholar

[33]

V. G. Maz'ja, "Sobolev Spaces,", Translated from the Russian by T. O. Shaposhnikova, (1985). Google Scholar

[34]

R. Monneau and S. Patrizi, Homogenization of the Peierls-Nabarro model for dislocation dynamics and the Orowan's law,, preprint, (). Google Scholar

[35]

A. B. Movchan, R. Bullough and J. R. Willis, Stability of a dislocation: Discrete model,, Eur. J. Appl. Math., 9 (1998), 373. doi: 10.1017/S0956792598003489. Google Scholar

[36]

F. R. N. Nabarro, Fifty-year study of the Peierls-Nabarro stress,, Material Science and Engineering A, 234-236 (1997), 234. doi: 10.1016/S0921-5093(97)00184-6. Google Scholar

[37]

G. Palatucci, O. Savin and A. Valdinoci, Local and global minimizers for a variational energy involving a fractional form,, preprint, (). Google Scholar

[38]

L. Silvestre, "Regularity of the Obstacle Problem for a Fractional Power of the Laplace Operator,", Ph.D thesis, (2005). Google Scholar

[39]

E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces,", Princeton Mathematical Series, (1971). Google Scholar

[40]

J. F. Toland, The Peierls-Nabarro and Benjamin-Ono equations,, J. Funct. Anal., 145 (1997), 136. doi: 10.1006/jfan.1996.3016. Google Scholar

[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. Google Scholar

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. Google Scholar

[2]

G. Alberti, G. Bouchitté and P. Seppecher, Phase transition with the line-tension effect,, Arch. Rational Mech. Anal., 144 (1998), 1. doi: 10.1007/s002050050111. Google Scholar

[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. doi: 10.1007/s00205-006-0418-5. Google Scholar

[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. Google Scholar

[5]

H. Brezis, "Analyse Fonctionnelle. Théorie et Applications,", Collection Mathématiques Appliquées pour la Maîtrise [Collection of Applied Mathematics for the Master's Degree], (1983). Google Scholar

[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. doi: 10.1098/rspa.1992.0176. Google Scholar

[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. doi: 10.1002/cpa.3160430804. Google Scholar

[8]

X. Cabré and Y. Sire, Non-linear equations for fractional Laplacians I: Regularity, maximum principles and Hamiltoniam estimates,, preprint, (). Google Scholar

[9]

X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions,, Comm. Pure Appl. Math., 58 (2005), 1678. doi: 10.1002/cpa.20093. Google Scholar

[10]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. Google Scholar

[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. doi: 10.1002/cpa.3160420502. Google Scholar

[12]

X. Chen, Generation, propagation, and annihilation of metastable patterns,, J. Differential Equ., 206 (2004), 399. doi: 10.1016/j.jde.2004.05.017. Google Scholar

[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. doi: 10.4171/JEMS/140. Google Scholar

[14]

J. Droniou and C. Imbert, Fractal first-order partial differential equations,, Arch. Ration. Mech. Anal., 182 (2006), 299. doi: 10.1007/s00205-006-0429-2. Google Scholar

[15]

C. Denoual, Dynamic dislocation modeling by combining Peierls Nabarro and Galerkin methods,, Phys. Rev. B, 70 (2004). doi: 10.1103/PhysRevB.70.024106. Google Scholar

[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. doi: 10.1007/s00161-009-0103-7. Google Scholar

[17]

S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems,, J. Dynamics Differential Equ., 14 (2002), 85. doi: 10.1023/A:1012980128575. Google Scholar

[18]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19,, American Mathematical Society, (1998). Google Scholar

[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. Google Scholar

[20]

A. Fino, H. Ibrahim and R. Monneau, The Peierls-Nabarro model as a limit of a Frenkel-Kontorova model,, preprint, (). Google Scholar

[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. doi: 10.3934/dcds.2009.23.785. Google Scholar

[22]

G. Fusco and J. K. Hale, Slow-motion manifolds, dormant instability, and singular perturbations,, J. Dynam. Differential Equations, 1 (1989), 75. doi: 10.1007/BF01048791. Google Scholar

[23]

A. Garroni and S. Müller, $\Gamma$-limit of a phase-field model of dislocations,, SIAM J. Math. Anal., 36 (2005), 1943. Google Scholar

[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, (1983). Google Scholar

[25]

M.d.M. González, Gamma convergence of an energy functional related to the fractional Laplacian,, Calc. Var. Partial Differential Equations, 36 (2009), 173. Google Scholar

[26]

C. P. Grant, Slow motion in one-dimensional Cahn-Morral systems,, SIAM J. Math. Anal., 26 (1995), 21. Google Scholar

[27]

J. R. Hirth and L. Lothe, "Theory of Dislocations,", Second edition, (1992). Google Scholar

[28]

C. Imbert and P. E. Souganidis, Phasefield theory for fractional diffusion-reaction equations and applications,, preprint, (). Google Scholar

[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. doi: 10.1016/S0362-546X(99)00245-X. Google Scholar

[30]

M. Kurzke, The gradient flow motion of boundary vortices,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 91. Google Scholar

[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. Google Scholar

[32]

E. H. Lieb and M. Loss, "Analysis," Second edition, Graduate Studies in Mathematics, 14,, American Mathematical Society, (2001). Google Scholar

[33]

V. G. Maz'ja, "Sobolev Spaces,", Translated from the Russian by T. O. Shaposhnikova, (1985). Google Scholar

[34]

R. Monneau and S. Patrizi, Homogenization of the Peierls-Nabarro model for dislocation dynamics and the Orowan's law,, preprint, (). Google Scholar

[35]

A. B. Movchan, R. Bullough and J. R. Willis, Stability of a dislocation: Discrete model,, Eur. J. Appl. Math., 9 (1998), 373. doi: 10.1017/S0956792598003489. Google Scholar

[36]

F. R. N. Nabarro, Fifty-year study of the Peierls-Nabarro stress,, Material Science and Engineering A, 234-236 (1997), 234. doi: 10.1016/S0921-5093(97)00184-6. Google Scholar

[37]

G. Palatucci, O. Savin and A. Valdinoci, Local and global minimizers for a variational energy involving a fractional form,, preprint, (). Google Scholar

[38]

L. Silvestre, "Regularity of the Obstacle Problem for a Fractional Power of the Laplace Operator,", Ph.D thesis, (2005). Google Scholar

[39]

E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces,", Princeton Mathematical Series, (1971). Google Scholar

[40]

J. F. Toland, The Peierls-Nabarro and Benjamin-Ono equations,, J. Funct. Anal., 145 (1997), 136. doi: 10.1006/jfan.1996.3016. Google Scholar

[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. Google Scholar

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