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On isotopy and unimodal inverse limit spaces
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 |
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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