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

María del Mar González; Regis Monneau

doi:10.3934/dcds.2012.32.1255

Previous Article
Box dimension and bifurcations of one-dimensional discrete dynamical systems
DCDS Home
This Issue
Next Article
On isotopy and unimodal inverse limit spaces

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
Related Papers

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 & 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. Google Scholar
[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. 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), 449-504. 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), 567-585. 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], Masson, Paris, 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), 669-682. 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), 983-997. 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), 1678-1732. 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), 1245-1260. 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), 523-576. doi: 10.1002/cpa.3160420502. Google Scholar
[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. 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), 1061-1104. 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), 299-331. 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), 024106. 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), 109-123. 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), 85-137. doi: 10.1023/A:1012980128575. Google Scholar
[18]	L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 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), 77-116. 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), 785-826. 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), 75-94. 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), 1943-1964. 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, Berlin, 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), 173-210. Google Scholar
[26]	C. P. Grant, Slow motion in one-dimensional Cahn-Morral systems, SIAM J. Math. Anal., 26 (1995), 21-34. Google Scholar
[27]	J. R. Hirth and L. Lothe, "Theory of Dislocations," Second edition, Malabar, Florida: Krieger, 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), 33-57. 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), no. 1, 91-112. 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), 337-366. Google Scholar
[32]	E. H. Lieb and M. Loss, "Analysis," Second edition, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001. Google Scholar
[33]	V. G. Maz'ja, "Sobolev Spaces," Translated from the Russian by T. O. Shaposhnikova, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 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), 373-396. 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), 67-76. 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, The University of Texas at Austin, 2005. Google Scholar
[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. Google Scholar
[40]	J. F. Toland, The Peierls-Nabarro and Benjamin-Ono equations, J. Funct. Anal., 145 (1997), 136-150. 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), 275-293. 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), 333-338. Google Scholar
[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. 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), 449-504. 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), 567-585. 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], Masson, Paris, 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), 669-682. 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), 983-997. 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), 1678-1732. 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), 1245-1260. 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), 523-576. doi: 10.1002/cpa.3160420502. Google Scholar
[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. 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), 1061-1104. 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), 299-331. 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), 024106. 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), 109-123. 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), 85-137. doi: 10.1023/A:1012980128575. Google Scholar
[18]	L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 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), 77-116. 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), 785-826. 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), 75-94. 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), 1943-1964. 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, Berlin, 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), 173-210. Google Scholar
[26]	C. P. Grant, Slow motion in one-dimensional Cahn-Morral systems, SIAM J. Math. Anal., 26 (1995), 21-34. Google Scholar
[27]	J. R. Hirth and L. Lothe, "Theory of Dislocations," Second edition, Malabar, Florida: Krieger, 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), 33-57. 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), no. 1, 91-112. 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), 337-366. Google Scholar
[32]	E. H. Lieb and M. Loss, "Analysis," Second edition, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001. Google Scholar
[33]	V. G. Maz'ja, "Sobolev Spaces," Translated from the Russian by T. O. Shaposhnikova, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 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), 373-396. 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), 67-76. 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, The University of Texas at Austin, 2005. Google Scholar
[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. Google Scholar
[40]	J. F. Toland, The Peierls-Nabarro and Benjamin-Ono equations, J. Funct. Anal., 145 (1997), 136-150. 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), 275-293. Google Scholar

[1]	Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319
[2]	Henri Berestycki, Nancy Rodríguez. A non-local bistable reaction-diffusion equation with a gap. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 685-723. doi: 10.3934/dcds.2017029
[3]	Yuan Gao, Jian-Guo Liu, Tao Luo, Yang Xiang. Revisit of the Peierls-Nabarro model for edge dislocations in Hilbert space. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3177-3207. doi: 10.3934/dcdsb.2020224
[4]	Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations & Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517
[5]	Kazuhisa Ichikawa, Mahemauti Rouzimaimaiti, Takashi Suzuki. Reaction diffusion equation with non-local term arises as a mean field limit of the master equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 115-126. doi: 10.3934/dcdss.2012.5.115
[6]	Zhuoran Du, Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407
[7]	Ciprian G. Gal, Maurizio Grasselli. The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 1009-1040. doi: 10.3934/dcds.2008.22.1009
[8]	Suting Wei, Jun Yang. Clustering phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2575-2616. doi: 10.3934/cpaa.2020113
[9]	Fang Li, Kimie Nakashima. Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1391-1420. doi: 10.3934/dcds.2012.32.1391
[10]	Jun Yang, Xiaolin Yang. Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation in higher dimensional domains. Communications on Pure & Applied Analysis, 2013, 12 (1) : 303-340. doi: 10.3934/cpaa.2013.12.303
[11]	Mohammad Hassan Farshbaf-Shaker, Takeshi Fukao, Noriaki Yamazaki. Singular limit of Allen--Cahn equation with constraint and its Lagrange multiplier. Conference Publications, 2015, 2015 (special) : 418-427. doi: 10.3934/proc.2015.0418
[12]	Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703
[13]	Shin-Ichiro Ei, Hiroshi Matsuzawa. The motion of a transition layer for a bistable reaction diffusion equation with heterogeneous environment. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 901-921. doi: 10.3934/dcds.2010.26.901
[14]	Maicon Sônego, Arnaldo Simal do Nascimento. Stable transition layer induced by degeneracy of the spatial inhomogeneities in the Allen-Cahn problem. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021185
[15]	Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935
[16]	Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127
[17]	Michio Urano, Kimie Nakashima, Yoshio Yamada. Transition layers and spikes for a reaction-diffusion equation with bistable nonlinearity. Conference Publications, 2005, 2005 (Special) : 868-877. doi: 10.3934/proc.2005.2005.868
[18]	Abraham Solar. Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5799-5823. doi: 10.3934/dcds.2019255
[19]	Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015
[20]	Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025

2020 Impact Factor: 1.392

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors