. .. L'équation-intégrodifférentielle, 229 6.3.2 Partie linéaire pour les modes I et II

.. .. Donnée-initiale,

, Equation de Peierls-Nabarro Dynamique vectorielle

, Quelques propriétés mathématiques de l'équation de Peierls-Nabarro Dynamique 237

. .. Résolvantes-de-l'équation-de-peiers-nabarro-dynamique,

. .. Description-de-r-?-i-et-r-?-ii,

, Un théorème d'existence et d'unicité dans le cas non-linéaire, p.248

, Une remarque sur les dérivées fractionnaires

, Résolution numérique de l'équation de Peierls-Nabarro Dynamique 253

.. .. Détails-de-la-discrétisation-spatiale,

. .. Trois-schémas, 262 8.3.3 La méthode bloc-par-bloc appliquée à la forme résolue

.. .. Une, 264 8.4.2 Description de la méthode pour un second membre variable, p.265

.. .. Dégénérescence,

F. Achleitner and C. Kuehn, Traveling waves for a bistable equation with nonlocal diffusion, Adv. Differential Equations, vol.20, issue.9, pp.887-936, 2015.

G. Allaire, Shape optimization by the homogenization method, Applied Mathematical Sciences, vol.146, 2002.
DOI : 10.1007/978-1-4684-9286-6

B. Alpert, L. Greengard, and T. Hagstrom, Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation, SIAM J. Numer. Anal, vol.37, issue.4, pp.1138-1164, 2000.

A. Anantharaman, R. Costaouec, C. L. Bris, F. Legoll, and F. Thomines, Introduction to numerical stochastic homogenization and the related computational challenges : some recent developments. In Multiscale modeling and analysis for materials simulation, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap, vol.22, pp.197-272, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00624354

S. Armstrong, T. Kuusi, and J. Mourrat, The additive structure of elliptic homogenization, Invent. Math, vol.208, issue.3, pp.999-1154, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01483468

S. Armstrong, T. Kuusi, and J. Mourrat, Quantitative stochastic homogenization and large-scale regularity, 2018.
DOI : 10.1007/978-3-030-15545-2
URL : https://hal.archives-ouvertes.fr/hal-01728747

S. Armstrong and J. Mourrat, Lipschitz regularity for elliptic equations with random coefficients, Arch. Ration. Mech. Anal, vol.219, issue.1, pp.255-348, 2016.
DOI : 10.1007/s00205-015-0908-4
URL : https://hal.archives-ouvertes.fr/ensl-01401892

S. Armstrong and Z. Shen, Lipschitz estimates in almost-periodic homogenization, Comm. Pure Appl. Math, vol.69, issue.10, pp.1882-1923, 2016.
DOI : 10.1002/cpa.21616
URL : https://hal.archives-ouvertes.fr/hal-01483384

S. Armstrong and C. Smart, Quantitative stochastic homogenization of convex integral functionals, Ann. Sci. Éc. Norm. Supér, vol.49, issue.4, pp.423-481, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01483473

K. E. Atkinson, The numerical solution of integral equations of the second kind, Cambridge Monographs on Applied and Computational Mathematics, 1997.

M. Avellaneda and F. Lin, Compactness methods in the theory of homogenization, Comm. Pure Appl. Math, vol.40, issue.6, pp.803-847, 1987.

M. Avellaneda and F. Lin, L p bounds on singular integrals in homogenization, Comm. Pure Appl. Math, vol.44, issue.8-9, pp.897-910, 1991.
DOI : 10.1002/cpa.3160440805

L. Banjai, M. López-fernández, and A. Schädle, Fast and oblivious algorithms for dissipative and two-dimensional wave equations, SIAM J. Numer. Anal, vol.55, issue.2, pp.621-639, 2017.
DOI : 10.1137/16m1070657

L. Banjai and C. Lubich, An error analysis of Runge-Kutta convolution quadrature, BIT, vol.51, issue.3, pp.483-496, 2011.

L. Banjai and C. Lubich, Runge-Kutta convolution coercivity and its use for timedependent boundary integral equations, 2017.

L. Banjai and A. Rieder, Convolution quadrature for the wave equation with a nonlinear impedance boundary condition, Math. Comp, vol.87, issue.312, pp.1783-1819, 2018.

P. Bella, A. Giunti, and F. Otto, Quantitative stochastic homogenization : local control of homogenization error through corrector, Mathematics and materials, vol.23, pp.301-327, 2017.

A. Bensoussan, J. Lions, and G. Papanicolaou, Asymptotic analysis for periodic structures, 2011.

S. Benzoni-gavage, Semi-discrete shock profiles for hyperbolic systems of conservation laws, Phys. D, vol.115, issue.1-2, pp.109-123, 1998.

M. Berezhnyy and L. Berlyand, Continuum limit for three-dimensional mass-spring networks and discrete Korn's inequality, J. Mech. Phys. Solids, vol.54, issue.3, pp.635-669, 2006.

J. Bergh and J. Löfström, Interpolation spaces. An introduction, 1976.

X. Blanc and M. Josien, From the Newton equation to the wave equation : the case of shock waves, Applied Mathematics Research, pp.338-385, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01314690

X. Blanc, M. Josien, and C. L. Bris, Approximation locale précisée dans des problèmes multi-échelles avec défauts localisés

X. Blanc, M. Josien, and C. L. Bris, Precised approximations in elliptic homogenization beyond the periodic setting
URL : https://hal.archives-ouvertes.fr/hal-01958207

X. Blanc, C. L. Bris, and P. Lions, On correctors for linear elliptic homogenization in the presence of local defects
URL : https://hal.archives-ouvertes.fr/hal-01697104

X. Blanc, C. L. Bris, and P. Lions, From the Newton equation to the wave equation in some simple cases, Netw. Heterog. Media, vol.7, issue.1, pp.1-41, 2012.

X. Blanc, C. L. Bris, and P. Lions, A possible homogenization approach for the numerical simulation of periodic microstructures with defects, Milan J. Math, vol.80, issue.2, pp.351-367, 2012.

X. Blanc, C. L. Bris, and P. Lions, Local profiles for elliptic problems at different scales : defects in, and interfaces between periodic structures, Comm. Partial Differential Equations, vol.40, issue.12, pp.2173-2236, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01143193

X. Blanc, F. Legoll, and A. Anantharaman, Asymptotic behavior of Green functions of divergence form operators with periodic coefficients, Appl. Math. Res. Express. AMRX, issue.1, pp.79-101, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01079280

A. Bonito, J. P. Borthagaray, R. Nochetto, E. Otarola, and A. Salgado, Numerical methods for fractional diffusion, 2017.

S. Börm, Efficient numerical methods for non-local operators, EMS Tracts in Mathematics. European Mathematical Society (EMS), vol.14, 2010.

B. Böttcher, R. Schilling, and J. Wang, Lévy matters. III, Lecture Notes in Mathematics, vol.2099, 2013.

R. N. Bracewell, The Fourier transform and its Applications, 2000.

J. Braun, Connecting atomistic and continuous models of elastodynamics, Arch. Ration. Mech. Anal, vol.224, issue.3, pp.907-953, 2017.

Y. Brenier, Approximation of a simple Navier-Stokes model by monotonic rearrangement, Discrete Contin. Dyn. Syst, vol.34, issue.4, pp.1285-1300, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00749578

H. Brezis, Functional analysis, Sobolev spaces and partial differential equations. Universitext, 2011.

A. Bueno-orovio, D. Kay, and K. Burrage, Fourier spectral methods for fractionalin-space reaction-diffusion equations, BIT, vol.54, issue.4, pp.937-954, 2014.

V. Bulatov, F. Abraham, L. Kubin, B. Devincre, and S. Yip, Connecting atomistic and mesoscale simulations of crystal plasticity, Nature, vol.391, issue.6668, p.669, 1998.

X. Cabré, N. Cónsul, and J. V. Mandé, Traveling wave solutions in a half-space for boundary reactions, Anal. PDE, vol.8, issue.2, pp.333-364, 2015.

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.31, issue.1, pp.23-53, 2014.

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II, Trans. Amer. Math. Soc, vol.367, issue.2, pp.911-941, 2015.

I. Catto, C. L. Bris, and P. Lions, The mathematical theory of thermodynamic limits : Thomas-Fermi type models, 1998.

T. Cazenave and A. Haraux, An introduction to semilinear evolution equations, volume 13 of Oxford Lecture Series in Mathematics and its Applications, 1998.

S. Chen, J. Shen, and L. Wang, Laguerre functions and their applications to tempered fractional differential equations on infinite intervals, J. Sci. Comput, vol.74, issue.3, pp.1286-1313, 2018.

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, vol.2, issue.1, pp.125-160, 1997.

A. Chmaj, Existence of traveling waves in the fractional bistable equation, Arch. Math. (Basel), vol.100, issue.5, pp.473-480, 2013.

J. Christian, The relation between dislocation velocity and stress, Scripta metallurgica, vol.4, issue.10, pp.811-814, 1970.

A. Cochard and J. Rice, A spectral method for numerical elastodynamic fracture analysis without spatial replication of the rupture event, Journal of the Mechanics and Physics of Solids, vol.45, issue.8, pp.1393-1418, 1997.

S. Day, L. Dalguer, N. Lapusta, and Y. Liu, Comparison of finite difference and boundary integral solutions to three-dimensional spontaneous rupture, Journal of Geophysical Research : Solid Earth, vol.110, issue.B12, 2005.

P. Deift, K. , and T. Mclaughlin, A continuum limit of the Toda lattice, vol.131, 1998.

C. , Dynamic dislocation modeling by combining Peierls-Nabarro and Galerkin methods, Physical Review B, vol.70, issue.2, p.24106, 2004.

C. , Modeling dislocation by coupling Peierls-Nabarro and element-free Galerkin methods, Computer methods in applied mechanics and engineering, vol.196, pp.1915-1923, 2007.

B. Devincre and L. Kubin, Mesoscopic simulations of dislocations and plasticity, Materials Science and Engineering : A, vol.234, pp.8-14, 1997.

K. S. Djaka, A. Villani, V. Taupin, L. Capolungo, and S. Berbenni, Field dislocation mechanics for heterogeneous elastic materials : a numerical spectral approach, Comput. Methods Appl. Mech. Engrg, vol.315, pp.921-942, 2017.
DOI : 10.1016/j.cma.2016.11.036
URL : https://hal.archives-ouvertes.fr/hal-01947367

G. Dolzmann and S. Müller, Estimates for Green's matrices of elliptic systems by L p theory, Manuscripta Math, vol.88, issue.2, pp.261-273, 1995.
DOI : 10.1007/bf02567822

J. Drouet, L. Dupuy, F. Onimus, and F. Mompiou, A direct comparison between in-situ transmission electron microscopy observations and dislocation dynamics simulations of interaction between dislocation and irradiation induced loop in a zirconium alloy, Scripta Materialia, vol.119, pp.71-75, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01729350

P. Wei-nan-e and . Ming, Cauchy-Born rule and the stability of crystalline solids : dynamic problems, Acta Math. Appl. Sin. Engl. Ser, vol.23, issue.4, pp.529-550, 2007.

L. Evans, Partial differential equations, Graduate Studies in Mathematics, vol.19, 2010.

L. Evans and R. Gariepy, Measure theory and fine properties of functions, Textbooks in Mathematics, 2015.
DOI : 10.1201/9780203747940

P. C. Fife and J. B. Mcleod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rat. Mech. Anal, vol.65, issue.4, pp.335-361, 1977.

C. A. Fletcher, Computational techniques for fluid dynamics. 1. Springer Series in Computational Physics, 1991.
DOI : 10.1007/978-3-642-97035-1

P. Geubelle and J. Rice, A spectral method for three-dimensional elastodynamic fracture problems, Journal of the Mechanics and Physics of Solids, vol.43, issue.11, pp.1791-1824, 1995.
DOI : 10.1016/0022-5096(95)00043-i
URL : http://esag.harvard.edu/rice/174_Geubelle_Rice_JMPS_95.pdf

M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol.105, 1983.

D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, 2001.
DOI : 10.1007/978-3-642-61798-0

J. J. Gilman, Dislocation motion in a viscous medium, Physical Review Letters, vol.20, issue.4, p.157, 1968.
DOI : 10.1103/physrevlett.20.157

D. Givoli, Non-reflecting boundary conditions, Journal of computational physics, vol.94, issue.1, pp.1-29, 1991.
DOI : 10.1016/0021-9991(91)90135-8

A. Gloria, S. Neukamm, and F. Otto, A regularity theory for random elliptic operators, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01093368

A. Gloria, S. Neukamm, and F. Otto, Quantification of ergodicity in stochastic homogenization : optimal bounds via spectral gap on Glauber dynamics, Invent. Math, vol.199, issue.2, pp.455-515, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01093405

R. Gracie and T. Belytschko, An adaptive concurrent multiscale method for the dynamic simulation of dislocations, International Journal for Numerical Methods in Engineering, vol.86, issue.4-5, pp.575-597, 2011.

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 2007.
DOI : 10.1115/1.3138251
URL : http://biomechanical.asmedigitalcollection.asme.org/data/journals/jbendy/25673/58_2.pdf

G. W. Griffiths and W. E. Schiesser, Traveling wave analysis of partial differential equations, 2012.

M. Grüter and K. Widman, The Green function for uniformly elliptic equations, Manuscripta Math, vol.37, issue.3, pp.303-342, 1982.

C. Gui and M. Zhao, Traveling wave solutions of Allen-Cahn equation with a fractional Laplacian, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.32, issue.4, pp.785-812, 2015.

W. Hackbusch, Hierarchical matrices : algorithms and analysis, Springer Series in Computational Mathematics, vol.49, 2015.

T. Hagstrom, Radiation boundary conditions for the numerical simulation of waves, Acta numerica, vol.8, pp.47-106, 1999.

E. Hairer, C. Lubich, and M. Schlichte, Fast numerical solution of nonlinear Volterra convolution equations, SIAM J. Sci. Statist. Comput, vol.6, issue.3, pp.532-541, 1985.

E. Hairer, C. Lubich, and G. Wanner, Geometric numerical integration, Springer Series in Computational Mathematics, vol.31, 2006.
URL : https://hal.archives-ouvertes.fr/hal-01403326

E. Hairer, S. Nørsett, and G. Wanner, Solving ordinary differential equations. I Nonstiff problems, Springer Series in Computational Mathematics, vol.8, 1993.

E. Hairer and G. Wanner, Solving ordinary differential equations. II Stiff and differential-algebraic problems, Springer Series in Computational Mathematics, vol.14, 1996.

F. B. Hildebrand, Introduction to numerical analysis, 1987.

J. P. Hirth and J. Lothe, Theory of dislocations, 1982.

B. L. Holian, H. Flaschka, and D. W. Mclaughlin, Shock waves in the Toda lattice : analysis, Phys. Rev. A, vol.24, issue.3, pp.2595-2623, 1981.

Y. Huang and A. Oberman, Numerical methods for the fractional Laplacian : a finite difference-quadrature approach, SIAM J. Numer. Anal, vol.52, issue.6, pp.3056-3084, 2014.

D. E. Hurtado and M. Ortiz, Finite element analysis of geometrically necessary dislocations in crystal plasticity, International Journal for Numerical Methods in Engineering, vol.93, issue.1, pp.66-79, 2013.

V. Jikov, S. Kozlov, and O. Ole?, Homogenization of differential operators and integral functionals, 1994.

M. Josien, Decomposition and pointwise estimates of periodic Green functions of some elliptic equations with periodic oscillatory coefficients
URL : https://hal.archives-ouvertes.fr/hal-01848268

M. Josien, Mathematical properties of the Weertman equation, Communications in Mathematical Sciences
URL : https://hal.archives-ouvertes.fr/hal-01589690

M. Josien, Y. Pellegrini, F. Legoll, and C. L. Bris, Fourier-based numerical approximation of the Weertman equation for moving dislocations, International Journal for Numerical Methods in Engineering, vol.113, pp.1827-1850, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01510158

M. Kanninen and C. Popelar, Advanced fracture mechanics, Oxford Engineering Science Series, vol.15, 1985.

R. P. , Generalized functions, 2004.

V. Karlin, V. G. Maz-ya, A. B. Movchan, J. R. Willis, and R. Bullough, Numerical solution of nonlinear hypersingular integral equations of the Peierls type in dislocation theory, SIAM J. Appl. Math, vol.60, issue.2, pp.664-678, 2000.

C. T. Kelley, Iterative methods for linear and nonlinear equations, Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), vol.16, 1995.

C. Kenig, F. Lin, and Z. Shen, Homogenization of elliptic systems with Neumann boundary conditions, J. Amer. Math. Soc, vol.26, issue.4, pp.901-937, 2013.

C. Kenig, F. Lin, and Z. Shen, Periodic homogenization of Green and Neumann functions, Comm. Pure Appl. Math, vol.67, issue.8, pp.1219-1262, 2014.

F. King, of Encyclopedia of Mathematics and its Applications, vol.1, 2009.

R. Kurtz, T. Farris, and C. Sun, The numerical solution of Cauchy singular integral equations with application to fracture, Int. J. Fract, vol.66, issue.2, pp.139-154, 1994.

N. Lapusta, J. Rice, Y. Ben-zion, and G. Zheng, Elastodynamic analysis for slow tectonic loading with spontaneous rupture episodes on faults with rate-and statedependent friction, Journal of Geophysical Research : Solid Earth, vol.105, issue.B10, pp.23765-23789, 2000.

P. D. Lax and C. D. Levermore, The small dispersion limit of the Korteweg-de Vries equation, I. Comm. Pure Appl. Math, vol.36, issue.3, pp.253-290, 1983.

C. and L. Bris, Systèmes multi-échelles, Mathématiques & Applications, vol.47, 2005.

F. Legoll and P. Rothé, On the numerical approximation of fluctuations in stochastic homogenization

L. Lej?ek, Peierls-Nabarro model of planar dislocation cores in bcc crystals, Czech. J. Phys, vol.22, issue.9, pp.802-812, 1972.

L. Lej?ek, Dissociated dislocations in the Peierls-Nabarro model, Czech. J. Phys. B, vol.26, issue.3, pp.294-299, 1976.

Y. Y. Li and M. Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Ration. Mech. Anal, vol.153, issue.2, pp.91-151, 2000.

T. N. Lien, D. D. Trong, A. Pham-ngoc, and . Dinh, Laguerre polynomials and the inverse Laplace transform using discrete data, J. Math. Anal. Appl, vol.337, issue.2, pp.1302-1314, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00098062

P. Linz, Analytical and numerical methods for Volterra equations, SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), vol.7, 1985.

M. López-fernández, C. Lubich, and A. Schädle, Adaptive, fast, and oblivious convolution in evolution equations with memory, SIAM J. Sci. Comput, vol.30, issue.2, pp.1015-1037, 2008.

M. López-fernández, C. Palencia, and A. Schädle, A spectral order method for inverting sectorial Laplace transforms, SIAM J. Numer. Anal, vol.44, issue.3, pp.1332-1350, 2006.

G. Lu, V. Bulatov, and N. Kioussis, A nonplanar Peierls-Nabarro model and its application to dislocation cross-slip, Philos. Mag, vol.83, pp.3539-3548, 2003.

G. Lu, N. Kioussis, V. Bulatov, and E. Kaxiras, Generalized-stacking-fault energy surface and dislocation properties of aluminum, Physical Review B, vol.62, issue.5, p.3099, 2000.

C. Lubich, On the multistep time discretization of linear initial-boundary value problems and their boundary integral equations, Numer. Math, vol.67, issue.3, pp.365-389, 1994.

C. Lubich, Convolution quadrature revisited, BIT, vol.44, issue.3, pp.503-514, 2004.

C. Lubich and A. Schädle, Fast convolution for nonreflecting boundary conditions, SIAM J. Sci. Comput, vol.24, issue.1, pp.161-182, 2002.

Z. Mao and J. Shen, Hermite spectral methods for fractional PDEs in unbounded domains, SIAM J. Sci. Comput, vol.39, issue.5, pp.1928-1950, 2017.

Y. Meyer, Ondelettes et opérateurs. II

, Opérateurs de Calderón-Zygmund, Actualités Mathématiques. Hermann, 1990.

J. Mianroodi, A. Hunter, I. Beyerlein, and B. Svendsen, Theoretical and computational comparison of models for dislocation dissociation and stacking fault/core formation in fcc crystals, J. Mech. Phys. Solids, 2016.

A. Mielke and L. Truskinovsky, From discrete visco-elasticity to continuum rateindependent plasticity : rigorous results, Arch. Ration. Mech. Anal, vol.203, issue.2, pp.577-619, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00784831

R. Miller, R. Phillips, G. Beltz, and M. Ortiz, A non-local formulation of the Peierls dislocation model, Journal of the Mechanics and Physics of Solids, vol.46, issue.10, pp.1845-1867, 1998.

S. Müller, Variational models for microstructure and phase transitions, Lecture Notes in Math, vol.1713, 1999.

N. I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity, 1977.

F. R. Nabarro, Dislocations in a simple cubic lattice, Proc. Phys. Soc, vol.59, issue.2, p.256, 1947.

Y. Nec, A. A. Nepomnyashchy, and A. A. Golovin, Front-type solutions of fractional Allen-Cahn equation, Phys. D, vol.237, issue.24, pp.3237-3251, 2008.

J. N. Newman, Approximations for the Bessel and Struve functions, Math. Comp, vol.43, issue.168, pp.551-556, 1984.

C. P. Niculescu and L. Persson, Convex functions and their applications, 2006.

R. Nochetto, E. Otárola, and A. Salgado, A PDE approach to space-time fractional parabolic problems, SIAM J. Numer. Anal, vol.54, issue.2, pp.848-873, 2016.

H. Noda and N. Lapusta, Three-dimensional earthquake sequence simulations with evolving temperature and pore pressure due to shear heating : Effect of heterogeneous hydraulic diffusivity, Journal of Geophysical Research : Solid Earth, vol.115, issue.B12, 2010.

H. Noda and N. Lapusta, Stable creeping fault segments can become destructive as a result of dynamic weakening, Nature, vol.493, issue.7433, p.518, 2013.

R. , Convolution operators and L(p, q) spaces, Duke Math. J, vol.30, pp.129-142, 1963.

G. C. Papanicolaou and S. R. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random fields, vol.I, pp.835-873, 1979.

R. Peierls, The size of a dislocation, Proceedings of the Physical Society, vol.52, issue.1, p.34, 1940.

Y. Pellegrini, Dynamic Peierls-Nabarro equations for elastically isotropic crystals, Phys. Rev. B, vol.81, p.24101, 2010.
URL : https://hal.archives-ouvertes.fr/cea-00412985

Y. Pellegrini, Equation of motion and subsonic-transonic transitions of rectilinear edge dislocations : A collective-variable approach, Phys. Rev. B, vol.90, p.54120, 2014.
URL : https://hal.archives-ouvertes.fr/cea-01062460

Y. Pellegrini and M. Josien, Numerical solutions of the multidimensional Weertman equation

L. Pillon, Modélisations du mouvement instationnaire et des interactions de dislocations, 2008.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical recipes, 2007.

P. Rosakis, Supersonic dislocation kinetics from an augmented Peierls model, Phys. Rev. Lett, vol.86, issue.1, p.95, 2001.

D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math, vol.22, issue.3, pp.312-355, 1976.

A. Schädle, M. López-fernández, and C. Lubich, Fast and oblivious convolution quadrature, SIAM J. Sci. Comput, vol.28, issue.2, pp.421-438, 2006.

B. Schweizer and M. Veneroni, The needle problem approach to non-periodic homogenization, Netw. Heterog. Media, vol.6, issue.4, pp.755-781, 2011.

D. Serre, Systems of conservation laws, vol.1, 1999.
URL : https://hal.archives-ouvertes.fr/ensl-01402415

Z. Shen, Bounds of Riesz transforms on L p spaces for second order elliptic operators, Ann. Inst. Fourier (Grenoble), vol.55, issue.1, pp.173-197, 2005.

Z. Shen, The Calderón-Zygmund lemma revisited, Lectures on the analysis of nonlinear partial differential equations. Part, vol.2, pp.203-224, 2012.

E. Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws, I. Math. Comp, vol.49, issue.179, pp.91-103, 1987.

A. Talbot, The accurate numerical inversion of laplace transforms, IMA Journal of Applied Mathematics, vol.23, issue.1, pp.97-120, 1979.

L. Tartar, An introduction to Sobolev spaces and interpolation spaces, vol.3, 2007.

L. Tartar, The general theory of homogenization, Lecture Notes of the Unione Matematica Italiana, vol.7, 2009.

T. P. Theodoulidis, Struve functions, 2012.

M. Toda, Theory of nonlinear lattices, Springer Series in Solid-State Sciences, vol.20, 1989.

J. Tolédano, Bases physiques de la plasticité des solides. Editions Ecole Polytechnique, 2007.

S. Venakides, P. Deift, and R. Oba, The Toda shock problem, Comm. Pure Appl. Math, vol.44, issue.8-9, pp.1171-1242, 1991.

A. I. Volpert, V. A. Volpert, and V. A. Volpert, Traveling wave solutions of parabolic systems, Translations of Mathematical Monographs, vol.140, 1994.

V. Volpert, Elliptic partial differential equations, vol.2, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01097226

B. Sydow, J. Hartford, and G. Wahnström, Atomistic simulations and PeierlsNabarro analysis of the Shockley partial dislocations in palladium, Comput. Mater. Sci, vol.15, issue.3, pp.367-379, 1999.

J. Wang and Q. Ma, Numerical techniques on improving computational efficiency of spectral boundary integral method, International Journal for Numerical Methods in Engineering, vol.102, issue.10, pp.1638-1669, 2015.

J. Weertman, Dislocations in uniform motion on slip or climb planes having periodic force laws, Mathematical Theory of Dislocations, pp.178-202, 1969.

J. Weertman, Stress dependence on the velocity of a dislocation moving on a viscously damped slip plane, pp.75-83, 1969.

Y. Xiang, H. Wei, P. Ming, and W. E. , A generalized Peierls-Nabarro model for curved dislocations and core structures of dislocation loops in Al and Cu, Acta Materialia, vol.56, issue.7, pp.1447-1460, 2008.

L. Y. Yap, Some remarks on convolution operators and L(p, q) spaces, Duke Math. J, vol.36, pp.647-658, 1969.
DOI : 10.1215/s0012-7094-69-03677-1

K. Yosida, Functional analysis, Classics in Mathematics, 1995.

V. Zhakhovsky, M. Budzevich, N. Inogamov, I. Oleynik, and C. White, Two-zone elastic-plastic single shock waves in solids, Physical review letters, vol.107, issue.13, p.135502, 2011.

X. Zhang, A. Acharya, N. J. Walkington, and J. Bielak, A single theory for some quasistatic, supersonic, atomic, and tectonic scale applications of dislocations, Journal of the Mechanics and Physics of Solids, vol.84, pp.145-195, 2015.

A. Zhu, C. Jin, D. Zhao, Y. Xiang, and J. Huang, A numerical scheme for generalized Peierls-Nabarro model of dislocations based on the fast multipole method and iterative grid redistribution, Commun. Comput. Phys, vol.18, issue.05, pp.1282-1312, 2015.

, La principale difficulté des preuves ci-dessous tient au fait que l'on effectue les preuves dans L ? (R), qui est un espace naturel où étudier les solutions de (4.15) (en effet, on s'intéresse ici à des solutions qui ne tendent pas vers 0 en +?), Les annexes de ce chapitre sont en anglais. Nous remercions Gilles Francfort pour ses suggestions en vue de simplifier les preuves de cette section

, Cette erreur, qui semble a priori faible, s'accumule cependant via les intégrales de (8.45). Finalement, cela engendre des erreurs appréciables dans la simulation de (8.2a) sur les temps longs

, malgré la documentation quasiment inexistante sur lesdites fonctions. Ce choix est fondé sur des tests numériques que nous avons effectués. Par ailleurs, précisons que les fonctions issues de [146] sont les seuls fichiers du code que nous n'avons pas écrits nous-mêmes, C'est pourquoi nous avons préféré utiliser les fonctions que l'on trouve dans, vol.146

, Remarques et extensions possibles Nous faisons ici quelques remarques et renvoyons à la publication en préparation, vol.24

, pé-riodique + défaut" que pour démontrer l'existence de correcteurs et d'un potentiel (à savoir la fonction B définie en (B.15)-(B.16) ci-dessous) fortement sous-linéaires. Ainsi, les conclusions du Théorème B.2.1 sont en fait valides sous les hypothèses suivantes, plus générales que celles utilisées ici : 1. la matrice a est elliptique, bornée, La démonstration du Théorème B.2.1 ne fait usage de l'hypothèse de la structure

, elle admet un correcteur w j , c'est-à-dire une solution de (B, vol.3

, correcteur est fortement sous-linéaire à l'infini, c'est-à-dire qu'il vérifie (B.8)

, il existe un potentiel B associé (i.e une solution antisymétrique de (B.15)-(B.16) ci-dessous), qui est lui aussi fortement sous-linéaire, c'est-à-dire qu'il vérifie (B.19)

, B.3.2 Autres remarques

, La preuve esquissée ici est faite dans le cas où a est scalaire. Toutefois, il est possible de travailler avec un coefficient matriciel, On obtient alors des résultats analogues

, Par conséquent, dans la mesure où l'existence des correcteurs w j est aussi prouvée dans [25] pour le cas des systèmes, il semble a priori possible de démontrer un résultat analogue au Théo-rème B.2.1 dans le cadre d'un système d'équations. Une telle adaptation n'a cependant pas été entreprise. Voir à ce sujet la Remarque 69 ci-dessous. Notons que, dans le cas d'une équation

, De la même manière que dans [94], il est possible d'approximer la fonction de Green G ? relative à l'Equation (B.1), ainsi que ses gradients x G ? et y G ? , et son gradient croisé x y G ?

, Toutefois, dans le cas d'une matrice périodique, Les estimations (B.11) et (B.12) sont des estimations à l'intérieur du domaine

, On peut aussi montrer dans le cadre du Théorème B.2.1 que, si f ? L p (?), pour tout p ? [2, +?[, alors R ? L p (?) ? C? ?r f L p (?) et R ? L p (? 1 ) ? C? ?r f L p (?)

, L'estimation sur R ? est immédiate vu le schéma de preuve ci-dessous. L'estimation sur R ? découle d'un Lemme de mesure à la Calderón-Zygmund

, Notre schéma de preuve suit celui des articles, vol.94

. L'idée-repose-sur-le-fait-que, pour ? = 0, l'équation est à coefficients constants, donc vérifie des estimations de régularité elliptique, à la fois de type Schauder (en normes C k,? )

, avec un second membre nul) : on obtient l'estimation