Well-posedness and numerical analysis of a one-dimensional non-local transport equation modelling dislocations dynamics

Ghorbel, A.; Monneau, R.

doi:10.1090/S0025-5718-10-02326-4

Well-posedness and numerical analysis of a one-dimensional non-local transport equation modelling dislocations dynamics
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by A. Ghorbel and R. Monneau;

Math. Comp. 79 (2010), 1535-1564

DOI: https://doi.org/10.1090/S0025-5718-10-02326-4

Published electronically: March 23, 2010

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Abstract:

We consider a situation where dislocations are parallel lines moving in a single plane. For this simple geometry, dislocations dynamics is modeled by a one-dimensional non-local transport equation. We prove a result of existence and uniqueness for all time of the continuous viscosity solution for this equation. A finite difference scheme is proposed to approximate the continuous viscosity solution. We also prove an error estimate result between the continuous solution and the discrete solution, and we provide some simulations.

References

O. Alvarez, P. Cardaliaguet, and R. Monneau, Existence and uniqueness for dislocation dynamics with nonnegative velocity, Interfaces Free Bound. 7 (2005), no. 4, 415–434. MR 2191694, DOI 10.4171/IFB/131
O. Alvarez, E. Carlini, R. Monneau, and E. Rouy, Convergence of a first order scheme for a non-local eikonal equation, Appl. Numer. Math. 56 (2006), no. 9, 1136–1146. MR 2244967, DOI 10.1016/j.apnum.2006.03.002
O. Alvarez, E. Carlini, R. Monneau, and E. Rouy, A convergent scheme for a non local Hamilton Jacobi equation modelling dislocation dynamics, Numer. Math. 104 (2006), no. 4, 413–444. MR 2249672, DOI 10.1007/s00211-006-0030-5
Olivier Alvarez, Philippe Hoch, Yann Le Bouar, and Régis Monneau, Résolution en temps court d’une équation de Hamilton-Jacobi non locale décrivant la dynamique d’une dislocation, C. R. Math. Acad. Sci. Paris 338 (2004), no. 9, 679–684 (French, with English and French summaries). MR 2065373, DOI 10.1016/j.crma.2004.03.007
Olivier Alvarez, Philippe Hoch, Yann Le Bouar, and Régis Monneau, Dislocation dynamics: short-time existence and uniqueness of the solution, Arch. Ration. Mech. Anal. 181 (2006), no. 3, 449–504. MR 2231781, DOI 10.1007/s00205-006-0418-5
Martino Bardi and Italo Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. With appendices by Maurizio Falcone and Pierpaolo Soravia. MR 1484411, DOI 10.1007/978-0-8176-4755-1
Guy Barles, Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 17, Springer-Verlag, Paris, 1994 (French, with French summary). MR 1613876
Guy Barles, Pierre Cardaliaguet, Olivier Ley, and Régis Monneau, Global existence results and uniqueness for dislocation equations, SIAM J. Math. Anal. 40 (2008), no. 1, 44–69. MR 2403312, DOI 10.1137/070682083
Guy Barles and Olivier Ley, Nonlocal first-order Hamilton-Jacobi equations modelling dislocations dynamics, Comm. Partial Differential Equations 31 (2006), no. 7-9, 1191–1208. MR 2254611, DOI 10.1080/03605300500361446
Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. MR 1118699, DOI 10.1090/S0273-0979-1992-00266-5
Michael G. Crandall and Pierre-Louis Lions, Condition d’unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 3, 183–186 (French, with English summary). MR 610314
Michael G. Crandall and Pierre-Louis Lions, On existence and uniqueness of solutions of Hamilton-Jacobi equations, Nonlinear Anal. 10 (1986), no. 4, 353–370. MR 836671, DOI 10.1016/0362-546X(86)90133-1
M. G. Crandall and P.-L. Lions, Two approximations of solutions of Hamilton-Jacobi equations, Math. Comp. 43 (1984), no. 167, 1–19. MR 744921, DOI 10.1090/S0025-5718-1984-0744921-8
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), no. 4, 1061–1104. MR 2443929, DOI 10.4171/JEMS/140
Nicolas Forcadel, Dislocation dynamics with a mean curvature term: short time existence and uniqueness, Differential Integral Equations 21 (2008), no. 3-4, 285–304. MR 2484010
A. Ghorbel, P. Hoch and R. Monneau, A numerical study for the homogenisation of one-dimensional models describing the motion of dislocations, Int. J. of Computing Science and Mathematics, Vol. 2, Nos. 1/2, 28-52, (2008).
A. Ghorbel and R. Monneau, Equation d’Hamilton-Jacobi non-locale modélisant la dynamique des dislocations, Proceedings of the 2nd TAM-TAM (Trends in Applied Mathematics in Tunisia, Algeria, Morocco), 322-328, (2005).
J.P. Hirth and J. Lothe, Theory of dislocations, Second Edition. Malabar, Florida: Krieger, (1992).
Stanley Osher and James A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79 (1988), no. 1, 12–49. MR 965860, DOI 10.1016/0021-9991(88)90002-2
D. Rodney, Y. Le Bouar and A. Finel, Phase field methods and dislocations, Acta Materialia 51, 17-30, (2003).