Archive for Rational Mechanics and Analysis

, Volume 196, Issue 1, pp 71–96 | Cite as

Global Existence for a System of Non-Linear and Non-Local Transport Equations Describing the Dynamics of Dislocation Densities

  • Marco Cannone
  • Ahmad El HajjEmail author
  • Régis Monneau
  • Francis Ribaud


In this paper, we study the global in time existence problem for the Groma-Balogh model describing the dynamics of dislocation densities. This model is a two-dimensional model where the dislocation densities satisfy a system of transport equations such that the velocity vector field is the shear stress in the material, solving the equations of elasticity. This shear stress can be expressed as some Riesz transform of the dislocation densities. The main tool in the proof of this result is the existence of an entropy for this system.


Dislocation Density Burger Vector Global Existence Orlicz Space Young Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Adams, R.A.: Sobolev Spaces. Pure and Applied Mathematics, Vol. 65. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York, 1975Google Scholar
  2. 2.
    Alvarez O., Cardaliaguet P., Monneau R.: Existence and uniqueness for dislocation dynamics with nonnegative velocity. Interfaces Free Bound. 7, 415–434 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alvarez O., Hoch P., Le Bouar Y., Monneau R.: Dislocation dynamics: short-time existence and uniqueness of the solution. Arch. Ration. Mech. Anal. 181, 449–504 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ambrosio L.: Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158, 227–260 (2004)zbMATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Ambrosio, L., Serfaty, S.: A gradient flow approach to an evolution problem arising in superconductivity (2007, preprint)Google Scholar
  6. 6.
    Barles G., Ley O.: Nonlocal first-order Hamilton-Jacobi equations modelling dislocations dynamics. Comm. Partial Differ. Equ. 31, 1191–1208 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Pure and Applied Mathematics, Vol. 129. Academic Press Inc., Boston, 1988Google Scholar
  8. 8.
    Biham O., Middleton A.A., Levine D.: Self-organization and a dynamical transition in traffic-flow models. Phys. Rev. A 46, R6124–R6127 (1992)CrossRefADSGoogle Scholar
  9. 9.
    Cannone M.: Ondelettes, Paraproduits et Navier-Stokes. Diderot Editeur, Paris, 1995Google Scholar
  10. 10.
    Chae D., Córdoba A., Córdoba D., Fontelos M.A.: Finite time singularities in a 1D model of the quasi-geostrophic equation. Adv. Math. 194, 203–223 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Constantin P., Majda A.J., Tabak E.: Formation of strong fronts in the 2D quasigeostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994)zbMATHCrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Constantin P., Majda A.J., Tabak E.G.: Singular front formation in a model for quasigeostrophic flow. Phys. Fluids 6, 9–11 (1994)zbMATHCrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Córdoba A., Córdoba D., Fontelos M.A.: Formation of singularities for a transport equation with nonlocal velocity. Ann. of Math. 162(2), 1377–1389 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Deshpande V.S., Needleman A., Vander Giessen E.: Finite strain discrete dislocation plasticity. J. Mech. Phys. Solids 51, 2057–2083 (2003)zbMATHCrossRefMathSciNetADSGoogle Scholar
  15. 15.
    DiPerna R.J., Lions P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)zbMATHCrossRefMathSciNetADSGoogle Scholar
  16. 16.
    EL-Azab A.: Statistical mechanics treatment of the evolution of dislocation distributions in single crystals. Phys. Rev. B 61, 11956–11966 (2000)CrossRefADSGoogle Scholar
  17. 17.
    El Hajj A.: Well-posedness theory for a nonconservative Burgers-type system arising in dislocation dynamics. SIAM J. Math. Anal. 39, 965–986 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    El Hajj A., Forcadel N.: A convergent scheme for a non-local coupled system modelling dislocations densities dynamics. Math. Comp. 77, 789–812 (2008)zbMATHCrossRefMathSciNetADSGoogle Scholar
  19. 19.
    Goudon T., Jabin P.-E., Vasseur A.: Hydrodynamic limit for the Vlasov-Navier-Stokes equations. I. Light particles regime. Indiana Univ. Math. J. 53, 1495–1515 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Goudon T., Jabin P.-E., Vasseur A.: Hydrodynamic limit for the Vlasov-Navier-Stokes equations. II. Fine particles regime. Indiana Univ. Math. J. 53, 1517–1536 (2004)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Goudon T., Nieto J., Poupaud F., Soler J.: Multidimensional high-field limit of the electrostatic Vlasov-Poisson-Fokker-Planck system. J. Differ. Equ. 213, 418–442 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Groma I.: Link between the microscopic and mesoscopic length-scale description of the collective behaviour of dislocations. Phys. Rev. B 56, 5807 (1997)CrossRefADSGoogle Scholar
  23. 23.
    Groma I., Balogh P.: Investigation of dislocation pattern formation in a two- dimensional self-consistent field approximation. Acta Mater 47, 3647–3654 (1999)CrossRefGoogle Scholar
  24. 24.
    Groma I., Csikor F., Zaiser M.: Spatial correlations and higher-order gradient terms in a continuum description of dislocation dynamics. Acta Mater 51, 1271–1281 (2003)CrossRefGoogle Scholar
  25. 25.
    Hirth, J., Lothe, J.: Theory of dislocations, 2nd edn. Krieger Publishing compagny, Florida 32950, 1982Google Scholar
  26. 26.
    Ibrahim, H.: Existence and uniqueness for a non-linear parabolic/Hamilton-Jacobi system describing the dynamics of dislocation densities. Annales de l’I.H.P, Analysis non linéaire (2007, to appear)Google Scholar
  27. 27.
    Lieberman, G.M.: Second order parabolic differential equations. World Scientific Publishing Co. Inc., River Edge, 1996Google Scholar
  28. 28.
    Masmoudi N., Zhang P.: Global solutions to vortex density equations arising from sup-conductivity. Ann. Inst. H. Poincaré Anal. Non Linéaire 22, 441–458 (2005)zbMATHCrossRefMathSciNetADSGoogle Scholar
  29. 29.
    Monneau, R.: A kinetic formulation of moving fronts and application to dislocations dynamics, (2006, preprint)Google Scholar
  30. 30.
    Nieto J., Poupaud F., Soler J.: High-field limit for the Vlasov-Poisson-Fokker-Planck system. Arch. Ration. Mech. Anal. 158, 29–59 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    O’Neil R.: Fractional integration in Orlicz spaces. I. Trans. Amer. Math. Soc. 115, 300–328 (1965)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, Vol. 44. Springer, New York, 1983Google Scholar
  33. 33.
    Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Monographs and Textbooks in Pure and Applied Mathematics, Vol. 146. Marcel Dekker Inc., New York, 1991Google Scholar
  34. 34.
    Serre, D.: Systems of conservation laws. I, II. Geometric structures, oscillations, and initial-boundary value problems. Translated from the 1996 French original (Ed. Sneddon, I.N.) Cambridge University Press, Cambridge, 1999–2000Google Scholar
  35. 35.
    Simon J.: Compact sets in the space L p(0,T ; B). Ann. Mat. Pura Appl. 146(4), 65–96 (1987)zbMATHMathSciNetGoogle Scholar
  36. 36.
    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, Vol. 43. Princeton University Press, Princeton, NJ, 1993. (with the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III)Google Scholar
  37. 37.
    Trudinger N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)zbMATHMathSciNetGoogle Scholar
  38. 38.
    Yefimov, S.: Discrete Dislocation and Nonlocal Crystal Plasticity Modelling. Netheerlands Institute for Metals Research, University of Groningen, 2004Google Scholar
  39. 39.
    Zaiser M., Hochrainer T.: Some steps towards a continuum representation of 3d dislocation systems. Scripta Mater. 54, 717–721 (2006)CrossRefGoogle Scholar
  40. 40.
    Zygmund, A.: Trigonometric Series, 2nd edn., Vols. I, II. Cambridge University Press, New York, 1959Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Marco Cannone
    • 1
  • Ahmad El Hajj
    • 1
    Email author
  • Régis Monneau
    • 1
  • Francis Ribaud
    • 1
  1. 1.Ecole Nationale des Ponts et Chaussees, CERMICSMarne-la-Vallee Cedex 2France

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