×

Nonlinear diffusion of dislocation density and self-similar solutions. (English) Zbl 1207.82049

Summary: We study a nonlinear pseudodifferential equation describing the dynamics of dislocations in crystals. The long time asymptotics of solutions is described by the self-similar profiles.

MSC:

82D25 Statistical mechanics of crystals
35B40 Asymptotic behavior of solutions to PDEs
35C06 Self-similar solutions to PDEs
74N05 Crystals in solids
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alvarez O., Hoch P., Le Bouar Y., Monneau R.: Dislocation dynamics: short time existence and uniqueness of the solution. Arch. Rat. Mech. Anal. 181, 449–504 (2006) · Zbl 1158.74335 · doi:10.1007/s00205-006-0418-5
[2] Amour L., Ben-Artzi M.: Global existence and decay for viscous Hamilton-Jacobi equations. Nonlinear Anal. 31, 621–628 (1998) · Zbl 1023.35049 · doi:10.1016/S0362-546X(97)00427-6
[3] Barles G., Chasseigne E., Imbert C.: On the Dirichlet problem for second-order elliptic integro-differential equations. Indiana Univ. Math. J. 57, 213–246 (2008) · Zbl 1139.47057 · doi:10.1512/iumj.2008.57.3315
[4] Barles G., Imbert C.: Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited. Ann. I.H.P., Anal Non-lin. 25, 567–585 (2008) · Zbl 1155.45004 · doi:10.1016/j.anihpc.2007.02.007
[5] Ben-Artzi M., Souplet Ph., Weissler F.B.: The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces. J. Math. Pures Appl. 81, 343–378 (2002) · Zbl 1046.35046 · doi:10.1016/S0021-7824(01)01243-0
[6] Castro A., Córdoba D.: Global existence, singularities and ill-posedness for a nonlocal flux. Adv. Math. 219, 1916–1936 (2008) · Zbl 1186.35002 · doi:10.1016/j.aim.2008.07.015
[7] 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) · Zbl 1128.76372 · doi:10.1016/j.aim.2004.06.004
[8] Constantin P., Lax P., Majda A.: A simple one-dimensional model for the three dimensional vorticity. Comm. Pure Appl. Math. 38, 715–724 (1985) · Zbl 0615.76029 · doi:10.1002/cpa.3160380605
[9] Constantin P., Majda A., Tabak E.: Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994) · Zbl 0809.35057 · doi:10.1088/0951-7715/7/6/001
[10] Córdoba A., Córdoba D., Fontelos M.A.: Formation of singularities for a transport equation with nonlocal velocity. Ann. Math. 162, 1377–1389 (2005) · Zbl 1101.35052 · doi:10.4007/annals.2005.162.1377
[11] Crandall M.G., Ishii H., Lions P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27, 1–67 (1992) · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5
[12] Deslippe J., Tedstrom R., Daw M.S., Chrzan D., Neeraj T., Mills M.: Dynamics scaling in a simple one-dimensional model of dislocation activity. Phil. Mag. 84, 2445–2454 (2004) · doi:10.1080/14786430410001690042
[13] Droniou J., Imbert C.: Fractal first order partial differential equations. Arch. Rat. Mech. Anal. 182, 299–331 (2006) · Zbl 1111.35144 · doi:10.1007/s00205-006-0429-2
[14] Forcadel N., Imbert C., Monneau R.: Homogenization of the dislocation dynamics and of some particle systems with two-body interactions. Disc. Contin. Dyn. Syst. Ser. A 23, 785–826 (2009) · Zbl 1154.35306 · doi:10.3934/dcds.2009.23.785
[15] Getoor R.K.: First passage times for symmetric stable processes in space. Trans. Amer. Math. Soc. 101, 75–90 (1961) · Zbl 0104.11203 · doi:10.1090/S0002-9947-1961-0137148-5
[16] Head A.K.: Dislocation group dynamics I. Similarity solutions od the n-body problem. Phil. Mag. 26, 43–53 (1972) · doi:10.1080/14786437208221018
[17] Head A.K.: Dislocation group dynamics II. General solutions of the n-body problem. Phil. Mag. 26, 55–63 (1972) · doi:10.1080/14786437208221019
[18] Head A.K.: Dislocation group dynamics III. Similarity solutions of the continuum approximation. Phil. Mag. 26, 65–72 (1972) · doi:10.1080/14786437208221020
[19] Head A.K., Louat N.: The distribution of dislocations in linear arrays. Austral. J. Phys. 8, 1–7 (1955) · Zbl 0068.40801
[20] Hirth J.R., Lothe L.: Theory of Dislocations. Second Ed. Malabar, FL:Krieger (1992) · Zbl 1365.82001
[21] Hörmander,: The Analysis of Linear Partial Differential Operators. Vol. 1, New York: Springer-Verlag, 1990
[22] Imbert C.: A non-local regularization of first order Hamilton-Jacobi equations. J. Differ. Eq. 211, 214–246 (2005) · Zbl 1073.35059 · doi:10.1016/j.jde.2004.06.001
[23] Imbert C., Monneau R., Rouy E.: Homogenization of first order equations, with (u/{\(\epsilon\)})-periodic Hamiltonians. Part II: application to dislocations dynamics. Comm. Part. Diff. Eq. 33, 479–516 (2008) · Zbl 1143.35005 · doi:10.1080/03605300701318922
[24] Jakobsen E.R., Karlsen K.H.: Continuous dependence estimates for viscosity solutions of integro-PDEs. J. Differ. Eq. 212, 278–318 (2005) · Zbl 1082.45008 · doi:10.1016/j.jde.2004.06.021
[25] Jakobsen E.R., Karlsen K.H.: A maximum principle for semicontinuous functions applicable to integro-partial differential equations. NoDEA Nonlin. Differ. Eqs. Appl. 13, 137–165 (2006) · Zbl 1105.45006 · doi:10.1007/s00030-005-0031-6
[26] Karch G., Miao C., Xu X.: On the convergence of solutions of fractal Burgers equation toward rarefaction waves. SIAM J. Math. Anal. 39, 1536–1549 (2008) · Zbl 1154.35080 · doi:10.1137/070681776
[27] Liskevich, V.A., Semenov, Yu.A.: Some problems on Markov semigroups. In: Schrödinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras, Math. Top. 11, Berlin: Akademie Verlag, 1996, pp. 163–217 · Zbl 0854.47027
[28] Muskhelishvili, N.I.: Singular Integral Equations. Groningen: P. Noordhoff, N. V., 1953 · Zbl 0051.33203
[29] Sayah A.: Équations d’Hamilton-Jacobi du premier ordre avec termes intégro-différentiels. I Unicité des solutions de viscosité, II Existence de solutions de viscosité. Comm. Part. Diff. Eq. 16, 1057–1093 (1991) · Zbl 0742.45004 · doi:10.1080/03605309108820789
[30] Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series 30, Princeton, NJ: Princeton University Press, 1970 · Zbl 0207.13501
[31] Tricomi F.G.: Integral Equations. New York-London, Interscience Publ. (1957) · Zbl 0078.09404
[32] Vázquez, J.L.: Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type. Oxford Lecture Series in Mathematics and its Applications 33, Oxford: Oxford University Press, 2006 · Zbl 1113.35004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.