×

Pinning of interfaces in random media. (English) Zbl 1231.35323

Summary: For a model for the propagation of a curvature sensitive interface in a time independent random medium, as well as for a linearized version which is commonly referred to as quenched Edwards-Wilkinson equation, we prove existence of a stationary positive supersolution at non-vanishing applied load. This leads to the emergence of a hysteresis that does not vanish for slow loading, even though the local evolution law is viscous (in particular, the velocity of the interface in the model is linear in the driving force).

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
74N20 Dynamics of phase boundaries in solids
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] BRAZOVSKII, S., & NATTERMANN, T. Pinning and sliding of driven elastic systems: from domain walls to charge density waves. Adv. Phys. 53 (2004), 177-252.
[2] CARDALIAGUET, P., LIONS P.-L., & SOUGANIDIS, P. E. A discussion about the homogenization of moving interfaces. J. Math. Pures Appl. 91 (2009), 339-363. · Zbl 1180.35070 · doi:10.1016/j.matpur.2009.01.014
[3] COVILLE, J., DIRR, N., & LUCKHAUS, S. Non-existence of positive stationary solutions for a class of semi-linear PDEs with random coefficients. Networks Heterogeneous Media 5 (2010), 745-763. · Zbl 1259.35231 · doi:10.3934/nhm.2010.5.745
[4] CRACIUN, B., & BHATTACHARYA, K. Effective motion of a curvature-sensitive interface through a heterogeneous medium. Interfaces Free Bound. 6 (2004), 151-173. · Zbl 1061.35148 · doi:10.4171/IFB/95
[5] DELAUNAY, C. Sur la surface de révolution dont la courbure moyenne est constante. J. Math. Pures Appl. 1 (1841), 309-320.
[6] DIRR, N., DONDL, P. W., GRIMMETT, G. R., HOLROYD, A. E., & SCHEUTZOW, M. Lipschitz percolation. Electron. Comm. Probab. 15 (2010), 14-21. · Zbl 1193.60115
[7] DIRR, N., KARALI, G., & YIP, N. K. Pulsating wave for mean curvature flow in inhomogeneous medium. Eur. J. Appl. Math. 19 (2008), 661-699. · Zbl 1185.53076 · doi:10.1017/S095679250800764X
[8] DIRR, N., & YIP, N. K. Pinning and de-pinning phenomena in front propagation in heterogeneous media. Interfaces Free Bound. 8 (2006), 79-109. · Zbl 1101.35074 · doi:10.4171/IFB/136
[9] GRIMMETT, G. R., & HOLROYD, A. E. Geometry of Lipschitz percolation. Ann. Inst. H. Poincaré Probab. Statist., to appear. · Zbl 1255.60167 · doi:10.1214/10-AIHP403
[10] KARDAR, M. Nonequilibrium dynamics of interfaces and lines. Phys. Rep. 301 (1998), 85-112.
[11] KLEEMANN, W. Dynamic phase transitions in ferroic systems with pinned domain wall. Oberwolfach Reports 1 (2004), 1587-1656.
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.