Abstract
We show the existence and uniqueness of solutions (either local or global for small data) for an equation arising in different aspects of surface growth. Following the work of Koch and Tataru we consider spaces critical with respect to scaling and we prove our results in the largest possible critical space such that weak solutions are defined, which turns out to be a Besov space. Similarly to 3D-Navier Stokes, the uniqueness of global weak solutions remains unfortunately open, unless the initial conditions are sufficiently small.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barabasi A.L., Stanley H.E.: Fractal Concepts in Surface Growth. Cambridge University Press, Cambridge (1995)
Blömker D., Flandoli F., Romito M.: Markovianity and ergodicity for a surface growth PDE. Ann. Probab. 37(1), 275–313 (2009)
Blömker, D., Gugg, C.: Thin-film-growth-models: on local solutions. In: Albeverio, S., et al. (eds.) Recent developments in stochastic analysis and related topics. Proceedings of the First Sino-German Conference on Stochastic Analysis (A Satellite Conference of ICM 2002), Beijing, China, 29 August–3 September 2002, pp. 66–77. World Scientific, River Edge (2004)
Blömker D., Gugg C., Raible M.: Thin-film-growth models: roughness and correlation functions. Eur. J. Appl. Math. 13(4), 385–402 (2002)
Blömker D., Romito M.: Regularity and blow up in a surface growth model. Dyn. Partial Differ. Equ. 6(3), 227–252 (2009)
Chow, P.-L.: Stochastic partial differential equations, In: Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series. Chapman & Hall/CRC Press, Boca Raton ix, p. 281 (2007)
Cuerno R., Vázquez L., Gago R.: Self-organized ordering of nanostructures produced by ion-beam sputtering. Phys. Rev. Lett. 94, 016102 (2005)
Da Prato G., Zabczyk J.: Stochastic equations in infinite dimensions Encyclopedia of Mathematics and Its Applications, vol 44. Cambridge University Press, Cambridge (1992)
Frisch T., Verga A.: Effect of step stiffness and diffusion anisotropy on the meandering of a growing vicinal surface. Phys. Rev. Lett. 96, 166104 (2006)
Germain, P., Pavlović, N., Staffilani G.: Regularity of solutions to the Navier-Stokes equations evolving from small data in BMO−1. Int. Math. Res. Not. IMRN. (21):Art. ID rnm087 (2007)
Halpin-Healy T., Zhang Y.C.: Kinetic roughening, stochastic growth, directed polymers and all that. Phys. Rep. 254, 215–415 (1995)
Hoppe R., Linz S., Litvinov W.: On solutions of certain classes of evolution equations for surface morphologies. Nonlinear Phenom. Complex Syst. 6, 582–591 (2003)
Hoppe R., Nash E.: A combined spectral element/finite element approach to the numerical solution of a nonlinear evolution equation describing amorphous surface growth of thin films. J. Numer. Math. 10(2), 127–136 (2002)
Hoppe, R.H., Nash, E.: Numerical solution of a nonlinear evolution equation describing amorphous surface growth of thin films. In: Feistauer, M., et al. (eds.) Numerical Mathematics and Advanced Applications. Proceedings of ENUMATH 2003, the 5th European Conference on Numerical Mathematics and Advanced Applications, Prague, Czech Republic, August 18–22, 2003, pp. 440–448. Springer, Berlin (2004)
Koch H., Tataru D.: Well-posedness for the Navier–Stokes equations. Adv. Math. 157(1), 22–35 (2001)
Koch H., Lamm T.: Geometric flows with rough initial data, preprint, 2009. arXiv:0902.1488 [math.AP]
Lemarié-Rieusset P.G.: Recent developments in the Navier-Stokes problem Chapman & Hall/CRC Research Notes in Mathematics Series, vol 431. CRC Press, Boca Raton (2002)
Liu K: Stability of infinite dimensional stochastic differential equations with applications Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 135. Chapman & Hall/CRC Press, Boca Raton, xi (2006)
Muñoz-García J., Cuerno R., Castro M.: Coupling of morphology to surface transport in ion-beam-irradiated surfaces: normal incidence and rotating targets. J. Phys. Condens. Matter 21(22), 224020 (2009)
Muñoz-García J., Gago R., Vázquez L., Sánchez-García J.A., Cuerno R.: Observation and modeling of interrupted pattern coarsening: surface nanostructuring by ion erosion. Phys. Rev. Lett. 104, 026101 (2010)
Nečas J., Růžička M., Šverák V.: On Leray’s self-similar solutions of the Navier–Stokes equations. Acta Math. 176(2), 283–294 (1996)
Prato G.D., Debussche A.: Stochastic Cahn-Hilliard equation. Nonlinear Anal. Theory Methods Appl. 26(2), 1–263 (1996)
Raible M., Linz S., Hänggi P.: Amorphous thin film growth: modeling and pattern formation. Adv. Solid State Phys. 41, 391–403 (2001)
Raible M., Linz S.J., Hänggi P.: Amorphous thin film growth: minimal deposition equation. Phys. Rev. E 62, 1691–1694 (2000)
Raible M., Mayr S., Linz S., Moske M., Hänggi P., Samwer K.: Amorphous thin film growth: theory compared with experiment. Europhys. Lett. 50, 61–67 (2000)
Stein E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. In: Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton (1993)
Stein O., Winkler M.: Amorphous molecular beam epitaxy: global solutions and absorbing sets. Eur. J. Appl. Math. 16(6), 767–798 (2005)
Walsh, J.B.: An introduction to stochastic partial differential equations. In: École d’été de probabilités de Saint-Flour, XIV—1984. Lecture Notes in Math., vol. 1180, pp. 265–439. Springer, Berlin (1986)
Wang, C.: Well-posedness for the heat flow of biharmonic maps with rough initial data. J. Geom. Anal. (2010). doi:10.1007/s12220-010-9195-3
Winkler M.: Global solutions in higher dimensions to a fourth order parabolic equation modeling epitaxial thin film growth. Z. Angew. Math. Phys. (ZAMP). 62(4), 575–608 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work has been partially supported by the GNAMPA project Studio delle singolarità di alcune equazioni legate a modelli idrodinamici. Part of the work was done at the Newton institute for Mathematical Sciences in Cambridge (UK), whose support is gratefully acknowledged, during the program “Stochastic partial differential equations”. The authors would like to thank Herbert Koch for explaining the method and for the several feedbacks on a preliminary version of the paper.
Rights and permissions
About this article
Cite this article
Blömker, D., Romito, M. Local existence and uniqueness in the largest critical space for a surface growth model. Nonlinear Differ. Equ. Appl. 19, 365–381 (2012). https://doi.org/10.1007/s00030-011-0133-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00030-011-0133-2