Dynamic pattern formation in Swift-Hohenberg equations
Authors:
Tung Hoang and Hyung Ju Hwang
Journal:
Quart. Appl. Math. 69 (2011), 603-612
MSC (2000):
Primary 35K50, 35B35, 92C15
DOI:
https://doi.org/10.1090/S0033-569X-2011-01260-1
Published electronically:
April 26, 2011
MathSciNet review:
2850747
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: In this paper we study nonlinear instability in the Swift-Hohenberg equation. The short-time pattern formation under the fully nonlinear setting is rigorously characterized. For unstable constant equilibria, we investigate the nonlinear dynamics of patterns which occur within the time interval of existence.
References
- Gunduz Caginalp and Paul Fife, Higher-order phase field models and detailed anisotropy, Phys. Rev. B (3) 34 (1986), no. 7, 4940–4943. MR 875362, DOI https://doi.org/10.1103/PhysRevB.34.4940
- M.C. Cross and P.C. Hohenberg, Pattern formation outside of equilibrium. Rev. Mod. Phys. 65 (1993), 851–1112.
- Peter Grindrod, Patterns and waves, Oxford Applied Mathematics and Computing Science Series, The Clarendon Press, Oxford University Press, New York, 1991. The theory and applications of reaction-diffusion equations. MR 1136256
- Y. Guo, H.J. Hwang, Pattern formation. I: The Keller-Segel model, J. Differential Equations 249 (2010), 1519–1530.
- Yan Guo and Hyung Ju Hwang, Pattern formation. II. The Turing instability, Proc. Amer. Math. Soc. 135 (2007), no. 9, 2855–2866. MR 2317962, DOI https://doi.org/10.1090/S0002-9939-07-08850-8
- Yan Guo, Chris Hallstrom, and Daniel Spirn, Dynamics near an unstable Kirchhoff ellipse, Comm. Math. Phys. 245 (2004), no. 2, 297–354. MR 2039699, DOI https://doi.org/10.1007/s00220-003-1017-z
- Hyung Ju Hwang and Yan Guo, On the dynamical Rayleigh-Taylor instability, Arch. Ration. Mech. Anal. 167 (2003), no. 3, 235–253. MR 1978583, DOI https://doi.org/10.1007/s00205-003-0243-z
- J. Swift and P.C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A 15 (1977), 319.
- R.E. La Quey, P.H. Mahajan, P.H. Rutherford, and W.M. Tang, Phys. Rev. Lett. 34 (1975), 391.
- T. Shang and G.L. Sivashinsky, Phys. J. 43 (1983), 459–466.
- Lambertus A. Peletier and Vivi Rottschäfer, Pattern selection of solutions of the Swift-Hohenberg equation, Phys. D 194 (2004), no. 1-2, 95–126. MR 2075667, DOI https://doi.org/10.1016/j.physd.2004.01.043
- Lambertus A. Peletier and Vivi Rottschäfer, Large time behaviour of solutions of the Swift-Hohenberg equation, C. R. Math. Acad. Sci. Paris 336 (2003), no. 3, 225–230 (English, with English and French summaries). MR 1968263, DOI https://doi.org/10.1016/S1631-073X%2803%2900021-9
- L. A. Peletier and J. F. Williams, Some canonical bifurcations in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst. 6 (2007), no. 1, 208–235. MR 2299639, DOI https://doi.org/10.1137/050647232
References
- G. Caginalp and P.C. Fife, Higher order phase field models and detailed anisotropy. Phys. Rev. B 34 (1986), 4940–4943. MR 875362 (88f:82019)
- M.C. Cross and P.C. Hohenberg, Pattern formation outside of equilibrium. Rev. Mod. Phys. 65 (1993), 851–1112.
- P. Grinrod, Patterns and Waves: The Theory and Applications of Reaction-Diffusion Equations, Oxford: Clarendon, 1991. MR 1136256 (92k:35145)
- Y. Guo, H.J. Hwang, Pattern formation. I: The Keller-Segel model, J. Differential Equations 249 (2010), 1519–1530.
- Y. Guo and H.J. Hwang, Pattern formation. II. The Turing instability, Proc. Amer. Math. Soc. 135 (2007), 2855–2866. MR 2317962 (2008a:35131)
- Y. Guo, C. Hallstrom, and D. Spirn, Dynamics near an unstable Kirchhoff ellipse, Comm. Math. Phys. 245 (2004), 297-354. MR 2039699 (2005m:76039)
- H-J. Hwang and Y. Guo, On the dynamical Rayleigh-Taylor instability. Arch. Ration. Mech. Anal. 167 (2003), no. 3, 235-253. MR 1978583 (2004f:76064)
- J. Swift and P.C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A 15 (1977), 319.
- R.E. La Quey, P.H. Mahajan, P.H. Rutherford, and W.M. Tang, Phys. Rev. Lett. 34 (1975), 391.
- T. Shang and G.L. Sivashinsky, Phys. J. 43 (1983), 459–466.
- L.A. Peletier and V. Rottschäfer, Pattern selection of solutions of the Swift-Hohenberg equation, Physica D 194 (2004), 95-126. MR 2075667 (2005h:35176)
- L.A. Peletier and V. Rottschäfer, Large time behavior of solutions of the Swift–Hohenberg equation, C. R. Acad. Sci. Paris Ser. I Math. 336 (2003), 225–230. MR 1968263 (2004b:35167)
- L.A. Peletier and J.F. Williams, Some canonical bifurcations in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst. 6 (2007), no. 1, 208-235. MR 2299639 (2007m:34051)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2000):
35K50,
35B35,
92C15
Retrieve articles in all journals
with MSC (2000):
35K50,
35B35,
92C15
Additional Information
Tung Hoang
Affiliation:
Department of Mathematics, State University of New York at Buffalo, New York
Email:
thoang@buffalo.edu
Hyung Ju Hwang
Affiliation:
Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, Korea
MR Author ID:
672369
Email:
hjhwang@postech.ac.kr
Received by editor(s):
March 24, 2010
Published electronically:
April 26, 2011
Article copyright:
© Copyright 2011
Brown University
