Hostname: page-component-7479d7b7d-68ccn Total loading time: 0 Render date: 2024-07-13T22:33:24.790Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite time blow-up of complex solutions of the conserved Kuramoto–Sivashinsky equation in ℝd and in the torus 𝕋d, d ⩾ 1

Published online by Cambridge University Press:  16 January 2019

Léo Agélas
Léo Agélas*
Affiliation:
Department of Mathematics, IFP Energies Nouvelles, 1-4, avenue de Bois-Préau, F-92852 Rueil-Malmaison, France (leo.agelas@ifpen.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider complex-valued solutions of the conserved Kuramoto–Sivashinsky equation which describes the coarsening of an unstable solid surface that conserves mass and that is parity symmetric. This equation arises in different aspects of surface growth. Up to now, the problem of existence and smoothness of global solutions of such equations remained open in ℝd and in the torus 𝕋d, d ⩾ 1. In this paper, we answer partially to this question. We prove the finite time blow-up of complex-valued solutions associated with a class of large initial data. More precisely, we show that there is complex-valued initial data that exists in every Besov space (and hence in every Lebesgue and Sobolev space), such that after a finite time, the complex-valued solution is in no Besov space (and hence in no Lebesgue or Sobolev space).

Keywords

Blow upSurface growthconserved Kuramoto–Sivashinsky
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 5 , October 2019 , pp. 1175 - 1188
Copyright
Copyright © Royal Society of Edinburgh 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1 Agélas, L.. Global regularity of solutions of equation modeling epitaxy thin film growth in ℝd, d = 1,2. J. Evol. Equ. 15 (2015), 89–106.Google Scholar
2Bahouri, H., Chemin, J.-Y. and Danchin, R.. Fourier analysis and nonlinear partial differential equations. Grundlehren der mathematischen Wissenschaften, vol. 343 (Springer, 2011).Google Scholar
3Bergh, J. and Löfström, J.. Interpolation Spaces. An Introduction (Berlin-Heidelberg-New York: Springer-Verlag, 1976).Google Scholar
4Blömker, D. and Gugg, C., Thin-Film-Growth-Models: on local solutions. Recent developments in Stochastic analysis and related topics, World Scientific, Singapore. Proceedings of the First Sino-German Conference on Stochastic Analysis, (Ed. Albeverio, S., Ma, Z. M., Röckner, M.),pp. 66–77 (2004).Google Scholar
5Blömker, D. and Romito, M.. Regularity and blow up in a surface growth model. Dynamics of PDE 6 (2009), 227–252.Google Scholar
6Blömker, D. and Romito, M.. Local existence and uniqueness in the largest critical space for a surface growth model. Nonlinear Differ. Equ. Appl. 19 (2012), 365–381.Google Scholar
7Blömker, D. and Romito, M.. Stochastic PDEs and lack of regularity. A surface growth equation with noise: existence, uniqueness, and blow-up. Jahresber Dtsch Math-Ver 117 (2015), 233–286.Google Scholar
8Blömker, D., Gugg, C. and Raible, M.. Thin-film-growth models: roughness and correlation functions. Eur. J. Appl. Math. 13 (2002), 385–402.Google Scholar
9Blömker, D., Flandoli, F. and Romito, M.. Markovianity and ergodicity for a surface growth PDE. Ann. Probab. 37 (2009), 275–313.Google Scholar
10Blömker, D., Nolde, C. and Robinson, J.. Rigorous numerical verification of uniqueness and smoothness in a surface growth model. J. Math. Anal. Appl. 429 (2015), 311–325.Google Scholar
11Cannone, M.. Harmonic analysis tools for solving the incompressible Navier-stokes equations. Handbook of Mathematical fluid dynamics, vol. III,pp. 161–244 (North-Holland, Amsterdam, 2004).Google Scholar
12Chemin, J.-Y.. Perfect incompressible fluids (Oxford: Clarendon Press, 1998).Google Scholar
13Frazier, M., Jawerth, B. and Weiss, G.. Littlewood-Paley theory and the study of function spaces. CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences,vol. 79 (Washington, DC; by the American Mathematical Society, Providence, RI (1991).Google Scholar
14Frisch, T. and Verga, A.. Effect of step stiffness and diffusion anisotropy on the meandering of a growing vicinal surface. Phys. Rev. Lett. 96 (2006), 166104.Google Scholar
15Furioli, G., Lemarié-Rieusset, P.-G. and Terraneo, E.. Sur l'unicité dans L 3(ℝ3) des solutions mild des équations de Navier-Stokes. [On the uniqueness in L 3(ℝ3) of mild solutions for the Navier-Stokes equations]. C. R. Acad. Sci. Paris Sér. I Math. 325, 1253–1256, (1997).Google Scholar
16Gazzola, F. and Grunau, H.-C.. Some new properties of biharmonic heat kernels. Nonlinear Anal. 70 (2009), 2965–2973.Google Scholar
17Giga, Y.. Weak and strong solutions of the Navier-Stokes initial value problem. Publ. RIMS, Kyoto Univ. 19 (1983), 887–910.Google Scholar
18Kato, T.. Strong L p −solutions of the Navier-Stokes equation in ℝm, with applications to weak solutions. Math. Z. 187.Google Scholar
19Kato, T. and Ponce, G.. Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure. Applied. Math 41 (1988), 891–907.Google Scholar
20Li, D. and Sinai, Y.. Blow ups of complex solutions of the 3d-Navier-Stokes system and renormalization group method. J. Eur. Math. Soc. 10 (2008), 267–313.Google Scholar
21Linz, S. J., Raible, M. and Hänggi, P.. Amorphous thin film growth: modeling and pattern formation. Adv. Solid State Phys. 41 (2001), 391–403.Google Scholar
22Montgomery-Smith, S.. Finite time blow up for a Navier-Stokes like equation. Proc. A.M.S., 129 (2001), 3017–3023.Google Scholar
23Raible, M., Linz, S. J. and Hänggi, P.. Amorphous thin film growth: minimal deposition equation. Phys. Rev. E 62 (2000a), 1691–1694.Google Scholar
24Raible, M., Mayr, S. G., Linz, S., Moske, M., Hänggi, P. and Samwer, K.. Amorphous thin film growth: theory compared with experiment. Europhys. Lett. 50 (2000b), 61–67.Google Scholar
25Schmeisser, H.-J. and Triebel, H.. Topics in Fourier analysis and function spaces, Mathematik und ihre Anwendungen in Physik und Technik [Mathematics and its Applications in Physics and Technology], vol. 42, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, (1987).Google Scholar
26Siegert, M. and Plischke, M.. Solid-on-solid models of molecular-beam epitaxy. Phys. Rev. E 50 (1994), 917–931.Google Scholar
27Stein, O. and Winkler, M.. Amorphous molecular beam epitaxy: global solutions and absorbing sets. Eur. J. Appl. Math. 16 (2005), 767–798.Google Scholar
28Triebel, H.. Theory of Function Spaces. Geest & Portig, Leipzig,vol. 78 (Basel: Birkhäuser, 1983).Google Scholar
29Winkler, M.. Global solutions in higher dimensions to a fourth order parabolic equation modeling epitaxial thin film growth. Z. Angew. Math. Phys. (ZAMP). 62 (2011), 575–608.Google Scholar