Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-22T12:12:07.917Z Has data issue: false hasContentIssue false
  • English
  • Français

Fourier splitting and dissipation of nonlinear dispersive waves

Published online by Cambridge University Press:  14 November 2011

J. L. Bona ,
F. Demengel  and
K. Promislow
J. L. Bona
Affiliation:
Department of Mathematics and Texas Institute for Computational and Applied Mathematics, University of Texas, Austin, TX 78712, USA
F. Demengel
Affiliation:
Department de Mathematique, University de Cergy-Pontoise, 2 Ave. Adophe Chauvin, Site de St. Martin, 95302 Cergy-Pontoise, France
K. Promislow
Affiliation:
Department of Mathematics and Statistics, Simon Fraser University, Burnaby BC, Canada V5A 1S6
Get access Rights & Permissions [Opens in a new window]

Abstract

Presented herein is a new method for analysing the long-time behaviour of solutions of nonlinear, dispersive, dissipative wave equations. The method is applied to the generalized Korteweg–de Vries equation posed on the entire real axis, with a homogeneous dissipative mechanism included. Solutions of such equations that commence with finite energy decay to zero as time becomes unboundedly large. In circumstances to be spelled out presently, we establish the existence of a universal asymptotic structure that governs the final stages of decay of solutions. The method entails a splitting of Fourier modes into long and short wavelengths which permits the exploitation of the Hamiltonian structure of the equation obtained by ignoring dissipation. We also develop a helpful enhancement of Schwartz's inequality. This approach applies particularly well to cases where the damping increases in strength sublinearly with wavenumber. Thus the present theory complements earlier work using centre-manifold and group-renormalization ideas to tackle the situation wherein the nonlinearity is quasilinear with regard to the dissipative mechanism.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 3 , 1999 , pp. 477 - 502
Copyright
Copyright © Royal Society of Edinburgh 1999

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

1Amick, C. J., Bona, J. L. and Schonbek, M. E.. Decay of solutions of some nonlinear wave equations. J. Diffl Eqns 81 (1989), 1–49.CrossRefGoogle Scholar
2Benjamin, T. B.. Lectures on nonlinear wave motion. In Nonlinear wave motion (ed. Newell, A. C.), pp. 3–47. Lectures in Applied Mathematics, vol. 15 (Providence, RI: AMS, 1974).Google Scholar
3Biler, P.. Asymptotic behavior in time of solutions to some equations generalizing the Korteweg–de Vries equation. Bull. Polish Acad. Sci. 32 (1984), 275–282.Google Scholar
4Bona, J. L.. On solitary waves and their role in the evolution of long waves. Applications of nonlinear analysis in the physical sciences (ed. Amann, H., Bazley, N. and Kirchgassner, K.), pp. 183–205 (London: Pitman, 1981).Google Scholar
5Bona, J. L., Dougalis, V., Karakashian, O. and McKinney, W.. Computations of blow-up and decay for periodic solutions of the generalized Korteweg–de Vries–Burgers equation. Appl Numer. Math. 10 (1992), 335–355.CrossRefGoogle Scholar
6Bona, J. L. and Luo, L.. More results on the decay of solutions to nonlinear, dispersive, wave equations. Discrete Contin. Dynam. Systems 1 (1995), 151–193.CrossRefGoogle Scholar
7Bona, J. L., Promislow, K. and Wayne, C. E.. On the asymptotic behavior of solutions to nonlinear, dispersive, dissipative wave equations. J. Math. Computers Simulation 37 (1994), 265–277.CrossRefGoogle Scholar
8Bona, J. L., Promislow, K. and Wayne, C. E.. Higher-order asymptotics of decaying solutions of some nonlinear dispersive, dissipative wave equations. Nonlinearity 8 (1995), 1179–1206.CrossRefGoogle Scholar
9Bona, J. L., Pritchard, W. G. and Scott, L. R.. An evaluation of a model equation for water waves Phil. Trans. R. Soc. Lond. A 302 (1981), 457–510.Google Scholar
10Bricmont, J., Kupiainen, A. and Lin, G.. Renormalization group and asymptotics of solutions of nonlinear parabolic equations. Commun. Pure Appl. Math. 47 (1994), 893–922.CrossRefGoogle Scholar
11Dix, D. B.. The dissipation of nonlinear dispersive waves: the case of asymptotically weak nonlinearity. Commun. PDE 17 (1992), 1665–1693.CrossRefGoogle Scholar
12Gardner, C. S., Greene, J. M., Kruskal, M. D. and Miura, R. M.. A method for solving the Korteweg-de Vries equation. Phys. Rev. Lett. 19 (1967), 1095–1097.CrossRefGoogle Scholar
13Gardner, C. S., Greene, J. M., Kruskal, M. D. and Miura, R. M.. Korteweg–de Vries equation and generalizations. VI. Methods for exact solution. Commun. Pure Appl. Math 27 (1974), 97–133.CrossRefGoogle Scholar
14Hammack, J. L.. A note on tsunamis: their generation and propagation in an ocean of uniform depth. J. Fluid Mech. 60 (1974), 769–799.CrossRefGoogle Scholar
15Hammack, J. L. and Segur, H.. The Korteweg–de Vries equation and water waves. Part 2. Comparison with experiments. J. Fluid Mech. 65 (1974), 289–314.CrossRefGoogle Scholar
16Kakutani, T. and Matsuuchi, K.. Effect of viscosity on long gravity waves. J. Phys. Soc. Jpn 39 (1975), 237–246.CrossRefGoogle Scholar
17Kato, T.. On the Cauchy problem for the (generalized) Korteweg–de Vries equation. In Stud. Appl. Math. (Suppl.) 8 (1983), 93–128 (New York: Academic Press).Google Scholar
18Mielke, A.. Locally invariant manifolds for quasilinear parabolic equations. Rocky Mountain J. Math. 21 (1991), 708–714.CrossRefGoogle Scholar
19Mahony, J. J. and Pritchard, W. G.. Wave reflexion from beaches. J. Fluid Mech. 101 (1980), 809–832.CrossRefGoogle Scholar
20Naumkin, P. and Shishmarev, I.. Nonlinear nonlocal equations in the theory of waves. Translations of Mathematics Monographs, vol. 133 (Providence, RI: AMS, 1994).CrossRefGoogle Scholar
21Ott, E. and Sudan, R. N.. Phys. Fluids 12 (1969), 2388.CrossRefGoogle Scholar
22Schechter, E.. Well-behaved evolutions and Trotter product formulas. PhD thesis, University of Chicago, 1978.Google Scholar
23Schonbek, M. E.. Large time behavior of solutions to the Navier–Stokes equations in Hm spaces. Commun. PDE 20 (1995), 103–117.CrossRefGoogle Scholar
24Schonbek, M. E.. Decay of solutions to parabolic conservation laws. Commun. PDE 7 (1980), 449–473.CrossRefGoogle Scholar
25Schonbek, M. E. and Rajopadhye, S.. Asymptotic behavior of solutions to the Korteweg–de Vries–Burgers system. Ann. Inst. H. Poincaré Analysis NonLinéaire 12 (1995), 425–457.CrossRefGoogle Scholar
26Wayne, C. E.. Invariant manifolds for parabolic partial differential equations on unbounded domains. Arch. Ration. Mech. Analysis 138 (1997), 279–306.CrossRefGoogle Scholar