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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
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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

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