Rigorous Numerics for ill-posed PDEs: Periodic Orbits in the Boussinesq Equation

Roberto Castelli; Marcio Gameiro; Jean Philippe Lessard

doi:10.1007/s00205-017-1186-0

Rigorous Numerics for ill-posed PDEs: Periodic Orbits in the Boussinesq Equation

Roberto Castelli
, Marcio Gameiro
, Jean Philippe Lessard^*

^*Corresponding author for this work

Mathematics

Research output: Contribution to Journal › Article › Academic › peer-review

Abstract

In this paper, we develop computer-assisted techniques for the analysis of periodic orbits of ill-posed partial differential equations. As a case study, our proposed method is applied to the Boussinesq equation, which has been investigated extensively because of its role in the theory of shallow water waves. The idea is to use the symmetry of the solutions and a Newton–Kantorovich type argument (the radii polynomial approach) to obtain rigorous proofs of existence of the periodic orbits in a weighted ℓ¹ Banach space of space-time Fourier coefficients with exponential decay. We present several computer-assisted proofs of the existence of periodic orbits at different parameter values.

Original language	English
Pages (from-to)	129-157
Number of pages	29
Journal	Archive for Rational Mechanics and Analysis
Volume	228
Issue number	1
DOIs	https://doi.org/10.1007/s00205-017-1186-0
Publication status	Published - 1 Apr 2018

Funding

Acknowledgements. Marcio Gameiro was partially supported by FAPESP grants 2013/07460-7, 2013/50382-7, 2016/08704-5, and 2016/21032-6 and by CNPq grants 305860/2013-5 and 310740/2016-9, Brazil. Jean-Philippe Lessard was partially supported by an NSERC Discovery Grant and by a FAPESP-CALDO grant.

Funders	Funder number
FAPESP-CALDO
Natural Sciences and Engineering Research Council of Canada
Fundação de Amparo à Pesquisa do Estado de São Paulo	2013/07460-7, 2016/21032-6, 2013/50382-7, 2016/08704-5
Conselho Nacional de Desenvolvimento Científico e Tecnológico	305860/2013-5, 310740/2016-9

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SDG 6 Clean Water and Sanitation

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Rigorous Numerics for ill-posed PDEs: Periodic Orbits in the Boussinesq Equation

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