Spontaneous Periodic Orbits in the Navier–Stokes Flow

Jan Bouwe van den Berg; Maxime Breden; Jean Philippe Lessard; Lennaert van Veen

doi:10.1007/s00332-021-09695-4

Spontaneous Periodic Orbits in the Navier–Stokes Flow

Jan Bouwe van den Berg, Maxime Breden, Jean Philippe Lessard^*, Lennaert van Veen

^*Corresponding author for this work

Mathematics

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Abstract

In this paper, a general method to obtain constructive proofs of existence of periodic orbits in the forced autonomous Navier–Stokes equations on the three-torus is proposed. After introducing a zero finding problem posed on a Banach space of geometrically decaying Fourier coefficients, a Newton–Kantorovich theorem is applied to obtain the (computer-assisted) proofs of existence. The required analytic estimates to verify the contractibility of the operator are presented in full generality and symmetries from the model are used to reduce the size of the problem to be solved. As applications, we present proofs of existence of spontaneous periodic orbits in the Navier–Stokes equations with Taylor–Green forcing.

Original language	English
Article number	41
Pages (from-to)	1-64
Number of pages	64
Journal	Journal of nonlinear science
Volume	31
Issue number	2
Early online date	26 Mar 2021
DOIs	https://doi.org/10.1007/s00332-021-09695-4
Publication status	Published - Apr 2021

Bibliographical note

Funding Information:
J. B. van den Berg: partially supported by NWO-VICI Grant 639033109. M. Breden: partially supported by a Lichtenberg Professorship grant of the VolkswagenStiftung awarded to C. Kuehn. J.-P. Lessard and L. van Veen: supported by NSERC.

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

Funding

J. B. van den Berg: partially supported by NWO-VICI Grant 639033109. M. Breden: partially supported by a Lichtenberg Professorship grant of the VolkswagenStiftung awarded to C. Kuehn. J.-P. Lessard and L. van Veen: supported by NSERC.

Keywords

Computer-assisted proofs
Navier–Stokes equations
Periodic orbits
Symmetry breaking

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10.1007/s00332-021-09695-4

Berg2021_Article_SpontaneousPeriodicOrbitsInTheNavier-StokesFlowFinal published version, 1.5 MBLicence: Article 25fa Dutch Copyright Act

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abstract = "In this paper, a general method to obtain constructive proofs of existence of periodic orbits in the forced autonomous Navier–Stokes equations on the three-torus is proposed. After introducing a zero finding problem posed on a Banach space of geometrically decaying Fourier coefficients, a Newton–Kantorovich theorem is applied to obtain the (computer-assisted) proofs of existence. The required analytic estimates to verify the contractibility of the operator are presented in full generality and symmetries from the model are used to reduce the size of the problem to be solved. As applications, we present proofs of existence of spontaneous periodic orbits in the Navier–Stokes equations with Taylor–Green forcing.",

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Spontaneous Periodic Orbits in the Navier–Stokes Flow

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Keywords

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