Spontaneous Periodic Orbits in the Navier–Stokes Flow

• Published in: Journal of nonlinear science Vol. 31; no. 2
• Main Authors: van den Berg, Jan Bouwe, Breden, Maxime, Lessard, Jean-Philippe, van Veen, Lennaert
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2021; Springer Nature B.V; Springer Verlag
• Subjects: Analysis; Analysis of PDEs; Banach spaces; Classical Mechanics; Computational fluid dynamics; Economic Theory/Quantitative Economics/Mathematical Methods; Existence theorems; Fluid flow; Fluid mechanics; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Physics; Mathematics; Mathematics and Statistics; Mechanics; Navier-Stokes equations; Orbits; Physics; Stokes flow; Theoretical; Toruses; Navier–Stokes equations; 35B06; Periodic orbits; 35B36; 65G20; 35B10; Computer-assisted proofs; 35Q30; 76D17; Symmetry breaking; symmetry breaking; periodic orbits; computer-assisted proofs; Navier-Stokes equations
• ISSN: 0938-8974; 1432-1467
• DOI: 10.1007/s00332-021-09695-4

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.