Spontaneous periodic orbits in the Navier-Stokes flow. 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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References in zbMATH (referenced in 8 articles , 1 standard article )
Showing results 1 to 8 of 8.
- Liu, Xuefeng; Nakao, Mitsuhiro T.; Oishi, Shin’ichi: Computer-assisted proof for the stationary solution existence of the Navier-Stokes equation over 3D domains (2022)
- Arioli, Gianni; Gazzola, Filippo; Koch, Hans: Uniqueness and bifurcation branches for planar steady Navier-Stokes equations under Navier boundary conditions (2021)
- Arioli, Gianni; Koch, Hans: A Hopf bifurcation in the planar Navier-Stokes equations (2021)
- Kalita, Piotr; Zgliczyński, Piotr: Rigorous FEM for one-dimensional Burgers equation (2021)
- Kepley, Shane; Zhang, Tianhao: A constructive proof of the Cauchy-Kovalevskaya theorem for ordinary differential equations (2021)
- Lessard, Jean-Philippe; James, J. D. Mireles: A rigorous implicit (C^1) Chebyshev integrator for delay equations (2021)
- van den Berg, Jan Bouwe; Breden, Maxime; Lessard, Jean-Philippe; van Veen, Lennaert: Spontaneous periodic orbits in the Navier-Stokes flow (2021)
- Breden, Maxime; Kuehn, Christian: Rigorous validation of stochastic transition paths (2019)