# Spontaneous periodic orbits in the Navier-Stokes flow

## TL;DR

This paper introduces a computer-assisted method to prove the existence of spontaneous periodic orbits in the three-dimensional Navier-Stokes equations with specific forcing, utilizing a zero-finding approach on Fourier coefficients.

## Contribution

It develops a general constructive framework using a Newton-Kantorovich theorem for proving periodic orbits in Navier-Stokes flows, incorporating symmetry reductions and analytic estimates.

## Key findings

- Proofs of spontaneous periodic orbits in Navier-Stokes with Taylor-Green forcing
- Application of a zero-finding problem on Fourier coefficients
- Use of symmetry to reduce computational complexity

## 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.