  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.
