A computer-assisted proof on the stability of the Kolmogorov flows of incompressible viscous fluid

摘要

There exists a great number of references of bifurcations and stability for the Navier–Stokes equations. Only a few, however, provide a rigorous result which guarantees stability or instability. Our aim is to present a rigorous theorem which proves the stability of certain solutions arising in what is called the Kolmogorov problem. We accomplish this by the verified computation. The eigenvalue problem arising in the Kolmogorov problem is not self-adjoint and, accordingly, it is quite difficult to treat theoretically. Our method is a numerical approach to deal with this difficulty and numerical examples are given as a demonstration.