| In this thesis, we prove a Liouville-type theorem for solutions of the two-dimensional Navier-Stokes equations, that solutions with globally bounded velocity (there are no conditions imposed on the pressure) are constant in space for each fixed time. In particular, in the steady-state case, solutions with bounded velocity are constant. The proof depends on a Harnack inequality argument applied to the vorticity of the solution, which satisfies a parabolic equation with bounded coefficients as well as certain local energy estimates. The proof in the steady-state case can be simplified due to an observation regarding the pressure in that case, and is presented separately for its independent interest. The main results of this work, contained in Theorems 1 and 8, are based on joint work with Nikolai Nadirashvili and Vladimir Sverak. |
