Lioiville theorems for the Navier-Stokes equations and applications

We study bounded ancient solutions of the Navier-Stokes equa-tions. These are solutions with bounded velocity defined in Rn × (−∞, 0). In two space dimensions we prove that such solutions are either constant or of the form u(x, t) = b(t), depending on the ex-act definition of admissible solutions. The general three dimensional problem seems to be out of reach of existing techniques, but partial results can be obtained in the case of axi-symmetric solutions. We apply these results to some scenarios of potential singularity forma-tion for axi-symmetric solutions, and obtain extensions of results in a recent paper by Chen, Strain, Tsai and Yau [4].