A Liouville theorem of Navier-Stokes equations with two periodic variables

Abstract

In this paper, we study the Liouville theorem of the stationary Navier-Stokes equations in R3. When the solution is periodic in two variables, we can prove that actually the solution is trivial (constant vector) under the assumption that one component of the velocity, vanishing at infinity, has finite Dirichlet integral and the other two components can have some growth with respect to the distance to the origin.