Liouville-Type Theorems for the Forced Euler Equations and the Navier-Stokes Equations

In this paper we study the Liouville-type properties for solutions to the steady incompressible Euler equations with forces in . If we assume "single signedness condition" on the force, then we can show that a solution (v, p) with is trivial, v = 0. For the solution of the steady Navier-Stokes equations, satisfying as , the condition , which is stronger than the important D-condition, , but both having the same scaling property, implies that v = 0. In the appendix we reprove Theorem 1.1 (Chae, Commun Math Phys 273:203-215, 2007), using the self-similar Euler equations directly.