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 R N . If we assume “single signedness condition” on the force, then we can show that a C 1 ( R N ) solution ( v , p ) with | v | 2 + | p | ∈ L q 2 ( R N ) , q ∈ ( 3 N N - 1 , ∞ ) is trivial, v = 0. For the solution of the steady Navier–Stokes equations, satisfying v ( x ) → 0 as | x | → ∞ , the condition ∫ R 3 | Δ v | 6 5 d x < ∞ , which is stronger than the important D-condition, ∫ R 3 | ∇ v | 2 d x < ∞ , 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.