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 $${\mathbb {R}^N}$$ . If we assume “single signedness condition” on the force, then we can show that a $${C^1 (\mathbb {R}^N)}$$ solution (v, p) with $${|v|^2+ |p| \in L^{\frac{q}{2}}(\mathbb {R}^N),\,q \in (\frac{3N}{N-1}, \infty)}$$ is trivial, v = 0. For the solution of the steady Navier–Stokes equations, satisfying $${v(x) \to 0}$$ as $${|x| \to \infty}$$ , the condition $${\int_{\mathbb {R}^3} |\Delta v|^{\frac{6}{5}} dx < \infty}$$ , which is stronger than the important D-condition, $${\int_{\mathbb {R}^3} |\nabla v|^2 dx < \infty}$$ , 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.