In this paper we study the Liouville-type properties for solutions to the steady incompressible Euler equations with forces in RN\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}^N}$$\end{document}. If we assume “single signedness condition” on the force, then we can show that a C1(RN)\documentclass[12pt]{minimal}
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\begin{document}$${C^1 (\mathbb {R}^N)}$$\end{document} solution (v, p) with |v|2+|p|∈Lq2(RN),q∈(3NN-1,∞)\documentclass[12pt]{minimal}
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\begin{document}$${|v|^2+ |p| \in L^{\frac{q}{2}}(\mathbb {R}^N),\,q \in (\frac{3N}{N-1}, \infty)}$$\end{document} is trivial, v = 0. For the solution of the steady Navier–Stokes equations, satisfying v(x)→0\documentclass[12pt]{minimal}
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\begin{document}$${v(x) \to 0}$$\end{document} as |x|→∞\documentclass[12pt]{minimal}
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\begin{document}$${|x| \to \infty}$$\end{document}, the condition ∫R3|Δv|65dx<∞\documentclass[12pt]{minimal}
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\begin{document}$${\int_{\mathbb {R}^3} |\Delta v|^{\frac{6}{5}} dx < \infty}$$\end{document}, which is stronger than the important D-condition, ∫R3|∇v|2dx<∞\documentclass[12pt]{minimal}
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\begin{document}$${\int_{\mathbb {R}^3} |\nabla v|^2 dx < \infty}$$\end{document}, 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.
