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Solutions with scaling-invariant bounds such as self-similar solutions, playan important role in the understanding of the regularity and asymptoticstructures of solutions to the Navier-Stokes equations. In this paper, we provethat any steady solution satisfying $|\Bu(x)|\leq C/|x|$ for any constant $C$in $\mathbb{R}^n\setminus \{0\}$ with $ n \geq 4$, must be zero. Our main ideais to analyze the velocity field and the total head pressure via weightedenergy estimates with suitable multipliers so that the proof is prettyelementary and short. These results not only give the Liouville-type theoremfor steady solutions in higher dimensions with neither smallness norself-similarity type assumptions, but also help to remove a class ofsingularities of solutions and give the optimal asymptotic behaviors ofsolutions at infinity in the exterior domains.

Rigidity of Steady Solutions to the Navier-Stokes Equations in High Dimensions and its applications