Note on the fast decay property of steady Navier–Stokes flows in the whole space

Abstract

Abstract We investigate the pointwise asymptotic behavior of solutions to the stationary Navier–Stokes equation in {\mathbb{R}^{n}} ( {n\geq 3} ). We show the existence of a unique solution {\{u,p\}} such that {|\nabla^{j}u(x)|=O(|x|^{1-n-j})} and {|\nabla^{k}p(x)|=O(|x|^{-n-k})} ( {j,k=0,1,\ldots} ) as {|x|\rightarrow\infty} , assuming the smallness of the external force and the rapid decay of its derivatives. The solution {\{u,p\}} decays more rapidly than the Stokes fundamental solution.