Self-similar solutions to the steady Navier-Stokes equations in arbitrarily high dimensions

Abstract

We study $(-1)$-homogeneous solutions of the steady Navier-Stokes equations in $\mathbb R^n\setminus\{0\}$ with $(-3)$-homogeneous, locally Lipschitz forces. Previous work [2] established the existence of $(-1)$-homogeneous solutions without any smallness assumption in dimensions $4\leq n\leq16$. In this paper, we remove the dimension restriction and obtain the existence of self-similar solutions in all higher dimensions. The new ingredient is a dimension-independent a priori estimate for the total head pressure on the sphere, obtained by adapting the source-flux method and the weighted identity of the pressure of Wang and Yang [28] to the homogeneous setting. This combines with the structural relation between the radial velocity and the total head pressure used in [2] to close the a priori estimates for the solutions in all dimensions.