On the existence of self-similar solutions to the steady Navier-Stokes equations in high dimensions

Abstract

We prove that the steady incompressible Navier-Stokes equations with any given $(-3)$-homogeneous, locally Lipschitz external force on ${\mathbb{R}}^n\setminus \{0\}$, $4\le n\le 16$, have at least one $(-1)$-homogeneous solution which is scale-invariant and regular away from the origin. The global uniqueness of the self-similar solution is obtained as long as the external force is small. The key observation is to exploit a nice relation between the radial component of the velocity and the total head pressure under the self-similarity assumption. It plays an essential role in establishing the energy estimates. If the external force has only the nonnegative radial component, we can prove the existence of $(-1)$-homogeneous solutions for all $n\ge 4$. The regularity of the solution follows from integral estimates of the positive part of the total head pressure, which is due to the maximum principle and a ``dimension-reduction" effect arising from the self-similarity.