Self-similar stationary Navier--Stokes flows in a half-space with arbitrarily large swirl forcing

Abstract

We study axisymmetric self-similar solutions of the stationary Navier--Stokes equations with $(-3)$-homogeneous forces and no-slip boundary conditions in a half-space. For a fixed force without swirl, we characterize the interval of amplitudes for which solutions exist and describe the velocity singularity at its finite endpoints. For each parameter in this interval, a solution exists after adding a swirl force of any size, and it is unique among solutions with bounded profiles when the swirl force is sufficiently small. The proof uses the nonnegative contribution of swirl to a Riccati equation and a comparison estimate. We also prove that, when the tangential curl of the force is sufficiently small, there exists a unique solution with small velocity profiles.

arXiv comment26 pages; comments are welcome!