Forced self-similar solutions to the stationary Navier--Stokes equations in a half-space

Abstract

We study axisymmetric self-similar solutions to the stationary Navier--Stokes equations in the half-space with the no-slip boundary condition, driven by an axisymmetric (-3)-homogeneous external force. If the tangential curl of the force on the unit sphere is sufficiently small, we prove the existence of a unique small solution; when the force is swirl-free, the solution is automatically swirl-free and unique. For a swirl-free external force $\boldsymbol{F}$, we introduce a scaling parameter $λ$ and consider the system with force $λ\boldsymbol{F}$; we prove that solutions exist precisely for $λ$ in an open interval containing zero. The same approach extends to solid cones with the no-slip boundary condition, where narrower opening angles allow the existence of solutions under larger external forces.