Singular stationary Navier-Stokes flows: examples and stability

Abstract

Landau solutions, when oriented along the vertical axis, represent a one parameter family of exact, self-similar, axisymmetric, swirl-free solutions to the 3D stationary Navier-Stokes equations forced by an upward facing point-source of momentum at the origin. They have an isolated singularity at the origin. Other singular steady state solutions can be derived from a similar framework. Some of these variants have been proposed as models for physical scenarios. For example, a solution first found by Squire has been proposed as a model of a fluid entrained to a radially discharging surface layer of oil. Another, Serrin's swirling vortex, exhibits qualitative features shared with some tornadoes, like a two-cell structure consisting of a central downdraft and peripheral updraft as well as swirl. The first objective of this paper is to provide a detailed analysis of these and other examples, especially when boundaries are present. In this direction we find several new, physically motivated classes of solutions and identify new connections between the physics literature and the mathematics literature. Many of these examples are formulated on the half-space, but much of the mathematical literature on singular steady-state solutions is for the whole-space. The second objective of this paper is to establish asymptotic stability for many of these solutions by formulating the problem in domains with boundaries. Our most general result requires a new approach to asymptotic stability that is based on eventual regularity. As a special case, we prove that a class of steady-state solutions motivated by Serrin's swirling vortex are stable under axisymmetric perturbations. This requires a novel observation because these solutions are too singular -- they are called ``Type III'' in the literature -- to be directly amenable to existing approaches.