Homogeneous Solutions of Stationary Navier–Stokes Equations with Isolated Singularities on the Unit Sphere. I. One Singularity

Abstract

We classify all (−1)−homogeneous axisymmetric no swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus the south pole, parameterize them as a two dimensional surface with boundary, and analyze their pressure profiles near the north pole. Then we prove that there is a curve of (−1)−homogeneous axisymmetric solutions with nonzero swirl, having the same smoothness property, emanating from every point of the interior and one part of the boundary of the solution surface. Moreover we prove that there is no such curve of solutions for any point on the other part of the boundary. We also establish asymptotic expansions for every (-1)-homogeneous axisymmetric solutions in a neighborhood of the singular point on the unit sphere.

Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. Ⅲ. Two singularities

“…If U c,γ (1) < −3, U i θ (1) < −3 for large i. Then by Theorem 1.4 in [1], U i φ must be constants for large i, the theorem is then proved.…”

“…Let U i = sin θu i for all i ∈ N. We have ||U i θ − U µ,γ θ || L ∞ (−1,1) → 0. By Theorem 1.3 of [1], U i (±1) must exists and is finite for every i. If U c,γ (−1) > 3, U i θ (−1) > 3 for large i.…”

“…A vector field u can be written as A vector field u is called axisymmetric if u r , u θ and u φ are independent of φ, and is called no-swirl if u φ = 0. Landau discovered in [4] a three parameter family of explicit (−1)-homogeneous solutions of the stationary NSE (1), which are axisymmetric and with no swirl. These solutions are now called Landau solutions.…”

“…We are interested in analyzing solutions which are smooth on S 2 minus finite points. In our first paper [1], we have classified all axisymmtric no-swirl solutions with one singularity, as a two dimensional surface with boundary, and have proved the existence of a curve of axisymmetric solutions with nonzero swirl emanating from the interior and one part of the boundary of the surface of no-swirl solutions. We have also proved that there is no such curve of solutions for any point on the other part of the boundary.…”

“…Since (c, γ) ∈ I k,l with (k, l) ∈ A 1 , we haveŪ θ (−1) < 3 andŪ θ (−1) = 2, orŪ θ (−1) = 2 with η 1 = 0, andŪ θ (1) > −3 andŪ θ (1) = −2, orŪ θ (1) = −2 with η 2 = 0. According to Theorem 1.3 in [1] and Lemma 2.3 in [2], we then havē…”