On the non-uniqueness of solutions of the axi-symmetric swirl-free Navier-Stokes equations, I

In this paper we construct numerically a new class of unstable self-similar solutions of the incompressible Navier-Stokes equations in $\mathbb{R}^3$. Our solutions are axially symmetric and homogeneous of degree $-1$ at $\infty$, and are unstable in the sense that the linearization around these solutions contains unstable modes. Solutions of this type have been discovered numerically by Guillod and \v{S}ver\'ak and Hou, Wang, and Yang, and have applications to proving non-uniqueness results. The main novelty in this paper is that we discover the existence of such solutions in the space of axially symmetric swirl-free (ASSF) vector fields. These approximate solutions are defined on all of $\mathbb R^3$ and achieve global pointwise residuals of order $10^{-10}$. We discuss the numerical construction of these solutions in detail, as well as their relevance to the problem of non-uniqueness of solutions of the incompressible Navier-Stokes equations in 3D, in the space of ASSF solutions.