Axisymmetric self-similar solutions to the MHD equations

TL;DR

This work characterizes axisymmetric self-similar solutions to the stationary MHD equations. By deriving an energy-type identity for the swirl component $B^{\theta}$ and exploiting self-similarity together with a bound on the radial velocity $u^{r}$, the authors show that any nontrivial swirl magnetic field must vanish, reducing the problem to Landau solutions of the Navier–Stokes equations. Consequently, in $\mathbb{R}^3\setminus\{0\}$, an axisymmetric self-similar solution with $B$ having only swirl and $u^{r}$ obeying $u^{r}<\frac{1}{3r}+\frac{2r}{3}$ on $\partial B_{1}$ is a Landau solution with $B\equiv0$. In the half-space $\mathbb{R}^3_+$ under no-slip or Navier slip boundary conditions, all axisymmetric self-similar solutions are trivial, i.e., $\mathbf{u}=\mathbf{B}=0$. These results tie MHD self-similarity to classical Landau flows and provide sharp rigidity under boundary constraints.

Abstract

We study the axisymmetric self-similar solutions $(\mathbf{u},\mathbf{B})$ to the stationary MHD equations, where $\mathbf{u}=u^r(r,z)\mathbf{e}_{r}+u^θ(r,z)\mathbf{e}_θ+u^z(r,z)\mathbf{e}_{z}$, $\mathbf{B}=B^θ(r,z)\mathbf{e}_θ$ in cylindrical coordinates, and $u^r$ satisfies $u^r<\frac{1}{3r}+\frac{2r}{3}$ on the unit sphere. Our first result shows that in $\mathbb{R}^3\setminus\{0\}$, $\mathbf{u}$ is a Landau solution and $\mathbf{B}\equiv0$. Our second result establishes the triviality of axisymmetric self-similar solutions in the half-space $\mathbb{R}^3_+$ with the no-slip boundary condition or the Navier slip boundary condition.

Theorems & Definitions (8)

• Theorem 1.1
• Corollary 1.1
• Remark 1.1
• Theorem 1.2
• Remark 1.2
• proof : Proof of Theorem \ref{['thm1']}
• proof : Proof of Theorem \ref{['thm2']}
• proof : Proof of Corollary \ref{['cor1']}