On axisymmetric self-similar solutions to the MHD system

1. Problem Statement

The paper investigates the stationary incompressible Magnetohydrodynamic (MHD) system with magnetic diffusion in $\mathbb{R}^3$. The system is given by:
$$\begin{cases} -\Delta u + (u \cdot \nabla)u - (B \cdot \nabla)B + \nabla p = 0, \\ -\Delta B + (u \cdot \nabla)B - (B \cdot \nabla)u = 0, \\ \nabla \cdot u = \nabla \cdot B = 0, \end{cases}$$
where $u$ is the fluid velocity, $B$ is the magnetic field, and $p$ is the pressure.

The specific focus is on axisymmetric self-similar solutions in two domains:

1. The whole space excluding the origin: $\Omega = \mathbb{R}^3 \setminus \{0\}$.
2. The half-space: $\Omega = \mathbb{R}^3_+$.

The solutions are assumed to be of the form:

• $u = u_r(r, z)\mathbf{e}_r + u_\theta(r, z)\mathbf{e}_\theta + u_z(r, z)\mathbf{e}_z$
• $B = B_\theta(r, z)\mathbf{e}_\theta$ (swirl-free magnetic field, purely azimuthal)

The goal is to characterize these solutions under specific boundary conditions and growth constraints, specifically determining if they must be trivial ($u=B=0$) or reduce to known Landau solutions for the Navier-Stokes equations.

2. Methodology

The author employs a combination of scaling analysis, integral identities, and maximum principle arguments adapted for self-similar functions.

• Self-Similarity Properties: The solutions are assumed to be $(-1)$-homogeneous for velocity/magnetic field and $(-2)$-homogeneous for pressure. This implies $u(\lambda x) = \lambda^{-1}u(x)$ and $B(\lambda x) = \lambda^{-1}B(x)$.
• Reduction to Scalar Equations: By exploiting the axisymmetry and the specific form of $B$, the magnetic equation reduces to a scalar equation for the swirl component $B_\theta$:
  $$-\Delta B_\theta + (u \cdot \nabla)B_\theta + \frac{1}{r^2}B_\theta - \frac{u_r}{r}B_\theta = 0$$
• Energy-Type Integral Estimates: The core of the proof involves multiplying the equation for $B_\theta$ by $r B_\theta$ and integrating over annular domains (or truncated half-space domains).
  • Integration by parts is used to handle the Laplacian and convective terms.
  • The self-similarity property is crucial in simplifying boundary integrals on spheres, often causing them to vanish or cancel out.
• Inequality Analysis: The proof relies on establishing a specific inequality involving the radial velocity component $u_r$. The author derives an integral identity of the form:
  $$\int r |\nabla B_\theta|^2 + \int |B_\theta|^2 \left( \frac{1}{2r} - \frac{3}{2}u_r \right) = 0$$
  By analyzing the sign of the term in the parentheses, the author forces $B_\theta$ to vanish.
• Boundary Condition Handling: For the half-space case, the author carefully analyzes the boundary terms arising from integration by parts under both no-slip ($u=0$) and Navier slip conditions, showing they vanish due to the specific structure of the boundary conditions and the self-similarity.

3. Key Contributions and Results

A. Results in $\mathbb{R}^3 \setminus \{0\}$

Theorem 1.2: If $(u, B)$ is a smooth axisymmetric self-similar solution in $\mathbb{R}^3 \setminus \{0\}$ satisfying the condition:
$$u_r(r, z) < \frac{1}{3r} + \frac{2r}{3} \quad \text{on the unit sphere } \partial B_1,$$
then:

1. The magnetic field vanishes identically: $B \equiv 0$.
2. The velocity field $u$ is a Landau solution (a known class of self-similar solutions to the Navier-Stokes equations).

Corollary 1.1: A more explicit, constant-based condition is derived. If $u_r(x) \leq \alpha$ on $\partial B_1$ with $\alpha < \frac{2\sqrt{2}}{3}$, the same conclusion holds ($B \equiv 0$, $u$ is Landau).

• Significance: This improves upon previous results (e.g., ZWW25) which required the entire velocity magnitude $|u|$ to be sufficiently small. This paper shows that controlling only the radial component $u_r$ with an explicit constant is sufficient.

B. Results in the Half-Space $\mathbb{R}^3_+$

Theorem 1.3: Consider the half-space with either no-slip or Navier slip boundary conditions. If the solution satisfies the same radial velocity bound on the intersection of the unit sphere and the half-space ($u_r < \frac{1}{3r} + \frac{2r}{3}$ on $\partial B_1 \cap \mathbb{R}^3_+$), then:
$$u \equiv 0 \quad \text{and} \quad B \equiv 0.$$

• Significance: This establishes the non-existence of non-trivial axisymmetric self-similar solutions in the half-space under these conditions, extending previous Navier-Stokes results (KMT18) to the full MHD system.

4. Significance and Impact

1. Rigidity of MHD Solutions: The paper provides strong rigidity results, showing that under axisymmetry and mild growth constraints, the magnetic field cannot sustain a non-trivial self-similar structure; it must decay to zero.
2. Reduction to Navier-Stokes: By proving $B \equiv 0$, the complex coupled MHD system effectively decouples into the Navier-Stokes equations, allowing the application of established classification theorems (Landau solutions).
3. Improved Conditions: The shift from requiring smallness of the full vector field $|u|$ to a specific bound on the radial component $u_r$ represents a significant technical refinement. It demonstrates that the radial flow is the critical factor preventing the existence of non-trivial magnetic fields in this context.
4. Boundary Condition Robustness: The results hold for both no-slip and Navier slip conditions, indicating that the triviality of solutions in the half-space is a robust feature of the MHD system, independent of the specific tangential stress condition at the boundary.

5. Conclusion

Shaoheng Zhang's work characterizes the landscape of axisymmetric self-similar solutions to the stationary MHD equations. The central finding is that non-trivial magnetic fields cannot exist in these self-similar configurations under the specified radial velocity bounds. Consequently, any such solution is either a Landau solution (in free space) or completely trivial (in the half-space). This bridges a gap in the literature between Navier-Stokes self-similarity theory and the more complex MHD system.