Liouville-type theorems and perturbation uniqueness for stationary incompressible MHD with directional fractional dissipation

We study Liouville-type theorems for stationary solutions of the incompressible magnetohydrodynamics equations in ${\mathbb{R}}^n$, with directional fractional dissipation $(-{\mathrm{\partial }}_i^2{)}^{\alpha }u$ and $(-{\mathrm{\partial }}_j^2{)}^{\beta }b$, where $n\ge 2$ and $0<\alpha ,\beta \le 1$. We prove that $u\equiv b\equiv 0$ by assuming that $u\in L^{p,\mathrm{\infty }}({\mathbb{R}}^n)$ and $b\in L^{q,\mathrm{\infty }}({\mathbb{R}}^n)$, with suitable conditions on $p$ and $q$. The proof uses a localized fractional energy identity and does not assume finiteness of the global homogeneous Sobolev energy. The method also applies to the case of full fractional dissipation. For the system with common-directional fractional dissipation, we prove stationary uniqueness for compactly supported perturbations of arbitrary smooth transverse shear backgrounds when $0<\alpha ,\beta <1$. We also establish a quantitative uniqueness criterion for perturbations of backgrounds with bounded gradients that are not necessarily compactly supported, for all $0<\alpha ,\beta \le 1$, assuming directional localization.