Besov regularity for the stationary incompressible MHD equations in bounded Lipschitz domains

We establish Besov regularity for the stationary incompressible magnetohydrodynamic (MHD) system on bounded Lipschitz domains $\Omega \subset \mathbb{R}^{3}$. For sufficiently small external forces, a unique solution $(\boldsymbol{u}, \boldsymbol{b}) \in H^{3/2}_{0,\sigma}(\Omega )^{3} \times H^{3/2}_{0,\sigma}(\Omega )^{3}$ is constructed via the contraction mapping principle. Novel weighted estimates for the Lorentz force $\mathbf{curl}\,\boldsymbol{b} \times \boldsymbol{b}$ yield enhanced regularity $(\boldsymbol{u}, \boldsymbol{b}) \in B^{s}_{\tau ,\tau}(\Omega )^{3} \times B^{s}_{\tau ,\tau}(\Omega )^{3}$ with $1/\tau = s/3 + 1/2$ and $s < 2$. Crucially, for $s > 3/2$, this Besov regularity exceeds the baseline $H^{3/2}$ Sobolev smoothness. It is also notable that the obtained regularity index $s < 2$ lies within the same critical adaptivity scale ($1/ \tau = s/3 + 1/2$) previously established for the stationary Navier-Stokes system (Eckhardt et al. in Appl. Anal. 97:466–485, 2017). This demonstrates that the additional magnetic coupling does not degrade the Besov regularity regime.