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.