Work

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

Year: 2026

Type: article

Abstract: 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.

Source: Advances in Continuous and Discrete Models

DOI: 10.1186/s13662-026-04054-7

Authors: Haojie Guo, Qiu Meng

Institution: Beihua University

Language: English

Topic: Navier-Stokes equation solutions

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $1,890

Funder: Jilin Scientific and Technological Development Program