Stability of the 3D Boussinesq-MHD system with fractional horizontal dissipation

Abstract This paper studies the stability of perturbations near equilibria for the 3D Boussinesq-magnetohydro-dynamic (MHD) system with fractional horizontal dissipation in the whole space \(\mathbb{R}^3\). Compared with the fully dissipative case, the presence of fractional dissipation and the absence of vertical damping introduce substantial analytical difficulties. To overcome these challenges, we design a set of new anisotropic estimates involving fractional derivatives. For initial data in \(H^k(\mathbb{R}^3)\) with \(k\ge 3\), global stability is established for all exponents \(\alpha_1,\alpha_2,\beta_1,\beta_2,\gamma_1,\gamma_2\in(0,1]\). In the \(H^2(\mathbb{R}^3)\) framework, global stability remains valid when the exponents belong to \((\frac14,\frac12)\) or \([\frac13,1]\).