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Abstract Physical experiments and numerical simulations have observed a remarkable stabilizing phenomenon: a background magnetic field stabilizes and dampens electrically conducting fluids. This paper intends to establish this phenomenon as a mathematically rigorous fact on a magnetohydrodynamic (MHD) system with anisotropic dissipation in$$\mathbb R^3$$ $R^3$. The velocity equation in this system is the 3D Navier–Stokes equation with dissipation only in the$$x_1$$ $x_1$-direction, while the magnetic field obeys the induction equation with magnetic diffusion in two horizontal directions. We establish that any perturbation near the background magnetic field (0, 1, 0) is globally stable in the Sobolev setting$$H^3({\mathbb {R}}^3)$$ $H^3\left(R^3\right)$. In addition, explicit decay rates in$$H^2({\mathbb {R}}^3)$$ $H^2\left(R^3\right)$are also obtained. For when there is no presence of a magnetic field, the 3D anisotropic Navier–Stokes equation is not well understood and the small data global well-posedness in$$\mathbb R^3$$ $R^3$remains an intriguing open problem. This paper reveals the mechanism of how the magnetic field generates enhanced dissipation and helps to stabilize the fluid. 

Abstract The global well-posedness on the 2D resistive MHD equations without kinematic dissipation remains an outstanding open problem. This is a critical problem. Any $L^p$-norm of the vorticity $$\omega $$ with $$1\le p<\infty $$ has been shown to be bounded globally (in time), but whether the $$L^\infty $$-norm of $$\omega $$ is globally bounded remains elusive. The global boundedness of $$\|\omega \|_{L^\infty }$$ yields the resolution of the aforementioned open problem. This paper examines the $$L^\infty $$-norm of $$\omega $$ from a different perspective. We construct a sequence of initial data near a special steady state to show that the $$L^\infty $$-norm of $$\omega $$ is actually mildly ill-posed.