Liouville-Type Theorems for the 3D Stationary MHD Equations

In this paper, we consider the Liouville-type theorems for the 3D stationary incompressible MHD equations. Using the Caccioppoli type estimate, we proved the smooth solutions (u, b) are identically equal to zero when \((u,b)\in L^{p}({\mathbb {R}}^{3}),\ p\in (\frac{3}{2},3).\) In addition, under an additional assumption in the setting of the Sobolev space of negative order \(\dot{H}^{-1}({\mathbb {R}}^{3}),\) we can extend the index \(p\in (3,+\infty ).\) In fact, our results combine with the result of Yuan and Xiao (J Math Anal Appl 491(2):124343, 2020) that \(p\in [2,\frac{9}{2}],\) which implies a very intriguing and novel result for the 3D stationary MHD equations with \( p\in (\frac{3}{2},+\infty ).\)