New Liouville type theorems for 3D steady incompressible MHD equations and Hall-MHD equations
Zhibing Zhang
TL;DR
This paper establishes Liouville-type theorems for three-dimensional stationary incompressible MHD and Hall-MHD equations using an energy-method framework combined with an annulus-based iteration. It introduces growth functionals $X_{p,\alpha}(R)$ and $Y_{q,\beta}(R)$ to capture decay/rigidity of $u$ and $B$, proves conventional triviality results (Theorems main1 and main2) under a suite of integrability-growth conditions, and extends these with logarithmic improvements (Theorems main3 and main4) by incorporating $\lambda$ and $\mu$ into the norms. A key technical backbone includes the Bogovskii operator for pressure, a Giaquinta-type iteration lemma, and precise control of nonlinear terms via energy estimates on tailored auxiliary functionals $J_1$–$J_4$, with the Hall-term treated by absorption. Collectively, the results broaden the scope of Liouville-type rigidity for MHD and Hall-MHD, generalizing prior works and providing sharper decay/regularity implications under weaker magnetic-field hypotheses.
Abstract
In this paper, we study Liouville type results for the three-dimensional stationary incompressible MHD equations and Hall-MHD equations. Using the energy method and an iteration argument, we establish Liouville type theorems if Lebesgue norms of the velocity and magnetic field on the annulus satisfy certain growth conditions. Furthermore, by establishing new energy estimates and developing some novel differential inequality techniques, we relax the growth conditions by logarithmic factors and obtain logarithmic improvement version of Liouville type theorems. For the MHD equations, the assumptions imposed on the magnetic field are weaker and wider than that of the velocity field in certain sense. Our results extend and improve several recent works.