Liouville Type Theorem for Three-dimensional Stationary MHD Model and Three-dimensional Stationary Hall-MHD Model in M1xed Lebesgue Spaces

The uniqueness of the trivial solution for the three-dimensional steady-state magnetohydrodynamics (MHD) model and the magnetohydrodynamics model with the Hall effect (Hall-MHD) is studied under the framework of mixed Lebesgue spaces, in which the constraint of the traditional finite Dirichlet integral condition is overcome. By resolving the critical challenge of pressure term estimation, a Liouville type theorem is established for both types of models. Specifically, if a smooth solution (u, b) belongs to the mixed Lebesgue spaces L^p\left(\mathbbR^3\right) with \boldsymbolp=\left(p_1, p_2, p_3\right) satisfying / p_1+1 / p_2+1 / p_3 \geqslant 2 / 3, then for the three-dimensional stationary MHD model under the condition p_1, p_2, p_3 \in3, +\infty), and for the three-dimensional stationary Hall-MHD model under p_1, p_2, p_3 \in4, +\infty), it is proved that u=b=0.