On Liouville-type theorems for the 2D stationary MHD equations

Abstract

We establish new Liouville-type theorems for the two-dimensional stationary magneto-hydrodynamic incompressible system assuming that the velocity and magnetic field have bounded Dirichlet integral. The key tool in our proof is observing that the stream function associated to the magnetic field satisfies a simple drift–diffusion equation for which a maximum principle is available.

New Liouville type theorems for the stationary Navier–Stokes, MHD, and Hall–MHD equations

“…Liouville-type theorems for the 2D MHD equations were recently obtained by Wang and Wang [13] and De Nitti et al [14]. The problem is more difficult than in the case B ≡ 0 because one cannot apply the maximum principle to the equation for vorticity.…”

“…The main point in the proof of theorem 1.2 in [13] is the second order gradient estimates of u and B. De Nitti et al [14] also dealt with solutions u, B, satisfying ´R2 |u| 2 + |B| 2 dx < ∞. Using the fact that the stream function for the magnetic field B satisfies an equation for which the maximum principle holds, they showed that u and B are constants provided that the norms ∥u∥ L p (R 2 ) 2 and ∥B∥ L q (R 2 ) 2 (for appropriate p, q) are suitably bounded (theorem 2.1).…”