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

This paper is devoted to investigating Liouville type properties of the two-dimensional stationary incompressible Magnetohydrodynamics equations. More precisely, we show that there are no non-trivial solutions to MHD equations either the Dirichlet integral or some L^p norm of the velocity-magnetic fields is suitably bounded, which generalize the well-known results for the 2D Navier–Stokes equations by Gilbarg and Weinberger (1978 Ann. Scuola Norm. Super. Pisa Cl. Sci. 5 381–404; Koch et al 2009 Acta Math. 203 83–105). Compared to the Navier–Stokes equations, there is no maximum principle for solutions to the MHD equation. To overcome this difficulty, we develop a different approach, which does not appeal to the special structure of the vorticity equation as Gilbarg and Weinberger (1978 Ann. Scuola Norm. Super. Pisa Cl. Sci. 5 381–404) did.