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

Abstract

For the two dimensional stationary MHD equations, we prove that Liouville type theorems hold if the velocity is growing at infinity, where the magnetic field is assumed to be bounded under a smallness condition. The key point is to overcome the nonlinear terms, since no maximum principle holds for the MHD case with respect to the Navier-Stokes equations. As a corollary, we obtain that all the solutions of the 2D Navier-Stokes equations satisfying ∇u ∈ L p (R 2 ) with 1 < p < ∞ are constants, which is sharp since the same argument fails in the case of ∇u ∈ L ∞ (R 2 ).

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

“…In recent years, many mathematicians attempted to bring a complete understanding to the Liouville problem for the MHD system; we cite the works of [1,3,14,22] (and the references contained therein) for interesting results. We also point out the recent contribution of Wang and Wang in [20], where they proved a Liouville theorem for the system in question under some smallness condition on the L 1 -norm of b.…”

Asymptotic behavior of the steady Navier-Stokes flow in the exterior domain

“…One main step is to obtain the higher regularity of the equation (1.1) via the condition v L p (Ω) < ∞. We follow the same route as the proof of Liouville type theorems, for example see [20,21], where the divergence equation, Poincaré-Sobolev inequality and iteration lemma are used.…”

Stability of the Couette flow under the 2D steady Navier–Stokes flow

“…The above results also can be generalized to the shear thickening flows, for example, see [5-7, 16, 32]. For the two-dimensional (2D) steady magnetohydrodynamic (MHD) equations, the similar Liouvile-type theorems were obtained by Y. Wang and the author in [30] by assuming (1.3) or 𝑢 ∈ 𝜒 0,𝑝 (ℝ 2 ) with 2 < 𝑝 ≤ ∞, where the smallness conditions of the magnetic field are added. See also the recent result in [29] for 𝑢 ∈ 𝜒 𝛼,∞ (ℝ 2 ) with 𝛼 < 1 3 by using the idea of [21] and energy estimates in an annular domain.…”