Liouville type Theorems for the 3D stationary Hall‐MHD equations

Abstract

We prove several Liouville type results for the stationary MHD and Hall-MHD equations. In particular, we show that the velocity and magnetic field, belonging to some Lorentz spaces or satisfying a priori decay assumption, must be zero.

On Liouville type theorems for the stationary MHD and the Hall-MHD systems in $\mathbb{R}^3$

“…As the Liouville type problem for the stationary Navier-Stokes equations has been studied extensively recently(see [1,3,4,8,11,13,14,15,18,19,20]), there also are many works on the Liouville type problem on the MHD equations (see [16,22,23] and references therein). Here, we focus on the results under the assumptions in terms of the potential functions.…”

Global regularity for the 3D Hall-MHD equations with low regularity axisymmetric data

“…Very recently, Danchin et al in [7] showed global well-posedness when the initial conditions in critical spaces Ḃ 3 p −1 p,1 , 1 p < ∞, and are small enough. More interesting results, we recommend [14,15,17,22].…”

Notes on Liouville type Theorems for the stationary compressible Navier-Stokes equations

“…The interested readers can also refer [9], which proved a different result. We note that the condition (1.3) is weaker than (1.2) in the sense that u can decay more slowly at infinity.…”

“…Theorem 1.1. Suppose that (ρ, u) is a smooth solution to (1.1) with ρ ∈ L ∞ (R 3 ), ∇u ∈ L 2 (R 3 ) and u ∈ L p,q (R 3 ) for 3 < p < 9 2 , 3 ≤ q ≤ ∞ or p = q = 3. Then u ≡ 0 and ρ = constant on R 3 .…”