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

De Nitti, Nicola; Hounkpe, Francis; Schulz, Simon

doi:10.48550/arxiv.2103.00551

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2021

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“…Due to the lack of maximum principle, there is not much progress in the study of MHD equation. For the two-dimensional MHD equations, Liouville type theorems were proved by assuming the smallness of the norm of the magnetic field in [31] and [30], and we refer to the recent paper in [10] by removing the smallness assumption. For the three-dimensional MHD equations, Chae-Weng [6] proved that if a smooth solution to (1.1) in R 3 with finite Dirichlet integral…”

Section: Introductionmentioning

confidence: 99%

Remarks on Liouville type theorems for the steady MHD and Hall-MHD equations

Chen,

Li,

Wang

2021

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In this note we investigate Liouville type theorems for the steady three dimensional MHD and Hall-MHD equations, and show that the velocity field u and the magnetic field B are vanishing provided that u, B ∈ L 6 (R 3 ) and u ∈ BM O −1 (R 3 ), which state that the velocity field plays an important role. Moreover, the similar result holds in the case of partial viscosity or diffusivity for the three dimensional MHD equations. R 3 |∇u| 2 dx is finite, which dates back to Leray's celebrated paper [20] and is explicitly written in Galdi's book ([12], Remark X. 9.4, pp.729; see also Tsai's book

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Section: Introductionmentioning

confidence: 99%