On the Vanishing of Some D-Solutions to the Stationary Magnetohydrodynamics System

  • Zijin Li
  • Xinghong PanEmail author


In this paper, we study the stationary magnetohydrodynamics system in \(\mathbb {R}^2\times \mathbb {T}\). We prove trivialness of D-solutions (the velocity field u and the magnetic field h) when they are swirl-free. Meanwhile, this Liouville type theorem also holds provided u is swirl-free and h is axially symmetric, or both u and h are axially symmetric. Our method is also valid for certain related boundary value problems in the slab \(\mathbb {R}^2\times [-\pi ,\,\pi ]\).


Incompressible Magnetohydrodynamics system Swirl-free Axially symmetric 

Mathematics Subject Classification

35Q30 76N10 



The authors wish to thank Prof. Qi S. Zhang in UC Riverside for his constant encouragement on this topic. Z. Li is supported by the Startup Foundation for Introducing Talent of NUIST (No. 2019r033). X. Pan is supported by Natural Science Foundation of Jiangsu Province (No. SBK2018041027) and National Natural Science Foundation of China (No. 11801268).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsNanjing University of Information Science and TechnologyNanjingChina
  2. 2.Department of MathematicsNanjing University of Aeronautics and AstronauticsNanjingChina

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