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Remarks on Liouville type theorems for the 3D stationary MHD equations

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The aim of this paper is to establish Liouville type results for the stationary MHD equations. In particular, we show that the velocity and magnetic field, belonging to some Lorentz spaces, must be zero. Moreover, we also obtain Liouville type theorem for the case of axially symmetric MHD equations. Our results generalize previous works by Schulz \cite{Sc2018} and Seregin-Wang \cite{SW2018}.

Abstract

The aim of this paper is to establish Liouville type results for the stationary MHD equations. In particular, we show that the velocity and magnetic field, belonging to some Lorentz spaces, must be zero. Moreover, we also obtain Liouville type theorem for the case of axially symmetric MHD equations. Our results generalize previous works by Schulz \cite{Sc2018} and Seregin-Wang \cite{SW2018}.

참고문헌 (19)

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  2. Dongho Chae, "On Liouville type theorems for the steady Navier-Stokes equations in R3", Journal of Differential Equations, 261(10), pp.5541~5560 (2016), Elsevier BV[DOI]
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  7. Guilong Gui, "Global well-posedness of the two-dimensional incompressible magnetohydrodynamics system with variable density and electrical conductivity", Journal of Functional Analysis, 267(5), pp.1488~1539 (2014), Elsevier BV[DOI]
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  9. Zhen Lei, "A Liouville theorem for the axially-symmetric Navier–Stokes equations", Journal of Functional Analysis, 261(8), pp.2323~2345 (2011), Elsevier BV[DOI]
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  14. Simon Schulz, "Liouville Type Theorem for the Stationary Equations of Magneto-Hydrodynamics", Acta Mathematica Scientia, 39(2), pp.491~497 (2019), Springer Science and Business Media LLC[DOI]
  15. G. Seregin, "Liouville type theorem for stationary Navier-Stokes equations", Nonlinearity, 29(8), pp.2191~2195 (2016)[DOI]
  16. G. Seregin, "Remarks on Liouville type theorems for steady-state Navier-Stokes equations", St. Petersburg Math. J, 30(2), pp.321~328 (2019)[DOI]
  17. G. Seregin, "A Liouville type theorem for steady-state Navier-Stokes equations", arXiv:1611.01563, ~
  18. G. Seregin, "Sucient conditions on Liouville type theorems for the 3D steady Navier-Stokes equations", St. Petersburg Math. J, 31(2), pp.387~393 (2020)[DOI]
  19. Wendong Wang, "Remarks on Liouville type theorems for the 3D steady axially symmetric Navier–Stokes equations", Journal of Differential Equations, 266(10), pp.6507~6524 (2019), Elsevier BV[DOI]

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