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Liouville Type Theorem for the Stationary Equations of Magneto-Hydrodynamics

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Abstract

We show that any smooth solution (u, H) to the stationary equations of magneto-hydrodynamics belonging to both spaces L6(ℝ3) and BMO−1(ℝ3) must be identically zero. This is an extension of previous results, all of which systematically required stronger integra-bility and the additional assumption ∇u,∇H ∈ L2(ℝ3), i.e., finite Dirichlet integral.

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References

  1. Bogovskiĭ M E. Solution of the first boundary value problem for an equation of continuity of an incompressible medium. Dokl Akad Nauk SSSR, 1979, 248(5): 1037–1040

    MathSciNet  Google Scholar 

  2. Chae D. Liouville type theorems for the Euler and Navier-Stokes equations. Adv Math, 2011, 228: 2855–2868

    Article  MathSciNet  MATH  Google Scholar 

  3. Chae D, Weng S. Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohy-drodynamic equations. Discrete Contin Dyn Syst, 2016, 36(10): 5267–5285

    Article  MathSciNet  MATH  Google Scholar 

  4. Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. Berlin: Springer-Verlag, 2001

    MATH  Google Scholar 

  5. Koch G, Nadirashvili N, Seregin G A, Šverák V. Liouville theorems for the Navier-Stokes equations and applications. Acta Math, 2009, 203: 83–105

    Article  MathSciNet  MATH  Google Scholar 

  6. Seregin G A. Lecture Notes on Regularity Theory for the Navier-Stokes Equations. Hackensack, NJ: World Scientific Publishing Co Pte Ltd, 2015

    MATH  Google Scholar 

  7. Seregin G A. Liouville type theorem for stationary Navier-Stokes equations. Nonlinearity, 2016, 29: 2191–2195

    Article  MathSciNet  MATH  Google Scholar 

  8. Stein E M. Singular Integrals and Differentiability Properties of Functions. Princeton, NJ: Princeton University Press, 1970

    MATH  Google Scholar 

  9. Zhang Z, Yang X, Qiu S. Remarks on Liouville type result for the 3D Hall-MHD system. J Part Diff Equ, 2015, 28(3): 286–290

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author wishes to thank Gui-Qiang Chen and Gregory Seregin for useful discussions.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Woodstock Road, OX2 6GG, Oxford, UK

    Simon Schulz

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  1. Simon Schulz
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Correspondence to Simon Schulz.

Additional information

The author is supported by the Engineering and Physical Sciences Research Council [EP/L015811/1].

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Schulz, S. Liouville Type Theorem for the Stationary Equations of Magneto-Hydrodynamics. Acta Math Sci 39, 491–497 (2019). https://doi.org/10.1007/s10473-019-0213-7

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  • DOI: https://doi.org/10.1007/s10473-019-0213-7

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