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Liouville theorem of the 3D stationary MHD system: for D-solutions converging to non-zero constant vectors

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Abstract

In this paper, we derive the Liouville theorem of D-solutions to the stationary MHD system under the asymptotic assumption: one of the velocity field and the magnetic field approaches zero and the other approaches a non zero constant vector at infinity. Our result extends the corresponding one of D-solutions to the Navier–Stokes equations when the velocity approaches a non zero constant vector at infinity.

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References

  1. Babenko, K.I.: On the stationary solutions of the problem of flow past a body of a viscous incompressible fluid. Mat. Sbornik 91, 3–26 (1973); English Transl.: Math. USSR Sbornik 20(1), 1–25 (1973)

  2. Carrillo, B., Pan, X., Zhang, Q.S., Zhao, N.: Decay and vanishing of some D-solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 237(3), 1383–1419 (2020)

    Article  MathSciNet  Google Scholar 

  3. Chae, D.: Liouville-type theorem for the forced Euler equations and the Navier–Stokes equations. Commun. Math. Phys. 326, 37–48 (2014)

    Article  MathSciNet  Google Scholar 

  4. Chae, D., Wolf, J.: On Liouville type theorem for the stationary Navier–Stokes equations. Calc. Var. Partial Differ. Equ. 58(3), Art. 111, 11 pp (2019)

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

    Article  MathSciNet  Google Scholar 

  6. Farwig, R., Sohr, H.: Weighted estimates for the Oseen equations and the Navier–Stokes equations in exterior domains. In: Theory of the Navier–Stokes equations. Ser. Adv. Math. Appl. Sci., vol. 47, pp. 11–30. World Sci. Publ., River Edge (1998)

  7. Finn, R.: On the steady-state solutions of the Navier–Stokes equations, III. Acta Math. 105(3–4), 197–244 (1961)

    Article  MathSciNet  Google Scholar 

  8. Galdi, G.P.: An Introducion to the Mathematical Theory of the Navier–Stokes Equations. Springer, Berlin (2011)

    Google Scholar 

  9. Gilbarg, D., Weinberger, H.F.: Asymptotic properties of steady plane solutions of the Navier–Stokes equations with bounded Dirichlet integral. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5(2), 381–404 (1978)

    MathSciNet  MATH  Google Scholar 

  10. Kozono, H., Terasawa, Y., Wakasugi, Y.: A remark on Liouville-type theorems for the stationary Navier–Stokes equations in three space dimensions. J. Funct. Anal. 272(2), 804–818 (2017)

    Article  MathSciNet  Google Scholar 

  11. Ladyzhenskaya, O.A.: Matematicheskie problemy dinamiki vzyakoi neszhimaemoi zhidkosti, Fizmatgiz, Moscow, 1961, 2nd edn, Nauka, Moscow, 1970. Translation: The Mathematical Theory of Viscous Imcompressible Flow. Gordon and Nreach, New York (1969)

  12. Laudau, L.D., Lifshitz, E.M.: Electrodynamics of Continuous Media, 2nd edn. Pergamon, New York (1984)

    Google Scholar 

  13. Leray, J.: Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique. J. Math. Pures Appl. 12, 1–82 (1933)

    MATH  Google Scholar 

  14. Li, Z., Pan, X.: On the vanishing of some D-solutions to the stationary magnetohydrodynamics system. J. Math. Fluid Mech. 21(52), 1–13 (2019)

    MathSciNet  Google Scholar 

  15. Lizorkin, P.I.: \(({L}_p,{L}_q)\)-multipliers of Fourier integrals. Soviet Math. Dokl. 4, 1420–1424 (1963)

    MATH  Google Scholar 

  16. Marcinkiewicz, J.: Sur les multiplicateurs des séries de Fourier. Stud. Math. 8, 78–91 (1939)

    Article  MathSciNet  Google Scholar 

  17. Mikhlin, S.G.: Fourier integrals and multiple singular integrals. Vestn. Leningrad. Univ. Ser. Mat. Meh. Astron. 12, 143–155 (1957)

    MathSciNet  MATH  Google Scholar 

  18. Pan, X., Li, Z.: Liouville theorem of axially symmetric Navier–Stokes equations with growing velocity at infinity. Nonlinear Anal. Real World Appl. 56, 103159 (2020)

    Article  MathSciNet  Google Scholar 

  19. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton. ISBN 0-691-08079-8 (1970)

  20. Seregin, G.: Liouville type theorem for stationary Navier–Stokes equations. Nonlinearity 29(8), 2191–2195 (2016)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Z. Li is supported by Natural Science Foundation of Jiangsu Province (No. BK20200803), National Natural Science Foundation of China (No. 12001285) and the Startup Foundation for Introducing Talent of NUIST (No. 2019r033). X. Pan is supported by Natural Science Foundation of Jiangsu Province (No. BK20180414) and National Natural Science Foundation of China (No. 11801268).

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Correspondence to Xinghong Pan.

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Li, Z., Pan, X. Liouville theorem of the 3D stationary MHD system: for D-solutions converging to non-zero constant vectors. Nonlinear Differ. Equ. Appl. 28, 12 (2021). https://doi.org/10.1007/s00030-021-00674-y

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  • DOI: https://doi.org/10.1007/s00030-021-00674-y

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