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Existence of Solutions to Steady MHD System with Multiply Connected Boundary

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Abstract

In this paper, we study the inhomogeneous boundary value problem for the steady MHD system of a viscous incompressible fluid in an arbitrary bounded multiply connected domain. We prove the existence of generalized solutions of the steady MHD system with some smallness conditions on the boundary values.

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References

  1. Borchers, W., Pileckas, K.: Note on the flux problem for stationary incompressible Navier-Stokes equations in domains with a multiply connected boundary. Acta Appl. Math. 37, 21–30 (1994)

    Article  MathSciNet  Google Scholar 

  2. Evans, L.C.: Partial Differential Equations, 2nd edn. Graduate Studies in Mathematics, vol. 19. Am. Math. Soc., Providence (2010)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Fujita, H.: On the existence and regularity of the steady-state solutions of the Navier-Stokes theorem. J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 9, 59–102 (1961)

    MathSciNet  MATH  Google Scholar 

  5. Galdi, G.P.: Steady-state problems. In: An Introduction to the Mathematical Theory of the Navier-Stokes Equations, 2nd edn. Springer Monographs in Mathematics Springer, New York (2011)

    Chapter  Google Scholar 

  6. Galdi, G.P.: Nonlinear steady problems. In: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II. Springer Tracts in Natural Philosophy, vol. 39. Springer, New York (1994)

    MATH  Google Scholar 

  7. Korobkov, M.V., Pileckas, K., Russo, R.: Solution of Leray’s problem for stationary Navier-Stokes equations in plane and axially symmetric spatial domains. Ann. Math. 181, 769–807 (2015)

    Article  MathSciNet  Google Scholar 

  8. Kozono, H., Yanagisawa, T.: Leray’s problem on the stationary Navier-Stokes equations with inhomogeneous boundary data. Math. Z. 262, 27–39 (2009)

    Article  MathSciNet  Google Scholar 

  9. Ladyženskaja, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Mathematics and Its Applications, vol. 2. Gordon and Breach, Science Publishers, New York-London-Paris (1969). Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu

    Google Scholar 

  10. Leray, J.: Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique (1933). NUMDAM

    MATH  Google Scholar 

  11. Pathak, H.K.: An Introduction to Nonlinear Analysis and Fixed Point Theory. Springer, Singapore (2018)

    Book  Google Scholar 

  12. Russo, A., Starita, G.: On the existence of steady-state solutions to the Navier-Stokes system for large fluxes. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 7, 171–180 (2008)

    MathSciNet  MATH  Google Scholar 

  13. Takeshita, A.: A remark on Leray’s inequality. Pac. J. Math. 157, 151–158 (1993)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

J. Wu was partially supported by NNSFC (Grant No. 11771043 and No. 11771045). X. Xu was partially supported by NNSFC (Grant No. 11771045 and No. 11871087) and the National Key R&D Programe of China (Grant No. 2020- YFA0712900).

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Authors and Affiliations

  1. College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen, 518060, P.R. China

    Jingbo Wu

  2. Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P.R. China

    Jingbo Wu & Xiaojing Xu

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  1. Jingbo Wu
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  2. Xiaojing Xu
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Correspondence to Xiaojing Xu.

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Wu, J., Xu, X. Existence of Solutions to Steady MHD System with Multiply Connected Boundary. Acta Appl Math 175, 2 (2021). https://doi.org/10.1007/s10440-021-00430-5

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  • DOI: https://doi.org/10.1007/s10440-021-00430-5

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