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On Slowly Rotating Axisymmetric Solutions of the Euler–Poisson Equations

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Abstract

We construct stationary axisymmetric solutions of the Euler–Poisson equations, which govern the internal structure of polytropic gaseous stars, with small constant angular velocity when the adiabatic exponent \({\gamma}\) belongs to \({(\frac65,\frac32]}\) . The problem is formulated as a nonlinear integral equation, and is solved by an iteration technique. By this method, not only do we get the existence, but we also clarify properties of the solutions such as the physical vacuum condition and the oblateness of the star surface.

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References

  1. Auchmuty, J.F.G.: The global branching of rotating stars. Arch. Ration. Mech. Anal. 114, 179–194, 1991. doi:10.1007/BF00375402

  2. Auchmuty, J.F.G.; Beals, R.: Variational solutions of some nonlinear free boundary problems. Arch. Ration. Mech. Anal. 43, 255–271, 1971. doi:10.1007/BF00250465

  3. Caffarelli, L.A.; Friedman, A.: The shape of axisymmetric rotating fluid. J. Funct. Anal. 35, 100–142, 1980. doi:10.1016/0022-1236(80)90082-8

  4. Chandrasekhar S.: The equilibrium of distorted polytropes (I). Mon. Not. R. Astron. Soc. 93, 390–405 (1933)

    Article  ADS  MATH  Google Scholar 

  5. Chandrasekhar S.: An Introduction to the Study of Stellar Structures. University of Chicago Press, Chicago (1938)

    Google Scholar 

  6. Chandrasekhar, S.: Ellipsoidal figures of equilibrium—an historical account. Commun. Pure Appl. Math. 20, 251–265, 1967. doi:10.1002/cpa.3160200203

  7. Chanillo, S.; Li, Y.Y.: On diameters of uniformly rotating stars. Commun. Math. Phys. 166(2), 417–430, 1994. doi:10.1007/BF02112323

  8. Chanillo, S.; Weiss, G.S.: A remark on the geometry of uniformly rotating stars. J. Differ. Equ. 253(2), 553–562, 2012. doi:10.1016/j.jde.2012.04.011

  9. FriedmanA.; Turkington B.: The oblateness of an axisymmetric rotating fluid. Indiana Univ. Math. J. 29(5), 777–792 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  10. Friedman, A.; Turkington, B.: Existence and dimensions of a rotating white dwarf. J. Differ. Equ. 42(3), 414–437, 1981. doi:10.1016/0022-0396(81)90114-5

  11. Jang, J.: Nonlinear instability theory of Lane–Emden stars. Commun. Pure Appl. Math. 67(9), 1418–1465 (2014). doi:10.1002/cpa.21499

  12. Jang, J.: Time-periodic approximations of the Euler–Poisson system near Lane–Emden stars. Anal. PDE 9, 1043–1078, 2016. doi:10.2140/apde.2016.9.1043

  13. Jang, J.; Masmoudi, N.: Well-posedness of compressible Euler equations in a physical vacuum. Commun. Pure Appl. Math. 68(1), 61–111, 2015. doi:10.1002/cpa.21517

  14. Joseph, D.D., Lundgren, T.S.: Quasilinear Dirichlet problems driven by positive sources. Arch. Ration. Mech. Anal. 49, 241–269, 1972/1973. doi:10.1007/BF00250508

  15. Kato T.: Perturbation Theory for Linear Operators. Springer, Berlin (1976)

    MATH  Google Scholar 

  16. Kovetz A.: Slowly rotating polytropes. Astrophys. J. 154, 999–1003 (1968)

    Article  ADS  Google Scholar 

  17. Li, Y.Y.: On uniformly rotating stars. Arch. Ration. Mech. Anal. 115, 367–393, 1991. doi:10.1007/BF00375280

  18. Luo, T.; Smoller, J.: Rotating fluids with self-gravitation in bounded domains. Arch. Ration. Mech. Anal. 173(3), 345–377, 2004. doi:10.1007/s00205-004-0319-4

  19. Luo, T.; Smoller, J.: Existence and non-linear stability of rotating star solutions of the compressible Euler–Poisson equations. Arch. Ration. Mech. Anal. 191, 447–496, 2009. doi:10.1007/s00205-007-0108-y

  20. Makino, T.: On spherically symmetric motions of a gaseous star governed by the Euler–Poisson equations. Osaka J. Math. 52, 545–580, 2015. https://projecteuclid.org/euclid.ojm/1427202902

  21. Makino, T.: An Application of the Nash–Moser theorem to the vacuum boundary problem of gaseous stars. J. Differ. Equ. 262(2), 803–843, 2017. doi:10.1016/j.jde.2016.09.042

  22. Milne E.A.: The equilibrium of a rotating star. Mon. Not. R. Astron. Soc. 83, 118–147 (1923)

    Article  ADS  Google Scholar 

  23. von Zeipel, H.: The radiative equilibrium of a slightly oblate rotating star. Mon. Not. R. Astron. Soc. 84(Suppl), 664–701, 1924

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Authors and Affiliations

  1. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089, USA

    Juhi Jang

  2. Korea Institute for Advanced Study, Seoul, Korea

    Juhi Jang

  3. Yamaguchi University, Yamaguchi, Japan

    Tetu Makino

Authors
  1. Juhi Jang
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  2. Tetu Makino
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Correspondence to Juhi Jang.

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Communicated by P. Rabinowitz

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Jang, J., Makino, T. On Slowly Rotating Axisymmetric Solutions of the Euler–Poisson Equations. Arch Rational Mech Anal 225, 873–900 (2017). https://doi.org/10.1007/s00205-017-1115-2

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  • DOI: https://doi.org/10.1007/s00205-017-1115-2

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