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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Auchmuty, J.F.G.: The global branching of rotating stars. Arch. Ration. Mech. Anal. 114, 179–194, 1991. doi:10.1007/BF00375402
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
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
Chandrasekhar S.: The equilibrium of distorted polytropes (I). Mon. Not. R. Astron. Soc. 93, 390–405 (1933)
Chandrasekhar S.: An Introduction to the Study of Stellar Structures. University of Chicago Press, Chicago (1938)
Chandrasekhar, S.: Ellipsoidal figures of equilibrium—an historical account. Commun. Pure Appl. Math. 20, 251–265, 1967. doi:10.1002/cpa.3160200203
Chanillo, S.; Li, Y.Y.: On diameters of uniformly rotating stars. Commun. Math. Phys. 166(2), 417–430, 1994. doi:10.1007/BF02112323
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
FriedmanA.; Turkington B.: The oblateness of an axisymmetric rotating fluid. Indiana Univ. Math. J. 29(5), 777–792 (1980)
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
Jang, J.: Nonlinear instability theory of Lane–Emden stars. Commun. Pure Appl. Math. 67(9), 1418–1465 (2014). doi:10.1002/cpa.21499
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
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
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
Kato T.: Perturbation Theory for Linear Operators. Springer, Berlin (1976)
Kovetz A.: Slowly rotating polytropes. Astrophys. J. 154, 999–1003 (1968)
Li, Y.Y.: On uniformly rotating stars. Arch. Ration. Mech. Anal. 115, 367–393, 1991. doi:10.1007/BF00375280
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
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
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
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
Milne E.A.: The equilibrium of a rotating star. Mon. Not. R. Astron. Soc. 83, 118–147 (1923)
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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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