Abstract.
In this paper, we study the steady solutions of Euler-Poisson equations in bounded domains with prescribed angular velocity. This models a rotating Newtonian star consisting of a compressible perfect fluid with given equation of state P=eSργ. When the domain is a ball and the angular velocity is constant, we obtain both existence and non-existence theorems, depending on the adiabatic gas constant γ. In addition we obtain some interesting properties of the solutions; e.g., monotonicity of the radius of the star with both angular velocity and central density. We also prove that the radius of a rotating spherically symmetric star, with given constant angular velocity and constant entropy, is uniformly bounded independent of the central density. This is physically striking and in sharp contrast to the case of the non-rotating star. For general domains and variable angular velocities, both an existence result for the isentropic equations of state and non-existence result for the non-isentropic equation of state are also obtained.
Similar content being viewed by others
References
Adler, R. Bazin, M., Schiffer, M.: Introduction to General Relativity. McGraw-Hill, New York (1975)
Auchmuty, G., Beals, R.: Variational solutions of some nonlinear free boundary problems. Arch. Rational Mech. Anal. 43, 255–271 (1971)
Caffarelli, L., Friedman, A.: The shape of axi-symmetric rotating fluid. J. Funct. Anal. 694, 109–142 (1980)
Castro, A., Shivaji, R.: Non-negative solutions for a class of radially symmetric non-positone problems. Proc. AMS 106, 735–740 (1989)
Chandrasekhar, S.: Introduction to the Stellar Structure. University of Chicago Press (1939)
Chanillo, S., Li, Y.Y.: On diameters of uniformly rotating stars. Comm. Math. Phys. 166, 417–430 (1994)
Evans, L.: Partial Differential Equations. Graduate Studies in Math., AMS, Providence, Rhode Island (1998)
Deng, Y., Liu, T.P., Yang, T., Yao, Z.: Solutions of Euler-Poisson equations for gaseous stars. Arch. Rational Mech. Anal. 164, 261–285 (2002)
Gidas, B., Ni, W., Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations in Rn. Comm. Math. Phys. 68, 209–243 (1975)
Gilbarg, D., Trudinger, N.: Elliptic Partial Differentail Equations of Second Order (2nd ed.) Springer (1983)
Li, Y. Y.: On uniformly rotating stars. Arch. Rational Mech. Anal. 115, 367–393 (1991)
Lions, P. L.: On the existence of positive solutions of semilinear elliptic equations. SIAM, Rev. 24, 441–467 (1982)
Makino, T.: Blowing up of the Euler-Poisson equation for the evolution of gaseous star. Transport Theory and Statistical Physics 21, 615–624 (1992)
Pohozaev, S. I.: Eigenfunctions of the equations Δu+λ f(u)=0. Sov. Math. Dok. 5, 1408–1411 (1965)
Rabinowitz, P.: Minimax methods in critical point theory with applications to differential equations. CBMS # 65, AMS (1986)
Smoller, J.: Shock Waves and Reaction-Diffusion Equations. Springer, Berlin, New York (1983)
Smoller, J., Wasserman, A.: Existence of positive solutions for semilinear elliptic equations in general domains. Arch. Rational Mech. Anal. 98, 229–249 (1987)
Weinberg, S.: Gravitation and Cosmology. John Wiley and Sons, New York, (1972)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. H. Rabinowitz
Part of this work was completed when Tao Luo was an assistant professor at the University of Michigan. Joel Smoller was supported in part by the NSF, contract number DMS-010-3998. We are grateful to the referee for his very interesting remarks and comments, which enabled a new section, Section 6, to be added in the final version of the paper.
Rights and permissions
About this article
Cite this article
Luo, T., Smoller, J. Rotating Fluids with Self-Gravitation in Bounded Domains. Arch. Rational Mech. Anal. 173, 345–377 (2004). https://doi.org/10.1007/s00205-004-0319-4
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-004-0319-4