# American Institute of Mathematical Sciences

July  2011, 29(3): 1085-1096. doi: 10.3934/dcds.2011.29.1085

## Euler-Poisson equations related to general compressible rotating fluids

 1 School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China, China

Received  March 2010 Revised  August 2010 Published  November 2010

This paper is mainly concerned with Euler-Poisson equations modeling Newtonian stars. We establish the existence of rotating star solutions for general compressible fluids with prescribed angular velocity law, which is the main point distinguished with the case with prescribed angular momentum per unit mass. The compactness of any minimizing sequence is established, which is important from the stability point of view.
Citation: Haigang Li, Jiguang Bao. Euler-Poisson equations related to general compressible rotating fluids. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1085-1096. doi: 10.3934/dcds.2011.29.1085
##### References:
 [1] J. F. G. Auchmuty, Existence of equilibrium figures, Arch. Rational Mech. Anal., 65 (1977), 249-261. doi: 10.1007/BF00280443. [2] J. F. G. Auchmuty, The global branching of rotating stars, Arch. Rational Mech. Anal., 114 (1991), 179-194. doi: 10.1007/BF00375402. [3] J. F. G. Auchmuty and R. Beals, Variation solutions of some non-linear free boundary problems, Arch. Rational Mech. Anal., 43 (1971), 255-271. doi: 10.1007/BF00250465. [4] J. F. G. Auchmuty and R. Beals, Models of rotating stars, Astrophysical J., 165 (1971), 79-82. doi: 10.1086/180721. [5] L. A. Caffarelli and A. Friedman, The shape of axisymmetric rotating fluid, J. Funct. Anal., 35 (1980), 109-142. doi: 10.1016/0022-1236(80)90082-8. [6] S. Chandrasekhar, "Introduction to the Stellar Structure," University of Chicago Press, 1939 [7] S. Chandrasekhar, "Ellipsoidal Figures of Equilibrium," Dover Publication Inc., New York, 1987 [8] S. Chanillo and Y. Y. Li, On diameters of uniformly rotating stars, Comm. Math. Phys., 166 (1994), 417-430. doi: 10.1007/BF02112323. [9] A. Friedman and B. Turkington, Asymptotic estimates for an axisymmetric rotating fluid, J. Fun. Anal., 37 (1980), 136-163. doi: 10.1016/0022-1236(80)90038-5. [10] A. Friedman and B. Turkington, The oblateness of an axisymmetric rotating fluid, Indiana Univ. Math. J., 29 (1980), 777-792. doi: 10.1512/iumj.1980.29.29056. [11] A. Friedman and B. Turkington, Existence and dimensions of a rotating white dwarf, J. Diff. Equations, 42 (1981), 414-437. doi: 10.1016/0022-0396(81)90114-5. [12] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer, 1983. [13] G. H. Hardy, J. E. Littlewood and G. Polya, "Inequality," Cambridge Univ. Press, Cambridge, 1934. [14] J. Jang, Nonlinear instability in gravitational Euler-Poisson system for $\gamma=\frac{6}{5}$, Arch. Rational Mech. Anal., 188 (2008), 265-307. doi: 10.1007/s00205-007-0086-0. [15] H. G. Li and J. G. Bao, Existence of the rotating stars with prescribed angular velocity law, to appear in Houston J. Math. [16] Y. Y. Li, On uniformly rotating stars, Arch. Rational Mech. Anal., 115 (1991), 367-393. doi: 10.1007/BF00375280. [17] P. L. Lions, The concentration-compactness principle in the calculus of variation, The locally case, part I, Ann. I. H. Anal. Nonli., 1 (1984), 109-145. [18] P. L. Lions, Minimization problems in $L^1(\bb R^3)$, J. Funct. Anal., 41 (1981), 236-275. doi: 10.1016/0022-1236(81)90089-6. [19] T. Luo and J. Smoller, Rotating fluids with self-gravitation in bounded domains, Arch. Rational Mech. Anal., 173 (2004), 345-377. [20] T. Luo and J. Smoller, Nonlinear dynamical stability of Newtonian rotating white dwarfs and supermassive stars, Comm. Math. Physics, 284 (2008), 425-457. doi: 10.1007/s00220-008-0569-3. [21] T. Luo and J. Smoller, Existence and nonlinear stability of rotating star solutions of the compressible Euler-Poisson equations, Arch. Rational Mech. Anal., 191 (2009), 447-496. doi: 10.1007/s00205-007-0108-y. [22] R. J. McCann, Stable rotating binary stars and fluid in a tube, Houston J. Math., 32 (2006), 603-632. [23] G. Rein, Reduction and a concentration-compactness principle for energy-Casimir functionals, SIAM J. Math. Anal., 33 (2001), 896-912. doi: 10.1137/P0036141001389275. [24] S. H. Shapiro and S. A. Teukolsky, "Black Holes, White Dwarfs, and Neutron Stars," WILEY-VCH, New York, 1983. doi: 10.1002/9783527617661. [25] J. L. Tassoul, "Theory of Rotating Stars," Princeton Univ. Press, New Jersey, 1978. [26] S. Weinberg, "Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity," John Wiley and Sons, Inc. New York, 1972.

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##### References:
 [1] J. F. G. Auchmuty, Existence of equilibrium figures, Arch. Rational Mech. Anal., 65 (1977), 249-261. doi: 10.1007/BF00280443. [2] J. F. G. Auchmuty, The global branching of rotating stars, Arch. Rational Mech. Anal., 114 (1991), 179-194. doi: 10.1007/BF00375402. [3] J. F. G. Auchmuty and R. Beals, Variation solutions of some non-linear free boundary problems, Arch. Rational Mech. Anal., 43 (1971), 255-271. doi: 10.1007/BF00250465. [4] J. F. G. Auchmuty and R. Beals, Models of rotating stars, Astrophysical J., 165 (1971), 79-82. doi: 10.1086/180721. [5] L. A. Caffarelli and A. Friedman, The shape of axisymmetric rotating fluid, J. Funct. Anal., 35 (1980), 109-142. doi: 10.1016/0022-1236(80)90082-8. [6] S. Chandrasekhar, "Introduction to the Stellar Structure," University of Chicago Press, 1939 [7] S. Chandrasekhar, "Ellipsoidal Figures of Equilibrium," Dover Publication Inc., New York, 1987 [8] S. Chanillo and Y. Y. Li, On diameters of uniformly rotating stars, Comm. Math. Phys., 166 (1994), 417-430. doi: 10.1007/BF02112323. [9] A. Friedman and B. Turkington, Asymptotic estimates for an axisymmetric rotating fluid, J. Fun. Anal., 37 (1980), 136-163. doi: 10.1016/0022-1236(80)90038-5. [10] A. Friedman and B. Turkington, The oblateness of an axisymmetric rotating fluid, Indiana Univ. Math. J., 29 (1980), 777-792. doi: 10.1512/iumj.1980.29.29056. [11] A. Friedman and B. Turkington, Existence and dimensions of a rotating white dwarf, J. Diff. Equations, 42 (1981), 414-437. doi: 10.1016/0022-0396(81)90114-5. [12] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer, 1983. [13] G. H. Hardy, J. E. Littlewood and G. Polya, "Inequality," Cambridge Univ. Press, Cambridge, 1934. [14] J. Jang, Nonlinear instability in gravitational Euler-Poisson system for $\gamma=\frac{6}{5}$, Arch. Rational Mech. Anal., 188 (2008), 265-307. doi: 10.1007/s00205-007-0086-0. [15] H. G. Li and J. G. Bao, Existence of the rotating stars with prescribed angular velocity law, to appear in Houston J. Math. [16] Y. Y. Li, On uniformly rotating stars, Arch. Rational Mech. Anal., 115 (1991), 367-393. doi: 10.1007/BF00375280. [17] P. L. Lions, The concentration-compactness principle in the calculus of variation, The locally case, part I, Ann. I. H. Anal. Nonli., 1 (1984), 109-145. [18] P. L. Lions, Minimization problems in $L^1(\bb R^3)$, J. Funct. Anal., 41 (1981), 236-275. doi: 10.1016/0022-1236(81)90089-6. [19] T. Luo and J. Smoller, Rotating fluids with self-gravitation in bounded domains, Arch. Rational Mech. Anal., 173 (2004), 345-377. [20] T. Luo and J. Smoller, Nonlinear dynamical stability of Newtonian rotating white dwarfs and supermassive stars, Comm. Math. Physics, 284 (2008), 425-457. doi: 10.1007/s00220-008-0569-3. [21] T. Luo and J. Smoller, Existence and nonlinear stability of rotating star solutions of the compressible Euler-Poisson equations, Arch. Rational Mech. Anal., 191 (2009), 447-496. doi: 10.1007/s00205-007-0108-y. [22] R. J. McCann, Stable rotating binary stars and fluid in a tube, Houston J. Math., 32 (2006), 603-632. [23] G. Rein, Reduction and a concentration-compactness principle for energy-Casimir functionals, SIAM J. Math. Anal., 33 (2001), 896-912. doi: 10.1137/P0036141001389275. [24] S. H. Shapiro and S. A. Teukolsky, "Black Holes, White Dwarfs, and Neutron Stars," WILEY-VCH, New York, 1983. doi: 10.1002/9783527617661. [25] J. L. Tassoul, "Theory of Rotating Stars," Princeton Univ. Press, New Jersey, 1978. [26] S. Weinberg, "Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity," John Wiley and Sons, Inc. New York, 1972.
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