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 & Continuous Dynamical Systems - A, 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. doi: 10.1007/BF00280443.

[2]

J. F. G. Auchmuty, The global branching of rotating stars,, Arch. Rational Mech. Anal., 114 (1991), 179. 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. doi: 10.1007/BF00250465.

[4]

J. F. G. Auchmuty and R. Beals, Models of rotating stars,, Astrophysical J., 165 (1971), 79. doi: 10.1086/180721.

[5]

L. A. Caffarelli and A. Friedman, The shape of axisymmetric rotating fluid,, J. Funct. Anal., 35 (1980), 109. 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., (1987).

[8]

S. Chanillo and Y. Y. Li, On diameters of uniformly rotating stars,, Comm. Math. Phys., 166 (1994), 417. doi: 10.1007/BF02112323.

[9]

A. Friedman and B. Turkington, Asymptotic estimates for an axisymmetric rotating fluid,, J. Fun. Anal., 37 (1980), 136. 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. 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. doi: 10.1016/0022-0396(81)90114-5.

[12]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", 2nd edition, (1983).

[13]

G. H. Hardy, J. E. Littlewood and G. Polya, "Inequality,", Cambridge Univ. Press, (1934).

[14]

J. Jang, Nonlinear instability in gravitational Euler-Poisson system for $\gamma=\frac{6}{5}$,, Arch. Rational Mech. Anal., 188 (2008), 265. 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. 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.

[18]

P. L. Lions, Minimization problems in $L^1(\bb R^3)$,, J. Funct. Anal., 41 (1981), 236. 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.

[20]

T. Luo and J. Smoller, Nonlinear dynamical stability of Newtonian rotating white dwarfs and supermassive stars,, Comm. Math. Physics, 284 (2008), 425. 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. 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.

[23]

G. Rein, Reduction and a concentration-compactness principle for energy-Casimir functionals,, SIAM J. Math. Anal., 33 (2001), 896. doi: 10.1137/P0036141001389275.

[24]

S. H. Shapiro and S. A. Teukolsky, "Black Holes, White Dwarfs, and Neutron Stars,", WILEY-VCH, (1983). doi: 10.1002/9783527617661.

[25]

J. L. Tassoul, "Theory of Rotating Stars,", Princeton Univ. Press, (1978).

[26]

S. Weinberg, "Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity,", John Wiley and Sons, (1972).

show all references

References:
[1]

J. F. G. Auchmuty, Existence of equilibrium figures,, Arch. Rational Mech. Anal., 65 (1977), 249. doi: 10.1007/BF00280443.

[2]

J. F. G. Auchmuty, The global branching of rotating stars,, Arch. Rational Mech. Anal., 114 (1991), 179. 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. doi: 10.1007/BF00250465.

[4]

J. F. G. Auchmuty and R. Beals, Models of rotating stars,, Astrophysical J., 165 (1971), 79. doi: 10.1086/180721.

[5]

L. A. Caffarelli and A. Friedman, The shape of axisymmetric rotating fluid,, J. Funct. Anal., 35 (1980), 109. 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., (1987).

[8]

S. Chanillo and Y. Y. Li, On diameters of uniformly rotating stars,, Comm. Math. Phys., 166 (1994), 417. doi: 10.1007/BF02112323.

[9]

A. Friedman and B. Turkington, Asymptotic estimates for an axisymmetric rotating fluid,, J. Fun. Anal., 37 (1980), 136. 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. 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. doi: 10.1016/0022-0396(81)90114-5.

[12]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", 2nd edition, (1983).

[13]

G. H. Hardy, J. E. Littlewood and G. Polya, "Inequality,", Cambridge Univ. Press, (1934).

[14]

J. Jang, Nonlinear instability in gravitational Euler-Poisson system for $\gamma=\frac{6}{5}$,, Arch. Rational Mech. Anal., 188 (2008), 265. 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. 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.

[18]

P. L. Lions, Minimization problems in $L^1(\bb R^3)$,, J. Funct. Anal., 41 (1981), 236. 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.

[20]

T. Luo and J. Smoller, Nonlinear dynamical stability of Newtonian rotating white dwarfs and supermassive stars,, Comm. Math. Physics, 284 (2008), 425. 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. 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.

[23]

G. Rein, Reduction and a concentration-compactness principle for energy-Casimir functionals,, SIAM J. Math. Anal., 33 (2001), 896. doi: 10.1137/P0036141001389275.

[24]

S. H. Shapiro and S. A. Teukolsky, "Black Holes, White Dwarfs, and Neutron Stars,", WILEY-VCH, (1983). doi: 10.1002/9783527617661.

[25]

J. L. Tassoul, "Theory of Rotating Stars,", Princeton Univ. Press, (1978).

[26]

S. Weinberg, "Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity,", John Wiley and Sons, (1972).

[1]

A. Alexandrou Himonas, Gerard Misiołek, Feride Tiǧlay. On unique continuation for the modified Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 515-529. doi: 10.3934/dcds.2007.19.515

[2]

Yongcai Geng. Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force. Communications on Pure & Applied Analysis, 2015, 14 (2) : 549-564. doi: 10.3934/cpaa.2015.14.549

[3]

Hong Cai, Zhong Tan. Stability of stationary solutions to the compressible bipolar Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4677-4696. doi: 10.3934/dcds.2017201

[4]

La-Su Mai, Kaijun Zhang. Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 981-1004. doi: 10.3934/dcds.2016.36.981

[5]

Manwai Yuen. Cylindrical blowup solutions to the isothermal Euler-Poisson equations. Conference Publications, 2011, 2011 (Special) : 1448-1456. doi: 10.3934/proc.2011.2011.1448

[6]

Jiang Xu, Ting Zhang. Zero-electron-mass limit of Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4743-4768. doi: 10.3934/dcds.2013.33.4743

[7]

Sasho Popov, Jean-Marie Strelcyn. The Euler-Poisson equations: An elementary approach to integrability conditions. Journal of Geometric Mechanics, 2018, 10 (3) : 293-329. doi: 10.3934/jgm.2018011

[8]

Masahiro Suzuki. Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics. Kinetic & Related Models, 2011, 4 (2) : 569-588. doi: 10.3934/krm.2011.4.569

[9]

Zhigang Wu, Weike Wang. Pointwise estimates of solutions for the Euler-Poisson equations with damping in multi-dimensions. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 1101-1117. doi: 10.3934/dcds.2010.26.1101

[10]

Xueke Pu. Quasineutral limit of the Euler-Poisson system under strong magnetic fields. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2095-2111. doi: 10.3934/dcdss.2016086

[11]

Shu Wang, Chundi Liu. Boundary Layer Problem and Quasineutral Limit of Compressible Euler-Poisson System. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2177-2199. doi: 10.3934/cpaa.2017108

[12]

Myoungjean Bae, Yong Park. Radial transonic shock solutions of Euler-Poisson system in convergent nozzles. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 773-791. doi: 10.3934/dcdss.2018049

[13]

Qiangchang Ju, Hailiang Li, Yong Li, Song Jiang. Quasi-neutral limit of the two-fluid Euler-Poisson system. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1577-1590. doi: 10.3934/cpaa.2010.9.1577

[14]

Yeping Li. Existence and some limit analysis of stationary solutions for a multi-dimensional bipolar Euler-Poisson system. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 345-360. doi: 10.3934/dcdsb.2011.16.345

[15]

Ming Mei, Yong Wang. Stability of stationary waves for full Euler-Poisson system in multi-dimensional space. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1775-1807. doi: 10.3934/cpaa.2012.11.1775

[16]

Zhong Tan, Yong Wang, Fanhui Xu. Large-time behavior of the full compressible Euler-Poisson system without the temperature damping. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1583-1601. doi: 10.3934/dcds.2016.36.1583

[17]

Corrado Lattanzio, Pierangelo Marcati. The relaxation to the drift-diffusion system for the 3-$D$ isentropic Euler-Poisson model for semiconductors. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 449-455. doi: 10.3934/dcds.1999.5.449

[18]

Yeping Li, Jie Liao. Stability and $ L^{p}$ convergence rates of planar diffusion waves for three-dimensional bipolar Euler-Poisson systems. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1281-1302. doi: 10.3934/cpaa.2019062

[19]

Masahiro Suzuki. Asymptotic stability of a boundary layer to the Euler--Poisson equations for a multicomponent plasma. Kinetic & Related Models, 2016, 9 (3) : 587-603. doi: 10.3934/krm.2016008

[20]

Elaine Cozzi, James P. Kelliher. Well-posedness of the 2D Euler equations when velocity grows at infinity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2361-2392. doi: 10.3934/dcds.2019100

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]