Euler-Poisson equations related to general compressible rotating fluids

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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

Abstract
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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.

Keywords: Euler-Poisson equations, White dwarfs., Given angular velocity, Rotating star.

Mathematics Subject Classification: Primary: 35J05; 35J20; Secondary: 85A1.

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. Google Scholar
[2]	J. F. G. Auchmuty, The global branching of rotating stars,, Arch. Rational Mech. Anal., 114 (1991), 179. doi: 10.1007/BF00375402. Google Scholar
[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. Google Scholar
[4]	J. F. G. Auchmuty and R. Beals, Models of rotating stars,, Astrophysical J., 165 (1971), 79. doi: 10.1086/180721. Google Scholar
[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. Google Scholar
[6]	S. Chandrasekhar, "Introduction to the Stellar Structure,", University of Chicago Press, (1939). Google Scholar
[7]	S. Chandrasekhar, "Ellipsoidal Figures of Equilibrium,", Dover Publication Inc., (1987). Google Scholar
[8]	S. Chanillo and Y. Y. Li, On diameters of uniformly rotating stars,, Comm. Math. Phys., 166 (1994), 417. doi: 10.1007/BF02112323. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", 2nd edition, (1983). Google Scholar
[13]	G. H. Hardy, J. E. Littlewood and G. Polya, "Inequality,", Cambridge Univ. Press, (1934). Google Scholar
[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. Google Scholar
[15]	H. G. Li and J. G. Bao, Existence of the rotating stars with prescribed angular velocity law,, to appear in Houston J. Math., (). Google Scholar
[16]	Y. Y. Li, On uniformly rotating stars,, Arch. Rational Mech. Anal., 115 (1991), 367. doi: 10.1007/BF00375280. Google Scholar
[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. Google Scholar
[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. Google Scholar
[19]	T. Luo and J. Smoller, Rotating fluids with self-gravitation in bounded domains,, Arch. Rational Mech. Anal., 173 (2004), 345. Google Scholar
[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. Google Scholar
[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. Google Scholar
[22]	R. J. McCann, Stable rotating binary stars and fluid in a tube,, Houston J. Math., 32 (2006), 603. Google Scholar
[23]	G. Rein, Reduction and a concentration-compactness principle for energy-Casimir functionals,, SIAM J. Math. Anal., 33 (2001), 896. doi: 10.1137/P0036141001389275. Google Scholar
[24]	S. H. Shapiro and S. A. Teukolsky, "Black Holes, White Dwarfs, and Neutron Stars,", WILEY-VCH, (1983). doi: 10.1002/9783527617661. Google Scholar
[25]	J. L. Tassoul, "Theory of Rotating Stars,", Princeton Univ. Press, (1978). Google Scholar
[26]	S. Weinberg, "Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity,", John Wiley and Sons, (1972). Google Scholar

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References:

[1]	J. F. G. Auchmuty, Existence of equilibrium figures,, Arch. Rational Mech. Anal., 65 (1977), 249. doi: 10.1007/BF00280443. Google Scholar
[2]	J. F. G. Auchmuty, The global branching of rotating stars,, Arch. Rational Mech. Anal., 114 (1991), 179. doi: 10.1007/BF00375402. Google Scholar
[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. Google Scholar
[4]	J. F. G. Auchmuty and R. Beals, Models of rotating stars,, Astrophysical J., 165 (1971), 79. doi: 10.1086/180721. Google Scholar
[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. Google Scholar
[6]	S. Chandrasekhar, "Introduction to the Stellar Structure,", University of Chicago Press, (1939). Google Scholar
[7]	S. Chandrasekhar, "Ellipsoidal Figures of Equilibrium,", Dover Publication Inc., (1987). Google Scholar
[8]	S. Chanillo and Y. Y. Li, On diameters of uniformly rotating stars,, Comm. Math. Phys., 166 (1994), 417. doi: 10.1007/BF02112323. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", 2nd edition, (1983). Google Scholar
[13]	G. H. Hardy, J. E. Littlewood and G. Polya, "Inequality,", Cambridge Univ. Press, (1934). Google Scholar
[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. Google Scholar
[15]	H. G. Li and J. G. Bao, Existence of the rotating stars with prescribed angular velocity law,, to appear in Houston J. Math., (). Google Scholar
[16]	Y. Y. Li, On uniformly rotating stars,, Arch. Rational Mech. Anal., 115 (1991), 367. doi: 10.1007/BF00375280. Google Scholar
[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. Google Scholar
[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. Google Scholar
[19]	T. Luo and J. Smoller, Rotating fluids with self-gravitation in bounded domains,, Arch. Rational Mech. Anal., 173 (2004), 345. Google Scholar
[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. Google Scholar
[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. Google Scholar
[22]	R. J. McCann, Stable rotating binary stars and fluid in a tube,, Houston J. Math., 32 (2006), 603. Google Scholar
[23]	G. Rein, Reduction and a concentration-compactness principle for energy-Casimir functionals,, SIAM J. Math. Anal., 33 (2001), 896. doi: 10.1137/P0036141001389275. Google Scholar
[24]	S. H. Shapiro and S. A. Teukolsky, "Black Holes, White Dwarfs, and Neutron Stars,", WILEY-VCH, (1983). doi: 10.1002/9783527617661. Google Scholar
[25]	J. L. Tassoul, "Theory of Rotating Stars,", Princeton Univ. Press, (1978). Google Scholar
[26]	S. Weinberg, "Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity,", John Wiley and Sons, (1972). Google Scholar

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