Abstract
We prove the existence of rotating star solutions which are steady-state solutions of the compressible isentropic Euler–Poisson (Euler–Poisson) equations in three spatial dimensions with prescribed angular momentum and total mass. This problem can be formulated as a variational problem of finding a minimizer of an energy functional in a broader class of functions having less symmetry than those functions considered in the classical Auchmuty–Beals paper. We prove the non-linear dynamical stability of these solutions with perturbations having the same total mass and symmetry as the rotating star solution. We also prove finite time stability of \(W^{1, \infty}(\mathbb {R}^{3})\) solutions where the perturbations are entropy-weak solutions of the Euler–Poisson equations. Finally, we give a uniform (in time) a priori estimate for entropy-weak solutions of the Euler–Poisson equations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ambrosio L. (2004) Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158(2): 227–260
Auchmuty G., Beals R. (1971) Variational solutions of some nonlinear free boundary problems. Arch. Rat. Mech. Anal. 43: 255–271
Auchmuty G. (1991) The global branching of rotating stars. Arch. Rat. Mech. Anal. 114: 179–194
Caffarelli L., Friedman A. (1980) The shape of axisymmetric rotating fluid. J. Funct. Anal. 694: 109–142
Chandrasekhar S. (1939) Introduction to the Stellar Structure. University of Chicago Press, Chicago
Chanillo S., Li Y.Y. (1994) On diameters of uniformly rotating stars. Comm. Math. Phys. 166(2): 417–430
Chen G.Q., Frid H. (1999) Large time behavior of entropy solutions of conservation laws. J. Differ. Equ. 152: 308–357
Chen G.Q., Frid H. (1991) Divegence-measure fields and hyperbolic conservation laws. Arch. Rat. Mech. Anal. 147: 89–118
Chen G.Q., Frid H. (2003) Extended divergence-measure fields and the Euler equations for gas dynamics. Comm. Math. Phys. 236(2): 251–280
Dafermos, C.M.: Entropy and the stability of classical solutions of hyperbolic system of conservation laws, Recent Mathematical Methods in Nonlinear Wave Propagation, Montecatini Terme, 1994, Lecture Notes in Math., vol. 1640, pp. 48–69. Springer, Berlin, 1996
Deng Y., Liu T.P., Yang T., Yao Z. (2002) Solutions of Euler–Poisson equations for gaseous stars. Arch. Ration. Mech. Anal. 164(3): 261–285
Deng Y., Xiang J., Yang T. (2003) Blowup phenomena of the Euler–Poisson equations for gaseous stars. J. of Math. Anal. and Appl. 286: 295–306
Di Perna R.J., Lions P.L. (1989) Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98: 511–547
Evans L., Gariepy R. (1992) Measure Theory and Fine Properties of Functions. CRC press, Boca Raton, FL
Friedman A., Turkington B. (1980) Asymptotic estimates for an axi-symmetric rotating fluid. J. Func. Anal. 37: 136–163
Friedman A., Turkington B. (1981) Existence and dimensions of a rotating white dwarf. J. Differ. Equ. 42: 414–437
Feireisl E. (2004) Dynamics of Viscous Compressible Fluids. Oxford University Press, New York
Jang, J.: Nonlinear instability in gravitational Euler–Poisson system for \({\gamma}=\frac{6}{5}\) . Arch. Ration. Mech. Anal., to appear
Guo Y., Rein G. (1999) Stable steady states in stellar dynamics. Arch. Rat. Mech. Anal. 147: 225–243
Guo Y., Rein G. (2002) Stable models of elliptical galaxies. Mon. Not. R. Astron. Soc. 344: 1296–1306
Gilbarg D., Trudinger N. (1983) Elliptic Partial Differentail Equations of Second Order, 2nd edn. Springer, Berlin
Lax, P.: Shock waves and entropy. Contribution to Nonlinear Functional Analysis. (Ed. Zarantonello E. A.) Academic Press, New York, 603–634, 1971
Lebovitz N.R.(1961) The virial tensor and its application to self-gravitating fluids. Astrophys. J. 134: 500–536
Lebovitz, N.R., Lifschitz, A.: Short-wavelength instabilities of Riemann ellipsoids, Philos. Trans. Roy. Soc. London Ser. A 354(1709), 927–950 (1996)
Li Y.Y. (1991) On uniformly rotating stars. Arch. Rat. Mech. Anal. 115(4): 367–393
Lieb E.H., Loss M. (1997) Analysis, American Mathematical Society. Providence, Rhode Island
Lin S.S. (1997) Stability of gaseous stars in spherically symmetric motions. SIAM J. Math. Anal. 28(3): 539–569
Lions P.L. (1984) The concentration-compactness principle in the calculus of variations. The locally compact case. Part I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1: 109–145
Luo T., Smoller J. (2004) Rotating fluids with self-gravitation in bounded domains. Arch. Rat. Mech. Anal. 173(3): 345–377
McCann R. (2006) Stable rotating binary stars and fluid in a tube. Houston J. Math. 32(2): 603–631
Makino T. (1992) Blowing up of the Euler–Poisson equation for the evolution of gaseous star, Transp. Theory Statist. Phys. 21: 615–624
Reed M., Simon B. (1975) Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness. Academic Press, NewYork
Rein G. (2001) Reduction and a concentration-compactness principle for energy-casimir functionals. SIAM J. Math. Anal.33(4): 896–912
Rein G. (2003) Non-linear stability of gaseous stars. Arch. Rat. Mech. Anal. 168(2): 115–130
Rudin W. (1976) Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, Auckland
Smoller J. (1994) Shock Waves and Reaction-Diffusion Equations, 2nd edn. Springer, Berlin
Tassoul J.L. (1978) Theory of Rotating Stars. Princeton University Press, Princeton
Wang D. (1999) Global solutions and stability for self-gravitating isentropic gases. J. Math. Anal. Appl. 229: 530–542
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Rabinowitz
Rights and permissions
About this article
Cite this article
Luo, T., Smoller, J. Existence and Non-linear Stability of Rotating Star Solutions of the Compressible Euler–Poisson Equations. Arch Rational Mech Anal 191, 447–496 (2009). https://doi.org/10.1007/s00205-007-0108-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-007-0108-y