Skip to main content
SpringerLink
Log in

Variational rotating solutions to non-isentropic Euler-Poisson equations with prescribed total mass

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

This paper proves the existence of variational rotating solutions to the compressible non-isentropic Euler-Poisson equations with prescribed total mass. This extends the result of the isentropic case (see Auchmuty and Beals (1971)) to the non-isentropic case. Compared with the previous result of variational rotating solutions in the non-isentropic case (see Wu (2015)), to keep the constraint of prescribed finite total mass, we establish a new variational structure of the non-isentropic Euler-Poisson equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Auchmuty G. The global branching of rotating stars. Arch Ration Mech Anal, 1991, 114: 179–193

    Article  MathSciNet  MATH  Google Scholar 

  2. Auchmuty J F G, Beals R. Variational solutions of some nonlinear free boundary problems. Arch Ration Mech Anal, 1971, 43: 255–271

    Article  MathSciNet  MATH  Google Scholar 

  3. Caffarelli L A, Friedman A. The shape of axisymmetric rotating fluid. J Funct Anal, 1980, 35: 109–142

    Article  MathSciNet  MATH  Google Scholar 

  4. Chandrasekhar S. An Introduction to the Study of Stellar Structure. New York: Dover Publications, 1957

    MATH  Google Scholar 

  5. Chanillo S, Li Y-Y. On diameters of uniformly rotating stars. Comm Math Phys, 1994, 166: 417–430

    Article  MathSciNet  MATH  Google Scholar 

  6. Chanillo S, Weiss G S. A remark on the geometry of uniformly rotating stars. J Differential Equations, 2012, 253: 553–562

    Article  MathSciNet  MATH  Google Scholar 

  7. Deng Y-B, Liu T-P, Yang T, et al. Solutions of Euler-Poisson equations for gaseous stars. Arch Ration Mech Anal, 2002, 164: 261–285

    Article  MathSciNet  MATH  Google Scholar 

  8. Deng Y-B, Yang T. Multiplicity of stationary solutions to the Euler-Poisson equations. J Differential Equations, 2006, 231: 252–289

    Article  MathSciNet  MATH  Google Scholar 

  9. Federbush P, Luo T, Smoller J. Existence of magnetic compressible fluid stars. Arch Ration Mech Anal, 2015, 215: 611–631

    Article  MathSciNet  MATH  Google Scholar 

  10. Friedman A, Turkington B. Asymptotic estimates for an axisymmetric rotating fluid. J Funct Anal, 1980, 37: 136–163

    Article  MathSciNet  MATH  Google Scholar 

  11. Friedman A, Turkington B. Existence and dimensions of a rotating white dwarf. J Differential Equations, 1981, 42: 414–437

    Article  MathSciNet  MATH  Google Scholar 

  12. Gu X-M, Lei Z. Local well-posedness of the three dimensional compressible Euler-Poisson equations with physical vacuum. J Math Pures Appl (9), 2016, 105: 662–723

    Article  MathSciNet  MATH  Google Scholar 

  13. Hadžić M, Jang J. Nonlinear stability of expanding star solutions of the radially symmetric mass-critical Euler-Poisson system. Comm Pure Appl Math, 2018, 71: 827–891

    Article  MathSciNet  MATH  Google Scholar 

  14. Jang J. Nonlinear instability in gravitational Euler-Poisson systems for \(\gamma = {6 \over 5}\). Arch Ration Mech Anal, 2008, 188: 265–307

    Article  MathSciNet  Google Scholar 

  15. Jang J. Nonlinear instability theory of Lane-Emden stars. Comm Pure Appl Math, 2014, 67: 1418–1465

    Article  MathSciNet  MATH  Google Scholar 

  16. Jang J, Makino T. On slowly rotating axisymmetric solutions of the Euler-Poisson equations. Arch Ration Mech Anal, 2017, 225: 873–900

    Article  MathSciNet  MATH  Google Scholar 

  17. Jang J, Makino T. On rotating axisymmetric solutions of the Euler-Poisson equations. J Differential Equations, 2019, 266: 3942–3972

    Article  MathSciNet  MATH  Google Scholar 

  18. Jang J, Strauss W A, Wu Y-L. Existence of rotating magnetic stars. Phys D, 2019, 397: 65–74

    Article  MathSciNet  Google Scholar 

  19. Jang J, Tice I. Instability theory of the Navier-Stokes-Poisson equations. Anal PDE, 2013, 6: 1121–1181

    Article  MathSciNet  MATH  Google Scholar 

  20. Lax P D. Functional Analysis. New York: John Wiley & Sons, 2002

    MATH  Google Scholar 

  21. Li Y-Y. On uniformly rotating stars. Arch Ration Mech Anal, 1991, 115: 367–393

    Article  MathSciNet  MATH  Google Scholar 

  22. Lieb E H, Yau H-T. The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Comm Math Phys, 1987, 112: 147–174

    Article  MathSciNet  MATH  Google Scholar 

  23. Lin S-S. Stability of gaseous stars in spherically symmetric motions. SIAM J Math Anal, 1997, 28: 539–569

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu X. A model of radiational gaseous stars. SIAM J Math Anal, 2018, 50: 6100–6155

    Article  MathSciNet  MATH  Google Scholar 

  25. Liu X. On the expanding configurations of viscous radiation gaseous stars: The isentropic model. Nonlinearity, 2019, 32: 2975–3011

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu X. On the expanding configurations of viscous radiation gaseous stars: Thermodynamic model. J Differential Equations, 2020, 268: 2717–2751

    Article  MathSciNet  MATH  Google Scholar 

  27. Luo T, Smoller J. Rotating fluids with self-gravitation in bounded domains. Arch Ration Mech Anal, 2004, 173: 345–377

    Article  MathSciNet  MATH  Google Scholar 

  28. Luo T, Smoller J. Nonlinear dynamical stability of Newtonian rotating and non-rotating white dwarfs and rotating supermassive stars. Comm Math Phys, 2008, 284: 425–457

    Article  MathSciNet  MATH  Google Scholar 

  29. Luo T, Smoller J. Existence and non-linear stability of rotating star solutions of the compressible Euler-Poisson equations. Arch Ration Mech Anal, 2009, 191: 447–496

    Article  MathSciNet  MATH  Google Scholar 

  30. Luo T, Xin Z-P, Zeng H-H. Well-posedness for the motion of physical vacuum of the three-dimensional compressible Euler equations with or without self-gravitation. Arch Ration Mech Anal, 2014, 213: 763–831

    Article  MathSciNet  MATH  Google Scholar 

  31. Luo T, Xin Z-P, Zeng H-H. Nonlinear asymptotic stability of the Lane-Emden solutions for the viscous gaseous star problem with degenerate density dependent viscosities. Comm Math Phys, 2016, 347: 657–702

    Article  MathSciNet  MATH  Google Scholar 

  32. Luo T, Xin Z-P, Zeng H-H. On nonlinear asymptotic stability of the Lane-Emden solutions for the viscous gaseous star problem. Adv Math, 2016, 291: 90–182

    Article  MathSciNet  MATH  Google Scholar 

  33. Makino T. On spherically symmetric motions of a gaseous star governed by the Euler-Poisson equations. Osaka J Math, 2015, 52: 545–580

    MathSciNet  MATH  Google Scholar 

  34. Makino T. An application of the Nash-Moser theorem to the vacuum boundary problem of gaseous stars. J Differential Equations, 2017, 262: 803–843

    Article  MathSciNet  MATH  Google Scholar 

  35. Rein G. Reduction and a concentration-compactness principle for energy-Casimir functionals. SIAM J Math Anal, 2001, 33: 896–912

    Article  MathSciNet  MATH  Google Scholar 

  36. Rein G. Non-linear stability of gaseous stars. Arch Ration Mech Anal, 2003, 168: 115–130

    Article  MathSciNet  MATH  Google Scholar 

  37. Rudin W. Principles of Mathematical Analysis, 3rd ed. International Series in Pure and Applied Mathematics. New York: McGraw-Hill, 1976

    MATH  Google Scholar 

  38. Strauss W A, Wu Y-L. Steady states of rotating stars and galaxies. SIAM J Math Anal, 2017, 49: 4865–4914

    Article  MathSciNet  MATH  Google Scholar 

  39. Strauss W A, Wu Y-L. Rapidly rotating stars. Comm Math Phys, 2019, 368: 701–721

    Article  MathSciNet  MATH  Google Scholar 

  40. Strauss W A, Wu Y-L. Rapidly rotating white dwarfs. Nonlinearity, 2020, 33: 4783–4798

    Article  MathSciNet  MATH  Google Scholar 

  41. Tassoul J-L. Theory of Rotating Stars. Princeton: Princeton University Press, 2015

    Book  Google Scholar 

  42. Wu Y-L. On rotating star solutions to the non-isentropic Euler-Poisson equations. J Differential Equations, 2015, 259: 7161–7198

    Article  MathSciNet  MATH  Google Scholar 

  43. Wu Y-L. Existence of rotating planet solutions to the Euler-Poisson equations with an inner hard core. Arch Ration Mech Anal, 2016, 219: 1–26

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11901208 and 11971009). The author is grateful to Professor Zhouping Xin and Professor Tao Luo for their constructive suggestions on this work.

Author information

Authors and Affiliations

  1. South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Guangzhou, 510631, China

    Yuan Yuan

Authors
  1. Yuan Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuan Yuan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yuan, Y. Variational rotating solutions to non-isentropic Euler-Poisson equations with prescribed total mass. Sci. China Math. 65, 2061–2078 (2022). https://doi.org/10.1007/s11425-021-1859-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-021-1859-8

Keywords

MSC(2020)

Search

Navigation