Skip to main content
SpringerLink
Log in

Global strong solutions to the 3D non-isentropic compressible Navier–Stokes-Poisson equations in bounded domains

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

This paper is concerned with initial-boundary-value problems to the 3D non-isentropic compressible Naiver-Stokes-Poisson equations, where the velocity admits slip boundary condition. For small initial energy, strong solutions are proved to exist globally in time. We overcome the difficulties caused by the domain by establishing the time-uniform higher-order norms of the absolute temperature. To this end, we first bound \(L^2(0,T;L^2)\)-norm of the Poisson term, then obtain \(L^p\)-norm of the gradient of the density by means of effective viscous flux. In particular, the exponential decay rate of the \(L^2\)-norm of solutions is obtained when the absolute temperature satisfies the Dirichlet boundary condition.

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. Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Commun. Pure Appl. Math. 17(1), 35–92 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bella, P.: Long time behavior of weak solutions to Navier-Stokes-Poisson system. J. Math. Fluid Mech. 14, 279–294 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bourguignon, J.P., Brezis, H.: Remarks on the Euler equation. J. Funct. Anal. 15, 341–363 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bresch, D., Desjardins, B.: On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids. J. Math. Pures Appl. 87(9), 57–90 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cai, G., Li. J.: Existence and exponential growth of global classical solutions to the compressible Navier-Stokes equations with slip boundary conditions in 3D bounded domains. arXiv:2102.06348

  6. Chen, Y. Z., Huang, B., Shi, X. D.: Global well-posedness of classical solutions to the compressible Navier-Stokes-Poisson equations with slip boundary conditions in 3D bounded domains. arXiv:2102.07938v2

  7. Cho, Y., Kim, H.: Strong solutions of the Naiver-Stokes equations for isentropic compressible fluids. J. Differ. Equ. 190, 504–523 (2003)

    Article  Google Scholar 

  8. Ducomet, B., Feireisl, E., Petzeltova, H., Straskraba, I.: Global in time solutions for compressible barotropic self-gravitating fluids. Discret Contin. Dyn. Syst. 11(1), 113–130 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford Univ. Press, Oxford (2004)

    MATH  Google Scholar 

  10. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, 2nd edn. Springer-Verlag, New York (2011)

    Book  MATH  Google Scholar 

  11. Hao, C.C., Li, H.L.: Global existence for compressible Navier-Stokes-Poisson equations in three and higher dimensions. J. Differ. Equ. 246, 4791–4812 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hoff, D.: Discontinuous solutions of the Navier-Stokes equations for multidimensional flows of heat-conducting fluids. Arch. Rational Mech. Anal. 139(4), 303–354 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hoff, D.: Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data. Trans. Am. Math. Soc. 303, 169 (1987)

    Article  MATH  Google Scholar 

  14. Huang, X.D., Li, J.: Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vaccum and large oscillations. Arch. Rational Mech. Anal. 227, 995–1059 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kazhikhov, A.V., Shelukhin, V.V.: Unique global solution with respect to time of initial boundary value problems for one-dimensional equations of a viscous gas. J. Appl. Math. Mech. 41, 273–282 (1977)

    Article  MathSciNet  Google Scholar 

  16. Li, J.K.: Global small solutions of heat conductive compressible Naiver-Stokes equations with vacuum: smallness on scaling invariant quantity. Arch. Rational Mech. Anal. 237, 899–919 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, H., Matsumura, A., Zhang, G.: Optimal decay rate of the compressible Navier-Stokes-Poisson system in \(\mathbb{R} ^3\). Arch. Rational Mech. Anal. 196, 681–713 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu, H.R., Luo, T., Zhong, H.: Global solutions to compressible Navier-Stokes-Poisson and Euler-Poisson equations of plasma on exterior domains. J. Differ. Equ. 269, 9936–10001 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, H.R., Zhong, H.: Global solutions to the initial boundary problem of 3-D compressible Navier-Stokes-Poisson on boundary domain. Zeitschrift für angewandte Math. Phys. 72(2), 1–30 (2021)

    Google Scholar 

  20. Matsumura, A., Nishida, T.: The initial value problem for the equation of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20, 67–104 (1980)

    MathSciNet  MATH  Google Scholar 

  21. Mellet, A., Vasseur, A.: Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations. SIAM J. Math. Anal. 39, 1344–1365 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa. 13(2), 115–162 (1959)

    MathSciNet  MATH  Google Scholar 

  23. Novotny, A., Straskraba, I.: Introduction to the mathematical theory of compressible flow. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  24. Tan, Z., Yang, T., Zhao, H.J., Zou, Q.Y.: Global solutions to the one-dimensional compressible Navier-Stokes-Poisson equations with large data. SIAM J. Math. Anal. 45, 537–571 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Tan, Z., Wu, G.C.: Global existence for the non-isentropic compressible Navier-Stokes-Poisson system in three and higher dimensions. Nonlinear Anal. 13(2), 650–664 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tan, Z., Zhang, X.: Decay rate of the non-isentropic Navier-Stokes-Poisson equations. J. Math. Anal. Appl. 400, 293–303 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. von Wahl, W.: Estimating \(\nabla u\) by \({\rm div}u\) and \({\rm curl}u\). Math. Methods Appl. Sci. 15, 123–143 (1992)

    MathSciNet  Google Scholar 

  28. Wen, H., Zhu, C.: Global solutions to the three-dimensional full compressible Navier-Stokes equations with vacuum at infinity in some classes of large data. SIAM J. Math. Anal. 49(1), 162–221 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yin, J.P., Tan, Z.: Local existence of the strong solutions for the full Navier-Stokes-Poisson equations. Nonlinear Anal. 71(7–8), 2397–2415 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yu, H.B., Zhang, P.X.: Global strong solutions to the 3D full compressible Navier-Stokes equations with density-temperature-dependent viscosities in bounded domains. J. Differ. Equ. 268, 7286–7310 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhang, G.J., Li, H.L., Zhu, C.J.: Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in \(\mathbb{R} ^3\). J. Differ. Equ. 250, 866–891 (2011)

    Article  MATH  Google Scholar 

  32. Zheng, X.X.: Global well-posedness for the compressible Navier-Stokes-Poisson system in the \(L^p\) framework. Nonlinear Anal. 75(10), 4156–4175 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

HY is supported by Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (Grant No. ZQN-901). XS is supported by National Natural Science Foundation of China (Grant Nos. 12061037 and 41801219) and High-level Personnel of Special Support Program of Xiamen University of Technology (Grant No. 4010520009).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Huaqiao University, Quanzhou, 362021, People’s Republic of China

    Huayu Chen & Haibo Yu

  2. School of Mathematical Sciences, Xiamen University of Techonology, Xiamen, 361024, People’s Republic of China

    Xin Si

Authors
  1. Huayu Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xin Si
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Haibo Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Haibo Yu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, H., Si, X. & Yu, H. Global strong solutions to the 3D non-isentropic compressible Navier–Stokes-Poisson equations in bounded domains. Z. Angew. Math. Phys. 74, 100 (2023). https://doi.org/10.1007/s00033-023-01999-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-023-01999-7

Keywords

Mathematics Subject Classification

Search

Navigation