Skip to main content
SpringerLink
Log in

Global Dynamics of 3D Compressible Viscous and Heat-Conducting Micropolar Fluids with Vacuum at Infinity

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

In this paper, we are concerned with the Cauchy problem of 3D viscous and heat-conducting micropolar fluids with far field vacuum. Compared with the case of non-vacuum at infinity (Huang and Li in Arch Ration Mech Anal 227:995–1059, 2018; Huang et al. in J Math Fluid Mech 23(1):50, 2021), due to \((\rho (t, x), \theta (t, x))\rightarrow (0, 0)\) as \(|x|\rightarrow \infty \), we don’t have useful energy equality (or inequality), which is very important to establish a priori estimates in Huang and Li (Arch Ration Mech Anal 227:995–1059, 2018) and Huang et al. (J Math Fluid Mech 23(1):50, 2021). Thus, a new assumption of a priori estimates and more complicated calculations will be needed. On the other hand, we need to deal with some strong nonlinear terms which come from the interactions of velocity and micro-rotation velocity. Finally, we show that the global existence and uniqueness of strong solutions provided that the initial energy is suitably small. In particular, large-time behavior and a exponential decay rate of the strong solution are obtained, which generalizes the incompressible case (Ye in Dicret Contin Dyn Syst Ser B 24:6725–6743, 2019) to the full compressible case.

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

Data availability

Data sharing is not applicable to this article as new data were created or analyzed in this study.

References

  1. Chen, M.: Global strong solutions for the viscous, micropolar, compressible flow. J. Part. Differ. Equ. 24, 158–164 (2011)

    MathSciNet  Google Scholar 

  2. Chen, M.: Blow up criterion for viscous, compressible micropolar fluids with vacuum. Nonlinear Anal. RWA 13, 850–859 (2012)

    Article  Google Scholar 

  3. Chen, M., Huang, B., Zhang, J.: Blowup criterion for the three-dimensional equations of compressible viscous micropolar fluids with vacuum. Nonlinear Anal. 79, 1–11 (2013)

    Article  MathSciNet  Google Scholar 

  4. Chen, M., Xu, X., Zhang, J.: Global weak solutions of 3D compressible micropolar fluids with discontinuous initial data and vacuum. Commun. Math. Sci. 13, 225–247 (2015)

    Article  MathSciNet  Google Scholar 

  5. Cho, Y., Kim, H.: Existence results for viscous polytropic fluids with vacuum. J. Differ. Equ. 228, 377–411 (2006)

    Article  MathSciNet  Google Scholar 

  6. Cho, Y., Kim, H.: On classical solutions of the compressible Navier–Stokes equations with nonnegative initial densities. Manuscr. Math. 120, 91–129 (2006)

    Article  MathSciNet  Google Scholar 

  7. Eringen, A.C.: Theory of micropolar fluids. J. Math. Mech. 16, 1–18 (1966)

    MathSciNet  Google Scholar 

  8. Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford University Press, Oxford (2004)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Huang, X., Li, J., Xin, Z.: Serrin-type criterion for the three-dimensional viscous compressible flows. SIAM J. Math. Anal. 43, 1872–1886 (2011)

    Article  MathSciNet  Google Scholar 

  11. Huang, X., Li, J., Xin, Z.: Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier–Stokes equations. Commun. Pure Appl. Math. 65, 549–585 (2012)

    Article  MathSciNet  Google Scholar 

  12. Huang, B., Liu, L., Zhang, L.: Global dynamics of 3-D compressible micropolar fluids with vacuum and large oscillations. J. Math. Fluid Mech. 23, 50 (2021)

    Article  MathSciNet  Google Scholar 

  13. Li, J.: Global small solutions of heat conductive compressible Navier–Stokes equations with vaccum: smallness on scaling invariant quantity. Arch. Ration. Mech. Anal. 237, 899–919 (2020)

    Article  MathSciNet  Google Scholar 

  14. Li, J., Xin, Z.: Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier–Stokes equations with vacuum. Ann. PDE 5, 37 (2019)

    Article  MathSciNet  Google Scholar 

  15. Li, J., Zhang, Y.: Local existence and uniqueness of heat conductive compressible Navier–Stokes equations in the presence of vacuum and without initial compatibility conditions. http://arxiv.org/abs/2108.10783

  16. Liang, Z.: Global strong solutions of Navier–Stokes equations for heat-conducting compressible fluids with vacuum at infinity. J. Math. Fluid Mech. 23, 22 (2021)

    Article  MathSciNet  Google Scholar 

  17. Lions, P.L.: Mathematical Topics in Fluid Mechanics, Compressible Models. Oxford University Press, Oxford (1998)

    Google Scholar 

  18. Liu, Q., Zhang, P.: Optimal time decay of the compressible micropolar fluids. J. Differ. Equ. 260(2016), 7634–7661 (2016)

    Article  MathSciNet  Google Scholar 

  19. Liu, Q., Zhang, P.: Long-time behavior of solution to the compressible micropolar fluids with external force. Nonlinear Anal. Real World Appl. 40, 361–376 (2018)

    Article  MathSciNet  Google Scholar 

  20. Łukaszewicz, G.: Micropolar Fluids, Theory and Applications, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston (1999)

    Google Scholar 

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

    MathSciNet  Google Scholar 

  22. Matsumura, A., Nishida, T.: Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Commun. Math. Phys. 89(4), 445–464 (1983)

    Article  MathSciNet  Google Scholar 

  23. Mujaković, N.: One-dimensional flow of a compressible viscous micropolar fluid: regularity of the solution. Radiat. Mater. 10, 181–193 (2001)

    MathSciNet  Google Scholar 

  24. Mujaković, N.: Global in time estimates for one-dimensional compressible viscous micropolar fluid model. Glas. Mat. Ser. III(40), 103–120 (2005)

    Article  MathSciNet  Google Scholar 

  25. Mujaković, N.: One-dimensional flow of a compressible viscous micropolar fluid: the Cauchy problem. Math. Commun. 10, 1–14 (2005)

    MathSciNet  Google Scholar 

  26. Mujaković, N.: Non-homogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: a local existence theorem. Ann. Univ. Ferrara Sez. VII Sci. Mat. 53, 361–379 (2007)

    Article  MathSciNet  Google Scholar 

  27. Novotny, A., Straskraba, I.: Introduction to the Mathematical Theory of Compressible Flow. Oxford University Press, Oxford (2004)

    Book  Google Scholar 

  28. Peng, H., Hou, X.: Global existence for a class of large solution to the three-dimensional micropolar fluid equations with vacuum. J. Math. Anal. Appl. 498, 124931 (2021)

    Article  MathSciNet  Google Scholar 

  29. Talenti, G.: Best constants in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)

    Article  MathSciNet  Google Scholar 

  30. Valli, A.: An existence theorem for compressible viscous fluids. Ann. Mat. Pura Appl. 130, 197–213 (1982)

    Article  MathSciNet  Google Scholar 

  31. 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, 162–221 (2017)

    Article  MathSciNet  Google Scholar 

  32. Wu, Z., Jiang, X.: Pointwise space-time estimates of non-isentropic compressible micropolar fluids. Z. Angew. Math. Phys. 72, 17 (2021)

    Article  MathSciNet  Google Scholar 

  33. Wu, Z., Wang, W.: The pointwise estimates of diffusion wave of the compressible micropolar fluids. J. Differ. Equ. 265, 2544–2576 (2018)

    Article  MathSciNet  Google Scholar 

  34. Xu, H., Zhang, J.: Regularity and uniqueness for the compressible full Navier–Stokes equations. J. Differ. Equ. 272, 46–73 (2021)

    Article  MathSciNet  Google Scholar 

  35. Ye, Z.: Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations. Discret. Contin. Dyn. Syst. Ser. B 24, 6725–6743 (2019)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics, Changchun Normal University, Changchun, 130032, People’s Republic of China

    Siqi Liu, Yang Liu & Nan Zhou

Authors
  1. Siqi Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nan Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yang Liu.

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

Liu, S., Liu, Y. & Zhou, N. Global Dynamics of 3D Compressible Viscous and Heat-Conducting Micropolar Fluids with Vacuum at Infinity. J Geom Anal 34, 243 (2024). https://doi.org/10.1007/s12220-024-01688-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-024-01688-5

Keywords

Mathematics Subject Classification

Search

Navigation