Skip to main content
SpringerLink
Log in

On the cauchy problem of 3D compressible, viscous, heat-conductive navier-stokes-Poisson equations subject to large and non-flat doping profile

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

In this paper, we study an initial value problem of the Navier-Stokes-Poisson equations for compressible, viscous, heat-conducting flows on the whole space \(\mathrm{{{\mathbb {R}}}^3}\). The global well-posedness of strong solutions subject to large and non-flat doping profile is established. The initial data is of small energy but possible large oscillations, and the initial density is allowed to contain vacuum states.

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. Chikami, N., Ogawa, T.: Well-posedness of the compressible Navier-Stokes-Poisson system in the critical Besov space. J. Evol. Equ. 17, 717–747 (2017)

    Article  MathSciNet  Google Scholar 

  2. Cho, Y., Choe, H., Kim, H.: Unique solvability of the initial boundary value problems for compressible viscous fluid. J. Math. Pures Appl. 83, 243–275 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Choe, H., Kim, H.: Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J. Differ. Equ. 190, 504–523 (2003)

    Article  MathSciNet  Google Scholar 

  5. Donatelli, D.: Local and global existence for the coupled Navier-Stokes-Poisson problem. Q. Appl. Math. 61, 345–361 (2003)

    Article  MathSciNet  Google Scholar 

  6. B. Ducomet, E. Feireisl, H. Petzeltová, I. Straškraba, Existence globale pour un fluide barotrope autogravitant (Global existence for compressible barotropic self-gravitating fluids), C. R. Acad. Sci., Sér. 1 Math. 322 (7) (2001) 627-632

  7. Ducomet, B., Feireisl, E., Petzeltová, H., Straškraba, I.: Global in time weak solutions for compressible barotropic self-gravitating fluids. Discrete Contin. Dyn. Syst. 11(1), 113–130 (2004)

    Article  MathSciNet  Google Scholar 

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

  9. Feireisl, E., Novotny, A., Petzeltová, H.: On the existence of globally defined weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. 3, 358–392 (2001)

    Article  MathSciNet  Google Scholar 

  10. G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol 1:Linearized Steady Problem, Springer-Verlag, New York, 1994

  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  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. L. Hisao, H.L. Li, Compressible Navier-Stokes-Poisson equations, Acta. Math. Sci. Ser. B (Engl. Ed.) 30 (2010) 1937-1948

  14. Hsiao, L., Ju, Q.C., Wang, S.: The asymptotic behaviour of global smooth solutions to the mulit-dimensional hydrodynamic model for semiconductors. Math. Meth. Appl. Sci. 26, 1187–1210 (2003)

    Article  Google Scholar 

  15. 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  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Huang, X.D., Li, J., Xin, Z.P.: 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 

  18. Kobayashi, T., Suzuki, T.: Weak solutions to the Navier-Stokes-Poisson equation. Adv. Math. Sci. Appl. 18, 141–168 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Li, H.L., Li, J., Xin, Z.P.: Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations. Comm. Math. Phys. 281(2), 401–444 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. J. Li, Z. Liang, On classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum. J.Math. Pures Appl. (9) 102(4) (2014) 640-671

  22. J. Li, Z.P. Xin, Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with vacuum. Ann. PDE 5(1) (2019) Paper No. 7, 37 pp

  23. Lions, P.L.: Math. Topics in Fluid Dynamics, vol. II. Compressible Models, Oxford Science Publication (1998)

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

    Article  MathSciNet  Google Scholar 

  25. Liu, T.P., Xin, Z.P., Yang, T.: Vacuum states for compressible flow. Discrete Contin. Dynam. Sys. 4(1), 1–32 (1998)

    Article  MathSciNet  Google Scholar 

  26. H.R. Liu, H. Zhong, Global solutions to the initial boundary problem of 3-D compressible Navier-Stokes-Poisson on bounded domains. Z. Angew. Math. Phys. 72 (2021), no. 2, Paper No. 78, 30 pp

  27. S.Q. Liu, X.Y. Xu, J.W. Zhang, Global well-posedness of strong solutions with large oscillations and vaccum to the compressible Navier-Stokes-Poisson equations subject to large and non-flat doping profile, J. Differ. Equ. 269 (2020) 8468-8508. (see also Corrigendum to “Global well-posedness of strong solutions with large oscillations and vacuum to the compressible Navier-Stokes-Poisson equations subject to large and non-flat doping profile” [J. Differ. Equ. 269 (10) (2020) 8468-8508]. J. Differential Equations 295 (2021), 292-295.)

  28. 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 

  29. Markowich, P.A., Ringhofer, C.A., Schmeiser, C.: Semiconductor Equations, Springer, (1990)

  30. Nash, J.: Le Problème de Cauchy pour les équations différentielles d’un fluide général. Bull. Soc. Math. France. 90, 487–497 (1962)

    Article  MathSciNet  Google Scholar 

  31. Nirenberg, L.: On the elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 13(3), 115–162 (1959)

    MathSciNet  MATH  Google Scholar 

  32. Serrin, J.: On the uniqueness of compressible fluid motion. Arch. Rational. Mech. Anal. 3, 271–288 (1959)

    Article  MathSciNet  Google Scholar 

  33. Tan, Z., Wang, Y.J.: Global existence and large-time behavior of weak solutions to the compressible magenetohydro-dynamic equations with Coulomb force. Nonlinear Anal. 71, 5866–5884 (2009)

    Article  MathSciNet  Google Scholar 

  34. Tan, Z., Wang, Y.J., Wang, Y.: Stability of steady states of the Navier-Stokes-Poisson equations with non-flat doping profile. SIAM J. Math. Anal. 47, 179–209 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  36. Tani, A.: On the first initial-boundary value problem of compressible viscous fluid motion. Publ. Res. Inst. Math. Sci. 13, 193–253 (1977)

    Article  Google Scholar 

  37. Valli, A.: Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method. Ann. Scuola. Norm. Super. Pisa CI. Sci. IV 10, 607–647 (1983)

    MathSciNet  MATH  Google Scholar 

  38. Wang, Y.J.: Decay of the Navier-Stokes-Poisson equations. J. Differ. Equ. 253, 273–297 (2012)

    Article  MathSciNet  Google Scholar 

  39. Xin, Z.: Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Comm. Pure Appl. Math. 51(3), 229–240 (1998)

    Article  MathSciNet  Google Scholar 

  40. 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, 866–1636 (2011)

    Article  Google Scholar 

  41. Zhang, Y.H., Tan, Z.: On the existence of solutions to the Navier-Stokes-Poisson equations of a two-dimensinal compressible flow. Math. Methods Appl. Sci. 30, 305–329 (2007)

    Article  MathSciNet  Google Scholar 

  42. A. Zlotnik, Uniform estimate and stabilization of symmetric solutions of a system of quasilinear equation, Differ. Uran. (Minsk) 36(5), 634-646 (2000); translation in differential equations, 36 (5), 701-716 (2000)

  43. Zhao, Z.Y., Li, Y.P.: Existence and optimal decay rates of the compressible non-isentropic Navier-Stokes-Poisson models with external forces. Nonlinear Anal. 75, 6130–6147 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, P. R. China

    Xinying Xu, Jianwen Zhang & Minghui Zhong

Authors
  1. Xinying Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianwen Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Minghui Zhong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xinying Xu.

Additional information

Communicated by L. Szekelyhidi.

Publisher's Note

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

The work was partly supported by the National Natural Science Foundation of China (Grant Nos. 11871407, 12071390, 12131007)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, X., Zhang, J. & Zhong, M. On the cauchy problem of 3D compressible, viscous, heat-conductive navier-stokes-Poisson equations subject to large and non-flat doping profile. Calc. Var. 61, 161 (2022). https://doi.org/10.1007/s00526-022-02280-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-022-02280-x

Mathematics Subject Classification

Search

Navigation