Skip to main content
SpringerLink
Log in

Local Existence and Uniqueness of Strong Solutions to the Two Dimensional Nonhomogeneous Incompressible Primitive Equations

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

In this article, we study the initial boundary value problem of the two-dimensional nonhomogeneous incompressible primitive equations and obtain the local existence and uniqueness of strong solutions. The initial vacuum is allowed.

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. Bresch D, Kazhikhov A V, Lemoine J. On the two-dimensional hydrostatic Navier-Stokes equations. SIAM J Math Anal, 2004, 36: 796–814

    Article  MathSciNet  Google Scholar 

  2. Bihari I. A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations. Acta Math Acad Sci Hungar, 1956, 7: 81–94

    Article  MathSciNet  Google Scholar 

  3. Choe H, Kim H. Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids. Comm Partial Differential Equations, 2003, 28: 1183–1201

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Cao C, Titi E S. Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics. Ann Math, 2007, 166: 245–267

    Article  MathSciNet  Google Scholar 

  6. Cao C, Titi E S. Global well-posedness of the 3D primitive equations with partial vertical turbulence mixing heat diffusion. Comm Math Phys, 2012, 310: 537–568

    Article  MathSciNet  Google Scholar 

  7. Cao C, Li J, Titi E S. Global well-posedness of strong solutions to the 3D primitive equations with horizontal eddy diffusivity. J Differential Equations, 2014, 257(11): 4108–4132

    Article  MathSciNet  Google Scholar 

  8. Cao C, Li J, Titi E S. Local and global well-posedness of strong solutions to the 3D primitive equations with vertical eddy diffusivity. Arch Ration Mech Anal, 2014, 214(1): 35–76

    Article  MathSciNet  Google Scholar 

  9. Cao C, Ibrahim S, Nakanishi K, et al. Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics. Commun Math Phys, 2015, 337(2): 473–482

    Article  MathSciNet  Google Scholar 

  10. Cao C, Li J, Titi E S. Global well-posedness of the three-dimensional primitive equations with only horizontal viscosity and diffusion. Comm Pure Appl Math, 2016, 69(8): 1492–1531

    Article  MathSciNet  Google Scholar 

  11. Cao C, Li J, Titi E S. Strong solutions to the 3D primitive equations with only horizontal dissipation: near H1 initial data. J Funct Anal, 2017, 272(11): 4606–4641

    Article  MathSciNet  Google Scholar 

  12. Ersoy M, Ngom T. Existence of a global weak solution to compressible primitive equations. C R Math Acad Sci Paris, 2012, 350: 379–382

    Article  MathSciNet  Google Scholar 

  13. Ersoy M, Ngom T, Sy M. Compressible primitive equations: formal derivation and stability of weak solutions. Nonlinearity, 2011, 24: 79–96

    Article  MathSciNet  Google Scholar 

  14. Gatapov B V, Kazhikhov A V. Existence of a global solution to one model problem of atmosphere dynamics. Sibirsk Mat Zh, 2005, 46: 1011–1020

    MATH  Google Scholar 

  15. Guillén-González F, Rodríguez-Bellido M A. On the strong solutions of the primitive equations in 2D domains. Nonlinear Anal Ser A: Theory Methods, 2002, 50: 621–646

    Article  MathSciNet  Google Scholar 

  16. Jiu Q, Li M, Wang F. Uniqueness of the global weak solutions to 2D compressible primitive equations. J Math Anal Appl, 2018, 461(2): 1653–1671

    Article  MathSciNet  Google Scholar 

  17. Tang T, Gao H. On the stability of weak solution for compressible primitive equations. Acta Appl Math, 2015, 140: 133–145

    Article  MathSciNet  Google Scholar 

  18. Li J, Titi E S. Existence and uniqueness of weak solutions to viscous primitive equations for a certain class of discontinuous initial data. SIAM J Math Anal, 2017, 49(1): 1–28

    Article  MathSciNet  Google Scholar 

  19. LaSalle J. Uniqueness theorems and successive approximations. Ann Math, 1949, 50: 722–730

    Article  MathSciNet  Google Scholar 

  20. Lions J L, Temam R, Wang S. New formulations for the primitive equations for the atmosphere and applications. Nonlinearity, 1992, 5(2): 237–288

    Article  MathSciNet  Google Scholar 

  21. Lions P L. Mathematical Topics in Fluid Mechanics, Vol 1: Incompressible Model. Oxford Lecture Series in Mathematic and Its Applications, Vol 3, Oxford Science Publications. New York: The Clarendon Press, Oxford University Press. 1966

    Google Scholar 

  22. Pedlosky J. Geophysical Fluid Dynamics. 2nd ed. New-York: Springer-Verlag, 1987

    Book  Google Scholar 

  23. Temam R, Ziane M. Some mathematical problems in geophysical fluid dynamics//Handbook of Mathematical Fluid Dynamics. 2004

  24. Ziane M. Regularity results for Stokes type systems. Appl Anal, 1995, 58: 263–292

    Article  MathSciNet  Google Scholar 

  25. Wang F, Dou C, Jiu Q. Global Weak solutions to 3D compressible primitive equations with density-dependent viscosity. J Math Phys, 2020, 61(2): 021507, 33

    Article  MathSciNet  Google Scholar 

  26. Guo B, Huang D. On the 3D viscous primitive equations of the large-scale atmosphere. Acta Math Sci, 2009, 29B(4): 846–866

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Professor Li Jinkai not only for posing the problem, but also for making many valuable suggestions.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

    Quansen Jiu

  2. Advanced Institute of Natural Sciences, Beijing Normal University at Zhuhai, Zhuhai, 519087, China

    Fengchao Wang

  3. School of Mathematical Sciences, Beijing Normal University and Key Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, China

    Fengchao Wang

Authors
  1. Quansen Jiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fengchao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fengchao Wang.

Additional information

Q.S. Jiu was partially supported by the National Natural Science Foundation of China (11671273 and 11931010), key research project of the Academy for Multidisciplinary Studies of CNU and Beijing Natural Science Foundation (1192001)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jiu, Q., Wang, F. Local Existence and Uniqueness of Strong Solutions to the Two Dimensional Nonhomogeneous Incompressible Primitive Equations. Acta Math Sci 40, 1316–1334 (2020). https://doi.org/10.1007/s10473-020-0510-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-020-0510-1

Key words

2010 MR Subject Classification

Search

Navigation