Skip to main content
SpringerLink
Log in

The Cauchy Problem for the Two Layer Viscous Shallow Water Equations

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

In this paper, the Cauchy problem for the two layer viscous shallow water equations is investigated with third-order surface-tension terms and a low regularity assumption on the initial data. The global existence and uniqueness of the strong solution in a hybrid Besov space are proved by using the Littlewood-Paley decomposition and Friedrichs’ regularization method.

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. Bahouri H, Chemin J-Y, Danchin R. Fourier Analysis and Nonlinear Partial Differential Equations. Berlin, Heidelberg: Springer, 2011

    Book  Google Scholar 

  2. Bresch D, Desjardins B. Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Commun Math Phy, 2003, 238: 211–223

    Article  MathSciNet  Google Scholar 

  3. Chen Q, Miao C, Zhang Z. On the well-posedness for the viscous shallow water equations. SIAM J Math Anal, 2008, 40(2): 443–474

    Article  MathSciNet  Google Scholar 

  4. Choboter P F, Swaters G E. Modeling equator-crossing currents on the ocean bottom. Canadian Applied Mathematics Quarterly, 2000, 8(4): 367–385

    Article  MathSciNet  Google Scholar 

  5. Cheng F, Xu C. Analytical smoothing effect of solutions for the Boussinesq equations. Acta Math Sci, 2019, 39B(1): 165–179

    Article  MathSciNet  Google Scholar 

  6. Danchin R. Global existence in critical spaces for compressible Navier-Stokes equations. Invent Math, 2000, 141(3): 579–614

    Article  MathSciNet  Google Scholar 

  7. Danchin R. Global existence in critical spaces for flows of compressible viscous and heat-conductive gases. Arch Rational Mech Anal, 2001, 160: 1–39

    Article  MathSciNet  Google Scholar 

  8. Danchin R. Fourier analysis methods for PDEs. Lecture Notes, 2005

  9. Dellar P J, Salmon R. Shallow water equations with a complete Coriolis force and topography. Phy Fluids, 2005, 17(10): 1–100

    Article  MathSciNet  Google Scholar 

  10. Ferrari S, Saleri F. A new two dimensional shallow-water model including pressure effects and slow varying bottom topography. Math Model Numer Anal, 2004, 38(2): 211–234

    Article  MathSciNet  Google Scholar 

  11. Guo Z, Jiu Q, Xin Z. Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients. SIAM J Math Anal, 2008, 39(5): 1402–1427

    Article  MathSciNet  Google Scholar 

  12. Hao C, Hsiao L, Li H. Cauchy problem for viscous rotating shallow water equations. J Differ Equ, 2013, 247(12): 3234–3257

    Article  MathSciNet  Google Scholar 

  13. Haspot B. Cauchy problem for viscous shallow water equations with a term of capillarity. Math Models Methods Appl Sci, 2010, 20: 1049–1087

    Article  MathSciNet  Google Scholar 

  14. Li H, Li J, Xin Z. Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations. Commun Math Phy, 2008, 281: 401–444

    Article  MathSciNet  Google Scholar 

  15. Muñoz-Ruiz M L. On a non-homogeneous bi-layer shallow water problem: smoothness and uniqueness results. Nonlinear Anal, 2004, 59(3): 253–282

    Article  MathSciNet  Google Scholar 

  16. Narbona-Reina G, Zabsonre J D, Fernández-Nieto E D, Bresch D. Derivation of a bilayer model for shallow water equations with viscosity. Cmes Comput Model Engin Ences, 2009, 43(1): 27–71

    MathSciNet  MATH  Google Scholar 

  17. Qin H, Xie C, Fang S. Remarks on regularity criteria for 3D generalized MHD equations and Boussinesq equations. Acta Math Sci, 2019, 39A(2): 316–328

    MathSciNet  MATH  Google Scholar 

  18. Roamba B, Zabsonre J D. A bidimensional bi-layer shallow-water model. Elec J Differ Equ, 2017, 168: 1–19

    MathSciNet  MATH  Google Scholar 

  19. Vallis G K. Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. Cambridge: Cambridge University Press, 1996

    MATH  Google Scholar 

  20. Wang W, Xu C. The cauchy problem for viscous shallow water equations. Revista Matematica Iberoamericana, 2005, 21(1): 1–24

    Article  MathSciNet  Google Scholar 

  21. Zabsonre J D, Narbona-Reina G. Existence of a global weak solution for a 2D viscous bi-layer Shallow Water model. Nonlinear Analysis: Real World Applications, 2009, 10(5): 2971–2984

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China

    Pengcheng Mu

  2. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Qiangchang Ju

Authors
  1. Pengcheng Mu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qiangchang Ju
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qiangchang Ju.

Additional information

Ju was supported by the NSFC (11571046, 11671225), the ISF-NSFC joint research program NSFC (11761141008) and the BJNSF (1182004).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mu, P., Ju, Q. The Cauchy Problem for the Two Layer Viscous Shallow Water Equations. Acta Math Sci 40, 1783–1807 (2020). https://doi.org/10.1007/s10473-020-0612-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-020-0612-9

Key words

2010 MR Subject Classification

Search

Navigation