Publication Cover
Applicable Analysis
An International Journal
Volume 95, 2016 - Issue 1
219
Views
9
CrossRef citations to date
0
Altmetric
Original Articles

Global existence and local well-posedness of the 2D viscous shallow water system in Sobolev spaces

&
Pages 78-96 | Received 05 Aug 2014, Accepted 10 Dec 2014, Published online: 07 Jan 2015

References

  • Bresch D, Desjardins B. Existence of global weak solutions for 2D viscous shallow water equations and convergence to the quasi-geostophic model. Commun. Math. Phys. 2003;238:211–233.
  • Bresch D, Desjardins B, Metivier G. Recent mathematical results and open problem about shallow water equations. In: Calgaro C, Coulombel J-F, Goudon T, editors. Analysis and simulation of fluid dynamics. Advances in mathematical fluid mechanics. Basel: Birkhäuser Verlag; 2006. p. 15–31.
  • Bui A-T. Existence and uniqueness of a classical solution of an initial boundary value problem of the theory of shallow waters. SIAM J. Math. Anal. 1981;12:229–241.
  • Kloeden P-E. Global existence of classic solution in the dissipative shallow water equations. SIAM J. Math. Anal. 1985;16:301–315.
  • Sondbye L. Global existence for Dirichlet problem for the viscous shallow water equations. J. Math. Anal. Appl. 1996;202:236–258.
  • Matsumura A, Nishida T. The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 1980;20:67–104.
  • Sondbye L. Global existence for the Cauchy problem for the viscous shallow water equations. Rocky Mountain J. Math. 1998;28:1135–1152.
  • Wang W, Xu C-J. The Cauchy problem for viscous shallow water equations. Rev. Mat. Iberoamericana. 2005;21:1–24.
  • Haspot B. Cauchy problem for viscous shallow water equations with a term of capillarity. In: Tadmor E, Liu J-G, Tzavaras AE, editors. Hyperbolic problems: theory, numerics and applications. Part 2. Vol. 67, Proceedings of symposia in applied mathematics. Providence (RI): American Mathematical Society; 2009. p. 625–634.
  • Chen Q, Miao C, Zhang Z. Well-posedness for the viscous shallow water equations in critical spaces. SIAM J. Math. Anal. 2008;40:443–474.
  • Liu Y, Yin Z. Global existence and well-posedness of 2D viscous shallow water system in Sobolev spaces with low regularity. arXiv:1411.0461.
  • Bahouri H, Chemin J-Y, Danchin R. Fourier analysis and nonlinear partial differential equations. Vol. 343, Grundlehren der mathematischen Wissenschaften. Berlin: Springer; 2011. p. 51–164.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.