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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bahouri H, Chemin J-Y, Danchin R. Fourier Analysis and Nonlinear Partial Differential Equations. Berlin, Heidelberg: Springer, 2011
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
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
Choboter P F, Swaters G E. Modeling equator-crossing currents on the ocean bottom. Canadian Applied Mathematics Quarterly, 2000, 8(4): 367–385
Cheng F, Xu C. Analytical smoothing effect of solutions for the Boussinesq equations. Acta Math Sci, 2019, 39B(1): 165–179
Danchin R. Global existence in critical spaces for compressible Navier-Stokes equations. Invent Math, 2000, 141(3): 579–614
Danchin R. Global existence in critical spaces for flows of compressible viscous and heat-conductive gases. Arch Rational Mech Anal, 2001, 160: 1–39
Danchin R. Fourier analysis methods for PDEs. Lecture Notes, 2005
Dellar P J, Salmon R. Shallow water equations with a complete Coriolis force and topography. Phy Fluids, 2005, 17(10): 1–100
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
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
Hao C, Hsiao L, Li H. Cauchy problem for viscous rotating shallow water equations. J Differ Equ, 2013, 247(12): 3234–3257
Haspot B. Cauchy problem for viscous shallow water equations with a term of capillarity. Math Models Methods Appl Sci, 2010, 20: 1049–1087
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
Muñoz-Ruiz M L. On a non-homogeneous bi-layer shallow water problem: smoothness and uniqueness results. Nonlinear Anal, 2004, 59(3): 253–282
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
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
Roamba B, Zabsonre J D. A bidimensional bi-layer shallow-water model. Elec J Differ Equ, 2017, 168: 1–19
Vallis G K. Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. Cambridge: Cambridge University Press, 1996
Wang W, Xu C. The cauchy problem for viscous shallow water equations. Revista Matematica Iberoamericana, 2005, 21(1): 1–24
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
Author information
Authors and Affiliations
Corresponding author
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
About this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-020-0612-9