On the Well-Posedness for the Viscous Shallow Water Equations

In this paper, we prove the existence and uniqueness of the solutions for the two-dimensional viscous shallow water equations with low regularity assumptions on the initial data as well as the initial height bounded away from zero.

[1]  R. Danchin.  Density-dependent incompressible viscous fluids in critical spaces , 2003, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[2]  Marco Cannone,et al.  Chapter 3 - Harmonic Analysis Tools for Solving the Incompressible Navier–Stokes Equations , 2005 .

[3]  R. Danchin.  LOCAL THEORY IN CRITICAL SPACES FOR COMPRESSIBLE VISCOUS AND HEAT-CONDUCTIVE GASES , 2001 .

[4]  Linda Sundbye.  Global Existence for the Dirichlet Problem for the Viscous Shallow Water Equations , 1996 .

[5]  J. Chemin,et al.  Théorèmes d’unicité pour le système de navier-stokes tridimensionnel , 1999 .

[6]  Martin Greiner,et al.  Wavelets , 2018, Complex..

[7]  Raphaël Danchin,et al.  Global existence in critical spaces for compressible Navier-Stokes equations , 2000 .

[8]  Hiroshi Fujita,et al.  On the Navier-Stokes initial value problem. I , 1964 .

[9]  R. Danchin.  Global Existence in Critical Spaces¶for Flows of Compressible¶Viscous and Heat-Conductive Gases , 2001 .

[10]  Chao-Jiang Xu,et al.  The Cauchy problem for viscous shallow water equations , 2005 .

[11]  Hajer Bahouri,et al.  Equations de transport relatives à des champs de vecteurs non-lipschitziens et mécanique des fluides , 1994 .

[12]  Yves Meyer,et al.  Wavelets, Paraproducts, and Navier-Stokes Equations , 1996 .

[13]  J. Chemin,et al.  Équations d'ondes quasilinéaires et estimations de Strichartz , 1999 .

[14]  Marco Cannone,et al.  Ondelettes, paraproduits et navier-stokes , 1995 .

[15]  Jean-Yves Chemin,et al.  Perfect Incompressible Fluids , 1998 .

[16]  various.  Current Developments in Mathematics , 2008 .

[17]  H. Triebel.  Theory Of Function Spaces , 1983 .

[18]  Linda Sundbye.  Global Existence for the Cauchy Problem for the Viscous Shallow Water Equations , 1998 .

[19]  Raphaël Danchin,et al.  On the uniqueness in critical spaces for compressible Navier-Stokes equations , 2005 .

[20]  Winfried Sickel,et al.  Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations , 1996, de Gruyter series in nonlinear analysis and applications.

[21]  J. Bony,et al.  Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires , 1980 .

[22]  N. Lerner,et al.  Flow of Non-Lipschitz Vector-Fields and Navier-Stokes Equations , 1995 .

[23]  Didier Bresch,et al.  Recent mathematical results and open problems about shallow water equations. Analysis and simulation of fluid dynamic , 2006 .