Cauchy problem for viscous shallow water equations Weike WANG

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Cauchy problem for viscous shallow water equations Weike WANG ? Department of Mathematics, Shanghai Jiao Tong University 200030 Shanghai, China Chao-Jiang XU Mathematiques UMR 6085, Universite de Rouen 76821 Mont Saint Aignan, France Abstract : In this paper, we study the Cauchy problems for viscous shallow water equations. We work in the Sobolev spaces of index s > 2, we obtain the local solutions for any initial data, and global solution for small initial data. Key words Shallow water equation, Littlewood-Paley decomposition, global solution. A.M.S. Classification 35Q, 76D 1 Introduction We consider in this work the Cauchy problems for viscous shallow water equations as follows: h(ut + (u · ?)u)? ?? · (h?u) + h?h = 0, (1.1) ht + div(hu) = 0, (1.2) u|t=0 = u0, h|t=0 = h0; (1.3) where h(x, t) is the height of fluid surface, u(x, t) = (u1(x, t), u2(x, t))t is the horizontal velocity field, x = (x1, x2) ? R2, and 0 < ? < 1 is the viscous coefficients. The equations form a quasi-linear hyperbolic-parabolic system.