Well-posedness for the fifth-order shallow water equations

The Cauchy problems for some kind of fifth-order shallow water equations partial derivative(t)u + alpha partial derivative(5)(x)u + beta partial derivative(3)(x)u + gamma partial derivative(u)(x) + F(u, partial derivative(x)u, partial derivative(2)(x)u) = 0, x, t is an element of R x R, are considered by the Fourier restriction norm method, where nonlinear terms F(u, partial derivative(x)u, partial derivative(2)(x)u) are mu partial derivative(x)(u(k)), k = 2,3, mu u partial derivative(2)(x)u or mu partial derivative(x)u partial derivative(2)(x)u respectively. The local well-posedness is established for data in H-s(R) with s > -7/4 for the Kawahara equation (F = mu partial derivative(x)(u(2))) and is established for data in H-s(R) with s >= -1/4 for the modified Kawahara equation (F = mu partial derivative(x)(u(3))), respectively. Moreover, the local result is established for data in H-s(R) with s > 0 if F = mu u partial derivative(2)(x)u and is established for data in HI(R) with s > -1/4 if F = mu partial derivative(x)u partial derivative(2)(x) u, respectively. (C) 2008 Elsevier Inc. All rights reserved.