Well-posedness for the fifth-order shallow water equations

The Cauchy problems for some kind of fifth-order shallow water equations∂tu+α∂x5u+β∂x3u+γ∂xu+F(u,∂xu,∂x2u)=0,x,t∈R×R, are considered by the Fourier restriction norm method, where nonlinear terms F(u,∂xu,∂x2u) are μ∂x(uk)μ∂x(uk), k=2,3k=2,3, μu∂x2u or μ∂xu∂x2u respectively. The local well-posedness is established for data in Hs(R)Hs(R) with s>−74 for the Kawahara equation (F=μ∂x(u2)F=μ∂x(u2)) and is established for data in Hs(R)Hs(R) with s⩾−14 for the modified Kawahara equation (F=μ∂x(u3)F=μ∂x(u3)), respectively. Moreover, the local result is established for data in Hs(R)Hs(R) with s>0s>0 if F=μu∂x2u and is established for data in Hs(R)Hs(R) with s>−14 if F=μ∂xu∂x2u, respectively.