Well-posedness for the fifth-order shallow water equations

Abstract

The Cauchy problems for some kind of fifth-order shallow water equations ∂ t u + α ∂ x 5 u + β ∂ x 3 u + γ ∂ x u + F ( u , ∂ x u , ∂ x 2 u ) = 0 , x , t ∈ R × R , are considered by the Fourier restriction norm method, where nonlinear terms F ( u , ∂ x u , ∂ x 2 u ) are μ ∂ x ( u k ) , k = 2 , 3 , μ u ∂ x 2 u or μ ∂ x u ∂ x 2 u respectively. The local well-posedness is established for data in H s ( R ) with s > − 7 4 for the Kawahara equation ( F = μ ∂ x ( u 2 ) ) and is established for data in H s ( R ) with s ⩾ − 1 4 for the modified Kawahara equation ( F = μ ∂ x ( u 3 ) ), respectively. Moreover, the local result is established for data in H s ( R ) with s > 0 if F = μ u ∂ x 2 u and is established for data in H s ( R ) with s > − 1 4 if F = μ ∂ x u ∂ x 2 u , respectively.