Strichartz estimates for dispersive equations and solvability of the Kawahara equation

We first establish a series of Strichartz estimates for a general class of linear dispersive equations by applying the theory of oscillatory integrals established by Kenig, Ponce and Vega. Next we use such estimates to study solvability of the Cauchy problem of the Kawahara equation ∂tu+au∂xu+β∂x3u+γ∂x5u=0 in the class C(R,Hs(R)). Local existence is proved for s>1/4 and global existence is proved for s⩾2.