Well-posedness and ill-posedness for a fifth-order shallow water wave equation

Abstract

In this paper we establish a new bilinear estimate in suitable Bourgain spaces by using a fundamental estimate on dyadic blocks for the Kawahara equation which was obtained by the [ k ; Z ] multiplier norm method of Tao (2001) [2]; then the local well-posedness of the Cauchy problem for a fifth-order shallow water wave equation in H s ( R ) with s > − 5 4 is obtained by the Fourier restriction norm method. And some ill-posedness in H s ( R ) with s < − 5 4 is derived from a general principle of Bejenaru and Tao.