Low regularity solutions of two fifth-order KdV type equations

Abstract

The Kawahara and modified Kawahara equations are fifth-order KdV type equations and have been derived to model many physical phenomena such as gravity-capillary waves and magneto-sound propagation in plasmas. This paper establishes the local well-posedness of the initial-value problem for Kawahara equation in $H^s(\mathbf{R})$ with $s>-\frac{7}{4}$ and the local well-posedness for the modified Kawahara equation in $H^s(\mathbf{R})$ with $s\ge -\frac{1}{4}$. To prove these results, we derive a fundamental estimate on dyadic blocks for the Kawahara equation through the $[k;Z]$ multiplier norm method of Tao \cite{Tao2001} and use this to obtain new bilinear and trilinear estimates in suitable Bourgain spaces.

17pages