Low regularity solutions of two fifth-order KDV type equations

• DOI: 10.1007/s11854-009-0009-0
• Publication Date: 2009-03-31T08:34:42Z
• Authors: Wengu Chen; Junfeng Li; Changxing Miao; Jiahong Wu
• Keywords: Mathematics - Analysis of PDEs; FOS: Mathematics; FOS: Physical sciences; Mathematical Physics (math-ph); 35Q53, 35B30, 76B45; 0101 mathematics; 01 natural sciences; Mathematical Physics; Analysis of PDEs (math.AP)

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 $s>-\frac74$ and the local well-posedness for the modified Kawahara equation in $H^s({\mathbf R})$ with $s\ge-\frac14$. 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.<br/>17pages<br/>