Low regularity solutions of two fifth-order KdV type equations

Details Low regularity solutions of two fifth-order KdV type equations Low regularity solutions of two fifth-order KdV type equations

2007-10-15

arxiv.org/abs/0710.2704v3

Journal D'Analyses Mathematique, Vol.107(2009)221-238

Mathematics

Analysis of PDEs

17pages

Scientific paper

10.1007/s11854-009-0009-0

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.

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