Abstract
The Kawahara and modified Kawahara equations are fifth-order KdV type equations that 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 the Kawahara equation in H s (R) with s ≥ − 7/4 and the local well-posedness for the modified Kawahara equation in H s (R) with s ≥ − 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 [14] and use this to obtain new bilinear and trilinear estimates in suitable Bourgain spaces.
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Chen, W., Li, J., Miao, C. et al. Low regularity solutions of two fifth-order KDV type equations. J Anal Math 107, 221–238 (2009). https://doi.org/10.1007/s11854-009-0009-0
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DOI: https://doi.org/10.1007/s11854-009-0009-0