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Low regularity solutions of two fifth-order KdV type equations. (English) Zbl 1181.35229
Summary: 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 ${H}^{s}\left(ℝ\right)$ with $s\ge -7/4$ and the local well-posedness for the modified Kawahara equation in ${H}^{s}\left(ℝ\right)$ with $s\ge -1/4$. To prove these results, we derive a fundamental estimate on dyadic blocks for the Kawahara equation through the $\left[k;Z\right]$ multiplier norm method of T. Tao [Am. J. Math. 123, No. 5, 839–908 (2001; Zbl 0998.42005)]. and use this to obtain new bilinear and trilinear estimates in suitable Bourgain spaces.

##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 35A01 Existence problems for PDE: global existence, local existence, non-existence 35A02 Uniqueness problems for PDE: global uniqueness, local uniqueness, non-uniqueness 35B45 A priori estimates for solutions of PDE
##### References:
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