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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 $H^s (\Bbb R)$ with $s\geq -7/4$ and the local well-posedness for the modified Kawahara equation in $H^s (\Bbb R)$ with $s\geq -1/4$. To prove these results, we derive a fundamental estimate on dyadic blocks for the Kawahara equation through the $[k; Z]$ multiplier norm method of {\it 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.

35Q53KdV-like (Korteweg-de Vries) equations
35A01Existence problems for PDE: global existence, local existence, non-existence
35A02Uniqueness problems for PDE: global uniqueness, local uniqueness, non-uniqueness
35B45A priori estimates for solutions of PDE
Full Text: DOI arXiv
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