Electron. J. Differential Equations, Vol. 2020 (2020), No. 24, pp. 1-10.

Almost optimal local well-posedness for modified Boussinesq equations

Dan-Andrei Geba, Bai Lin

Abstract:
In this article, we investigate a class of modified Boussinesq equations, for which we provide first an alternate proof of local well-posedness in the space $(H^s\cap L^\infty)\times (H^s\cap L^\infty)(\mathbb{R})$ ($s\geq 0$) to the one obtained by Constantin and Molinet [7]. Secondly, we show that the associated flow map is not smooth when considered from $H^s\times H^s(\mathbb{R})$ into $H^s(\mathbb{R})$ for s<0, thus providing a threshold for the regularity needed to perform a Picard iteration for these equations.

Submitted June 12, 2019. Published March 19, 2020.
Math Subject Classifications: 35B30, 35Q55.
Key Words: Modified Boussinesq equation; well-posedness; ill-posedness.
DOI: 10.58997/ejde.2020.24

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Dan-Andrei Geba
Department of Mathematics
University of Rochester
Rochester, NY 14627, USA
email: dangeba@math.rochester.edu
Bai Lin
Department of Mathematics
University of Rochester
Rochester, NY 14627, USA
email: blin13@ur.rochester.edu

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