Almost optimal local well-posedness for modified Boussinesq equations

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 (Hs ∩ L∞) x (Hs ∩ L∞)(ℝ) (s ≥ 0) to the one obtained by Constantin and Molinet [7]. Secondly, we show that the associated flow map is not smooth when considered from Hs x Hs(ℝ) into Hs(ℝ) for s < 0, thus providing a threshold for the regularity needed to perform a Picard iteration for these equations.