Dispersive estimates for linearized water wave type equations in \mathbb R^d

2021/06/04 by Deneke, Tilahun, Dufera, Tamirat T., Tesfahun, Achenef
#35A01 #35Q35 #5Q53 #76B03 #76B15 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2106.02717

Abstract

We derive a L¹_x (\mathbb R^d)-L^∞_x ( \mathbb R^d) decay estimate of order \mathcal O ( t^-d/2) for the linear propagators exp ( ± it √( |D|(1+ β|D|²) \tanh |D | ) ), β∈ \0, 1\. D = -i∇, with a loss of 3d/4 or d/4-derivatives in the case β=0 or β=1, respectively. These linear propagators are known to be associated with the linearized water wave equations, where the parameter β measures surface tension effects. As an application we prove low regularity well-posedness for a Whitham-Boussinesq type system in \mathbb R^d, d≥ 2. This generalizes a recent result by Dinvay, Selberg and the third author where they proved low regularity well-posedness in \mathbb R and \mathbb R².

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