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An optimal decay estimate for the linearized water wave equation in 2D


Author: Aynur Bulut
Journal: Proc. Amer. Math. Soc. 144 (2016), 4733-4742
MSC (2010): Primary 35Q35, 35Q55; Secondary 76B15
Published electronically: July 22, 2016
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Abstract: We obtain a decay estimate for solutions to the linear dispersive equation $ iu_t-(-\Delta )^{1/4}u=0$ for $ (t,x)\in \mathbb{R}\times \mathbb{R}$. This corresponds to a factorization of the linearized water wave equation $ u_{tt}+(-\Delta )^{1/2}u=0$. In particular, by making use of the Littlewood-Paley decomposition and stationary phase estimates, we obtain decay of order $ \vert t\vert^{-1/2}$ for solutions corresponding to data $ u(0)=\varphi $, assuming only bounds on $ \lVert \varphi \rVert _{H_x^1(\mathbb{R})}$ and $ \lVert x\partial _x\varphi \rVert _{L_x^2(\mathbb{R})}$. As another application of these ideas, we give an extension to equations of the form $ iu_t-(-\Delta )^{\alpha /2}u=0$ for a wider range of $ \alpha $.


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Additional Information

Aynur Bulut
Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109-1043
Email: abulut@umich.edu

DOI: https://doi.org/10.1090/proc/12894
Received by editor(s): November 18, 2014
Received by editor(s) in revised form: June 14, 2015
Published electronically: July 22, 2016
Communicated by: Catherine Sulem
Article copyright: © Copyright 2016 American Mathematical Society