An optimal decay estimate for the linearized water wave equation in 2D
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- by Aynur Bulut
- Proc. Amer. Math. Soc. 144 (2016), 4733-4742
- DOI: https://doi.org/10.1090/proc/12894
- 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 $|t|^{-1/2}$ 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 $\alpha$.References
- T. Alazard, N. Burq, and C. Zuily, On the water-wave equations with surface tension, Duke Math. J. 158 (2011), no. 3, 413–499. MR 2805065, DOI 10.1215/00127094-1345653
- T. Alazard and J. M. Delort, Global solutions and asymptotic behavior for two dimensional gravity water waves. Preprint (2013), arXiv:1305.4090.
- T. Alazard and J. M. Delort, Sobolev estimates for two dimensional gravity water waves. Preprint (2013), arXiv:1307.3836.
- J. Beichman, Nonstandard Dispersive Estimates and Linearized Water Waves. Ph.D. Thesis (2013) University of Michigan.
- J. Beichman, Nonstandard estimates for a class of 1D dispersive equations and applications to linearized water waves. Preprint (2014). arXiv:1409.8088.
- Hans Christianson, Vera Mikyoung Hur, and Gigliola Staffilani, Strichartz estimates for the water-wave problem with surface tension, Comm. Partial Differential Equations 35 (2010), no. 12, 2195–2252. MR 2763354, DOI 10.1080/03605301003758351
- P. Germain, N. Masmoudi, and J. Shatah, Global solutions for the gravity water waves equation in dimension 3, Ann. of Math. (2) 175 (2012), no. 2, 691–754. MR 2993751, DOI 10.4007/annals.2012.175.2.6
- J. Hunter, M. Ifrim, and D. Tataru, Two dimensional water waves in holomorphic coordinates. Preprint. (2014), arXiv:1401.1252.
- M. Ifrim and D. Tataru, Two dimensional water waves in holomorphic coordinates. II: global solutions. Preprint. (2014), arXiv:1404.7583.
- Alexandru D. Ionescu and Fabio Pusateri, Nonlinear fractional Schrödinger equations in one dimension, J. Funct. Anal. 266 (2014), no. 1, 139–176. MR 3121725, DOI 10.1016/j.jfa.2013.08.027
- Alexandru D. Ionescu and Fabio Pusateri, Global solutions for the gravity water waves system in 2d, Invent. Math. 199 (2015), no. 3, 653–804. MR 3314514, DOI 10.1007/s00222-014-0521-4
- A. Ionescu and F. Pusateri, Global analysis of a model for capillary water waves in 2D. Preprint (2014), arXiv:1406.6042.
- Sijue Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math. 177 (2009), no. 1, 45–135. MR 2507638, DOI 10.1007/s00222-009-0176-8
- Sijue Wu, Global wellposedness of the 3-D full water wave problem, Invent. Math. 184 (2011), no. 1, 125–220. MR 2782254, DOI 10.1007/s00222-010-0288-1
Bibliographic Information
- Aynur Bulut
- Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109-1043
- MR Author ID: 913497
- Email: abulut@umich.edu
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4733-4742
- MSC (2010): Primary 35Q35, 35Q55; Secondary 76B15
- DOI: https://doi.org/10.1090/proc/12894
- MathSciNet review: 3544525