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Well-posedness of water wave model with viscous effects


Authors: Rafael Granero-Belinchón and Stefano Scrobogna
Journal: Proc. Amer. Math. Soc. 148 (2020), 5181-5191
MSC (2010): Primary 35Q35, 35R35, 35Q31, 35L25
DOI: https://doi.org/10.1090/proc/15219
Published electronically: September 18, 2020
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Abstract: Starting from the paper by Dias, Dyachenko, and Zakharov
(Physics Letters A, 2008) on viscous water waves, we derive a model that describes water waves with viscosity moving in deep water with or without surface tension effects. This equation takes the form of a nonlocal fourth order wave equation and retains the main contributions to the dynamics of the free surface. Then, we prove the well-posedness in Sobolev spaces of such an equation.


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Rafael Granero-Belinchón
Affiliation: Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Avenida. Los Castros s/n, Santander, Spain
ORCID: 0000-0003-2752-8086
Email: rafael.granero@unican.es

Stefano Scrobogna
Affiliation: IMUS, University of Seville, 41012 Sevilla, Spain; and Basque Center of Applied Mathematics, Mazarredo 14, 48009, Bilbao, Spain
MR Author ID: 1229361
Email: sscrobogna@bcamath.org

DOI: https://doi.org/10.1090/proc/15219
Received by editor(s): November 5, 2019
Received by editor(s) in revised form: March 31, 2020
Published electronically: September 18, 2020
Additional Notes: The first author was funded by the grant MTM2017-89976-P from the Spanish government.
The research of the second author was supported by the Basque Government through the BERC 2018-2021 program and by the Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and through project MTM2017-82184-R funded by (AEI/FEDER, UE) and “DESFLU” and by the European Research Council through the Starting Grant project H2020-EU.1.1.-639227 FLUID-INTERFACE
Communicated by: Catherine Sulem
Article copyright: © Copyright 2020 American Mathematical Society