Global well-posedness and decay for viscous water wave models

The motion of the free surface of an incompressible fluid is a very active research area. Most of these works examine the case of an inviscid fluid. However, in several practical applications, there are instances where the viscous damping needs to be considered. In this paper, we derive and study a new asymptotic model for the motion of unidirectional viscous water waves. In particular, we establish the global well-posedness in Sobolev spaces. Furthermore, we also establish the global well-posedness and decay of a fourth order partial differential equation modeling bidirectional water waves with viscosity moving in deep water with or without surface tension effects.The research of S.S. is supported by the European Research Council through the Starting Grant project H2020-EU.1.1.-639227. R.G-B was supported by the project ”Mathematical Analysis of Fluids and Applications” with reference PID2019-109348GA-I00/AEI/10.13039/501100011033 and acronym “MAFyA” funded by Agencia Estatal de Investigación and the Ministerio de Ciencia, Innovación y Universidades (MICIU). Project supported by a 2021 Leonardo Grant for Researchers and Cultural Creators, BBVA Foundation. The BBVA Foundation accepts no responsability for the opinions, statements and contents included in the project and/or the results thereof, which are entirely the responsability of the authors