On the mathematical description of time-dependent surface water waves

TL;DR

This paper surveys the mathematical theory of water waves, covering modeling, well-posedness results, reduced models, and their rigorous justification across different regimes.

Contribution

It provides a comprehensive overview of recent developments in the mathematical analysis and modeling of water waves, including rigorous validation of simplified models.

Findings

Overview of local and global well-posedness results

Discussion of reduced models in various regimes

Rigorous justification of model validity

Abstract

This article provides a survey on some main results and recent developments in the mathematical theory of water waves. More precisely, we briefly discuss the mathematical modeling of water waves and then we give an overview of local and global well-posedness results for the model equations. Moreover, we present reduced models in various parameter regimes for the approximate description of the motion of typical wave profiles and discuss the mathematically rigorous justification of the validity of these models.