Long time existence results for solutions of water waves equations

International audienceWe present in this report various results obtained during the last years by several authors about the problem of long time existence of solutions of water waves and related equations with initial data that are small, smooth, and decaying at infinity. After recalling some facts about local existence theory, we focus mainly on global existence theorems for gravity waves equations proved by Ionescu-Pusateri, Alazard-Delort and Ifrim-Tataru. We describe some of the ideas of the proofs of these theorems, and conclude the paper mentioning related results