Hadamard well-posedness of the gravity water waves system

TL;DR

This paper proves that solutions to the gravity water waves system depend continuously on initial data in a strong topology, establishing well-posedness in the sense of Hadamard for fluid domains with general bottoms.

Contribution

It extends the well-posedness of the gravity water waves system to a strong topology, completing the Hadamard well-posedness result for general bottom geometries.

Findings

Solutions depend continuously on initial data in the strong topology.

Completes a well-posedness result in the sense of Hadamard.

Applicable to water waves in fluid domains with general bottoms.

Abstract

We consider in this article the system of (pure) gravity water waves in any dimension and in fluid domains with general bottoms. The unique solvability of the problem was established by Alazard-Burq-Zuily [Invent. Math, 198 (2014), no. 1, 71--163] at a low regularity level where the initial surface is $C^{\frac{3}{2}+}$ in terms of Sobolev embeddings, which allows the existence of free surfaces with unbounded curvature. Our result states that the solutions obtained above depend continuously on initial data in the strong topology where they are constructed. This completes a well-posedness result in the sense of Hadamard.