Low regularity well-posedness for two-dimensional deep gravity water waves with constant vorticity

TL;DR

This work proves local well-posedness for two-dimensional gravity water waves with constant vorticity in infinite depth at low regularity, showing solvability for data in the Sobolev scale $\mathcal{H}^s$ with $s\ge \frac{3}{4}$. The authors develop a robust paradifferential framework in holomorphic coordinates, derive a differentiated system that diagonalizes into good variables, and construct paradifferential normal-form corrections to eliminate quadratic and near-quadratic resonances. A balanced energy functional $E_s$ is created, with norm equivalence to the homogeneous Sobolev-type space and an energy estimate that avoids dependence on the strongest control norm, enabling Gronwall-type control at low regularity. The approach sharpens previous regularity thresholds by $\frac14$ derivative and extends the Ai–Ifrim–Tataru technology to the nonzero constant vorticity setting, providing a rigorous foundation for the local theory of 2D deep-water waves with vorticity and potential pathways toward rough data and further refinements.

Abstract

We consider the two dimensional gravity water waves with nonzero constant vorticity in infinite depth. We show that for $s\geq \frac{3}{4}$, the water waves system is locally well-posed in $\mathcal{H}^{s}$, which is the nonzero constant vorticity counterpart of the breakthrough work of Ai-Ifrim-Tataru in [4]. It is also a $\frac{1}{4}$ improvement in Sobolev regularity compared to the previous result of Ifrim-Tataru in [17].

Figures (1)

• Figure 1.1: The fluid domain.

Theorems & Definitions (57)

• Theorem 1.1: MR3869381
• Theorem 1.2: MR4462478
• Theorem 1.3
• Theorem 1.4
• Remark 1
• Remark 2
• Lemma 2.1: ai2023dimensional
• Lemma 2.2: ai2023dimensional
• Lemma 2.3: ai2023dimensional
• Lemma 2.4: ai2023dimensional