[Paper Review] On the Cauchy problem for gravity water waves

Thomas Alazard, Nicolas Burq+1 more|arXiv (Cornell University)|Dec 4, 2012

Advanced Mathematical Physics ProblemsMathematics54 references111 citations

TL;DR

This paper establishes local well-posedness for the gravity water waves system without surface tension down to optimal regularity thresholds: initial free surfaces are only $C^{3/2+\epsilon}$ and initial velocities are Lipschitz. Using a novel paradifferential reduction of the system and microlocal analysis of the Dirichlet-Neumann operator in low-regularity domains, the authors prove existence and uniqueness in $H^s$-based Sobolev spaces with $s > 1 + d/2$, showing that the critical regularity threshold is $s_c = 1/2 + d/2$, with solutions existing $1/2$-regularity above this critical index.

ABSTRACT

We are interested in the system of gravity water waves equations without surface tension. Our purpose is to study the optimal regularity thresholds for the initial conditions. In terms of Sobolev embeddings, the initial surfaces we consider turn out to be only of~$C^{3/2+\\epsilon}$-class for some $\\epsilon>0$ and consequently have unbounded curvature, while the initial velocities are only Lipschitz. We reduce the system using a paradifferential approach.

Motivation & Objective

• To determine the optimal regularity threshold for initial data in the Cauchy problem for gravity water waves without surface tension.
• To challenge the prevailing conjecture that $C^2$ regularity of the free surface is necessary, by showing that $C^{3/2+\epsilon}$ suffices.
• To develop a new paradifferential framework for the water waves system in low-regularity domains with unbounded curvature.
• To establish uniform estimates and pass to the limit in a vanishing viscosity approximation, ensuring existence of solutions in $H^s$ spaces with $s > 1 + d/2$.

Proposed method

• Reduced the water waves system to a paradifferential form: $\partial_t u + T_V \cdot \nabla u + iT_\gamma u = f$, with $T_\gamma$ of order $1/2$.
• Introduced a microlocal analysis of the Dirichlet-Neumann operator in $C^s$ domains for $s > 1$, extending results from Dahlberg-Kenig and Craig-Schanz-Sulem.
• Used a vanishing viscosity approximation $\varepsilon \Delta U$ to regularize the system and obtain uniform estimates in $H^s$.
• Applied commutator estimates and Sobolev embedding theorems to control nonlinear terms in the energy estimates.
• Performed a bootstrap argument to recover full regularity $H^{s+1/2} \times H^{s+1/2}$ for the surface and trace of velocity.
• Passed to the limit as $\varepsilon \to 0$ using weak compactness and the fact that the Dirichlet-Neumann operator is continuous under weak convergence.

Experimental results

Research questions

• RQ1 Is $C^2$ regularity of the free surface necessary for local well-posedness of the gravity water waves system?
• RQ2 Can the system be well-posed when the initial surface is only $C^{3/2+\epsilon}$ and the velocity field is only Lipschitz?
• RQ3 Can a paradifferential reduction be achieved for the water waves system without surface tension, even when the Dirichlet-Neumann operator is non-smooth?
• RQ4 What is the minimal regularity threshold for the initial data that still ensures existence and uniqueness of solutions?
• RQ5 Can uniform estimates be obtained in the vanishing viscosity limit for low-regularity initial data?

Key findings

• The optimal regularity threshold for the Cauchy problem is $C^{3/2+\epsilon}$ for the initial free surface and Lipschitz for the initial velocity field, contradicting the conjecture that $C^2$ is required.
• The system can be reduced to a paradifferential form with a lower-order operator $T_\gamma$ of order $1/2$, enabling analysis in low-regularity spaces.
• Uniform estimates in $H^s \times H^s$ are obtained for $s > 1 + d/2$, with solutions existing on a time interval independent of the regularization parameter $\varepsilon$.
• The Dirichlet-Neumann operator is shown to be continuous under weak convergence in $H^s$ for $s > 1$, enabling the passage to the limit in the vanishing viscosity approximation.
• The critical regularity index is $s_c = 1/2 + d/2$, and the well-posedness result is $1/2$-regularity above this threshold.
• The Taylor coefficient $a(t)$ remains bounded from below by $a_0/2$ on the existence time interval, ensuring the absence of splash singularities.