[Paper Review] Paralinearization of the Dirichlet to Neumann operator, and regularity of three-dimensional water waves

Thomas Alazard, Guy Métivier|arXiv (Cornell University)|Jan 19, 2009

Navier-Stokes equation solutionsMathematics42 references88 citations

TL;DR

This paper establishes the a priori $C^∞$ regularity of three-dimensional doubly periodic diamond waves in water waves by introducing an exact paralinearization formula for the Dirichlet-to-Neumann operator. It proves that solutions satisfying a refined diophantine condition on Fourier modes are smooth, without requiring smallness assumptions, resolving a key obstacle in the analysis of 3D pure gravity waves due to small divisors and non-ellipticity.

ABSTRACT

This paper is concerned with a priori $C^\\infty$ regularity for three-dimensional doubly periodic travelling gravity waves whose fundamental domain is a symmetric diamond. The existence of such waves was a long standing open problem solved recently by Iooss and Plotnikov. The main difficulty is that, unlike conventional free boundary problems, the reduced boundary system is not elliptic for three-dimensional pure gravity waves, which leads to small divisors problems. Our main result asserts that sufficiently smooth diamond waves which satisfy a diophantine condition are automatically $C^\\infty$. In particular, we prove that the solutions defined by Iooss and Plotnikov are $C^\\infty$. Two notable technical aspects are that (i) no smallness condition is required and (ii) we obtain an exact paralinearization formula for the Dirichlet to Neumann operator.

Motivation & Objective

• To establish a priori $C^\infty$ regularity for three-dimensional doubly periodic gravity waves, known as diamond waves, which are solutions to the water wave equations with a symmetric diamond-shaped fundamental domain.
• To overcome the major analytical challenge posed by the non-ellipticity of the reduced boundary system in 3D pure gravity waves, which leads to small divisors problems.
• To prove that sufficiently smooth diamond waves satisfying a refined diophantine condition on their Fourier modes are automatically $C^\infty$, even without smallness assumptions on the wave amplitude.
• To develop and apply an exact paralinearization formula for the Dirichlet-to-Neumann operator, enabling the use of paradifferential calculus in a non-elliptic setting.

Proposed method

• The authors derive an exact paralinearization formula for the Dirichlet-to-Neumann operator, which is central to handling the nonlinearity and non-ellipticity of the system.
• They employ paradifferential calculus and paracomposition techniques to reduce the full system to a form amenable to regularity analysis, particularly near the characteristic variety.
• A key step involves transforming the problem via a change of variables to a flat boundary setting, allowing the use of standard Sobolev and Hölder space estimates.
• The proof relies on a refined diophantine condition on Fourier frequencies $k_1, k_2$, ensuring that $\left|k_2 - \left(\nu k_1^2 + \kappa_0 + \frac{\kappa_1}{k_1^2}\right)\right| \geq \frac{1}{k_1^{2+\delta}}$ for $\delta < 1$, which controls small divisors.
• Elliptic regularity is applied away from the characteristic variety, and a second reduction step is used to propagate regularity globally.
• The analysis combines energy estimates with the structure of the Dirichlet-to-Neumann operator's symbol to deduce higher-order Sobolev regularity, ultimately implying $C^\infty$ smoothness.

Experimental results

Research questions

• RQ1 Can $C^\infty$ regularity be established for three-dimensional doubly periodic gravity waves without assuming small amplitude or smallness in the solution?
• RQ2 How can the non-elliptic nature of the boundary system for 3D pure gravity waves be overcome in the regularity analysis?
• RQ3 What role does the diophantine condition on Fourier modes play in controlling small divisors and ensuring smoothness?
• RQ4 Can an exact paralinearization formula for the Dirichlet-to-Neumann operator be constructed and used to prove regularity in non-elliptic settings?
• RQ5 Does the regularity of solutions constructed by Iooss and Plotnikov via a Nash-Moser scheme automatically imply $C^\infty$ smoothness?

Key findings

• The main result establishes that any $H^{12}$ diamond wave satisfying the refined diophantine condition $\left|k_2 - \left(\nu k_1^2 + \kappa_0 + \frac{\kappa_1}{k_1^2}\right)\right| \geq \frac{1}{k_1^{2+\delta}}$ for $\delta < 1$ is automatically $C^\infty$.
• The paper provides an exact paralinearization formula for the Dirichlet-to-Neumann operator, which is a novel technical contribution enabling the analysis in the non-elliptic regime.
• No smallness condition on the wave amplitude is required, distinguishing this result from many classical regularity theorems in nonlinear PDE.
• The solutions constructed by Iooss and Plotnikov are shown to be $C^\infty$, confirming their smoothness despite the absence of smallness assumptions.
• The method avoids the use of the hodograph transform and instead relies on paradifferential calculus and energy estimates in Sobolev spaces.
• The result extends to capillary-gravity waves under surface tension, where ellipticity is restored and $C^\infty$ regularity follows from standard elliptic theory and paralinearization.