A paradifferential reduction for the gravity-capillary waves system at low regularity and applications

TL;DR

This paper develops a paradifferential reduction of the gravity-capillary waves system at low regularity, enabling new well-posedness and blow-up criteria in Sobolev spaces with minimal regularity assumptions.

Contribution

It introduces a novel paradifferential reduction at low Sobolev regularity for gravity-capillary waves, facilitating analysis of well-posedness and blow-up criteria.

Findings

Reduction to a quasilinear dispersive equation of order 3/2

Blow-up criterion involving Lipschitz norm and surface regularity

A priori estimates and solution map contraction in Sobolev spaces

Abstract

We consider in this article the system of gravity-capillary waves in all dimensions and under the Zakharov/Craig-Sulem formulation. Using a paradifferential approach introduced by Alazard-Burq-Zuily, we symmetrize this system into a quasilinear dispersive equation whose principal part is of order $3/2$. The main novelty, compared to earlier studies, is that this reduction is performed at the Sobolev regularity of quasilinear pdes: $H^s(R^d)$ with $s\textgreater{}3/2+d/2$, $d$ being the dimension of the free surface. From this reduction, we deduce a blow-up criterion involving solely the Lipschitz norm of the velocity trace and the $C^{5/2+}$-norm of the free surface. Moreover, we obtain an a priori estimate in the $H^s$-norm and the contraction of the solution map in the $H^{s-3/2}$-norm using the control of a Strichartz norm. These results have been applied in establishing a local well-posedness theory for non-Lipschitz initial velocity in our companion paper.