Paralinearization of free boundary problems in fluid dynamics

Abstract

A classical topic in the mathematical theory of hydrodynamics is to study the evolution of the free surface separating air from an incompressible perfect fluid. The goal of this survey is to examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. These equations are different in nature, dispersive or parabolic, but we will see that they can be studied using related tools. In particular, we will discuss a paradifferential approach to these problems.

arXiv commentsurvey article

References in corpus (13)

• Large time existence for 3D water-waves and asymptotics
• On the Water Waves Equations with Surface Tension
• Breakdown of smoothness for the Muskat problem
• Fully nonlinear long-waves models in presence of vorticity
• Gravity capillary standing water waves
• Sharp commutator estimates via harmonic extensions
• Enhanced existence time of solutions to evolution equations of Whitham type
• A geometric proof of the Quasi-linearity of the water-waves system
• Propagation of singularities for gravity-capillary water waves
• Blow-up conditions for gravity water-waves
• Toolbox of Para-differential Calculus on Compact Lie Groups
• On the Cauchy Problem of Spherical Capillary Water Waves
• The Hele-Shaw semi-flow