Regularity Structure, Vorticity Layer and Convergence Rates of Inviscid Limit of Free Surface Navier-Stokes Equations with or without Surface Tension

TL;DR

This paper analyzes the inviscid limit of free surface Navier-Stokes equations, revealing the formation of vorticity layers and how surface tension influences the regularity and convergence rates of solutions.

Contribution

It establishes the existence and nature of vorticity layers under various initial conditions and determines how surface tension affects the regularity structure of solutions.

Findings

Existence of strong or weak vorticity layers depending on initial vorticity differences.

Different convergence rates for tangential and normal derivatives.

Surface tension alters the regularity structure of the solutions.

Abstract

In this paper, we study the inviscid limit of the free surface incompressible Navier-Stokes equations with or without surface tension. By delicate estimates, we prove the weak boundary layer of the velocity of the free surface Navier-Stokes equations and the existence of strong or weak vorticity layer for different conditions. When the limit of the difference between the initial Navier-Stokes vorticity and the initial Euler vorticity is nonzero, or the tangential projection on the free surface of the Euler strain tensor multiplying by normal vector is nonzero, there exists a strong vorticity layer. Otherwise, the vorticity layer is weak. We estimate convergence rates of tangential derivatives and the first order standard normal derivative in energy norms, we show that not only tangential derivatives and standard normal derivative have different convergence rates, but also their convergence rates are different for different Euler boundary data. Moreover, we determine regularity structure of the free surface Navier-Stokes solutions with or without surface tension, surface tension changes regularity structure of the solutions.