On decay properties of solutions to the Stokes equations with surface tension and gravity in the half space

TL;DR

This paper investigates the decay behavior of solutions to the Stokes equations with surface tension and gravity in a half-space, using complex analysis and Fourier transform techniques to analyze low-frequency asymptotics.

Contribution

It provides a detailed analysis of the asymptotic behavior of the zero points of the Lopatinskii determinant and decomposes the solution to establish decay rates, advancing understanding of fluid dynamics with surface effects.

Findings

Low frequency solutions behave like a convolution involving a heat kernel and oscillatory exponential.

The zero points of the Lopatinskii determinant have specific asymptotics as frequency approaches zero.

The remainder part of the solution decays faster than the main residue part.

Abstract

In this paper, we proved decay properties of solutions to the Stokes equations with surface tension and gravity in the half space $\mathbf{R}_{+}^{N}=\{(x',x_N)\mid x'\in\mathbf{R}^{N-1},\ x_N>0\}$ $(N\geq 2)$. In order to prove the decay properties, we first show that the zero points $\lambda_\pm$ of Lopatinskii determinant for some resolvent problem associated with the Stokes equations have the asymptotics: $\lambda_\pm=\pm i c_g^{1/2}|\xi'|^{1/2} -2|\xi'|^2+O(|\xi'|^{5/2})$ as $|\xi'|\to0$, where $c_g>0$ is the gravitational acceleration and $\xi'\in\mathbf{R}^{N-1}$ is the tangential variable in the Fourier space. We next shift the integral path in the representation formula of the Stokes semi-group to the complex left half-plane by Cauchy's integral theorem, and then it is decomposed into closed curves enclosing $\lambda_\pm$ and the remainder part. We finally see, by the residue theorem, that the low frequency part of the solution to the Stokes equations behaves like the convolution of the $(N-1)$-dimensional heat kernel and $\mathcal{F}_{\xi'}^{-1}[e^{\pm i c_g^{1/2}|\xi'|^{1/2}t}](x')$ formally, where $\mathcal{F}_{\xi'}^{-1}$ is the inverse Fourier transform with respect to $\xi'$. However, main task in our approach is to show that the remainder part in the above decomposition decay faster than the residue part.