Journal of the Mathematical Society of Japan

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

Hirokazu SAITO and Yoshihiro SHIBATA

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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 $|\xi'|\to 0$, 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 $\mathcal{F}_{\xi'}^{-1}[e^{\pm ic_{g}^{1/2}|\xi'|^{1/2}t}](x')$ 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.

Article information

Source
J. Math. Soc. Japan, Volume 68, Number 4 (2016), 1559-1614.

Dates
First available in Project Euclid: 24 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1477327226

Digital Object Identifier
doi:10.2969/jmsj/06841559

Mathematical Reviews number (MathSciNet)
MR3564444

Zentralblatt MATH identifier
1358.35131

Subjects
Primary: 35Q35: PDEs in connection with fluid mechanics
Secondary: 76D07: Stokes and related (Oseen, etc.) flows

Keywords
Stokes equations half-space problem surface tension gravity decay properties

Citation

SAITO, Hirokazu; SHIBATA, Yoshihiro. On decay properties of solutions to the Stokes equations with surface tension and gravity in the half space. J. Math. Soc. Japan 68 (2016), no. 4, 1559--1614. doi:10.2969/jmsj/06841559. https://projecteuclid.org/euclid.jmsj/1477327226


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