Open Access
October, 2016 On decay properties of solutions to the Stokes equations with surface tension and gravity in the half space
Hirokazu SAITO, Yoshihiro SHIBATA
J. Math. Soc. Japan 68(4): 1559-1614 (October, 2016). DOI: 10.2969/jmsj/06841559

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.

Citation

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Hirokazu SAITO. Yoshihiro SHIBATA. "On decay properties of solutions to the Stokes equations with surface tension and gravity in the half space." J. Math. Soc. Japan 68 (4) 1559 - 1614, October, 2016. https://doi.org/10.2969/jmsj/06841559

Information

Published: October, 2016
First available in Project Euclid: 24 October 2016

zbMATH: 1358.35131
MathSciNet: MR3564444
Digital Object Identifier: 10.2969/jmsj/06841559

Subjects:
Primary: 35Q35
Secondary: 76D07

Keywords: decay properties , gravity , half-space problem , Stokes equations , Surface tension

Rights: Copyright © 2016 Mathematical Society of Japan

Vol.68 • No. 4 • October, 2016
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