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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.

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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

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Vol.68 • No. 4 • October, 2016
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