## 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

#### 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

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

#### 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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