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

摘要

In this paper, we proved decay properties of solutions to the Stokesequations with surface tension and gravity in the half space$\mathbf{R}_{+}^{N}=\{(x',x_N)\mid x'\in\mathbf{R}^{N-1},\ x_N>0\}$ $(N\geq2)$. In order to prove the decay properties, we first show that the zero points$\lambda_\pm$ of Lopatinskii determinant for some resolvent problem associatedwith the Stokes equations have the asymptotics: $\lambda_\pm=\pm ic_g^{1/2}|\xi'|^{1/2} -2|\xi'|^2+O(|\xi'|^{5/2})$ as $|\xi'|\to0$, where$c_g>0$ is the gravitational acceleration and $\xi'\in\mathbf{R}^{N-1}$ is thetangential variable in the Fourier space. We next shift the integral path inthe representation formula of the Stokes semi-group to the complex lefthalf-plane by Cauchy's integral theorem, and then it is decomposed into closedcurves enclosing $\lambda_\pm$ and the remainder part. We finally see, by theresidue theorem, that the low frequency part of the solution to the Stokesequations behaves like the convolution of the $(N-1)$-dimensional heat kerneland $\mathcal{F}_{\xi'}^{-1}[e^{\pm i c_g^{1/2}|\xi'|^{1/2}t}](x')$ formally,where $\mathcal{F}_{\xi'}^{-1}$ is the inverse Fourier transform with respectto $\xi'$. However, main task in our approach is to show that the remainderpart in the above decomposition decay faster than the residue part.