Resolvent estimates for the Stokes operator on an infinite layer

Abstract

In this paper we prove unique solvability of the generalized Stokes resolvent equations in an infinite layer $\Omega=\mathbb R^{n-1}\times (-1,1)$, $n\geq 2$, in $L^q$ Sobolev spaces, $1<q<\infty$, with nonslip boundary condition $u|_{\partial\Omega}=0$. The unique solvability is proved for every $\lambda\in {\mathbb{C}} \setminus (-\infty,-\pi^2/4]$, where $-\frac{\pi^2}4$ is the least upper bound of the spectrum of Dirichlet realization of the Laplacian and the Stokes operator in $\Omega$. Moreover, we provide uniform estimates of the solutions for large spectral parameter $\lambda$ as well as $\lambda$ close to $-\frac{\pi^2}4$. Because of the special geometry of the domain, a partial Fourier transformation is used to calculate the solution explicitly. Then Fourier multiplier theorems are used to estimate the solution operator.