## Differential and Integral Equations

- Differential Integral Equations
- Volume 18, Number 10 (2005), 1081-1110.

### Resolvent estimates for the Stokes operator on an infinite layer

Helmut Abels and Michael Wiegner

#### 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.

#### Article information

**Source**

Differential Integral Equations, Volume 18, Number 10 (2005), 1081-1110.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060107

**Mathematical Reviews number (MathSciNet)**

MR2162625

**Zentralblatt MATH identifier**

1212.35343

**Subjects**

Primary: 35Q35: PDEs in connection with fluid mechanics

Secondary: 35C05: Solutions in closed form 35J55 35P15: Estimation of eigenvalues, upper and lower bounds 76D07: Stokes and related (Oseen, etc.) flows

#### Citation

Abels, Helmut; Wiegner, Michael. Resolvent estimates for the Stokes operator on an infinite layer. Differential Integral Equations 18 (2005), no. 10, 1081--1110. https://projecteuclid.org/euclid.die/1356060107