Differential and Integral Equations

Resolvent estimates for the Stokes operator on an infinite layer

Helmut Abels and Michael Wiegner

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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

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

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

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