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2005 Resolvent estimates for the Stokes operator on an infinite layer
Helmut Abels, Michael Wiegner
Differential Integral Equations 18(10): 1081-1110 (2005).


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.


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Helmut Abels. Michael Wiegner. "Resolvent estimates for the Stokes operator on an infinite layer." Differential Integral Equations 18 (10) 1081 - 1110, 2005.


Published: 2005
First available in Project Euclid: 21 December 2012

zbMATH: 1212.35343
MathSciNet: MR2162625

Primary: 35Q35
Secondary: 35C05 , 35J55 , 35P15 , 76D07

Rights: Copyright © 2005 Khayyam Publishing, Inc.


Vol.18 • No. 10 • 2005
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