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The Resolvent Problem and $${{H^{\infty}}}$$ -calculus of the Stokes Operator in Unbounded Cylinders with Several Exits to Infinity

The Resolvent Problem and $${{H^{\infty}}}$$ -calculus of the Stokes Operator in Unbounded... It is proved that the Stokes operator in L q -space on an infinite cylindrical domain of $${{\mathbb{R}^{n}}}$$ , $${{n \geq 3}}$$ , with several exits to infinity generates a bounded and exponentially decaying analytic semigroup and admits a bounded $${{H^{\infty}}}$$ -calculus. For the resolvent estimates, the Stokes resolvent system with a prescribed divergence in an infinite straight cylinder with bounded cross-section $${{\Sigma}}$$ is studied in L q $${{(\mathbb{R}; L^{r}_{\omega} (\Sigma))}}$$ where $${{1 < q,r < \infty}}$$ and $${{\omega \, \epsilon \, A_{r}(\mathbb{R}^{n-1})}}$$ is an arbitrary Muckenhoupt weight. The proofs use cut-off techniques and the theory of Schauder decomposition of UMD spaces based on $${{\mathcal{R}}}$$ -boundedness of operator families and on square function estimates involving Muckenhoupt weights. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

The Resolvent Problem and $${{H^{\infty}}}$$ -calculus of the Stokes Operator in Unbounded Cylinders with Several Exits to Infinity

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References (26)

Publisher
Springer Journals
Copyright
Copyright © 2007 by Birkhäuser Verlag, Basel
Subject
Mathematics; Analysis
ISSN
1424-3199
eISSN
1424-3202
DOI
10.1007/s00028-007-0300-4
Publisher site
See Article on Publisher Site

Abstract

It is proved that the Stokes operator in L q -space on an infinite cylindrical domain of $${{\mathbb{R}^{n}}}$$ , $${{n \geq 3}}$$ , with several exits to infinity generates a bounded and exponentially decaying analytic semigroup and admits a bounded $${{H^{\infty}}}$$ -calculus. For the resolvent estimates, the Stokes resolvent system with a prescribed divergence in an infinite straight cylinder with bounded cross-section $${{\Sigma}}$$ is studied in L q $${{(\mathbb{R}; L^{r}_{\omega} (\Sigma))}}$$ where $${{1 < q,r < \infty}}$$ and $${{\omega \, \epsilon \, A_{r}(\mathbb{R}^{n-1})}}$$ is an arbitrary Muckenhoupt weight. The proofs use cut-off techniques and the theory of Schauder decomposition of UMD spaces based on $${{\mathcal{R}}}$$ -boundedness of operator families and on square function estimates involving Muckenhoupt weights.

Journal

Journal of Evolution EquationsSpringer Journals

Published: Aug 1, 2007

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