Research reports

Years: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Shape Holomorphy of the stationary Navier-Stokes Equations

by A. Cohen and Ch. Schwab and J. Zech

(Report number 2016-45)

Abstract
We consider the stationary Stokes and Navier-Stokes Equations for viscous, incompressible flow in parameter dependent bounded domains \({\mathrm{D}}_T\), subject to homogeneous Dirichlet ("no-slip") boundary conditions on \(\partial {{\mathrm{D}}_T}\). Here, \({\mathrm{D}}_T\) is the image of a given fixed nominal Lipschitz domain \(\hat{\mathrm{D}}\subseteq\mathbb{R}^d\), \(d\in\{2,3\}\), under a map \(T:\mathbb{R}^d\to\mathbb{R}^d\). We establish shape holomorphy of Leray solutions which is to say, holomorphy of the map \(T\mapsto (\hat u_T,\hat p_T)\) where \((\hat u_T,\hat p_T)\in H^1_0(\hat{\mathrm{D}})^d\times L^2(\hat{\mathrm{D}})\) denotes the pullback of the corresponding weak solutions and \(T\) varies in \(W^{k,\infty}\) with \(k\in\{1,2\}\), depending on the type of pullback. We consider in particular parametrized families \(\{T_{\boldsymbol{y}}:{\boldsymbol{y}}\in U\}\subseteq W^{1,\infty}\) of domain mappings, with parameter domain \(U=[-1,1]^\mathbb{N}\) and with affine dependence of \(T_{\boldsymbol{y}}\) on \({\boldsymbol{y}}\). The presently obtained shape holomorphy implies summability results and \(n\)-term approximation rate bounds for gpc ("generalized polynomial chaos") expansions for the corresponding parametric solution map \({\boldsymbol{y}}\mapsto ({\hat u({\boldsymbol{y}})} , \hat p({\boldsymbol{y}}))\in H^1_0(\hat{\mathrm{D}})^d\times L^2(\hat{\mathrm{D}})\).

Keywords: Shape-holomorphy, Navier-Stokes Equations, Uncertainty Quantification, Parametric Operator Equations

@Techreport{CSZ16_682,
  author = {A. Cohen and Ch. Schwab and J. Zech},
  title = {Shape Holomorphy of the stationary Navier-Stokes Equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-45},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-45.pdf },
  year = {2016}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).