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Shape holomorphy of the stationary Navier-Stokes equations. (English) Zbl 1390.35227

Summary: 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 reference Lipschitz domain \(\hat{\mathrm D}\subseteq\mathbb{R}^d\), \(d\in\{2,3\}\), under a map \(T:\mathbb{R}^d\rightarrow\mathbb{R}^d\). We establish shape holomorphy of Leray solutions which is to say, holomorphy of the map \(T\mapsto (\hat u_T,\hat p_T)\), where \((\hat u_T,\hat p_T)\in H^1_0(\hat{D})^d\times L^2(\hat{D})\) denotes the pullback of the corresponding weak solutions in \(\mathrm{D}=T(\hat{D})\) and \(T\) varies in (subsets of) \(W^{k,\infty}\) with \(k\in\{1,2\}\), depending on the type of pullback. We consider, in particular, parametrized families \(\{T_{\boldsymbol y}\,:\,{\boldsymbol y}\in U\}\subseteq W^{1,\infty}(\hat{D})^d\) of domain mappings with parameter domain \(U=[-1,1]^\mathbb{N}\) and with affine dependence of \(T_{\boldsymbol y}\) on \({y}\). The presently obtained shape holomorphy implies summability results and \(n\)-term approximation rate bounds for “generalized polynomial chaos” expansions for the corresponding parametric solution map \({\boldsymbol y}\mapsto ({\hat u({\boldsymbol y})}, \hat p({\boldsymbol y}))\in H^1_0(\hat{D})^d\times L^2(\hat{D})\).

MSC:

35Q30 Navier-Stokes equations

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