Shape Holomorphy of the Stationary Navier-Stokes Equations
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 $\hat{\rm D}\subseteq\mathbb{R}^d$, $d\in\{2,3\}$, 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 $(\hat u_T,\hat p_T)\in H^1_0(\hat{D})^d\times L^2(\hat{D})$ 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 $\{T_{y}\,:\,{y}\in U\}\subseteq W^{1,\infty}(\hat{D})^d$ of domain mappings with parameter domain $U=[-1,1]^\mathbb{N}$ and with affine dependence of $T_{y}$ 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 ${y}\mapsto ({\hat u({y})} , \hat p({y}))\in H^1_0(\hat{D})^d\times L^2(\hat{D})$.