Shape Holomorphy of the stationary Navier-Stokes Equations *

Abstract

We consider the stationary Stokes and Navier-Stokes Equations for viscous, incompressible flow in parameter dependent bounded domains D T , subject to homogeneous Dirichlet (" no-slip ") boundary conditions on ∂D T. Here, D T is the image of a given fixed nominal Lipschitz domainˆDdomainˆ domainˆD ⊆ R d , d ∈ {2, 3}, under a map T : R d → R d. We establish shape holomorphy of Leray solutions which is to say, holomorphy of the map T → (ˆ u T , ˆ p T) where (ˆ u T , ˆ p T) ∈ H 1 0 (ˆ D) d ×L 2 (ˆ D) denotes the pullback of the corresponding weak solutions and T varies in W k,∞ with k ∈ {1, 2}, depending on the type of pullback. We consider in particular parametrized families {T y : y ∈ U } ⊆ W 1,∞ of domain mappings, with parameter domain U = [−1, 1] 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 gpc (" generalized polynomial chaos ") expansions for the corresponding parametric solution map y → (ˆ u y , ˆ p y) ∈ H 1 0 (ˆ D) d × L 2 (ˆ D)