Fréchet Differentiability of Unsteady Incompressible Navier-Stokes Flow with Respect to Domain Variations of Low Regularity by Using a General Analytical Framework

We consider shape optimization problems governed by the unsteady Navier-Stokes equations by applying the method of mappings, where the problem is transformed to a reference domain Ωref and the physical domain is given by Ω = τ(Ωref) with a domain transformation τ ∈ W (Ωref). We show the Fréchet-differentiability of τ 7→ (v, p)(τ) in a neighborhood of τ = id under as low regularity requirements on Ωref and τ as possible. We propose a general analytical framework beyond the implicit function theorem to show the Fréchet-differentiability of the transformationto-state mapping conveniently. It can be applied to other shape optimization or optimal control problems and takes care of the usual norm discrepancy needed for nonlinear problems to show differentiability of the state equation and invertibility of the linearized operator. By applying the framework to the unsteady Navier-Stokes equations, we show that for Lipschitz domains Ωref and arbitrary r > 1, s > 0 the mapping τ ∈ (W1,∞ ∩W)(Ωref) 7→ (v, p)(τ) ∈ (W (0, T ;V ) + W (0, T ; H0))× (L2(0, T ;L0) +W 1,1(0, T ; cl(H1)∗ (L0))) is Fréchet-differentiable at τ = id and the mapping τ ∈ (W1,∞ ∩W)(Ωref) 7→ (v, p)(τ) ∈ (L2(0, T ; H0) ∩ C([0, T ]; L2) × (L2(0, T ;L0) + W 1,1(0, T ; cl(H1)∗ (L 2 0)) ∗) is Fréchet-differentiable on a neighborhood of id, where V ⊂ H0(Ωref) is the subspace of solenoidal functions and W (0, T ;V ) is the usual space of weak solutions. A crucial role in the analysis plays the handling of the incompressibility condition and the low time regularity of the pressure for weak solutions.