FRECHET DIFFERENTIABILITY OF UNSTEADY INCOMPRESSIBLE NAVIER-STOKES FLOW WITH RESPECT TO DOMAIN VARIATIONS OF LOW REGULARITY BY USING A GENERAL ANALYTICAL FRAMEWORK

摘要

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 Omega(ref) and the physical domain is given by Omega = tau (Omega(ref)) with a domain transformation tau is an element of W-1,W-infinity(Omega(ref)). We show the Frechet differentiability of tau -> (v, p) (tau) in a neighborhood of tau = id under as low regularity requirements on Omega(ref) and tau as possible. We propose a general analytical framework beyond the implicit function theorem to show the Frechet differentiability of the transformation to-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 Omega(ref) and arbitrary r > 1, s > 0 the mapping tau is an element of (W-1,W-infinity (W-1,W-infinity boolean AND W-1+s,W-r) (Omega(ref)) -> (v, p)(tau) is an element of(W (0, T;V) + W(0, T;HI01)) x (L-2 (0,T; L-0(2)) + W-1,W-1(0,T;cl((H1)*)(L-0(2)))*) is Frechet differentiable at tau = id and the mapping tau is an element of (W1-infinity boolean AND (W-1+s,W-r) (Omega(ref)) -> (v, p) (tau) is an element of (L-2(0, T; H-0(1)) boolean AND C([0, T]; L-2) x (L-2(0, T; L-0(2)) + W-1,W-1(0,T;c1((H1)*)(L-0(2)))*) is Frechet differentiable on a neighborhood of id, where V subset of H-0(1)(Omega(ref)) 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.