×

Some problems of shape optimization arising in stationary fluid motion. (English) Zbl 1058.49033

Summary: We investigate the existence of a drag-minimizing shape for two classes of optimal design problems of fluid mechanics, namely the vector Burgers equations and the Navier-Stokes equations. It is known that the two-dimensional Navier-Stokes problem of shape optimization has a solution in any class of domains with at most \(l\) holes. We show, for the Burgers equation in three dimensions, that the existence of a minimizer still holds in the classes \({\mathcal O}_{c,r}(C)\) and \({\mathcal W}_w (C)\) introduced by Bucur and Zolésio. These classes are defined by means of capacitary constraints at the boundary. For the 3D Navier-Stokes equations we prove some results of existence of drag-minimizing shape, under additional assumptions on the class of domains to be considered. We also discuss how these assumptions critically depend on the definition of weak solutions for Navier-Stokes equations and, more specifically, on the characterization of the spaces in which it is possible to prove the uniqueness for the linear Stokes problem.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
49J10 Existence theories for free problems in two or more independent variables
76D55 Flow control and optimization for incompressible viscous fluids
35Q30 Navier-Stokes equations
35Q53 KdV equations (Korteweg-de Vries equations)
PDFBibTeX XMLCite