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Shape derivative of sharp functionals governed by Navier-Stokes flow. (English) Zbl 0937.35130
Jäger, W. (ed.) et al., Partial differential equations: theory and numerical solution. Proceedings of the ICM’98 satellite conference, Prague, Czech Republic, August 10-16, 1998. Boca Raton, FL: Chapman & Hall/CRC. Chapman Hall/CRC Res. Notes Math. 406, 49-63 (2000).
The Navier-Stokes equations \(-\nu\Delta u+Du.u +\nabla p=f\) in \(\Omega\), \(\text{div} u=0\) in \(\Omega\), \(u=0\) on \(\partial\Omega\) are adapted to obtain a transported equation (not written here) which allows to transfer the vector fields from a perturbed domain \(\Omega_s\) to the initial one \(\Omega_0\) by preserving the divergence-free property: the solutions \(u_s\in V^1(\Omega_s)\) of the original equations are transferred into the solutions \(u^s\in V^1( \Omega_0)\) of the transported equation. (We abbreviate \(V^1(\Omega)=\{u\in H^1 (\Omega)\cap H^1_0(\Omega),\text{div} u=0\}.)\)
Then the shape variations of various functionals \(\Omega\to y(\Omega)\) can be established with mild regularity assumptions. By using a tangential calculus, even the resultant of the forces and the moment of the forces \[ F=\int_\Gamma\bigl(-\sigma(u)n+ pn\bigr)d\Gamma,\quad M=\int_\Gamma x\times\bigl(-\sigma(u)n+pn \bigr)d\Gamma, \] exerted by the fluid on a piece \(\Gamma\subset \partial\Omega\) of the boundary, become shape differentiable functionals.
For the entire collection see [Zbl 0923.00019].

35Q30 Navier-Stokes equations
49J20 Existence theories for optimal control problems involving partial differential equations
49Q10 Optimization of shapes other than minimal surfaces
35B37 PDE in connection with control problems (MSC2000)