## 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].

### MSC:

 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)

### Keywords:

Navier-Stokes equation; shape functional; shape variations