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Incompressible Maxwell-Boussinesq approximation: Existence, uniqueness and shape sensitivity

Consiglieri L., Necasova S., Sokolowski J.

Control and Cybernetics

2009

Vol. 38, no 4A

1193-1215

We prove the existence and uniqueness of weak solutions to the variational formulation of the Maxwell-Boussinesq approximation problem. Some further regularity in W1,2+δ, δ > 0, is obtained for the weak solutions. The shape sensitivity analysis by the boundary variations technique is performed for the weak solutions. As a result, the existence of the strong material derivatives for the weak solutions of the problem is shown. The result can be used to establish the shape differentiability for a broad class of shape functionals for the models of Fourier-Navier-Stokes flows under the electromagnetic field.

Shape differentiability of the Neumann problem of the Laplace equation in the half-space

Amrouche C., Necasova S., Sokołowski J.

Control and Cybernetics

2008

Vol. 37, no 4

747-769

We deal with the existence of the material derivative of the Laplace equation with the Neumann boundary condition in the half space. We consider two different perturbations of domains to get the existence of weak Gateaux material derivative and the existence of Fréchet material derivatives.