New approach to the incompressible Maxwell-Boussinesq approximation: Existence, uniqueness and shape sensitivity

The Boussinesq approximation to the Fourier-Navier-Stokes (F-N-S) flows under the electromagnetic field is considered. Such a model is the so-called Maxwell-Boussinesq approximation. We propose a new approach to the problem. We prove the existence and uniqueness of weak solutions to the variational formulation of the model. 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.