Optimizing the heat transfer is an important application of fluid mechanics. We address the problem of optimizing the surface roughness such that the heat energy transferred through the boundary becomes maximal/minimal. Furthermore, we deal with existence and uniqueness of strong solutions of the Boussinesq equations with Dirichlet boundary conditions for u, θ in arbitrary domains Ω ⊆ R3. These results will be used to formulate regularity criteria for weak solutions of the Boussinesq equations.
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