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The Navier-Stokes equations coupled with the heat equation: Analysis and control. (English) Zbl 0901.49003
Summary: We are concerned with the time-dependent Navier-Stokes equations coupled with the heat equation under the Boussinesq’s approximation. We study the regularity of the strong solutions and we consider an optimal control problem associated to these equations. The problem consists in minimizing a functional involving the turbulence within the flow, the control being the heat flux through the boundary of the domain occupied by the fluid. We prove existence of optimal controls and derive some first-order optimality conditions.

MSC:
49J20 Existence theories for optimal control problems involving partial differential equations
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
49N60 Regularity of solutions in optimal control
49K20 Optimality conditions for problems involving partial differential equations
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