Local Null Controllability of the Smagorinsky-Boussinesq Model

This work deals with local null controllability for the complete Smagorinsky-Boussinesq-type system with distributed controls supported in small sets. We consider the term \((\nu _{0}+\nu _{1}\Vert \nabla y\Vert ^{2})Dy:\nabla y\) in the right hand side of the temperature equation, that makes the system more realistic, difficult to analyze and control. In this equation that describes an incompressible fluid flow non-Newtonians coupled to thermal dynamics, we find local and nonlocal nonlinearities: the recurring transport terms and a turbulent viscosity that depends on the global in space energy dissipated by the mean flow. The proof of local null controllability is based on well-known arguments: Carleman estimates and Liusternik’s Inverse Mapping Theorem.