Abstract
We study the exact controllability of finite dimensional Galerkin approximation of a Navier-Stokes type system describing doubly diffusive convection with Soret effect in a bounded smooth domain in Rd (d = 2, 3) with controls on the boundary. The doubly diffusive convection system with Soret effect involves a difficult coupling through second order terms. The Galerkin approximations are introduced undercertain assumptions on the Galerkin basis related to the linear independence of suitable traces of its elements over the boundary. By Using Hilbert uniqueness method in combination with a fixed point argument, we prove that the finite dimensional Galerkin approximations are exactly controllable.
Highlights
Control of fluid flows modeled by Navier-Stokes equations has received considerable attention due to its importance in practice and to the theoretical and computational challenges it poses
The doubly diffusive system with Soret effect involves a difficult coupling through second order terms
Significant work has been devoted to studying the stability and physics of doubly diffusive convection with and without Soret effect, see for e.g. [3, 20, 26, 22, 24, 14]
Summary
Control of fluid flows modeled by Navier-Stokes equations has received considerable attention due to its importance in practice and to the theoretical and computational challenges it poses. We consider controllability of a doubly diffusive convection with Soret effect modeled by a coupled Navier-Stokes type partial differential equations. Optimal boundary control of doubly diffusive flows with Soret effect is studied in [21]. In [5], global exact controllability for the two-dimensional Navier-Stokes equations in a manifold without boundary is proved. In [17, 18], exact controllability of finite dimensional Galerkin approximations of Navier-Stokes equations are proved. We investigate the exact controllability for the doubly diffusive convection with Soret effect modeled by the Navier-Stokes system approached by Galerkin approximations. The proof uses the Hilbert uniqueness method to study the exact boundary controllability of linear system and a fixed point method
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