Optimal control of vortices in non-homogeneous Navier-Stokes flows
Control of vortices even of incompressible fluid has been investigated by several authors as part of their investigations of the Navier-Stokes fluid. Here the authors attempt to control the formation of vortices of incompressible Navier-Stokes equation with Dirichlet boundary conditions. Thus \(\text{div}(y)= 0\) in \(Q\), where \(Q= \Omega\times[0.T]\). \(\Omega\) is the spatial domain. The authors choose the cost functional: \[ j(y, u)= J_1(y)+ \beta J_2(u).\tag{1} \] The functional \(J_2(u)= {1\over 2}\int^T_0 \int_\Omega|u(t, x)|^2\,dx\,dt\) commonly accounts for energy of control, but there are many ways of penalizing some feature of this flow. Here the authors wish to penalize formation of vortices. One way is to choose the functional: \[ J_2(u)={1\over 2} \int^T_0 \int_\Omega |\text{curl}(u(t, x))|^2\,dx\,dt.\tag{2} \] The authors prefer to deal instead with a cost functional containing in the integrand the determinant of the gradient of \(y\), which is known to be closely related to the integrand with respect to the curl, which is shown above as (2). The authors offer Lagrangian formulation of the optimality problem, and formulate the adjoint equation. The functional analytic background is \(L^2\). Some problems arise with convexity, but that does not affect the authors' progress. Making assumptions of differentiability of the state, the authors use (repeatedly) the Kuhn-Tucker theorem to prove regularity in the vicinity of the local optimum. The authors suggest a future attempt, extending their analysis to three dimensions.
scientific article; zbMATH DE number 5146772