A note on optimal control problem for a hemivariational inequality modeling fluid flow
Let \(\Omega \) be a bounded simply connected domain in \({\mathbb{R}}^d\), \(d=2,3\), with connected boundary \(\Gamma \) of class \(C^2\). Consider the following system of stationary Navier-Stokes equations: \(- \nu\, \mathrm{rot} \,\mathrm{rot} , u + \mathrm{rot}, u \times u + \nabla h =f \;\text{ in } \;\Omega\); \(\mathrm{div}\, u = 0 \;\text{ in } \;\Omega\); \(u_{\tau} = u - u_{n}n= 0 \;\text{ on } \;\Gamma\); \(h(x)\in \partial j(x,u_{n}(x)), \;x\in \Gamma\), where \(u\) denotes the velocity field, \(h\) is the total head of the fluid (given by \(h=\text{pressure } +\frac{1}{2}|u|^2\)), \(\nu >0\) denotes the kinematic viscosity, \(u_n\) and \(u_{\tau}\) represent the normal and tangential components of the velocity vector, and \(\partial j(x,\cdot)\) is the Clarke subdifferential of \(j(x,\cdot )\), which is assumed to be locally Lipschitz. The system describes the steady state flow of the fluid occupying \(\Omega \) subjected to external forces with density \(f\). Since \(j(x, \cdot )\) is not assumed to be a convex function, the weak formulation of the problem above is a hemivariational inequality in \(u\), for which existence and uniqueness have been established in a previous paper by the first author and \textit{A. Ochal} [J. Math. Anal. Appl. 306, No. 1, 197--217 (2005; Zbl 1109.35089)]. Here the author provides a result on the continuous dependence of the solution on \(f\in L^2(\Omega ; {\mathbb{R}}^d)\), and an existence result for an optimal control problem governed by the hemivariational inequality mentioned above.
scientific article; zbMATH DE number 6388604