Normal solutions to boundary value problems for stationary systems of the Navier-Stokes type
In the present article, we consider equations with quadratic nonlinearity in a Hilbert space which describe boundary value problems for stationary Navier-Stokes equations for a viscous incompressible fluid. The problem of finding a normal solution can be interpreted from a physical viewpoint, for instance, as the problem of finding a flow with the least vorticity. From a mathematical viewpoint, the posed problem represents an extremal problem for a nonlinear equation in a Hilbert space and its study is carried out by means of control theory for singular distributed systems. The main goal of the article is to construct and justify necessary optimality conditions for a solution and also, on using these conditions, to obtain extremal principles like the Pontryagin maximum principle. Moreover, for the so-called slow normal solutions to a nonlinear equation, a uniqueness theorem is obtained and a variational inequality is derived which is equivalent to the extremal problem in question.
necessary optimality conditions for a solution
scientific article; zbMATH DE number 923550