In this paper, we study existence, dependence, and optimal control results concerning solutions to a class of hemivariational inequalities for stationary Navier–Stokes equations but without making use of the theory of pseudo-monotone operators. To do so, we consider a classical assumption, due to Rauch, which constrains us to make a slight change on the definition of a solution. The Rauch assumption, although it insures the existence of a solution, does not allow the conclusion that the non-convex functional is locally Lipschitz. Moreover, two dependence results are proved, one with respect to changes of the boundary condition and the other with respect to the density of external forces. The later one will be used to prove the existence of an optimal control to the distributed parameter optimal control problem where the control is represented by the external forces.

在本文中，我们研究了固定的Navier–Stokes方程的一类半变分不等式解的存在性，依赖性和最优控制结果，但没有使用伪单调算符的理论。为此，由于劳赫（Rauch），我们考虑了一个经典假设，该假设约束我们对解决方案的定义进行一些细微更改。Rauch假设虽然确保了解决方案的存在，但不能得出这样的结论，即非凸函数是局部Lipschitz。此外，证明了两个依赖性结果，一个关于边界条件的变化，另一个关于外力的密度。