This paper is devoted to the study of a class of hemivariational inequalities for the time-dependent Navier-Stokes equations, including both boundary hemivariational inequalities and domain hemivariational inequalities. The hemivariational inequalities are analyzed in the framework of an abstract hemivariational inequality. Solution existence for the abstract hemivariational inequality is explored through a limiting procedure for a temporally semi-discrete scheme based on the backward Euler difference of the time derivative, known as the Rothe method. It is shown that solutions of the Rothe scheme exist, they contain a weakly convergent subsequence as the time step-size approaches zero, and any weak limit of the solution sequence is a solution of the abstract hemivariational inequality. It is further shown that under certain conditions, a solution of the abstract hemivariational inequality is unique and the solution of the abstract hemivariational inequality depends continuously on the problem data. The results on the abstract hemivariational inequality are applied to hemivariational inequalities associated with the time-dependent Navier-Stokes equations.
Fang, C., Han, W., Migórski, S., & Sofonea, M. (2016). A class of hemivariational inequalities for nonstationary Navier-Stokes equations. Nonlinear Analysis: Real World Applications, 31, 257–276. https://doi.org/10.1016/j.nonrwa.2016.02.005