Evolutionary Variational Inequalities with Viscosity

The variational inequalities studied in this chapter are evolutionary since they involve the time derivative of the solution and an initial condition. Their main feature lies in the fact that they are governed by two bilinear forms, one of which involves only the time derivative of the solution, consequently, using the terminology arising in the mechanics of continua, we call then evolutionary variational inequality with viscosity. We start with a basic existence and uniqueness result then we prove regularity and convergence results. We also consider evolutionary quasivariational inequalities and history-dependent evolutionary variational inequalities with viscosity for which we prove existence and uniqueness results. The results presented in this chapter have interest in and of themselves, as they may be applied directly in the study of frictional contact problems involving viscoelastic materials with short memory. Also, they are useful in the study of evolutionary variational inequalities via the regularization method, as it will be shown in the next chapter. As usual, everywhere in this chapter X denotes a real Hilbert space with inner product (·,·)X and norm $||\cdot||\ X$ and, moreover, [0, T] denotes the time interval of interest, where T > 0.