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We investigate a new diffuse-interface model that describes creeping two-phase flows (i.e., flows exhibiting a low Reynolds number), especially flows that permeate a porous medium. The system of equations consists of a Brinkman equation for the volume averaged velocity field and a convective Cahn–Hilliard equation with dynamic boundary conditions for the phase field, which describes the location of the two fluids within the domain. The dynamic boundary conditions are incorporated to model the interaction of the fluids with the wall of the container more precisely. In particular, they allow for a dynamic evolution of the contact angle between the interface separating the fluids and the boundary, and for a convection-induced motion of the corresponding contact line. For our model, we first prove the existence of global-in-time weak solutions in the case where regular potentials are used in the Cahn–Hilliard subsystem. In this case, we can further show the uniqueness of the weak solution under suitable additional assumptions. We further prove the existence of weak solutions in the case of singular potentials. Therefore, we regularize such singular potentials by a Moreau–Yosida approximation, such that the results for regular potentials can be applied, and eventually pass to the limit in this approximation scheme. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the existence, optimal control and feedback control for a class of nonlinear impulsive evolution equations involving weakly continuous operators. We first obtain the existence result of the evolution equation without impulse in any subinterval by Rothe method, and construct a solution of impulsive evolution equation by using this result. Moreover, we obtain the existence of optimal control pairs for the optimal control problem and feasible pairs for the feedback control system by establishing several sufficient assumptions. An application on a quasistatic frictional contact problem is provided. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the study of parabolic variational inequality. Under appropriate assumptions on the main functions, we obtain the existence of weak solutions after the construction of the penalized Young measure by Galerkin's method and the penalty method. The passage to the limit follows relying on the theory of Young measures. [ABSTRACT FROM AUTHOR]