Filters

2016 - 20182

More filters

20192
Reset filters

Settings

Author: Frigeri, Sergio × Author: Sprekels, Jürgen × End date: 2019 × Start date: 2010 × Type: Text × WGL Institute: WIAS ×

Search Results

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    Strong solutions to nonlocal 2D Cahn-Hilliard-Navier-Stokes systems with nonconstant viscosity, degenerate mobility and singular potential
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Frigeri, Sergio; Gal, Ciprian G.; Grasselli, Maurizio; Sprekels, Jürgen
    We consider a nonlinear system which consists of the incompressible Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. This is a diffuse interface model which describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluids having the same density. We suppose that the viscosity depends smoothly on the order parameter as well as the mobility. Moreover, we assume that the mobility is degenerate at the pure phases and that the potential is singular (e.g. of logarithmic type). This system is endowed with no-slip boundary condition for the (average) velocity and homogeneous Neumann boundary condition for the chemical potential. Thus the total mass is conserved. In the two-dimensional case, this problem was already analyzed in some joint papers of the first three authors. However, in the present general case, only the existence of a global weak solution, the (conditional) weak-strong uniqueness and the existence of the global attractor were proven. Here we are able to establish the existence of a (unique) strong solution through an approximation procedure based on time discretization. As a consequence, we can prove suitable uniform estimates which allow us to show some smoothness of the global attractor. Finally, we discuss the existence of strong solutions for the convective nonlocal Cahn-Hilliard equation, with a given velocity field, in the three dimensional case as well.
  • Thumbnail Image
    Item
    Optimal distributed control of two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular potential
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Frigeri, Sergio; Grasselli, Maurizio; Sprekels, Jürgen
    In this paper, we consider a two-dimensional diffuse interface model for the phase separation of an incompressible and isothermal binary fluid mixture with matched densities. This model consists of the NavierStokes equations, nonlinearly coupled with a convective nonlocal CahnHilliard equation. The system rules the evolution of the (volume-averaged) velocity u of the mixture and the (relative) concentration difference ' of the two phases. The aim of this work is to study an optimal control problem for such a system, the control being a time-dependent external force v acting on the fluid. We first prove the existence of an optimal control for a given tracking type cost functional. Then we study the differentiability properties of the control-to-state map v 7! [u; '], and we establish first-order necessary optimality conditions. These results generalize the ones obtained by the first and the third authors jointly with E. Rocca in [19]. There the authors assumed a constant mobility and a regular potential with polynomially controlled growth. Here, we analyze the physically more relevant case of a degenerate mobility and a singular (e.g., logarithmic) potential. This is made possible by the existence of a unique strong solution which was recently proved by the authors and C. G. Gal in [14].