Asymptotic limits and optimal control for the Cahn–Hilliard system with convection and dynamic boundary conditions

Abstract

In this paper, we study initial-boundary value problems for the Cahn-Hilliard system with convection and nonconvex potential, where dynamic boundary conditions are assumed for both the associated order parameter and the corresponding chemical potential. While recent works addressed the case of viscous Cahn-Hilliard systems, the 'pure' nonviscous case is investigated here. In its first part, the paper deals with the asymptotic behavior of the solutions as time approaches infinity. It is shown that the ω-limit of any trajectory can be characterized in terms of stationary solutions, provided the initial data are sufficiently smooth. The second part of the paper deals with the optimal control of the system by the fluid velocity. Results concerning existence and first-order necessary optimality conditions are proved. Here, we have to restrict ourselves to the case of everywhere defined smooth potentials. In both parts of the paper, we start from corresponding known results for the viscous case, derive sufficiently strong estimates that are uniform with respect to the (positive) Asymptotic limits and control of convective Cahn-Hilliard systems viscosity parameter, and then let the viscosity tend to zero to establish the sought results for the nonviscous case.

On a transmission problem for equation and dynamic boundary condition of Cahn–Hilliard type with nonsmooth potentials

“…We refer to [29] for more detailed information on the comparison between these models. In addition, we mention the contributions [6,8,15,21] related to the well-posedness, [9,12,13,16,17,20] for the study of long time behavior and the optimal control problems, [7,14] for numerical analysis and [24] for the maximal regularity theory. Comparing the large number of known results on the previous model [15,21], we are only aware of the recent papers [18,29] that analyze the well-posedness of the system (1.1)-(1.6) with (1.12).…”

Well-posedness and optimal control for a Cahn–Hilliard–Oono system with control in the mass term

“…A very general approach for distributed control problems for possibly fractional equations of CH-type is carried out in the papers [20,21,22], with an extension of the analysis to double obstacle potentials like f 2obs in (1.7) via deep quench approximation. The coupling of CH equations in the bulk with dynamic boundary conditions has been investigated in [14,15,23], and the presence of a convective term with the velocity vector taken as control has been dealt with in [16,17,19,30] (see also the references in the quoted contributions).…”

Longtime behavior for a generalized Cahn-Hilliard system with fractional operators

“…The latter are obtained, e.g., by adding viscosity or memory contributions as well as convective terms; another possibility is coupling (1.1)-(1.3) with other equations, like heat type equations or fluid dynamics equations, or introducing non-local-in-space terms; finally, one can replace the classical Neumann boundary conditions by other ones, e.g., the dynamic boundary conditions. Without any claim of completeness, by starting from [37], we can quote, e.g., [1,2,5,[7][8][9][19][20][21][22][23]30,33,34] for the study of the trajectories and related topics, and [12-18, 22, 24-29, 31, 36] for the existence of global or exponential attractors and their properties. However, if nonlocal terms are considered in these papers, they are not defined as fractional powers in the spectral sense of the operators involved.…”