ALEX 2018 Workshop: Abstracts

Well-posedness, regularity, and optimal control of general Cahn–Hilliard systems with fractional operators

Pierluigi Colli${}^{\mathrm{(}\mathrm{1}\mathrm{)}}$, Gianni Gilardi${}^{\mathrm{(}\mathrm{1}\mathrm{)}}$, and Jürgen Sprekels${}^{\mathrm{(}\mathrm{2}\mathrm{)}}$

(1) Dipartimento di Matematica “Felice Casorati”, Università di Pavia (Italy) and

Research Associate at the IMATI–C.N.R. di Pavia (Italy)

(2) Department of Mathematics, Humboldt-Universität zu Berlin (Germany) and

Weierstraß-Institut Berlin (Germany)

In this lecture, we consider general systems of Cahn–Hilliard type of the form

${\partial }_{t}y+A^{2r}\mu =0,$ (1)

$\tau {\partial }_{t}y+B^{2\sigma }y+f_1^{\prime }(y)+f_2^{\prime }(y)=\mu +u,$ (2)

$y(0)=y_0.$ (3)

Here, $A$ and $B$ are linear, unbounded, selfadjoint, and positive operators having compact resolvents, and $A^{2r}$ and $B^{2\sigma }$, where $r>0$ and $\sigma >0$, denote fractional powers in the spectral sense of $A$ and $B$, respectively. The unknowns $\mu$ and $y$ stand for the chemical potential and the order parameter in an isothermal phase separation process taking place in a container in $\mathrm{I}{\mathrm{R}}^3$, while $u$ denotes a distributed control function. Moreover, the functions $f_1$ and $f_2$ are such that $f=f_1+f_2$ is a double-well potential; in this connection, $f_1$ is a convex function, and $f_2$ is typically a smooth concave perturbation.

In our analysis, we report about results for the system (1)–(3) concerning existence, uniqueness, regularity, and optimal control that have recently been established in the papers [1, 2, 3].