ALEX 2018 Workshop: Abstracts

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

Pierluigi Colli(1), Gianni Gilardi(1), and Jürgen Sprekels(2)

(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

ty+A2rμ=0, (1)
τty+B2σy+f1(y)+f2(y)=μ+u, (2)
y(0)=y0. (3)

Here, A and B are linear, unbounded, selfadjoint, and positive operators having compact resolvents, and A2r and B2σ, where r>0 and σ>0, denote fractional powers in the spectral sense of A and B, respectively. The unknowns μ and y stand for the chemical potential and the order parameter in an isothermal phase separation process taking place in a container in IR3, while u denotes a distributed control function. Moreover, the functions f1 and f2 are such that f=f1+f2 is a double-well potential; in this connection, f1 is a convex function, and f2 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].

References

  • 1 P. Colli, G. Gilardi, and J. Sprekels, Well-posedness and regularity for a generalized fractional Cahn–Hilliard system, Preprint arXiv:1804.11290 [math. AP](2018), pp. 1-35, and WIAS Preprint No. 2509.
  • 2 – , Optimal distributed control of a generalized fractional Cahn–Hilliard system, Preprint arXiv:1807.03218 [math. AP](2018), pp. 1-35, and WIAS Preprint No. 2519.
  • 3 – , Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double-obstacle potentials, Unpublished preprint 2018, pp. 1-31.