Solvability and sliding mode control for the viscous Cahn–Hilliard system with a possibly singular potential

Abstract

<p style='text-indent:20px;'>In the present contribution we study a viscous Cahn–Hilliard system where a further leading term in the expression for the chemical potential <inline-formula><tex-math id="M1">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> is present. This term consists of a subdifferential operator <inline-formula><tex-math id="M2">\begin{document}$ S $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M3">\begin{document}$ L^2(\Omega) $\end{document}</tex-math></inline-formula> (where <inline-formula><tex-math id="M4">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is the domain where the evolution takes place) acting on the difference of the phase variable <inline-formula><tex-math id="M5">\begin{document}$ \varphi $\end{document}</tex-math></inline-formula> and a given state <inline-formula><tex-math id="M6">\begin{document}$ {\varphi^*} $\end{document}</tex-math></inline-formula>, which is prescribed and may depend on space and time. We prove existence and continuous dependence results in case of both homogeneous Neumann and Dirichlet boundary conditions for the chemical potential <inline-formula><tex-math id="M7">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>. Next, by assuming that <inline-formula><tex-math id="M8">\begin{document}$ S = \rho\;{\rm{sign}} $\end{document}</tex-math></inline-formula>, a multiple of the <inline-formula><tex-math id="M9">\begin{document}$ \;{\rm{sign}} $\end{document}</tex-math></inline-formula> operator, and for smoother data, we first show regularity results. Then, in the case of Dirichlet boundary conditions for <inline-formula><tex-math id="M10">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> and under suitable conditions on <inline-formula><tex-math id="M11">\begin{document}$ \rho $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M12">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>, we also prove the sliding mode property, that is, that <inline-formula><tex-math id="M13">\begin{document}$ \varphi $\end{document}</tex-math></inline-formula> is forced to join the evolution of <inline-formula><tex-math id="M14">\begin{document}$ {\varphi^*} $\end{document}</tex-math></inline-formula> in some time <inline-formula><tex-math id="M15">\begin{document}$ T^* $\end{document}</tex-math></inline-formula> lower than the given final time <inline-formula><tex-math id="M16">\begin{document}$ T $\end{document}</tex-math></inline-formula>. We point out that all our results hold true for a very general and possibly singular multi-well potential acting on <inline-formula><tex-math id="M17">\begin{document}$ \varphi $\end{document}</tex-math></inline-formula>.</p>