Stochastic Cahn-Hilliard equation: analysis and applications

The project is a study of the stochastic Cahn-Hilliard equation, in terms of mathematical analysis and applications. We will focus on the stochastic convective Cahn-Hilliard equation, where a velocity field acting on the system is taken into account, and study some applications to stochastic phase-field modelling for tumor growth. In the first work package we analyse the stochastic Cahn-Hilliard equation with a convective perturbation in divergence form. We expect to prove well-posedness and regularity, and to perform an optimal velocity control, which amounts to minimizing a cost functional by controlling the velocity field. In the second work package we study a class of stochastic phase-field models for tumor growth, coupling the stochastic Cahn-Hilliard equation (for the difference in volume fractions between necrotic and healthy cells) with a reaction-diffusion equation for the nutrient (glucose). We expect to prove well-posedness and to perform an optimal control problem arising from applications: by administrating an amount of drug, one wants to reach a target configuration of the tumoral region at the end of the treatment in such a way that the amount of given drug is minimal. The approach is based on a generalized variational setting to SPDEs. We will rely on mono- tone and convex analysis in order to deal with rapidly-growing potentials, and we will employ techniques from functional analysis and probability as fixed point arguments, differentiability in Banach spaces and stochastic compactness. The originality of the project is evident at different levels. First of all, the stochastic con- vective Cahn-Hilliard equation is fundamental in phase-transition, as it allows to handle more complex models involving also an evolution equation for the velocity variable, with applications to physics and biology. Despite its crucial role, the equation still lacks of a rigorous mathemat- ical analysis: the proposed study is the first step in this direction, and will be necessary for a deeper treatment of stochastic phase-field models. Secondly, phase-field models for tumor growth have been studied only in the deterministic setting. The possibility of taking into account the uncertainty of the biological phenomenon is crucial, as it is due to unpredictable oscillations at a microscopic level. The second proposed work package will thus provide a more accurate description of the real phenomenon, and will pave the way to studying more accurate models for tumor growth. Due to the interdisciplinary aspect, the project will highly benefit from the collaboration with international researchers: Ulisse Stefanelli (University of Vienna), Carlo Marinelli (University College London), Elisabetta Rocca (University of Pavia) and Carlo Orrieri (University of Trento). This will allow to strengthen the cooperation between Vienna and my country of origin. 1

The main objective of the project was to provide a mathematical analysis of some selected models arising in Physics, Engineering, and Biology in the context of diffuse-interface modelling, by taking into account possible random perturbations. Such models describe the evolution of two-phase materials (e.g. metallic alloys, human tissues) and are based on the so-called Cahn-Hilliard equation. The main novelty brought by the project was to account for unpredictable effects described by stochastic perturbations. In first part of the project some preliminary results were obtained on the stochastic Cahn-Hilliard equation subject to velocity contributions. The second part of the project focused then on direct applications to biology and Medicine, and analysed stochastic diffuse interface models for tumor growth. Here, typical questions arising in tumor treatment were considered: what is the optimal chemotherapy treatment of a patient if the tumor/healhy cells undergo a diffuse interface evolution? Explicit results were given in terms of optimal control of stochastic PDEs.