MANEQUI · Mathematical studies on critical non-equilibrium phenomena via mean field theories and theories of nonlinear partial differential equations

Periodic Report Summary 1 - MANEQUI (Mathematical studies on critical non-equilibrium phenomena via mean field theories and theories of nonlinear partial differential equations)

The MaNEqui project originated with the aim of uniting, in a cooperative and synergetic way, the work of researchers and students from 5 European institutions, of which 2 are in Italy (Napoli Federico II University and Second University of Naples) and 3 are in Greece (FORTH Heraklion, University of Crete and University of the Aegean) and the work of the research group of Professor Takashi Suzuki at Osaka University, the University of Miyazaki, Fukuoka University and Tokyo University of Marine Sciences. The focus of the project is in mathematical analysis, particularly nonlinear elliptic and parabolic partial differential equations, including applications to problems of particular relevance in various fields of Physics, Biology and Medical Sciences, such as material sciences, chemotaxis, tumour growth, in a marked interdisciplinary way. Several important critical phenomena in material science, such as phase transition, phase separation, shape memory, and so forth, are formulated by making use of thermodynamics theories. As a consequence, the associated nonlinear partial differential equations are endowed with a variational structure. The point vortex system with many intensities and phase field theories involving free energies are typical examples. Several models in Biology share similar structures, in the sense that they are provided with coarse points of view, which mathematically correspond to nonlocal terms. The modelling of tumor growths based on chemical reactions is a typical example in this context. The objectives of the MaNEqui project include clarifying certain non-standard dynamics deriving from stable and unstable stationary solutions, essentially non-equilibrium phenomena such as nucleation, propagation of interface, etc., using various significant and new analytic tools such as regularity theory, methods of asymptotic expansion, perturbation theory and calculus of variations. Within the first two years of the project, significant progress has been made towards the achievement of the scientific objectives, and reported in 7 published or in-press articles, 7 submitted preprints and a technical report, as well as in two workshops held in Kyoto. More precisely, T. Ricciardi at Napoli Federico II, R. Takahashi and T. Suzuki at Osaka University and H. Ohtsuka at the University of Miyazaki successfully pursued their ongoing study on improved Moser-Trudinger inequalities containing probability measures related to mean field equations arising in equilibrium turbulence. The best constant in the inequality, which from the physical point of view corresponds to a critical temperature, was identified. Related blow-up results concerning the associated elliptic equation with exponential nonlinearity have been derived in collaboration with a new young participant, G. Zecca. G. Pisante at SUN, joined by some researchers in Pisa and France, made significant progress in the study of evolution equation in material sciences, particularly in developing a deeper understanding of the behavior of surface type energies (both local and non-local) and how they interact with bulk type energies. In order to describe free boundary problems such as the confined plasma problem, epitaxially growing films and evolving cavities in elastic materials, several variational and topological techniques to study the behavior of critical points of integral functionals characterized by the presence of two competing energies have been developed. Such techniques are now being applied, jointly with the research group at Osaka University, to the investigation on how the topology of the domain is reflected on the properties of the solutions, with particular interest on the solutions of equations of mean field type. N. Kavallaris at the University of the Aegean, G. Karali at FORTH, E. Latos at the University of Crete, T. Suzuki at Osaka University and Y. Yamada at Fukuoka University successfully engaged in the study of Reaction-Diffusion (RD) systems. In particular, they investigated a non-local RD system arising in cell biology. They nearly entirely clarified its qualitative behavior. Using energy methods and the corresponding shadow and ODE systems the complete transient and asymptotic dynamics of the prey-predator system has been achieved [LSY]. Such a rigorous approach is new in the literature of the RD systems. Its relevance in this context has been emphasized by G. Karali, T. Suzuki and Y. Yamada in their work on the long-time behavior of solutions. The training of young researchers is a relevant aspect of the Project. The young researchers F. Farroni from Naples Federico II, N. Chatzitzisis and I.M. Goumas from the University of Crete, greatly benefited from their long-term stays at the stimulating Suzuki Lab at Osaka University, by strengthening their knowledge on nonlinear analysis, as well as by being exposed to a new culture. The expected final results include completion of the afore-mentioned scientific objectives, further training of young researchers, the strengthening of the European research groups on the interdisciplinary themes of the Project and the consolidation of the existing successful collaboration with the Japanese team, thus enforcing the research network so that it can last well beyond the duration of the Project. Project website: http://www.dma.unina.it/manequi/ Contacts: Prof. Tonia Ricciardi Scientific responsible of MaNEqui Dipartimento di Matematica e Applicazioni Università di Napoli Federico II Via Cintia, 80126 Napoli-Italy tonia.ricciardi@unina.it

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