The aim of this research project is to join together well established researchers who are interested in mathematical models both under the applicative and under the analytical point of view.
The models we are going to consider are all amenable to a natural asymptotic study, falling in one of two classes. Some models contain a `small' parameter, which is sent to zero in order to achieve a simplification of an otherwise too complex mathematical scheme; this is the case when we investigate the electric response of bio-materials or the behavioUr of a magneto-viscoelastic body, where concurrent different effects need to be considered, sometimes in an intricate geometry. A second kind of asymptotics is simply the study of the large time behavioUr of solutions, for example of soliton type solutions or solutions of equations with nonlinear convection.
In this framework, our project focuses on various specific themes: new materials including materials with memory with or without magnetic effects, biological tissues also in the presence of electrical effects, current through lattices or even crowd evolution when the crowd itself is modelled as collective entity.
A description of the mathematical methods employed as well as of their interest, under the applicative and analytical points of view, is provided in the following Sections.
In Section 2.A and 2.E the international connections of the participants referring to this project are given: they are testified also by the enclosed publications in Section 3 [ P1, P3, P5-6, P9, P11, P14, O1, O6, O8-9, O11-14] and by many Preprints in Section 2. E, after the List of External Collaborators involved in the present project.
The mathematical physics interest of the proposed research subjects falls into one of the main themes: thermodynamics, diffusion and dynamics, all with applications.
REFERENCES: SEE SECT 2.E & PUBBL. (P =PRINCIPAL INVESTIGATOR, O=OTHER PARTICIPANTS)
A) The common feature of the proposed models is the periodic microstructure, which calls for a homogenization procedure, in order to provide a simpler macroscopic description, avoiding the technical difficulties due to the intricate microscopic geometry. The presence of active interfaces is a further mathematical difficulty in tackling such problems.Most of these models are proposed in the literature, but often with little or no mathematical understanding. Therefore, a deeper analysis is required, starting from the study of existence and uniqueness of the systems of PDEs describing the physical models under investigation, up to their homogenization.It is our opinion that achieving these goals could provide a mathematical motivation of the experimental formula, possibly introducing new and hopefully more efficient macroscopic models.
B) We obtained a preliminary characterisation of the macroscopic behaviour, from the point of view of diffusion, of media exhibiting aFokker-Planck microscopic behaviour (or a mixed Fokker-Planck and Fick behaviour) [09d]. We want to consider the case when the Fokker-Planck coefficient may degenerate to zero in some subregions.
C1) The sought-after asymptotic expansion should give not only the stabilisation rate (to zero) of the solution, but the limiting shape of the solution itself, when rescaled exactly by the power of time given by that rate. A connected result is a uniqueness theorem under optimal or at least good enough assumptions.
C2) Blow up in graphs has been already studied, mostly in the case of finite graphs and semilinear equations. We look at the setting of nonlinear diffusion equations in infinite graphs, aiming at results connecting sharply the on linearities of the equation and the properties of the graph (e.g., the degree function) to the onset of blow up in a finite time.
D) MATERIALS WITH MEMORY: NEW AND BIOLOGICAL MATERIALS
We wish to introduce new innovatie models of the human spine further to proceed in the well established study on materials with memory (see ref.s)
E) MATRIX SOLUTIONS OF THE NON-COMMUTATIVE EVOLUTION EQUATIONS VIA RECURSION METHOD.
1) Matrix solutions of the noncommutative KdV and mKdV hierarchies via recursion methods: This project is already far advanced. In [12s], soliton solutions are constructed for the hierarchies of the non commutative potential KdV, the noncommutative KdV and the noncommutative modified KdV (mKdV) with essential use of Bäcklund transformations.
Most of the intended applications to the classical counterparts of these hierarchies have been completed. Our applications include regularity theorems for solutions of the classical hierarchies and countable superpositions of solitary waves. The main remaining point is to deepen our understanding for matrix solutions. In [12s] we made first steps into this direction by extending results of Goncharenko on multisolitons of the noncommutative KdV equation. Our next goal will be to investigate the matrix solutions of the mKdV hierarchy, which should be richer than the solutions of the KdV, in analogy to the scalar case. We are particularly interested in matrix breather solutions.
2) Computational problems related to symmetry properties: In [13s] and [17s] the hereditary property is obtained for the recursion operators of the noncommutative Burgers equation and the noncommutative KdV. The proof of this property requires computations, which are already challenging in the classical setting but much more involved in the noncommutative situation. From the eighties on, the problem of computer-aided verification of heriditariness had attracted the interest of several researchers (Ablowitz, Fuchssteiner, Oevel). The transition from the commutative to the noncommutative case is not only interesting because of increasing complexity, but also since the study of the noncommutative setting leads to discovering and exploiting richer structural properties. This project will be conducted in collaboration with Mauro Lo Schiavo (Roma La Sapienza). He has already implemented and tested algorithms to investigate properties of the recursion operators of the Burgers equation and the KdV [18s].
F) DIFFUSION AND CURRENT ON LATTICES
The rigorous derivation of an effective equation for the motion of driven particles through an array of small obstacle is a not a trivial task due to the presence of a nonlinear transport term accounting for the presence of drift in the system. In absence of drift the limit equations have been found in [11e].The breakthrough of our research will be that of treating the nonlinear term and, hopefully, developing techniques useful to address an even more ambitious task, namely, the problem of the behaviour of two different particle species undergoing a simultaneous exclusion dynamics with different rates. For such a problem we can rely at the moment only on numerical simulations performed in [16e] which shows a variety of not trivial and unexpected behaviours.