Mathematical Models and Methods in Applied Sciences

We develop a computational model for the martensitic first-order structural phase transformation in a single crystal thin film, and we use this model to study the effect of spatial compositional fluctuation, spatial temporal noise, and the loss of stability of the metastable phase at temperatures sufficiently far from the transformation temperature.

Our intention in this paper is to publicize and extend somewhat important work of Plotnikov⁷ on the asymptotic limits of solutions of viscous regularizations of a nonlinear diffusion PDE with a cubic nonlinearity. Since the formal limit PDE is in general ill-posed, we expect that the limit solves instead a corresponding diffusion equation with hysteresis effects. We employ entropy/entropy flux pairs to prove various assertions consistent with this expectation.

The bio-heat transfer equation is a macroscopic model for describing the heat transfer in microvascular tissue. In Ref. 8 the authors applied homogenization techniques to derive the bio-heat transfer equation as asymptotic result of boundary value problems which provide a microscopic description for microvascular tissue. Here those results are generalized to a geometrical setting where the regions of blood are allowed to be connected, which covers more biologically relevant geometries. Moreover, asymptotic corrector results are derived under weaker assumptions.

We describe the mortar element method for the two-dimensional edge finite elements of class H(curl). These finite elements are currently used for the discretization of various models coming from the Maxwell equations and using them in a mortar framework has several interesting applications in the electromagnetic and electrotechnical domains. We develop some technical tools necessary to perform the numerical analysis of this approach. Then, we prove some optimal approximation results and illustrate the theory by some numerical experiences.

This paper deals with the competition between the immune system and an agressive host such as tumor. We are particularly interested in the asymptotic behavior of the system when the time goes to infinity: we prove that this competition will absolutely finish either by the suppression of the harmful substances or by the inhibition of the immune system. The model which we introduce is suitable to describe the disease evolution for a long time scale, because it takes the capacity of the body to repair cells damage into account: body cells produce new cells to replace dead ones. We also examine the evolution of the solutions if we assume that the immune system has a normal healthy state that it always tries to reach.

This paper deals with a critical analysis and some developments related to the mathematical literature on multiscale modelling of multicellular systems involving tumor immune cells competition at the cellular level. The analysis is focused on the development of mathematical methods of the classical kinetic theory to model the above physical system and to recover macroscopic equation from the microscopic description. Various hints are given toward research perspectives, with special attention on the modelling of the interplay of microscopic (at the cellular level) biological and mechanical variables on the overall evolution of the system. Indeed the final aim of this paper consists of organizing the various contributions available in the literature into a mathematical framework suitable to generate a mathematical theory for complex biological systems.