关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller–Segel model and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow up of solutions. The presentation is organized in three parts. The first part focuses on a survey of some sample models, namely the original model and some of its developments, such as flux limited models, or models derived according to similar concepts. The second part is devoted to the qualitative analysis of analytic problems, such as the existence of solutions, blow-up and asymptotic behavior. The third part deals with the derivation of macroscopic models from the underlying description, delivered by means of kinetic theory methods. This approach leads to the derivation of classical models as well as that of new models, which might deserve attention as far as the related analytic problems are concerned. Finally, an overview of the entire contents leads to suggestions for future research activities.

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

Quantum theory of open systems

Statistical Mechanics. — A Set of Lectures

This paper presents a critical review of selected topics related to the modelling of cancer onset, evolution and growth, with the aim of illustrating, to a wide applied mathematical readership, some of the novel mathematical problems in the field. This review attempts to capture, from the appropriate literature, the main issues involved in the modelling of phenomena related to cancer dynamics at all scales which characterise this highly complex system: from the molecular scale up to that of tissue. The last part of the paper discusses the challenge of developing a mathematical biological theory of tumour onset and evolution.

On the foundations of cancer modelling: Selected topics, speculations, and perspectives

This paper proposes a review and critical analysis of the asymptotic limit methods focused on the derivation of macroscopic equations for a class of equations modeling complex multicellular systems by methods of the kinetic theory for active particles. Cellular interactions generate both modification of biological functions and proliferative/destructive events. The asymptotic analysis deals with suitable parabolic, hyperbolic, and mixed limits. The review includes the derivation of the classical Keller–Segel model and flux limited models that prevent non-physical blow up of solutions.

On the asymptotic theory from microscopic to macroscopic growing tissue models: an overview with perspectives

This paper deals with the analysis of the asymptotic limit towards the derivation of hyperbolic macroscopic equations for a class of equations modeling complex multicellular systems. Cellular interactions generate both modification of biological functions and proliferating destructive events related to growth of tumor cells in competition with the immune system. The asymptotic analysis refers to the hyperbolic limit to show how the macroscopic tissue behavior can be described by linear and nonlinear hyperbolic systems which seem the most natural in this context.

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Multicellular biological growing systems: hyperbolic limits towards macroscopic description

Multidimensional high-field limit of the electrostatic Vlasov-Poisson-Fokker-Planck system.

This paper deals with the multiscale modeling of vehicular traffic according to a kinetic
theory approach, where the microscopic state of vehicles is described by position, velocity
and activity, namely a variable suitable to model the quality of the driver-vehicle micro-system. Interactions at the microscopic scale are modeled by methods of game theory, thus leading to the derivation of mathematical models within the framework of the kinetic theory. Macroscopic equations are derived by asymptotic limits from the underlying description at the lower scale. This approach shows the hypothesis under which macroscopic models known in the literature can be derived and how new models can be developed.

On the multiscale modeling of vehicular traffic: From kinetic to hydrodynamics

In this paper we investigate the existence of a unique global mild solution in H 1 ( R 3 ) of the initial-boundary value problem associated with the logarithmic Schrodinger equation i ∂ t ψ = − D Δ ψ + σ log ( | ψ | 2 ) ψ , with D > 0 and σ ∈ R ∖ { 0 } .

J. Nieto

Papers

On the asymptotic theory from microscopic to macroscopic growing tissue models: an overview with perspectives

Multicellular biological growing systems: hyperbolic limits towards macroscopic description

Multidimensional high-field limit of the electrostatic Vlasov-Poisson-Fokker-Planck system.

On the multiscale modeling of vehicular traffic: From kinetic to hydrodynamics

Global H1 solvability of the 3D logarithmic Schrödinger equation