Juanjo Nieto

MULTICELLULAR BIOLOGICAL GROWING SYSTEMS: HYPERBOLIC LIMITS TOWARDS MACROSCOPIC DESCRIPTION

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.

ON THE ASYMPTOTIC THEORY FROM MICROSCOPIC TO MACROSCOPIC GROWING TISSUE MODELS: AN OVERVIEW WITH 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.

Complexity and mathematical tools toward the modelling of multicellular growing systems

This paper deals with a multiscale modelling approach to complex biological systems constituted by several interacting entities. The methodology is based on mathematical kinetic theory for active particles and is focused on the modelling of complex multicellular systems under therapeutic actions at the cellular level and mutations with onset of new populations. Asymptotic hyperbolic methods are developed to derive models at the macroscopic scale of tissues from the underlying description at the level of cells for a open system with variable number of populations.

ON A DISPERSIVE MODEL FOR THE UNZIPPING OF DOUBLE-STRANDED DNA MOLECULES

The paper deals with the analysis of a nonlinear Fokker–Planck equation modeling the mechanical unzipping of double-stranded DNA under the influence of an applied force. The dependent variable is the probability density of unzipping m base pairs. The nonlinear Fokker–Planck equation we propose here is obtained when we couple the model proposed in [D. K. Lubensky and D. R. Nelson, Pulling pinned polymers and unzipping DNA, Phys. Rev. Lett.85 (2000) 1572–1575] with a transcendental equation for the applied force. The resulting model incorporates nonlinear effects in a different way than the usual models in kinetic theory. We show the well-posedness of this model. For that we require a combination of techniques coming from second-order kinetic equations and compensated compactness arguments in conservation laws.

Hyperbolic versus Parabolic Asymptotics in Kinetic Theory toward Fluid Dynamic Models

In this work we are interested in the hyperbolic limits in kinetic theory. We propose a nonstandard scaling to be understood as a sort of intermediate hyperbolic limit, which connects the (macroscopic) hyperbolic limiting behavior of the physical system with the microscopic properties usually obtained under parabolic scalings. We present our main result by means of a general kinetic frame for the intermediate hyperbolic limit which covers some well-known examples in kinetic theory (Vlasov--Poisson--Fokker--Planck systems and linear relaxation for Boltzmann-type equations in semiconductor theory, among others). We will also apply our methods to deal with the Kac approach to Boltzmann operators.

On the time evolution of the mean-field polaron

In this paper a mean-field theory for the evolution of an electron in a crystal is proposed in the framework of the Schrodinger formalism. The well-posedness of the problem as well as the conservation laws associated to the invariances of the Action Functional of the problem and the stability of the minimal energy solution are studied.

ABOUT THE KINETIC DESCRIPTION OF FRACTIONAL DIFFUSION EQUATIONS MODELING CHEMOTAXIS

In this paper, we are interested in the microscopic description of fractional diffusion chemotactic models. We will use the kinetic framework of collisional equations having a heavy-tailed distribution as equilibrium state and take an adequate hydrodynamic scaling to deduce the fractional Keller–Segel system for the cell dynamics. In addition, we use this frame to deduce some models for chemotaxis with fractional diffusion including biological effects and non-standard drift terms.

On the Relativistic BGK-Boltzmann Model: Asymptotics and Hydrodynamics

The generalization of the BGK relaxation model to the special relativity setting is revisited here. We deal with several issues related to this relativistic kinetic model which seem to have been overlooked in the previous physical literature, including the unique determination of associated physical parameters, classical, ultra-relativistic and hydrodynamical limits, maximum entropy principles and the analysis of the linearized operator.

VANISHING VISCOSITY REGIMES AND NONSTANDARD SHOCK RELATIONS FOR SEMICONDUCTOR SUPERLATTICES MODELS

This paper is concerned with the analysis of asymptotic problems from discrete drift-diffusion models describing charge transport in semiconductor superlattices. The regimes we are interested in lead to balance laws. However, the nonconservative structure of the discrete system might produce defect measure terms in the limit process. These defect terms, concentrated on the shock discontinuities, can be related to nonstandard jump relations (in contrast with the usual Rankine–Hugoniot law) when considering discontinuous solutions and wave fronts.

ABOUT UNIQUENESS OF WEAK SOLUTIONS TO FIRST ORDER QUASI-LINEAR EQUATIONS

In this paper we give a criterion to discriminate the entropy solution to quasi-linear equations of first order among weak solutions. This uniqueness statement is a generalization of Oleiniks criterion, which makes reference to the measure of the increasing character of weak solutions. The link between Oleiniks criterion and the entropy condition due to Kruzhkov is also clarified. An application of this analysis to the convergence of the particle method for conservation laws is also given by using the Filippov characteristics.