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Juan Calvo

Publications and source records attributed to Juan Calvo.

15 recordsLinked to original sources
arXiv2024

Hybrid methods in reaction-diffusion equations

Simulation of stochastic spatially-extended systems is a challenging problem. The fundamental quantities in these models are individual entities such as molecules, cells, or animals, which move and react in a random manner. In big systems, accounting for each individual is inefficient. If the number of entities is large enough, random effects are negligible, and often partial differential equations (PDEs) are used in which the fluctuations are neglected. When the system is heterogeneous, so that the number of individuals is large in certain regions and small in others, the PDE description becomes inaccurate in certain regions. To overcome this problem, the so-called hybrid schemes have been proposed that couple a stochastic description in parts of the domain with its mean field limit in the others. In this chapter, we review the different formulations of this approach and our recent contributions to overcome several of the limitations of previous schemes, including the extension of the concept to multiscale models of cell populations.

q-bio.QM↗
arXiv2023

Long-time asymptotic of the Lifshitz-Slyozov equation with nucleation

We consider the Lifshitz-Slyozov model with inflow boundary conditions of nucleation type. We show that for a collection of representative rate functions the size distributions approach degenerate states concentrated at zero size for sufficiently large times. The proof relies on monotonicity properties of some quantities associated to an entropy functional. Moreover, we give numerical evidence on the fact that the convergence rate to the goal state is algebraic in time. Besides their mathematical interest, these results can be relevant for the interpretation of experimental data.

math.AP↗
arXiv2023

Modeling compartmentalization within intracellular signaling pathway

We present a new modeling approach for G protein coupled receptors signaling systems, that take into account the compartmentalization of receptors and their effectors, both at plasma membrane and in dynamic intra-cellular vesicles called endosomes. The first building block of the model is about compartment dynamics. It takes into account creation of de-novo endosomes, i.e. endocytosis, recycling of endosomes back to plasma membrane, degradation through transfer into lysosomes as well as endosomes fusion through coagulation dynamics. The second building block is biochemical reactions into each compartments and the transfer of molecules between the dynamical compartments. In this work, we prove sufficient conditions to obtain exponentially ergodicity for the size distribution of intracellular compartments. We futher design a finite volume scheme to simulate our model and show two application cases for receptor trafficking and spatially biased second effector signaling.

math.AP↗
arXiv2020

The Initial-boundary value problem for the Lifshitz-Slyozov equation with non-smooth rates at the boundary

We prove existence and uniqueness of solutions to the initial-boundary value problem for the Lifshitz--Slyozov equation (a nonlinear transport equation on the half-line), focusing on the case of kinetic rates with unbounded derivative at the origin. Our theory covers in particular those cases with rates behaving as power laws at the origin, for which an inflow behavior is expected and a boundary condition describing nucleation phenomena needs to be imposed. The method we introduce here to prove existence is based on a formulation in terms of characteristics, with a careful analysis on the behavior near the singular boundary. As a byproduct we provide a general theory for linear continuity equations on a half-line with transport fields that degenerate at the boundary. We also address both the maximality and the uniqueness of inflow solutions to the Lifshitz--Slyozov model, exploiting monotonicity properties of the associated transport equation.

math.AP↗
arXiv2020

Anisotropic tempered diffusion equations

We introduce a functional framework which is specially suited to formulate several classes of anisotropic evolution equations of tempered diffusion type. Under an amenable set of hypothesis involving a very natural potential function, these models can be shown to belong to the entropy solution framework devised by 4, 5, therefore ensuring well-posedness. We connect the properties of this potential with those of the associated cost function, thus providing a link with optimal transport theory and a supply of new examples of relativistic cost functions. Moreover, we characterize the anisotropic spreading properties of these models and we determine the Rankine-Hugoniot conditions that rule the temporal evolution of jump hypersurfaces under the given anisotropic flows.

math.AP↗
arXiv2019

Global weak solutions to the relativistic BGK equation

In this paper the global existence of weak solutions to the relativistic BGK model for the relativistic Boltzmann equation is analyzed. The proof relies on the strong compactness of the density, velocity and temperature under minimal assumptions on the control of some moments of the initial condition together with the initial entropy.

math.AP↗
arXiv2018

Long-time asymptotics for polymerization models

This study is devoted to the long-term behavior of nucleation, growth and fragmentation equations, modeling the spontaneous formation and kinetics of large polymers in a spatially homogeneous and closed environment. Such models are, for instance, commonly used in the biophysical community in order to model in vitro experiments of fibrillation. We investigate the interplay between four processes: nucleation, polymeriza-tion, depolymerization and fragmentation. We first revisit the well-known Lifshitz-Slyozov model, which takes into account only polymerization and depolymerization, and we show that, when nucleation is included, the system goes to a trivial equilibrium: all polymers fragmentize, going back to very small polymers. Taking into account only polymerization and fragmentation, modeled by the classical growth-fragmentation equation, also leads the system to the same trivial equilibrium, whether or not nucleation is considered. However, also taking into account a depolymer-ization reaction term may surprisingly stabilize the system, since a steady size-distribution of polymers may then emerge, as soon as polymeriza-tion dominates depolymerization for large sizes whereas depolymerization dominates polymerization for smaller ones-a case which fits the classical assumptions for the Lifshitz-Slyozov equations, but complemented with fragmentation so that " Ostwald ripening " does not happen.

math.AP↗
arXiv2015

Analysis of a class of degenerate parabolic equations with saturation mechanisms

We analyze a family of degenerate parabolic equations with linear growth Lagrangian having the form $u_t=÷(φ(u)ψ(\nabla u/u))$. Here $|ψ|\le 1$ and saturates at infinity. We present a simple and natural set of assumptions on the functions $ψ,φ$, under which: 1) these equations fall in the framework provided by \cite{ACMEllipticFLDE, ACMMRelat} and hence they are well posed, 2) we can ensure finite propagation speed for these models, 3) a Rankine--Hugoniot analysis on traveling fronts is also performed. On the particular case of $φ(u)=u$ we get more detailed information on the spreading rate of compactly supported solutions and some interesting connections with optimal mass transportation theory.

math.AP↗
arXiv2014

Qualitative behavior for flux-saturated mechanisms: Traveling waves, waiting time and smoothing effects

This paper is devoted to the analysis of qualitative properties of flux-saturated type operators in dimension one. Specifically, we study regularity properties and smoothing effects, discontinuous interfaces, the existence of traveling wave profiles, sub- and super-solutions and waiting time features. The aim of the paper is to better understand these kind of phenomena throughout two prototypic operators: The relativistic heat equation and the porous media flux-limited equation. As an important consequence of our results we deduce that solutions to the one-dimensional relativistic heat equation become smooth inside their support on the long time run.

math.AP↗
arXiv2012

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.

math-ph↗
arXiv2011

On a nonlinear flux--limited equation arising in the transport of morphogens

Motivated by a mathematical model for the transport of morphogenes in biological systems, we study existence and uniqueness of entropy solutions for a mixed initial-boundary value problem associated with a nonlinear flux--limited diffusion system. From a mathematical point of view the problem behaves more as an hyperbolic system that a parabolic one.

math.AP↗
arXiv2009

Virial inequalities for steady states in relativistic galactic dynamics

It is well known that steady states of the Vlasov-Poisson system, a widely used model in non-relativistic galactic dynamics, have negative energy. In this paper we derive the analogous property for two relativistic generalizations of the Vlasov-Poisson system: The Nordström-Vlasov system and the Einstein-Vlasov system. In the first case we show that the energy of steady states is bounded by their total rest mass; in the second case, where we also assume spherical symmetry, we prove an inequality which involves not only the energy and the rest mass, but also the central redshift. In both cases the proof makes use of integral inequalities satisfied by time depedent solutions and which are derived using the vector fields multipliers method.

math-ph↗
arXiv2009

Dispersive behavior in Galactic Dynamics

The purpose of this paper is to study the relations between different concepts of dispersive solution for the Vlasov-Poisson system in the gravitational case. Moreover we give necessary conditions for the existence of partially and totally dispersive solutions and a sufficient condition for the occurence of statistical dispersion. These conditions take the form of inequalities involving the energy, the mass and the momentum of the solution. Examples of dispersive and non-dispersive solutions--steady states, periodic solutions and virialized solutions--are also considered.

math-ph↗