arXiv subjects

J. J. L. Velázquez

Publications and source records attributed to J. J. L. Velázquez.

17 recordsLinked to original sources

arXiv2026

Localization properties of a free boundary problem for cell polarization

We investigate localization in a time-dependent contrained obstacle problem for the density of an active protein on a cell membrane. This free boundary problem has been derived in Logioti et al. [17] as a model for cell polarization under the influence of an external signal. We show that in the limit of small mass there is instantaneous localization, that is the mass of the active protein concentrates in the set where the signal is maximal for any positive time $t>0$. We give a precise a characterization of this concentration process on the appropriate slow time scale and illustrate this for a few examples of the external signal.

math.AP↗

arXiv2026

Flux solutions for stochastic chemical systems with sources and sinks

In this paper we study a class of stochastic chemical systems that, in general, do not satisfy the property of detailed balance nor the property of complex balance. These systems are obtained by adding sources and sinks to conservative chemical systems. This procedure is a way to define rigorously stochastic chemical systems in contact with reservoirs. We prove that these systems are non-explosive Markov chains and we prove that they converge to a steady state as time tends to infinity. The stationary solution are out of equilibrium solutions. We conclude the paper by applying our results in order to describe fluxes of molecules through some membrane channels.

math.PR↗

arXiv2026

On the onset of correlations in Wave Turbulence close to singularities

In this paper we describe in a formal way how the derivation of the turbulent wave equation for the Schrödinger equation breaks down for times close to the self similar blow up of the wave turbulence kinetic equation. To this end, we study how the derivation of the cumulants hierarchy can not be approximated using solutions of the wave turbulence kinetic equation near the blow up time. It tuns out that near the blow up time the kinetic equation has to be replaced by a hierarchy of equations which is equivalent to a random field, defined for times $t\in (-\infty, \infty)$ and satisfying a nonlinear non autonomous Schrödinger equation.

math.AP↗

arXiv2025

Critical lack of equilibrium in stochastic kinetic proofreading

In this paper we study the relation between the property of detailed balance and the ability of discriminating between different ligands for a class of stochastic models of kinetic proofreading. We prove the existence of a critical amount of lack of detailed balance that the kinetic proofreading models must have in order to have strong specificity for a value of the binding energy $σ$. We also prove that the fact that a kinetic proofreading model has a lack of detailed balance that is larger than the critical one does not necessarily yield strong discrimination properties. Indeed, there exist different sets of chemical rates, leading to the same amount of lack of detailed balance, that have strong discrimination property in some cases and not in others.

math.PR↗

arXiv2025

The adaptation property in non-equilibrium chemical systems

The goal of this paper is to understand if the property of adaptation, which is a typical property of many biochemical systems, can be achieved only by biological systems that actively consume energy or if it can be achieved also by passive systems. We prove that, unless the conserved quantities of a signalling system satisfy a very specific factorization assumption, adaptation cannot be achieved in a robust manner (i.e. in a stable manner under perturbations of the chemical reaction rates) by systems that satisfy the detailed balance property, hence that are passive, and that do not exchange substances or energy with the environment. We also prove that robust adaptation can be achieved by systems that satisfy the detailed balance property, but exchange substances with the environment.

math.CA↗

arXiv2025

The detailed balance property and chemical systems out of equilibrium

The detailed balance property is a fundamental property that must be satisfied in all the macroscopic systems with a well defined temperature at each point. On the other hand, many biochemical networks work in non-equilibrium conditions and they can be effectively modelled using sets of equations in which the detailed balance condition fails. In this paper we study a class of "out of equilibrium" chemical networks that can be obtained freezing the concentration of some substances in chemical networks for which the detailed balance property holds. In particular, we prove that any chemical system with bidirectional chemical reactions can be extended to a system having additional substances and for which the detailed balance property holds.

math.CA↗

arXiv2023

Description of Chemical Systems by means of Response Functions

In this paper we introduce a formalism that allows to describe the response of a part of a biochemical system in terms of renewal equations. In particular, we examine under which conditions the interactions between the different parts of a chemical system, described by means of linear ODEs, can be represented in terms of renewal equations. We show also how to apply the formalism developed in this paper to some particular types of linear and non-linear ODEs, modelling some biochemical systems of interest in biology (for instance, some time-dependent versions of the classical Hopfield model of kinetic proofreading). We also analyse some of the properties of the renewal equations that we are interested in, as the long-time behaviour of their solution. Furthermore, we prove that the kernels characterising the renewal equations derived by biochemical system with reactions that satisfy the detail balance condition belong to the class of completely monotone functions.

math.AP↗

arXiv2015

On self-similar solutions to a kinetic equation arising in weak turbulence theory for the nonlinear Schrödinger equation

We construct a family of self-similar solutions with fat tails to a quadratic kinetic equation. This equation describes the long time behaviour of weak solutions with finite mass to the weak turbulence equation associated to the nonlinear Schrödinger equation. The solutions that we construct have finite mass, but infinite energy. In J. Stat. Phys. 159:668-712, self-similar solutions with finite mass and energy were constructed. Here we prove upper and lower exponential bounds on the tails of these solutions.

math-ph↗

arXiv2014

On the transfer of energy towards infinity in the theory of weak turbulence for the nonlinear Schrödinger equation

We study the mathematical properties of a kinetic equation which describes the long time behaviour of solutions to the weak turbulence equation associated to the cubic nonlinear Schrödinger equation. In particular, we give a precise definition of weak solutions and prove global existence of solutions for all initial data with finite mass. We also prove that any nontrivial initial datum yields the instantaneous onset of a condensate, i.e. a Dirac mass at the origin for any positive time. Furthermore we show that the only stationary solutions with finite total measure are Dirac masses at the origin. We finally construct solutions with finite energy, which is transferred to infinity in a self-similar manner.

math-ph↗

arXiv2014

On the blow up and condensation of supercritical solution of the Nordheim equation for bosons

In this paper we prove that the solutions of the isotropic, spatially homogeneous Nordheim equation for bosons, with bounded initial, data blow up in finite time in the $L^\infty$ norm if the values of the energy and particle density are in the range of values where the corresponding equilibria contains a Dirac mass. We also prove that, in the weak solutions, whose initial data are measures with values of particle and energy densities satisfying the previous condition, a Dirac measure at the origin forms in finite time.

math-ph↗

arXiv2013

Finite time blow-up and condensation for the bosonic Nordheim equation

The homogeneous bosonic Nordheim equation is a kinetic equation describing the dynamics of the distribution of particles in the space of moments for a homogeneous, weakly interacting, quantum gas of bosons. We show the existence of classical solutions of the homogeneous bosonic Nordheim equation that blow up in finite time. We also prove finite time condensation for a class of weak solutions of the kinetic equation.

math-ph↗

arXiv2013

Uniqueness of self-similar solutions to Smoluchowski's coagulation equations for kernels that are close to constant

We consider self-similar solutions to Smoluchowski's coagulation equation for kernels $K=K(x,y)$ that are homogeneous of degree zero and close to constant in the sense that \[ -\eps \leq K(x,y)-2 \leq \eps \Big(\Big(\frac{x}{y}\Big)^α + \Big(\frac{y}{x}\Big)^α\Big) \] for $α\in [0,1)$. We prove that self-similar solutions with given mass are unique if $\eps$ is sufficiently small which is the first such uniqueness result for kernels that are not solvable. Our proof relies on a contraction argument in a norm that measures the distance of solutions with respect to the weak topology of measures.

math.AP↗

arXiv2013

Existence of solutions describing accumulation in a thin-film flow

We consider a third order non-autonomous ODE that arises as a model of fluid accumulation in a two dimensional thin-film flow driven by surface tension and gravity. With the appropriate matching conditions, the equation describes the inner structure of solutions around a stagnation point. In this paper we prove the existence of solutions that satisfy this problem. In order to prove the result we first transform the equation into a four dimensional dynamical system. In this setting the problem consists of finding heteroclinic connections that are the intersection of a two dimensional centre-stable manifold and a three-dimensional centre-unstable one. We then use a shooting argument that takes advantage of the information of the flow in the far-field, part of the analysis also requires the understanding of oscillatory solutions with large amplitude. The far-field is represented by invariant three-dimensional subspaces and the flow on them needs to be understood, most of the necessary results in this regard are obtained in \cite{CV}. This analysis focuses on the understanding of oscillatory solutions and some results are used in the current proof, although the structure of oscillations is somewhat more complicated.

math.DS↗

arXiv2011

Asymptotics of self-similar solutions to coagulation equations with product kernel

We consider mass-conserving self-similar solutions for Smoluchowski's coagulation equation with kernel $K(ξ,η)= (ξη)^λ$ with $λ\in (0,1/2)$. It is known that such self-similar solutions $g(x)$ satisfy that $x^{-1+2λ} g(x)$ is bounded above and below as $x \to 0$. In this paper we describe in detail via formal asymptotics the qualitative behavior of a suitably rescaled function $h(x)=h_λ x^{-1+2λ} g(x)$ in the limit $λ\to 0$. It turns out that $h \sim 1+ C x^{λ/2} \cos(\sqrtλ \log x)$ as $x \to 0$. As $x$ becomes larger $h$ develops peaks of height $1/λ$ that are separated by large regions where $h$ is small. Finally, $h$ converges to zero exponentially fast as $x \to \infty$. Our analysis is based on different approximations of a nonlocal operator, that reduces the original equation in certain regimes to a system of ODE.

math.AP↗

arXiv2011

Ant foraging and minimal paths in simple graphs

Ants are known to be able to find paths of minimal length between the nest and food sources. The deposit of pheromones while they search for food and their chemotactical response to them has been proposed as a crucial element in the mechanism for finding minimal paths. We investigate both individual and collective behavior of ants in some simple networks representing basic mazes. The character of the graphs considered is such that it allows a fully rigorous mathematical treatment via analysis of some markovian processes in terms of which the evolution can be represented. Our analytical and computational results show that in order for the ants to follow shortest paths between nest and food, it is necessary to superimpose to the ants' random walk the chemotactic reinforcement. It is also needed a certain degree of persistence so that ants tend to move preferably without changing their direction much. It is also important the number of ants, since we will show that the speed for finding minimal paths increases very fast with it.

q-bio.PE↗

arXiv2010

Measure valued solutions of the 2D Keller-Segel system

We study the solutions of the two-dimensional Keller-Segel system describing chemotaxis. The Keller-Segel system as well as the properties of the blow-up set has been extensively studied. In this paper we obtain generalized solutions for the Keller-Segel system in the sense of measures as the limit of two-different regularizations of it. We will show that the resulting limit measures, that are in some suitable sense global weak solutions of the Keller-Segel system, depend in the regularization.

math.AP↗

arXiv2010

Classical non mass preserving solutions of coagulation equations

In this paper we construct classical solutions of a family of coagulation equations with homogeneous kernels that exhibit the behaviour known as gelation. This behaviour consists in the loss of mass due to the fact that some of the particles can become infinitely large in finite time.

math-ph↗