arXiv subjects

Juan J. L. Velázquez

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

At least 37 records · Page 2Linked to original sources

arXiv2024

Multi-parameter Hopf bifurcations of rimming flows

In order to investigate the emergence of periodic oscillations of rimming flows, we study analytically the stability of steady states for the model of (Benilov, Kopteva, O'Brien, 2005), which describes the dynamics of a thin fluid film coating the inner wall of a rotating cylinder and includes effects of surface tension, gravity, and hydrostatic pressure. We apply multi-parameter perturbation methods to eigenvalues of Fréchet derivatives and prove the transition of a pair of conjugate eigenvalues from the stable to the unstable complex half-plane under appropriate variations of parameters. In order to prove rigorously the corresponding branching of periodic solutions from critical equilibria, we extend the multi-parameter Hopf-bifurcation theory to the case of infinite-dimensional dynamical systems.

math.AP↗

arXiv2024

A stochastic version of the Hopfield-Ninio kinetic proofreading model

In this paper we study a simple stochastic version of the Hopfield-Ninio kinetic proofreading model. The model is characterized by means of two parameters, the unbinding time, which depends on the binding energy between a ligand and a receptor, and the number of times $M \geq 1$ that a ligand attaches to a receptor. We prove that, under suitable assumptions on M, our model has an extreme specificity, i.e. it is capable to discriminate between different ligands, and a high sensitivity, i.e. the response of the system does not change in a significant manner for ranges of ligands varying within several orders of magnitude. Additional quantities like the amount of energy used by the network or the time required to yield a response will be also computed. We also show that our results are robust, i.e., they do not depend on the specific choice of parameters that we make in this paper.

math.PR↗

arXiv2024

On an inhomogeneous coagulation model with a differential sedimentation kernel

We study an inhomogeneous coagulation equation that contains a transport term in the spatial variable modeling the sedimentation of clusters. We prove local existence of mass conserving solutions for a class of coagulation kernels for which in the space homogeneous case instantaneous gelation (i.e., instantaneous loss of mass) occurs. Our result holds true in particular for sum-type kernels of homogeneity greater than one, for which solutions do not exist at all in the spatially homogeneous case. Moreover, our result covers kernels that in addition vanish on the diagonal, which have been used to describe the onset of rain and the behavior of air bubbles in water.

math.AP↗

arXiv2024

Properties of some dynamical systems for three collapsing inelastic particles

In this article we continue the study of the collapse of three inelastic particles in dimension $d \geq 2$, complementing the results we obtained in its companion paper. We focus on the particular case of the nearly-linear inelastic collapse, when the order of collisions becomes eventually the infinite repetition of the period $0-1$, $0-2$, under the assumption that the relative velocities of the particles (with respect to the central particle $0$) do not vanish at the time of collapse. Taking as starting point the full dynamical system that describes two consecutive collisions of the nearly-linear collapse, we derive formally a two-dimensional dynamical system, called the two-collision mapping. This mapping governs the evolution of the variables of the full dynamical system. We show in particular that in the so-called Zhou-Kadanoff regime, the orbits of the two-collision mapping can be described in full detail. We study rigorously the two-collision mapping, proving that the Zhou-Kadanoff regime is stable and locally attracting in a certain region of the phase space of the two-collision mapping. We describe all the fixed points of the two-collision mapping in the case when the norms of the relative velocities tend to the same positive limit. We establish conjectures to characterize the orbits that verify the Zhou-Kadanoff regime, motivated by numerical simulations, and we prove these conjectures for a simplified version of the two-collision mapping.

math-ph↗

arXiv2024

Collapse of inelastic hard spheres in dimension $d \geq 2$

We investigate the collapse of three inelastic particles in dimension $d \geq 2$. We obtain general results of convergence and asymptotics concerning the variables of the dynamical system describing a collapsing system of particles. We prove a complete classification of the singularities when a collapse of three particles takes place, obtaining only two possible orders of collisions between the particles. In the first case we recover that the particles arrange in a nearly-linear chain, already studied by Zhou and Kadanoff, and in the second case we obtain that the particles arrange in a triangle, and we show that, after sufficiently many collisions, the particles collide according to a unique order of collisions, which is periodic. Finally, we construct an initial configuration leading to a nearly-linear collapse, stable under perturbations, and such that the angle between the particles at the time of collapse can be chosen a priori, with an arbitrary precision.

math-ph↗

arXiv2024

Characterizing the detailed balance property by means of measurements in chemical networks

In this paper we study how to determine if a linear biochemical network satisfies the detailed balance condition, without knowing the details of all the reactions taking place in the network. To this end, we use the formalism of response functions $R_{ij} (t) $ that measure how the system reacts to the injection of the substance $j$ at time $t=0$, by measuring the concentration of the substance $i \neq j$ for $t >0$. In particular, we obtain a condition involving two reciprocal measurements (i.e.~$R_{ij}(t), \, R_{ji}(t)$) that is necessary, but not sufficient for the detailed balance condition to hold in the network. Moreover, we prove that this necessary condition is also sufficient if a topological condition is satisfied, as well as a stability property that guarantees that the chemical rates are not fine-tuned.

math-ph↗

arXiv2024

Interface behavior for the solutions of a mass conserving free boundary problem modeling cell polarization

We consider a parabolic non-local free boundary problem that has been derived as a limit of a bulk-surface reaction-diffusion system which models cell polarization. In previous papers, we have established well-posedness of this problem and derived conditions on the initial data that imply continuity of the free boundary as $t\to 0$. In this paper we extend the qualitative study of the free boundary by considering axisymmetric data. Under additional monotonicity assumptions on the data we prove global continuity of the free boundary. On the other hand, if the initial data violate a "no-fattening" condition we show that the free boundary can oscillate as $t \to 0$.

math.AP↗

arXiv2024

The three dimensional magneto-hydrostatic equations with Grad-Rubin boundary value

In this work, we study the well-posedness of the three dimensional magneto-hydrostatic equation under Grad-Rubin boundary value conditions. The proof relies on a fixed point argument to construct solutions to an elliptic-hyperbolic problem in a perturbative regime by means of pseudo-differential operators with symbols with limited regularity in Hölder spaces. As a byproduct, the employed technique in this work is more flexible and simplifies the arguments of the proof for the previous two-dimensional setting.

math.AP↗

arXiv2024

Compactness and existence theory for a general class of stationary radiative transfer equations

In this paper, we study the steady-states of a large class of stationary radiative transfer equations in a $C^1$ convex bounded domain. Namely, we consider the case in which both absorption-emission and scattering coefficients depend on the local temperature $T$ and the radiation frequency $ν.$ The radiative transfer equation determines the temperature of the material at each point. The main difficulty in proving existence of solutions is to obtain compactness of the sequence of integrals along lines that appear in several exponential terms. We prove a new compactness result suitable to deal with such a non-local operator containing integrals on a line segment. On the other hand, to obtain the existence theory of the full equation with both absorption and scattering terms we combine the compactness result with the construction of suitable Green functions for a class of non-local equations.

math.AP↗

arXiv2023

On the diffusion approximation of the stationary radiative transfer equation with absorption and emission

We study the situation in which the distribution of temperature a body is due to its interaction with radiation. We consider the boundary value problem for the stationary radiative transfer equation under the assumption of the local thermodynamic equilibrium. We study the diffusion equilibrium approximation in the absence of scattering. We consider absorption coefficient independent of the frequency $ ν$ (the so-called Grey approximation) and the limit when the photons' mean free path tends to zero, i.e. the absorption coefficient tends to infinity. We show that the densitive of radiative energy $ u $, which is proportional to the fourth power of the temperature due to the Stefan-Boltzmann law, solves in the limit an elliptic equation where the boundary value can be determined uniquely in terms of the original boundary condition. We derive formally with the method of matched asymptotic expansions the boundary condition for the limit problem and we prove rigorously the convergence to the solution of the limit problem with a careful analysis of some non-local integral operators. The method developed here allows to prove all the results using only maximum principle arguments.

math.AP↗

arXiv2023

Coagulation equations for non-spherical clusters

In this work, we study the long time asymptotics of a coagulation model which describes the evolution of a system of particles characterized by their volume and surface area. The aggregation mechanism takes place in two stages: collision and fusion of particles. During the collision stage, the two particles merge at a contact point. The newly formed particle has volume and area equal to the sum of the respective quantities of the two colliding particles. After collision, the fusion phase begins and during it the geometry of the interacting particles is modified in such a way that the volume of the total system is preserved and the surface area is reduced. During their evolution, the particles must satisfy the isoperimetric inequality. Therefore, the distribution of particles in the volume and area space is supported in the region where $\{a\geq (36π)^{\frac{1}{3}}v^{\frac{2}{3}}\}$. We assume the coagulation kernel has a weak dependence on the area variable. We prove existence of self-similar profiles for some choices of the functions describing the fusion rate for which the particles have a shape that is close to spherical. On the other hand, for other fusion mechanisms and suitable choices of initial data, we show that the particle distribution describes a system of ramified-like particles.

math.AP↗

arXiv2023

On the temperature distribution of a body heated by radiation

In this paper, we study the temperature distribution of a body when the heat is transmitted only by radiation. The heat transmitted by convection and conduction is ignored. We consider the stationary radiative transfer equation in the local thermodynamic equilibrium. We prove that the stationary radiative transfer equation coupled with the non-local temperature equation is well-posed in a generic case when emission-absorption or scattering of interacting radiation is considered. The emission-absorption and the scattering coefficients are assumed to be general and they can depend on the frequency of radiation. We also establish an entropy production formula of the system, which is used to prove the uniqueness of solutions for an incoming radiation with constant temperature.

math.AP↗

arXiv2023

Coagulation equations with source leading to anomalous self-similarity

We study the long-time behaviour of the solutions to Smoluchowski coagulation equations with a source term of small clusters. The source drives the system out-of-equilibrium, leading to a rich range of different possible long-time behaviours, including anomalous self-similarity. The coagulation kernel is non-gelling, homogeneous, with homogeneity $γ\leq -1 $, and behaves like $x^{γ+λ} y^{-λ} $ when $y \ll x$ with $γ+2λ> 1 $. Our analysis shows that the long-time behaviour of the solutions depends on the parameters $γ$ and $λ$. More precisely, we argue that the long-time behaviour is self-similar, although the scaling of the self-similar solutions depends on the sign of $γ+λ$ and on whether $γ=-1$ or $γ< -1$. In all these cases, the scaling differs from the usual one that has been previously obtained when $γ+2λ<1$ or $γ+2λ\geq 1, γ>-1$. In the last part of the paper, we present some conjectures supporting the self-similar ansatz also for the critical case $γ+2λ=1, γ\leq -1 $.

math-ph↗

arXiv2023

Qualitative properties of solutions to a non-local free boundary problem modeling cell polarization

We consider a parabolic non-local free boundary problem that has been derived as a limit of a bulk-surface reaction-diffusion system which models cell polarization. The authors have justified the well-posedness of this problem and have further proved uniqueness of solutions and global stability of steady states. In this paper we investigate qualitative properties of the free boundary. We present necessary and sufficient conditions for the initial data that imply continuity of the support at $t = 0$. If one of these assumptions fail, then jumps of the support take place. In addition we provide a complete characterization of the jumps for a large class of initial data.

math.AP↗

arXiv2023

Fast fusion in a two-dimensional coagulation model

In this work, we study a particular system of coagulation equations characterized by two values, namely volume $v$ and surface area $a$. Compared to the standard one-dimensional models, this model incorporates additional information about the geometry of the particles. We describe the coagulation process as a combination between collision and fusion of particles. We prove that we are able to recover the standard one-dimensional coagulation model when fusion happens quickly and that we are able to recover an equation in which particles interact and form a ramified-like system in time when fusion happens slowly.

math.AP↗

arXiv2023

Rotating solutions to the incompressible Euler-Poisson equation with external particle

We consider a two-dimensional, incompressible fluid body, together with self-induced interactions. The body is perturbed by an external particle with small mass. The whole configuration rotates uniformly around the common center of mass. We construct solutions, which are stationary in a rotating coordinate system, using perturbative methods. In addition, we consider a large class of internal motions of the fluid. The angular velocity is related to the position of the external particle and is chosen to satisfy a non-resonance condition.

math.AP↗

arXiv2022

Long-time asymptotics for coagulation equations with injection that do not have stationary solutions

In this paper we study a class of coagulation equations including a source term that injects in the system clusters of size of order one. The coagulation kernel is homogeneous, of homogeneity $γ< 1$, such that $K(x,y)$ is approximately $x^{γ+ λ} y^{-λ}$, when $x$ is larger than $y$. We restrict the analysis to the case $γ+ 2 λ\geq 1 $. In this range of exponents, the transport of mass toward infinity is driven by collisions between particles of different sizes. This is in contrast with the case when $γ+ 2 λ<1$. In that case, the transport of mass toward infinity is due to the collision between particles of comparable sizes. In the case $γ+2λ\geq 1$, the interaction between particles of different sizes leads to an additional transport term in the coagulation equation that approximates the solution of the original coagulation equation with injection for large times. We prove the existence of a class of self-similar solutions for suitable choices of $γ$ and $λ$ for this class of coagulation equations with transport. We prove that for the complementary case such self-similar solutions do not exist.

math.AP↗

arXiv2022

Vanishing angular singularity limit to the hard-sphere Boltzmann equation

In this note we study Boltzmann's collision kernel for inverse power law interactions $U_s(r)=1/r^{s-1}$ for $s>2$ in dimension $ d=3 $. We prove the limit of the non-cutoff kernel to the hard-sphere kernel and give precise asymptotic formulas of the singular layer near $θ\simeq 0$ in the limit $ s\to \infty $. Consequently, we show that solutions to the homogeneous Boltzmann equation converge to the respective solutions.

math.AP↗