arXiv subjects

Juan J. L. Velázquez

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

At least 73 records · Page 4Linked to original sources

arXiv2019

Self-similar asymptotics for a modified Maxwell-Boltzmann equation in systems subject to deformations

In this paper we study a generalized class of Maxwell-Boltzmann equations which in addition to the usual collision term contains a linear deformation term described by a matrix A. This class of equations arises, for instance, from the analysis of homoenergetic solutions for the Boltzmann equation considered by many authors since 1950s. Our main goal is to study a large time asymptotics of solutions under assumption of smallness of the matrix A. The main result of the paper is formulated in Theorem 2.1. Informally stated, this Theorem says that, for sufficiently small norm of A, any non-negative solution with finite second moment tends to a self-similar solution of relatively simple form for large values of time. This is what we call "the self-similar asymptotics". We also prove that the higher order moments of the self-similar profile are finite under further smallness condition on the matrix A.

math-ph↗

arXiv2019

Temporal oscillations in Becker-Doering equations with atomization

We prove that time-periodic solutions arise via Hopf bifurcation in a finite closed system of coagulation-fragmentation equations. The system we treat is a variant of the Becker-Doering equations, in which clusters grow or shrink by addition or deletion of monomers. To this is added a linear atomization reaction for clusters of maximum size. The structure of the system is motivated by models of gas evolution oscillators in physical chemistry, which exhibit temporal oscillations under certain input/output conditions.

math.DS↗

arXiv2019

Long time asymptotics for homoenergetic solutions of the Boltzmann equation. Hyperbolic-dominated case

In this paper we continue the formal analysis of the long-time asymptotics of the homoenergetic solutions for the Boltzmann equation that we began in [18]. They have the form $f\left( x,v,t\right) =g\left(v-L\left( t\right) x,t\right) $ where $L\left( t\right) =A\left(I+tA\right) ^{-1}$ where $A$ is a constant matrix. Homoenergetic solutions satisfy an integro-differential equation which contains, in addition to the classical Boltzmann collision operator, a linear hyperbolic term. Depending on the properties of the collision kernel the collision and the hyperbolic terms might be of the same order of magnitude as $t\to\infty$, or the collision term could be the dominant one for large times, or the hyperbolic term could be the largest. The first case has been rigorously studied in [17]. Formal asymptotic expansions in the second case have been obtained in [18]. All the solutions obtained in this case can be approximated by Maxwellian distributions with changing temperature. In this paper we focus in the case where the hyperbolic terms are much larger than the collision term for large times (hyperbolic-dominated behavior). In the hyperbolic-dominated case it does not seem to be possible to describe in a simple way all the long time asymptotics of the solutions, but we discuss several physical situations and formulate precise conjectures. We give explicit formulas for the relationship between density, temperature and entropy for these solutions. These formulas differ greatly from the ones at equilibrium.

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arXiv2019

Analysis of a thin film approximation for two-fluid Taylord-Couette flows

In this work we study the evolution of the interface between two different fluids in two concentric cylinders when the velocity is given by the Navier-Stokes equation and one of the fluids is thin. We present a formal asymptotic derivation of the evolution equation for the interface under different scaling assumptions for the surface tension. We then study the different types of the stationary solutions and travelling waves for the resulting equation. In particular, we state a global well posedness result and using Center Manifold Theory, we obtain detailed information about the long time asymptotics of the solutions of the problem.

math.AP↗

arXiv2019

A bulk-surface reaction-diffusion system for cell polarization

We propose a model for cell polarization as a response to an external signal which results in a system of PDEs for different variants of a protein on the cell surface and interior respectively. We study stationary states of this model in certain parameter regimes in which several reaction rates on the membrane as well as the diffusion coefficient within the cell are large. It turns out that in suitable scaling limits steady states converge to solutions of some obstacle type problems. For these limiting problems we prove the onset of polarization if the total mass of protein is sufficiently small. For some variants we can even characterize precisely the critical mass for which polarization occurs.

math.AP↗

arXiv2018

Self-similar solutions to coagulation equations with time-dependent tails: the case of homogeneity one

We prove the existence of a one-parameter family of self-similar solutions with time dependent tails for Smoluchowski's coagulation equation, for a class of kernels $K(x,y)$ which are homogeneous of degree one and satisfy $K(x,1)\to k_0>0$ as $x\to 0$. In particular, we establish the existence of a critical $ρ_*>0$ with the property that for all $ρ\in(0,ρ_*)$ there is a positive and differentiable self-similar solution with finite mass $M$ and decay $A(t)x^{-(2+ρ)}$ as $x\to\infty$, with $A(t)=e^{M(1+ρ)t}$. Furthermore, we show that (weak) self-similar solutions in the class of positive measures cannot exist for large values of the parameter $ρ$.

math.AP↗

arXiv2018

On the homogenization of random stationary elliptic operators in divergence form

In this note we comment on the homogenization of a random elliptic operator in divergence form $-\nabla \cdot a\nabla$, where the coefficient field $a$ is distributed according to a stationary, but not necessarily ergodic, probability measure $P$. We generalize the well-known case for $P$ stationary and ergodic by showing that the operator $-\nabla \cdot a(\frac{\cdot}{\varepsilon})\nabla$ almost surely homogenizes to a constant-coefficient, random operator $-\nabla \cdot A_h\nabla$. Furthermore, we use a disintegration formula for $P$ with respect to a family of ergodic and stationary probability measures to show that the law of $A_h$ may be obtained by using the standard homogenization results on each probability measure of the previous family. We finally provide a more explicit formula for $A_h$ in the case of coefficient fields which are a function of a stationary Gaussian field.

math.AP↗

arXiv2018

Long time asymptotics for homoenergetic solutions of the Boltzmann equation. Collision-dominated case

In this paper we present a formal analysis of the long-time asymptotics of a particular class of solutions of the Boltzmann equation, known as homoenergetic solutions, which have the form $f\left( x,v,t\right)=g\left( v-L\left( t\right) x,t\right)$ where $L\left( t\right) =A\left(I+tA\right) ^{-1}$ with the matrix $A$ describing a shear flow or a dilatation or a combination of both. We began this study in \cite{JNV1}. Homoenergetic solutions satisfy an integro-differential equation which contains, in addition to the classical Boltzmann collision operator, a linear hyperbolic term. In \cite{JNV1} it has been proved rigorously the existence of self-similar solutions which describe the change of the average energy of the particles of the system in the case in which there is a balance between the hyperbolic and the collision term. In this paper we focus in homoenergetic solutions for which the collision term is much larger than the hyperbolic term (collision-dominated behavior). In this case the long time asymptotics for the distribution of velocities is given by a time dependent Maxwellian distribution with changing temperature.

math-ph↗

arXiv2018

Hopf Bifurcation in Structural Population Models

We study a nonlinear PDE problem motivated by the peculiar patterns arising in myxobacteria, namely counter-migrating cell density waves. We rigorously prove the existence of Hopf bifurcations for some specific values of the parameters of the system. This shows the existence of periodic solutions for the systems under consideration.

math.AP↗

arXiv2018

From a non-Markovian system to the Landau equation

In this paper, we prove that in macroscopic times of order one, the solutions to the truncated BBGKY hierarchy (to second order) converge in the weak coupling limit to the solution of the nonlinear spatially homogeneous Landau equation. The truncated problem describes the formal leading order behavior of the underlying particle dynamics and can be reformulated as a non-Markovian hyperbolic equation, which converges to the Markovian evolution described by the parabolic Landau equation. The analysis in this paper is motivated by Bogolyubov's derivation of the kinetic equation by means of a multiple time scale analysis of the BBGKY hierarchy.

math-ph↗

arXiv2018

The two-particle correlation function for systems with long-range interactions

In this paper, we study the truncated two-particle correlation function in particle systems with long range interactions. For Coulombian and soft potentials, we define and give well-posedness results for the equilibrium correlations. In the Coulombian case, we prove the onset of the Debye screening length in the equilibrium correlations, for suitable velocity distributions. Additionally, we give precise estimates on the effective range of interaction between particles. In the case of soft potential interaction the equilibrium correlations and their fluxes in the space of velocities are shown to be linearly stable.

math-ph↗

arXiv2018

Self-similar asymptotic behavior for the solutions of a linear coagulation equation

In this paper we consider the long time asymptotics of a linear version of the Smoluchowski equation which describes the evolution of a tagged particle moving at constant speed in a random distribution of fixed particles. The volumes $v$ of the particles are independently distributed according to a probability distribution which decays asymptotically as a power law $v^{-σ}$. The validity of the equation has been rigorously proved in \cite{NoV} for values of the exponent $σ>3$. The solutions of this equation display a rich structure of different asymptotic behaviours according to the different values of the exponent $σ$. Here we show that for $\frac{5}{3}<σ<2$ the linear Smoluchowski equation is well posed and that there exists a unique self-similar profile which is asymptotically stable.

math.AP↗

arXiv2018

Homogenization for the Poisson equation in randomly perforated domains under minimal assumptions on the size of the holes

This paper deals with the homogenization of the Poisson equation in a bounded domain of $\mathbb{R}^d$, $d>2$, which is perforated by a random number of small spherical holes with random radii and positions. We show that for a class of stationary short-range correlated measures for the centres and radii of the holes, we recover in the homogenized limit an averaged analogue of the "strange term" obtained by Cioranescu and Murat in the periodic case [D. Cioranescu and F. Murat, \textit{Un term étrange venu d'ailleurs} (1986)]. We stress that we only require that the random radii have finite $(d-2)$-moment, which is the minimal assumption in order to ensure that the average of the capacity of the balls is finite. Under this assumption, there are holes which overlap with probability one. However, we show that homogenization occurs and that the clustering holes do not have any effect in the resulting homogenized equation.

math.AP↗

arXiv2018

Nonuniqueness for the kinetic Fokker-Planck equation with inelastic boundary conditions

We describe the structure of solutions of the kinetic Fokker-Planck equations in domains with boundaries near the singular set in one-space dimension. We study in particular the behaviour of the solutions of this equation for inelastic boundary conditions which are characterized by means of a coefficient $r$ describing the amount of energy lost in the collisions of the particles with the boundaries of the domain. A peculiar feature of this problem is the onset of a critical exponent rc which follows from the analysis of McKean (cf. [26]) of the properties of the stochastic process associated to the Fokker-Planck equation under consideration. In this paper, we prove rigorously that the solutions of the considered problem are nonunique if $r < r_c$ and unique if $r_{c}<r\leq 1$. In particular, this nonuniqueness explains the different behaviours found in the physics literature for numerical simulations of the stochastic differential equation associated to the Fokker-Planck equation. In the proof of the results of this paper we use several asymptotic formulas and computations in the companion paper [16].

math.AP↗

arXiv2018

On the structure of the singular set for the kinetic Fokker-Planck equations in domains with boundaries

In this paper we compute asymptotics of solutions of the kinetic Fokker-Planck equation with inelastic boundary conditions which indicate that the solutions are nonunique if $r < r_c$. The nonuniqueness is due to the fact that different solutions can interact in a different manner with a Dirac mass which appears at the singular point $(x,v)=(0,0)$. In particular, this nonuniqueness explains the different behaviours found in the physics literature for numerical simulations of the stochastic differential equation associated to the kinetic Fokker-Planck equation. The asymptotics obtained in this paper will be used in a companion paper [34] to prove rigorously nonuniqueness of solutions for the kinetic Fokker-Planck equation with inelastic boundary conditions.

math.AP↗

arXiv2018

Self-similar solutions to coagulation equations with time-dependent tails: the case of homogeneity smaller than one

We prove the existence of a one-parameter family of self-similar solutions with time-dependent tails for Smoluchowski's coagulation equation, for a class of rate kernels $K(x,y)$ which are homogeneous of degree $γ\in(-\infty,1)$ and satisfy $K(x,1)\sim x^{-a}$ as $x\to 0$, for $a=1-γ$. In particular, for small values of a parameter $ρ>0$ we establish the existence of a positive self-similar solution with finite mass and asymptotics $A(t)x^{-(2+ρ)}$ as $x\to\infty$, with $A(t)\simρt^\fracρ{1-γ}$.

math.AP↗

arXiv2017

Self-similar profiles for homoenergetic solutions of the Boltzmann equation: particle velocity distribution and entropy

In this paper we study a class of solutions of the Boltzmann equation which have the form $f\left( x,v,t\right) =g\left( v-L\left( t\right) x,t\right) $ where $L\left( t\right) =A\left( I+tA\right) ^{-1}$ with the matrix $A$ describing a shear flow or a dilatation or a combination of both. These solutions are known as homoenergetic solutions. We prove existence of homoenergetic solutions for a large class of initial data. For different choices for the matrix $A$ and for different homogeneities of the collision kernel, we characterize the long time asymptotics of the velocity distribution for the corresponding homoenergetic solutions. For a large class of choices of $A$ we then prove rigorously the existence of self-similar solutions of the Boltzmann equation. The latter are non Maxwellian distributions and describe far-from-equilibrium flows. For Maxwell molecules we obtain exact formulas for the $H$-function for some of these flows. These formulas show that in some cases, despite being very far from equilibrium, the relationship between density, temperature and entropy is exactly the same as in the equilibrium case. We make conjectures about the asymptotics of homoenergetic solutions that do not have self-similar profiles.

math-ph↗

arXiv2017

Self-similar lifting and persistent touch-down points in the thin-film equation

We study self-similar solutions of the thin-film equation, with mobility exponent m in (0,4], that describe the lifting of an isolated touch-down point given by an initial profile of the form |x|. This provides a mechanism for non-uniqueness of the thin-film equation with m in (2,4), since solutions with a persistent touch-down point also exist in this case. In order to prove existence of the self-similar solutions, we need to study a four-dimensional continuous dynamical system. The proof consists of a shooting argument based on the identification of invariant regions and on suitable energy formulas.

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