Ricardo Jose Alonso

Texas A&M University at Qatar | TAMU Qatar · Arts and Sciences

We study an inverse source problem for the acoustic wave equation in a random waveguide. The goal is to estimate the source of waves from measurements of the acoustic pressure at a remote array of sensors. The waveguide effect is due to boundaries that trap the waves and guide them in a preferred (range) direction, the waveguide axis, along which t...

In this work we study the radiative transfer equation in the forward-peaked regime in free space. Specifically, it is shown that the equation is well-posed by proving instantaneous regularization of weak solutions for arbitrary initial datum in L 1. Classical techniques for hypo-elliptic operators, such as averaging lemma, are used in the argument....

We study long range propagation of electromagnetic waves in random waveguides with rectangular cross-section and perfectly conducting boundaries. The waveguide is filled with an isotropic linear dielectric material, with randomly fluctuating electric permittivity. The fluctuations are weak, but they cause significant cumulative scattering over long...

In this paper we study the properties of a doubly nonlinear diffusion equation arising in shallow water flow models. Existence, uniqueness, some regularity results and conditions for positivity of classical solutions are presented for the zero Dirichlet initial/boundary value problem. The Faedo Galerkin method is used to approximate the solution an...

We investigate the long time behavior of a system of viscoelastic particles modeled with the homogeneous Boltzmann equation. We prove the existence of a universal Maxwellian intermediate asymptotic state and explicit the rate of convergence towards it. Exponential lower pointwise bounds and propagation of regularity are also studied. These results...

This paper gives an affirmative answer to the question of the global existence of Boltzmann equations without angular cutoff in the $$L^\infty $$ L ∞ -setting. In particular, we show that when the initial data is close to equilibrium and the perturbation is small in $$L^2 \cap L^\infty $$ L 2 ∩ L ∞ with a polynomial decay tail, the Boltzmann equati...

This manuscript focus on an extensive survey with new techniques on the problem of solving the Boltzmann flow by bringing a unified approach to the Cauchy problem to homogeneous kinetic equations with Boltzmann-like collision operators under integrability assumption of the scattering profile in the particle-particle interaction mechanism. The work...

We prove uniqueness of self-similar profiles for the one-dimensional inelastic Boltzmann equation with moderately hard potentials. The result provides the first uniqueness statement for self-similar profiles of inelastic Boltzmann models allowing for strong inelasticity besides the explicitly solvable case of Maxwell interactions. This results can...

The goal of the present paper is to establish existence and uniqueness result for the coupled system of space homogeneous Boltzmann equations modelling a gas mixture composed of monatomic and polyatomic gases. We first prove estimates on statistical moments of the vector valued collision operator that accounts for different possible interactions am...

We present some essential properties of solutions to the homogeneous Landau-Fermi-Dirac equation for moderately soft potentials. Uniform in time estimates for statistical moments, Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage...

In this paper, we give an overview of the results established in Alonso (http://arxiv.org/org/abs/2008.05173, 2020) which provides the first rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres in 3D. In particular, we obtain a new system of hydrodynamic equations describing granular flows and prove e...

This document presents a priori estimates related to statistical moments and in-tegrability properties for solutions of systems of monatomic gas mixtures modelled with the homogeneous Boltzmann equation with long range interactions for hard potentials. We detail the conditions for the generation and propagation of polynomial and exponential moments...

In this paper, we give an overview of the results established in [3] which provides the first rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres in 3D. In particular, we obtain a new system of hydrodynamic equations describing granular flows and prove existence of classical solutions to the aforemen...

In this work, we discuss a situation which could lead to both wave turbulence and collective behavior kinetic equations. The wave turbulence kinetic models appear in the kinetic limit when the wave equations have local differential operators. Viewing wave equations on the lattice as chains of anharmonic oscillators and replacing the local different...

In this manuscript we go over the main ideas of the \(L^{r}\)-theory, with \(r\in [1,\infty ]\), for the homogeneous Boltzmann equation with Maxwell and hard potential kernels. A discussion for the cutoff and non-cutoff cases is presented in a concise manner. In the non-cutoff case the whole range of singularity is addressed.

We present in this document some essential properties of solutions to the homogeneous Landau-Fermi-Dirac equation for moderately soft potentials. Uniform in time estimates for statistical moments, $L^{p}$-norm generation and Sobolev regularity are shown using a combination of techniques that include recent developments concerning level set analysis...

In this paper, we present new estimates for the entropy dissipation of the Landau–Fermi–Dirac equation (with hard or moderately soft potentials) in terms of a weighted relative Fisher information adapted to this equation. Such estimates are used for studying the large time behaviour of the equation, as well as for providing new a priori estimates (...

In this document we discuss the long time behaviour for the homogeneous Landau-Fermi-Dirac equation in the hard potential case. Uniform in time estimates for statistical moments and Sobolev regularity are presented and used to prove exponential relaxation of non degenerate distributions to the Fermi-Dirac statistics. All these results are valid for...

In this paper, we present new estimates for the entropy dissipation of the Landau-Fermi-Dirac equation (with hard or moderately soft potentials) in terms of a weighted relative Fisher information adapted to this equation. Such estimates are used for studying the large time behaviour of the equation, as well as for providing new a priori estimates (...

We study the well-posedness and regularity theory for the Radiative Transfer equation in the peaked regime posed in the half-space. An average lemma for the transport equation in the half-space is established and used to generate interior regularity for solutions of the model. The averaging also shows a fractional regularisation gain up to the boun...

This paper fills in a gap of establishing global weighted $L^\infty$-solutions to the Boltzmann equation without angular cutoff. In order to overcome the difficulties arising from the singular cross-section and the low regularity, a De Giorgi type argument well developed for diffusion equations is crafted in this kinetic context with the help of th...

In this paper, we provide the first rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres with small inelasticity. The hydrodynamic system that we obtain is an incompressible Navier-Stokes-Fourier system with self-consistent forcing terms and is thus the first hydrodynamic system that properly describe...

In this manuscript we go over the main ideas of the Lr - theory, with r ∈ [1, ∞], for the homogeneous Boltzmann equation, Maxwell and hard potential models. A discussion for both, the cutoff and non-cutoff cases is presented in a concise manner. In the non-cutoff case the whole range of singularity is presented.

We study the well-posedness and regularity theory for the Radiative Transfer equation in the peaked regime posed in the half-space. An average lemma for the transport equation in the half-space is stablished and used to generate interior regularity for solutions of the model. The averaging also shows a fractional regularization gain up to the bound...

We present in this document a short discussion about the time asymptotic behaviour of dissipative systems with large number of particles. Two classical examples of non conservative phenomena are brought to the discussion, viscoelastic and reactive particles. Non linear techniques based on entropy and spectral analysis are used to describe in rigour...

In this document we discuss the long time behaviour for the homogeneous Landau-Fermi-Dirac equation in the hard potential case. Uniform in time estimates for statistical moments and Sobolev regularity are presented and used to prove exponential relaxation of non degenerate distributions to the Fermi-Dirac statistics. All these results are valid und...

In this document we discuss the long time behaviour for the homogeneous Landau-Fermi-Dirac equation in the hard potential case. Uniform in time estimates for statistical moments and Sobolev regularity are presented and used to prove exponential relaxation of non degenerate distributions to the Fermi-Dirac statistics. All these results are valid for...

In this note we prove that, under some minimal regularity assumptions on the initial datum, solutions to the spatially homogenous Boltzmann and Landau equations for hard potentials uniformly propagate the Fisher information. The proof of such a result is based upon some explicit pointwise lower bound on solutions to Boltzmann equation and strong di...

We consider the homogeneous Boltzmann equation for Maxwell and hard potentials, without cutoff, and study the appearance and propagation of Lp-norms, including polynomial and exponential weights. Propagation of Sobolev regularity with such weights is also considered. Classical and novel ideas are combined to elaborate an elementary argument that pr...

The Boltzmann equation without angular cutoff is considered when the initial data is a per turbation of a global Maxwellian with algebraic decay in the velocity variable. A well-posedness theory in the perturbation framework is obtained for strong angular singularity by combining the analysis in moment propagation, spectrum gap and the regularizing...

We consider the homogeneous Boltzmann equation for Maxwell and hard potentials, without cutoff, and study the appearance and propagation of Lp-norms, including polynomial and exponential weights. Propagation of Sobolev regularity with such weights is also considered. Classical and novel ideas are combined to elaborate an elementary argument that pr...

We consider the spatially homogeneous Boltzmann equation for ballistic annihilation in dimension d 2. Such model describes a system of ballistic hard spheres that, at the moment of interaction, either annihilate with probability $\alpha$ $\in$ (0, 1) or collide elastically with probability 1 -- $\alpha$. Such equation is highly dissipative in the s...

We present in this document the Lebesgue and Sobolev propagation of exponential tails for solutions of the homogeneous Boltzmann equation for hard and Maxwell interactions. In addition, we show the $L^{p}$-integrability creation of such tails in the case of hard interactions. The document also presents a result on exponentially-fast convergence to...

In this paper we study the approximation properties of the spectral conservative method for the elastic and inelastic Boltzmann problem introduced by the authors in \cite{GT09}. The method is based on the Fourier transform of the collisional operator and a Lagrangian optimization correction used for conservation of mass, momentum and energy. We pre...

We solve the Cauchy problem for a kinetic quantum Boltzmann model that approximates the evolution of a radial distribution of quasiparticles in a dilute gas of bosons at very low temperature with a cubic kinetic transition probability kernel. We classify some relevant qualitative properties of such solutions which include the propagation and creati...

We study generation and propagation properties of Mittag-Leffler moments for solutions of the spatially homogeneous Boltzmann equation for scattering collision kernels corresponding to hard potentials without angular Grad's cutoff assumption, i.e. the angular part of the scattering kernel is non-integrable with prescribed singularity rate. These ki...

The main goal of this notes is to present the reader an introduction to modern kinetic theory. We cover some of the influential result in the area and give a baseline for research initiation in this topic. The list of reference is, by no means, exhaustive, yet, it is a good initial step for further reading and cross–reference. These notes are divid...

This manuscript investigates the following aspects of the one dimensional dissipative Boltzmann equation associated to variable hard-spheres kernel: (1) we show the optimal cooling rate of the model by a careful study of the system satisfied by the solution's moments, (2) give existence and uniqueness of measure solutions, and (3) prove the existen...

In this paper we analyze a system of PDEs recently introduced in [P. Amorim, {\it Modeling ant foraging: a {chemotaxis} approach with pheromones and trail formation}], in order to describe the dynamics of ant foraging. The system is made of convection-diffusion-reaction equations, and the coupling is driven by chemotaxis mechanisms. We establish th...

We investigate the large time behavior of the solutions of a Vlasov–Fokker–Planck equation where particles are subjected to a confining external potential and a self–consistent potential intended to describe the interaction of the particles with their environment. The environment is seen as a medium vibrating in a direction transverse to particles'...

We study an inverse source problem for the acoustic wave equation in a random waveguide. The goal is to estimate the source of waves from measurements of the acoustic pressure at a remote array of sensors. The random waveguide is a model of perturbed ideal waveguides which have flat boundaries and are filled with known media that do not change with...

Dynamic, cross-linked, biological fiber networks play major roles in cell and tissue function. They are challenging structures to model due to the vast number of components and the complexity of the interactions within the structure. We present here a particle-based model for fiber networks inspired from flocking theory, where fibers are modeled as...

We give a detailed analysis of long range cumulative scattering effects from rough boundaries in waveguides. We assume small random fluctuations of the boundaries and obtain a quantitative statistical description of the wave field. The method of solution is based on coordinate changes that straighten the boundaries. The resulting problem is similar...

We study the creation and propagation of exponential moments of solutions to the spatially homogeneous $d$-dimensional Boltzmann equation. In particular, when the collision kernel is of the form $|v-v_*|^\beta b(\cos(\theta))$ for $\beta \in (0,2]$ with $\cos(\theta)= |v-v_*|^{-1}(v-v_*)\cdot \sigma$ and $\sigma \in \mathbb{S}^{d-1}$, and assuming...

We study the uniqueness and regularity of the steady states of the diffusively driven Boltzmann equation in the physically relevant case where the restitution coefficient depends on the impact velocity including, in particular, the case of viscoelastic hard-spheres. We adopt a strategy which is novel in several aspects, in particular, the study of...

We give a detailed analysis of long range cumulative scattering effects from rough boundaries in waveguides. We assume small random fluctuations of the boundaries and obtain a quantitative statistical description of the wave field. The method of solution is based on coordinate changes that straighten the boundaries. The resulting problem is similar...

This short note complements the recent paper of the authors (Alonso, Gamba in J. Stat. Phys. 137(5–6):1147–1165, 2009). We revisit the results on propagation of regularity and stability using L p estimates for the gain and loss collision operators which had the exponent range misstated for the loss operator. We show here the correct range of expon...

We revisit our recent contribution (SIAM J. Math. Analysis, 42 (2010) 2499--2538) and give two simpler proofs of the so-called Haff's law for granular gases (with non-necessarily constant restitution coefficient). The first proof is based upon the use of entropy and asserts that Haff's law holds whenever the initial datum is of finite entropy. The...

We revisit the gain of integrability property of the gain part of the Boltzmann collision operator. This property implies the W k l,r regularity propagation for solutions of the associated space homogeneous initial value problem. We present a new method to prove the gain of integrability that simplifies the technicalities of previous approaches by...

Echoes from small reflectors buried in heavy clutter are weak and difficult to distinguish from the medium backscatter. Detection and imaging with sensor arrays in such media requires filtering out the unwanted backscatter and enhancing the echoes from the reflectors that we wish to locate. We consider a filtering and detection approach based on th...

In this short note we revisit the gain of integrability property of the gain part of the Boltzmann collision operator. This property implies the W l,r k regularity propagation for solutions of the associated space homogeneous initial value problem. We present a new method to prove the gain of integrability that simplifies the technicalities of prev...

We extend the Lp-theory of the Boltzmann collision operator by using classical techniques based in the Carleman representation and Fourier analysis, allied to new ideas that exploit the radial symmetry of this operator. We are then able to greatly simplify existent technical proofs in this theory, extend the range, and obtain explicit sharp constan...

We prove the so-called generalized Haff's law yielding the optimal algebraic cooling rate of the temperature of a granular gas described by the homogeneous Boltzmann equation for inelastic interactions with non constant restitution coefficient. Our analysis is carried through a careful study of the infinite system of moments of the solution to the...

This paper focuses on the study of existence and uniqueness of distributional and classical solutions to the Cauchy Boltzmann problem for the soft potential case assuming S n−1 integrability of the angular part of the collision kernel (Grad cut-off assumption). For this purpose we revisit the Kaniel–Shinbrot iteration technique to present an elemen...

We study integrability properties of a general version of the Boltzmann collision operator for hard and soft potentials in n-dimensions. A reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution, where the weight is given by an operator invariant under rotations. Using a symmet...

In this paper, we study basic properties of the diffusive wave approximation of the shallow water equations (DSW). This equation is a doubly non-linear diffusion equation arising in shallow water flow models. It has been used as a model to simulate water flow driven mainly by gravitational forces and dominated by shear stress, that is, under unifor...

We consider the n-dimensional space homogeneous Boltzmann equation for elastic collisions for variable hard potentials with Grad (angular) cutoff. We prove sharp moment inequalities, the propagation of L 1-Maxwellian weighted estimates, and consequently, the propagation L ∞-Maxwellian weighted estimates to all derivatives of the initial value probl...

The Cauchy problem for the inelastic Boltzmann equation is studied for small data. Existence and uniqueness of mild and weak solutions is obtained for sufficiently small data that lies in the space of functions bounded by Maxwellians. The technique used to derive the result is the well known iteration process of Kaniel and Shinbrot.

We consider the $n$-dimensional space homogeneous Boltzmann equation for elastic collisions for variable hard potentials with Grad (angular) cutoff. We prove sharp moment inequalities, the propagation of $L^1$-Maxwellian weighted estimates, and consequently, the propagation $L^\infty$-Maxwellian weighted estimates to all derivatives of the initial...