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Ricardo J. Alonso

Publications and source records attributed to Ricardo J. Alonso.

At least 19 recordsLinked to original sources
arXiv2025

Conditional integrability and stability for the homogeneous Boltzmann equation with very soft potentials

We introduce a practical criterion that justifies the propagation and appearance of $L^{p}$-norms for the solutions to the spatially homogeneous Boltzmann equation with very soft potentials without cutoff. Such criterion also provides a new conditional stability result for classical solutions to the equation. All results are quantitative. Our approach is inspired by a recent analogous result for the Landau equation derived in arXiv:2306.15729 and generalises existing conditional results related to higher integrability properties and stability of solutions to the Boltzmann equation with very soft potentials.

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arXiv2023

The Cauchy problem for Boltzmann bi-linear systems: The mixing of monatomic and polyatomic gases

From a unified vision of vector valued solutions in weighted Banach spaces, this manuscript establishes the existence and uniqueness for space homogeneous Boltzmann bi-linear systems with conservative collisional forms arising in complex gas dynamical structures. This broader vision is directly applied to dilute multi-component gas mixtures composed of both monatomic and polyatomic gases. Such models can be viewed as extensions of scalar Boltzmann binary elastic flows, as much as monatomic gas mixtures with disparate masses and single polyatomic gases, providing a unify approach for vector valued solutions in weighted vector Banach spaces. Novel aspects of this work include developing the extension of a general ODE theory in vector valued weighted Banach spaces, precise lower bounds for the collision frequency in terms of the weighted Banach norm, energy identities and the consequently angular or compact manifold averaging lemmas which provide coerciveness resulting into global in time stability, a new combinatorics estimate for $p$-binomial forms producing sharper estimates for the $k$-moments of bi-linear collisional forms. These techniques enable the Cauchy problem improvement that resolves the model with initial data corresponding to strictly positive and bounded initial vector valued mass and total energy, in addition to only $2^+$ moment determined by the hard potential rates discrepancy, a result comparable in generality to the classical Cauchy theory of the scalar homogeneous Boltzmann equation.

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arXiv2023

The Boltzmann equation for hard potentials with integrable angular transition: Coerciveness, exponential tails rates, and Lebesgue integrability

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 focuses on the relevant hard potential case where the solution properties are studied with a modern take. While many of the discussed results can be found the literature spread over several papers along the years, we bring a complete program that includes a new approach to the existence and uniqueness theorem securing the well-posedness theory, to moments estimates, and integrability propagation for the homogeneous Boltzmann flow. In particular, a detailed calculation of classical polynomial moments upper bounds as function of the coerciveness is described, which characterized the rate of exponential moments obtained by summability of the polynomial ones. In addition, a proof of uniform propagation of $L^\infty$ regularity under general integrable scattering kernels is performed. Along the way, constants appearing in estimates are carefully calculated, improving most of previous exiting results in the literature. For the non expert reader we also include a general discussion of the basic elements of the Boltzmann model and important key results for the understanding of the mathematical discussion of the equation and include an extensive set of references that enrich and motivate further discussions on Boltzmann flows for broader gas modeling configuration such as gas mixtures systems, polyatomic gases, multilinear collisional forms such as, ternary or quartic, to those derived from symmetry braking quantum mean field theories or weak turbulence models from spectral energy waves in classical fluid.

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arXiv2022

One-dimensional inelastic Boltzmann equation: Regularity \& uniqueness of self-similar profiles for moderately hard potentials

We prove uniqueness of self-similar profiles for the one-dimensional inelastic Boltzmann equation with moderately hard potentials, that is with collision kernel of the form | $\bullet$ | $γ$ for $γ$ > 0 small enough (explicitly quantified). Our 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 (corresponding to $γ$ = 0). Our approach relies on a perturbation argument from the corresponding Maxwell model through a careful study of the associated linearised operator. In particular, a part of the paper is devoted to the trend to equilibrium for the Maxwell model in suitable weighted Sobolev spaces, an extension of results which are known to hold in weaker topologies. Our results can be seen as a first step towards a full proof, in the one-dimensional setting, of a conjecture in Ernst \& Brito (2002) regarding the determination of the long-time behaviour of solutions to inelastic Boltzmann equation.

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arXiv2021

From Boltzmann equation for granular gases to a modified Navier-Stokes-Fourier system

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 aforementioned system. One of the main issue is to identify the correct relation between the restitution coefficient (which quantifies the rate of energy loss at the microscopic level) and the Knudsen number which allows us to obtain non trivial hydrodynamic behavior. In such a regime, we construct strong solutions to the inelastic Boltzmann equation, near thermal equilibrium whose role is played by the so-called homogeneous cooling state. We prove then the uniform exponential stability with respect to the Knudsen number of such solutions, using a spectral analysis of the linearized problem combined with technical a priori nonlinear estimates. Finally, we prove that such solutions converge, in a specific weak sense, towards some hydrodynamic limit that depends on time and space variables only through macroscopic quantities that satisfy a suitable modification of the incompressible Navier-Stokes-Fourier system.

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arXiv2021

Fluid dynamic limit of Boltzmann equation for granular hard--spheres in a nearly elastic regime

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 describes rapid granular flows. To do that, we write our Boltzmann equation in nondimensional form introducing the dimensionless Knudsen number which is intended to tend to 0. The difficulties are then manyfold, the first one coming from the fact that the original Boltzmann equation is free-cooling and thus requires a self-similar change of variables to work with an equation that has an homogeneous steady state. The latter is not explicit and is heavy-tailed, which is a major obstacle to adapt energy estimates and spectral analysis. One of the main challenges here is to understand the relation between the restitution coefficient (which quantifies the loss of energy at the microscopic level) and the Knudsen number. This is done identifying the correct nearly elastic regime to capture nontrivial hydrodynamic behavior. We are then able to prove exponential stability uniformly with respect to the Knudsen number of the solution of our rescaled Boltzmann equation in a close to equilibrium regime. Finally, we prove that our solution to the Boltzmann equation converges in some very specific weak sense towards some hydrodynamic solution which depends on time and space variables only through macroscopic quantities. Such macroscopic quantities are solutions to a suitable modification of the incompressible Navier-Stokes-Fourier system which appears to be new in this context.

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arXiv2020

About the use of entropy production for the Landau-Fermi-Dirac equation

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 the soft potential case). An important feature of such estimates is that they are uniform with respect to the quantum parameter. Consequently, the same estimations are recovered for the classical limit, that is the Landau equation.

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arXiv2019

Convergence and error estimates for the Lagrangian based Conservative Spectral method for Boltzmann Equations

We develop error estimates for the semi-discrete conservative spectral method for the approximation of the elastic and inelastic space homogeneous Boltzmann equation introduced by the authors in \cite{GT09}. In addition we study the long time convergence of such semi-discrete solution to equilibrium Maxwellian distribution that conserves the mass, momentum and energy associated to the initial data. The numerical method is based on the Fourier transform of the collisional operator and a Lagrangian optimization correction that enforces the collision invariants, namely conservation of mass, momentum and energy in the elastic case, and just mass and momentum in the inelastic one. We present a detailed semi-discrete analysis on convergence of the proposed numerical method which includes the $L^{1}-L^{2}$ theory for the scheme. This analysis allows us to present, additionally, convergence in Sobolev spaces and convergence to equilibrium for the numerical approximation. The results of this work answer a long standing open problem posed by Cercignani et al. in \cite[Chapter 12]{CIP} about finding error estimates for a numerical scheme associated to the Boltzmann equation, as well as showing the semi-discrete numerical solution converges to the equilibrium Maxwellian distribution associated to the initial value problem.

math.NA↗
arXiv2019

Uniform estimates on the Fisher information for solutions to Boltzmann and Landau equations

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 diffusion properties for the Landau equation. We include an application of this result related to emergence and propagation of exponential tails for the solution's gradient. These results complement estimates provided in the literature.

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arXiv2017

On Mittag-Leffler moments for the Boltzmann equation for hard potentials without cutoff

We establish the $L^1$ weighted propagation properties for solutions of the Boltzmann equation with hard potentials and non-integrable angular components in the collision kernel. Our method identifies null forms by angular averaging and deploys moment estimates of solutions to the Boltzmann equation whose summability is achieved by introducing the new concept of Mittag-Leffler moments - extensions of $L^1$ exponentially weighted norms. Such $L^1$ weighted norms of solutions to the Boltzmann equation are, both, generated and propagated in time and the characterization of their corresponding Mittag-Leffler weights depends on the angular singularity and potential rates in the collision kernel. These estimates are a fundamental step in order to obtain $L^\infty$ exponentially weighted estimates for solutions of the Boltzmann equation being developed in a follow up work.

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arXiv2013

Boltzmann Model for viscoelastic particles: asymptotic behavior, pointwise lower bounds and regularity

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 can be seen as the generalization of several classical facts holding for the pseudo-Maxwellian and constant normal restitution models.

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arXiv2011

Uniqueness and regularity of steady states of the Boltzmann equation for viscoelastic hard-spheres driven by a thermal bath

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 regularity does not requires a priori knowledge of the time-dependent problem. Furthermore, the uniqueness result is obtained in the small thermalization regime by studying the so-called quasi-elastic limit for the problem. An important new aspect lies in the fact that no entropy functional inequality is needed in the limiting process.

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arXiv2011

Two proofs of Haff's law for dissipative gases: the use of entropy and the weakly inelastic regime

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 second proof uses only the moments of the solutions and holds in some weakly inelasticity regime which has to be clearly defined whenever the restitution coefficient is non-constant

math.AP↗
arXiv2010

Free cooling and high-energy tails of granular gases with variable restitution coefficient

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 Boltzmann equation for granular gases and precise Lp estimates in the selfsimilar variables. In the process, we generalize several results on the Boltzmann collision operator obtained recently for homogeneous granular gases with constant restitution coefficient to a broader class of physical restitution coefficients that depend on the collision impact velocity. This generalization leads to the so-called L1-exponential tails theorem. for this model.

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arXiv2010

Convolution inequalities for the Boltzmann collision operator

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 symmetrization technique in $L^p$ we prove a Young's inequality for hard potentials, which is sharp for Maxwell molecules in the $L^2$ case. Further, we find a new Hardy-Littlewood-Sobolev type of inequality for Boltzmann collision integrals with soft potentials. The same method extends to radially symmetric, non-increasing potentials that lie in some $L^{s}_{weak}$ or $L^{s}$. The method we use resembles a Brascamp, Lieb and Luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting. Consequently, it is closely connected to the classical analysis of Young and Hardy-Littlewood-Sobolev inequalities. In all cases, the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section (in the spirit of Grad cut-off). As an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions.

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arXiv2010

Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section

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 elementary proof of existence and uniqueness results that includes large data near a local Maxwellian regime with possibly infinite initial mass. We study the propagation of regularity using a recent estimate for the positive collision operator given in [3], by E. Carneiro and the authors, that permits to study such propagation without additional conditions on the collision kernel. Finally, an $L^{p}$-stability result (with $1\leq p\leq\infty$) is presented assuming the aforementioned condition.

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arXiv2009

Estimates for the Boltzmann collision operator via radial symmetry and Fourier transform

We extend the $L^p$-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 constants in some convolution-like inequalities for the gain part of the Boltzmann collision operator.

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