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Bertrand Lods

Publications and source records attributed to Bertrand Lods.

At least 19 recordsLinked to original sources
arXiv2024

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.

math.AP↗
arXiv2024

Quantitative relaxation towards equilibrium for solutions to the Boltzmann-Fermi-Dirac equation with cutoff hard potentials

We provide the first quantitative result of convergence to equilibrium in the context of the spatially homogeneous Boltzmann-Fermi-Dirac equation associated to hard potentials interactions under angular cut-off assumption, providing an explicit - algebraic - rate of convergence to Fermi-Dirac steady solutions. This result complements the quantitative convergence result of Liu and Lu and is based upon new uniform-in-time-and-$\varepsilon$ $L^{\infty}$ bound on the solutions.

math.AP↗
arXiv2023

A priori estimates for solutions to Landau equation under Prodi-Serrin like criteria

In this paper, we introduce Prodi-Serrin like criteria which enable to provide a priori estimates for the solutions to the spatially homogeneous Landau equation for all classical soft potentials and dimensions $d \geq 3$. The physical case of Coulomb interaction in dimension $d=3$ is included in our analysis, which generalizes the work of \cite{silvestre}. Our approach is quantitative and does not require a preliminary knowledge of elaborate tools for nonlinear parabolic equations.

math.AP↗
arXiv2023

Hydrodynamic limits for kinetic equations preserving mass, momentum and energy: a spectral and unified approach in the presence of a spectral gap

Triggered by the fact that, in the hydrodynamic limit, several different kinetic equations of physical interest all lead to the same Navier-Stokes-Fourier system, we develop in the paper an abstract framework which allows to explain this phenomenon. The method we develop can be seen as a significant improvement of known approaches for which we fully exploit some structural assumptions on the linear and nonlinear collision operators as well as a good knowledge of the Cauchy theory for the limiting equation. {In particular, we fully exploit the fact that the collision operator is preserving both momentum and kinetic energy.} We adopt a perturbative framework in a Hilbert space setting and first develop a general and fine spectral analysis of the linearized operator and its associated semigroup. Then, we introduce a splitting adapted to the various regimes (kinetic, acoustic, hydrodynamic) present in the kinetic equation which allows, by a fixed point argument, to construct a solution to the kinetic equation and prove the convergence towards suitable solutions to the Navier-Stokes-Fourier system. Our approach is robust enough to treat, in the same formalism, the case of the Boltzmann equation with hard and moderately soft potentials, with and without cut-off assumptions, as well as the Landau equation for hard and moderately soft potentials in presence of a spectral gap. New well-posedness and strong convergence results are obtained within this framework. In particular, for initial data with algebraic decay with respect to the velocity variable, our approach provides the first result concerning the strong Navier-Stokes limit from Boltzmann equation without Grad cut-off assumption or Landau equation. The method developed in the paper is also robust enough to apply, at least at the linear level, to quantum kinetic equations for Fermi-Dirac or Bose-Einstein particles.

math.AP↗
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.

math.AP↗
arXiv2022

Convergence rate to equilibrium for conservative scattering models on the torus: a new tauberian approach

The object of this paper is to provide a new and systematic tauberian approach to quantitative long time behaviour of $C_{0}$-semigroups $\left(\mathcal{V}(t)\right)_{t \geq0}$ in $L^{1}(\mathbb{T}^{d}\times \mathbb{R}^{d})$ governing conservative linear kinetic equations on the torus with general scattering kernel $ {k}(v,v')$ and degenerate (i.e. not bounded away from zero) collision frequency $\sigma(v)=\int_{\mathbb{R}^{d}} {k}(v',v)m(\mathrm{d} v')$, (with $m(\mathrm{d} v)$ being absolutely continuous with respect to the Lebesgue measure). We show in particular that if $N_{0}$ is the maximal integer $s \geq0$ such that $$\frac{1}{\sigma(\cdot)}\int_{\mathbb{R}^{d}}k(\cdot,v)\sigma^{-s}(v)m(\mathrm{d} v) \in L^{\infty}(\mathbb{R}^{d})$$ then, for initial datum $f$ such that $\mathrm{d} s\int_{\mathbb{T}^{d}\times \mathbb{R}^{d}}|f(x,v)|\sigma^{-N_{0}}(v)\mathrm{d} x m(\mathrm{d} v) <\infty$ it holds $$\left\|\mathcal{V}(t)f-\varrho_{f}\Psi\right\|_{L^{1}}=\dfrac{{\epsilon}_{f}(t)}{(1+t)^{N_{0}-1}}, \qquad \varrho_{f}:= \int_{\mathbb{R}^{d}}f(x,v)m(\mathrm{d} v)$$ where $\Psi$ is the unique invariant density of $\left(\mathcal{V}(t)\right)_{t \geq0}$ and $\lim_{t\to\infty}{\epsilon}_{f}(t)=0$. We in particular provide a new criteria of the existence of invariant density. The proof relies on the explicit computation of the time decay of each term of the Dyson-Phillips expansion of $\mathcal{V}(t)$ and on suitable smoothness and integrability properties of the trace on the imaginary axis of Laplace transform of remainders of large order of this Dyson-Phillips expansion. Our construction resorts also on collective compactness arguments and provides various technical results of independent interest. Finally, as a by-product of our analysis, we derive essentially sharp ``subgeometric'' convergence rate for Markov semigroups associated to general transition kernels.

math.AP↗
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.

math.AP↗
arXiv2021

On eventual compactness of collisionless kinetic semigroups with velocities bounded away from zero

In this paper, we consider the long time behaviour of collisionless kinetic equation with stochastic diffuse boundary operators for velocities bounded away from zero. We show that under suitable reasonable conditions, the semigroup is eventually compact. In particular, without any irreducibility assumption, the semigroup converges exponentially to the spectral projection associated to the zero eigenvalue as $t \to \infty.$ This contrasts drastically to the case allowing arbitrarily slow velocities for which the absence of a spectral gap yields at most algebraic rate of convergence to equilibrium. Some open questions are also mentioned.

math.AP↗
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.

math.AP↗
arXiv2021

About the Landau-Fermi-Dirac equation with moderately soft potentials

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 the spirit of De Giorgi and refined entropy-entropy dissipation functional inequalities for the Landau collision operator which are extended to the case in question here. As a consequence of the analysis, we prove algebraic relaxation of non degenerate distributions towards the Fermi-Dirac statistics under a weak non saturation condition for the initial datum. All quantitative estimates are uniform with respect to the quantum parameter. They therefore also hold for the classical limit, that is the Landau equation.

math.AP↗
arXiv2021

Convergence Rate To Equilibrium For Collisionless Transport Equations With Diffuse Boundary Operators: A New Tauberian Approach

This paper provides a new tauberian approach to the study of quantitative time asymptotics of collisionless transport semigroups with general diffuse boundary operators. We obtain an (almost) optimal algebraic rate of convergence to equilibrium under very general assumptions on the initial datum and the boundary operator. The rate is prescribed by the maximal gain of integrability that the boundary operator is able to induce. The proof relies on a representation of the collisionless transport semigroups by a (kind of) Dyson-Phillips series and on a fine analysis of the trace on the imaginary axis of Laplace transform of remainders (of large order) of this series. Our construction is systematic and is based on various preliminary results of independent interest.

math.AP↗
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.

math.AP↗
arXiv2020

Contractivity for Smoluchowski's coagulation equation with solvable kernels

We show that the Smoluchowski coagulation equation with the solvable kernels $K(x,y)$ equal to $2$, $x+y$ or $xy$ is contractive in suitable Laplace norms. In particular, this proves exponential convergence to a self-similar profile in these norms. These results are parallel to similar properties of Maxwell models for Boltzmann-type equations, and extend already existing results on exponential convergence to self-similarity for Smoluchowski's coagulation equation.

math.AP↗
arXiv2019

Invariant density & time asymptotics for collisionless kinetic equations with partly diffuse boundary operators

This paper deals with collisionless transport equations in bounded open domains $Ω\subset \R^{d}$ $(d\geq 2)$ with $\mathcal{C}^{1}$ boundary $\partial Ω$, orthogonally invariant velocity measure $\bm{m}(\d v)$ with support $V\subset \R^{d}$ and stochastic partly diffuse boundary operators $\mathsf{H}$ relating the outgoing and incoming fluxes. Under very general conditions, such equations are governed by stochastic $C_{0}$-semigroups $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ on $% L^{1}(Ω\times V,\d x \otimes \bm{m}(\d v)).$ We give a general criterion of irreducibility of $% \left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ and we show that, under very natural assumptions, if an invariant density exists then $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ converges strongly (not simply in Cesarò means) to its ergodic projection. We show also that if no invariant density exists then $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ is \emph{sweeping} in the sense that, for any density $φ$, the total mass of $ U_{\mathsf{H}}(t)φ$ concentrates near suitable sets of zero measure as $ t\rightarrow +\infty .$ We show also a general weak compactness theorem of interest for the existence of invariant densities. This theorem is based on several results on smoothness and transversality of the dynamical flow associated to $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}.$

math.AP↗
arXiv2019

A Kac model for kinetic annihilation

In this paper we consider the stochastic dynamics of a finite system of particles in a finite volume (Kac-like particle system) which annihilate with probability $α\in (0,1)$ or collide elastically with probability $1-α$. We first establish the well-posedness of the particle system which exhibits no conserved quantities. We rigorously prove that, in some thermodynamic limit, a suitable hierarchy of kinetic equations is recovered for which tensorized solution to the homogenous Boltzmann with annihilation is a solution. For bounded collision kernels, this shows in particular that propagation of chaos holds true. Furthermore, we make conjectures about the limit behaviour of the particle system when hard-sphere interactions are taken into account.

math-ph↗
arXiv2019

Long time dynamics for the Landau-Fermi-Dirac equation with hard potentials

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 rather general initial datum. An important feature of the estimates is the independence with respect to the quantum parameter. Consequently, in the classical limit the same estimates are recovered for the Landau equation.

math.AP↗
arXiv2019

Kinetic description of a Rayleigh Gas with annihilation

In this paper, we consider the dynamics of a tagged point particle in a gas of moving hard-spheres that are non-interacting among each other. This model is known as the ideal Rayleigh gas. We add to this model the possibility of annihilation (ideal Rayleigh gas with annihilation), requiring that each obstacle is either annihilating or elastic, which determines whether the tagged particle is elastically reflected or removed from the system. We provide a rigorous derivation of a linear Boltzmann equation with annihilation from this particle model in the Boltzmann-Grad limit. Moreover, we give explicit estimates for the error in the kinetic limit by estimating the contributions of the configurations which prevent the Markovianity. The estimates show that the system can be approximated by the Boltzmann equation on an algebraically long time scale in the scaling parameter.

math-ph↗