The goal of this paper is to extend the existence result of measure-valued solution to the Boltzmann equation in elastic interaction, given by Morimoto-Wang-Yang, to the inelastic Boltzmann equation with moderately soft potentials, also as an extensive work of our preceding result in the inelastic Maxwellian molecules case. We prove the existence and uniqueness of measure-valued solution under Grad's angular cutoff assumption, as well as the existence of non-cutoff solution, for both finite and infinite energy initial datum, by a delicate compactness argument. In addition, the moments propagation and energy dissipation properties are justified for the obtained measure-valued solution as well.
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 show the existence of smooth stationary solutions for the inelastic Boltzmann equation under the thermalization induced by a host-medium with a fixed distribution. This is achieved by controlling the Lp-norms, the moments and the regularity of the solutions for the Cauchy problem together with arguments related to a dynamical proof for the existence of stationary states.
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 line…
We consider the spatially inhomogeneous Boltzmann equation for inelastic hard-spheres, with constant restitution coefficient $α\in(0,1)$, under the thermalization induced by a host medium with a fixed Maxwellian distribution and any fixed $e\in(0,1]$. When the restitution coefficient $α$ is close to 1 we prove existence and uniqueness of global solutions considering the close-to-equilibrium regime. We also study the long-time behaviour of these solutions and prove a convergence to equilibrium with an exponential rate.
We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres, in the framework of so-called constant normal restitution coefficients. We prove the existence of self-similar solutions, and we give pointwise estimates on their tail. We also give general estimates on the tail and the regularity of generic solutions. In particular we prove Haff 's law on the rate of decay of temperature, as well as the algebraic decay of singularities. The proofs are based on the regularity study of a rescaled problem, with the help of the regularity properties of the gain part of the Boltzm…
We develop the Cauchy theory of the spatially homogeneous inelastic Boltzmann equation for hard spheres, for a general form of collision rate which includes in particular variable restitution coefficients depending on the kinetic energy and the relative velocity as well as the sticky particles model. We prove (local in time) non-concentration estimates in Orlicz spaces, from which we deduce weak stability and existence theorem. Strong stability together with uniqueness and instantaneous appearance of exponential moments are proved under additional smoothness assumption on the initial datum, fo…
We study the high-energy asymptotics of the steady velocity distributions for model systems of granular media in various regimes. The main results obtained are integral estimates of solutions of the hard-sphere Boltzmann equations, which imply that the velocity distribution functions $f(v)$ behave in a certain sense as $C\exp(-r|v|^s)$ for $|v|$ large. The values of $s$, which we call {\em the orders of tails}, range from $s=1$ to $s=2$, depending on the model of external forcing. The method we use is based on the moment inequalities and careful estimating of constants in the integral form of …
We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres (with constant restitution coefficient $α\in (0,1)$) under the thermalization induced by a host medium with a fixed Maxwellian distribution. We prove that the solution to the associated initial-value problem converges exponentially fast towards the unique equilibrium solution. The proof combines a careful spectral analysis of the linearised semigroup as well as entropy estimates. The trend towards equilibrium holds in the weakly inelastic regime in which $α$ is close to $1$, and the rate of convergence is expli…
We consider the spatially homogeneous Boltzmann equation for inelastic hard-spheres (with constant restitution coefficient $α\in (0,1)$) under the thermalization induced by a host medium with a fixed Maxwellian distribution. We prove uniqueness of the stationary solution (with given mass) in the weakly inelastic regime; i.e., for any inelasticity parameter $α\in (α_0,1)$, with some constructive $α_0 \in [0, 1)$. Our analysis is based on a perturbative argument which uses the knowledge of the stationary solution in the elastic limit and quantitative estimates of the convergence of stationary so…
We consider the linear dissipative Boltzmann equation describing inelastic interactions of particles with a fixed background. For the simplified model of Maxwell molecules first, we give a complete spectral analysis, and deduce from it the optimal rate of exponential convergence to equilibrium. Moreover we show the convergence to the heat equation in the diffusive limit and compute explicitely the diffusivity. Then for the physical model of hard spheres we use a suitable entropy functional for which we prove explicit inequality between the relative entropy and the production of entropy to get …
A uniform approach is introduced to study the existence of measure valued solutions to the homogeneous Boltzmann equation for both hard potential with finite energy, and soft potential with finite or infinite energy, by using Toscani metric. Under the non-angular cutoff assumption on the cross-section, the solutions obtained are shown to be in the Schwartz space in the velocity variable as long as the initial data is not a single Dirac mass without any extra moment condition for hard potential, and with the boundedness on moments of any order for soft potential.
We consider the spatially homogeneous Boltzmann equation for {\em inelastic hard spheres}, in the framework of so-called {\em constant normal restitution coefficients} $α\in [0,1]$. In the physical regime of a small inelasticity (that is $α\in [α_*,1)$ for some constructive $α_*>0$) we prove uniqueness of the self-similar profile for given values of the restitution coefficient $α\in [α_*,1)$, the mass and the momentum; therefore we deduce the uniqueness of the self-similar solution (up to a time translation). Moreover, if the initial datum lies in $L^1_3$, and under some smallness condition on…
We consider a space-homogeneous gas of {\it inelastic hard spheres}, with a {\it diffusive term} representing a random background forcing (in the framework of so-called {\em constant normal restitution coefficients} $α\in [0,1]$ for the inelasticity). In the physical regime of a small inelasticity (that is $α\in [α_*,1)$ for some constructive $α_* \in [0,1)$) we prove uniqueness of the stationary solution for given values of the restitution coefficient $α\in [α_*,1)$, the mass and the momentum, and we give various results on the linear stability and nonlinear stability of this stationary solut…
In this work, we are interested in the spectrum of the diffusively excited granular gases equation, in a space inhomogeneous setting, linearized around an homogeneous equilibrium. We perform a study which generalizes to a non-hilbertian setting and to the inelastic case the seminal work of Ellis and Pinsky about the spectrum of the linearized Boltzmann operator. We first give a precise localization of the spectrum, which consists in an essential part lying on the left of the imaginary axis and a discrete spectrum, which is also of nonnegative real part for small values of the inelasticity para…
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 or…
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 value problem associated to the afore mentioned problem. More specifically, we extend to all derivatives of the initial value problem associated to this class of Boltzmann equations corresponding sharp moment (Povzner) inequalities and time propagati…
We show that the Euclidean Wasserstein distance is contractive for inelastic homogeneous Boltzmann kinetic equations in the Maxwellian approximation and its associated Kac-like caricature. This property is as a generalization of the Tanaka theorem to inelastic interactions. Consequences are drawn on the asymptotic behavior of solutions in terms only of the Euclidean Wasserstein distance.
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 …
We study the Boltzmann equation for a space-homogeneous gas of inelastic hard spheres, with a diffusive term representing a random background forcing. Under the assumption that the initial datum is a nonnegative $L^2$ function, with bounded mass and kinetic energy (second moment), we prove the existence of a solution to this model, which instantaneously becomes smooth and rapidly decaying. Under a weak additional assumption of bounded third moment, the solution is shown to be unique. We also establish the existence (but not uniqueness) of a stationary solution. In addition we show that the hig…
In this paper, we will introduce a precise classification of characteristic functions in the Fourier space according to the moment constraint in the physical space of any order. Based on this, we construct measure valued solutions to the homogeneous Boltzmann equation with the exact moment condition as the initial data.
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…
We introduce and discuss spatially homogeneous Maxwell-type models of the nonlinear Boltzmann equation undergoing binary collisions with a random component. The random contribution to collisions is such that the usual collisional invariants of mass, momentum and energy do not hold pointwise, even if they all hold in the mean. Under this assumption it is shown that, while the Boltzmann equation has the usual conserved quantities, it possesses a steady state with power-like tails for certain random variables. A similar situation occurs in kinetic models of economy recently considered by two of t…
We investigate in this article the long-time behaviour of the solutions to the energy-dependant, spatially-homogeneous, inelastic Boltzmann equation for hard spheres. This model describes a diluted gas composed of hard spheres under statistical description, that dissipates energy during collisions. We assume that the gas is "anomalous", in the sense that energy dissipation increases when temperature decreases. This allows the gas to cool down in finite time. We study existence and uniqueness of blow up profiles for this model, together with the trend to equilibrium and the cooling law associat…
We establish an instantaneous smoothing property for decaying solutions on the half-line $(0,+\infty)$ of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of $H^1$ stable manifolds of such equations, showing that $L^2_{loc}$ solutions that remain sufficiently small in $L^\infty$ (i) decay exponentially, and (ii) are $C^\infty$ for $t>0$, hence lie eventually in the $H^1$ stable manifold constructed by Pogan and …
The purpose of this paper is to introduce a new characterization of the characteristic functions for the study on the measure valued solution to the homogeneous Boltzmann equation so that it precisely captures the moment constraint in physics. This significantly improves the previous result by Cannone-Karch [CPAM 63(2010), 747-778] in the sense that the new characterization gives a complete description of infinite energy solutions for the Maxwellian cross section. In addition, the global in time smoothing effect of the infinite energy solution except for a single Dirac mass initial datum is ju…
The spatially homogeneous Boltzmann equation with hard potentials is considered for measure valued initial data having finite mass and energy. We prove the existence of \emph{weak measure solutions}, with and without angular cutoff on the collision kernel; the proof in particular makes use of an approximation argument based on the Mehler transform. Moment production estimates in the usual form and in the exponential form are obtained for these solutions. Finally for the Grad angular cutoff, we also establish uniqueness and strong stability estimate on these solutions.
We characterize probability measure with finite moment of any order in terms of the symmetric difference operators of their Fourier transforms. By using our new characterization, we prove the continuity $f(t,v)\in C((0, \infty),L^1_{2k-2 +α})$, where $f(t, v)$ stands for the density of unique measure-valued solution $(F_t)_{t\ge0}$ of the Cauchy problem for the homogeneous non-cutoff Boltzmann equation, with Maxwellian molecules, corresponding to a probability measure initial datum $F_0$ satisfying \[ \int |v|^{2k-2+α} dF_0(v) < \infty, 0\leq α< 2,k= 2, 3, 4,\cdots \] provided that $F_0$ is no…
Recently linear dissipative models of the Boltzmann equation have been introduced. In this work, we consider the problem of constructiing suitable hydrodynamic approximations for such models where the mean velocity and the temperature of inelastic particles appear as independent variables.
In this paper, we consider the spatially inhomogeneous diffusively driven inelastic Boltzmann equation in different cases: the restitution coefficient can be constant or can depend on the impact velocity (which is a more physically relevant case), including in particular the case of viscoelastic hard spheres. In the weak thermalization regime, i.e. when the diffusion parameter is sufficiently small, we prove existence of global solutions considering the close-to-equilibrium regime as well as the weakly inhomogeneous regime in the case of a constant restitution coefficient. It is the very first…
In this work, we prove global existence of solutions for second order differential problems in a general framework. More precisely, we consider second order differential inclusions involving proximal normal cone to a set-valued map. This set-valued map is supposed to take admissible values (so in particular uniformly prox-regular values, which may be non-smooth and non-convex). Moreover we require the solution to satisfy an impact law, appearing in the description of mechanical systems with inelastic shocks.
