We prove the stability of L-1 self-similar profiles under the hard-to-Maxwell potential limit for the one-dimensional inelastic Boltzmann equation with moderately hard potentials, which, in turn, leads to the uniqueness of such profiles for hard potential collision kernels of the form |& sdot;|(gamma) with gamma > 0 sufficiently small (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 gamma = 0). Our approach relies on a perturbation argument from the corresponding Maxwell model and a careful study of the associated linearized operator recently derived in the companion paper [1]. The results can be seen as a first step towards a complete proof, in the one-dimensional setting, of a conjecture in [2] regarding the determination of the long-time behavior of solutions to the inelastic Boltzmann equation, at least in a regime of moderately hard potentials.

We study the dynamic relaxation to equilibrium of the 1D dissipative Boltzmann equation with Maxwell interactions in classical H^s Sobolev spaces. In addition, we present a spectral shrinkage analysis and spectral gap estimates for the linearised 1D dissipative Boltzmann operator with such interactions. Based on this study, we explore the convergence in H^s and L^1 spaces for the linear and nonlinear models. This study extends classical results found in the literature given for spaces with weak topologies.

This document presents an elementary approach using $\varepsilon$-Poincar\'e inequality to prove generation of $L^{p}$-bounds, $p\in(1,\infty)$, for the homogeneous Landau equation with moderate soft potentials $\gamma\in[-2,0)$. The critical case $\gamma=-2$ uses an interpolation approach in the realm of Lorentz spaces and entropy. Alternatively, a direct approach using the Hardy-Littlewood-Sobolev (HLS) inequality and entropy is also presented. On this basis, the generation of pointwise bounds $p=\infty$ is deduced from a De Giorgi argument.

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.

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 | $\bullet$ | $\gamma$ for $\gamma$ > 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 $\gamma$ = 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.

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, $$L^{p}$$ -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.

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.

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.

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 sense that all observables, hence solutions, vanish as time progresses. Following a contribution , by two of the authors, considering well posedness of the steady self-similar profile in the regime of small annihilation rate $\alpha$ $\ll$ 1, we prove here that such self-similar profile is the intermediate asymptotic attractor to the annihilation dynamics with explicit universal algebraic rate. This settles the issue about universality of the annihilation rate for this model brought in the applied literature.

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.

This manuscript investigates the following aspects of the one-dimensional dissipative Boltzmann equation associated to a 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) we give existence and uniqueness of measure solutions, and (3) we prove the existence of a nontrivial self-similar profile, i.e., homogeneous cooling state, after appropriate scaling of the equation. The latter issue is based on compactness tools in the set of Borel measures. More specifically, we apply a dynamical fixed point theorem on a suitable stable set, for the model dynamics, of Borel measures.

We prove the uniqueness of the self-similar profile solution for a modified Boltzmann equation describing probabilistic ballistic annihilation. Such a model describes a system of hard spheres such that, whenever two particles meet, they either annihilate with probability α∈(0,1) or they undergo an elastic collision with probability 1−α. The existence of a self-similar profile for α smaller than an explicit threshold value α_1 has been obtained in our previous contribution [6]. We complement here our analysis of such a model by showing that, for some α♯ explicit, the self-similar profile is unique for α∈(0,α♯).

We point out a few mistakes in our earlier paper [2] and propose some corrections. Our main results remain valid provided some more restrictive assumptions.

We show the existence of a self-similar solution for a modified Boltzmann equation describing probabilistic ballistic annihilation. Such a model describes a system of hard spheres such that, whenever two particles meet, they either annihilate with probability α∈(0,1) or they undergo an elastic collision with probability 1−α. For such a model, the number of particles, the linear momentum and the kinetic energy are not conserved. We show that, for α smaller than some explicit threshold value α⁎, a self-similar solution exists.

We consider here a Kac equation with a Gaussian thermostat in the case of a non-cutoff cross section. Under the sole assumptions of finite mass and finite energy for the initial data, we prove the existence of a global in time solution for which mass and energy are preserved. Then, via Fourier transform techniques, we show that this solution is smooth, unique and converges to the corresponding stationary state.

We study the stationary Kac equation in the presence of an external force field and a Gaussian thermostat and investigate the behaviour of a solution to this equation for varying field strength. We prove that for a weak field, the stationary density is continuous, but for a strong field the density has a singularity. Monte Carlo simulations illustrating the same are also presented.

The existence of self-similar solutions with a finite first moment is established for the Oort–Hulst–Safronov coagulation equation when the coagulation kernel is given by $a(y,y_*)=y^\lambda+y_*^\lambda$ for some $\lambda\in (0,1)$. The corresponding self-similar profiles are compactly supported and have a discontinuity at the edge of their support.

In this paper, we are interested in the derivation of macroscopic equations from kinetic ones using a moment method in a relativistic framework. More precisely, we establish the general form of moments that are compatible with the Lorentz invariance and derive a hierarchy of relativistic moment systems from a Boltzmann kinetic equation. The proof is based on the representation theory of Lie algebras. We then extend this derivation to the classical case and general families of moments that obey the Galilean invariance are also constructed. It is remarkable that the set of formal classical limits of the so-obtained relativistic moment systems is not identical to the set of classical moments quoted in Ref. 21 and one could use a new physically relevant criterion to derive suitable moment systems in the classical case. Finally, the ultra-relativistic limit is considered.

The existence of self-similar solutions with a finite first moment is established for the Oort-Hulst-Safronov coagulation equation when the coagulation kernel is given by a(y,y ) = y + y for some 2 (0, 1). The corresponding self-similar profiles are compactly supported and have a discontinuity at the edge of their support.