Related papers: Convolution inequalities for the Boltzmann collisi…
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…
We study the quantitative pointwise behavior of the solutions of the linearized Boltzmann equation for hard potentials, Maxwellian molecules and soft potentials, with Grad's angular cutoff assumption. More precisely, for solutions inside…
The linearized Boltzmann collision operator is fundamental in many studies of the Boltzmann equation and its main properties are of substantial importance. The decomposition into a sum of a positive multiplication operator, the collision…
The purpose of this note is to demonstrate the announced result in [Loher, The Strong Harnack inequality for the Boltzmann equation, S\'eminaire Laurent Schwartz proceeding] by filling the gap in the proof sketch. We prove the semi-local…
The Hardy-Littlewood maximal operator satisfies the classical Sawyer-type estimate $$ \left \Vert \frac{Mf}{v}\right \Vert_{L^{1,\infty}(uv)} \leq C_{u,v} \Vert f \Vert_{L^{1}(u)}, $$ where $u\in A_1$ and $uv\in A_{\infty}$. We prove a…
We study a three-weight inequality for the superposition of the Hardy operator and the Copson operator, namely \begin{equation*} \bigg(\int_a^b \bigg(\int_t^b \bigg(\int_a^s f(\tau)^p v(\tau) \,d\tau \bigg)^\frac{q}{p} u(s) \,ds…
We derive the 3D spatially homogeneous Boltzmann's equation with moderately soft potentials and singular angular interaction, from an interacting particles system. The collision kernel is of the form $B(z,\sigma)=|z|^{\gamma}b\left(…
We give quantitative estimates on the asymptotics of the linearized Boltzmann collision operator and its associated equation from angular cutoff to non cutoff. On one hand, the results disclose the link between the hyperbolic property…
The dynamics of dilute electrons can be modeled by the fundamental one-species Vlasov-Poisson-Boltzmann system which describes mutual interactions of the electrons through collisions in the self-consistent electrostatic field. For cutoff…
Motivated by the open problem of large-data global existence for the non-cutoff Boltzmann equation, we introduce a model equation that in some sense disregards the anisotropy of the Boltzmann collision kernel. We refer to this model…
We consider the spatially inhomogeneous non-cutoff Boltzmann equation with moderately soft potentials and any singularity parameter $s\in (0,1)$, i.e. with $\gamma+2s\in(0,2]$ on the whole space $\mathbb{R}^3$. We prove that if the initial…
We consider functions L_p-integrable with Jacobi weights on [-1,1] and prove Hardy--Littlewood type inequalities for fractional integrals. As applications, we obtain the sharp (L_p, L_q) Ulyanov-type inequalities for the Ditzian--Totik…
Let $1\leq p <\infty$ and $0 < q,r < \infty$. We characterize validity of the inequality for the composition of the Hardy operator, \begin{equation*} \bigg(\int_a^b \bigg(\int_a^x \bigg(\int_a^t f(s)ds \bigg)^q u(t) dt \bigg)^{\frac{r}{q}}…
This work is devoted to the analysis of the linear Boltzmann equation in a bounded domain, in the presence of a force deriving from a potential. The collision operator is allowed to be degenerate in the following two senses: (1) the…
We investigate the large time behavior of solutions to the spatially homogeneous linear Boltzmann equation from a semigroup viewpoint. Our analysis is performed in some (weighted) $L^{1}$-spaces. We deal with both the cases of hard and soft…
Let $t\in(0,\infty)$, $p\in(1,\infty)$, $q\in[1,\infty]$, $w\in A_p$ and $v\in A_q$. We introduce the weighted amalgam space $(L^p,L^q)_t(\mathbb R^n)$ and show some properties of it. Some estimates on these spaces for the classical…
In this paper, we investigate the problems of Cauchy theory and exponential stability for the inhomogeneous Boltzmann equation without angular cut-off. We only deal with the physical case of hard potentials type interactions (with a…
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,…
Boundary effects are central to the dynamics of the dilute particles governed by Boltzmann equation. In this paper, we study both the diffuse reflection and the specular reflection boundary value problems for Boltzmann equation with soft…
In this dissertation we explore the $[L^{\mathrm{p}},\ L^{q}]$-boundedness of certain integral operators on weighted spaces on cones in ${\mathbb R}^{n}.$ These integral operators are of the type $\displaystyle \int_{V}k(x,\ y)f(y)dy$…