In this paper, we prove some a priori stability estimates (in weighted Sobolev spaces) for the spatially homogeneous Boltzmann equation without angular cutoff (covering every physical collision kernels). These estimates are conditioned to some regularity estimates on the solutions, and therefore reduce the stability and uniqueness issue to the one of proving suitable regularity bounds on the solutions. We then prove such regularity bounds for a class of interactions including the so-called (non cutoff and non mollified) hard potentials and moderately soft potentials. In particular, we obtain the first result of global existence and uniqueness for these long-range interactions.
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…
For the spatially homogeneous Boltzmann equation with cutoff hard potentials it is shown that solutions remain bounded from above, uniformly in time, by a Maxwellian distribution, provided the initial data have a Maxwellian upper bound. The main technique is based on a comparison principle that uses a certain dissipative property of the linear Boltzmann equation. Implications of the technique to propagation of upper Maxwellian bounds in the spatially-inhomogeneous case are discussed.
We prove that the Cauchy problem associated to the radially symmetric spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules enjoys the same Gelfand-Shilov regularizing effect as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator.
By means of Banach fixed point theorem , the uniqueness of Boltzmann Equation generalizaled Solutions in Sobolev spaces in $L^1(\mathbb{R}^+\times \mathbb{R}^n\times \mathbb{R}^n)$, can be proved as well as Boltzmann Equation renormalized solutions uniqueness
We consider the rate of convergence of solutions of spatially inhomogenous Boltzmann equations, with hard sphere potentials, to some equilibriums, called Maxwellians. Maxwellians are spatially homogenous static Maxwell velocity distributions with different temperatures and mean velocities. We study solutions in weighted space $L^{1}(\mathbb{R}^{3}\times \mathbb{T}^3)$. We prove a conjecture of C. Villani: assume the solution is sufficiently localized and sufficiently smooth, then the solution, in $L^{1}$-space, converges to a Maxwellian, exponentially fast in time.
For the spatially homogeneous Boltzmann equation with hard po- tentials and Grad's cutoff (e.g. hard spheres), we give quantitative estimates of exponential convergence to equilibrium, and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation, on which we provide a lower bound. Our approach is based on establishing spectral gap-like estimates valid near the equilibrium, and then connecting the latter to the quantitative nonlinear theory. This leads us to an explicit study of the linearized Boltzmann collision operator in functional spaces larger…
We consider the spatially homogeneous Boltzmann equation for regularized soft potentials and Grad's angular cutoff. We prove that uniform (in time) bounds in $L^1 ((1 + |v|^s)dv)$ and $H^k$ norms, $s, k \ge 0$ hold for its solution. The proof is based on the mixture of estimates of polynomial growth in time of those norms together with the quantitative results of relaxation to equilibrium in $L^1$ obtained by the so-called "entropy-entropy production" method in the context of dissipative systems with slowly growing a priori bounds (see reference [14]).
In this paper, we prove the propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation. These $L^\infty$ bounds have been known to be a challenging open problem in relativistic kinetic theory. To accomplish this, we establish two types of estimates for the gain part of the collision operator: first, we prove a potential type estimate and a relativistic hyper-surface integral estimate. We then combine those estimates using the relativistic counter-part of the Carleman representation to derive uniform control of the gain term for the relativistic collision …
The paper is concerned with the spatially homogeneous isotropic Boltzmann equation for Bose-Einstein particles with quantum collision kernel where the interaction potential $ϕ({\bf x})$ can be approximately written as the delta function plus a certain attractive potential such that the Fourier transform $\widehatϕ$ of $ϕ$ behaves like $0 \le \widehatϕ(ξ) \le {\rm const.} |ξ|^η$ for $|ξ|<<1$ for some constant $η\ge 1$. We prove that in this case, there is no condensation in finite time for all temperatures and all solutions, and thus it is completely different from the case $\widehatϕ(ξ) \ge {\…
We prove an inequality on the Kantorovich-Rubinstein distance --which can be seen as a particular case of a Wasserstein metric-- between two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, but with a moderate angular singularity. Our method is in the spirit of [7]. We deduce some well-posedness and stability results in the physically relevant cases of hard and moderately soft potentials. In the case of hard potentials, we relax the regularity assumption of [6], but we need stronger assumptions on the tail of the distribution (namely some exponential decay). We…
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 li…
In this paper, we consider a class of spatially homogeneous Boltzmann equation without angular cutoff. We prove that any radial symmetric weak solution of the Cauchy problem become analytic for positive time.
We solve the Cauchy problem associated to the space homogeneous Boltzmann equation with an angle-potential singular concentration modeling the collision kernel, proposed in 2013 by Bobylev and Potapenko. The potential under consideration ranges from Coulomb to hard spheres cases. However, the motivation of such a collision kernel is to treat the case of Coulomb potentials, on which this particular form of collision operator is well defined. We also show that the scaled angle-potential singular concentration in a grazing collisions limit makes the Boltzmann operator converge in the sense of dis…
We prove an inequality on the Wasserstein distance with quadratic cost between two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, from which we deduce some uniqueness results. In particular, we obtain a local (in time) well-posedness result in the case of (possibly very) soft potentials. A global well-posedeness result is shown for all regularized hard and soft potentials without angular cutoff. Our uniqueness result seems to be the first one applying to a strong angular singularity, except in the special case of Maxwell molecules. Our proof relies on the ide…
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 study the spatially homogeneous relativistic Boltzmann equation for massless particles in an FLRW background with scattering kernels in a certain range of soft and hard potentials. We obtain the future global existence of small solutions in a weighted $L^1\cap L^\infty$ space.
In this paper, we consider the spatially homogeneous Boltzmann equation without angular cutoff. We prove that every $L^1$ weak solution to the Cauchy problem with finite moments of all order acquires the $C^\infty$ regularity in the velocity variable for the positive time.
In this paper, we establish optimal hypoelliptic estimates for a class of kinetic equations, which are simplified linear models for the spatially inhomogeneous Boltzmann equation without angular cutoff.
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 this paper we consider a modified quantum Boltzmann equation with the quantum effect measured by a continuous parameter $δ$ that can decrease from $δ=1$ for the Fermi-Dirac particles to $δ=0$ for the classical particles. In case of soft potentials, for the corresponding Cauchy problem in the whole space or in the torus, we establish the global existence and uniqueness of non-negative mild solutions in the function space $L^{\infty}_{T}L^{\infty}_{v,x}\cap L^{\infty}_{T}L^{\infty}_{x}L^1_v$ with small defect mass, energy and entropy but allowed to have large amplitude up to the possibly maxi…
In this paper, we study spatially homogeneous solutions of the Boltzmann equation in special relativity and in Robertson-Walker spacetimes. We obtain an analogue of the Povzner inequality in the relativistic case and use it to prove global existence theorems. We show that global solutions exist for a certain class of collision cross sections of the hard potential type in Minkowski space and in spatially flat Robertson-Walker spacetimes.
The classification of equilibria for the spatially homogeneous Boltzmann equation for Fermi-Dirac particles is proved for any $n$-dimensional velocity space with $n\ge 2$. The same classification has been proven in \cite{Lu2001} for $n=3$. Now the proof for $n\ge 2$ is based on a recent result on a characterization of Euclidean balls for all dimensions $\ge 2$.
In this work, we study the Cauchy problem for the radially symmetric spatially homogeneous Boltzmann equation with Debye-Yukawa potential. We prove that this Cauchy problem enjoys the same smoothing effect as the Cauchy problem defined by the evolution equation associated to a fractional logarithmic harmonic oscillator. To be specific, we can prove the solution of the Cauchy problem belongs to Shubin spaces.
In this paper we consider the non-cutoff Boltzmann equation in spatially inhomogeneous case. We prove the propagation of Gevrey regularity for the so-called smooth Maxwellian decay solutions to the Cauchy problem of spatially inhomogeneous Boltzmann equation, and obtain Gevrey regularity of order $1/(2s)$ in the velocity variable $v$ and order 1 in the space variable $x$. The strategy relies on our recent results for spatially homogeneous case (J. Diff. Equ. 253(4) (2012), 1172-1190. DOI: 10.1016/j.jde.2012.04.023). Rather, we need much more intricate analysis additionally in order to handle w…
We prove a local-in-time existence and uniqueness theorem for a smooth classical solution to the spatially homogeneous Boltzmann equation with cutoff soft potentials. Our proof is based on a series of bilinear estimates for the integrability and Sobolev regularity of the associated collision operator. While the global-in-time existence is left inconclusive, we give a lower bound of the maximal time of existence and a necessary condition for finite time extinction of existence.
Departing from the weak solution, we prove the uniqueness, smoothing estimates and the global dynamics for the non cutoff spatially homogeneous Boltzmann equation with moderate soft potentials. Our results show that the behavior of the solution(including the production of regularity and the longtime behavior) can be {\it characterized quantitatively} by the initial data at the large velocities, i.e.(i). initially polynomial decay at the large velocities in $L^1$ space will induce the finite smoothing estimates in weighted Sobolev spaces and the polynomial convergence rate (including the lower …
We consider a class of distribution functions having the form $ f(v,t)=t^{-dat} F(v e^{-at} )$, $ a=\mathrm{const.} $, where $ v $ and $ t $ denote the particle velocity and the time. This class of self-similar solutions to the spatially homogeneous Boltzmann equation (BE) for Maxwell molecules was studied by Bobylev and Cercignani in early 2000s. The solutions are positive, but have an infinite second moment (energy). However, the same class of distribution functions with finite energy appears to be closely connected with quite different class of group-invariant solutions of the spatially inh…
In this paper, we study the Gevrey regularity of spatially homogeneous Boltzmann equation without angular cutoff. We prove the propagation of Gevrey regularity for $C^\infty$ solutions with the Maxwellian decay to the Cauchy problem of spatially homogeneous Boltzmann equation. The idea we use here is based on the framework of Morimoto's recent paper (See Morimoto: J. Pseudo-Differ. Oper. Appl. (2010) 1: 139-159, DOI:10.1007/s11868-010-0008-z), but we extend the range of the index $γ$ satisfying $γ+ 2s \in (-1,1)$, $s\in (0,1/2)$ and in this case we consider the kinetic factor in the form of $Φ…
In this article we study the Gevrey regularization effect for the spatially inhomogeneous Boltzmann equation without angular cutoff. This equation is partially elliptic in the velocity direction and degenerates in the spatial variable. We consider the nonlinear Cauchy problem for the fluctuation around the Maxwellian distribution and prove that any solution with mild regularity will become smooth in Gevrey class at positive time, with Gevrey index depending on the angular singularity. Our proof relies on the symbolic calculus for the collision operator and the global subelliptic estimate for t…
We consider an approximation of the linearised equation of the homogeneous Boltzmann equation that describes the distribution of quasiparticles in a dilute gas of bosons at low temperature. The corresponding collision frequency is neither bounded from below nor from above. We prove the existence and uniqueness of solutions satisfying the conservation of energy. We show that these solutions converge to the corresponding stationary state, at an algebraic rate as time tends to infinity.
We develop the regularity theory of the spatially homogeneous Boltzmann equation with cut-off and hard potentials (for instance, hard spheres), by (i) revisiting the Lp-theory to obtain constructive bounds, (ii) establishing propagation of smoothness and singularities, (iii) obtaining estimates about the decay of the sin- gularities of the initial datum. Our proofs are based on a detailed study of the "regularity of the gain operator". An application to the long-time behavior is presented.
Numerical approximation of the Boltzmann equation is a challenging problem due to its high-dimensional, nonlocal, and nonlinear collision integral. Over the past decade, the Fourier-Galerkin spectral method has become a popular deterministic method for solving the Boltzmann equation, manifested by its high accuracy and potential of being further accelerated by the fast Fourier transform. Albeit its practical success, the stability of the method is only recently proved by Filbet, F. & Mouhot, C. in [$ Trans. Amer. Math. Soc.$ 363, no. 4 (2011): 1947-1980.] by utilizing the "spreading" property …
For the homogeneous Boltzmann equation with (cutoff or non cutoff) hard potentials, we prove estimates of propagation of Lp norms with a weight $(1+ |x|^2)^q/2$ ($1 < p < +\infty$, $q \in \R\_+$ large enough), as well as appearance of such weights. The proof is based on some new functional inequalities for the collision operator, proven by elementary means.
The unique global strong solution in the Chemin-Lerner type space to the Cauchy problem on the Boltzmann equation for hard potentials is constructed in perturbation framework. Such solution space is of critical regularity with respect to spatial variable, and it can capture the intrinsic property of the Botlzmann equation. For the proof of global well-posedness, we develop some new estimates on the nonlinear collision term through the Littlewood-Paley theory.
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…
In this paper we prove the strong and time-averaged strong convergence to equilibrium for solutions (with general initial data) of the spatially homogeneous Boltzmann equation for Fermi-Dirac particles. The assumption on the collision kernel includes the Coulomb potential with a weaker angular cutoff. The proof is based on moment estimates, entropy dissipation inequalities, regularity of the collision gain operator, and a new observation that many collision kernels are larger than or equal to some completely positive kernels, which enables us to avoid dealing with the convergence problem of th…
We study a class of elliptic competition-diffusion systems of long range segregation models for two and more competing species. The existence and uniqueness of the solution are shown. We prove that as the competition rate goes to infinity the solution converges, along with suitable sequences, to a spatially long range segregated state satisfying some free boundary problems.
We consider the spatially homogeneous Boltzmann equation for hard potentials without cutoff. We prove that an exponential moment of order $ρ=\min\{2γ/(2-ν),2\}$, with the usual notation, is immediately created. This is stronger than what happens in the case with cutoff. We also show that exponential moments of order $ρ\in (0,2]$ are propagated.
In a general $C^1$ domain, we study the perturbative Cauchy theory for the Boltzmann equation with Maxwell boundary conditions with an accommodation coefficient $α$ in $(\sqrt{2/3},1]$, and discuss this threshold. We consider polynomial or stretched exponential weights $m(v)$ and prove existence, uniqueness and exponential trend to equilibrium around a global Maxwellian in $L^\infty_{x,v}(m)$. Of important note is the fact that the methods do not involve contradiction arguments.
In this work, we study the Cauchy problem for the spatially homogeneous non-cutoff Boltzamnn equation with Maxwellian molecules. We prove that this Cauchy problem enjoys Gelfand-Shilov regularizing effect, that means the smoothing properties is same as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator. The power of this fractional is exactly the singular index of non-cutoff collisional kernel of Boltzmann operator. So that we get the regularity of solution in the Gevery class with the sharp power and the optimal exponential decay of solutions. …
We study the Boltzmann equation near a global Maxwellian in the case of bounded domains. We consider the boundary conditions to be either specular reflections or Maxwellian diffusion. Starting from the reference work of Guo in $L^\infty_{x,v}\left(\left(1+|v|\right)^βe^{|v|^2/4}\right)$, we prove existence, uniqueness, continuity and positivity of solutions for less restrictive weights in the velocity variable; namely, polynomials and stretch exponentials. The methods developed here are constructive.
The uniform shear flow for the rarefied gas is governed by the time-dependent spatially homogeneous Boltzmann equation with a linear shear force. The main feature of such flow is that the temperature may increase in time due to the shearing motion that induces viscous heat and the system becomes far from equilibrium. For Maxwell molecules, we establish the unique existence, regularity, shear-rate-dependent structure and non-negativity of self-similar profiles for any small shear rate. The non-negativity is justified through the large time asymptotic stability even in spatially inhomogeneous pe…
In this paper, we study the Cauchy problem for the linear spatially homogeneous Boltzamnn equation with Debye-Yukawa potential. Using the spectral decomposition of the linear operator, we prove that, for an initial datum in the sense of distribution which contains the dual of the Sobolev spaces, there exists a unique solution which belongs to a more regular Sobolev space for any positive time. We also study the sharp regularity of the solution.
In this paper, we prove the propagation of $L^p$ upper bounds for the spatially homogeneous relativistic Boltzmann equation for any $1<p<\infty$. We consider the case of relativistic \textit{hard ball} with Grad's angular cutoff. Our proof is based on a detailed study of the interrelationship between the relative momenta, the regularity and the $L^p$ estimates for the gain operator, the development of the relativistic Carleman representation, and several estimates on the relativistic hypersurface $E^{v_*}_{v'-v}$. We also derive a Pythagorean theorem for the \textit{relative momenta} $g(v,v_*)…
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 propagat…
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.
It is expected in physics that the homogeneous quantum Boltzmann equation with Fermi-Dirac or Bose-Einstein statistics and with Maxwell-Boltzmann operator (neglecting effect of the statistics) for the weak coupled gases will converge to the homogeneous Fokker-Planck-Landau equation as the Planck constant $\hbar$ tends to zero. In this paper and the upcoming work \cite{HLP2}, we will provide a mathematical justification on this semi-classical limit. Key ingredients into the proofs are the new framework to catch the {\it weak projection gradient}, which is motivated by Villani \cite{V1} to ident…
Uniqueness is proven for two 3-d inverse problems of the determination of the spatially distributed sound speed in the frequency dependent acoustic PDE. The main new point is the assumption that only the modulus of the scattered complex valued wave field is measured on a certain set.
The purpose of this paper is to study the evolution of moving interacting particles on the mesoscopic scale. We will introduce an uncertainty principle and a new priori bound for the evolution of particles subject to a general mesoscopic interaction and use this setting to demonstrate evidence for the universal dispersion of particles in the absence of the conventional methods like velocity averaging lemmas or Strichartz-type estimates. We will develop a scattering theory in generality and use its frame work to show the existence and uniqueness of a class scattering solutions to the Boltzmann …
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.
