相关论文: On the Boltzmann-Grad Limit for the classical hard…

Despite the progress achieved by kinetic theory, its rigorous theoretical foundations still remain unsolved to date. This concerns in particular the search of possible exact kinetic equations and, specifically, the conjecture proposed by…

In this paper the problem is posed of the prescription of the so-called Boltzmann-Grad (BG) limit ($\mathcal{L}_{BG}$) for the $N-$body system of smooth hard-spheres which undergo unary, binary as well as multiple elastic instantaneous…

In the paper the possible approaches to the rigorous derivation of the Boltzmann kinetic equation with hard sphere collisions from underlying dynamics are considered. In particular, a formalism for the description of the evolution of…

The kinetic theory of Maxwell and Boltzmann has been the subject of major scientific controversies. The alleged incompatibility between the reversible nature of the equations of classical mechanics and the increase of entropy, which, in the…

The present article proposes a rigorous derivation of the Boltzmann equation in the half-space. We show an analog of the Lanford's theorem in this domain, with specular reflection boundary condition, stating the convergence in the low…

We consider a classical system of point particles interacting by means of a short range potential. We prove that, in the low--density (Boltzmann--Grad) limit, the system behaves, for short times, as predicted by the associated Boltzmann…

Obtaining a rigorous justification of the kinetic theory of gases from Newton's second law of dynamics for a large number of identical spheres interacting by elastic binary collisions, is a problem formulated by Hilbert in 1900 (Hilbert's…

In (Commun Pure Appl Math 2(4):331-407, 1949), Grad proposed a Hermite series expansion for approximating solutions to kinetic equations that have an unbounded velocity space. However, for initial boundary value problems, poorly imposed…

We provide a rigorous derivation of Boltzmann's kinetic equation from the hard sphere system for rarefied gas, which is valid for arbitrarily long times, as long as the solution to the Boltzmann equation exists. This extends Lanford's…

The BBGKY hierarchy of equations for a particle interacting with ideal gas is analyzed in terms of irreducible many-particle correlations between gas atoms and the particle's motion. The transition to the hard-sphere interaction is…

Boltzmann's equation provides a microscopic model for the evolution of dilute classical gases. A fundamental problem in mathematical physics is to rigorously derive Boltzmann's equation starting from Newton's laws. In the 1970s, Oscar…

It is shown that early suggested derivation of the Boltzmann kinetic equation for dilute hard sphere gas from the time-reversible BBGKY equations is incorrect since in fact a priori substitutes for them definite irreversible equations.…

We review various contributions on the fundamental work of Lanford deriving the Boltzmann equation from hard-sphere dynamics in the low density limit. We focus especially on the assumptions made on the initial data and on how they encode…

In this paper we present a rigorous derivation of the Boltzmann equation in a compact domain with diffuse reflection boundary conditions. We consider a system of $N$ hard spheres of diameter $\epsilon$ in a box $\Lambda := [0, 1] \times…

We develop a rigorous theory of hard-sphere dynamics in the kinetic regime, away from thermal equilibrium. In the low density limit, the empirical density obeys a law of large numbers and the dynamics is governed by the Boltzmann equation.…

The kinetic theory description of a low density gas of hard spheres or disks, confined between two parallel plates separated a distance smaller than twice the diameter of the particles, is addressed starting from the Liouville equation of…

One of the most challenging and fascinating issues in mathematical and theoretical physics concerns identifying the common logic, if any, which underlies the physical world. More precisely, this involves the search of the possibly-unique…

We provide a rigorous derivation of the Boltzmann equation as the mesoscopic limit of systems of hard spheres, or Newtonian particles interacting via a short-range potential, as the number of particles $N$ goes to infinity and the…

The local balance equations for the density, momentum, and energy of a dilute gas of elastic or inelastic hard spheres, strongly confined between two parallel hard plates are obtained. The starting point is a Boltzmann-like kinetic…

We derive the linear acoustic and Stokes-Fourier equations as the limiting dynamics of a system of N hard spheres of diameter $\epsilon$ in two space dimensions, when N $\rightarrow$ $\infty$, $\epsilon$ $\rightarrow$ 0, N $\epsilon$ =…