Local validity of the boltzmann equation

Shinbrot, Marvin; Neunzert, Helmut

doi:10.1002/mma.1670060135

Cited by 12 publications

(9 citation statements)

References 9 publications

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“…Therefore the series (1.8) is converging for λ~1 small, uniformly in N and d, (this implies (2.9). The statement (2.8) follows by a direct inspection of the convergence term by term (see [3] and also [4] for a revisited version).…”

Section: Kφi(x)) = L(φ T (X))+ T (Tin-um-ydw-ridmentioning

confidence: 91%

Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum

Pulvirenti

1987

Commun.Math. Phys.

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We consider a system of N hard spheres in the Boltzmann-Grad limit (i.e. d -• 0, N -* oo, Nd 2 -• λ "* > 0, where d is the diameter of the spheres). If λ is sufficiently large, and if the joint distribution densities factorize at time zero, with the one particle distribution decaying sufficiently rapidly in space and velocities, we prove that the time evolved one-particle distribution converges for all times to the solution of the Boltzmann equation with the same initial datum. This result improves and is based on a previous paper [1], valid only in two dimensions.

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Section: Kφi(x)) = L(φ T (X))+ T (Tin-um-ydw-ridmentioning

confidence: 91%

Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum

Pulvirenti

1987

Commun.Math. Phys.

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“…In addition, the proper definition must be made for the limit operator L * . In fact, in order that the sequences f [7,9,11,12,13,14,15,16,17,18,19,20,21,22]However, the validity of the Boltzmann equation for general situations remains dubious.…”

Section: B Boltzmann-grad Limit and The Lanford Conjecturementioning

confidence: 99%

“…If the RG ordering and the previous assumptions hold locally (i.e., in the whole phasespace Γ 1 and at least in a finite time interval I o1 ≡ [t o , t 1 ]), the Boltzmann equation ( 5) is expected to be locally valid in the same domain [17,19,20] at least in an asymptotic sense. This means, introducing an arbitrary monotonic decreasing sequence of infinitesimal parameters {ε} ≡ {ε i > 0, i ∈ N} , that the sequence {f 1 (ε, x 1 , t)} defined in terms of them is expected to converge in a weak (asymptotic) sense for ε → 0.…”

Section: A Asymptotic Kinetic Theoriesmentioning

confidence: 99%

“…In fact, in order that the sequences f . [7,9,11,12,13,14,15,16,17,18,19,20,21,22] As a consequence, it was found that L * can be defined in the sense of weak* convergence for the sequence f (N ) s [7]. The proof of the weak convergence of f (N ) s in this sense was first reached in the seminal work of Lanford (Lanford, 1974 [7]) who was able to prove also the local validity of the Boltzmann equation in a finite time interval…”

Section: B Boltzmann-grad Limit and The Lanford Conjecturementioning

confidence: 99%

“…sense it is necessary to determine their time evolution. According to Lanford and previous authors this can be achieved by constructing and explicit solution of the corresponding equation of the BBGKY hierarchy, to be represented explicitly by a time-series expansion for each distribution f (N ) s [7,9,11,12,13,14,15,16,17,18,19,20,21,22]. As a consequence, it was found that L * can be defined in the sense of weak* convergence for the sequence f (N ) s[7].…”

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confidence: 99%

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On the Existence of the Boltzmann-Grad Limit for a System of Hard Smooth Spheres

Tessarotto¹,

Nicolini²

2008

AIP Conference Proceedings

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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 Grad (Grad, 1972) and developed in a seminal work by Lanford (Lanford, 1974) that kinetic equations -such as the Boltzmann equation for a gas of classical hard spheres -might result exact in an appropriate asymptotic limit, usually denoted as Boltzmann-Grad limit. The Lanford conjecture has actually had a profound influence on the scientific community, giving rise to a whole line of original research in kinetic theory and mathematical physics. Nevertheless, several aspects of the theory remain to be addressed and clarified.In fact, its validity has been proven for the Boltzmann equation only at most in a weak sense, i.e., if the Boltzmann-Grad limit is defined according to the weak * convergence. While it is doubtful whether the result applies for arbitrary times and for general situations (and in particular more generally for classical systems of particles interacting via binary forces), it remains completely unsolved the issue whether the conjecture might be valid also in a stronger sense (strong Lanford conjecture). This paper will point out a physical model providing a counter-example to the strong Lanford conjecture, representing a straightforward generalization of the classical model based on a gas of hard-smooth spheres. In particular we claim that that the one-particle limit function, defined in the sense of the strong Boltzmann-Grad limit, does not generally satisfy the BBGKY (or Boltzmann) hierarchy. The result is important for the theoretical foundations of kinetic theory.

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Theorems on the Solutions of The Boltzmann Equation

Cercignani

1988

The Boltzmann Equation and Its Applications

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Local validity of the boltzmann equation

Abstract: The validity of the Boltzmann equation is proved, locally in time, by showing that its solution is a limit of solutions of the BBGKY hierarchy.

Cited by 12 publications

References 9 publications

Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum

Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum

On the Existence of the Boltzmann-Grad Limit for a System of Hard Smooth Spheres

Theorems on the Solutions of The Boltzmann Equation

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